Subgroups — C₃×#60373 #10678980

Order 768.

5058 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#577	C₂	1	2	yes	maximal, complemented
3	C₃×#3489	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3012	C₂	1	6	yes	maximal, complemented
5	C₃×#3009	C₂	1	6	yes	maximal, complemented
6	C₃×#3007	C₂	1	3	yes	maximal, complemented
7	C₃×#3003	C₂	1	3	yes	maximal, complemented
8	C₃×#3002	C₂	1	3	yes	maximal, complemented
9	C₂×C₃×#450	C₂	1	3	yes	maximal, complemented
10	C₃×#2897	C₂	1	2	yes	maximal, complemented
11	C₃×#2880	C₂	1	1	yes	maximal, characteristic, complemented
12	#60373	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃×H(5,𝔽₂)	—	2	16	no	—
15	C₂×C₃×H(5,𝔽₂)	—	2	3	no	—
16	C₂×C₃×H(5,𝔽₂)	—	2	1	no	—
17	C₂×C₃×H(5,𝔽₂)	C₂²	1	6	yes	—
18	C₂×C₃×H(5,𝔽₂)	C₂²	1	2	yes	—
19	C₂²×C₃×#41	—	2	1	no	—
20	C₂²×C₃×#41	C₂²	1	3	yes	—
21	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
22	C₂³×C₃×D₄	—	2	3	no	—
23	C₂³×C₃×D₄	C₂²	1	3	yes	—
24	C₃×#577	—	2	16	no	—
25	C₂×C₃×#137	C₂²	1	6	yes	complemented
26	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
27	C₂×C₃×#135	C₂²	1	6	yes	complemented
28	C₂²×C₃×SD₁₆	C₂²	1	6	yes	complemented
29	C₂×C₃×#131	C₂²	1	2	yes	complemented
30	C₃×D₄×Q₈	—	2	1	no	—
31	C₃×#548	—	2	1	no	—
32	C₃×#547	—	2	3	no	—
33	C₃×#546	—	2	3	no	—
34	C₃×#537	C₂²	1	3	yes	complemented
35	C₃×#536	C₂²	1	3	yes	complemented
36	C₃×#525	C₂²	1	3	yes	complemented
37	C₂×C₃×C₂²⋊Q₈	C₂²	1	3	yes	complemented
38	C₃×#520	C₂²	1	1	yes	characteristic, complemented
39	C₃×#452	C₂²	1	6	yes	complemented
40	C₃×#452	C₂²	1	12	yes	complemented
41	C₃×#451	C₂²	1	3	yes	complemented
42	C₃×#450	C₂²	1	3	yes	complemented
43	C₃×#450	C₂²	1	6	yes	complemented
44	C₃×#450	C₂²	1	12	yes	complemented
45	C₃×#450	C₂²	1	6	yes	complemented
46	C₃×#449	C₂²	1	6	yes	complemented
47	C₃×#448	C₂²	1	1	yes	characteristic, complemented
48	C₃×#447	C₂²	1	3	yes	complemented
49	C₃×#447	C₂²	1	6	yes	complemented
50	C₃×#418	C₂²	1	6	yes	complemented
51	C₃×#417	C₂²	1	6	yes	complemented
52	C₃×#416	C₂²	1	6	yes	complemented
53	C₂×C₃×#102	C₂²	1	6	yes	complemented
54	C₃×#408	C₂²	1	6	yes	complemented
55	C₃×#407	C₂²	1	3	yes	complemented
56	C₂×C₃×C₂²⋊C₈	C₂²	1	3	yes	complemented
57	C₂²×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
58	C₂×#577	C₂×C₃	1	2	yes	complemented
59	#3489	C₂×C₃	1	1	yes	characteristic, complemented
60	#3012	C₂×C₃	1	6	yes	complemented
61	#3009	C₂×C₃	1	6	yes	complemented
62	#3007	C₂×C₃	1	3	yes	complemented
63	#3003	C₂×C₃	1	3	yes	complemented
64	#3002	C₂×C₃	1	3	yes	complemented
65	C₂×#450	C₂×C₃	1	3	yes	complemented
66	#2897	C₂×C₃	1	2	yes	complemented
67	#2880	C₂×C₃	1	1	yes	characteristic, complemented
68	C₂⁵×C₃	—	2	3	no	—
69	C₃×2₋¹⁺⁴	—	2	8	no	—
70	C₃×H(5,𝔽₂)	—	2	16	no	—
71	C₃×H(5,𝔽₂)	—	2	48	no	—
72	C₃×H(5,𝔽₂)	—	2	24	no	—
73	C₃×H(5,𝔽₂)	—	2	8	no	—
74	C₃×H(5,𝔽₂)	—	2	24	no	—
75	C₃×H(5,𝔽₂)	D₄	1	16	yes	complemented
76	C₂×C₃×#41	—	2	16	no	—
77	C₂×C₃×#41	—	2	24	no	—
78	C₂×C₃×#41	—	2	8	no	—
79	C₂×C₃×#41	—	2	6	no	—
