Subgroups — #79511

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5541	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#435	C₂	1	1	yes	maximal, characteristic
4	#5137	C₂	1	1	yes	maximal, characteristic, complemented
5	#5120	C₂	1	1	yes	maximal, characteristic, complemented
6	#5116	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#816	C₂	1	1	yes	maximal, characteristic, complemented
8	#4849	C₂	1	1	yes	maximal, characteristic, complemented
9	#4829	C₂	1	1	yes	maximal, characteristic, complemented
10	#4739	C₂	1	1	yes	maximal, characteristic, complemented
11	#4739	C₂	1	1	yes	maximal, characteristic, complemented
12	#4723	C₂	1	1	yes	maximal, characteristic, complemented
13	#4721	C₂	1	1	yes	maximal, characteristic, complemented
14	#4716	C₂	1	1	yes	maximal, characteristic, complemented
15	#4703	C₂	1	1	yes	maximal, characteristic, complemented
16	#3312	—	3	1	no	maximal, Sylow
17	C₂²×Dic₆	C₂²	1	1	yes	characteristic
18	C₂²×Dic₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
22	#935	—	2	1	no	—
23	D₄×Dic₃	—	2	1	no	—
24	#926	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	#924	C₂²	1	1	yes	characteristic
27	#924	C₂²	1	1	yes	characteristic
28	#924	C₂²	1	1	yes	characteristic
29	#924	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
31	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
32	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
33	#889	—	2	1	no	—
34	#888	—	2	1	no	—
35	#879	—	2	1	no	—
36	#878	—	2	1	no	—
37	#878	—	2	1	no	—
38	#877	—	2	1	no	—
39	#870	C₂²	1	1	yes	characteristic
40	#869	C₂²	1	1	yes	characteristic
41	C₄×Dic₆	C₂²	1	1	yes	characteristic
42	C₄×Dic₆	C₂²	1	1	yes	characteristic
43	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₃×#97	C₂²	1	1	yes	characteristic
47	#818	C₂²	1	1	yes	characteristic
48	#817	C₂²	1	1	yes	characteristic
49	#817	C₂²	1	1	yes	characteristic
50	#816	C₂²	1	1	yes	characteristic
51	#816	C₂²	1	1	yes	characteristic
52	#816	C₂²	1	1	yes	characteristic
53	#816	C₂²	1	1	yes	characteristic
54	#816	C₂²	1	1	yes	characteristic
55	#816	C₂²	1	1	yes	characteristic
56	#816	C₂²	1	1	yes	characteristic
57	#816	C₂²	1	1	yes	characteristic
58	#816	C₂²	1	1	yes	characteristic
59	#816	C₂²	1	1	yes	characteristic
60	#552	—	3	1	no	—
61	D₄×Q₈	—	3	1	no	—
62	#491	—	3	1	no	—
63	#485	—	3	1	no	—
64	#483	—	3	1	no	—
65	#479	—	3	1	no	—
66	#470	—	3	1	no	—
67	#467	—	3	1	no	—
68	#462	—	3	1	no	—
69	#451	—	3	1	no	—
70	#451	—	3	1	no	—
71	#441	—	3	1	no	—
72	#435	D₃	1	1	yes	characteristic
73	#419	—	3	1	no	—
74	C₂×#103	—	3	1	no	—
75	C₂×C₃×D₄	D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂²×Dic₃	—	2	1	no	—
79	C₂²×Dic₃	—	2	1	no	—
80	C₂²×Dic₃	—	2	1	no	—
81	C₂×Dic₆	—	2	2	no	—
82	C₂×Dic₆	—	2	2	no	—
83	C₂×Dic₆	—	2	1	no	—
84	C₂×Dic₆	—	2	1	no	—
85	C₂×Dic₆	—	4	1	no	—
86	C₂×Dic₆	—	4	1	no	—
87	C₂×Dic₆	C₂³	1	1	yes	characteristic
88	C₂×Dic₆	C₂³	1	1	yes	characteristic
89	C₂×Dic₆	C₂³	1	1	yes	characteristic
90	C₂×Dic₆	C₂³	1	1	yes	characteristic
91	C₃×M₁₆	—	4	1	no	—
92	C₂×C₃×C₈	—	4	1	no	—
93	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
97	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
98	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
99	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
