Subgroups — Q₈×#14524050 #31720247

Order 17152.

1882 conjugacy classes of subgroups.

The subgroup lattice has 262 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	3	yes	maximal, complemented
2	#22569158	C₂	1	9	yes	maximal, complemented
3	#22569144	C₂	1	9	yes	maximal, complemented
4	C₂×Q₈×C₆₇⋊C₈	C₂	1	3	yes	maximal, complemented
5	#22568777	C₂	1	3	yes	maximal
6	C₄×#14524050	C₂	1	3	yes	maximal
7	Q₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524050	C₂²	1	3	yes	—
9	C₂×#14524050	C₂²	1	1	yes	characteristic
10	#16863527	C₂²	1	18	yes	—
11	#16863527	C₂²	1	18	yes	—
12	Q₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×C₆₇⋊C₈	C₂²	1	9	yes	—
14	Q₈×C₆₇⋊C₈	C₂²	1	9	yes	—
15	Q₈×C₆₇⋊C₈	C₂²	1	3	yes	—
16	Q₈×C₆₇⋊C₈	C₂²	1	6	yes	—
17	#16863423	C₂²	1	18	yes	—
18	#16863420	C₂²	1	9	yes	—
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	9	yes	—
20	#16863352	C₂²	1	9	yes	—
21	#16863352	C₂²	1	9	yes	—
22	C₄×#11761612	C₂²	1	9	yes	—
23	C₂×C₄×C₆₇⋊C₈	C₂²	1	9	yes	—
24	Q₈²×C₆₇	C₄	1	1	yes	characteristic
25	#557×C₆₇	C₄	1	3	yes	—
26	#554×C₆₇	C₄	1	9	yes	—
27	D₄×Q₈×C₆₇	C₄	1	3	yes	—
28	#533×C₆₇	C₂²	1	9	yes	—
29	C₄×#41×C₆₇	C₂²	1	3	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
31	#14524050	—	2	12	no	—
32	#14524050	Q₈	1	8	yes	direct product factor
33	C₂×#11761612	C₂³	1	9	yes	—
34	C₂×#11761612	C₂³	1	3	yes	—
35	C₂²×C₆₇⋊C₈	C₂³	1	9	yes	—
36	C₂²×C₆₇⋊C₈	C₂³	1	3	yes	—
37	C₄⋊(C₆₇⋊C₈)	C₂³	1	3	yes	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	9	yes	—
39	C₄⋊(C₆₇⋊C₈)	C₂³	1	18	yes	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	18	yes	—
41	#14523954	C₂³	1	18	yes	—
42	C₄×C₆₇⋊C₈	C₂³	1	3	yes	—
43	C₄×C₆₇⋊C₈	C₂³	1	9	yes	—
44	C₄×C₆₇⋊C₈	C₂³	1	9	yes	—
45	C₄×C₆₇⋊C₈	C₂³	1	9	yes	—
46	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₆₇	C₂×C₄	1	3	yes	—
48	C₄⋊Q₈×C₆₇	C₂×C₄	1	9	yes	—
49	C₄⋊Q₈×C₆₇	C₂×C₄	1	9	yes	—
50	#125×C₆₇	C₂×C₄	1	18	yes	—
51	C₂²⋊Q₈×C₆₇	C₂×C₄	1	18	yes	—
52	C₂²⋊Q₈×C₆₇	C₂×C₄	1	18	yes	—
53	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
54	C₄×Q₈×C₆₇	C₂×C₄	1	9	yes	—
55	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
56	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
57	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₆₇	C₂³	1	9	yes	—
59	C₄×Q₈×C₆₇	C₂³	1	6	yes	—
60	C₄×D₄×C₆₇	C₂×C₄	1	9	yes	—
61	C₄×D₄×C₆₇	C₂×C₄	1	9	yes	—
62	#117×C₆₇	C₂³	1	9	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	9	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
65	#11761612	—	2	18	no	—
66	#11761612	C₂×Q₈	1	12	yes	complemented
67	C₂×C₆₇⋊C₈	—	2	18	no	—
68	C₂×C₆₇⋊C₈	C₂×Q₈	1	12	yes	complemented
