Subgroups — Q₈×#14523962 #30872406

Order 17152.

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×#11761612	C₂	1	2	yes	maximal
2	#22561199	C₂	1	6	yes	maximal
3	#22560205	C₂	1	3	yes	maximal
4	C₄×#14523962	C₂	1	3	yes	maximal
5	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863527	—	2	6	no	—
7	Q₈×C₆₇⋊C₈	—	2	2	no	—
8	#16863420	C₂²	1	6	yes	—
9	C₂×#14523962	C₂²	1	3	yes	—
10	C₂×#14523962	C₂²	1	1	yes	characteristic
11	C₄×#11761612	C₂²	1	6	yes	—
12	#16863165	C₂²	1	6	yes	—
13	#16863165	C₂²	1	6	yes	—
14	Q₈²×C₆₇	—	2	2	no	—
15	Q₈²×C₆₇	—	2	6	no	—
16	C₂×C₄⋊Q₈×C₆₇	C₄	1	6	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#11761612	C₂³	1	6	yes	—
22	C₂×#11761612	C₂³	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
25	#14523962	—	2	6	no	—
26	#14523962	Q₈	1	4	yes	direct product factor
27	#14523954	—	2	6	no	—
28	C₄×C₆₇⋊C₈	—	2	6	no	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₆₇	C₂×C₄	1	6	yes	—
31	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₆₇	—	2	12	no	—
33	C₄⋊Q₈×C₆₇	—	2	12	no	—
34	C₄⋊Q₈×C₆₇	—	2	6	no	—
35	C₄⋊Q₈×C₆₇	—	2	6	no	—
36	C₄×Q₈×C₆₇	—	2	6	no	—
37	C₄×Q₈×C₆₇	—	2	2	no	—
38	C₄×Q₈×C₆₇	—	2	6	no	—
39	C₄×Q₈×C₆₇	—	2	6	no	—
40	C₄×Q₈×C₆₇	D₄	1	2	yes	—
41	C₄×Q₈×C₆₇	D₄	1	6	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
48	#11761612	—	2	2	no	—
49	#11761612	—	2	6	no	—
50	#11761612	—	2	6	no	—
51	#11761612	C₂×Q₈	1	4	yes	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	6	no	—
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂×Q₈×C₆₇	—	2	12	no	—
56	C₂×Q₈×C₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	—	2	2	no	—
58	C₂×Q₈×C₆₇	—	8	6	no	—
59	C₂×Q₈×C₆₇	C₂²⋊C₄	1	3	yes	—
60	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
62	C₂×Q₈×C₆₇	#41	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₆₇	—	2	12	no	—
69	C₄⋊C₄×C₆₇	—	2	6	no	—
70	C₄⋊C₄×C₆₇	—	2	6	no	—
71	C₄⋊C₄×C₆₇	—	2	6	no	—
72	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
73	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
74	C₄²×C₆₇	—	2	6	no	—
75	C₄²×C₆₇	—	2	6	no	—
76	C₄²×C₆₇	C₂×D₄	1	6	yes	—
77	C₄²×C₆₇	C₂×D₄	1	6	yes	—
78	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₆₇	—	2	4	no	—
80	Q₈×C₆₇	—	2	4	no	—
81	Q₈×C₆₇	—	2	6	no	—
82	Q₈×C₆₇	—	8	12	no	—
83	Q₈×C₆₇	#101	1	4	yes	complemented
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	6	no	—
87	C₂×C₄×C₆₇	—	2	6	no	—
88	C₂×C₄×C₆₇	—	4	3	no	—
89	C₂×C₄×C₆₇	—	4	6	no	—
90	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
92	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
93	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
94	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
98	C₆₇⋊C₈	—	2	4	no	—
99	C₆₇⋊C₈	—	4	6	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	—	2	1	no	—
102	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
104	C₂²×C₆₇	#519	1	1	yes	characteristic
105	C₄×C₆₇	—	2	4	no	—
106	C₄×C₆₇	—	2	3	no	—
107	C₄×C₆₇	—	2	6	no	—
108	C₄×C₆₇	—	4	6	no	—
109	C₄×C₆₇	C₂×#101	1	6	yes	—
110	C₄×C₆₇	#548	1	2	yes	—
111	C₄×C₆₇	D₄×Q₈	1	2	yes	—
112	Q₈×#101	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	—	2	1	no	—
114	C₂×C₆₇	—	2	1	no	—
115	C₂×C₆₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₆₇	C₂²×#101	1	1	yes	characteristic
117	C₂×C₆₇	#2888	1	1	yes	characteristic
118	C₂×Q₈²	D₆₇	1	1	yes	characteristic
119	Q₈×M₁₆	—	67	2	no	—
120	#1997	—	67	6	no	—
121	#1801	—	67	3	no	—
122	C₄×#101	—	67	3	no	—
123	C₆₇	Q₈×#101	1	1	yes	characteristic, complemented, Sylow