80	C₂×C₃×#41	—	2	2	no	—
81	C₂×C₃×#41	—	2	3	no	—
82	C₂×C₃×#41	—	2	3	no	—
83	C₂×C₃×#41	—	2	3	no	—
84	C₂×C₃×#41	D₄	1	3	yes	—
85	C₂×C₃×#41	D₄	1	1	yes	characteristic
86	C₂×C₃×#41	C₂³	1	6	yes	—
87	C₂×C₃×#41	C₂³	1	6	yes	—
88	C₂×C₃×#41	C₂³	1	3	yes	—
89	C₂×C₃×#41	C₂³	1	1	yes	characteristic
90	C₂×C₃×#41	C₂³	1	3	yes	complemented
91	C₂²×C₃×Q₈	—	2	1	no	—
92	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
93	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
94	C₂²×C₃×Q₈	C₂³	1	3	yes	complemented
95	C₂²×C₃×D₄	—	2	48	no	—
96	C₂²×C₃×D₄	—	2	3	no	—
97	C₂²×C₃×D₄	—	2	12	no	—
98	C₂²×C₃×D₄	—	2	6	no	—
99	C₂²×C₃×D₄	—	2	6	no	—
100	C₂²×C₃×D₄	—	2	24	no	—
101	C₂²×C₃×D₄	—	2	3	no	—
102	C₂²×C₃×D₄	—	2	3	no	—
103	C₂²×C₃×D₄	—	2	3	no	—
104	C₂²×C₃×D₄	D₄	1	3	yes	—
105	C₂²×C₃×D₄	C₂³	1	3	yes	—
106	C₂²×C₃×D₄	C₂³	1	6	yes	—
107	C₂²×C₃×D₄	C₂³	1	6	yes	—
108	C₂³×C₃×C₄	—	2	3	no	—
109	C₂³×C₃×C₄	C₂³	1	3	yes	—
110	C₃×#137	—	2	24	no	—
111	C₃×AGL(1,ℤ/8)	—	2	24	no	—
112	C₃×#135	—	2	24	no	—
113	C₂×C₃×Q₁₆	C₂³	1	6	yes	complemented
114	C₂×C₃×SD₁₆	—	2	24	no	—
115	C₂×C₃×SD₁₆	C₂³	1	6	yes	complemented
116	C₂×C₃×SD₁₆	C₂³	1	12	yes	complemented
117	C₂×C₃×SD₁₆	C₂³	1	2	yes	complemented
118	C₂×C₃×D₈	C₂³	1	6	yes	complemented
119	C₃×#131	—	2	8	no	—
120	C₂×C₃×M₁₆	C₂³	1	6	yes	complemented
121	C₂²×C₃×C₈	C₂³	1	6	yes	complemented
122	C₃×C₄⋊Q₈	—	2	3	no	—
123	C₃×#124	—	2	3	no	—
124	C₃×#123	—	2	6	no	—
125	C₃×#123	—	2	3	no	—
126	C₃×C₂²⋊Q₈	—	2	3	no	—
127	C₃×C₂²⋊Q₈	—	2	3	no	—
128	C₃×C₂²⋊Q₈	—	2	3	no	—
129	C₃×C₂²⋊Q₈	C₂³	1	3	yes	—
130	C₃×C₂²⋊Q₈	C₂³	1	6	yes	complemented
131	C₃×C₂²⋊Q₈	C₂³	1	3	yes	complemented
132	C₃×#121	—	2	6	no	—
133	C₃×#121	—	2	3	no	—
134	C₃×#121	C₂³	1	1	yes	characteristic
135	C₃×#121	C₂³	1	3	yes	complemented
136	C₃×#120	—	2	3	no	—
137	C₃×C₄×Q₈	—	2	1	no	—
138	C₃×C₄×D₄	—	2	3	no	—
139	C₃×C₄×D₄	—	2	3	no	—
140	C₃×C₄×D₄	—	2	3	no	—
141	C₃×#117	C₂³	1	3	yes	complemented
142	C₂×C₃×C₄⋊C₄	—	2	3	no	—
143	C₂×C₃×C₄⋊C₄	C₂³	1	3	yes	complemented
144	C₂×C₃×C₂²⋊C₄	—	2	3	no	—
145	C₃×#103	C₂³	1	2	yes	complemented
146	C₃×#103	C₂³	1	6	yes	complemented
147	C₃×#102	C₂³	1	6	yes	complemented
148	C₃×#102	C₂³	1	12	yes	complemented
149	C₃×#102	C₂³	1	6	yes	complemented
150	C₃×C₂²⋊C₈	C₂³	1	1	yes	characteristic
151	C₃×C₂²⋊C₈	C₂³	1	3	yes	—
152	C₃×C₂²⋊C₈	C₂³	1	6	yes	complemented
153	C₃×C₂²⋊C₈	C₂³	1	6	yes	complemented
154	C₂×2₋¹⁺⁴	C₂²×C₃	1	1	yes	characteristic, complemented
155	C₂×H(5,𝔽₂)	—	2	16	no	—
156	C₂×H(5,𝔽₂)	—	2	3	no	—
157	C₂×H(5,𝔽₂)	—	2	1	no	—
158	C₂×H(5,𝔽₂)	C₂²×C₃	1	6	yes	—
159	C₂×H(5,𝔽₂)	C₂²×C₃	1	2	yes	—
160	C₂²×#41	—	2	1	no	—
161	C₂²×#41	C₂²×C₃	1	3	yes	—
162	C₂²×#41	C₂²×C₃	1	1	yes	characteristic
163	C₂³×D₄	—	2	3	no	—
164	C₂³×D₄	C₂²×C₃	1	3	yes	—
165	#577	—	2	16	no	—
166	C₂×#137	C₂²×C₃	1	6	yes	complemented
167	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	6	yes	complemented
168	C₂×#135	C₂²×C₃	1	6	yes	complemented
169	C₂²×SD₁₆	C₂²×C₃	1	6	yes	complemented
170	C₂×#131	C₂²×C₃	1	2	yes	complemented
171	D₄×Q₈	—	2	1	no	—
172	#548	—	2	1	no	—
