Subgroups — Q₈×#13093941 #26232840

Order 12032.

1198 conjugacy classes of subgroups.

The subgroup lattice has 264 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, complemented
2	#20244378	C₂	1	6	yes	maximal, complemented
3	#20243380	C₂	1	3	yes	maximal
4	C₄×#13093941	C₂	1	3	yes	maximal
5	C₂×D₄×Q₈×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×#13093941	C₂²	1	3	yes	—
10	C₂×#13093941	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	C₂³×Q₈×C₄₇	C₄	1	2	yes	—
15	D₄×Q₈×C₄₇	—	2	2	no	—
16	D₄×Q₈×C₄₇	—	2	6	no	—
17	C₂×C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
18	C₂×C₂²⋊Q₈×C₄₇	C₄	1	6	yes	—
19	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₄₇	C₂²	1	3	yes	—
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	#13093941	—	2	6	no	—
26	#13093941	Q₈	1	4	yes	direct product factor
27	#13093935	—	2	6	no	—
28	C₄×C₄₇⋊C₈	—	2	6	no	—
29	C₂²×Q₈×C₄₇	—	2	6	no	—
30	C₂²×Q₈×C₄₇	—	2	2	no	—
31	C₂²×Q₈×C₄₇	—	4	8	no	—
32	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×Q₈×C₄₇	C₂×C₄	1	6	yes	—
34	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂×C₄	1	6	yes	—
36	C₄⋊Q₈×C₄₇	—	2	6	no	—
37	C₄⋊Q₈×C₄₇	—	2	6	no	—
38	C₂²⋊Q₈×C₄₇	—	2	12	no	—
39	C₂²⋊Q₈×C₄₇	—	2	12	no	—
40	C₄×Q₈×C₄₇	D₄	1	2	yes	—
41	C₄×Q₈×C₄₇	D₄	1	6	yes	—
42	C₄×D₄×C₄₇	—	2	6	no	—
43	C₄×D₄×C₄₇	—	2	6	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
49	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂×Q₈×C₄₇	—	2	4	no	—
51	C₂×Q₈×C₄₇	—	2	6	no	—
52	C₂×Q₈×C₄₇	—	2	6	no	—
53	C₂×Q₈×C₄₇	—	4	12	no	—
54	C₂×Q₈×C₄₇	—	4	24	no	—
55	C₂×Q₈×C₄₇	—	4	8	no	—
56	C₂×Q₈×C₄₇	C₂²⋊C₄	1	3	yes	—
57	C₂×Q₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×D₄×C₄₇	—	2	2	no	—
59	C₂×D₄×C₄₇	—	2	2	no	—
60	C₂×D₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂×D₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	—	2	6	no	—
63	C₂²×C₄×C₄₇	—	2	6	no	—
64	C₂²×C₄×C₄₇	—	4	12	no	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₄₇	—	2	12	no	—
70	C₄⋊C₄×C₄₇	—	2	6	no	—
71	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
72	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
73	C₂²⋊C₄×C₄₇	—	2	12	no	—
74	C₄²×C₄₇	C₂×D₄	1	6	yes	—
75	C₄²×C₄₇	C₂×D₄	1	6	yes	—
76	#10625329	—	2	2	no	—
77	#10625329	—	2	6	no	—
78	#10625329	—	2	6	no	—
79	#10625329	C₂×Q₈	1	4	yes	complemented
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	6	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	4	2	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
87	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₄₇	—	2	6	no	—
89	Q₈×C₄₇	—	4	12	no	—
90	Q₈×C₄₇	—	4	12	no	—
91	Q₈×C₄₇	#100	1	4	yes	complemented
92	D₄×C₄₇	—	2	4	no	—
93	D₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	6	no	—
95	C₂×C₄×C₄₇	—	2	6	no	—
96	C₂×C₄×C₄₇	—	4	3	no	—
97	C₂×C₄×C₄₇	—	4	12	no	—
98	C₂×C₄×C₄₇	—	4	12	no	—
99	C₂×C₄×C₄₇	—	4	6	no	—
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
107	C₄₇⋊C₈	—	2	4	no	—
108	C₄₇⋊C₈	—	4	6	no	—
109	Q₈×#100	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	1	no	—
111	C₂²×C₄₇	—	2	1	no	—
112	C₂²×C₄₇	—	2	4	no	—
113	C₂²×C₄₇	—	4	4	no	—
114	C₂²×C₄₇	—	4	4	no	—
115	C₂²×C₄₇	—	4	2	no	—
116	C₂²×C₄₇	—	4	2	no	—
117	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₂²×C₄₇	#519	1	1	yes	characteristic
120	C₄×C₄₇	—	2	3	no	—
121	C₄×C₄₇	—	2	6	no	—
122	C₄×C₄₇	—	4	6	no	—
123	C₄×C₄₇	C₂×#100	1	6	yes	—
124	C₄×C₄₇	#548	1	2	yes	—
125	C₄×C₄₇	D₄×Q₈	1	2	yes	—
126	C₂×D₄×Q₈	D₄₇	1	1	yes	characteristic
127	Q₈×M₁₆	—	47	2	no	—
128	#1997	—	47	6	no	—
129	#1800	—	47	3	no	—
130	C₄×#100	—	47	3	no	—
131	C₂×C₄₇	—	2	1	no	—
132	C₂×C₄₇	—	2	1	no	—
133	C₂×C₄₇	—	4	4	no	—
134	C₂×C₄₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄₇	C₂²×#100	1	1	yes	characteristic
136	C₂×C₄₇	#2889	1	1	yes	characteristic
137	C₂³×Q₈	Dic₄₇	1	2	yes	—
138	D₄×Q₈	—	2	2	no	—
139	D₄×Q₈	—	2	6	no	—
140	C₂×C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
141	C₂×C₂²⋊Q₈	Dic₄₇	1	6	yes	—
142	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
