Subgroups — C₃×#61051 #10679658

Order 768.

2734 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3485	C₂	1	2	yes	maximal, complemented
4	C₃×#3283	C₂	1	4	yes	maximal, complemented
5	C₃×#3280	C₂	1	4	yes	maximal, complemented
6	C₃×#3267	C₂	1	8	yes	maximal, complemented
7	C₃×#3156	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3059	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#3050	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3042	C₂	1	2	yes	maximal, complemented
11	C₃×#3009	C₂	1	2	yes	maximal, complemented
12	C₃×#3005	C₂	1	2	yes	maximal, complemented
13	C₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃×#2929	C₂	1	1	yes	maximal, characteristic, complemented
15	#61051	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₃×H(5,𝔽₂)	C₂²	1	2	yes	—
17	C₂²×C₃×#41	C₂²	1	2	yes	complemented
18	C₃×#576	—	2	8	no	—
19	C₂×C₃×#137	C₂²	1	2	yes	complemented
20	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
21	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
22	C₂×C₃×#135	—	2	8	no	—
23	C₂×C₃×#135	C₂²	1	1	yes	characteristic
24	C₂×C₃×#135	C₂²	1	4	yes	complemented
25	C₂×C₃×#135	C₂²	1	4	yes	complemented
26	C₂×C₃×#135	C₂²	1	2	yes	complemented
27	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₃×SD₁₆	C₂²	1	2	yes	complemented
29	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
30	C₂²×C₃×D₈	C₂²	1	2	yes	complemented
31	C₂×C₃×#131	C₂²	1	1	yes	characteristic
32	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
33	C₃×#547	—	2	4	no	—
34	C₃×#547	C₂²	1	8	yes	complemented
35	C₃×#546	—	2	4	no	—
36	C₃×#546	C₂²	1	8	yes	complemented
37	C₃×#536	C₂²	1	2	yes	—
38	C₃×#536	C₂²	1	2	yes	complemented
39	C₃×#534	—	2	4	no	—
40	C₃×#525	C₂²	1	1	yes	characteristic
41	C₃×#525	C₂²	1	2	yes	complemented
42	C₂×C₃×#123	—	2	4	no	—
43	C₃×#518	C₂²	1	2	yes	complemented
44	C₃×#497	C₂²	1	4	yes	complemented
45	C₃×#495	C₂²	1	2	yes	complemented
46	C₃×#494	C₂²	1	2	yes	complemented
47	C₃×#470	C₂²	1	4	yes	complemented
48	C₃×#467	C₂²	1	4	yes	complemented
49	C₃×#467	C₂²	1	2	yes	complemented
50	C₃×#466	C₂²	1	2	yes	complemented
51	C₃×#465	C₂²	1	4	yes	complemented
52	C₃×#463	C₂²	1	8	yes	complemented
53	C₃×#461	C₂²	1	8	yes	complemented
54	C₃×#452	C₂²	1	8	yes	complemented
55	C₃×#452	C₂²	1	8	yes	complemented
56	C₃×#450	C₂²	1	8	yes	complemented
57	C₃×#449	C₂²	1	8	yes	complemented
58	C₃×#442	C₂²	1	4	yes	complemented
59	C₃×C₄×D₈	C₂²	1	4	yes	complemented
60	C₃×#435	C₂²	1	4	yes	complemented
61	C₃×C₈×D₄	C₂²	1	4	yes	complemented
62	C₃×#427	C₂²	1	1	yes	characteristic, complemented
63	C₃×#424	C₂²	1	1	yes	characteristic, complemented
64	C₃×#419	C₂²	1	2	yes	complemented
65	C₃×#418	C₂²	1	2	yes	complemented
66	C₃×#416	C₂²	1	2	yes	complemented
67	C₃×#408	C₂²	1	1	yes	characteristic
68	C₃×#407	C₂²	1	1	yes	characteristic, complemented
69	C₃×#405	C₂²	1	1	yes	characteristic, complemented
70	C₂×#576	C₂×C₃	1	1	yes	characteristic, complemented
71	C₂²×#135	C₂×C₃	1	1	yes	characteristic, complemented
72	#3485	C₂×C₃	1	2	yes	complemented
73	#3283	C₂×C₃	1	4	yes	complemented
74	#3280	C₂×C₃	1	4	yes	complemented
