Subgroups — #79959

Order 384.

348 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5578	C₂	1	1	yes	maximal, characteristic
2	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#5024	C₂	1	1	yes	maximal, characteristic, complemented
4	#4839	C₂	1	1	yes	maximal, characteristic
5	#4839	C₂	1	1	yes	maximal, characteristic
6	#4837	C₂	1	1	yes	maximal, characteristic
7	#4831	C₂	1	1	yes	maximal, characteristic, complemented
8	#4828	C₂	1	1	yes	maximal, characteristic, complemented
9	#4828	C₂	1	1	yes	maximal, characteristic, complemented
10	#4708	C₂	1	1	yes	maximal, characteristic
11	D₃×#97	C₂	1	1	yes	maximal, characteristic
12	#4701	C₂	1	1	yes	maximal, characteristic, complemented
13	#4701	C₂	1	1	yes	maximal, characteristic, complemented
14	#4689	C₂	1	1	yes	maximal, characteristic
15	#4679	C₂	1	1	yes	maximal, characteristic, complemented
16	#2961	—	3	1	no	maximal, Sylow
17	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
18	#946	C₄	1	1	yes	characteristic
19	#944	C₄	1	1	yes	characteristic
20	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
22	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
23	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
24	#897	C₄	1	1	yes	characteristic
25	#896	C₄	1	1	yes	characteristic
26	#896	C₄	1	1	yes	characteristic
27	#889	C₄	1	1	yes	characteristic
28	#889	C₄	1	1	yes	characteristic
29	#888	C₄	1	1	yes	characteristic
30	#872	C₂²	1	1	yes	characteristic
31	#872	C₂²	1	1	yes	characteristic
32	C₄²×D₃	C₂²	1	1	yes	characteristic
33	C₄×Dic₆	C₂²	1	1	yes	characteristic
34	C₄×Dic₆	C₂²	1	1	yes	characteristic
35	C₄×Dic₆	C₂²	1	1	yes	characteristic
36	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃×#97	C₂²	1	1	yes	characteristic
40	C₃×#97	C₂²	1	1	yes	characteristic
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#815	C₂²	1	1	yes	characteristic
43	#815	C₂²	1	1	yes	characteristic
44	#815	C₂²	1	1	yes	characteristic
45	#815	C₂²	1	1	yes	characteristic
46	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₃⋊C₈	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	C₈×Dic₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₃	C₂²	1	1	yes	characteristic
54	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
55	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
57	#803	C₂²	1	1	yes	characteristic
58	#803	C₂²	1	1	yes	characteristic
59	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
60	#533	—	3	1	no	—
61	#446	—	3	1	no	—
62	#446	—	3	1	no	—
63	#446	—	3	1	no	—
64	#446	—	3	1	no	—
65	#446	—	3	1	no	—
66	#446	—	3	1	no	—
67	#446	—	3	1	no	—
68	#446	D₃	1	1	yes	characteristic, complemented
69	#424	—	3	1	no	—
70	#424	—	3	1	no	—
71	#423	—	3	1	no	—
72	#405	—	3	1	no	—
73	#405	—	3	1	no	—
74	C₂×#97	—	3	1	no	—
75	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
80	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
81	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
82	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
86	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
88	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
89	C₃×C₄²	C₂³	1	1	yes	characteristic
90	C₃×C₄²	C₂³	1	1	yes	characteristic
91	C₃×C₄²	C₂³	1	1	yes	characteristic
92	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
93	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
94	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
95	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
97	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
98	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
99	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
100	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
101	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
102	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
103	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
104	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
105	Dic₃⋊C₄	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	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
110	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
111	C₄×Dic₃	C₂³	1	1	yes	characteristic
112	C₄×Dic₃	C₂³	1	1	yes	characteristic
113	C₄×Dic₃	C₂³	1	1	yes	characteristic
114	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
115	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
116	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
117	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
118	ΓL(1,𝔽₂₅)	—	2	2	no	—
119	ΓL(1,𝔽₂₅)	—	2	2	no	—
120	D₃×C₈	—	2	2	no	—
121	D₃×C₈	Q₈	1	4	yes	complemented
122	C₂×M₁₆	—	3	1	no	—
123	C₂×M₁₆	—	3	1	no	—
124	C₂²×C₈	—	3	1	no	—
125	C₂²×C₈	—	3	1	no	—
126	C₄⋊Q₈	—	3	1	no	—
127	C₄⋊Q₈	—	3	1	no	—
128	#125	—	3	1	no	—
129	#125	—	3	1	no	—
130	C₂²⋊Q₈	—	3	1	no	—
131	C₂²⋊Q₈	—	3	1	no	—
132	C₂²⋊Q₈	—	3	1	no	—
133	C₂²⋊Q₈	—	3	1	no	—
134	C₄×Q₈	—	3	1	no	—
135	C₄×Q₈	—	3	1	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
138	#117	—	3	1	no	—
139	#117	—	3	1	no	—
140	C₂×C₄²	—	3	1	no	—
