Subgroups — C₃×#60428 #10679035

Order 768.

2866 conjugacy classes of subgroups.

The subgroup lattice has 572 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	2	yes	maximal, complemented
2	C₃×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#3045	C₂	1	6	yes	maximal, complemented
5	C₃×#3041	C₂	1	6	yes	maximal, complemented
6	C₃×#3038	C₂	1	3	yes	maximal, complemented
7	C₃×#3036	C₂	1	3	yes	maximal, complemented
8	C₃×#3035	C₂	1	3	yes	maximal, complemented
9	C₂×C₃×#459	C₂	1	3	yes	maximal, complemented
10	C₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#2897	C₂	1	2	yes	maximal, complemented
12	#60428	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×H(5,𝔽₂)	C₂²	1	2	yes	—
14	C₂×C₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
15	C₃×#576	—	2	16	no	—
16	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
17	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
18	C₂×C₃×#135	C₂²	1	6	yes	complemented
19	C₂²×C₃×D₈	C₂²	1	6	yes	complemented
20	C₂×C₃×#131	C₂²	1	2	yes	—
21	C₃×#550	—	2	1	no	—
22	C₃×#550	—	2	1	no	—
23	C₃×D₄²	—	2	3	no	—
24	C₃×D₄²	—	2	3	no	—
25	C₃×#535	C₂²	1	6	yes	complemented
26	C₃×#532	C₂²	1	3	yes	complemented
27	C₂×C₃×#127	C₂²	1	3	yes	complemented
28	C₃×#520	—	2	1	no	—
29	C₃×#520	C₂²	1	1	yes	characteristic
30	C₃×#520	C₂²	1	3	yes	complemented
31	C₃×#518	—	2	3	no	—
32	C₃×#518	C₂²	1	3	yes	complemented
33	C₃×C₄×#41	—	2	1	no	—
34	C₃×C₄×#41	C₂²	1	1	yes	characteristic
35	C₃×C₄×#41	C₂²	1	3	yes	complemented
36	C₂×C₃×C₄×D₄	—	2	3	no	—
37	C₂×C₃×C₄×D₄	C₂²	1	3	yes	complemented
38	C₃×#464	C₂²	1	6	yes	complemented
39	C₃×#463	C₂²	1	6	yes	complemented
40	C₃×#463	C₂²	1	12	yes	complemented
41	C₃×#462	C₂²	1	1	yes	characteristic, complemented
42	C₃×#461	C₂²	1	3	yes	complemented
43	C₃×#460	C₂²	1	3	yes	complemented
44	C₃×#460	C₂²	1	6	yes	complemented
45	C₃×#459	C₂²	1	3	yes	complemented
46	C₃×#459	C₂²	1	6	yes	complemented
47	C₃×#459	C₂²	1	12	yes	complemented
48	C₃×#459	C₂²	1	6	yes	complemented
49	C₃×#424	C₂²	1	6	yes	complemented
50	C₃×#423	C₂²	1	3	yes	complemented
51	C₂×C₃×C₄⋊C₈	C₂²	1	3	yes	complemented
52	C₃×#418	C₂²	1	6	yes	complemented
53	C₃×#417	C₂²	1	6	yes	complemented
54	C₃×#416	C₂²	1	6	yes	complemented
55	C₂×C₃×#102	C₂²	1	6	yes	complemented
56	C₂×#576	C₂×C₃	1	2	yes	complemented
57	#3498	C₂×C₃	1	1	yes	characteristic, complemented
58	C₄×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
59	#3045	C₂×C₃	1	6	yes	complemented
60	#3041	C₂×C₃	1	6	yes	complemented
61	#3038	C₂×C₃	1	3	yes	complemented
