Subgroups — #79834

Order 384.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5580	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
4	#5186	C₂	1	1	yes	maximal, characteristic, complemented
5	#5185	C₂	1	1	yes	maximal, characteristic, complemented
6	#5030	C₂	1	1	yes	maximal, characteristic, complemented
7	#5026	C₂	1	1	yes	maximal, characteristic, complemented
8	#4863	C₂	1	1	yes	maximal, characteristic
9	#4857	C₂	1	1	yes	maximal, characteristic, complemented
10	#4811	C₂	1	1	yes	maximal, characteristic, complemented
11	#4806	C₂	1	1	yes	maximal, characteristic
12	#4798	C₂	1	1	yes	maximal, characteristic, complemented
13	#4796	C₂	1	1	yes	maximal, characteristic, complemented
14	#4710	C₂	1	1	yes	maximal, characteristic, complemented
15	#4705	C₂	1	1	yes	maximal, characteristic, complemented
16	#3117	—	3	1	no	maximal, Sylow
17	C₂×D₃×Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
21	#947	C₂²	1	1	yes	characteristic
22	#946	C₂²	1	1	yes	characteristic
23	Q₈×Dic₃	C₂²	1	1	yes	characteristic
24	C₂×#216	C₂²	1	1	yes	characteristic
25	C₂×#215	C₂²	1	1	yes	characteristic
26	C₂×#204	C₂²	1	1	yes	characteristic
27	#898	—	2	1	no	—
28	#895	C₂²	1	1	yes	characteristic
29	#894	—	2	1	no	—
30	D₃×C₄⋊C₄	—	2	1	no	—
31	D₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	#890	C₂²	1	1	yes	characteristic
33	#887	—	2	1	no	—
34	#887	C₂²	1	1	yes	characteristic
35	C₄×D₁₂	—	2	1	no	—
36	C₄×D₁₂	C₂²	1	1	yes	characteristic
37	#872	—	2	1	no	—
38	C₄²×D₃	—	2	1	no	—
39	C₄²×D₃	C₂²	1	1	yes	characteristic
40	C₄×Dic₆	—	2	1	no	—
41	C₄×Dic₆	C₂²	1	1	yes	characteristic
42	C₃×#106	C₂²	1	1	yes	characteristic
43	C₃×#103	C₂²	1	1	yes	characteristic
44	C₃×#103	C₂²	1	1	yes	characteristic
45	C₃×#97	C₂²	1	1	yes	characteristic
46	#835	C₂²	1	1	yes	characteristic
47	#835	C₂²	1	1	yes	characteristic
48	#821	C₂²	1	1	yes	characteristic
49	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#818	C₂²	1	1	yes	characteristic
52	#816	C₂²	1	1	yes	characteristic
53	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
55	#810	C₂²	1	1	yes	characteristic
56	#809	C₂²	1	1	yes	characteristic
57	#808	C₂²	1	1	yes	characteristic
58	#807	C₂²	1	1	yes	characteristic
59	#803	C₂²	1	1	yes	characteristic
60	#529	—	3	1	no	—
61	C₂×C₄×Q₈	—	3	1	no	—
62	#484	—	3	1	no	—
63	#481	—	3	1	no	—
64	#479	—	3	1	no	—
65	#479	—	3	1	no	—
66	#464	—	3	1	no	—
67	#464	—	3	1	no	—
68	#451	—	3	1	no	—
69	#448	—	3	1	no	—
70	#441	—	3	1	no	—
71	#441	D₃	1	1	yes	characteristic, complemented
72	#440	—	3	1	no	—
73	#440	—	3	1	no	—
74	#432	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
77	D₃×Q₈	—	2	2	no	—
78	D₃×Q₈	—	2	2	no	—
79	C₂×D₁₂	—	2	1	no	—
80	C₂×D₁₂	C₂³	1	1	yes	characteristic
81	C₂×C₄×D₃	—	2	1	no	—
82	C₂×C₄×D₃	—	2	1	no	—
83	C₂×C₄×D₃	—	2	1	no	—
84	C₂×C₄×D₃	D₄	1	1	yes	characteristic
85	C₂×C₄×D₃	D₄	1	1	yes	characteristic
86	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
87	C₂×Dic₆	—	2	1	no	—
88	C₂×Dic₆	C₂³	1	1	yes	characteristic
89	C₃×Q₁₆	—	2	2	no	—
90	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
92	C₃×C₄⋊C₄	—	2	1	no	—
93	C₃×C₄⋊C₄	—	2	1	no	—
94	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
95	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	—	2	1	no	—
98	C₃×C₄²	C₂³	1	1	yes	characteristic