75	#3267	C₂×C₃	1	8	yes	complemented
76	#3156	C₂×C₃	1	1	yes	characteristic, complemented
77	#3059	C₂×C₃	1	1	yes	characteristic, complemented
78	#3050	C₂×C₃	1	1	yes	characteristic, complemented
79	#3042	C₂×C₃	1	2	yes	complemented
80	#3009	C₂×C₃	1	2	yes	complemented
81	#3005	C₂×C₃	1	2	yes	complemented
82	#2948	C₂×C₃	1	1	yes	characteristic, complemented
83	#2929	C₂×C₃	1	1	yes	characteristic, complemented
84	C₃×H(5,𝔽₂)	—	2	8	no	—
85	C₃×H(5,𝔽₂)	—	4	4	no	—
86	C₂×C₃×#41	—	2	2	no	—
87	C₂×C₃×#41	—	2	4	no	—
88	C₂×C₃×#41	—	2	8	no	—
89	C₂×C₃×#41	—	4	4	no	—
90	C₂×C₃×#41	C₂³	1	2	yes	—
91	C₂×C₃×#41	C₂³	1	8	yes	—
92	C₂×C₃×#41	C₂³	1	4	yes	—
93	C₂×C₃×#41	C₂³	1	1	yes	characteristic
94	C₂²×C₃×Q₈	C₂³	1	2	yes	—
95	C₂²×C₃×D₄	—	2	2	no	—
96	C₂²×C₃×D₄	—	2	4	no	—
97	C₂²×C₃×D₄	—	2	2	no	—
98	C₂²×C₃×D₄	C₂³	1	4	yes	—
99	C₂²×C₃×D₄	C₂³	1	2	yes	—
100	C₂³×C₃×C₄	—	2	2	no	—
101	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
102	C₃×#137	—	4	4	no	—
103	C₃×AGL(1,ℤ/8)	—	2	8	no	—
104	C₃×AGL(1,ℤ/8)	—	2	16	no	—
105	C₃×#135	—	2	16	no	—
106	C₃×#135	—	2	16	no	—
107	C₃×#135	—	2	8	no	—
108	C₃×#135	—	2	4	no	—
109	C₂×C₃×Q₁₆	—	2	4	no	—
110	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
111	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
112	C₂×C₃×Q₁₆	C₂³	1	1	yes	characteristic
113	C₂×C₃×Q₁₆	C₂³	1	4	yes	—
114	C₂×C₃×SD₁₆	—	2	8	no	—
115	C₂×C₃×SD₁₆	C₂³	1	8	yes	—
116	C₂×C₃×SD₁₆	C₂³	1	2	yes	—
117	C₂×C₃×SD₁₆	C₂³	1	4	yes	—
118	C₂×C₃×SD₁₆	C₂³	1	4	yes	—
119	C₂×C₃×D₈	—	2	8	no	—
120	C₂×C₃×D₈	D₄	1	8	yes	complemented
121	C₂×C₃×D₈	C₂³	1	1	yes	characteristic
122	C₂×C₃×D₈	C₂³	1	4	yes	—
123	C₂×C₃×D₈	C₂³	1	2	yes	—
124	C₂×C₃×D₈	C₂³	1	2	yes	—
125	C₃×#131	—	2	4	no	—
126	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
127	C₂×C₃×M₁₆	C₂³	1	2	yes	—
128	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
129	C₂²×C₃×C₈	—	2	4	no	—
130	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
131	C₂²×C₃×C₈	C₂³	1	2	yes	—
132	C₂²×C₃×C₈	C₂³	1	2	yes	—
133	C₂²×C₃×C₈	C₂³	1	2	yes	—
134	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
135	C₃×C₄⋊Q₈	—	2	2	no	—
136	C₃×C₄⋊Q₈	C₂³	1	4	yes	—
137	C₃×#124	—	2	2	no	—
138	C₃×#124	C₂³	1	4	yes	—
139	C₃×#123	—	2	4	no	—
140	C₃×#123	—	2	2	no	—
141	C₃×#123	—	2	8	no	—
142	C₃×#123	—	2	4	no	—
143	C₃×#123	—	2	8	no	—
144	C₃×#123	—	2	8	no	—
145	C₃×#123	D₄	1	2	yes	—
146	C₃×C₂²⋊Q₈	—	2	4	no	—
147	C₃×C₂²⋊Q₈	D₄	1	2	yes	—
148	C₃×C₂²⋊Q₈	C₂³	1	8	yes	—
149	C₃×C₂²⋊Q₈	C₂³	1	4	yes	—
150	C₃×#121	—	2	8	no	—
151	C₃×#121	—	2	8	no	—
152	C₃×#121	D₄	1	2	yes	—
153	C₃×#121	C₂³	1	4	yes	—
154	C₃×#120	—	2	2	no	—
155	C₃×#120	—	2	8	no	—
156	C₃×#120	D₄	1	2	yes	—
157	C₃×C₄×D₄	—	2	4	no	—
158	C₃×C₄×D₄	—	2	4	no	—
159	C₃×C₄×D₄	C₂³	1	8	yes	—
160	C₃×C₄×D₄	C₂³	1	4	yes	—
161	C₃×#117	C₂³	1	2	yes	—
162	C₃×#117	C₂³	1	1	yes	characteristic
163	C₂×C₃×C₄⋊C₄	—	2	8	no	—
164	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
165	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
166	C₃×#107	C₂³	1	2	yes	—
167	C₃×#106	C₂³	1	2	yes	—
168	C₃×C₄⋊C₈	C₂³	1	4	yes	—
169	C₃×#103	C₂³	1	4	yes	—
170	C₃×#103	C₂³	1	8	yes	—
171	C₃×#102	C₂³	1	4	yes	—