143	C₂×C₄×D₄	C₂×D₄₇	1	3	yes	—
144	#446	—	94	6	no	—
145	C₈×Q₈	—	94	2	no	—
146	#423	—	47	6	no	—
147	C₂×#100	—	47	1	no	—
148	C₂×#100	—	47	3	no	—
149	C₄×M₁₆	—	47	6	no	—
150	#343	—	47	6	no	—
151	#343	—	47	6	no	—
152	C₄₇	Q₈×#100	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	2	6	no	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂²×Q₈	—	4	8	no	—
156	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
157	C₂²×Q₈	C₂×Dic₄₇	1	6	yes	—
158	C₂²×D₄	C₂²×D₄₇	1	1	yes	characteristic
159	C₂³×C₄	C₂×Dic₄₇	1	6	yes	—
160	C₂×M₁₆	—	47	2	no	—
161	C₂×M₁₆	—	47	6	no	—
162	C₄⋊Q₈	—	2	6	no	—
163	C₄⋊Q₈	—	2	6	no	—
164	C₂²⋊Q₈	—	2	12	no	—
165	C₂²⋊Q₈	—	2	12	no	—
166	C₄×Q₈	C₂²⋊D₄₇	1	2	yes	—
167	C₄×Q₈	C₂²⋊D₄₇	1	6	yes	—
168	C₄×D₄	—	2	6	no	—
169	C₄×D₄	—	2	6	no	—
170	C₂×C₄⋊C₄	C₂×Dic₄₇	1	6	yes	—
171	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
172	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
173	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	6	yes	—
174	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
175	C₄⋊C₈	—	94	6	no	—
176	C₄⋊C₈	—	94	6	no	—
177	#100	—	47	4	no	—
178	#100	—	94	6	no	—
179	#97	—	94	6	no	—
180	C₄×C₈	—	94	6	no	—
181	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	6	no	—
184	C₂×Q₈	—	2	6	no	—
185	C₂×Q₈	—	4	12	no	—
186	C₂×Q₈	—	4	24	no	—
187	C₂×Q₈	—	4	8	no	—
188	C₂×Q₈	C₂²⋊Dic₄₇	1	3	yes	—
189	C₂×Q₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	Q₈×D₄₇	1	2	yes	—
193	C₂×D₄	#10625354	1	2	yes	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	—	2	6	no	—
196	C₂²×C₄	—	4	12	no	—
197	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
200	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
201	M₁₆	—	47	4	no	—
202	M₁₆	—	94	2	no	—
203	M₁₆	—	94	6	no	—
204	M₁₆	—	94	6	no	—
205	C₂×C₈	—	94	2	no	—
206	C₂×C₈	—	94	6	no	—
207	C₄⋊C₄	—	2	12	no	—
208	C₄⋊C₄	—	2	6	no	—
209	C₄⋊C₄	C₂²⋊Dic₄₇	1	6	yes	—
210	C₄⋊C₄	C₂²⋊Dic₄₇	1	6	yes	—
211	C₂²⋊C₄	—	2	12	no	—
212	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
213	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	Q₈×Dic₄₇	1	4	yes	—
219	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
220	Q₈	—	2	6	no	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	12	no	—
223	Q₈	#13093941	1	4	yes	direct product factor
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	C₂×C₄	—	2	6	no	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	4	12	no	—
230	C₂×C₄	—	4	12	no	—
231	C₂×C₄	—	4	6	no	—
232	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
233	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
234	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
236	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
237	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
238	C₂×C₄	#13094072	1	2	yes	—
239	C₈	—	94	4	no	—
240	C₈	—	188	6	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
249	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
250	C₂²	#15689601	1	1	yes	characteristic
251	C₄	—	2	3	no	—
252	C₄	—	2	6	no	—
253	C₄	—	4	6	no	—
254	C₄	C₂×#13093941	1	6	yes	—
255	C₄	#15689784	1	2	yes	—
256	C₄	Q₈×C₂²⋊D₄₇	1	2	yes	—
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	4	4	no	—
260	C₂	Q₈×C₂²⋊Dic₄₇	1	1	yes	characteristic
261	C₂	C₂²×#13093941	1	1	yes	characteristic
262	C₂	#20251871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×D₄×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₇	C₂³
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×D₄×Q₈	D₄₇
hypocenter	C₄₇	Q₈×#100
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#10625329	C₂	semidirect
2. #20244378	C₂	semidirect
26. #13093941	Q₈	direct (N × H)
79. #10625329	C₂×Q₈	semidirect
91. Q₈×C₄₇	#100	semidirect
152. C₄₇	Q₈×#100	semidirect
223. Q₈	#13093941	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
#20243380	2
C₄×#13093941	2
C₂×D₄×Q₈×C₄₇	2
Q₈×#100	47

Subgroups — Q8×#13093941 #26232840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13093941 #26232840