62	#3036	C₂×C₃	1	3	yes	complemented
63	#3035	C₂×C₃	1	3	yes	complemented
64	C₂×#459	C₂×C₃	1	3	yes	complemented
65	#2906	C₂×C₃	1	1	yes	characteristic, complemented
66	#2897	C₂×C₃	1	2	yes	complemented
67	C₃×H(5,𝔽₂)	—	2	4	no	—
68	C₃×H(5,𝔽₂)	—	2	8	no	—
69	C₃×H(5,𝔽₂)	—	2	8	no	—
70	C₃×H(5,𝔽₂)	D₄	1	8	yes	complemented
71	C₂×C₃×#41	—	2	1	no	—
72	C₂×C₃×#41	—	2	2	no	—
73	C₂×C₃×#41	C₂³	1	6	yes	—
74	C₂×C₃×#41	C₂³	1	3	yes	—
75	C₂×C₃×#41	C₂³	1	1	yes	characteristic
76	C₂²×C₃×D₄	—	2	3	no	—
77	C₂²×C₃×D₄	—	2	6	no	—
78	C₂²×C₃×D₄	—	2	3	no	—
79	C₂²×C₃×D₄	—	2	3	no	—
80	C₂²×C₃×D₄	C₂³	1	6	yes	—
81	C₂²×C₃×D₄	C₂³	1	3	yes	—
82	C₂³×C₃×C₄	—	2	3	no	—
83	C₃×AGL(1,ℤ/8)	—	2	24	no	—
84	C₃×AGL(1,ℤ/8)	—	2	24	no	—
85	C₃×#135	—	2	24	no	—
86	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
87	C₂×C₃×SD₁₆	C₂³	1	6	yes	—
88	C₂×C₃×SD₁₆	C₂³	1	6	yes	—
89	C₂×C₃×D₈	—	2	24	no	—
90	C₂×C₃×D₈	C₂³	1	6	yes	—
91	C₂×C₃×D₈	C₂³	1	12	yes	—
92	C₃×#131	—	2	8	no	—
93	C₂×C₃×M₁₆	C₂³	1	6	yes	—
94	C₂²×C₃×C₈	C₂³	1	6	yes	—
95	C₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
96	C₃×#127	—	2	3	no	—
97	C₃×#127	—	2	3	no	—
98	C₃×#127	C₂³	1	3	yes	—
99	C₃×#127	C₂³	1	6	yes	—
100	C₃×#124	C₂³	1	6	yes	—
101	C₃×#121	—	2	3	no	—
102	C₃×#121	—	2	3	no	—
103	C₃×#121	—	2	6	no	—
104	C₃×#121	—	2	6	no	—
105	C₃×#120	—	2	12	no	—
106	C₃×C₄×Q₈	—	2	1	no	—
107	C₃×C₄×Q₈	D₄	1	1	yes	characteristic
108	C₃×C₄×Q₈	D₄	1	1	yes	characteristic
109	C₃×C₄×Q₈	C₂³	1	1	yes	characteristic
110	C₃×C₄×Q₈	C₂³	1	3	yes	—
111	C₃×C₄×D₄	—	2	3	no	—
112	C₃×C₄×D₄	—	2	3	no	—
113	C₃×C₄×D₄	—	2	3	no	—
114	C₃×C₄×D₄	—	2	3	no	—
115	C₃×C₄×D₄	—	2	3	no	—
116	C₃×C₄×D₄	—	2	6	no	—
117	C₃×C₄×D₄	—	2	3	no	—
118	C₃×C₄×D₄	—	2	3	no	—
119	C₃×C₄×D₄	—	2	3	no	—
120	C₃×C₄×D₄	—	2	3	no	—
121	C₃×C₄×D₄	D₄	1	3	yes	—
122	C₃×C₄×D₄	D₄	1	3	yes	—
123	C₃×C₄×D₄	C₂³	1	3	yes	—
124	C₃×C₄×D₄	C₂³	1	6	yes	—
125	C₃×C₄×D₄	C₂³	1	3	yes	—
126	C₃×#117	—	2	3	no	—
127	C₃×#117	—	2	3	no	—
128	C₃×#117	C₂³	1	3	yes	—
129	C₃×#117	C₂³	1	3	yes	—
130	C₂×C₃×C₄⋊C₄	—	2	3	no	—
131	C₂×C₃×C₄⋊C₄	C₂³	1	3	yes	—
132	C₂×C₃×C₂²⋊C₄	—	2	3	no	—
133	C₂×C₃×C₂²⋊C₄	—	2	6	no	—
134	C₂×C₃×C₄²	—	2	3	no	—
135	C₂×C₃×C₄²	C₂³	1	3	yes	—
136	C₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
137	C₃×C₄⋊C₈	C₂³	1	3	yes	—
138	C₃×C₄⋊C₈	C₂³	1	6	yes	—
139	C₃×C₄⋊C₈	C₂³	1	6	yes	—
140	C₃×#103	C₂³	1	2	yes	—
141	C₃×#103	C₂³	1	6	yes	—
142	C₃×#102	C₂³	1	6	yes	—
143	C₃×#102	C₂³	1	12	yes	—
144	C₃×#102	C₂³	1	6	yes	—
145	C₂×H(5,𝔽₂)	C₂²×C₃	1	2	yes	—
146	C₂×H(5,𝔽₂)	C₂²×C₃	1	1	yes	characteristic
147	#576	—	2	16	no	—
