Subgroups — #79363

Order 384.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5519	C₂	1	1	yes	maximal, characteristic
2	#5518	C₂	1	1	yes	maximal, characteristic
3	C₃×#433	C₂	1	1	yes	maximal, characteristic
4	#4851	C₂	1	1	yes	maximal, characteristic, complemented
5	#4850	C₂	1	1	yes	maximal, characteristic, complemented
6	#4829	C₂	1	1	yes	maximal, characteristic, complemented
7	#4829	C₂	1	1	yes	maximal, characteristic, complemented
8	#4723	C₂	1	1	yes	maximal, characteristic
9	#4723	C₂	1	1	yes	maximal, characteristic
10	#4722	C₂	1	1	yes	maximal, characteristic
11	#4721	C₂	1	1	yes	maximal, characteristic
12	#4703	C₂	1	1	yes	maximal, characteristic, complemented
13	#4702	C₂	1	1	yes	maximal, characteristic, complemented
14	#4683	C₂	1	1	yes	maximal, characteristic, complemented
15	#4681	C₂	1	1	yes	maximal, characteristic, complemented
16	#3242	—	3	1	no	maximal, Sylow
17	C₃×#117	C₂²	1	1	yes	characteristic
18	#926	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	#924	C₂²	1	1	yes	characteristic
21	#924	C₂²	1	1	yes	characteristic
22	#890	—	2	1	no	—
23	#889	—	2	1	no	—
24	#889	—	2	1	no	—
25	#888	—	2	1	no	—
26	#879	—	2	1	no	—
27	#879	—	2	1	no	—
28	#878	—	2	1	no	—
29	#878	—	2	1	no	—
30	#870	C₂²	1	1	yes	characteristic
31	#870	C₂²	1	1	yes	characteristic
32	#870	C₂²	1	1	yes	characteristic
33	#869	C₂²	1	1	yes	characteristic
34	#869	C₂²	1	1	yes	characteristic
35	#869	C₂²	1	1	yes	characteristic
36	C₄×Dic₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₆	C₂²	1	1	yes	characteristic
38	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×#97	C₂²	1	1	yes	characteristic
43	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
44	#818	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	#817	C₂²	1	1	yes	characteristic
49	#817	C₂²	1	1	yes	characteristic
50	#817	C₂²	1	1	yes	characteristic
51	#817	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	#544	—	3	1	no	—
61	#541	—	3	1	no	—
62	#501	—	3	1	no	—
63	#499	—	3	1	no	—
64	#491	—	3	1	no	—
65	#487	—	3	1	no	—
66	#485	—	3	1	no	—
67	#485	—	3	1	no	—
68	#484	—	3	1	no	—
69	#483	—	3	1	no	—
70	#479	—	3	1	no	—
71	#479	—	3	1	no	—
72	#477	—	3	1	no	—
73	#475	—	3	1	no	—
74	#433	D₃	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
76	C₂²×Dic₃	—	2	1	no	—
77	C₂×Dic₆	—	2	1	no	—
78	C₂×Dic₆	—	2	1	no	—
79	C₂×Dic₆	C₂³	1	1	yes	characteristic
80	C₂×Dic₆	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂³	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	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
86	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
88	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
89	C₃×C₄²	C₂³	1	1	yes	characteristic
90	C₃×C₄²	C₂³	1	1	yes	characteristic
91	#217	—	2	1	no	—
92	#217	—	2	1	no	—
93	#217	—	2	1	no	—
94	C₄⋊Dic₃	—	2	1	no	—
95	C₄⋊Dic₃	—	2	1	no	—
96	C₄⋊Dic₃	—	2	1	no	—
97	C₄⋊Dic₃	—	2	1	no	—
98	C₄⋊Dic₃	—	2	1	no	—
99	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
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	Dic₃⋊C₄	—	2	1	no	—
106	Dic₃⋊C₄	—	2	1	no	—
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⋊Q₈	—	3	1	no	—
119	C₄⋊Q₈	—	3	1	no	—
120	C₄⋊Q₈	—	3	1	no	—
121	C₄⋊Q₈	—	6	1	no	—
122	#126	—	6	1	no	—
123	#126	—	6	1	no	—
124	#125	—	3	1	no	—
125	#125	—	3	1	no	—
126	#125	—	3	1	no	—
127	#125	—	6	1	no	—
128	#125	—	6	1	no	—
129	#125	—	6	1	no	—
130	C₂²⋊Q₈	—	3	1	no	—
131	C₂²⋊Q₈	—	3	1	no	—
132	C₂²⋊Q₈	—	6	1	no	—
133	C₂²⋊Q₈	—	6	1	no	—
134	C₄×Q₈	—	3	1	no	—
135	C₄×Q₈	—	3	1	no	—
136	#117	—	3	1	no	—
137	#117	C₂×D₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	3	1	no	—
139	#107	—	3	1	no	—
140	#107	—	3	1	no	—
141	#107	—	3	1	no	—
142	#107	—	3	1	no	—
143	#106	—	3	1	no	—
144	#106	—	3	1	no	—
145	#106	—	3	1	no	—
146	#106	—	3	1	no	—
147	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
149	#103	—	3	1	no	—
150	#103	—	3	1	no	—
151	#103	—	3	1	no	—
152	#103	—	3	1	no	—