141	C₄⋊C₈	—	3	1	no	—
142	C₄⋊C₈	—	3	1	no	—
143	C₄⋊C₈	—	3	1	no	—
144	C₄⋊C₈	—	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	C₂×D₃	1	1	yes	characteristic
151	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
152	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
153	#97	—	3	1	no	—
154	#97	—	3	1	no	—
155	#97	—	3	1	no	—
156	#97	—	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	C₂×D₃	1	1	yes	characteristic
165	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
166	C₃×Q₈	—	4	1	no	—
167	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
174	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
175	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
176	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
177	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
178	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
179	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
180	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
181	C₄×D₃	—	4	1	no	—
182	C₄×D₃	—	4	1	no	—
183	C₄×D₃	C₂×Q₈	1	2	yes	—
184	C₄×D₃	#41	1	2	yes	—
185	Dic₆	—	4	1	no	—
186	Dic₆	—	4	1	no	—
187	Dic₆	—	4	1	no	—
188	C₃×C₈	—	2	1	no	—
189	C₃×C₈	—	2	1	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	C₂×Q₈	1	2	yes	—
192	C₃⋊C₈	—	2	1	no	—
193	C₃⋊C₈	—	2	1	no	—
194	C₃⋊C₈	—	2	1	no	—
195	C₃⋊C₈	C₂×Q₈	1	2	yes	—
196	C₂×Q₈	—	3	1	no	—
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	3	1	no	—
199	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
200	C₂²×C₄	—	3	1	no	—
201	C₂²×C₄	—	3	1	no	—
202	C₂²×C₄	—	3	1	no	—
203	M₁₆	—	6	2	no	—
204	M₁₆	—	6	2	no	—
205	C₂×C₈	—	3	1	no	—
206	C₂×C₈	—	3	4	no	—
207	C₂×C₈	—	3	1	no	—
208	C₂×C₈	—	3	1	no	—
209	C₂×C₈	—	3	1	no	—
210	C₂×C₈	—	6	2	no	—
211	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
213	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
214	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
215	C₄⋊C₄	—	3	1	no	—
216	C₄⋊C₄	—	3	1	no	—
217	C₄⋊C₄	—	3	1	no	—
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	1	no	—
220	C₄⋊C₄	—	3	1	no	—
221	C₄⋊C₄	—	3	1	no	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	3	1	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	3	1	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
229	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
230	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	3	1	no	—
233	C₂²⋊C₄	—	3	1	no	—
234	C₂²⋊C₄	—	3	1	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	3	1	no	—
237	C₄²	—	3	1	no	—
238	C₄²	—	3	1	no	—
239	C₄²	—	3	1	no	—
240	C₄²	C₂²×D₃	1	1	yes	characteristic
241	C₄²	C₂²×D₃	1	1	yes	characteristic
242	C₄²	C₂²×D₃	1	1	yes	characteristic
243	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
244	C₂×D₃	—	2	1	no	—
245	C₂×D₃	—	2	1	no	—
246	C₂×D₃	C₄×Q₈	1	2	yes	—
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	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
254	C₃×C₄	C₂×#41	1	1	yes	characteristic
255	Dic₃	—	2	1	no	—
256	Dic₃	—	2	1	no	—
257	Dic₃	—	2	1	no	—
258	Dic₃	—	2	1	no	—
259	Dic₃	—	2	1	no	—
260	Dic₃	—	2	1	no	—
261	Dic₃	C₄×Q₈	1	2	yes	—
262	C₂³	—	3	1	no	—
263	Q₈	—	4	1	no	—
264	Q₈	—	12	1	no	—
265	Q₈	—	12	1	no	—
266	Q₈	—	12	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	2	no	—
269	C₂×C₄	—	3	2	no	—
270	C₂×C₄	—	3	1	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	1	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	12	1	no	—
277	C₂×C₄	—	12	1	no	—
278	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
284	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	3	2	no	—
289	C₈	—	6	1	no	—
290	C₈	—	6	1	no	—
291	C₈	—	6	1	no	—
292	C₈	D₃×Q₈	1	2	yes	—
293	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
294	C₂×C₃	#568	1	1	yes	characteristic
295	C₂×C₃	#568	1	1	yes	characteristic
296	D₃	—	2	2	no	—
297	C₂²	—	3	2	no	—
298	C₂²	—	6	1	no	—
299	C₂²	—	6	1	no	—
300	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	3	2	no	—
308	C₄	—	6	1	no	—
309	C₄	—	6	1	no	—
310	C₄	—	6	1	no	—
311	C₄	—	6	1	no	—
312	C₄	—	6	1	no	—
313	C₄	—	6	1	no	—
314	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
315	C₄	D₃×#41	1	1	yes	characteristic
316	C₃	#2961	1	1	yes	characteristic, complemented, Sylow
317	C₂	—	6	2	no	—
318	C₂	C₄×D₃×Q₈	1	1	yes	characteristic
319	C₂	#5745	1	1	yes	characteristic
320	C₂	#5750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#446	C₂
prime radical	C₂²×D₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#446	D₃
hypocenter	C₃	#2961
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₃×#446	C₂	semidirect
3. #5024	C₂	semidirect
7. #4831	C₂	semidirect
8. #4828	C₂	semidirect
9. #4828	C₂	semidirect
12. #4701	C₂	semidirect
13. #4701	C₂	semidirect
15. #4679	C₂	semidirect
68. #446	D₃	semidirect
121. D₃×C₈	Q₈	semidirect
316. C₃	#2961	semidirect

Maximal subgroups

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

subgroup	index
#5578	2
C₃×#446	2
#5024	2
#4839	2
#4837	2
#4831	2
#4828	2
#4708	2
D₃×#97	2
#4701	2
#4689	2
#4679	2
#2961	3