148	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	6	yes	complemented
149	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	6	yes	complemented
150	C₂×#135	C₂²×C₃	1	6	yes	complemented
151	C₂²×D₈	C₂²×C₃	1	6	yes	complemented
152	C₂×#131	C₂²×C₃	1	2	yes	—
153	#550	—	2	1	no	—
154	#550	—	2	1	no	—
155	D₄²	—	2	3	no	—
156	D₄²	—	2	3	no	—
157	#535	C₂²×C₃	1	6	yes	complemented
158	#532	C₂²×C₃	1	3	yes	complemented
159	C₂×#127	C₂²×C₃	1	3	yes	complemented
160	#520	—	2	1	no	—
161	#520	C₂²×C₃	1	1	yes	characteristic
162	#520	C₂²×C₃	1	3	yes	complemented
163	#518	—	2	3	no	—
164	#518	C₂²×C₃	1	3	yes	complemented
165	C₄×#41	—	2	1	no	—
166	C₄×#41	C₂²×C₃	1	1	yes	characteristic
167	C₄×#41	C₂²×C₃	1	3	yes	complemented
168	C₂×C₄×D₄	—	2	3	no	—
169	C₂×C₄×D₄	C₂²×C₃	1	3	yes	complemented
170	#464	C₂²×C₃	1	6	yes	complemented
171	#463	C₂²×C₃	1	6	yes	complemented
172	#463	C₂²×C₃	1	12	yes	complemented
173	#462	C₂²×C₃	1	1	yes	characteristic, complemented
174	#461	C₂²×C₃	1	3	yes	complemented
175	#460	C₂²×C₃	1	3	yes	complemented
176	#460	C₂²×C₃	1	6	yes	complemented
177	#459	C₂²×C₃	1	3	yes	complemented
178	#459	C₂²×C₃	1	6	yes	complemented
179	#459	C₂²×C₃	1	12	yes	complemented
180	#459	C₂²×C₃	1	6	yes	complemented
181	#424	C₂²×C₃	1	6	yes	complemented
182	#423	C₂²×C₃	1	3	yes	complemented
183	C₂×C₄⋊C₈	C₂²×C₃	1	3	yes	complemented
184	#418	C₂²×C₃	1	6	yes	complemented
185	#417	C₂²×C₃	1	6	yes	complemented
186	#416	C₂²×C₃	1	6	yes	complemented
187	C₂×#102	C₂²×C₃	1	6	yes	complemented
188	C₂⁴×C₃	—	2	3	no	—
189	C₂⁴×C₃	—	2	6	no	—
190	C₃×#41	—	2	8	no	—
191	C₃×#41	—	2	6	no	—
192	C₃×#41	—	2	12	no	—
193	C₃×#41	—	2	12	no	—
194	C₃×#41	—	4	8	no	—
195	C₃×#41	C₂×D₄	1	4	yes	—
196	C₃×#41	C₂×D₄	1	12	yes	complemented
197	C₃×#41	#41	1	4	yes	—
198	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
199	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
200	C₂×C₃×Q₈	C₂⁴	1	1	yes	characteristic
201	C₂×C₃×D₄	—	2	3	no	—
202	C₂×C₃×D₄	—	2	24	no	—
203	C₂×C₃×D₄	—	2	6	no	—
204	C₂×C₃×D₄	—	2	3	no	—
205	C₂×C₃×D₄	—	2	12	no	—
206	C₂×C₃×D₄	—	2	3	no	—
207	C₂×C₃×D₄	—	2	3	no	—
208	C₂×C₃×D₄	—	2	6	no	—
209	C₂×C₃×D₄	—	2	6	no	—
210	C₂×C₃×D₄	—	2	6	no	—
211	C₂×C₃×D₄	—	2	6	no	—
212	C₂×C₃×D₄	—	2	12	no	—
213	C₂×C₃×D₄	—	4	24	no	—
214	C₂×C₃×D₄	—	8	6	no	—
215	C₂×C₃×D₄	—	8	6	no	—
216	C₂×C₃×D₄	C₂×D₄	1	3	yes	—
217	C₂×C₃×D₄	C₂×D₄	1	12	yes	complemented
218	C₂×C₃×D₄	C₂⁴	1	6	yes	—
219	C₂×C₃×D₄	C₂⁴	1	3	yes	—
220	C₂²×C₃×C₄	—	2	3	no	—
221	C₂²×C₃×C₄	—	2	3	no	—
222	C₂²×C₃×C₄	—	2	3	no	—
223	C₂²×C₃×C₄	—	2	6	no	—
224	C₂²×C₃×C₄	—	2	3	no	—
