Subgroups — C₂²×D₃²×Q₈ #11919178

Order 1152.

12953 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₃×Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×D₃²×Q₈	C₂	1	48	yes	maximal, direct product factor
4	C₂²×C₄×D₃²	C₂	1	3	yes	maximal
5	C₂²×#3797	C₂	1	3	yes	maximal, complemented
6	C₂²×D₃×Dic₆	C₂	1	6	yes	maximal, complemented
7	C₂³×D₃×Q₈	—	3	2	no	maximal
8	C₂³×D₃²	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
10	C₂×Q₈×C₃⋊D₃	C₂²	1	24	yes	complemented
11	C₂²×C₄×C₃⋊D₃	C₂²	1	3	yes	—
12	C₂²×#634	C₂²	1	3	yes	complemented
13	C₂×C₃×D₃×Q₈	C₂²	1	48	yes	complemented
14	C₂²×C₃×C₄×D₃	C₂²	1	6	yes	—
15	C₂²×C₃×Dic₆	C₂²	1	6	yes	complemented
16	C₂²×#627	C₂²	1	6	yes	complemented
17	C₂²×#624	C₂²	1	3	yes	—
18	C₂²×D₃×Dic₃	C₂²	1	6	yes	—
19	D₃²×Q₈	C₂²	1	256	yes	direct product factor
20	C₂×C₄×D₃²	C₂²	1	72	yes	—
21	C₂×#3797	C₂²	1	72	yes	complemented
22	C₂×D₃×Dic₆	C₂²	1	144	yes	complemented
23	C₂³×C₃×Q₈	—	3	2	no	—
24	C₂²×D₃×Q₈	—	3	2	no	—
25	C₂²×D₃×Q₈	—	3	96	no	—
26	C₂²×D₃×Q₈	—	3	6	no	—
27	C₂²×D₃×Q₈	—	3	6	no	—
28	C₂²×D₃×Q₈	—	6	1	no	—
29	C₂²×D₃×Q₈	D₃	1	2	yes	direct product factor
30	C₂³×C₄×D₃	—	3	6	no	—
31	C₂³×Dic₆	—	3	6	no	—
32	C₂³×C₃⋊D₃	C₂³	1	1	yes	characteristic
33	C₂³×C₃×D₃	C₂³	1	2	yes	—
34	C₂²×D₃²	Q₈	1	16	yes	direct product factor
35	C₂²×D₃²	C₂³	1	12	yes	—
36	C₂×C₃²×Q₈	C₂³	1	12	yes	complemented
37	C₂²×C₃²×C₄	C₂³	1	3	yes	—
38	C₂²×C₃⋊Dic₃	C₂³	1	3	yes	—
39	Q₈×C₃⋊D₃	C₂³	1	64	yes	complemented
40	C₂×C₄×C₃⋊D₃	C₂³	1	36	yes	—
41	C₂×#634	C₂³	1	36	yes	complemented
42	C₂²×C₃×Dic₃	C₂³	1	6	yes	—
43	C₃×D₃×Q₈	C₂³	1	128	yes	complemented
44	C₂×C₃×C₄×D₃	C₂³	1	72	yes	—
45	C₂×C₃×Dic₆	C₂³	1	72	yes	complemented
46	C₂×#627	C₂³	1	72	yes	complemented
47	C₂×#624	C₂³	1	36	yes	—
48	C₂×D₃×Dic₃	C₂³	1	72	yes	—
49	C₄×D₃²	C₂³	1	192	yes	—
50	#3797	C₂³	1	192	yes	complemented
51	D₃×Dic₆	C₂³	1	384	yes	complemented
52	C₂⁴×Q₈	—	9	1	no	Sylow
53	C₂⁴×D₃	—	3	2	no	—
54	C₂²×C₃×Q₈	—	2	1	no	—
55	C₂²×C₃×Q₈	—	3	48	no	—
56	C₂²×C₃×Q₈	—	3	6	no	—
57	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	complemented
58	C₂³×C₃×C₄	—	3	6	no	—