172	C₃×#102	C₂³	1	8	yes	—
173	C₃×C₂²⋊C₈	C₂³	1	4	yes	—
174	C₃×C₂²⋊C₈	C₂³	1	4	yes	—
175	C₃×#97	C₂³	1	2	yes	—
176	C₃×C₄×C₈	C₂³	1	2	yes	—
177	C₂×H(5,𝔽₂)	C₂²×C₃	1	2	yes	—
178	C₂²×#41	C₂²×C₃	1	2	yes	complemented
179	#576	—	2	8	no	—
180	C₂×#137	C₂²×C₃	1	2	yes	complemented
181	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	—
182	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	4	yes	complemented
183	C₂×#135	—	2	8	no	—
184	C₂×#135	C₂²×C₃	1	1	yes	characteristic
185	C₂×#135	C₂²×C₃	1	4	yes	complemented
186	C₂×#135	C₂²×C₃	1	4	yes	complemented
187	C₂×#135	C₂²×C₃	1	2	yes	complemented
188	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
189	C₂²×SD₁₆	C₂²×C₃	1	2	yes	complemented
190	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
191	C₂²×D₈	C₂²×C₃	1	2	yes	complemented
192	C₂×#131	C₂²×C₃	1	1	yes	characteristic
193	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
194	#547	—	2	4	no	—
195	#547	C₂²×C₃	1	8	yes	complemented
196	#546	—	2	4	no	—
197	#546	C₂²×C₃	1	8	yes	complemented
198	#536	C₂²×C₃	1	2	yes	—
199	#536	C₂²×C₃	1	2	yes	complemented
200	#534	—	2	4	no	—
201	#525	C₂²×C₃	1	1	yes	characteristic
202	#525	C₂²×C₃	1	2	yes	complemented
203	C₂×#123	—	2	4	no	—
204	#518	C₂²×C₃	1	2	yes	complemented
205	#497	C₂²×C₃	1	4	yes	complemented
206	#495	C₂²×C₃	1	2	yes	complemented
207	#494	C₂²×C₃	1	2	yes	complemented
208	#470	C₂²×C₃	1	4	yes	complemented
209	#467	C₂²×C₃	1	4	yes	complemented
210	#467	C₂²×C₃	1	2	yes	complemented
211	#466	C₂²×C₃	1	2	yes	complemented
212	#465	C₂²×C₃	1	4	yes	complemented
213	#463	C₂²×C₃	1	8	yes	complemented
214	#461	C₂²×C₃	1	8	yes	complemented
215	#452	C₂²×C₃	1	8	yes	complemented
216	#452	C₂²×C₃	1	8	yes	complemented
217	#450	C₂²×C₃	1	8	yes	complemented
218	#449	C₂²×C₃	1	8	yes	complemented
219	#442	C₂²×C₃	1	4	yes	complemented
220	C₄×D₈	C₂²×C₃	1	4	yes	complemented
221	#435	C₂²×C₃	1	4	yes	complemented
222	C₈×D₄	C₂²×C₃	1	4	yes	complemented
223	#427	C₂²×C₃	1	1	yes	characteristic, complemented
224	#424	C₂²×C₃	1	1	yes	characteristic, complemented
225	#419	C₂²×C₃	1	2	yes	complemented
226	#418	C₂²×C₃	1	2	yes	complemented
227	#416	C₂²×C₃	1	2	yes	complemented
228	#408	C₂²×C₃	1	1	yes	characteristic
229	#407	C₂²×C₃	1	1	yes	characteristic, complemented
230	#405	C₂²×C₃	1	1	yes	characteristic, complemented
231	C₂⁴×C₃	—	2	2	no	—
232	C₂⁴×C₃	—	2	4	no	—
233	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
234	C₃×#41	—	2	2	no	—
235	C₃×#41	—	2	2	no	—
236	C₃×#41	—	2	16	no	—
237	C₃×#41	—	2	8	no	—
238	C₃×#41	—	2	4	no	—
239	C₃×#41	—	2	4	no	—
240	C₃×#41	—	4	8	no	—
241	C₃×#41	—	4	4	no	—
242	C₃×#41	—	4	16	no	—
243	C₃×#41	—	4	8	no	—
244	C₂×C₃×Q₈	—	2	2	no	—
245	C₂×C₃×Q₈	—	2	4	no	—
246	C₂×C₃×Q₈	—	2	4	no	—
247	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
248	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
249	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
250	C₂×C₃×D₄	—	2	8	no	—
251	C₂×C₃×D₄	—	2	8	no	—
252	C₂×C₃×D₄	—	2	2	no	—
253	C₂×C₃×D₄	—	2	4	no	—
254	C₂×C₃×D₄	—	2	4	no	—
255	C₂×C₃×D₄	—	2	2	no	—
256	C₂×C₃×D₄	—	2	4	no	—
257	C₂×C₃×D₄	—	2	4	no	—
