Subgroups — C₃×C₄×C₂²⋊Q₈ #84166

Order 384.

656 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
4	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#387	C₂	1	2	yes	maximal, complemented
7	C₃×#385	C₂	1	2	yes	maximal
8	C₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×C₂²⋊C₄	C₂	1	2	yes	maximal
12	C₄×C₂²⋊Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₃×C₂²⋊Q₈	C₄	1	8	yes	direct product factor
18	C₃×C₄×Q₈	—	2	4	no	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₃×C₄²	—	2	4	no	—
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂²	1	2	yes	—
31	C₂×C₃×C₄²	C₂²	1	2	yes	—
32	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
34	C₃×#95	C₂²	1	2	yes	—
35	C₃×#95	C₂²	1	2	yes	—
36	C₃×#95	C₂²	1	2	yes	—
37	C₃×#95	C₂²	1	2	yes	—
38	C₃×#95	C₂²	1	2	yes	—
39	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
40	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
42	#391	C₂×C₃	1	1	yes	characteristic, complemented
43	#389	C₂×C₃	1	1	yes	characteristic, complemented
44	#387	C₂×C₃	1	2	yes	complemented
45	#385	C₂×C₃	1	2	yes	—
46	#380	C₂×C₃	1	1	yes	characteristic
47	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
48	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
49	C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	—
50	C₂⁴×C₃	C₂³	1	1	yes	characteristic
51	C₂×C₃×Q₈	—	2	2	no	—
52	C₂×C₃×Q₈	—	2	2	no	—
53	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	—	2	2	no	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
62	C₂²×C₃×C₄	Q₈	1	4	yes	complemented
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	C₂³	1	2	yes	—
65	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₂²×C₃×C₄	C₂³	1	2	yes	—
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₃×C₄⋊C₄	—	2	4	no	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄⋊C₄	—	2	4	no	—
76	C₃×C₄⋊C₄	—	2	2	no	—
77	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
78	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
79	C₃×C₂²⋊C₄	—	2	4	no	—
80	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	—	2	4	no	—
85	C₃×C₄²	—	2	4	no	—
86	C₃×C₄²	D₄	1	4	yes	—
87	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
88	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
89	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
91	C₂²⋊Q₈	C₃×C₄	1	8	yes	direct product factor
92	C₄×Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
95	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
100	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
101	C₂×C₄²	—	2	4	no	—
102	C₂×C₄²	C₂²×C₃	1	2	yes	—
103	C₂×C₄²	C₂²×C₃	1	2	yes	—
104	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
105	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
107	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
108	#95	C₂²×C₃	1	2	yes	—
109	#95	C₂²×C₃	1	2	yes	—
110	#95	C₂²×C₃	1	2	yes	—
111	#95	C₂²×C₃	1	2	yes	—
112	#95	C₂²×C₃	1	2	yes	—
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	—	2	1	no	—
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	—	2	1	no	—
117	C₂³×C₃	C₂²×C₄	1	2	yes	—
118	C₂³×C₃	C₂×Q₈	1	2	yes	—
119	C₂³×C₃	#41	1	2	yes	—
120	C₂³×C₃	C₂⁴	1	1	yes	characteristic
121	C₃×Q₈	—	2	8	no	—
122	C₂×C₃×C₄	—	2	8	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	8	no	—
136	C₂×C₃×C₄	—	2	8	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
144	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
145	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
146	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
147	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
148	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
149	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
150	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
151	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
152	C₂×C₃×C₄	#41	1	2	yes	—
153	C₂×C₃×C₄	#41	1	2	yes	—
154	C₂×C₃×C₄	#41	1	2	yes	—
155	C₂⁴	C₂³×C₃	1	1	yes	characteristic
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
167	C₂²×C₄	C₃×Q₈	1	4	yes	complemented
