Subgroups — #79968

Order 384.

314 conjugacy classes of subgroups.

The subgroup lattice has 314 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	#4840	C₂	1	1	yes	maximal, characteristic
5	#4840	C₂	1	1	yes	maximal, characteristic
6	#4835	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	#4709	C₂	1	1	yes	maximal, characteristic
11	#4705	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	#4690	C₂	1	1	yes	maximal, characteristic
15	#4679	C₂	1	1	yes	maximal, characteristic, complemented
16	#2987	—	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	#897	C₄	1	1	yes	characteristic
21	#896	C₄	1	1	yes	characteristic
22	#896	C₄	1	1	yes	characteristic
23	#889	C₄	1	1	yes	characteristic
24	#889	C₄	1	1	yes	characteristic
25	#888	C₄	1	1	yes	characteristic
26	#872	C₂²	1	1	yes	characteristic
27	#872	C₂²	1	1	yes	characteristic
28	C₄²×D₃	C₂²	1	1	yes	characteristic
29	C₄×Dic₆	C₂²	1	1	yes	characteristic
30	C₄×Dic₆	C₂²	1	1	yes	characteristic
31	C₄×Dic₆	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃×#97	C₂²	1	1	yes	characteristic
36	C₃×#97	C₂²	1	1	yes	characteristic
37	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
38	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
39	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
40	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
41	(C₂×D₃)⋊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	#433	—	3	1	no	—
70	#433	—	3	1	no	—
71	#433	—	3	1	no	—
72	#433	—	3	1	no	—
73	#432	—	3	1	no	—
74	#432	—	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	C₄⋊Q₈	—	3	1	no	—
119	C₄⋊Q₈	—	3	1	no	—
120	#125	—	3	1	no	—
121	#125	—	3	1	no	—
122	C₂²⋊Q₈	—	3	1	no	—
123	C₂²⋊Q₈	—	3	1	no	—
124	C₂²⋊Q₈	—	3	1	no	—
125	C₂²⋊Q₈	—	3	1	no	—
126	C₄×Q₈	—	3	1	no	—
127	C₄×Q₈	—	3	1	no	—
128	C₄×Q₈	—	3	1	no	—
129	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
130	#117	—	3	1	no	—
131	#117	—	3	1	no	—
132	C₂×C₄²	—	3	1	no	—
133	C₄⋊C₈	—	3	1	no	—
134	C₄⋊C₈	—	3	1	no	—
135	C₄⋊C₈	—	3	1	no	—
136	C₄⋊C₈	—	3	1	no	—
137	C₄⋊C₈	—	3	1	no	—
138	C₄⋊C₈	—	3	1	no	—
139	C₄⋊C₈	—	3	1	no	—
140	C₄⋊C₈	—	3	1	no	—
141	C₄⋊C₈	—	3	1	no	—
142	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
143	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
145	C₂²⋊C₈	—	3	1	no	—
146	C₂²⋊C₈	—	3	1	no	—
147	C₂²⋊C₈	—	3	1	no	—
148	C₂²⋊C₈	—	3	1	no	—
149	#97	—	3	1	no	—
150	#97	—	3	1	no	—
151	#97	—	3	1	no	—
152	#97	—	3	1	no	—
153	#97	—	3	1	no	—
154	#97	—	3	1	no	—
155	#97	C₂×D₃	1	1	yes	characteristic
156	#97	C₂×D₃	1	1	yes	characteristic
157	C₄×C₈	—	3	1	no	—
158	C₄×C₈	—	3	1	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	C₂×D₃	1	1	yes	characteristic
161	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
162	C₃×Q₈	—	4	1	no	—
163	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
170	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
171	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
172	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
