Subgroups — #79982

Order 384.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5585	C₂	1	1	yes	maximal, characteristic
2	#5567	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	#4848	C₂	1	1	yes	maximal, characteristic
5	#4847	C₂	1	1	yes	maximal, characteristic, complemented
6	#4845	C₂	1	1	yes	maximal, characteristic, complemented
7	#4845	C₂	1	1	yes	maximal, characteristic, complemented
8	#4842	C₂	1	1	yes	maximal, characteristic, complemented
9	#4829	C₂	1	1	yes	maximal, characteristic, complemented
10	#4716	C₂	1	1	yes	maximal, characteristic, complemented
11	#4714	C₂	1	1	yes	maximal, characteristic, complemented
12	#4713	C₂	1	1	yes	maximal, characteristic
13	#4710	C₂	1	1	yes	maximal, characteristic, complemented
14	#4697	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#204	C₂	1	1	yes	maximal, characteristic, complemented
16	#3371	—	3	1	no	maximal, Sylow
17	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
18	#947	—	2	1	no	—
19	#944	—	2	1	no	—
20	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
21	C₂×#204	C₂²	1	1	yes	characteristic
22	C₂×#204	C₂²	1	1	yes	characteristic
23	#894	—	2	1	no	—
24	#894	—	2	1	no	—
25	#893	—	2	1	no	—
26	#889	—	2	1	no	—
27	#889	—	2	1	no	—
28	#887	—	2	1	no	—
29	#875	C₂²	1	1	yes	characteristic
30	#875	C₂²	1	1	yes	characteristic
31	#874	C₂²	1	1	yes	characteristic
32	C₄×D₁₂	C₂²	1	1	yes	characteristic
33	C₄×D₁₂	C₂²	1	1	yes	characteristic
34	#870	C₂²	1	1	yes	characteristic
35	#870	C₂²	1	1	yes	characteristic
36	#869	C₂²	1	1	yes	characteristic
37	C₄×Dic₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₆	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×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₃×#97	C₂²	1	1	yes	characteristic
45	C₃×#97	C₂²	1	1	yes	characteristic
46	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
47	#821	C₂²	1	1	yes	characteristic
48	#821	C₂²	1	1	yes	characteristic
49	#821	C₂²	1	1	yes	characteristic
50	#821	C₂²	1	1	yes	characteristic
51	#821	C₂²	1	1	yes	characteristic
52	#821	C₂²	1	1	yes	characteristic
53	#818	C₂²	1	1	yes	characteristic
54	#817	C₂²	1	1	yes	characteristic
55	#817	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	#559	—	3	1	no	—
61	#557	—	3	1	no	—
62	#490	—	3	1	no	—
63	#489	—	3	1	no	—
64	#488	—	3	1	no	—
65	#479	—	3	1	no	—
66	#478	—	3	1	no	—
67	#476	—	3	1	no	—
68	#464	—	3	1	no	—
69	#464	—	3	1	no	—
70	#460	—	3	1	no	—
71	#446	D₃	1	1	yes	characteristic, complemented
72	#441	—	3	1	no	—
73	#440	—	3	1	no	—
74	C₄×SD₁₆	—	3	1	no	—
75	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
76	C₂×D₁₂	—	2	1	no	—
77	C₂×D₁₂	—	2	1	no	—
78	C₂×D₁₂	C₂³	1	1	yes	characteristic
79	C₂×D₁₂	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₃	—	2	1	no	—
81	C₂×C₄×D₃	—	2	1	no	—
82	C₂×Dic₆	—	2	1	no	—
83	C₂×Dic₆	C₂³	1	1	yes	characteristic
84	C₂×Dic₆	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	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