153	#103	—	3	1	no	—
154	#103	—	3	1	no	—
155	#103	—	3	1	no	—
156	#103	—	3	1	no	—
157	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
159	#97	C₂×D₃	1	1	yes	characteristic
160	C₄×C₈	C₂×D₃	1	1	yes	characteristic
161	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
170	C₂×Dic₃	—	2	1	no	—
171	C₂×Dic₃	—	2	1	no	—
172	C₂×Dic₃	—	2	1	no	—
173	C₂×Dic₃	—	2	1	no	—
174	C₂×Dic₃	—	2	1	no	—
175	C₂×Dic₃	—	2	1	no	—
176	C₂×Dic₃	—	2	1	no	—
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	—	8	1	no	—
179	Dic₆	—	2	1	no	—
180	Dic₆	—	2	1	no	—
181	Dic₆	—	2	1	no	—
182	Dic₆	—	2	1	no	—
183	Dic₆	—	8	1	no	—
184	Dic₆	—	8	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	C₂×Q₈	—	3	1	no	—
190	C₂×Q₈	—	3	1	no	—
191	C₂×Q₈	—	6	1	no	—
192	C₂×Q₈	—	6	1	no	—
193	C₂²×C₄	—	6	1	no	—
194	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
195	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
196	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
199	C₄⋊C₄	—	3	1	no	—
200	C₄⋊C₄	—	3	1	no	—
201	C₄⋊C₄	—	3	1	no	—
202	C₄⋊C₄	—	3	1	no	—
203	C₄⋊C₄	—	3	1	no	—
204	C₄⋊C₄	—	3	1	no	—
205	C₄⋊C₄	—	6	1	no	—
206	C₄⋊C₄	—	6	1	no	—
207	C₄⋊C₄	—	6	1	no	—
208	C₄⋊C₄	—	6	1	no	—
209	C₄⋊C₄	—	6	1	no	—
210	C₄⋊C₄	—	6	1	no	—
211	C₄⋊C₄	—	6	1	no	—
212	C₄⋊C₄	—	6	1	no	—
213	C₄⋊C₄	—	6	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	D₁₂	1	1	yes	characteristic
221	C₄⋊C₄	D₁₂	1	1	yes	characteristic
222	C₂²⋊C₄	—	6	1	no	—
223	C₂²⋊C₄	—	6	1	no	—
224	C₂²⋊C₄	—	6	1	no	—
225	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
226	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
227	C₄²	—	6	1	no	—
228	C₄²	—	6	1	no	—
229	C₄²	—	6	1	no	—
230	C₄²	C₂²×D₃	1	1	yes	characteristic
231	C₄²	C₂²×D₃	1	1	yes	characteristic
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
242	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
243	Dic₃	—	4	1	no	—
244	Dic₃	—	4	1	no	—
245	Dic₃	—	4	1	no	—
246	Dic₃	—	4	1	no	—
247	Dic₃	—	4	1	no	—
248	Dic₃	—	4	1	no	—
249	Dic₃	—	4	1	no	—
250	Dic₃	—	4	1	no	—
251	C₂³	C₂×D₁₂	1	1	yes	characteristic
252	Q₈	—	6	1	no	—
253	Q₈	—	6	1	no	—
254	Q₈	—	6	1	no	—
255	Q₈	—	6	1	no	—
256	Q₈	—	24	1	no	—
257	Q₈	—	24	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	6	1	no	—
263	C₂×C₄	—	6	1	no	—
264	C₂×C₄	—	6	1	no	—
265	C₂×C₄	—	6	1	no	—
266	C₂×C₄	—	6	1	no	—
267	C₂×C₄	—	6	1	no	—
268	C₂×C₄	—	24	1	no	—
269	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
270	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
271	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
272	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
273	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₂×C₃	—	4	1	no	—
280	C₂×C₃	#536	1	1	yes	characteristic
281	C₂×C₃	#577	1	1	yes	characteristic
282	C₂×C₃	#578	1	1	yes	characteristic
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	C₂²×D₁₂	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	12	1	no	—
293	C₄	—	12	1	no	—
294	C₄	—	12	1	no	—
295	C₄	—	12	1	no	—
296	C₄	—	12	1	no	—
297	C₄	—	12	1	no	—
298	C₄	—	12	1	no	—
299	C₄	—	12	1	no	—
300	C₄	#1010	1	1	yes	characteristic
301	C₄	#1010	1	1	yes	characteristic
302	C₃	#3242	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	4	1	no	—
304	C₂	#5526	1	1	yes	characteristic
305	C₂	#5751	1	1	yes	characteristic
306	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×#433	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	#433	D₃
hypocenter	C₃	#3242
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
4. #4851	C₂	semidirect
5. #4850	C₂	semidirect
6. #4829	C₂	semidirect
7. #4829	C₂	semidirect
12. #4703	C₂	semidirect
13. #4702	C₂	semidirect
14. #4683	C₂	semidirect
15. #4681	C₂	semidirect
302. C₃	#3242	semidirect

Maximal subgroups

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

subgroup	index
#5519	2
#5518	2
C₃×#433	2
#4851	2
#4850	2
#4829	2
#4723	2
#4722	2
#4721	2
#4703	2
#4702	2
#4683	2
#4681	2
#3242	3