124	Q₈²	—	2	2	no	—
125	Q₈²	—	2	6	no	—
126	C₂×C₄⋊Q₈	Dic₆₇	1	6	yes	—
127	C₂×C₄⋊Q₈	C₂×D₆₇	1	3	yes	—
128	C₂×C₄×Q₈	Dic₆₇	1	2	yes	—
129	C₂×C₄×Q₈	C₂×D₆₇	1	3	yes	—
130	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
131	#446	—	134	6	no	—
132	C₈×Q₈	—	134	2	no	—
133	#423	—	67	6	no	—
134	C₂×#101	—	67	1	no	—
135	C₂×#101	—	67	3	no	—
136	C₄×M₁₆	—	67	6	no	—
137	#343	—	67	6	no	—
138	#343	—	67	6	no	—
139	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
140	C₂²×Q₈	C₂×Dic₆₇	1	6	yes	—
141	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
142	C₂×M₁₆	—	67	2	no	—
143	C₂×M₁₆	—	67	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⋊D₆₇	1	2	yes	—
153	C₄×Q₈	C₂²⋊D₆₇	1	6	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
155	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
156	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
157	C₂×C₄⋊C₄	C₂²×D₆₇	1	3	yes	—
158	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
159	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
160	C₄⋊C₈	—	134	6	no	—
161	C₄⋊C₈	—	134	6	no	—
162	#101	—	67	4	no	—
163	#101	—	134	6	no	—
164	#97	—	134	6	no	—
165	C₄×C₈	—	134	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⋊Dic₆₇	1	3	yes	—
172	C₂×Q₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
173	C₂×Q₈	Q₈×D₆₇	1	2	yes	—
174	C₂×Q₈	#11761629	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
176	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
178	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
179	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
180	M₁₆	—	67	4	no	—
181	M₁₆	—	134	6	no	—
182	M₁₆	—	134	6	no	—
183	M₁₆	—	134	2	no	—
184	C₂×C₈	—	134	6	no	—
185	C₂×C₈	—	134	2	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⋊Dic₆₇	1	6	yes	—
191	C₄⋊C₄	C₂²⋊Dic₆₇	1	6	yes	—
192	C₄²	—	2	6	no	—
193	C₄²	—	2	6	no	—
194	C₄²	C₂×C₂²⋊D₆₇	1	6	yes	—
195	C₄²	C₂×C₂²⋊D₆₇	1	6	yes	—
196	C₂³	C₂³×Dic₆₇	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	#14523962	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⋊Dic₆₇	1	3	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	6	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	3	yes	—
211	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
213	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
215	C₂×C₄	#14524091	1	2	yes	—
216	C₈	—	268	6	no	—
217	C₈	—	134	4	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
221	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
222	C₂²	#16863874	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×#14523962	1	6	yes	—
228	C₄	#16864057	1	2	yes	—
229	C₄	Q₈×C₂²⋊D₆₇	1	2	yes	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	Q₈×C₂²⋊Dic₆₇	1	1	yes	characteristic
233	C₂	C₂²×#14523962	1	1	yes	characteristic
234	C₂	#22568688	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₆₇
commutator, prime radical	C₂³×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×Q₈²×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×Q₈²	D₆₇
hypocenter	C₆₇	Q₈×#101
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
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. #14523962	Q₈	direct (N × H)
83. Q₈×C₆₇	#101	semidirect
123. C₆₇	Q₈×#101	semidirect
201. Q₈	#14523962	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×#11761612	2
#22561199	2
#22560205	2
C₄×#14523962	2
C₂×Q₈²×C₆₇	2
Q₈×#101	67

Subgroups — Q8×#14523962 #30872406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14523962 #30872406