89	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
90	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
91	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
92	C₃×C₄²	C₂³	1	1	yes	characteristic
93	C₃×C₄²	C₂³	1	1	yes	characteristic
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	(C₂×D₃)⋊C₄	—	2	1	no	—
96	(C₂×D₃)⋊C₄	—	2	1	no	—
97	(C₂×D₃)⋊C₄	—	2	1	no	—
98	(C₂×D₃)⋊C₄	—	2	1	no	—
99	(C₂×D₃)⋊C₄	—	2	1	no	—
100	(C₂×D₃)⋊C₄	—	2	1	no	—
101	C₄⋊Dic₃	—	2	1	no	—
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	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	C₄×Dic₃	—	2	1	no	—
115	C₄×Dic₃	—	2	1	no	—
116	C₄×Dic₃	—	2	1	no	—
117	C₄×Dic₃	—	2	1	no	—
118	Dic₁₂	—	2	2	no	—
119	#204	—	2	2	no	—
120	#204	—	2	2	no	—
121	C₂×Q₁₆	—	3	1	no	—
122	C₂×SD₁₆	—	3	1	no	—
123	C₂×SD₁₆	—	3	1	no	—
124	C₄⋊Q₈	—	3	1	no	—
125	C₄⋊Q₈	—	6	1	no	—
126	#127	—	3	1	no	—
127	#125	—	3	1	no	—
128	#125	—	3	1	no	—
129	#125	—	6	1	no	—
130	#125	—	6	1	no	—
131	#124	—	3	1	no	—
132	#124	—	3	1	no	—
133	#124	—	6	1	no	—
134	#123	—	6	1	no	—
135	#123	—	6	1	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×Q₈	—	3	1	no	—
139	C₄×Q₈	—	3	1	no	—
140	C₄×Q₈	—	6	1	no	—
141	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
142	C₄×D₄	—	3	1	no	—
143	C₄×D₄	—	3	1	no	—
144	C₄×D₄	—	6	1	no	—
145	#107	—	3	1	no	—
146	#106	—	3	1	no	—
147	#106	—	3	1	no	—
148	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
151	#103	—	3	1	no	—
152	#103	—	3	1	no	—
153	#103	—	3	1	no	—
154	#103	—	3	1	no	—
155	#102	—	3	1	no	—
156	#102	—	3	1	no	—
157	#102	—	3	1	no	—
158	#102	—	3	1	no	—
159	#102	—	3	1	no	—
160	#102	—	3	1	no	—
161	#97	C₂×D₃	1	1	yes	characteristic
162	#97	C₂×D₃	1	1	yes	characteristic
163	C₄×C₈	C₂×D₃	1	1	yes	characteristic
164	C₂²×D₃	—	2	1	no	—
165	C₂²×D₃	—	2	1	no	—
166	C₃×Q₈	—	4	1	no	—
167	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	D₁₂	—	2	1	no	—
181	D₁₂	—	2	1	no	—
182	D₁₂	—	2	1	no	—
183	D₁₂	—	2	1	no	—
184	D₁₂	—	4	2	no	—
185	D₁₂	—	8	1	no	—
186	C₄×D₃	—	8	1	no	—
187	C₄×D₃	—	8	1	no	—
188	Dic₆	—	2	1	no	—
189	Dic₆	—	2	1	no	—
190	Dic₆	—	2	1	no	—
191	Dic₆	—	4	2	no	—
192	Dic₆	#41	1	2	yes	—
193	C₃×C₈	—	2	1	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	—	2	1	no	—
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	3	1	no	—
199	C₂×Q₈	—	6	1	no	—
200	C₂×Q₈	D₁₂	1	1	yes	characteristic
201	C₂×D₄	—	3	1	no	—
202	C₂×D₄	—	3	1	no	—
203	C₂×D₄	—	6	1	no	—
204	C₂×D₄	—	6	1	no	—
205	C₂²×C₄	—	6	1	no	—
206	C₂²×C₄	—	6	1	no	—
207	Q₁₆	—	6	2	no	—
208	SD₁₆	—	6	2	no	—
209	SD₁₆	—	6	2	no	—
210	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
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	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	—	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	—	6	1	no	—
225	C₄⋊C₄	—	6	1	no	—
226	C₄⋊C₄	—	6	1	no	—
227	C₄⋊C₄	D₁₂	1	1	yes	characteristic
228	C₄⋊C₄	D₁₂	1	1	yes	characteristic
229	C₄⋊C₄	D₁₂	1	1	yes	characteristic
230	C₂²⋊C₄	—	6	1	no	—