59	C₂³×Dic₃	—	3	6	no	—
60	C₂×D₃×Q₈	—	3	48	no	—
61	C₂×D₃×Q₈	—	3	512	no	—
62	C₂×D₃×Q₈	—	3	144	no	—
63	C₂×D₃×Q₈	—	3	144	no	—
64	C₂×D₃×Q₈	—	6	24	no	—
65	C₂×D₃×Q₈	C₂×D₃	1	48	yes	direct product factor
66	C₂²×C₄×D₃	—	3	6	no	—
67	C₂²×C₄×D₃	—	3	6	no	—
68	C₂²×C₄×D₃	—	3	6	no	—
69	C₂²×C₄×D₃	—	3	144	no	—
70	C₂²×C₄×D₃	—	6	3	no	—
71	C₂²×C₄×D₃	C₂×D₃	1	6	yes	—
72	C₂²×Dic₆	—	3	6	no	—
73	C₂²×Dic₆	—	3	12	no	—
74	C₂²×Dic₆	—	3	144	no	—
75	C₂²×Dic₆	—	6	3	no	—
76	C₂²×Dic₆	C₂×D₃	1	6	yes	complemented
77	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
78	C₂²×C₃⋊D₃	C₂×Q₈	1	8	yes	complemented
79	C₂²×C₃⋊D₃	C₂⁴	1	6	yes	—
80	C₂²×C₃×D₃	C₂×Q₈	1	16	yes	complemented
81	C₂²×C₃×D₃	C₂⁴	1	12	yes	—
82	C₂×D₃²	C₂×Q₈	1	96	yes	direct product factor
83	C₂×D₃²	C₂⁴	1	16	yes	—
84	C₃²×Q₈	C₂⁴	1	16	yes	complemented
85	C₂×C₃²×C₄	C₂⁴	1	18	yes	—
86	C₂×C₃⋊Dic₃	C₂⁴	1	18	yes	—
87	C₄×C₃⋊D₃	C₂⁴	1	48	yes	—
88	#634	C₂⁴	1	48	yes	complemented
89	C₂×C₃×Dic₃	C₂⁴	1	36	yes	—
90	C₃×C₄×D₃	C₂⁴	1	96	yes	—
91	C₃×Dic₆	C₂⁴	1	96	yes	complemented
92	#627	C₂⁴	1	96	yes	complemented
93	#624	C₂⁴	1	48	yes	—
94	D₃×Dic₃	C₂⁴	1	96	yes	—
95	C₂³×Q₈	—	3	2	no	—
96	C₂³×Q₈	—	9	6	no	—
97	C₂³×Q₈	—	9	48	no	—
98	C₂³×Q₈	—	9	3	no	—
99	C₂³×Q₈	—	9	1	no	—
100	C₂⁴×C₄	—	9	3	no	—
101	C₂⁴×C₃	—	3	2	no	—
102	C₂³×D₃	—	3	2	no	—
103	C₂³×D₃	—	3	32	no	—
104	C₂³×D₃	—	3	24	no	—
105	C₂³×D₃	—	6	1	no	—
106	C₂³×D₃	C₂²×D₃	1	2	yes	—
107	C₂×C₃×Q₈	—	2	12	no	—
108	C₂×C₃×Q₈	—	3	128	no	—
109	C₂×C₃×Q₈	—	3	72	no	—
110	C₂×C₃×Q₈	C₂²×D₃	1	24	yes	complemented
111	C₂²×C₃×C₄	—	2	3	no	—
112	C₂²×C₃×C₄	—	3	6	no	—
113	C₂²×C₃×C₄	—	3	72	no	—
114	C₂²×C₃×C₄	C₂²×D₃	1	6	yes	—
115	C₂²×Dic₃	—	3	6	no	—
116	C₂²×Dic₃	—	3	72	no	—
117	C₂²×Dic₃	—	6	3	no	—
118	C₂²×Dic₃	C₂²×D₃	1	6	yes	—
119	D₃×Q₈	—	3	128	no	—
120	D₃×Q₈	—	3	384	no	—
121	D₃×Q₈	—	3	384	no	—
122	D₃×Q₈	—	6	64	no	—
123	D₃×Q₈	C₂²×D₃	1	128	yes	direct product factor
124	C₂×C₄×D₃	—	3	72	no	—
125	C₂×C₄×D₃	—	3	72	no	—
126	C₂×C₄×D₃	—	3	72	no	—
127	C₂×C₄×D₃	—	3	384	no	—
128	C₂×C₄×D₃	—	6	36	no	—
129	C₂×C₄×D₃	C₂²×D₃	1	72	yes	—
130	C₂×Dic₆	—	3	72	no	—
131	C₂×Dic₆	—	3	144	no	—
132	C₂×Dic₆	—	3	384	no	—
