Subgroups — #79470

Order 384.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5543	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	C₂×#817	C₂	1	1	yes	maximal, characteristic, complemented
8	#4851	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	#4722	C₂	1	1	yes	maximal, characteristic, complemented
12	#4721	C₂	1	1	yes	maximal, characteristic, complemented
13	#4721	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	#3344	—	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	#935	—	2	1	no	—
21	D₄×Dic₃	—	2	1	no	—
22	#926	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₃	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	#890	—	2	1	no	—
32	#889	—	2	1	no	—
33	#879	—	2	1	no	—
34	#878	—	2	1	no	—
35	#878	—	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	#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	#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	#554	—	3	1	no	—
61	#552	—	3	1	no	—
62	#501	—	3	1	no	—
63	#485	—	3	1	no	—
64	#484	—	3	1	no	—
65	#483	—	3	1	no	—
66	#483	—	3	1	no	—
67	#479	—	3	1	no	—
68	#475	—	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	C₂×#106	—	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	C₂²×Dic₃	—	2	1	no	—
80	C₂²×Dic₃	—	2	1	no	—
81	C₂×Dic₆	—	2	1	no	—
82	C₂×Dic₆	C₂³	1	1	yes	characteristic
83	C₂×Dic₆	C₂³	1	1	yes	characteristic
84	C₃×M₁₆	—	2	2	no	—
85	C₂×C₃×C₈	—	2	2	no	—
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	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
90	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
91	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
92	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
93	C₃×C₄²	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
102	C₄⋊Dic₃	—	2	1	no	—
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	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
108	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
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	Dic₃⋊C₄	—	2	1	no	—
116	Dic₃⋊C₄	—	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	#125	—	3	1	no	—
127	#125	—	6	1	no	—
128	#125	—	6	1	no	—
129	#123	—	6	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	C₂²⋊Q₈	—	6	1	no	—
135	C₂²⋊Q₈	—	6	1	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×D₄	—	6	1	no	—
139	C₄×D₄	C₂×D₃	1	1	yes	characteristic
140	#117	—	3	1	no	—
141	#117	—	6	1	no	—
142	C₂×C₄⋊C₄	—	3	1	no	—
143	C₂×C₄⋊C₄	—	3	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	#106	—	3	1	no	—
149	#106	—	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	—	2	2	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	C₂×Dic₃	—	8	1	no	—
186	C₂×Dic₃	—	8	1	no	—
187	Dic₆	—	2	1	no	—
188	Dic₆	—	2	1	no	—
189	Dic₆	—	2	1	no	—
190	Dic₆	—	2	1	no	—
191	Dic₆	—	8	1	no	—
192	C₃×C₈	—	2	1	no	—
193	C₃×C₈	—	2	1	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	#41	1	2	yes	—
196	C₂×Q₈	—	3	1	no	—
197	C₂×Q₈	—	3	1	no	—
198	C₂×Q₈	—	6	1	no	—
199	C₂×D₄	D₁₂	1	1	yes	characteristic
200	C₂²×C₄	—	6	1	no	—
201	C₂²×C₄	—	6	1	no	—
202	C₂²×C₄	—	6	1	no	—
203	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
205	Q₁₆	—	6	2	no	—
206	M₁₆	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
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	—	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	—	6	1	no	—
228	C₄⋊C₄	—	6	1	no	—
229	C₄⋊C₄	—	6	1	no	—
230	C₄⋊C₄	D₁₂	1	1	yes	characteristic
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	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
237	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
238	C₄²	—	6	1	no	—
239	C₄²	—	6	1	no	—
240	C₄²	—	6	1	no	—
241	C₄²	C₂²×D₃	1	1	yes	characteristic
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	—	2	1	no	—
247	C₂²×C₃	#137	1	2	yes	—
248	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
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	C₂×#41	1	1	yes	characteristic
254	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
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	Dic₃	—	4	1	no	—
262	Dic₃	—	4	1	no	—
263	C₂³	C₂×D₁₂	1	1	yes	characteristic
264	C₂³	C₂×D₁₂	1	1	yes	characteristic
265	Q₈	—	6	1	no	—
266	Q₈	—	6	1	no	—
267	Q₈	—	6	1	no	—
268	Q₈	—	6	1	no	—
269	Q₈	—	24	1	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	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	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	12	2	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	C₂³×D₃	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	#237	1	2	yes	—
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	—	4	1	no	—
297	C₂×C₃	#547	1	1	yes	characteristic
298	C₂×C₃	C₂×#137	1	1	yes	characteristic
299	C₂×C₃	#577	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	#909	1	2	yes	—
306	C₂²	C₂²×D₁₂	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	12	1	no	—
312	C₄	—	12	1	no	—
313	C₄	—	12	1	no	—
314	C₄	—	12	1	no	—
315	C₄	—	12	1	no	—
316	C₄	—	12	1	no	—
317	C₄	—	12	1	no	—
318	C₄	—	12	1	no	—
319	C₄	C₂×#237	1	1	yes	characteristic
320	C₄	#1010	1	1	yes	characteristic
321	C₃	#3344	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	2	2	no	—
323	C₂	—	4	1	no	—
324	C₂	#5555	1	1	yes	characteristic
325	C₂	C₂×#909	1	1	yes	characteristic
326	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×#435	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×D₄	D₄
hypercenter	#435	D₃
hypocenter	C₃	#3344
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. C₂×#817	C₂	semidirect
8. #4851	C₂	semidirect
9. #4829	C₂	semidirect
10. #4723	C₂	semidirect
11. #4722	C₂	semidirect
12. #4721	C₂	semidirect
13. #4721	C₂	semidirect
14. #4716	C₂	semidirect
15. #4702	C₂	semidirect
321. C₃	#3344	semidirect

Maximal subgroups

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

subgroup	index
#5543	2
#5541	2
C₃×#435	2
#5137	2
#5121	2
#5116	2
C₂×#817	2
#4851	2
#4829	2
#4723	2
#4722	2
#4721	2
#4716	2
#4702	2
#3344	3