258	C₂×C₃×D₄	—	4	4	no	—
259	C₂×C₃×D₄	—	4	8	no	—
260	C₂×C₃×D₄	—	4	16	no	—
261	C₂×C₃×D₄	—	4	8	no	—
262	C₂×C₃×D₄	C₂×D₄	1	2	yes	—
263	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
264	C₂×C₃×D₄	C₂×D₄	1	8	yes	—
265	C₂×C₃×D₄	C₂×D₄	1	4	yes	—
266	C₂×C₃×D₄	C₂⁴	1	2	yes	—
267	C₂×C₃×D₄	C₂⁴	1	4	yes	—
268	C₂²×C₃×C₄	—	2	2	no	—
269	C₂²×C₃×C₄	—	2	2	no	—
270	C₂²×C₃×C₄	—	2	2	no	—
271	C₂²×C₃×C₄	—	2	8	no	—
272	C₂²×C₃×C₄	—	2	4	no	—
273	C₂²×C₃×C₄	—	2	2	no	—
274	C₂²×C₃×C₄	—	2	8	no	—
275	C₂²×C₃×C₄	—	4	8	no	—
276	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
277	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
278	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
279	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
280	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
281	C₃×Q₁₆	—	2	4	no	—
282	C₃×Q₁₆	—	2	4	no	—
283	C₃×Q₁₆	—	2	2	no	—
284	C₃×Q₁₆	—	4	4	no	—
285	C₃×SD₁₆	—	2	16	no	—
286	C₃×SD₁₆	—	2	8	no	—
287	C₃×SD₁₆	—	2	4	no	—
288	C₃×SD₁₆	—	4	4	no	—
289	C₃×D₈	—	2	8	no	—
290	C₃×D₈	—	2	2	no	—
291	C₃×D₈	C₂×D₄	1	8	yes	—
292	C₃×D₈	C₂×D₄	1	8	yes	—
293	C₃×M₁₆	—	2	2	no	—
294	C₃×M₁₆	—	2	2	no	—
295	C₃×M₁₆	—	2	4	no	—
296	C₂×C₃×C₈	—	2	2	no	—
297	C₂×C₃×C₈	—	2	4	no	—
298	C₂×C₃×C₈	—	2	4	no	—
299	C₂×C₃×C₈	—	2	4	no	—
300	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
301	C₂×C₃×C₈	C₂⁴	1	2	yes	—
302	C₂×C₃×C₈	C₂⁴	1	2	yes	—
303	C₂×C₃×C₈	C₂⁴	1	2	yes	—
304	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
305	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
306	C₃×C₄⋊C₄	—	2	4	no	—
307	C₃×C₄⋊C₄	—	2	8	no	—
308	C₃×C₄⋊C₄	—	2	8	no	—
309	C₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
310	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
311	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
312	C₃×C₂²⋊C₄	—	2	2	no	—
313	C₃×C₂²⋊C₄	—	2	8	no	—
314	C₃×C₂²⋊C₄	—	2	4	no	—
315	C₃×C₂²⋊C₄	—	2	4	no	—
316	C₃×C₂²⋊C₄	—	2	8	no	—
317	C₃×C₂²⋊C₄	—	2	8	no	—
318	C₃×C₂²⋊C₄	C₂×D₄	1	2	yes	—
319	C₃×C₂²⋊C₄	C₂×D₄	1	4	yes	—
320	C₃×C₂²⋊C₄	C₂×D₄	1	4	yes	—
321	C₃×C₄²	—	2	2	no	—
322	C₃×C₄²	C₂⁴	1	2	yes	—
323	H(5,𝔽₂)	—	2	8	no	—
324	H(5,𝔽₂)	—	4	4	no	—
325	C₂×#41	—	2	2	no	—
326	C₂×#41	—	2	4	no	—
327	C₂×#41	—	2	8	no	—
328	C₂×#41	—	4	4	no	—
329	C₂×#41	C₂³×C₃	1	2	yes	—
330	C₂×#41	C₂³×C₃	1	8	yes	—
331	C₂×#41	C₂³×C₃	1	4	yes	—
332	C₂×#41	C₂³×C₃	1	1	yes	characteristic
333	C₂²×Q₈	C₂³×C₃	1	2	yes	—
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	4	no	—
336	C₂²×D₄	—	2	2	no	—
337	C₂²×D₄	C₂³×C₃	1	4	yes	—
338	C₂²×D₄	C₂³×C₃	1	2	yes	—
339	C₂³×C₄	—	2	2	no	—
340	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
341	#137	—	4	4	no	—
342	AGL(1,ℤ/8)	—	2	8	no	—
343	AGL(1,ℤ/8)	—	2	16	no	—
344	#135	—	2	16	no	—
345	#135	—	2	16	no	—
346	#135	—	2	8	no	—
347	#135	—	2	4	no	—
348	C₂×Q₁₆	—	2	4	no	—
349	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
350	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
351	C₂×Q₁₆	C₂³×C₃	1	1	yes	characteristic
352	C₂×Q₁₆	C₂³×C₃	1	4	yes	—
353	C₂×SD₁₆	—	2	8	no	—
354	C₂×SD₁₆	C₂³×C₃	1	8	yes	—
355	C₂×SD₁₆	C₂³×C₃	1	2	yes	—
356	C₂×SD₁₆	C₂³×C₃	1	4	yes	—
