Subgroups — #78556

Order 384.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5528	C₂	1	1	yes	maximal, characteristic
2	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	#5128	C₂	1	1	yes	maximal, characteristic, complemented
4	#5127	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#67	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#67	C₂	1	1	yes	maximal, characteristic, complemented
7	#5000	C₂	1	1	yes	maximal, characteristic, complemented
8	#4709	C₂	1	1	yes	maximal, characteristic, complemented
9	#4706	C₂	1	1	yes	maximal, characteristic, complemented
10	#4705	C₂	1	1	yes	maximal, characteristic, complemented
11	#4701	C₂	1	1	yes	maximal, characteristic, complemented
12	#4689	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
16	#2966	—	3	1	no	maximal, Sylow
17	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
19	C₃×#117	C₂²	1	1	yes	characteristic
20	C₄×#67	C₂²	1	1	yes	characteristic
21	C₄×#67	C₂²	1	1	yes	characteristic
22	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
23	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
24	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
25	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
26	#898	C₄	1	1	yes	characteristic
27	#896	C₄	1	1	yes	characteristic
28	#894	C₄	1	1	yes	characteristic
29	#889	C₄	1	1	yes	characteristic
30	#885	C₄	1	1	yes	characteristic
31	#884	C₄	1	1	yes	characteristic
32	#883	C₄	1	1	yes	characteristic
33	#879	C₄	1	1	yes	characteristic
34	C₄×D₁₂	C₂²	1	1	yes	characteristic
35	#872	C₂²	1	1	yes	characteristic
36	C₄²×D₃	C₂²	1	1	yes	characteristic
37	C₄×Dic₆	C₂²	1	1	yes	characteristic
38	C₃×#97	C₂²	1	1	yes	characteristic
39	C₃×#97	C₂²	1	1	yes	characteristic
40	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#830	C₂²	1	1	yes	characteristic
43	#830	C₂²	1	1	yes	characteristic
44	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
47	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
48	#815	C₂²	1	1	yes	characteristic
49	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
53	C₈×Dic₃	C₂²	1	1	yes	characteristic
54	C₈×Dic₃	C₂²	1	1	yes	characteristic
55	C₈×Dic₃	C₂²	1	1	yes	characteristic
56	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
57	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
58	#803	C₂²	1	1	yes	characteristic
59	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
60	#529	—	3	1	no	—
61	#446	—	3	1	no	—
62	C₈×Q₈	—	3	1	no	—
63	#436	—	3	1	no	—
64	#435	—	3	1	no	—
65	#435	—	3	1	no	—
66	C₈×D₄	—	3	1	no	—
67	C₈×D₄	—	3	1	no	—
68	C₈×D₄	—	3	1	no	—
69	#433	—	3	1	no	—
70	#433	—	3	1	no	—
71	#432	—	3	1	no	—
72	#405	—	3	1	no	—
73	#405	D₃	1	1	yes	characteristic, complemented
74	C₂×C₄×C₈	—	3	1	no	—
75	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
76	C₂×#67	C₂×C₄	1	1	yes	characteristic
77	C₂×#67	C₂×C₄	1	1	yes	characteristic
78	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
82	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
83	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
84	C₃×M₁₆	—	2	2	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
90	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	C₃×C₄²	C₂³	1	1	yes	characteristic
96	#217	C₂×C₄	1	1	yes	characteristic
97	#217	C₂×C₄	1	1	yes	characteristic
98	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
99	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
100	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
101	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