133	C₂×Dic₆	—	6	36	no	—
134	C₂×Dic₆	C₂²×D₃	1	72	yes	complemented
135	C₂²×C₃²	C₂²×Q₈	1	4	yes	complemented
136	C₂²×C₃²	C₂⁵	1	3	yes	—
137	C₂×C₃⋊D₃	C₂²×Q₈	1	24	yes	complemented
138	C₂×C₃⋊D₃	C₂⁵	1	4	yes	—
139	C₂×C₃×D₃	C₂²×Q₈	1	48	yes	complemented
140	C₂×C₃×D₃	C₂⁵	1	8	yes	—
141	D₃²	C₂²×Q₈	1	64	yes	direct product factor
142	C₃²×C₄	C₂⁵	1	12	yes	—
143	C₃⋊Dic₃	C₂⁵	1	12	yes	—
144	C₃×Dic₃	C₂⁵	1	24	yes	—
145	C₂⁵	—	9	1	no	—
146	C₂²×Q₈	—	3	48	no	—
147	C₂²×Q₈	—	3	6	no	—
148	C₂²×Q₈	—	9	6	no	—
149	C₂²×Q₈	—	9	144	no	—
150	C₂²×Q₈	—	9	256	no	—
151	C₂²×Q₈	—	9	72	no	—
152	C₂²×Q₈	—	9	24	no	—
153	C₂²×Q₈	—	9	3	no	—
154	C₂²×Q₈	D₃²	1	1	yes	characteristic, direct product factor
155	C₂³×C₄	—	3	6	no	—
156	C₂³×C₄	—	9	3	no	—
157	C₂³×C₄	—	9	72	no	—
158	C₂³×C₄	—	9	6	no	—
159	C₂³×C₄	—	9	3	no	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	—	3	16	no	—
162	C₂³×C₃	—	3	12	no	—
163	C₂³×C₃	C₂³×D₃	1	2	yes	—
164	C₂²×D₃	—	3	16	no	—
165	C₂²×D₃	—	3	12	no	—
166	C₂²×D₃	—	3	192	no	—
167	C₂²×D₃	—	3	32	no	—
168	C₂²×D₃	—	6	8	no	—
169	C₂²×D₃	—	6	6	no	—
170	C₂²×D₃	D₃×Q₈	1	16	yes	direct product factor
171	C₂²×D₃	C₂³×D₃	1	12	yes	—
172	C₃×Q₈	—	2	16	no	—
173	C₃×Q₈	—	3	96	no	—
174	C₃×Q₈	C₂³×D₃	1	32	yes	complemented
175	C₂×C₃×C₄	—	2	18	no	—
176	C₂×C₃×C₄	—	3	36	no	—
177	C₂×C₃×C₄	—	3	96	no	—
178	C₂×C₃×C₄	C₂³×D₃	1	36	yes	—
179	C₂×Dic₃	—	3	36	no	—
180	C₂×Dic₃	—	3	96	no	—
181	C₂×Dic₃	—	6	18	no	—
182	C₂×Dic₃	C₂³×D₃	1	36	yes	—
183	C₄×D₃	—	3	96	no	—
184	C₄×D₃	—	3	96	no	—
185	C₄×D₃	—	3	96	no	—
186	C₄×D₃	—	6	48	no	—
187	C₄×D₃	C₂³×D₃	1	96	yes	—
188	Dic₆	—	3	96	no	—
189	Dic₆	—	3	192	no	—
190	Dic₆	—	6	48	no	—
191	Dic₆	C₂³×D₃	1	96	yes	complemented
192	C₂×C₃²	C₂³×Q₈	1	6	yes	complemented
193	C₂×C₃²	C₂⁶	1	1	yes	characteristic
194	C₃⋊D₃	C₂³×Q₈	1	8	yes	complemented
195	C₃×D₃	C₂³×Q₈	1	16	yes	complemented
196	C₂⁴	—	3	2	no	—
197	C₂⁴	—	9	12	no	—
198	C₂⁴	—	9	16	no	—
199	C₂⁴	—	9	1	no	—
200	C₂×Q₈	—	3	128	no	—
201	C₂×Q₈	—	3	72	no	—
202	C₂×Q₈	—	9	72	no	—
203	C₂×Q₈	—	9	384	no	—
204	C₂×Q₈	—	9	192	no	—
205	C₂×Q₈	—	9	64	no	—
206	C₂×Q₈	—	9	36	no	—