225	C₂²×C₃×C₄	—	2	6	no	—
226	C₂²×C₃×C₄	—	8	6	no	—
227	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
228	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
229	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
230	C₃×Q₁₆	—	2	4	no	—
231	C₃×SD₁₆	—	2	12	no	—
232	C₃×SD₁₆	—	2	12	no	—
233	C₃×D₈	—	2	12	no	—
234	C₃×D₈	—	2	24	no	—
235	C₃×M₁₆	—	2	12	no	—
236	C₂×C₃×C₈	—	2	12	no	—
237	C₂×C₃×C₈	C₂⁴	1	2	yes	—
238	C₂×C₃×C₈	C₂⁴	1	6	yes	—
239	C₃×C₄⋊C₄	—	2	3	no	—
240	C₃×C₄⋊C₄	—	2	3	no	—
241	C₃×C₄⋊C₄	—	2	1	no	—
242	C₃×C₄⋊C₄	—	2	1	no	—
243	C₃×C₄⋊C₄	—	2	1	no	—
244	C₃×C₄⋊C₄	C₂×D₄	1	3	yes	—
245	C₃×C₄⋊C₄	C₂×D₄	1	3	yes	—
246	C₃×C₄⋊C₄	C₂⁴	1	3	yes	—
247	C₃×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
248	C₃×C₂²⋊C₄	—	2	3	no	—
249	C₃×C₂²⋊C₄	—	2	3	no	—
250	C₃×C₂²⋊C₄	—	2	6	no	—
251	C₃×C₂²⋊C₄	—	2	3	no	—
252	C₃×C₂²⋊C₄	—	2	6	no	—
253	C₃×C₂²⋊C₄	—	2	6	no	—
254	C₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
255	C₃×C₄²	—	2	1	no	—
256	C₃×C₄²	—	2	3	no	—
257	C₃×C₄²	—	2	3	no	—
258	C₃×C₄²	C₂×D₄	1	3	yes	—
259	C₃×C₄²	C₂×D₄	1	3	yes	—
260	C₃×C₄²	C₂⁴	1	1	yes	characteristic
261	C₃×C₄²	C₂⁴	1	3	yes	—
262	H(5,𝔽₂)	—	2	4	no	—
263	H(5,𝔽₂)	—	2	8	no	—
264	H(5,𝔽₂)	—	2	8	no	—
265	H(5,𝔽₂)	C₃×D₄	1	8	yes	complemented
266	C₂×#41	—	2	1	no	—
267	C₂×#41	—	2	2	no	—
268	C₂×#41	C₂³×C₃	1	6	yes	—
269	C₂×#41	C₂³×C₃	1	3	yes	—
270	C₂×#41	C₂³×C₃	1	1	yes	characteristic
271	C₂²×D₄	—	2	3	no	—
272	C₂²×D₄	—	2	6	no	—
273	C₂²×D₄	—	2	3	no	—
274	C₂²×D₄	—	2	3	no	—
275	C₂²×D₄	C₂³×C₃	1	6	yes	—
276	C₂²×D₄	C₂³×C₃	1	3	yes	—
277	C₂³×C₄	—	2	3	no	—
278	AGL(1,ℤ/8)	—	2	24	no	—
279	AGL(1,ℤ/8)	—	2	24	no	—
280	#135	—	2	24	no	—
281	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
282	C₂×SD₁₆	C₂³×C₃	1	6	yes	—
283	C₂×SD₁₆	C₂³×C₃	1	6	yes	—
284	C₂×D₈	—	2	24	no	—
285	C₂×D₈	C₂³×C₃	1	6	yes	—
286	C₂×D₈	C₂³×C₃	1	12	yes	—
287	#131	—	2	8	no	—
288	C₂×M₁₆	C₂³×C₃	1	6	yes	—
289	C₂²×C₈	C₂³×C₃	1	6	yes	—
290	C₄⋊Q₈	C₂³×C₃	1	1	yes	characteristic
291	#127	—	2	3	no	—
292	#127	—	2	3	no	—
293	#127	C₂³×C₃	1	3	yes	—
294	#127	C₂³×C₃	1	6	yes	—
295	#124	C₂³×C₃	1	6	yes	—
296	#121	—	2	3	no	—
297	#121	—	2	3	no	—
298	#121	—	2	6	no	—
299	#121	—	2	6	no	—
300	#120	—	2	12	no	—
301	C₄×Q₈	—	2	1	no	—
302	C₄×Q₈	C₃×D₄	1	1	yes	characteristic
303	C₄×Q₈	C₃×D₄	1	1	yes	characteristic
304	C₄×Q₈	C₂³×C₃	1	1	yes	characteristic
305	C₄×Q₈	C₂³×C₃	1	3	yes	—
306	C₄×D₄	—	2	3	no	—
307	C₄×D₄	—	2	3	no	—
308	C₄×D₄	—	2	3	no	—
309	C₄×D₄	—	2	3	no	—
310	C₄×D₄	—	2	3	no	—
311	C₄×D₄	—	2	6	no	—
312	C₄×D₄	—	2	3	no	—
313	C₄×D₄	—	2	3	no	—
314	C₄×D₄	—	2	3	no	—