173	#547	—	2	3	no	—
174	#546	—	2	3	no	—
175	#537	C₂²×C₃	1	3	yes	complemented
176	#536	C₂²×C₃	1	3	yes	complemented
177	#525	C₂²×C₃	1	3	yes	complemented
178	C₂×C₂²⋊Q₈	C₂²×C₃	1	3	yes	complemented
179	#520	C₂²×C₃	1	1	yes	characteristic, complemented
180	#452	C₂²×C₃	1	6	yes	complemented
181	#452	C₂²×C₃	1	12	yes	complemented
182	#451	C₂²×C₃	1	3	yes	complemented
183	#450	C₂²×C₃	1	3	yes	complemented
184	#450	C₂²×C₃	1	6	yes	complemented
185	#450	C₂²×C₃	1	12	yes	complemented
186	#450	C₂²×C₃	1	6	yes	complemented
187	#449	C₂²×C₃	1	6	yes	complemented
188	#448	C₂²×C₃	1	1	yes	characteristic, complemented
189	#447	C₂²×C₃	1	3	yes	complemented
190	#447	C₂²×C₃	1	6	yes	complemented
191	#418	C₂²×C₃	1	6	yes	complemented
192	#417	C₂²×C₃	1	6	yes	complemented
193	#416	C₂²×C₃	1	6	yes	complemented
194	C₂×#102	C₂²×C₃	1	6	yes	complemented
195	#408	C₂²×C₃	1	6	yes	complemented
196	#407	C₂²×C₃	1	3	yes	complemented
197	C₂×C₂²⋊C₈	C₂²×C₃	1	3	yes	complemented
198	C₂⁴×C₃	—	2	24	no	—
199	C₂⁴×C₃	—	2	3	no	—
200	C₂⁴×C₃	—	2	6	no	—
201	C₂⁴×C₃	—	2	6	no	—
202	C₂⁴×C₃	—	8	6	no	—
203	C₂⁴×C₃	—	8	6	no	—
204	C₂⁴×C₃	C₂×D₄	1	3	yes	—
205	C₃×#41	—	2	24	no	—
206	C₃×#41	—	2	48	no	—
207	C₃×#41	—	2	24	no	—
208	C₃×#41	—	2	16	no	—
209	C₃×#41	—	2	4	no	—
210	C₃×#41	—	2	12	no	—
211	C₃×#41	—	2	12	no	—
212	C₃×#41	—	2	12	no	—
213	C₃×#41	—	2	12	no	—
214	C₃×#41	—	2	12	no	—
215	C₃×#41	—	2	12	no	—
216	C₃×#41	—	4	12	no	—
217	C₃×#41	C₂×D₄	1	24	yes	—
218	C₃×#41	C₂×D₄	1	8	yes	—
219	C₂×C₃×Q₈	—	2	4	no	—
220	C₂×C₃×Q₈	—	2	4	no	—
221	C₂×C₃×Q₈	—	2	1	no	—
222	C₂×C₃×Q₈	—	2	3	no	—
223	C₂×C₃×Q₈	—	2	12	no	—
224	C₂×C₃×Q₈	—	4	4	no	—
225	C₂×C₃×Q₈	C₂×D₄	1	3	yes	—
226	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
227	C₂×C₃×Q₈	C₂⁴	1	3	yes	—
228	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
229	C₂×C₃×D₄	—	2	24	no	—
230	C₂×C₃×D₄	—	2	24	no	—
231	C₂×C₃×D₄	—	2	96	no	—
232	C₂×C₃×D₄	—	2	24	no	—
233	C₂×C₃×D₄	—	2	48	no	—
234	C₂×C₃×D₄	—	2	24	no	—
235	C₂×C₃×D₄	—	2	6	no	—
236	C₂×C₃×D₄	—	2	6	no	—
237	C₂×C₃×D₄	—	2	12	no	—
238	C₂×C₃×D₄	—	2	48	no	—
239	C₂×C₃×D₄	—	2	12	no	—
240	C₂×C₃×D₄	—	2	24	no	—
241	C₂×C₃×D₄	—	2	1	no	—
242	C₂×C₃×D₄	—	2	3	no	—
243	C₂×C₃×D₄	—	2	3	no	—
244	C₂×C₃×D₄	—	2	6	no	—
245	C₂×C₃×D₄	—	2	3	no	—
246	C₂×C₃×D₄	—	2	6	no	—
247	C₂×C₃×D₄	—	2	12	no	—
248	C₂×C₃×D₄	C₂×D₄	1	6	yes	—
249	C₂×C₃×D₄	C₂×D₄	1	3	yes	—
250	C₂×C₃×D₄	C₂×D₄	1	24	yes	—
251	C₂×C₃×D₄	C₂×D₄	1	3	yes	—
252	C₂×C₃×D₄	C₂⁴	1	1	yes	characteristic
253	C₂×C₃×D₄	C₂⁴	1	6	yes	—
254	C₂²×C₃×C₄	—	2	24	no	—
255	C₂²×C₃×C₄	—	2	1	no	—
256	C₂²×C₃×C₄	—	2	6	no	—
257	C₂²×C₃×C₄	—	2	6	no	—
258	C₂²×C₃×C₄	—	2	12	no	—
259	C₂²×C₃×C₄	—	2	3	no	—
260	C₂²×C₃×C₄	—	2	3	no	—
261	C₂²×C₃×C₄	—	2	3	no	—
262	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
263	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
264	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
265	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
266	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
