Subgroups — #79475

Order 384.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5542	C₂	1	1	yes	maximal, characteristic
2	#5541	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	#5121	C₂	1	1	yes	maximal, characteristic, complemented
6	#5116	C₂	1	1	yes	maximal, characteristic, complemented
7	#5106	C₂	1	1	yes	maximal, characteristic, complemented
8	#4850	C₂	1	1	yes	maximal, characteristic, complemented
9	#4829	C₂	1	1	yes	maximal, characteristic, complemented
10	#4723	C₂	1	1	yes	maximal, characteristic, complemented
11	#4723	C₂	1	1	yes	maximal, characteristic, complemented
12	#4723	C₂	1	1	yes	maximal, characteristic, complemented
13	#4722	C₂	1	1	yes	maximal, characteristic, complemented
14	#4716	C₂	1	1	yes	maximal, characteristic, complemented
15	#4702	C₂	1	1	yes	maximal, characteristic, complemented
16	#3349	—	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×C₄×D₄	C₂²	1	1	yes	characteristic
20	#936	—	2	1	no	—
21	#935	—	2	1	no	—
22	#926	C₂²	1	1	yes	characteristic
23	#926	C₂²	1	1	yes	characteristic
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	#889	—	2	1	no	—
32	#887	—	2	1	no	—
33	#879	—	2	1	no	—
34	#879	—	2	1	no	—
35	#879	—	2	1	no	—
36	#877	—	2	1	no	—
37	#870	C₂²	1	1	yes	characteristic
38	#869	C₂²	1	1	yes	characteristic
39	C₄×Dic₆	C₂²	1	1	yes	characteristic
40	C₄×Dic₆	C₂²	1	1	yes	characteristic
41	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₃×#97	C₂²	1	1	yes	characteristic
45	#818	C₂²	1	1	yes	characteristic
46	#818	C₂²	1	1	yes	characteristic
47	#818	C₂²	1	1	yes	characteristic
48	#818	C₂²	1	1	yes	characteristic
49	#818	C₂²	1	1	yes	characteristic
50	#817	C₂²	1	1	yes	characteristic
51	#817	C₂²	1	1	yes	characteristic
52	#817	C₂²	1	1	yes	characteristic
53	#817	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	#556	—	3	1	no	—
61	#552	—	3	1	no	—
62	#501	—	3	1	no	—
63	#485	—	3	1	no	—
64	#485	—	3	1	no	—
65	#485	—	3	1	no	—
66	#484	—	3	1	no	—
67	#479	—	3	1	no	—
68	#477	—	3	1	no	—
69	#470	—	3	1	no	—
70	#467	—	3	1	no	—
71	#441	—	3	1	no	—
72	#435	D₃	1	1	yes	characteristic
73	#428	—	3	1	no	—
74	#427	—	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	C₂×Dic₆	—	2	1	no	—
80	C₂×Dic₆	C₂³	1	1	yes	characteristic
81	C₂×Dic₆	C₂³	1	1	yes	characteristic
82	C₃×M₁₆	—	2	2	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
88	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
89	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
90	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
91	C₃×C₄²	C₂³	1	1	yes	characteristic
92	#217	—	2	1	no	—
93	#217	—	2	1	no	—
94	#217	—	2	1	no	—
95	#217	—	2	1	no	—
96	#217	—	2	1	no	—
97	#217	—	2	1	no	—
98	#217	—	2	1	no	—
99	C₄⋊Dic₃	—	2	1	no	—
100	C₄⋊Dic₃	—	2	1	no	—
101	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
102	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
103	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
106	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
107	Dic₃⋊C₄	—	2	1	no	—
108	Dic₃⋊C₄	—	2	1	no	—
109	Dic₃⋊C₄	—	2	1	no	—
110	Dic₃⋊C₄	—	2	1	no	—
111	Dic₃⋊C₄	—	2	1	no	—
112	Dic₃⋊C₄	—	2	1	no	—
113	Dic₃⋊C₄	—	2	1	no	—
114	Dic₃⋊C₄	—	2	1	no	—
115	C₄×Dic₃	—	2	1	no	—
116	C₄×Dic₃	—	2	1	no	—
117	C₄×Dic₃	—	2	1	no	—
118	C₄×Dic₃	—	2	1	no	—
119	C₄×Dic₃	—	2	1	no	—
120	Dic₁₂	—	2	2	no	—
121	C₂×Q₁₆	—	3	1	no	—
122	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	3	1	no	—
125	#126	—	6	1	no	—
126	#126	—	6	1	no	—
127	#126	—	6	1	no	—
128	#125	—	3	1	no	—
129	#125	—	6	1	no	—
130	#124	—	6	1	no	—
131	#123	—	6	1	no	—
132	C₂²⋊Q₈	—	3	1	no	—
133	C₂²⋊Q₈	—	3	1	no	—
134	C₂²⋊Q₈	—	3	1	no	—
135	C₂²⋊Q₈	—	3	1	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×Q₈	—	6	1	no	—
139	C₄×D₄	C₂×D₃	1	1	yes	characteristic
140	#117	—	3	1	no	—
141	#117	—	3	1	no	—
142	#117	—	3	1	no	—
143	#117	—	6	1	no	—
144	C₂×C₄⋊C₄	—	3	1	no	—
145	#107	—	3	1	no	—
146	#107	—	3	1	no	—
147	#107	—	3	1	no	—
148	#107	—	3	1	no	—
149	#107	—	3	1	no	—
150	#106	—	3	1	no	—
151	#106	—	3	1	no	—
152	#106	—	3	1	no	—
153	#106	—	3	1	no	—
154	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
155	#103	—	3	1	no	—
156	#103	—	3	1	no	—
157	#103	—	3	1	no	—
158	#103	—	3	1	no	—
159	#103	—	3	1	no	—
160	#103	—	3	1	no	—
