Subgroups — Q₈×#13093943 #26232853

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#10625329	C₂	1	2	yes	maximal
2	#20244378	C₂	1	6	yes	maximal
3	#20243384	C₂	1	3	yes	maximal
4	C₄×#13093943	C₂	1	3	yes	maximal
5	C₂×Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689254	—	2	6	no	—
7	Q₈×C₄₇⋊C₈	—	2	2	no	—
8	#15689147	C₂²	1	6	yes	—
9	C₂×#13093943	C₂²	1	3	yes	—
10	C₂×#13093943	C₂²	1	1	yes	characteristic
11	C₄×#10625329	C₂²	1	6	yes	—
12	#15688892	C₂²	1	6	yes	—
13	#15688892	C₂²	1	6	yes	—
14	Q₈²×C₄₇	—	2	2	no	—
15	Q₈²×C₄₇	—	2	6	no	—
16	C₂×C₄⋊Q₈×C₄₇	C₄	1	6	yes	—
17	C₂×C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
19	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#10625329	C₂³	1	6	yes	—
22	C₂×#10625329	C₂³	1	2	yes	—
23	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
25	#13093943	—	2	6	no	—
26	#13093943	Q₈	1	4	yes	direct product factor
27	#13093935	—	2	6	no	—
28	C₄×C₄₇⋊C₈	—	2	6	no	—
29	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₄₇	C₂×C₄	1	6	yes	—
31	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₄₇	—	2	12	no	—
33	C₄⋊Q₈×C₄₇	—	2	12	no	—
34	C₄⋊Q₈×C₄₇	—	2	6	no	—
35	C₄⋊Q₈×C₄₇	—	2	6	no	—
36	C₄×Q₈×C₄₇	—	2	6	no	—
37	C₄×Q₈×C₄₇	—	2	2	no	—
38	C₄×Q₈×C₄₇	—	2	6	no	—
39	C₄×Q₈×C₄₇	—	2	6	no	—
40	C₄×Q₈×C₄₇	D₄	1	2	yes	—
41	C₄×Q₈×C₄₇	D₄	1	6	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
48	C₂×Q₈×C₄₇	—	2	4	no	—
49	C₂×Q₈×C₄₇	—	2	12	no	—
50	C₂×Q₈×C₄₇	—	2	2	no	—
51	C₂×Q₈×C₄₇	—	2	2	no	—
52	C₂×Q₈×C₄₇	—	8	6	no	—
53	C₂×Q₈×C₄₇	C₂²⋊C₄	1	3	yes	—
54	C₂×Q₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂×Q₈×C₄₇	#41	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	—	2	12	no	—
63	C₄⋊C₄×C₄₇	—	2	6	no	—
64	C₄⋊C₄×C₄₇	—	2	6	no	—
65	C₄⋊C₄×C₄₇	—	2	6	no	—
66	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
67	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
68	C₄²×C₄₇	—	2	6	no	—
69	C₄²×C₄₇	—	2	6	no	—
70	C₄²×C₄₇	C₂×D₄	1	6	yes	—
71	C₄²×C₄₇	C₂×D₄	1	6	yes	—
72	#10625329	—	2	2	no	—
73	#10625329	—	2	6	no	—
74	#10625329	—	2	6	no	—
75	#10625329	C₂×Q₈	1	4	yes	—
76	C₂×C₄₇⋊C₈	—	2	2	no	—
77	C₂×C₄₇⋊C₈	—	2	6	no	—
78	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₄₇	—	2	4	no	—
80	Q₈×C₄₇	—	2	4	no	—
81	Q₈×C₄₇	—	2	6	no	—
82	Q₈×C₄₇	—	8	12	no	—
83	Q₈×C₄₇	#101	1	4	yes	complemented
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	6	no	—
87	C₂×C₄×C₄₇	—	2	6	no	—
88	C₂×C₄×C₄₇	—	4	3	no	—
89	C₂×C₄×C₄₇	—	4	6	no	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
92	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
93	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
94	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
98	C₄₇⋊C₈	—	2	4	no	—
99	C₄₇⋊C₈	—	4	6	no	—
100	Q₈×#101	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	—	2	1	no	—
102	C₂²×C₄₇	—	2	1	no	—
103	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
105	C₂²×C₄₇	#519	1	1	yes	characteristic
106	C₄×C₄₇	—	2	4	no	—
107	C₄×C₄₇	—	2	3	no	—
108	C₄×C₄₇	—	2	6	no	—
109	C₄×C₄₇	—	4	6	no	—
110	C₄×C₄₇	C₂×#101	1	6	yes	—
111	C₄×C₄₇	#548	1	2	yes	—
112	C₄×C₄₇	D₄×Q₈	1	2	yes	—
113	C₂×Q₈²	D₄₇	1	1	yes	characteristic
114	Q₈×M₁₆	—	47	2	no	—
115	#1997	—	47	6	no	—
116	#1801	—	47	3	no	—
117	C₄×#101	—	47	3	no	—
118	C₂×C₄₇	—	2	1	no	—
119	C₂×C₄₇	—	2	1	no	—
120	C₂×C₄₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄₇	C₂²×#101	1	1	yes	characteristic
122	C₂×C₄₇	#2888	1	1	yes	characteristic
123	Q₈²	—	2	2	no	—
124	Q₈²	—	2	6	no	—
125	C₂×C₄⋊Q₈	Dic₄₇	1	6	yes	—
126	C₂×C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
127	C₂×C₄×Q₈	Dic₄₇	1	2	yes	—
128	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
129	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