168	C₂²×C₄	C₂³×C₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃	1	2	yes	—
172	C₂²×C₄	C₂³×C₃	1	2	yes	—
173	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
183	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	4	no	—
190	C₄²	—	2	4	no	—
191	C₄²	C₃×D₄	1	4	yes	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	C₄×Q₈	1	4	yes	complemented
199	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
200	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
201	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
202	C₂²×C₃	C₂×#41	1	1	yes	characteristic
203	C₂²×C₃	C₂×#41	1	1	yes	characteristic
204	C₂²×C₃	C₂×#41	1	1	yes	characteristic
205	C₂²×C₃	C₂×#41	1	1	yes	characteristic
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	4	2	no	—
213	C₃×C₄	—	4	2	no	—
214	C₃×C₄	C₄×D₄	1	4	yes	—
215	C₃×C₄	C₂²⋊Q₈	1	4	yes	direct product factor
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	C₂²×C₃×C₄	1	2	yes	—
221	C₂³	C₂×C₃×Q₈	1	2	yes	—
222	C₂³	C₃×#41	1	2	yes	—
223	C₂³	C₂⁴×C₃	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
247	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
248	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
249	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
250	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
251	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
252	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
253	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
254	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
255	C₂×C₄	C₃×#41	1	2	yes	—
256	C₂×C₄	C₃×#41	1	2	yes	—
257	C₂×C₄	C₃×#41	1	2	yes	—
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
260	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
261	C₂×C₃	C₄×#41	1	1	yes	characteristic
262	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂×C₃	#525	1	1	yes	characteristic
264	C₂×C₃	#529	1	1	yes	characteristic
265	C₂×C₃	#533	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	C₃×C₄×Q₈	1	4	yes	complemented
273	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
274	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
275	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
276	C₂²	C₂×C₃×#41	1	1	yes	characteristic
277	C₂²	C₂×C₃×#41	1	1	yes	characteristic
278	C₂²	C₂×C₃×#41	1	1	yes	characteristic
279	C₂²	C₂×C₃×#41	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	4	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	C₃×C₄×D₄	1	4	yes	—
289	C₄	C₃×C₂²⋊Q₈	1	4	yes	direct product factor
290	C₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	—	2	4	no	—
292	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
293	C₂	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
294	C₂	C₃×C₄×#41	1	1	yes	characteristic
295	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
296	C₂	C₃×#525	1	1	yes	characteristic
297	C₂	C₃×#529	1	1	yes	characteristic
298	C₂	C₃×#533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×Q₈	C₂	semidirect
4. C₃×#391	C₂	semidirect
5. C₃×#389	C₂	semidirect
6. C₃×#387	C₂	semidirect
9. C₃×C₄×C₄⋊C₄	C₂	semidirect
10. C₃×C₄×C₄⋊C₄	C₂	semidirect
12. C₄×C₂²⋊Q₈	C₃	direct (N × H)
17. C₃×C₂²⋊Q₈	C₄	direct (N × H)
40. C₂×C₄×Q₈	C₂×C₃	semidirect
42. #391	C₂×C₃	semidirect
43. #389	C₂×C₃	semidirect
44. #387	C₂×C₃	semidirect
47. C₄×C₄⋊C₄	C₂×C₃	semidirect
48. C₄×C₄⋊C₄	C₂×C₃	semidirect
53. C₂×C₃×Q₈	C₂×C₄	semidirect
62. C₂²×C₃×C₄	Q₈	semidirect
77. C₃×C₄⋊C₄	C₂×C₄	semidirect
78. C₃×C₄⋊C₄	C₂×C₄	semidirect
91. C₂²⋊Q₈	C₃×C₄	direct (N × H)
158. C₂×Q₈	C₂×C₃×C₄	semidirect
167. C₂²×C₄	C₃×Q₈	semidirect
182. C₄⋊C₄	C₂×C₃×C₄	semidirect
183. C₄⋊C₄	C₂×C₃×C₄	semidirect
198. C₂²×C₃	C₄×Q₈	semidirect
215. C₃×C₄	C₂²⋊Q₈	direct (N × H)
272. C₂²	C₃×C₄×Q₈	semidirect
289. C₄	C₃×C₂²⋊Q₈	direct (N × H)
290. C₃	C₄×C₂²⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₂²×C₃×C₄²	2
C₃×#391	2
C₃×#389	2
C₃×#387	2
C₃×#385	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
C₄×C₂²⋊Q₈	3

Subgroups — C3×C4×C22⋊Q8 #84166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₂²⋊Q₈ #84166