207	C₂×Q₈	C₂×D₃²	1	12	yes	direct product factor
208	C₂²×C₄	—	3	6	no	—
209	C₂²×C₄	—	3	72	no	—
210	C₂²×C₄	—	9	36	no	—
211	C₂²×C₄	—	9	192	no	—
212	C₂²×C₄	—	9	72	no	—
213	C₂²×C₄	—	9	3	no	—
214	C₂²×C₄	—	9	36	no	—
215	C₂²×C₄	C₂×D₃²	1	3	yes	—
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	—	2	3	no	—
218	C₂²×C₃	—	3	48	no	—
219	C₂²×C₃	—	3	8	no	—
220	C₂²×C₃	C₂×D₃×Q₈	1	8	yes	complemented
221	C₂²×C₃	C₂⁴×D₃	1	6	yes	—
222	C₂×D₃	—	3	48	no	—
223	C₂×D₃	—	3	8	no	—
224	C₂×D₃	—	3	128	no	—
225	C₂×D₃	—	6	4	no	—
226	C₂×D₃	—	6	24	no	—
227	C₂×D₃	C₂×D₃×Q₈	1	48	yes	direct product factor
228	C₂×D₃	C₂⁴×D₃	1	8	yes	—
229	C₃×C₄	—	2	12	no	—
230	C₃×C₄	—	3	24	no	—
231	C₃×C₄	C₂⁴×D₃	1	24	yes	—
232	Dic₃	—	3	24	no	—
233	Dic₃	—	6	12	no	—
234	Dic₃	C₂⁴×D₃	1	24	yes	—
235	C₃²	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
236	C₂³	—	3	16	no	—
237	C₂³	—	3	12	no	—
238	C₂³	—	9	16	no	—
239	C₂³	—	9	96	no	—
240	C₂³	—	9	8	no	—
241	C₂³	—	9	6	no	—
242	C₂³	C₂²×D₃²	1	1	yes	characteristic
243	Q₈	—	3	96	no	—
244	Q₈	—	9	96	no	—
245	Q₈	—	9	48	no	—
246	Q₈	C₂²×D₃²	1	16	yes	direct product factor
247	C₂×C₄	—	3	36	no	—
248	C₂×C₄	—	3	96	no	—
249	C₂×C₄	—	9	48	no	—
250	C₂×C₄	—	9	96	no	—
251	C₂×C₄	—	9	18	no	—
252	C₂×C₄	—	9	48	no	—
253	C₂×C₄	C₂²×D₃²	1	18	yes	—
254	C₂×C₃	—	2	6	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	3	16	no	—
257	C₂×C₃	C₂²×D₃×Q₈	1	12	yes	complemented
258	C₂×C₃	C₂⁵×D₃	1	2	yes	—
259	D₃	—	3	16	no	—
260	D₃	—	6	8	no	—
261	D₃	C₂²×D₃×Q₈	1	16	yes	direct product factor
262	C₂²	—	3	48	no	—
263	C₂²	—	3	8	no	—
264	C₂²	—	9	64	no	—
265	C₂²	—	9	24	no	—
266	C₂²	—	9	4	no	—
267	C₂²	D₃²×Q₈	1	4	yes	direct product factor
268	C₂²	C₂³×D₃²	1	3	yes	—
269	C₄	—	3	24	no	—
270	C₄	—	9	12	no	—
271	C₄	C₂³×D₃²	1	12	yes	—
272	C₃	—	2	1	no	—
273	C₃	C₂³×D₃×Q₈	1	2	yes	complemented
274	C₂	—	3	16	no	—
275	C₂	—	9	8	no	—
276	C₂	C₂×D₃²×Q₈	1	6	yes	direct product factor
277	C₂	C₂⁴×D₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃²
commutator, cosocle	C₂×C₃²	C₂⁶
Frattini	C₂	C₂⁴×D₃²
Fitting	C₂²×C₃²×Q₈	C₂²
socle	C₂³×C₃²	C₂⁴
prime radical	C₂³×D₃²	C₂²
hypercenter	C₂²×Q₈	D₃²