267	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
268	C₃×Q₁₆	—	2	12	no	—
269	C₃×SD₁₆	—	2	4	no	—
270	C₃×SD₁₆	—	2	12	no	—
271	C₃×SD₁₆	—	2	24	no	—
272	C₃×D₈	—	2	12	no	—
273	C₃×M₁₆	—	2	12	no	—
274	C₂×C₃×C₈	—	2	12	no	—
275	C₂×C₃×C₈	C₂⁴	1	2	yes	—
276	C₂×C₃×C₈	C₂⁴	1	6	yes	—
277	C₃×C₄⋊C₄	—	2	3	no	—
278	C₃×C₄⋊C₄	—	2	6	no	—
279	C₃×C₄⋊C₄	—	2	3	no	—
280	C₃×C₄⋊C₄	—	2	3	no	—
281	C₃×C₄⋊C₄	C₂⁴	1	3	yes	—
282	C₃×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
283	C₃×C₂²⋊C₄	—	2	3	no	—
284	C₃×C₂²⋊C₄	—	2	6	no	—
285	C₃×C₂²⋊C₄	—	2	3	no	—
286	C₃×C₂²⋊C₄	—	2	3	no	—
287	C₃×C₂²⋊C₄	—	2	1	no	—
288	C₃×C₂²⋊C₄	—	2	3	no	—
289	C₃×C₄²	—	2	3	no	—
290	C₂⁵	—	2	3	no	—
291	2₋¹⁺⁴	—	2	8	no	—
292	H(5,𝔽₂)	—	2	16	no	—
293	H(5,𝔽₂)	—	2	48	no	—
294	H(5,𝔽₂)	—	2	24	no	—
295	H(5,𝔽₂)	—	2	8	no	—
296	H(5,𝔽₂)	—	2	24	no	—
297	H(5,𝔽₂)	C₃×D₄	1	16	yes	complemented
298	C₂×#41	—	2	16	no	—
299	C₂×#41	—	2	24	no	—
300	C₂×#41	—	2	8	no	—
301	C₂×#41	—	2	6	no	—
302	C₂×#41	—	2	2	no	—
303	C₂×#41	—	2	3	no	—
304	C₂×#41	—	2	3	no	—
305	C₂×#41	—	2	3	no	—
306	C₂×#41	C₃×D₄	1	3	yes	—
307	C₂×#41	C₃×D₄	1	1	yes	characteristic
308	C₂×#41	C₂³×C₃	1	6	yes	—
309	C₂×#41	C₂³×C₃	1	6	yes	—
310	C₂×#41	C₂³×C₃	1	3	yes	—
311	C₂×#41	C₂³×C₃	1	1	yes	characteristic
312	C₂×#41	C₂³×C₃	1	3	yes	complemented
313	C₂²×Q₈	—	2	1	no	—
314	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
315	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
316	C₂²×Q₈	C₂³×C₃	1	3	yes	complemented
317	C₂²×D₄	—	2	48	no	—
318	C₂²×D₄	—	2	3	no	—
319	C₂²×D₄	—	2	12	no	—
320	C₂²×D₄	—	2	6	no	—
321	C₂²×D₄	—	2	6	no	—
322	C₂²×D₄	—	2	24	no	—
323	C₂²×D₄	—	2	3	no	—
324	C₂²×D₄	—	2	3	no	—
325	C₂²×D₄	—	2	3	no	—
326	C₂²×D₄	C₃×D₄	1	3	yes	—
327	C₂²×D₄	C₂³×C₃	1	3	yes	—
328	C₂²×D₄	C₂³×C₃	1	6	yes	—
329	C₂²×D₄	C₂³×C₃	1	6	yes	—
330	C₂³×C₄	—	2	3	no	—
331	C₂³×C₄	C₂³×C₃	1	3	yes	—
332	#137	—	2	24	no	—
333	AGL(1,ℤ/8)	—	2	24	no	—
334	#135	—	2	24	no	—
335	C₂×Q₁₆	C₂³×C₃	1	6	yes	complemented
336	C₂×SD₁₆	—	2	24	no	—
337	C₂×SD₁₆	C₂³×C₃	1	6	yes	complemented
338	C₂×SD₁₆	C₂³×C₃	1	12	yes	complemented
339	C₂×SD₁₆	C₂³×C₃	1	2	yes	complemented
340	C₂×D₈	C₂³×C₃	1	6	yes	complemented
341	#131	—	2	8	no	—
342	C₂×M₁₆	C₂³×C₃	1	6	yes	complemented
343	C₂²×C₈	C₂³×C₃	1	6	yes	complemented
344	C₄⋊Q₈	—	2	3	no	—
345	#124	—	2	3	no	—
346	#123	—	2	6	no	—
347	#123	—	2	3	no	—
348	C₂²⋊Q₈	—	2	3	no	—
349	C₂²⋊Q₈	—	2	3	no	—
350	C₂²⋊Q₈	—	2	3	no	—
351	C₂²⋊Q₈	C₂³×C₃	1	3	yes	—
352	C₂²⋊Q₈	C₂³×C₃	1	6	yes	complemented
353	C₂²⋊Q₈	C₂³×C₃	1	3	yes	complemented
354	#121	—	2	6	no	—
355	#121	—	2	3	no	—
356	#121	C₂³×C₃	1	1	yes	characteristic
357	#121	C₂³×C₃	1	3	yes	complemented
358	#120	—	2	3	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×D₄	—	2	3	no	—
361	C₄×D₄	—	2	3	no	—
362	C₄×D₄	—	2	3	no	—
363	#117	C₂³×C₃	1	3	yes	complemented
364	C₂×C₄⋊C₄	—	2	3	no	—
365	C₂×C₄⋊C₄	C₂³×C₃	1	3	yes	complemented
366	C₂×C₂²⋊C₄	—	2	3	no	—
367	#103	C₂³×C₃	1	2	yes	complemented
368	#103	C₂³×C₃	1	6	yes	complemented
369	#102	C₂³×C₃	1	6	yes	complemented