99	#216	—	2	2	no	—
100	#215	—	2	2	no	—
101	(C₂×D₃)⋊C₄	—	2	1	no	—
102	(C₂×D₃)⋊C₄	—	2	1	no	—
103	(C₂×D₃)⋊C₄	—	2	1	no	—
104	(C₂×D₃)⋊C₄	—	2	1	no	—
105	(C₂×D₃)⋊C₄	—	2	1	no	—
106	C₄⋊Dic₃	—	2	1	no	—
107	C₄⋊Dic₃	—	2	1	no	—
108	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
109	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
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×Dic₃	D₄	1	1	yes	characteristic
119	C₄×Dic₃	D₄	1	1	yes	characteristic
120	C₄×Dic₃	C₂³	1	1	yes	characteristic
121	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
122	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
123	#204	—	2	2	no	—
124	C₂²×Q₈	—	3	1	no	—
125	C₂×Q₁₆	—	3	1	no	—
126	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic
127	C₂×SD₁₆	—	3	1	no	—
128	C₂×SD₁₆	—	3	1	no	—
129	#126	—	6	1	no	—
130	#125	—	3	1	no	—
131	#124	—	3	1	no	—
132	#123	—	6	1	no	—
133	C₂²⋊Q₈	—	3	1	no	—
134	#121	—	3	1	no	—
135	C₄×Q₈	—	3	1	no	—
136	C₄×Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×Q₈	—	6	1	no	—
139	C₄×Q₈	—	6	1	no	—
140	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
141	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
142	C₄×D₄	—	3	1	no	—
143	C₄×D₄	—	6	1	no	—
144	#117	—	6	1	no	—
145	C₂×C₄⋊C₄	—	3	1	no	—
146	C₂×C₄⋊C₄	—	6	1	no	—
147	C₂×C₄²	—	3	1	no	—
148	C₂×C₄²	—	6	1	no	—
149	#107	—	3	1	no	—
150	#107	—	3	1	no	—
151	#106	—	3	1	no	—
152	#106	C₂×D₃	1	1	yes	characteristic
153	C₄⋊C₈	—	3	1	no	—
154	C₄⋊C₈	—	3	1	no	—
155	#103	—	3	1	no	—
156	#103	—	3	1	no	—
157	#103	—	3	1	no	—
158	#103	—	3	1	no	—
159	#103	C₂×D₃	1	1	yes	characteristic
160	#103	C₂×D₃	1	1	yes	characteristic
161	#102	—	3	1	no	—
162	#102	—	3	1	no	—
163	C₂²⋊C₈	—	3	1	no	—
164	C₂²⋊C₈	—	3	1	no	—
165	#97	—	3	1	no	—
166	#97	C₂×D₃	1	1	yes	characteristic
167	C₂²×D₃	—	2	1	no	—
168	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
169	C₃×Q₈	—	2	1	no	—
170	C₃×Q₈	—	2	1	no	—
171	C₃×Q₈	—	2	1	no	—
172	C₃×Q₈	#41	1	2	yes	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
180	C₂×Dic₃	—	2	1	no	—
181	C₂×Dic₃	—	2	1	no	—
182	C₂×Dic₃	—	2	1	no	—
183	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
184	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
185	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
186	D₁₂	—	2	1	no	—
187	D₁₂	—	2	1	no	—
188	D₁₂	—	4	2	no	—
189	C₄×D₃	—	2	4	no	—
190	C₄×D₃	—	2	2	no	—
191	C₄×D₃	—	2	1	no	—
192	C₄×D₃	—	2	1	no	—
193	C₄×D₃	—	4	1	no	—
194	C₄×D₃	—	4	2	no	—
195	C₄×D₃	—	4	2	no	—
196	Dic₆	—	2	4	no	—
197	Dic₆	—	2	1	no	—
198	Dic₆	#41	1	2	yes	—
199	C₃×C₈	—	2	1	no	—
200	C₃×C₈	—	2	1	no	—
201	C₃⋊C₈	—	2	1	no	—
202	C₃⋊C₈	—	2	1	no	—
203	C₂×Q₈	—	3	1	no	—
204	C₂×Q₈	—	6	1	no	—
205	C₂×Q₈	—	6	2	no	—
206	C₂×Q₈	—	6	2	no	—
207	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
208	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
209	C₂×D₄	—	3	1	no	—
210	C₂×D₄	—	6	1	no	—
211	C₂²×C₄	—	3	1	no	—
212	C₂²×C₄	—	3	1	no	—
213	C₂²×C₄	—	3	1	no	—
214	C₂²×C₄	—	6	1	no	—
215	C₂²×C₄	—	6	1	no	—
216	C₂²×C₄	—	6	1	no	—
217	Q₁₆	—	2	2	no	—
218	Q₁₆	—	6	2	no	—
219	SD₁₆	—	6	2	no	—
220	SD₁₆	—	6	2	no	—
221	C₂×C₈	—	3	1	no	—
222	C₂×C₈	—	3	1	no	—
223	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₄⋊C₄	—	3	1	no	—
229	C₄⋊C₄	—	6	1	no	—
230	C₄⋊C₄	—	6	1	no	—