357	C₂×SD₁₆	C₂³×C₃	1	4	yes	—
358	C₂×D₈	—	2	8	no	—
359	C₂×D₈	C₃×D₄	1	8	yes	complemented
360	C₂×D₈	C₂³×C₃	1	1	yes	characteristic
361	C₂×D₈	C₂³×C₃	1	4	yes	—
362	C₂×D₈	C₂³×C₃	1	2	yes	—
363	C₂×D₈	C₂³×C₃	1	2	yes	—
364	#131	—	2	4	no	—
365	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
366	C₂×M₁₆	C₂³×C₃	1	2	yes	—
367	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
368	C₂²×C₈	—	2	4	no	—
369	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
370	C₂²×C₈	C₂³×C₃	1	2	yes	—
371	C₂²×C₈	C₂³×C₃	1	2	yes	—
372	C₂²×C₈	C₂³×C₃	1	2	yes	—
373	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
374	C₄⋊Q₈	—	2	2	no	—
375	C₄⋊Q₈	C₂³×C₃	1	4	yes	—
376	#124	—	2	2	no	—
377	#124	C₂³×C₃	1	4	yes	—
378	#123	—	2	4	no	—
379	#123	—	2	2	no	—
380	#123	—	2	8	no	—
381	#123	—	2	4	no	—
382	#123	—	2	8	no	—
383	#123	—	2	8	no	—
384	#123	C₃×D₄	1	2	yes	—
385	C₂²⋊Q₈	—	2	4	no	—
386	C₂²⋊Q₈	C₃×D₄	1	2	yes	—
387	C₂²⋊Q₈	C₂³×C₃	1	4	yes	—
388	C₂²⋊Q₈	C₂³×C₃	1	8	yes	—
389	#121	—	2	8	no	—
390	#121	—	2	8	no	—
391	#121	C₃×D₄	1	2	yes	—
392	#121	C₂³×C₃	1	4	yes	—
393	#120	—	2	2	no	—
394	#120	—	2	8	no	—
395	#120	C₃×D₄	1	2	yes	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	C₂³×C₃	1	8	yes	—
399	C₄×D₄	C₂³×C₃	1	4	yes	—
400	#117	C₂³×C₃	1	2	yes	—
401	#117	C₂³×C₃	1	1	yes	characteristic
402	C₂×C₄⋊C₄	—	2	8	no	—
403	C₂×C₂²⋊C₄	—	2	4	no	—
404	C₂×C₂²⋊C₄	—	2	8	no	—
405	#107	C₂³×C₃	1	2	yes	—
406	#106	C₂³×C₃	1	2	yes	—
407	C₄⋊C₈	C₂³×C₃	1	4	yes	—
408	#103	C₂³×C₃	1	4	yes	—
409	#103	C₂³×C₃	1	8	yes	—
410	#102	C₂³×C₃	1	4	yes	—
411	#102	C₂³×C₃	1	8	yes	—
412	C₂²⋊C₈	C₂³×C₃	1	4	yes	—
413	C₂²⋊C₈	C₂³×C₃	1	4	yes	—
414	#97	C₂³×C₃	1	2	yes	—
415	C₄×C₈	C₂³×C₃	1	2	yes	—
416	C₂³×C₃	—	2	2	no	—
417	C₂³×C₃	—	2	4	no	—
418	C₂³×C₃	—	2	8	no	—
419	C₂³×C₃	—	2	2	no	—
420	C₂³×C₃	—	4	2	no	—
421	C₂³×C₃	—	4	8	no	—
422	C₂³×C₃	—	8	4	no	—
423	C₂³×C₃	—	8	8	no	—
424	C₂³×C₃	C₂²×D₄	1	2	yes	—
425	C₂³×C₃	C₂²×D₄	1	2	yes	—
426	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
427	C₃×Q₈	—	2	2	no	—
428	C₃×Q₈	—	2	2	no	—
429	C₃×Q₈	—	2	4	no	—
430	C₃×Q₈	—	2	4	no	—
431	C₃×Q₈	—	2	2	no	—
432	C₃×Q₈	—	4	4	no	—
433	C₃×D₄	—	2	4	no	—
434	C₃×D₄	—	2	8	no	—
435	C₃×D₄	—	2	2	no	—
436	C₃×D₄	—	2	2	no	—
437	C₃×D₄	—	2	4	no	—
438	C₃×D₄	—	2	2	no	—
439	C₃×D₄	—	4	4	no	—
440	C₃×D₄	—	4	8	no	—
441	C₃×D₄	—	4	8	no	—
442	C₃×D₄	—	4	16	no	—
443	C₃×D₄	—	4	4	no	—
444	C₃×D₄	—	8	4	no	—
445	C₃×D₄	C₂²×D₄	1	8	yes	—
446	C₂×C₃×C₄	—	2	2	no	—
447	C₂×C₃×C₄	—	2	4	no	—
448	C₂×C₃×C₄	—	2	2	no	—
449	C₂×C₃×C₄	—	2	2	no	—
450	C₂×C₃×C₄	—	2	2	no	—
451	C₂×C₃×C₄	—	2	2	no	—
452	C₂×C₃×C₄	—	2	4	no	—
453	C₂×C₃×C₄	—	2	4	no	—
454	C₂×C₃×C₄	—	2	2	no	—
455	C₂×C₃×C₄	—	4	16	no	—
456	C₂×C₃×C₄	—	4	8	no	—
457	C₂×C₃×C₄	—	4	4	no	—
458	C₂×C₃×C₄	—	4	4	no	—
459	C₂×C₃×C₄	—	8	8	no	—
460	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
461	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
462	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
463	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
464	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
465	C₂×C₃×C₄	H(5,𝔽₂)	1	2	yes	—
466	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