315	C₄×D₄	—	2	3	no	—
316	C₄×D₄	C₃×D₄	1	3	yes	—
317	C₄×D₄	C₃×D₄	1	3	yes	—
318	C₄×D₄	C₂³×C₃	1	3	yes	—
319	C₄×D₄	C₂³×C₃	1	6	yes	—
320	C₄×D₄	C₂³×C₃	1	3	yes	—
321	#117	—	2	3	no	—
322	#117	—	2	3	no	—
323	#117	C₂³×C₃	1	3	yes	—
324	#117	C₂³×C₃	1	3	yes	—
325	C₂×C₄⋊C₄	—	2	3	no	—
326	C₂×C₄⋊C₄	C₂³×C₃	1	3	yes	—
327	C₂×C₂²⋊C₄	—	2	3	no	—
328	C₂×C₂²⋊C₄	—	2	6	no	—
329	C₂×C₄²	—	2	3	no	—
330	C₂×C₄²	C₂³×C₃	1	3	yes	—
331	C₄⋊C₈	C₂³×C₃	1	1	yes	characteristic
332	C₄⋊C₈	C₂³×C₃	1	3	yes	—
333	C₄⋊C₈	C₂³×C₃	1	6	yes	—
334	C₄⋊C₈	C₂³×C₃	1	6	yes	—
335	#103	C₂³×C₃	1	2	yes	—
336	#103	C₂³×C₃	1	6	yes	—
337	#102	C₂³×C₃	1	6	yes	—
338	#102	C₂³×C₃	1	12	yes	—
339	#102	C₂³×C₃	1	6	yes	—
340	C₂³×C₃	—	2	3	no	—
341	C₂³×C₃	—	2	12	no	—
342	C₂³×C₃	—	2	6	no	—
343	C₂³×C₃	—	4	12	no	—
344	C₂³×C₃	—	8	3	no	—
345	C₂³×C₃	—	8	3	no	—
346	C₂³×C₃	—	8	6	no	—
347	C₂³×C₃	—	8	6	no	—
348	C₂³×C₃	C₂²×D₄	1	3	yes	—
349	C₃×Q₈	—	2	1	no	—
350	C₃×Q₈	—	2	2	no	—
351	C₃×Q₈	—	2	2	no	—
352	C₃×Q₈	#121	1	4	yes	—
353	C₃×Q₈	C₂²×D₄	1	2	yes	—
354	C₃×D₄	—	2	12	no	—
355	C₃×D₄	—	2	12	no	—
356	C₃×D₄	—	2	6	no	—
357	C₃×D₄	—	2	3	no	—
358	C₃×D₄	—	2	6	no	—
359	C₃×D₄	—	4	6	no	—
360	C₃×D₄	—	4	12	no	—
361	C₃×D₄	—	4	12	no	—
362	C₃×D₄	—	4	6	no	—
363	C₃×D₄	—	8	6	no	—
364	C₃×D₄	—	8	6	no	—
365	C₃×D₄	#121	1	12	yes	—
366	C₃×D₄	C₂²×D₄	1	6	yes	—
367	C₂×C₃×C₄	—	2	1	no	—
368	C₂×C₃×C₄	—	2	12	no	—
369	C₂×C₃×C₄	—	2	6	no	—
370	C₂×C₃×C₄	—	2	1	no	—
371	C₂×C₃×C₄	—	2	3	no	—
372	C₂×C₃×C₄	—	2	2	no	—
373	C₂×C₃×C₄	—	4	3	no	—
374	C₂×C₃×C₄	—	4	12	no	—
375	C₂×C₃×C₄	—	4	6	no	—
376	C₂×C₃×C₄	—	8	3	no	—
377	C₂×C₃×C₄	—	8	6	no	—
378	C₂×C₃×C₄	—	8	3	no	—
379	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
380	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
381	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
382	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
383	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
384	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
385	C₂×C₃×C₄	C₂×#41	1	6	yes	—
386	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
387	C₃×C₈	—	2	2	no	—
388	C₃×C₈	—	2	6	no	—
389	C₂⁴	—	2	3	no	—
390	C₂⁴	—	2	6	no	—
391	#41	—	2	8	no	—
392	#41	—	2	6	no	—
393	#41	—	2	12	no	—
394	#41	—	2	12	no	—
395	#41	—	4	8	no	—
396	#41	C₂×C₃×D₄	1	4	yes	—
397	#41	C₂×C₃×D₄	1	12	yes	complemented
398	#41	C₃×#41	1	4	yes	—
399	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
400	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
401	C₂×Q₈	C₂⁴×C₃	1	1	yes	characteristic
402	C₂×D₄	—	2	3	no	—