370	#102	C₂³×C₃	1	12	yes	complemented
371	#102	C₂³×C₃	1	6	yes	complemented
372	C₂²⋊C₈	C₂³×C₃	1	1	yes	characteristic
373	C₂²⋊C₈	C₂³×C₃	1	3	yes	—
374	C₂²⋊C₈	C₂³×C₃	1	6	yes	complemented
375	C₂²⋊C₈	C₂³×C₃	1	6	yes	complemented
376	C₂³×C₃	—	2	12	no	—
377	C₂³×C₃	—	2	24	no	—
378	C₂³×C₃	—	2	6	no	—
379	C₂³×C₃	—	2	6	no	—
380	C₂³×C₃	—	2	6	no	—
381	C₂³×C₃	—	2	24	no	—
382	C₂³×C₃	—	2	1	no	—
383	C₂³×C₃	—	4	3	no	—
384	C₂³×C₃	—	4	6	no	—
385	C₂³×C₃	—	4	6	no	—
386	C₂³×C₃	—	8	6	no	—
387	C₂³×C₃	—	8	6	no	—
388	C₂³×C₃	—	8	6	no	—
389	C₂³×C₃	—	8	6	no	—
390	C₂³×C₃	—	8	48	no	—
391	C₂³×C₃	C₂²×D₄	1	3	yes	—
392	C₂³×C₃	C₂²×D₄	1	3	yes	—
393	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
394	C₃×Q₈	—	2	4	no	—
395	C₃×Q₈	—	2	6	no	—
396	C₃×Q₈	—	2	2	no	—
397	C₃×Q₈	—	2	6	no	—
398	C₃×Q₈	—	4	6	no	—
399	C₃×Q₈	#120	1	4	yes	—
400	C₃×Q₈	C₂²×D₄	1	4	yes	—
401	C₃×D₄	—	2	24	no	—
402	C₃×D₄	—	2	24	no	—
403	C₃×D₄	—	2	12	no	—
404	C₃×D₄	—	2	24	no	—
405	C₃×D₄	—	2	12	no	—
406	C₃×D₄	—	2	6	no	—
407	C₃×D₄	—	2	24	no	—
408	C₃×D₄	—	2	6	no	—
409	C₃×D₄	—	2	2	no	—
410	C₃×D₄	—	4	6	no	—
411	C₃×D₄	—	4	2	no	—
412	C₃×D₄	—	8	3	no	—
413	C₃×D₄	#120	1	12	yes	—
414	C₃×D₄	C₂²×D₄	1	12	yes	—
415	C₂×C₃×C₄	—	2	4	no	—
416	C₂×C₃×C₄	—	2	24	no	—
417	C₂×C₃×C₄	—	2	6	no	—
418	C₂×C₃×C₄	—	2	2	no	—
419	C₂×C₃×C₄	—	2	6	no	—
420	C₂×C₃×C₄	—	2	24	no	—
421	C₂×C₃×C₄	—	2	6	no	—
422	C₂×C₃×C₄	—	2	2	no	—
423	C₂×C₃×C₄	—	2	6	no	—
424	C₂×C₃×C₄	—	2	1	no	—
425	C₂×C₃×C₄	—	2	3	no	—
426	C₂×C₃×C₄	—	2	3	no	—
427	C₂×C₃×C₄	—	4	6	no	—
428	C₂×C₃×C₄	—	4	6	no	—
429	C₂×C₃×C₄	—	8	3	no	—
430	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
431	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
432	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
433	C₂×C₃×C₄	C₂²×D₄	1	12	yes	—
434	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
435	C₃×C₈	—	2	2	no	—
436	C₃×C₈	—	2	6	no	—
437	C₂⁴	—	2	24	no	—
438	C₂⁴	—	2	3	no	—
439	C₂⁴	—	2	6	no	—
440	C₂⁴	—	2	6	no	—
441	C₂⁴	—	8	6	no	—
442	C₂⁴	—	8	6	no	—
443	C₂⁴	C₂×C₃×D₄	1	3	yes	—
444	#41	—	2	24	no	—
445	#41	—	2	48	no	—
446	#41	—	2	24	no	—
447	#41	—	2	16	no	—
448	#41	—	2	4	no	—
449	#41	—	2	12	no	—
450	#41	—	2	12	no	—
451	#41	—	2	12	no	—
452	#41	—	2	12	no	—
453	#41	—	2	12	no	—
454	#41	—	2	12	no	—
455	#41	—	4	12	no	—
456	#41	C₂×C₃×D₄	1	24	yes	—
457	#41	C₂×C₃×D₄	1	8	yes	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×Q₈	—	2	4	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	3	no	—
462	C₂×Q₈	—	2	12	no	—
463	C₂×Q₈	—	4	4	no	—
464	C₂×Q₈	C₂×C₃×D₄	1	3	yes	—
465	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
466	C₂×Q₈	C₂⁴×C₃	1	3	yes	—
467	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
468	C₂×D₄	—	2	24	no	—
469	C₂×D₄	—	2	24	no	—
470	C₂×D₄	—	2	96	no	—
471	C₂×D₄	—	2	24	no	—
472	C₂×D₄	—	2	48	no	—
473	C₂×D₄	—	2	24	no	—
474	C₂×D₄	—	2	6	no	—
475	C₂×D₄	—	2	6	no	—
476	C₂×D₄	—	2	12	no	—
477	C₂×D₄	—	2	48	no	—
478	C₂×D₄	—	2	12	no	—
479	C₂×D₄	—	2	24	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	2	3	no	—
482	C₂×D₄	—	2	3	no	—