231	C₄⋊C₄	—	6	1	no	—
232	C₄⋊C₄	—	6	1	no	—
233	C₄⋊C₄	—	6	1	no	—
234	C₄⋊C₄	—	6	1	no	—
235	C₄⋊C₄	—	6	1	no	—
236	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
237	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
238	C₂²⋊C₄	—	6	1	no	—
239	C₂²⋊C₄	—	6	1	no	—
240	C₂²⋊C₄	—	6	1	no	—
241	C₂²⋊C₄	—	6	1	no	—
242	C₂²⋊C₄	—	6	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	3	1	no	—
246	C₄²	—	3	1	no	—
247	C₄²	—	3	1	no	—
248	C₄²	—	6	1	no	—
249	C₄²	—	6	1	no	—
250	C₄²	—	6	1	no	—
251	C₄²	C₂²×D₃	1	1	yes	characteristic
252	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×D₃	—	2	1	no	—
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	4	1	no	—
256	C₂×D₃	—	4	1	no	—
257	C₂×D₃	—	4	1	no	—
258	C₂×D₃	#137	1	2	yes	—
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	4	1	no	—
264	C₃×C₄	—	4	1	no	—
265	C₃×C₄	C₂×#41	1	1	yes	characteristic
266	C₃×C₄	C₂×#41	1	1	yes	characteristic
267	Dic₃	—	2	1	no	—
268	Dic₃	—	2	2	no	—
269	Dic₃	—	2	1	no	—
270	Dic₃	—	2	1	no	—
271	Dic₃	—	4	1	no	—
272	Dic₃	—	4	1	no	—
273	C₂³	—	3	1	no	—
274	C₂³	—	6	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	3	2	no	—
279	Q₈	—	6	4	no	—
280	Q₈	—	6	1	no	—
281	Q₈	#235	1	2	yes	—
282	D₄	—	6	1	no	—
283	D₄	—	6	1	no	—
284	D₄	—	12	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	3	1	no	—
290	C₂×C₄	—	3	1	no	—
291	C₂×C₄	—	3	1	no	—
292	C₂×C₄	—	6	2	no	—
293	C₂×C₄	—	6	4	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	6	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	6	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	—	12	1	no	—
300	C₂×C₄	—	12	2	no	—
301	C₂×C₄	—	12	2	no	—
302	C₂×C₄	D₃×D₄	1	1	yes	characteristic
303	C₂×C₄	D₃×D₄	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	6	1	no	—
308	C₈	—	6	1	no	—
309	C₂×C₃	#525	1	1	yes	characteristic
310	C₂×C₃	C₂×#137	1	1	yes	characteristic
311	C₂×C₃	#575	1	1	yes	characteristic
312	D₃	—	2	2	no	—
313	D₃	—	8	1	no	—
314	C₂²	—	3	2	no	—
315	C₂²	—	6	1	no	—
316	C₂²	—	6	1	no	—
317	C₂²	—	12	1	no	—
318	C₂²	—	12	1	no	—
319	C₂²	—	12	1	no	—
320	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	6	1	no	—
328	C₄	—	6	1	no	—
329	C₄	—	6	1	no	—
330	C₄	—	6	2	no	—
331	C₄	—	12	1	no	—
332	C₄	—	12	1	no	—
333	C₄	C₂×#235	1	1	yes	characteristic
334	C₄	D₃×#41	1	1	yes	characteristic
335	C₃	#3117	1	1	yes	characteristic, complemented, Sylow
336	C₂	—	6	2	no	—
337	C₂	—	24	1	no	—
338	C₂	#5547	1	1	yes	characteristic
339	C₂	D₃×#137	1	1	yes	characteristic
340	C₂	#5777	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#441	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#895	C₂²
hypercenter	#441	D₃
hypocenter	C₃	#3117
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
2. C₄×D₃×Q₈	C₂	semidirect
3. C₃×#441	C₂	semidirect
4. #5186	C₂	semidirect
5. #5185	C₂	semidirect
6. #5030	C₂	semidirect
7. #5026	C₂	semidirect
9. #4857	C₂	semidirect
10. #4811	C₂	semidirect
12. #4798	C₂	semidirect
13. #4796	C₂	semidirect
14. #4710	C₂	semidirect
15. #4705	C₂	semidirect
71. #441	D₃	semidirect
335. C₃	#3117	semidirect

Maximal subgroups

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

subgroup	index
#5580	2
C₄×D₃×Q₈	2
C₃×#441	2
#5186	2
#5185	2
#5030	2
#5026	2
#4863	2
#4857	2
#4811	2
#4806	2
#4798	2
#4796	2
#4710	2
#4705	2
#3117	3