403	C₂×D₄	—	2	24	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	2	3	no	—
406	C₂×D₄	—	2	12	no	—
407	C₂×D₄	—	2	3	no	—
408	C₂×D₄	—	2	3	no	—
409	C₂×D₄	—	2	6	no	—
410	C₂×D₄	—	2	6	no	—
411	C₂×D₄	—	2	6	no	—
412	C₂×D₄	—	2	6	no	—
413	C₂×D₄	—	2	12	no	—
414	C₂×D₄	—	4	24	no	—
415	C₂×D₄	—	8	6	no	—
416	C₂×D₄	—	8	6	no	—
417	C₂×D₄	C₂×C₃×D₄	1	3	yes	—
418	C₂×D₄	C₂×C₃×D₄	1	12	yes	complemented
419	C₂×D₄	C₂⁴×C₃	1	6	yes	—
420	C₂×D₄	C₂⁴×C₃	1	3	yes	—
421	C₂²×C₄	—	2	3	no	—
422	C₂²×C₄	—	2	3	no	—
423	C₂²×C₄	—	2	3	no	—
424	C₂²×C₄	—	2	6	no	—
425	C₂²×C₄	—	2	3	no	—
426	C₂²×C₄	—	2	6	no	—
427	C₂²×C₄	—	8	6	no	—
428	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
429	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
430	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
431	Q₁₆	—	2	4	no	—
432	SD₁₆	—	2	12	no	—
433	SD₁₆	—	2	12	no	—
434	D₈	—	2	12	no	—
435	D₈	—	2	24	no	—
436	M₁₆	—	2	12	no	—
437	C₂×C₈	—	2	12	no	—
438	C₂×C₈	C₂⁴×C₃	1	2	yes	—
439	C₂×C₈	C₂⁴×C₃	1	6	yes	—
440	C₄⋊C₄	—	2	3	no	—
441	C₄⋊C₄	—	2	3	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	C₂×C₃×D₄	1	3	yes	—
446	C₄⋊C₄	C₂×C₃×D₄	1	3	yes	—
447	C₄⋊C₄	C₂⁴×C₃	1	3	yes	—
448	C₄⋊C₄	C₂⁴×C₃	1	1	yes	characteristic
449	C₂²⋊C₄	—	2	3	no	—
450	C₂²⋊C₄	—	2	3	no	—
451	C₂²⋊C₄	—	2	6	no	—
452	C₂²⋊C₄	—	2	3	no	—
453	C₂²⋊C₄	—	2	6	no	—
454	C₂²⋊C₄	—	2	6	no	—
455	C₂²⋊C₄	C₂×C₃×D₄	1	6	yes	—
456	C₄²	—	2	1	no	—
457	C₄²	—	2	3	no	—
458	C₄²	—	2	3	no	—
459	C₄²	C₂×C₃×D₄	1	3	yes	—
460	C₄²	C₂×C₃×D₄	1	3	yes	—
461	C₄²	C₂⁴×C₃	1	1	yes	characteristic
462	C₄²	C₂⁴×C₃	1	3	yes	—
463	C₂²×C₃	—	2	3	no	—
464	C₂²×C₃	—	2	3	no	—
465	C₂²×C₃	—	2	6	no	—
466	C₂²×C₃	—	4	3	no	—
467	C₂²×C₃	—	4	3	no	—
468	C₂²×C₃	—	4	6	no	—
469	C₂²×C₃	—	4	6	no	—
470	C₂²×C₃	—	4	6	no	—
471	C₂²×C₃	—	8	12	no	—
472	C₂²×C₃	—	16	12	no	—
473	C₂²×C₃	C₂×#121	1	6	yes	—
474	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
475	C₃×C₄	—	2	1	no	—
476	C₃×C₄	—	2	3	no	—
477	C₃×C₄	—	2	2	no	—
478	C₃×C₄	—	2	3	no	—
479	C₃×C₄	—	4	1	no	—
480	C₃×C₄	—	4	2	no	—
481	C₃×C₄	—	4	3	no	—
482	C₃×C₄	C₂×#121	1	6	yes	—
483	C₃×C₄	#576	1	2	yes	complemented
484	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
485	C₃×C₄	C₂²×#41	1	1	yes	characteristic
486	C₂³	—	2	3	no	—
487	C₂³	—	2	12	no	—
488	C₂³	—	2	6	no	—
489	C₂³	—	4	12	no	—
490	C₂³	—	8	3	no	—
491	C₂³	—	8	3	no	—
492	C₂³	—	8	6	no	—
493	C₂³	—	8	6	no	—
494	C₂³	C₂²×C₃×D₄	1	3	yes	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	2	no	—
497	Q₈	—	2	2	no	—
498	Q₈	C₃×#121	1	4	yes	—
499	Q₈	C₂²×C₃×D₄	1	2	yes	—
500	D₄	—	2	12	no	—
501	D₄	—	2	6	no	—