483	C₂×D₄	—	2	6	no	—
484	C₂×D₄	—	2	3	no	—
485	C₂×D₄	—	2	6	no	—
486	C₂×D₄	—	2	12	no	—
487	C₂×D₄	C₂×C₃×D₄	1	6	yes	—
488	C₂×D₄	C₂×C₃×D₄	1	3	yes	—
489	C₂×D₄	C₂×C₃×D₄	1	24	yes	—
490	C₂×D₄	C₂×C₃×D₄	1	3	yes	—
491	C₂×D₄	C₂⁴×C₃	1	1	yes	characteristic
492	C₂×D₄	C₂⁴×C₃	1	6	yes	—
493	C₂²×C₄	—	2	24	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	6	no	—
496	C₂²×C₄	—	2	6	no	—
497	C₂²×C₄	—	2	12	no	—
498	C₂²×C₄	—	2	3	no	—
499	C₂²×C₄	—	2	3	no	—
500	C₂²×C₄	—	2	3	no	—
501	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
502	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
503	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
504	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
506	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
507	Q₁₆	—	2	12	no	—
508	SD₁₆	—	2	4	no	—
509	SD₁₆	—	2	12	no	—
510	SD₁₆	—	2	24	no	—
511	D₈	—	2	12	no	—
512	M₁₆	—	2	12	no	—
513	C₂×C₈	—	2	12	no	—
514	C₂×C₈	C₂⁴×C₃	1	2	yes	—
515	C₂×C₈	C₂⁴×C₃	1	6	yes	—
516	C₄⋊C₄	—	2	3	no	—
517	C₄⋊C₄	—	2	6	no	—
518	C₄⋊C₄	—	2	3	no	—
519	C₄⋊C₄	—	2	3	no	—
520	C₄⋊C₄	C₂⁴×C₃	1	3	yes	—
521	C₄⋊C₄	C₂⁴×C₃	1	1	yes	characteristic
522	C₂²⋊C₄	—	2	3	no	—
523	C₂²⋊C₄	—	2	6	no	—
524	C₂²⋊C₄	—	2	3	no	—
525	C₂²⋊C₄	—	2	3	no	—
526	C₂²⋊C₄	—	2	1	no	—
527	C₂²⋊C₄	—	2	3	no	—
528	C₄²	—	2	3	no	—
529	C₂²×C₃	—	2	3	no	—
530	C₂²×C₃	—	2	1	no	—
531	C₂²×C₃	—	2	1	no	—
532	C₂²×C₃	—	2	3	no	—
533	C₂²×C₃	—	2	3	no	—
534	C₂²×C₃	—	2	3	no	—
535	C₂²×C₃	—	2	12	no	—
536	C₂²×C₃	—	2	3	no	—
537	C₂²×C₃	—	4	12	no	—
538	C₂²×C₃	—	4	24	no	—
539	C₂²×C₃	—	4	6	no	—
540	C₂²×C₃	—	4	6	no	—
541	C₂²×C₃	—	4	1	no	—
542	C₂²×C₃	—	4	1	no	—
543	C₂²×C₃	—	4	1	no	—
544	C₂²×C₃	—	8	24	no	—
545	C₂²×C₃	—	8	24	no	—
546	C₂²×C₃	C₂×#120	1	6	yes	—
547	C₂²×C₃	#577	1	2	yes	complemented
548	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
549	C₃×C₄	—	2	4	no	—
550	C₃×C₄	—	2	3	no	—
551	C₃×C₄	—	2	1	no	—
552	C₃×C₄	—	4	3	no	—
553	C₃×C₄	—	4	3	no	—
554	C₃×C₄	—	4	1	no	—
555	C₃×C₄	C₂×#120	1	6	yes	—
556	C₃×C₄	C₂³×D₄	1	2	yes	—
557	C₂³	—	2	12	no	—
558	C₂³	—	2	24	no	—
559	C₂³	—	2	6	no	—
560	C₂³	—	2	6	no	—
561	C₂³	—	2	6	no	—
562	C₂³	—	2	24	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	4	3	no	—
565	C₂³	—	4	6	no	—
566	C₂³	—	4	6	no	—
567	C₂³	—	8	6	no	—
568	C₂³	—	8	6	no	—
569	C₂³	—	8	6	no	—
570	C₂³	—	8	6	no	—
571	C₂³	—	8	48	no	—
572	C₂³	C₂²×C₃×D₄	1	3	yes	—
573	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
574	C₂³	C₂²×C₃×D₄	1	3	yes	—
575	Q₈	—	2	4	no	—
576	Q₈	—	2	2	no	—
577	Q₈	—	2	6	no	—
578	Q₈	—	2	6	no	—
579	Q₈	—	4	6	no	—
580	Q₈	C₃×#120	1	4	yes	—
581	Q₈	C₂²×C₃×D₄	1	4	yes	—
582	D₄	—	2	24	no	—
583	D₄	—	2	24	no	—
584	D₄	—	2	12	no	—
585	D₄	—	2	24	no	—
586	D₄	—	2	12	no	—
587	D₄	—	2	6	no	—
588	D₄	—	2	2	no	—
589	D₄	—	2	6	no	—
590	D₄	—	2	24	no	—
591	D₄	—	4	2	no	—
592	D₄	—	4	6	no	—
593	D₄	—	8	3	no	—
594	D₄	C₃×#120	1	12	yes	—
595	D₄	C₂²×C₃×D₄	1	12	yes	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	24	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	6	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	6	no	—