130	#446	—	94	6	no	—
131	C₈×Q₈	—	94	2	no	—
132	#423	—	47	6	no	—
133	C₂×#101	—	47	1	no	—
134	C₂×#101	—	47	3	no	—
135	C₄×M₁₆	—	47	6	no	—
136	#343	—	47	6	no	—
137	#343	—	47	6	no	—
138	C₄₇	Q₈×#101	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
140	C₂²×Q₈	C₂×Dic₄₇	1	6	yes	—
141	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
142	C₂×M₁₆	—	47	2	no	—
143	C₂×M₁₆	—	47	6	no	—
144	C₄⋊Q₈	—	2	12	no	—
145	C₄⋊Q₈	—	2	12	no	—
146	C₄⋊Q₈	—	2	6	no	—
147	C₄⋊Q₈	—	2	6	no	—
148	C₄×Q₈	—	2	6	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	6	no	—
151	C₄×Q₈	—	2	6	no	—
152	C₄×Q₈	C₂²⋊D₄₇	1	2	yes	—
153	C₄×Q₈	C₂²⋊D₄₇	1	6	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₄₇	1	6	yes	—
155	C₂×C₄⋊C₄	C₂×Dic₄₇	1	6	yes	—
156	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
157	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
158	C₂×C₄²	C₂×Dic₄₇	1	6	yes	—
159	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
160	C₄⋊C₈	—	94	6	no	—
161	C₄⋊C₈	—	94	6	no	—
162	#101	—	47	4	no	—
163	#101	—	94	6	no	—
164	#97	—	94	6	no	—
165	C₄×C₈	—	94	6	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	2	12	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	8	6	no	—
171	C₂×Q₈	C₂²⋊Dic₄₇	1	3	yes	—
172	C₂×Q₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
173	C₂×Q₈	Q₈×D₄₇	1	2	yes	—
174	C₂×Q₈	#10625354	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
176	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
178	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
179	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
180	M₁₆	—	47	4	no	—
181	M₁₆	—	94	2	no	—
182	M₁₆	—	94	6	no	—
183	M₁₆	—	94	6	no	—
184	C₂×C₈	—	94	2	no	—
185	C₂×C₈	—	94	6	no	—
186	C₄⋊C₄	—	2	12	no	—
187	C₄⋊C₄	—	2	6	no	—
188	C₄⋊C₄	—	2	6	no	—
189	C₄⋊C₄	—	2	6	no	—
190	C₄⋊C₄	C₂²⋊Dic₄₇	1	6	yes	—
191	C₄⋊C₄	C₂²⋊Dic₄₇	1	6	yes	—
192	C₄²	—	2	6	no	—
193	C₄²	—	2	6	no	—
194	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
195	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
196	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	2	4	no	—
199	Q₈	—	2	6	no	—
200	Q₈	—	8	12	no	—
201	Q₈	#13093943	1	4	yes	direct product factor
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	4	3	no	—
207	C₂×C₄	—	4	6	no	—
208	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
211	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
213	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
214	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
215	C₂×C₄	#13094072	1	2	yes	—
216	C₈	—	94	4	no	—
217	C₈	—	188	6	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
221	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
222	C₂²	#15689601	1	1	yes	characteristic
223	C₄	—	2	4	no	—
224	C₄	—	2	3	no	—
225	C₄	—	2	6	no	—
226	C₄	—	4	6	no	—
227	C₄	C₂×#13093943	1	6	yes	—
228	C₄	#15689784	1	2	yes	—
229	C₄	Q₈×C₂²⋊D₄₇	1	2	yes	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	Q₈×C₂²⋊Dic₄₇	1	1	yes	characteristic
233	C₂	C₂²×#13093943	1	1	yes	characteristic
234	C₂	#20251867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator, prime radical	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×Q₈²×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×Q₈²	D₄₇
hypocenter	C₄₇	Q₈×#101
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
26. #13093943	Q₈	direct (N × H)
83. Q₈×C₄₇	#101	semidirect
138. C₄₇	Q₈×#101	semidirect
201. Q₈	#13093943	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#10625329	2
#20244378	2
#20243384	2
C₄×#13093943	2
C₂×Q₈²×C₄₇	2
Q₈×#101	47

Subgroups — Q8×#13093943 #26232853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13093943 #26232853