502	D₄	—	2	3	no	—
503	D₄	—	2	6	no	—
504	D₄	—	2	12	no	—
505	D₄	—	4	6	no	—
506	D₄	—	4	12	no	—
507	D₄	—	4	6	no	—
508	D₄	—	4	12	no	—
509	D₄	—	8	6	no	—
510	D₄	—	8	6	no	—
511	D₄	C₃×#121	1	12	yes	—
512	D₄	C₂²×C₃×D₄	1	6	yes	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	6	no	—
515	C₂×C₄	—	2	12	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	3	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	4	3	no	—
520	C₂×C₄	—	4	6	no	—
521	C₂×C₄	—	4	12	no	—
522	C₂×C₄	—	8	6	no	—
523	C₂×C₄	—	8	3	no	—
524	C₂×C₄	—	8	3	no	—
525	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
526	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
527	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
528	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
529	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
530	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
531	C₂×C₄	C₂×C₃×#41	1	6	yes	—
532	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
533	C₈	—	2	2	no	—
534	C₈	—	2	6	no	—
535	C₂×C₃	—	2	6	no	—
536	C₂×C₃	—	4	6	no	—
537	C₂×C₃	—	8	6	no	—
538	C₂×C₃	C₂²×#121	1	1	yes	characteristic
539	C₂×C₃	C₂×#576	1	1	yes	characteristic
540	C₂×C₃	#3585	1	1	yes	characteristic
541	C₂²	—	2	3	no	—
542	C₂²	—	2	3	no	—
543	C₂²	—	2	6	no	—
544	C₂²	—	4	3	no	—
545	C₂²	—	4	3	no	—
546	C₂²	—	4	6	no	—
547	C₂²	—	4	6	no	—
548	C₂²	—	4	6	no	—
549	C₂²	—	8	12	no	—
550	C₂²	—	16	12	no	—
551	C₂²	C₂×C₃×#121	1	6	yes	—
552	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
553	C₄	—	2	2	no	—
554	C₄	—	2	3	no	—
555	C₄	—	2	1	no	—
556	C₄	—	2	3	no	—
557	C₄	—	4	2	no	—
558	C₄	—	4	3	no	—
559	C₄	—	4	1	no	—
560	C₄	C₂×C₃×#121	1	6	yes	—
561	C₄	C₃×#576	1	2	yes	complemented
562	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
563	C₄	C₂²×C₃×#41	1	1	yes	characteristic
564	C₃	#60428	1	1	yes	characteristic, direct product factor, Sylow
565	C₂	—	2	6	no	—
566	C₂	—	4	6	no	—
567	C₂	—	8	6	no	—
568	C₂	C₂²×C₃×#121	1	1	yes	characteristic
569	C₂	C₂×C₃×#576	1	1	yes	characteristic
570	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 ◁ C₃×C₄×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#576	C₂	semidirect
2. C₃×#3498	C₂	semidirect
3. C₃×C₄×H(5,𝔽₂)	C₂	semidirect
4. C₃×#3045	C₂	semidirect
5. C₃×#3041	C₂	semidirect
6. C₃×#3038	C₂	semidirect
7. C₃×#3036	C₂	semidirect
8. C₃×#3035	C₂	semidirect
9. C₂×C₃×#459	C₂	semidirect
10. C₃×#2906	C₂	semidirect
11. C₃×#2897	C₂	semidirect
12. #60428	C₃	direct (N × H)
16. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×C₃×AGL(1,ℤ/8)	C₂²	semidirect