102	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
103	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
104	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
105	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
106	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
107	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
108	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
109	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
110	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
111	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
112	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
113	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
114	C₄×Dic₃	C₂³	1	1	yes	characteristic
115	C₄×Dic₃	C₂³	1	1	yes	characteristic
116	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
117	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
118	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
119	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
120	ΓL(1,𝔽₂₅)	—	2	2	no	—
121	D₃×C₈	—	2	2	no	—
122	D₃×C₈	—	2	2	no	—
123	D₃×C₈	—	2	2	no	—
124	C₂×M₁₆	—	3	1	no	—
125	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
126	C₂²×C₈	—	3	1	no	—
127	C₂²×C₈	—	3	1	no	—
128	C₂²×C₈	—	3	1	no	—
129	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
130	#126	—	3	1	no	—
131	#126	—	3	1	no	—
132	#125	—	3	1	no	—
133	#124	—	3	1	no	—
134	#123	—	3	1	no	—
135	#123	—	3	1	no	—
136	C₂²⋊Q₈	—	3	1	no	—
137	#121	—	3	1	no	—
138	C₄×Q₈	—	3	1	no	—
139	C₄×D₄	—	3	1	no	—
140	C₄×D₄	—	3	1	no	—
141	C₄×D₄	—	3	1	no	—
142	#117	—	3	1	no	—
143	#117	C₂×D₃	1	1	yes	characteristic
144	C₂×C₄²	—	3	1	no	—
145	C₄⋊C₈	—	3	1	no	—
146	C₄⋊C₈	—	3	1	no	—
147	C₄⋊C₈	—	3	1	no	—
148	C₄⋊C₈	—	3	1	no	—
149	C₄⋊C₈	—	3	1	no	—
150	C₄⋊C₈	—	3	1	no	—
151	C₂²⋊C₈	—	3	1	no	—
152	C₂²⋊C₈	—	3	1	no	—
153	C₂²⋊C₈	—	3	1	no	—
154	C₂²⋊C₈	—	3	1	no	—
155	C₂²⋊C₈	—	3	1	no	—
156	C₂²⋊C₈	—	3	1	no	—
157	#97	—	3	1	no	—
158	#97	—	3	1	no	—
159	#97	C₂×D₃	1	1	yes	characteristic
160	#97	C₂×D₃	1	1	yes	characteristic
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	—	3	1	no	—
164	C₄×C₈	—	3	1	no	—
165	C₄×C₈	C₂×D₃	1	1	yes	characteristic
166	C₄×C₈	C₂×D₃	1	1	yes	characteristic
167	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
168	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
169	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
178	#67	—	2	2	no	—
179	#67	—	2	2	no	—
180	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
181	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
182	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
183	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
184	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
185	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
186	D₁₂	—	2	2	no	—
187	C₄×D₃	—	2	2	no	—
188	C₄×D₃	—	2	1	no	—
189	C₄×D₃	—	2	1	no	—
190	C₄×D₃	—	2	1	no	—
191	C₄×D₃	—	2	1	no	—
192	C₄×D₃	—	2	2	no	—
193	Dic₆	—	2	2	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	#41	1	2	yes	—
197	C₃×C₈	#41	1	2	yes	—
198	C₃⋊C₈	—	2	1	no	—
199	C₃⋊C₈	—	2	1	no	—
200	C₃⋊C₈	—	2	1	no	—
201	C₃⋊C₈	—	2	1	no	—
202	C₂×Q₈	—	3	1	no	—
203	C₂×D₄	—	3	1	no	—
204	C₂×D₄	—	3	1	no	—
205	C₂×D₄	—	3	1	no	—
206	C₂²×C₄	—	3	1	no	—
207	C₂²×C₄	—	3	1	no	—
208	C₂²×C₄	—	3	1	no	—
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
211	M₁₆	—	2	2	no	—
212	M₁₆	—	6	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	3	1	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	3	1	no	—
217	C₂×C₈	—	3	1	no	—
218	C₂×C₈	—	6	2	no	—