100	C₃×C₄²	C₂³	1	1	yes	characteristic
101	#217	—	2	1	no	—
102	#217	—	2	1	no	—
103	#217	—	2	1	no	—
104	#217	—	2	1	no	—
105	#217	—	2	1	no	—
106	C₄⋊Dic₃	—	2	1	no	—
107	C₄⋊Dic₃	—	2	1	no	—
108	C₄⋊Dic₃	—	2	1	no	—
109	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
110	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
111	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
112	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
113	Dic₃⋊C₄	—	2	1	no	—
114	Dic₃⋊C₄	—	2	1	no	—
115	Dic₃⋊C₄	—	2	1	no	—
116	Dic₃⋊C₄	—	2	1	no	—
117	Dic₃⋊C₄	—	2	1	no	—
118	Dic₃⋊C₄	—	2	1	no	—
119	Dic₃⋊C₄	—	2	1	no	—
120	Dic₃⋊C₄	—	2	1	no	—
121	C₄×Dic₃	—	2	1	no	—
122	C₄×Dic₃	—	2	1	no	—
123	C₄×Dic₃	—	2	1	no	—
124	Dic₁₂	—	2	2	no	—
125	Dic₁₂	—	4	1	no	—
126	Dic₁₂	—	4	1	no	—
127	C₂²×Q₈	—	3	1	no	—
128	C₂²×Q₈	—	3	1	no	—
129	C₂×Q₁₆	—	3	1	no	—
130	C₂×Q₁₆	—	3	1	no	—
131	C₂×Q₁₆	—	3	1	no	—
132	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
133	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
134	C₄⋊Q₈	—	3	1	no	—
135	C₄⋊Q₈	—	6	1	no	—
136	#126	—	6	1	no	—
137	#125	—	3	1	no	—
138	#125	—	6	1	no	—
139	#123	—	6	1	no	—
140	C₂²⋊Q₈	—	3	1	no	—
141	C₂²⋊Q₈	—	3	1	no	—
142	C₂²⋊Q₈	—	3	1	no	—
143	C₂²⋊Q₈	—	3	1	no	—
144	C₂²⋊Q₈	—	6	1	no	—
145	C₂²⋊Q₈	—	6	1	no	—
146	C₄×Q₈	—	3	1	no	—
147	C₄×Q₈	—	3	1	no	—
148	C₄×D₄	—	6	1	no	—
149	C₄×D₄	C₂×D₃	1	1	yes	characteristic
150	#117	—	3	1	no	—
151	#117	—	6	1	no	—
152	C₂×C₄⋊C₄	—	3	1	no	—
153	#107	—	3	1	no	—
154	#106	—	3	1	no	—
155	#106	—	3	1	no	—
156	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
157	#103	—	3	1	no	—
158	#103	—	3	1	no	—
159	#103	—	3	1	no	—
160	#103	—	3	1	no	—
161	#103	—	3	1	no	—
162	#103	—	3	1	no	—
163	#103	—	3	1	no	—
164	#103	—	3	1	no	—
165	#103	—	3	1	no	—
166	#103	—	3	1	no	—
167	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
168	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
169	#97	C₂×D₃	1	1	yes	characteristic
170	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
171	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
172	C₃×D₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
182	C₂×Dic₃	—	2	1	no	—
183	C₂×Dic₃	—	2	1	no	—
184	C₂×Dic₃	—	2	1	no	—
185	C₂×Dic₃	—	2	1	no	—
186	C₂×Dic₃	—	2	1	no	—
187	C₂×Dic₃	—	2	1	no	—
188	C₂×Dic₃	—	2	1	no	—
189	C₂×Dic₃	—	2	1	no	—
190	C₂×Dic₃	—	4	2	no	—
191	C₂×Dic₃	—	4	2	no	—
192	C₂×Dic₃	—	4	2	no	—
193	Dic₆	—	2	1	no	—
194	Dic₆	—	2	1	no	—
195	Dic₆	—	2	1	no	—
196	Dic₆	—	2	1	no	—
197	Dic₆	—	2	1	no	—
198	Dic₆	—	2	1	no	—
199	Dic₆	—	2	1	no	—
200	Dic₆	—	4	2	no	—
201	Dic₆	—	4	2	no	—
202	Dic₆	#41	1	2	yes	—
203	C₃×C₈	—	2	1	no	—
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₂×Q₈	—	3	1	no	—
208	C₂×Q₈	—	3	1	no	—
209	C₂×Q₈	—	3	1	no	—
210	C₂×Q₈	—	3	1	no	—
211	C₂×Q₈	—	6	2	no	—
212	C₂×Q₈	—	6	2	no	—
213	C₂×Q₈	—	6	1	no	—
214	C₂×Q₈	—	6	1	no	—
215	C₂×Q₈	—	12	1	no	—
216	C₂×Q₈	—	12	1	no	—
217	C₂×D₄	D₁₂	1	1	yes	characteristic
218	C₂²×C₄	—	6	1	no	—
219	C₂²×C₄	—	6	1	no	—
220	C₂²×C₄	—	6	1	no	—
221	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
223	Q₁₆	—	6	2	no	—
224	Q₁₆	—	12	1	no	—
225	Q₁₆	—	12	1	no	—
226	M₁₆	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
232	C₄⋊C₄	—	3	1	no	—
233	C₄⋊C₄	—	3	1	no	—
234	C₄⋊C₄	—	3	1	no	—