hypocenter	C₃²	C₂⁴×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₃⋊D₃	C₂	semidirect
2. C₂²×C₃×D₃×Q₈	C₂	semidirect
3. C₂×D₃²×Q₈	C₂	direct (N × H)
5. C₂²×#3797	C₂	semidirect
6. C₂²×D₃×Dic₆	C₂	semidirect
9. C₂²×C₃²×Q₈	C₂²	semidirect
10. C₂×Q₈×C₃⋊D₃	C₂²	semidirect
12. C₂²×#634	C₂²	semidirect
13. C₂×C₃×D₃×Q₈	C₂²	semidirect
15. C₂²×C₃×Dic₆	C₂²	semidirect
16. C₂²×#627	C₂²	semidirect
19. D₃²×Q₈	C₂²	direct (N × H)
21. C₂×#3797	C₂²	semidirect
22. C₂×D₃×Dic₆	C₂²	semidirect
29. C₂²×D₃×Q₈	D₃	direct (N × H)
34. C₂²×D₃²	Q₈	direct (N × H)
36. C₂×C₃²×Q₈	C₂³	semidirect
39. Q₈×C₃⋊D₃	C₂³	semidirect
41. C₂×#634	C₂³	semidirect
43. C₃×D₃×Q₈	C₂³	semidirect
45. C₂×C₃×Dic₆	C₂³	semidirect
46. C₂×#627	C₂³	semidirect
50. #3797	C₂³	semidirect
51. D₃×Dic₆	C₂³	semidirect
57. C₂²×C₃×Q₈	C₂×D₃	semidirect
65. C₂×D₃×Q₈	C₂×D₃	direct (N × H)
76. C₂²×Dic₆	C₂×D₃	semidirect
78. C₂²×C₃⋊D₃	C₂×Q₈	semidirect
80. C₂²×C₃×D₃	C₂×Q₈	semidirect
82. C₂×D₃²	C₂×Q₈	direct (N × H)
84. C₃²×Q₈	C₂⁴	semidirect
88. #634	C₂⁴	semidirect
91. C₃×Dic₆	C₂⁴	semidirect
92. #627	C₂⁴	semidirect
110. C₂×C₃×Q₈	C₂²×D₃	semidirect
123. D₃×Q₈	C₂²×D₃	direct (N × H)
134. C₂×Dic₆	C₂²×D₃	semidirect
135. C₂²×C₃²	C₂²×Q₈	semidirect
137. C₂×C₃⋊D₃	C₂²×Q₈	semidirect
139. C₂×C₃×D₃	C₂²×Q₈	semidirect
141. D₃²	C₂²×Q₈	direct (N × H)
154. C₂²×Q₈	D₃²	direct (N × H)
170. C₂²×D₃	D₃×Q₈	direct (N × H)
174. C₃×Q₈	C₂³×D₃	semidirect
191. Dic₆	C₂³×D₃	semidirect
192. C₂×C₃²	C₂³×Q₈	semidirect
194. C₃⋊D₃	C₂³×Q₈	semidirect
195. C₃×D₃	C₂³×Q₈	semidirect
207. C₂×Q₈	C₂×D₃²	direct (N × H)
220. C₂²×C₃	C₂×D₃×Q₈	semidirect
227. C₂×D₃	C₂×D₃×Q₈	direct (N × H)
235. C₃²	C₂⁴×Q₈	semidirect
246. Q₈	C₂²×D₃²	direct (N × H)
257. C₂×C₃	C₂²×D₃×Q₈	semidirect
261. D₃	C₂²×D₃×Q₈	direct (N × H)
267. C₂²	D₃²×Q₈	direct (N × H)
273. C₃	C₂³×D₃×Q₈	semidirect
276. C₂	C₂×D₃²×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
C₂²×Q₈×C₃⋊D₃	2
C₂²×C₃×D₃×Q₈	2
C₂×D₃²×Q₈	2
C₂²×C₄×D₃²	2
C₂²×#3797	2
C₂²×D₃×Dic₆	2
C₂³×D₃×Q₈	3

Subgroups — C22×D32×Q8 #11919178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₃²×Q₈ #11919178