219	C₂×C₈	—	6	2	no	—
220	C₂×C₈	—	6	2	no	—
221	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	3	1	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	—	3	1	no	—
229	C₄⋊C₄	—	3	1	no	—
230	C₄⋊C₄	—	3	1	no	—
231	C₄⋊C₄	—	3	1	no	—
232	C₄⋊C₄	—	3	1	no	—
233	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
234	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
235	C₂²⋊C₄	—	3	1	no	—
236	C₂²⋊C₄	—	3	1	no	—
237	C₂²⋊C₄	—	3	1	no	—
238	C₂²⋊C₄	—	3	1	no	—
239	C₂²⋊C₄	—	3	1	no	—
240	C₂²⋊C₄	—	3	1	no	—
241	C₂²⋊C₄	—	3	1	no	—
242	C₂²⋊C₄	—	3	1	no	—
243	C₂²⋊C₄	C₄×D₃	1	1	yes	characteristic
244	C₂²⋊C₄	C₄×D₃	1	1	yes	characteristic
245	C₄²	—	3	1	no	—
246	C₄²	—	3	1	no	—
247	C₄²	—	3	1	no	—
248	C₄²	—	3	1	no	—
249	C₄²	C₂²×D₃	1	1	yes	characteristic
250	C₄²	C₂²×D₃	1	1	yes	characteristic
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	2	1	no	—
257	C₂×D₃	—	2	1	no	—
258	C₂×D₃	—	2	1	no	—
259	C₂×D₃	—	2	1	no	—
260	C₂×D₃	#131	1	2	yes	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	C₂×#41	1	1	yes	characteristic
267	C₃×C₄	C₂×#41	1	1	yes	characteristic
268	Dic₃	—	2	1	no	—
269	Dic₃	—	2	1	no	—
270	Dic₃	—	2	1	no	—
271	Dic₃	—	2	1	no	—
272	Dic₃	—	2	1	no	—
273	Dic₃	#131	1	2	yes	—
274	C₂³	—	3	1	no	—
275	C₂³	—	3	1	no	—
276	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
277	Q₈	—	6	2	no	—
278	D₄	—	6	2	no	—
279	D₄	—	6	2	no	—
280	D₄	—	6	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	3	1	no	—
284	C₂×C₄	—	3	1	no	—
285	C₂×C₄	—	3	1	no	—
286	C₂×C₄	—	3	1	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	6	2	no	—
290	C₂×C₄	—	6	2	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	—	6	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
298	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
299	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	6	1	no	—
304	C₈	—	6	1	no	—
305	C₈	—	6	1	no	—
306	C₈	—	6	1	no	—
307	C₈	#235	1	2	yes	—
308	C₈	#235	1	2	yes	—
309	C₂×C₃	—	4	1	no	—
310	C₂×C₃	C₄×#41	1	1	yes	characteristic
311	C₂×C₃	C₂×#131	1	1	yes	characteristic
312	C₂×C₃	C₂×#131	1	1	yes	characteristic
313	D₃	—	2	2	no	—
314	D₃	—	4	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	3	2	no	—
319	C₂²	—	6	1	no	—
320	C₂²	—	6	1	no	—
321	C₂²	—	6	1	no	—
322	C₂²	—	6	1	no	—
323	C₂²	—	6	1	no	—
324	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	3	2	no	—
331	C₄	—	6	1	no	—
332	C₄	—	6	1	no	—
333	C₄	—	6	1	no	—
334	C₄	—	6	1	no	—
335	C₄	—	6	1	no	—
336	C₄	C₂×#235	1	1	yes	characteristic
337	C₄	C₂×#235	1	1	yes	characteristic
338	C₃	#2966	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	4	1	no	—
340	C₂	—	6	2	no	—
341	C₂	—	12	1	no	—
342	C₂	C₄×#235	1	1	yes	characteristic
343	C₂	D₃×#131	1	1	yes	characteristic
344	C₂	D₃×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#405	C₂
prime radical	#884	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#405	D₃
hypocenter	C₃	#2966
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. C₃×#405	C₂	semidirect
3. #5128	C₂	semidirect
4. #5127	C₂	semidirect
5. C₈×#67	C₂	semidirect
6. C₈×#67	C₂	semidirect
7. #5000	C₂	semidirect
8. #4709	C₂	semidirect
9. #4706	C₂	semidirect
10. #4705	C₂	semidirect
11. #4701	C₂	semidirect
12. #4689	C₂	semidirect
13. C₈×D₁₂	C₂	semidirect
14. C₄×D₃×C₈	C₂	semidirect
15. C₈×Dic₆	C₂	semidirect
73. #405	D₃	semidirect
338. C₃	#2966	semidirect

Maximal subgroups

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

subgroup	index
#5528	2
C₃×#405	2
#5128	2
#5127	2
C₈×#67	2
#5000	2
#4709	2
#4706	2
#4705	2
#4701	2
#4689	2
C₈×D₁₂	2
C₄×D₃×C₈	2
C₈×Dic₆	2
#2966	3