467	C₃×C₈	—	2	2	no	—
468	C₃×C₈	—	2	2	no	—
469	C₃×C₈	—	2	2	no	—
470	C₃×C₈	C₂²×D₄	1	2	yes	—
471	C₃×C₈	C₂²×D₄	1	2	yes	—
472	C₂⁴	—	2	2	no	—
473	C₂⁴	—	2	4	no	—
474	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
475	#41	—	2	2	no	—
476	#41	—	2	2	no	—
477	#41	—	2	16	no	—
478	#41	—	2	8	no	—
479	#41	—	2	4	no	—
480	#41	—	2	4	no	—
481	#41	—	4	8	no	—
482	#41	—	4	4	no	—
483	#41	—	4	16	no	—
484	#41	—	4	8	no	—
485	C₂×Q₈	—	2	2	no	—
486	C₂×Q₈	—	2	4	no	—
487	C₂×Q₈	—	2	4	no	—
488	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
489	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
490	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
491	C₂×D₄	—	2	8	no	—
492	C₂×D₄	—	2	8	no	—
493	C₂×D₄	—	2	2	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	4	no	—
499	C₂×D₄	—	4	4	no	—
500	C₂×D₄	—	4	8	no	—
501	C₂×D₄	—	4	16	no	—
502	C₂×D₄	—	4	8	no	—
503	C₂×D₄	C₂×C₃×D₄	1	2	yes	—
504	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
505	C₂×D₄	C₂×C₃×D₄	1	8	yes	—
506	C₂×D₄	C₂×C₃×D₄	1	4	yes	—
507	C₂×D₄	C₂⁴×C₃	1	2	yes	—
508	C₂×D₄	C₂⁴×C₃	1	4	yes	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	2	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	8	no	—
513	C₂²×C₄	—	2	4	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	8	no	—
516	C₂²×C₄	—	4	8	no	—
517	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
518	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
522	Q₁₆	—	2	4	no	—
523	Q₁₆	—	2	4	no	—
524	Q₁₆	—	2	2	no	—
525	Q₁₆	—	4	4	no	—
526	SD₁₆	—	2	16	no	—
527	SD₁₆	—	2	8	no	—
528	SD₁₆	—	2	4	no	—
529	SD₁₆	—	4	4	no	—
530	D₈	—	2	8	no	—
531	D₈	—	2	2	no	—
532	D₈	C₂×C₃×D₄	1	8	yes	—
533	D₈	C₂×C₃×D₄	1	8	yes	—
534	M₁₆	—	2	2	no	—
535	M₁₆	—	2	2	no	—
536	M₁₆	—	2	4	no	—
537	C₂×C₈	—	2	2	no	—
538	C₂×C₈	—	2	4	no	—
539	C₂×C₈	—	2	4	no	—
540	C₂×C₈	—	2	4	no	—
541	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
542	C₂×C₈	C₂⁴×C₃	1	2	yes	—
543	C₂×C₈	C₂⁴×C₃	1	2	yes	—
544	C₂×C₈	C₂⁴×C₃	1	2	yes	—
545	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
546	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	8	no	—
549	C₄⋊C₄	—	2	8	no	—
550	C₄⋊C₄	C₂×C₃×D₄	1	4	yes	—
551	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
552	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
553	C₂²⋊C₄	—	2	2	no	—
554	C₂²⋊C₄	—	2	8	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	8	no	—
558	C₂²⋊C₄	—	2	8	no	—
559	C₂²⋊C₄	C₂×C₃×D₄	1	2	yes	—
560	C₂²⋊C₄	C₂×C₃×D₄	1	4	yes	—
561	C₂²⋊C₄	C₂×C₃×D₄	1	4	yes	—
562	C₄²	—	2	2	no	—
563	C₄²	C₂⁴×C₃	1	2	yes	—
564	C₂²×C₃	—	2	2	no	—
565	C₂²×C₃	—	2	2	no	—
566	C₂²×C₃	—	2	4	no	—
567	C₂²×C₃	—	2	2	no	—
568	C₂²×C₃	—	2	2	no	—
569	C₂²×C₃	—	2	2	no	—
570	C₂²×C₃	—	2	8	no	—
571	C₂²×C₃	—	4	4	no	—
572	C₂²×C₃	—	4	8	no	—
573	C₂²×C₃	—	4	4	no	—
574	C₂²×C₃	—	4	2	no	—
575	C₂²×C₃	—	8	8	no	—
576	C₂²×C₃	—	16	8	no	—
577	C₂²×C₃	D₄²	1	2	yes	—
578	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
579	C₃×C₄	—	2	4	no	—
580	C₃×C₄	—	2	2	no	—
581	C₃×C₄	—	2	2	no	—
582	C₃×C₄	—	4	4	no	—
583	C₃×C₄	—	4	2	no	—
584	C₃×C₄	—	4	4	no	—
585	C₃×C₄	D₄²	1	2	yes	—
586	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