602	C₂×C₄	—	2	6	no	—
603	C₂×C₄	—	2	24	no	—
604	C₂×C₄	—	2	6	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	3	no	—
607	C₂×C₄	—	2	3	no	—
608	C₂×C₄	—	4	6	no	—
609	C₂×C₄	—	4	6	no	—
610	C₂×C₄	—	8	3	no	—
611	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
612	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
613	C₂×C₄	C₂²×C₃×D₄	1	12	yes	—
614	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
615	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
616	C₈	—	2	2	no	—
617	C₈	—	2	6	no	—
618	C₂×C₃	—	2	2	no	—
619	C₂×C₃	—	2	6	no	—
620	C₂×C₃	—	4	3	no	—
621	C₂×C₃	—	4	12	no	—
622	C₂×C₃	—	8	1	no	—
623	C₂×C₃	C₂²×#120	1	1	yes	characteristic
624	C₂×C₃	C₂×#577	1	1	yes	characteristic
625	C₂×C₃	#3585	1	1	yes	characteristic
626	C₂²	—	2	1	no	—
627	C₂²	—	2	3	no	—
628	C₂²	—	2	3	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	2	3	no	—
631	C₂²	—	2	12	no	—
632	C₂²	—	2	3	no	—
633	C₂²	—	2	3	no	—
634	C₂²	—	4	12	no	—
635	C₂²	—	4	24	no	—
636	C₂²	—	4	6	no	—
637	C₂²	—	4	6	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	1	no	—
641	C₂²	—	8	24	no	—
642	C₂²	—	8	24	no	—
643	C₂²	C₂×C₃×#120	1	6	yes	—
644	C₂²	C₃×#577	1	2	yes	complemented
645	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
646	C₄	—	2	4	no	—
647	C₄	—	2	3	no	—
648	C₄	—	2	1	no	—
649	C₄	—	4	3	no	—
650	C₄	—	4	1	no	—
651	C₄	—	4	3	no	—
652	C₄	C₂×C₃×#120	1	6	yes	—
653	C₄	C₂³×C₃×D₄	1	2	yes	—
654	C₃	#60373	1	1	yes	characteristic, direct product factor, Sylow
655	C₂	—	2	2	no	—
656	C₂	—	2	6	no	—
657	C₂	—	4	3	no	—
658	C₂	—	4	12	no	—
659	C₂	—	8	1	no	—
660	C₂	C₂²×C₃×#120	1	1	yes	characteristic
661	C₂	C₂×C₃×#577	1	1	yes	characteristic
662	C₂	C₃×#3585	1	1	yes	characteristic

	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 ◁ C₂²×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×H(5,𝔽₂)	C₂	semidirect
2. C₂×C₃×#577	C₂	semidirect
3. C₃×#3489	C₂	semidirect
4. C₃×#3012	C₂	semidirect
5. C₃×#3009	C₂	semidirect
6. C₃×#3007	C₂	semidirect
7. C₃×#3003	C₂	semidirect
8. C₃×#3002	C₂	semidirect
9. C₂×C₃×#450	C₂	semidirect
10. C₃×#2897	C₂	semidirect
11. C₃×#2880	C₂	semidirect
12. #60373	C₃	direct (N × H)
13. C₂×C₃×2₋¹⁺⁴	C₂²	semidirect
25. C₂×C₃×#137	C₂²	semidirect
26. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
27. C₂×C₃×#135	C₂²	semidirect
28. C₂²×C₃×SD₁₆	C₂²	semidirect
29. C₂×C₃×#131	C₂²	semidirect
34. C₃×#537	C₂²	semidirect
35. C₃×#536	C₂²	semidirect
36. C₃×#525	C₂²	semidirect
37. C₂×C₃×C₂²⋊Q₈	C₂²	semidirect
38. C₃×#520	C₂²	semidirect
39. C₃×#452	C₂²	semidirect
40. C₃×#452	C₂²	semidirect
41. C₃×#451	C₂²	semidirect
42. C₃×#450	C₂²	semidirect
43. C₃×#450	C₂²	semidirect
44. C₃×#450	C₂²	semidirect
45. C₃×#450	C₂²	semidirect
46. C₃×#449	C₂²	semidirect
47. C₃×#448	C₂²	semidirect
48. C₃×#447	C₂²	semidirect
49. C₃×#447	C₂²	semidirect
50. C₃×#418	C₂²	semidirect
51. C₃×#417	C₂²	semidirect
52. C₃×#416	C₂²	semidirect
53. C₂×C₃×#102	C₂²	semidirect
54. C₃×#408	C₂²	semidirect