173	C₂×Dic₃	C₂²×C₄	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	C₄×D₃	—	2	1	no	—
178	C₄×D₃	—	2	1	no	—
179	C₄×D₃	—	4	1	no	—
180	C₄×D₃	—	4	1	no	—
181	Dic₆	—	4	1	no	—
182	Dic₆	—	4	1	no	—
183	Dic₆	—	4	1	no	—
184	C₃×C₈	—	2	1	no	—
185	C₃×C₈	—	2	1	no	—
186	C₃×C₈	—	2	1	no	—
187	C₃×C₈	—	2	1	no	—
188	C₃⋊C₈	—	2	1	no	—
189	C₃⋊C₈	—	2	1	no	—
190	C₃⋊C₈	—	2	1	no	—
191	C₃⋊C₈	—	2	1	no	—
192	C₂×Q₈	—	3	1	no	—
193	C₂×Q₈	—	3	1	no	—
194	C₂×Q₈	—	3	1	no	—
195	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	3	1	no	—
199	C₂×C₈	—	3	1	no	—
200	C₂×C₈	—	3	1	no	—
201	C₂×C₈	—	3	1	no	—
202	C₂×C₈	—	3	1	no	—
203	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
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	C₄⋊C₄	—	3	1	no	—
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	—	3	1	no	—
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	C₄×D₃	1	1	yes	characteristic
221	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
222	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
223	C₂²⋊C₄	—	3	1	no	—
224	C₂²⋊C₄	—	3	1	no	—
225	C₂²⋊C₄	—	3	1	no	—
226	C₂²⋊C₄	—	3	1	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	3	1	no	—
229	C₄²	—	3	1	no	—
230	C₄²	—	3	1	no	—
231	C₄²	—	3	1	no	—
232	C₄²	C₂²×D₃	1	1	yes	characteristic
233	C₄²	C₂²×D₃	1	1	yes	characteristic
234	C₄²	C₂²×D₃	1	1	yes	characteristic
235	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
236	C₂×D₃	—	2	1	no	—
237	C₂×D₃	—	2	1	no	—
238	C₂×D₃	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
246	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
247	Dic₃	—	2	1	no	—
248	Dic₃	—	2	1	no	—
249	Dic₃	—	2	1	no	—
250	Dic₃	—	2	1	no	—
251	Dic₃	—	2	1	no	—
252	Dic₃	—	2	1	no	—
253	Dic₃	—	2	1	no	—
254	C₂³	—	3	1	no	—
255	Q₈	—	4	1	no	—
256	Q₈	—	12	1	no	—
257	Q₈	—	12	1	no	—
258	Q₈	—	12	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	1	no	—
263	C₂×C₄	—	3	1	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	3	1	no	—
266	C₂×C₄	—	6	1	no	—
267	C₂×C₄	—	6	1	no	—
268	C₂×C₄	—	12	1	no	—
269	C₂×C₄	—	12	1	no	—
270	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	6	1	no	—
282	C₈	—	6	1	no	—
283	C₈	—	6	1	no	—
284	C₈	—	6	1	no	—
285	C₂×C₃	#519	1	1	yes	characteristic
286	C₂×C₃	#568	1	1	yes	characteristic
287	C₂×C₃	#568	1	1	yes	characteristic
288	D₃	—	4	1	no	—
289	C₂²	—	6	1	no	—
290	C₂²	—	6	1	no	—
291	C₂²	—	6	1	no	—
292	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	6	1	no	—
300	C₄	—	6	1	no	—
301	C₄	—	6	1	no	—
302	C₄	—	6	1	no	—
303	C₄	—	6	1	no	—
304	C₄	—	6	1	no	—
305	C₄	—	6	1	no	—
306	C₄	#1007	1	1	yes	characteristic
307	C₄	#1010	1	1	yes	characteristic
308	C₃	#2987	1	1	yes	characteristic, complemented, Sylow
309	C₂	—	12	1	no	—
310	C₂	#5572	1	1	yes	characteristic
311	C₂	#5745	1	1	yes	characteristic
312	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	#2987
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
308. C₃	#2987	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
#4840	2
#4835	2
#4831	2
#4828	2
#4709	2
#4705	2
#4701	2
#4690	2
#4679	2
#2987	3