235	C₄⋊C₄	—	3	1	no	—
236	C₄⋊C₄	—	6	1	no	—
237	C₄⋊C₄	—	6	1	no	—
238	C₄⋊C₄	—	6	1	no	—
239	C₄⋊C₄	—	6	1	no	—
240	C₄⋊C₄	—	6	1	no	—
241	C₄⋊C₄	—	6	1	no	—
242	C₄⋊C₄	—	6	1	no	—
243	C₄⋊C₄	—	6	1	no	—
244	C₄⋊C₄	—	6	1	no	—
245	C₄⋊C₄	—	6	1	no	—
246	C₄⋊C₄	—	6	1	no	—
247	C₄⋊C₄	D₁₂	1	1	yes	characteristic
248	C₂²⋊C₄	—	6	1	no	—
249	C₂²⋊C₄	—	6	1	no	—
250	C₂²⋊C₄	—	6	1	no	—
251	C₂²⋊C₄	—	6	1	no	—
252	C₂²⋊C₄	—	6	1	no	—
253	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
254	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
255	C₄²	—	6	1	no	—
256	C₄²	—	6	1	no	—
257	C₄²	—	6	1	no	—
258	C₄²	C₂²×D₃	1	1	yes	characteristic
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	#137	1	2	yes	—
265	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	C₂×#41	1	1	yes	characteristic
271	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
272	Dic₃	—	2	2	no	—
273	Dic₃	—	4	1	no	—
274	Dic₃	—	4	1	no	—
275	Dic₃	—	4	1	no	—
276	Dic₃	—	4	1	no	—
277	Dic₃	—	4	1	no	—
278	Dic₃	—	4	1	no	—
279	Dic₃	—	4	1	no	—
280	C₂³	C₂×D₁₂	1	1	yes	characteristic
281	C₂³	C₂×D₁₂	1	1	yes	characteristic
282	Q₈	—	3	2	no	—
283	Q₈	—	6	1	no	—
284	Q₈	—	6	1	no	—
285	Q₈	—	6	1	no	—
286	Q₈	—	6	1	no	—
287	Q₈	—	6	1	no	—
288	Q₈	—	6	1	no	—
289	Q₈	—	6	1	no	—
290	Q₈	—	12	2	no	—
291	Q₈	—	12	2	no	—
292	D₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	6	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	—	6	1	no	—
300	C₂×C₄	—	6	1	no	—
301	C₂×C₄	—	6	1	no	—
302	C₂×C₄	—	6	1	no	—
303	C₂×C₄	—	6	1	no	—
304	C₂×C₄	—	6	1	no	—
305	C₂×C₄	—	12	2	no	—
306	C₂×C₄	—	12	2	no	—
307	C₂×C₄	—	12	2	no	—
308	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
309	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
310	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
311	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	4	1	no	—
319	C₂×C₃	#546	1	1	yes	characteristic
320	C₂×C₃	C₂×#137	1	1	yes	characteristic
321	C₂×C₃	#578	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	#909	1	2	yes	—
328	C₂²	C₂²×D₁₂	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	6	2	no	—
334	C₄	—	12	1	no	—
335	C₄	—	12	1	no	—
336	C₄	—	12	1	no	—
337	C₄	—	12	1	no	—
338	C₄	—	12	1	no	—
339	C₄	—	12	1	no	—
340	C₄	—	12	1	no	—
341	C₄	#1004	1	1	yes	characteristic
342	C₄	D₃×#41	1	1	yes	characteristic
343	C₃	#3312	1	1	yes	characteristic, complemented, Sylow
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	#5554	1	1	yes	characteristic
347	C₂	C₂×#909	1	1	yes	characteristic
348	C₂	#5753	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#435	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×D₄	D₄
hypercenter	#435	D₃
hypocenter	C₃	#3312
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
4. #5137	C₂	semidirect
5. #5120	C₂	semidirect
6. #5116	C₂	semidirect
7. C₂×#816	C₂	semidirect
8. #4849	C₂	semidirect
9. #4829	C₂	semidirect
10. #4739	C₂	semidirect
11. #4739	C₂	semidirect
12. #4723	C₂	semidirect
13. #4721	C₂	semidirect
14. #4716	C₂	semidirect
15. #4703	C₂	semidirect
343. C₃	#3312	semidirect

Maximal subgroups

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

subgroup	index
#5541	2
D₄×Dic₆	2
C₃×#435	2
#5137	2
#5120	2
#5116	2
C₂×#816	2
#4849	2
#4829	2
#4739	2
#4723	2
#4721	2
#4716	2
#4703	2
#3312	3