55. C₃×#407	C₂²	semidirect
56. C₂×C₃×C₂²⋊C₈	C₂²	semidirect
57. C₂²×H(5,𝔽₂)	C₂×C₃	semidirect
58. C₂×#577	C₂×C₃	semidirect
59. #3489	C₂×C₃	semidirect
60. #3012	C₂×C₃	semidirect
61. #3009	C₂×C₃	semidirect
62. #3007	C₂×C₃	semidirect
63. #3003	C₂×C₃	semidirect
64. #3002	C₂×C₃	semidirect
65. C₂×#450	C₂×C₃	semidirect
66. #2897	C₂×C₃	semidirect
67. #2880	C₂×C₃	semidirect
75. C₃×H(5,𝔽₂)	D₄	semidirect
90. C₂×C₃×#41	C₂³	semidirect
94. C₂²×C₃×Q₈	C₂³	semidirect
113. C₂×C₃×Q₁₆	C₂³	semidirect
115. C₂×C₃×SD₁₆	C₂³	semidirect
116. C₂×C₃×SD₁₆	C₂³	semidirect
117. C₂×C₃×SD₁₆	C₂³	semidirect
118. C₂×C₃×D₈	C₂³	semidirect
120. C₂×C₃×M₁₆	C₂³	semidirect
121. C₂²×C₃×C₈	C₂³	semidirect
130. C₃×C₂²⋊Q₈	C₂³	semidirect
131. C₃×C₂²⋊Q₈	C₂³	semidirect
135. C₃×#121	C₂³	semidirect
141. C₃×#117	C₂³	semidirect
143. C₂×C₃×C₄⋊C₄	C₂³	semidirect
145. C₃×#103	C₂³	semidirect
146. C₃×#103	C₂³	semidirect
147. C₃×#102	C₂³	semidirect
148. C₃×#102	C₂³	semidirect
149. C₃×#102	C₂³	semidirect
152. C₃×C₂²⋊C₈	C₂³	semidirect
153. C₃×C₂²⋊C₈	C₂³	semidirect
154. C₂×2₋¹⁺⁴	C₂²×C₃	semidirect
166. C₂×#137	C₂²×C₃	semidirect
167. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
168. C₂×#135	C₂²×C₃	semidirect
169. C₂²×SD₁₆	C₂²×C₃	semidirect
170. C₂×#131	C₂²×C₃	semidirect
175. #537	C₂²×C₃	semidirect
176. #536	C₂²×C₃	semidirect
177. #525	C₂²×C₃	semidirect
178. C₂×C₂²⋊Q₈	C₂²×C₃	semidirect
179. #520	C₂²×C₃	semidirect
180. #452	C₂²×C₃	semidirect
181. #452	C₂²×C₃	semidirect
182. #451	C₂²×C₃	semidirect
183. #450	C₂²×C₃	semidirect
184. #450	C₂²×C₃	semidirect
185. #450	C₂²×C₃	semidirect
186. #450	C₂²×C₃	semidirect
187. #449	C₂²×C₃	semidirect
188. #448	C₂²×C₃	semidirect
189. #447	C₂²×C₃	semidirect
190. #447	C₂²×C₃	semidirect
191. #418	C₂²×C₃	semidirect
192. #417	C₂²×C₃	semidirect
193. #416	C₂²×C₃	semidirect
194. C₂×#102	C₂²×C₃	semidirect
195. #408	C₂²×C₃	semidirect
196. #407	C₂²×C₃	semidirect
197. C₂×C₂²⋊C₈	C₂²×C₃	semidirect
297. H(5,𝔽₂)	C₃×D₄	semidirect
312. C₂×#41	C₂³×C₃	semidirect
316. C₂²×Q₈	C₂³×C₃	semidirect
335. C₂×Q₁₆	C₂³×C₃	semidirect
337. C₂×SD₁₆	C₂³×C₃	semidirect
338. C₂×SD₁₆	C₂³×C₃	semidirect
339. C₂×SD₁₆	C₂³×C₃	semidirect
340. C₂×D₈	C₂³×C₃	semidirect
342. C₂×M₁₆	C₂³×C₃	semidirect
343. C₂²×C₈	C₂³×C₃	semidirect
352. C₂²⋊Q₈	C₂³×C₃	semidirect
353. C₂²⋊Q₈	C₂³×C₃	semidirect
357. #121	C₂³×C₃	semidirect
363. #117	C₂³×C₃	semidirect
365. C₂×C₄⋊C₄	C₂³×C₃	semidirect
367. #103	C₂³×C₃	semidirect
368. #103	C₂³×C₃	semidirect
369. #102	C₂³×C₃	semidirect
370. #102	C₂³×C₃	semidirect
371. #102	C₂³×C₃	semidirect
374. C₂²⋊C₈	C₂³×C₃	semidirect
375. C₂²⋊C₈	C₂³×C₃	semidirect
547. C₂²×C₃	#577	semidirect
644. C₂²	C₃×#577	semidirect
654. C₃	#60373	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×H(5,𝔽₂)	2
C₂×C₃×#577	2
C₃×#3489	2
C₃×#3012	2
C₃×#3009	2
C₃×#3007	2
C₃×#3003	2
C₃×#3002	2
C₂×C₃×#450	2
C₃×#2897	2
C₃×#2880	2
#60373	3

Subgroups — C3×#60373 #10678980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60373 #10678980