587	C₃×C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
588	C₂³	—	2	4	no	—
589	C₂³	—	2	2	no	—
590	C₂³	—	2	8	no	—
591	C₂³	—	2	2	no	—
592	C₂³	—	4	2	no	—
593	C₂³	—	4	8	no	—
594	C₂³	—	8	4	no	—
595	C₂³	—	8	8	no	—
596	C₂³	C₂²×C₃×D₄	1	2	yes	—
597	C₂³	C₂²×C₃×D₄	1	2	yes	—
598	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
599	Q₈	—	2	2	no	—
600	Q₈	—	2	2	no	—
601	Q₈	—	2	4	no	—
602	Q₈	—	2	4	no	—
603	Q₈	—	2	2	no	—
604	Q₈	—	4	4	no	—
605	D₄	—	2	4	no	—
606	D₄	—	2	2	no	—
607	D₄	—	2	4	no	—
608	D₄	—	2	2	no	—
609	D₄	—	2	8	no	—
610	D₄	—	2	2	no	—
611	D₄	—	4	4	no	—
612	D₄	—	4	8	no	—
613	D₄	—	4	8	no	—
614	D₄	—	4	16	no	—
615	D₄	—	4	4	no	—
616	D₄	—	8	4	no	—
617	D₄	C₂²×C₃×D₄	1	8	yes	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	2	4	no	—
625	C₂×C₄	—	2	4	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	4	16	no	—
628	C₂×C₄	—	4	8	no	—
629	C₂×C₄	—	4	4	no	—
630	C₂×C₄	—	4	4	no	—
631	C₂×C₄	—	8	8	no	—
632	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
633	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
634	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
635	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
636	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
637	C₂×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
638	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
639	C₈	—	2	2	no	—
640	C₈	—	2	2	no	—
641	C₈	—	2	2	no	—
642	C₈	C₂²×C₃×D₄	1	2	yes	—
643	C₈	C₂²×C₃×D₄	1	2	yes	—
644	C₂×C₃	—	2	2	no	—
645	C₂×C₃	—	4	2	no	—
646	C₂×C₃	—	4	2	no	—
647	C₂×C₃	—	4	8	no	—
648	C₂×C₃	—	8	2	no	—
649	C₂×C₃	C₂×D₄²	1	1	yes	characteristic
650	C₂×C₃	#3584	1	2	yes	—
651	C₂²	—	2	2	no	—
652	C₂²	—	2	2	no	—
653	C₂²	—	2	2	no	—
654	C₂²	—	2	8	no	—
655	C₂²	—	2	2	no	—
656	C₂²	—	2	4	no	—
657	C₂²	—	2	2	no	—
658	C₂²	—	4	4	no	—
659	C₂²	—	4	8	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	4	no	—
662	C₂²	—	8	8	no	—
663	C₂²	—	16	8	no	—
664	C₂²	C₃×D₄²	1	2	yes	—
665	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
666	C₄	—	2	2	no	—
667	C₄	—	2	2	no	—
668	C₄	—	2	4	no	—
669	C₄	—	4	2	no	—
670	C₄	—	4	4	no	—
671	C₄	—	4	4	no	—
672	C₄	C₃×D₄²	1	2	yes	—
673	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
674	C₄	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
675	C₃	#61051	1	1	yes	characteristic, direct product factor, Sylow
676	C₂	—	2	2	no	—
677	C₂	—	4	2	no	—
678	C₂	—	4	2	no	—
679	C₂	—	4	8	no	—
680	C₂	—	8	2	no	—
681	C₂	C₂×C₃×D₄²	1	1	yes	characteristic
682	C₂	C₃×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#525 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×#576	C₂	semidirect
2. C₂²×C₃×#135	C₂	semidirect
3. C₃×#3485	C₂	semidirect
4. C₃×#3283	C₂	semidirect
5. C₃×#3280	C₂	semidirect
6. C₃×#3267	C₂	semidirect
7. C₃×#3156	C₂	semidirect
8. C₃×#3059	C₂	semidirect
9. C₃×#3050	C₂	semidirect
10. C₃×#3042	C₂	semidirect
11. C₃×#3009	C₂	semidirect
12. C₃×#3005	C₂	semidirect
13. C₃×#2948	C₂	semidirect
14. C₃×#2929	C₂	semidirect
15. #61051	C₃	direct (N × H)
17. C₂²×C₃×#41	C₂²	semidirect
19. C₂×C₃×#137	C₂²	semidirect
21. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
24. C₂×C₃×#135	C₂²	semidirect
25. C₂×C₃×#135	C₂²	semidirect
26. C₂×C₃×#135	C₂²	semidirect
27. C₂²×C₃×Q₁₆	C₂²	semidirect
28. C₂²×C₃×SD₁₆	C₂²	semidirect
30. C₂²×C₃×D₈	C₂²	semidirect
32. C₂³×C₃×C₈	C₂²	semidirect
34. C₃×#547	C₂²	semidirect
36. C₃×#546	C₂²	semidirect
38. C₃×#536	C₂²	semidirect
41. C₃×#525	C₂²	semidirect
43. C₃×#518	C₂²	semidirect
44. C₃×#497	C₂²	semidirect
45. C₃×#495	C₂²	semidirect
46. C₃×#494	C₂²	semidirect
47. C₃×#470	C₂²	semidirect
48. C₃×#467	C₂²	semidirect
49. C₃×#467	C₂²	semidirect
50. C₃×#466	C₂²	semidirect
51. C₃×#465	C₂²	semidirect
52. C₃×#463	C₂²	semidirect
53. C₃×#461	C₂²	semidirect
54. C₃×#452	C₂²	semidirect
55. C₃×#452	C₂²	semidirect
56. C₃×#450	C₂²	semidirect
57. C₃×#449	C₂²	semidirect
58. C₃×#442	C₂²	semidirect
59. C₃×C₄×D₈	C₂²	semidirect
60. C₃×#435	C₂²	semidirect
61. C₃×C₈×D₄	C₂²	semidirect
62. C₃×#427	C₂²	semidirect
63. C₃×#424	C₂²	semidirect
64. C₃×#419	C₂²	semidirect
65. C₃×#418	C₂²	semidirect
66. C₃×#416	C₂²	semidirect
68. C₃×#407	C₂²	semidirect
69. C₃×#405	C₂²	semidirect
70. C₂×#576	C₂×C₃	semidirect
71. C₂²×#135	C₂×C₃	semidirect
72. #3485	C₂×C₃	semidirect
73. #3283	C₂×C₃	semidirect
74. #3280	C₂×C₃	semidirect
75. #3267	C₂×C₃	semidirect
76. #3156	C₂×C₃	semidirect
77. #3059	C₂×C₃	semidirect
78. #3050	C₂×C₃	semidirect
79. #3042	C₂×C₃	semidirect
80. #3009	C₂×C₃	semidirect
81. #3005	C₂×C₃	semidirect
82. #2948	C₂×C₃	semidirect
83. #2929	C₂×C₃	semidirect
120. C₂×C₃×D₈	D₄	semidirect
178. C₂²×#41	C₂²×C₃	semidirect
180. C₂×#137	C₂²×C₃	semidirect
182. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
185. C₂×#135	C₂²×C₃	semidirect
186. C₂×#135	C₂²×C₃	semidirect
187. C₂×#135	C₂²×C₃	semidirect
188. C₂²×Q₁₆	C₂²×C₃	semidirect
189. C₂²×SD₁₆	C₂²×C₃	semidirect
191. C₂²×D₈	C₂²×C₃	semidirect
193. C₂³×C₈	C₂²×C₃	semidirect
195. #547	C₂²×C₃	semidirect
197. #546	C₂²×C₃	semidirect
199. #536	C₂²×C₃	semidirect
202. #525	C₂²×C₃	semidirect
204. #518	C₂²×C₃	semidirect
205. #497	C₂²×C₃	semidirect
206. #495	C₂²×C₃	semidirect
207. #494	C₂²×C₃	semidirect
208. #470	C₂²×C₃	semidirect
209. #467	C₂²×C₃	semidirect
210. #467	C₂²×C₃	semidirect
211. #466	C₂²×C₃	semidirect
212. #465	C₂²×C₃	semidirect
213. #463	C₂²×C₃	semidirect
214. #461	C₂²×C₃	semidirect
215. #452	C₂²×C₃	semidirect
216. #452	C₂²×C₃	semidirect
217. #450	C₂²×C₃	semidirect
218. #449	C₂²×C₃	semidirect
219. #442	C₂²×C₃	semidirect
220. C₄×D₈	C₂²×C₃	semidirect
221. #435	C₂²×C₃	semidirect
222. C₈×D₄	C₂²×C₃	semidirect
223. #427	C₂²×C₃	semidirect
224. #424	C₂²×C₃	semidirect
225. #419	C₂²×C₃	semidirect
226. #418	C₂²×C₃	semidirect
227. #416	C₂²×C₃	semidirect
229. #407	C₂²×C₃	semidirect
230. #405	C₂²×C₃	semidirect
359. C₂×D₈	C₃×D₄	semidirect
675. C₃	#61051	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#576	2
C₂²×C₃×#135	2
C₃×#3485	2
C₃×#3283	2
C₃×#3280	2
C₃×#3267	2
C₃×#3156	2
C₃×#3059	2
C₃×#3050	2
C₃×#3042	2
C₃×#3009	2
C₃×#3005	2
C₃×#2948	2
C₃×#2929	2
#61051	3

Subgroups — C3×#61051 #10679658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#61051 #10679658