18. C₂×C₃×#135	C₂²	semidirect
19. C₂²×C₃×D₈	C₂²	semidirect
25. C₃×#535	C₂²	semidirect
26. C₃×#532	C₂²	semidirect
27. C₂×C₃×#127	C₂²	semidirect
30. C₃×#520	C₂²	semidirect
32. C₃×#518	C₂²	semidirect
35. C₃×C₄×#41	C₂²	semidirect
37. C₂×C₃×C₄×D₄	C₂²	semidirect
38. C₃×#464	C₂²	semidirect
39. C₃×#463	C₂²	semidirect
40. C₃×#463	C₂²	semidirect
41. C₃×#462	C₂²	semidirect
42. C₃×#461	C₂²	semidirect
43. C₃×#460	C₂²	semidirect
44. C₃×#460	C₂²	semidirect
45. C₃×#459	C₂²	semidirect
46. C₃×#459	C₂²	semidirect
47. C₃×#459	C₂²	semidirect
48. C₃×#459	C₂²	semidirect
49. C₃×#424	C₂²	semidirect
50. C₃×#423	C₂²	semidirect
51. C₂×C₃×C₄⋊C₈	C₂²	semidirect
52. C₃×#418	C₂²	semidirect
53. C₃×#417	C₂²	semidirect
54. C₃×#416	C₂²	semidirect
55. C₂×C₃×#102	C₂²	semidirect
56. C₂×#576	C₂×C₃	semidirect
57. #3498	C₂×C₃	semidirect
58. C₄×H(5,𝔽₂)	C₂×C₃	semidirect
59. #3045	C₂×C₃	semidirect
60. #3041	C₂×C₃	semidirect
61. #3038	C₂×C₃	semidirect
62. #3036	C₂×C₃	semidirect
63. #3035	C₂×C₃	semidirect
64. C₂×#459	C₂×C₃	semidirect
65. #2906	C₂×C₃	semidirect
66. #2897	C₂×C₃	semidirect
70. C₃×H(5,𝔽₂)	D₄	semidirect
148. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
149. C₂×AGL(1,ℤ/8)	C₂²×C₃	semidirect
150. C₂×#135	C₂²×C₃	semidirect
151. C₂²×D₈	C₂²×C₃	semidirect
157. #535	C₂²×C₃	semidirect
158. #532	C₂²×C₃	semidirect
159. C₂×#127	C₂²×C₃	semidirect
162. #520	C₂²×C₃	semidirect
164. #518	C₂²×C₃	semidirect
167. C₄×#41	C₂²×C₃	semidirect
169. C₂×C₄×D₄	C₂²×C₃	semidirect
170. #464	C₂²×C₃	semidirect
171. #463	C₂²×C₃	semidirect
172. #463	C₂²×C₃	semidirect
173. #462	C₂²×C₃	semidirect
174. #461	C₂²×C₃	semidirect
175. #460	C₂²×C₃	semidirect
176. #460	C₂²×C₃	semidirect
177. #459	C₂²×C₃	semidirect
178. #459	C₂²×C₃	semidirect
179. #459	C₂²×C₃	semidirect
180. #459	C₂²×C₃	semidirect
181. #424	C₂²×C₃	semidirect
182. #423	C₂²×C₃	semidirect
183. C₂×C₄⋊C₈	C₂²×C₃	semidirect
184. #418	C₂²×C₃	semidirect
185. #417	C₂²×C₃	semidirect
186. #416	C₂²×C₃	semidirect
187. C₂×#102	C₂²×C₃	semidirect
196. C₃×#41	C₂×D₄	semidirect
217. C₂×C₃×D₄	C₂×D₄	semidirect
265. H(5,𝔽₂)	C₃×D₄	semidirect
397. #41	C₂×C₃×D₄	semidirect
418. C₂×D₄	C₂×C₃×D₄	semidirect
483. C₃×C₄	#576	semidirect
561. C₄	C₃×#576	semidirect
564. C₃	#60428	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
C₃×#3498	2
C₃×C₄×H(5,𝔽₂)	2
C₃×#3045	2
C₃×#3041	2
C₃×#3038	2
C₃×#3036	2
C₃×#3035	2
C₂×C₃×#459	2
C₃×#2906	2
C₃×#2897	2
#60428	3

Subgroups — C3×#60428 #10679035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60428 #10679035