231	C₂²⋊C₄	—	6	1	no	—
232	C₂²⋊C₄	—	6	1	no	—
233	C₂²⋊C₄	—	6	1	no	—
234	C₂²⋊C₄	—	6	1	no	—
235	C₂²⋊C₄	—	6	1	no	—
236	C₄²	—	6	1	no	—
237	C₄²	—	6	1	no	—
238	C₄²	—	6	1	no	—
239	C₄²	—	6	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	C₂²×D₄	1	1	yes	characteristic
244	C₂×D₃	—	4	1	no	—
245	C₂×D₃	—	4	1	no	—
246	C₂×D₃	—	4	1	no	—
247	C₂×D₃	—	4	1	no	—
248	C₂×D₃	—	4	1	no	—
249	C₂×D₃	—	4	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	C₂×#41	1	1	yes	characteristic
257	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
258	Dic₃	—	2	2	no	—
259	Dic₃	—	4	1	no	—
260	Dic₃	—	4	1	no	—
261	Dic₃	—	4	1	no	—
262	Dic₃	—	4	1	no	—
263	Dic₃	—	4	1	no	—
264	C₂³	—	6	1	no	—
265	C₂³	—	6	1	no	—
266	Q₈	—	3	2	no	—
267	Q₈	—	4	1	no	—
268	Q₈	—	6	1	no	—
269	Q₈	—	6	1	no	—
270	Q₈	—	6	1	no	—
271	Q₈	—	12	2	no	—
272	D₄	—	6	1	no	—
273	D₄	—	6	1	no	—
274	D₄	—	6	1	no	—
275	D₄	—	6	1	no	—
276	D₄	—	12	2	no	—
277	D₄	—	24	1	no	—
278	C₂×C₄	—	6	1	no	—
279	C₂×C₄	—	6	1	no	—
280	C₂×C₄	—	6	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	1	no	—
284	C₂×C₄	—	24	1	no	—
285	C₂×C₄	—	24	1	no	—
286	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
287	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
288	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
289	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
290	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
291	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₂×C₃	#548	1	1	yes	characteristic
298	C₂×C₃	#575	1	1	yes	characteristic
299	C₂×C₃	#577	1	1	yes	characteristic
300	D₃	—	8	1	no	—
301	D₃	—	8	1	no	—
302	C₂²	—	12	1	no	—
303	C₂²	—	12	1	no	—
304	C₂²	—	12	1	no	—
305	C₂²	—	12	1	no	—
306	C₂²	—	12	1	no	—
307	C₂²	—	12	1	no	—
308	C₂²	C₂²×D₁₂	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	6	2	no	—
316	C₄	—	12	1	no	—
317	C₄	—	12	1	no	—
318	C₄	—	12	1	no	—
319	C₄	—	12	1	no	—
320	C₄	—	12	1	no	—
321	C₄	#1007	1	1	yes	characteristic
322	C₄	D₃×#41	1	1	yes	characteristic
323	C₃	#3371	1	1	yes	characteristic, complemented, Sylow
324	C₂	—	24	1	no	—
325	C₂	—	24	1	no	—
326	C₂	#5576	1	1	yes	characteristic
327	C₂	#5748	1	1	yes	characteristic
328	C₂	#5751	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#446	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#874	C₂²
hypercenter	#446	D₃
hypocenter	C₃	#3371
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #5567	C₂	semidirect
3. C₃×#446	C₂	semidirect
5. #4847	C₂	semidirect
6. #4845	C₂	semidirect
7. #4845	C₂	semidirect
8. #4842	C₂	semidirect
9. #4829	C₂	semidirect
10. #4716	C₂	semidirect
11. #4714	C₂	semidirect
13. #4710	C₂	semidirect
14. #4697	C₂	semidirect
15. C₄×#204	C₂	semidirect
71. #446	D₃	semidirect
323. C₃	#3371	semidirect

Maximal subgroups

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

subgroup	index
#5585	2
#5567	2
C₃×#446	2
#4848	2
#4847	2
#4845	2
#4842	2
#4829	2
#4716	2
#4714	2
#4713	2
#4710	2
#4697	2
C₄×#204	2
#3371	3