161	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
162	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
163	#97	C₂×D₃	1	1	yes	characteristic
164	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
165	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
166	C₃×D₄	—	4	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
176	C₂×Dic₃	—	2	1	no	—
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	1	no	—
181	C₂×Dic₃	—	2	1	no	—
182	C₂×Dic₃	—	2	1	no	—
183	C₂×Dic₃	—	2	1	no	—
184	C₂×Dic₃	—	4	2	no	—
185	Dic₆	—	2	1	no	—
186	Dic₆	—	2	1	no	—
187	Dic₆	—	2	1	no	—
188	Dic₆	—	2	1	no	—
189	Dic₆	—	4	2	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	—	2	1	no	—
192	C₃×C₈	—	2	1	no	—
193	C₃×C₈	#41	1	2	yes	—
194	C₂×Q₈	—	3	1	no	—
195	C₂×Q₈	—	3	1	no	—
196	C₂×Q₈	—	6	1	no	—
197	C₂×D₄	D₁₂	1	1	yes	characteristic
198	C₂²×C₄	—	6	1	no	—
199	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
201	Q₁₆	—	6	2	no	—
202	M₁₆	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
206	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
207	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	1	no	—
210	C₄⋊C₄	—	3	1	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	1	no	—
213	C₄⋊C₄	—	3	1	no	—
214	C₄⋊C₄	—	6	1	no	—
215	C₄⋊C₄	—	6	1	no	—
216	C₄⋊C₄	—	6	1	no	—
217	C₄⋊C₄	—	6	1	no	—
218	C₄⋊C₄	—	6	1	no	—
219	C₄⋊C₄	—	6	1	no	—
220	C₄⋊C₄	—	6	1	no	—
221	C₄⋊C₄	—	6	1	no	—
222	C₄⋊C₄	—	6	1	no	—
223	C₄⋊C₄	—	6	1	no	—
224	C₄⋊C₄	D₁₂	1	1	yes	characteristic
225	C₂²⋊C₄	—	6	1	no	—
226	C₂²⋊C₄	—	6	1	no	—
227	C₂²⋊C₄	—	6	1	no	—
228	C₂²⋊C₄	—	6	1	no	—
229	C₂²⋊C₄	—	6	1	no	—
230	C₂²⋊C₄	—	6	1	no	—
231	C₂²⋊C₄	—	6	1	no	—
232	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
233	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
234	C₄²	—	6	1	no	—
235	C₄²	—	6	1	no	—
236	C₄²	—	6	1	no	—
237	C₄²	—	6	1	no	—
238	C₄²	—	6	1	no	—
239	C₄²	C₂²×D₃	1	1	yes	characteristic
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	C₂×#41	1	1	yes	characteristic
252	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
253	Dic₃	—	4	1	no	—
254	Dic₃	—	4	1	no	—
255	Dic₃	—	4	1	no	—
256	Dic₃	—	4	1	no	—
257	Dic₃	—	4	1	no	—
258	Dic₃	—	4	1	no	—
259	Dic₃	—	4	1	no	—
260	Dic₃	—	4	1	no	—
261	C₂³	C₂×D₁₂	1	1	yes	characteristic
262	C₂³	C₂×D₁₂	1	1	yes	characteristic
263	Q₈	—	6	1	no	—
264	Q₈	—	6	1	no	—
265	Q₈	—	6	1	no	—
266	Q₈	—	6	1	no	—
267	Q₈	—	12	2	no	—
268	D₄	—	4	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	6	1	no	—
274	C₂×C₄	—	6	1	no	—
275	C₂×C₄	—	6	1	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	6	1	no	—
278	C₂×C₄	—	6	1	no	—
279	C₂×C₄	—	6	1	no	—
280	C₂×C₄	—	6	1	no	—
281	C₂×C₄	—	12	2	no	—
282	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
283	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
284	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
285	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
286	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	#237	1	2	yes	—
291	C₂×C₃	—	4	1	no	—
292	C₂×C₃	—	4	1	no	—
293	C₂×C₃	#547	1	1	yes	characteristic
294	C₂×C₃	#575	1	1	yes	characteristic
295	C₂×C₃	#578	1	1	yes	characteristic
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	C₂²×D₁₂	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	12	1	no	—
308	C₄	—	12	1	no	—
309	C₄	—	12	1	no	—
310	C₄	—	12	1	no	—
311	C₄	—	12	1	no	—
312	C₄	—	12	1	no	—
313	C₄	—	12	1	no	—
314	C₄	—	12	1	no	—
315	C₄	C₂×#237	1	1	yes	characteristic
316	C₄	#1010	1	1	yes	characteristic
317	C₃	#3349	1	1	yes	characteristic, complemented, Sylow
318	C₂	—	4	1	no	—
319	C₂	—	4	1	no	—
320	C₂	#5555	1	1	yes	characteristic
321	C₂	#5748	1	1	yes	characteristic
322	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	#3349
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. #5121	C₂	semidirect
6. #5116	C₂	semidirect
7. #5106	C₂	semidirect
8. #4850	C₂	semidirect
9. #4829	C₂	semidirect
10. #4723	C₂	semidirect
11. #4723	C₂	semidirect
12. #4723	C₂	semidirect
13. #4722	C₂	semidirect
14. #4716	C₂	semidirect
15. #4702	C₂	semidirect
317. C₃	#3349	semidirect

Maximal subgroups

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

subgroup	index
#5542	2
#5541	2
C₃×#435	2
#5137	2
#5121	2
#5116	2
#5106	2
#4850	2
#4829	2
#4723	2
#4722	2
#4716	2
#4702	2
#3349	3