69	C₂×C₆₇⋊C₈	C₂⁴	1	3	yes	—
70	C₂×C₆₇⋊C₈	C₂⁴	1	9	yes	—
71	C₂×C₆₇⋊C₈	C₂⁴	1	1	yes	characteristic
72	C₂×C₆₇⋊C₈	C₂⁴	1	3	yes	—
73	#41×C₆₇	C₂×Q₈	1	4	yes	—
74	#41×C₆₇	#41	1	4	yes	—
75	C₂×Q₈×C₆₇	—	2	12	no	—
76	C₂×Q₈×C₆₇	C₂²×C₄	1	9	yes	—
77	C₂×Q₈×C₆₇	C₂²×C₄	1	6	yes	—
78	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×D₄×C₆₇	C₂²×C₄	1	3	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	9	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	9	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
84	C₄⋊C₄×C₆₇	C₂²×C₄	1	3	yes	—
85	C₄⋊C₄×C₆₇	C₂²×C₄	1	18	yes	—
86	C₄⋊C₄×C₆₇	C₂²×C₄	1	9	yes	—
87	C₄⋊C₄×C₆₇	C₂²×C₄	1	9	yes	—
88	C₄⋊C₄×C₆₇	C₂²×C₄	1	9	yes	—
89	C₄⋊C₄×C₆₇	C₂²×C₄	1	18	yes	—
90	C₂²⋊C₄×C₆₇	C₂²×C₄	1	18	yes	—
91	C₄²×C₆₇	C₂²×C₄	1	9	yes	—
92	C₄²×C₆₇	C₂²×C₄	1	9	yes	—
93	C₄²×C₆₇	C₂⁴	1	3	yes	—
94	C₄²×C₆₇	C₂⁴	1	9	yes	—
95	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
96	Q₈×C₆₇	—	2	18	no	—
97	Q₈×C₆₇	—	4	6	no	—
98	Q₈×C₆₇	C₄×Q₈	1	4	yes	—
99	Q₈×C₆₇	#131	1	4	yes	complemented
100	D₄×C₆₇	C₄×Q₈	1	12	yes	—
101	C₂×C₄×C₆₇	—	2	18	no	—
102	C₂×C₄×C₆₇	—	4	9	no	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
104	C₂×C₄×C₆₇	C₂³×C₄	1	9	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	9	yes	—
106	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
108	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
109	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
110	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
111	C₆₇⋊C₈	—	2	3	no	—
112	C₆₇⋊C₈	—	2	9	no	—
113	C₆₇⋊C₈	C₂²×Q₈	1	2	yes	—
114	C₆₇⋊C₈	C₂²×Q₈	1	6	yes	—
115	C₂²×C₆₇	—	2	3	no	—
116	C₂²×C₆₇	—	2	3	no	—
117	C₂²×C₆₇	C₂×C₄×Q₈	1	6	yes	—
118	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
119	C₄×C₆₇	—	2	9	no	—
120	C₄×C₆₇	—	2	3	no	—
121	C₄×C₆₇	—	2	9	no	—
122	C₄×C₆₇	C₂×C₄×Q₈	1	6	yes	—
123	C₄×C₆₇	C₂×#131	1	6	yes	—
124	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
125	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
126	Q₈×#131	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	—	2	6	no	—
128	C₂×C₆₇	C₂²×C₄×Q₈	1	1	yes	characteristic
129	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
130	C₂×C₆₇	#3573	1	1	yes	characteristic
131	Q₈×#41	D₆₇	1	1	yes	characteristic
132	Q₈×M₁₆	—	67	3	no	—
133	#2962	—	67	9	no	—
134	#2960	—	67	9	no	—
135	C₂×C₈×Q₈	—	67	3	no	—
136	#2906	—	67	3	no	—
137	C₄×#131	—	67	3	no	—
138	C₆₇	Q₈×#131	1	1	yes	characteristic, complemented, Sylow
139	C₂×#131	—	67	1	no	—
140	C₂×#131	—	67	3	no	—
141	Q₈²	Dic₆₇	1	1	yes	characteristic
142	#557	Dic₆₇	1	3	yes	—
143	#554	Dic₆₇	1	9	yes	—
144	D₄×Q₈	Dic₆₇	1	3	yes	—
145	#533	C₂×D₆₇	1	9	yes	—
146	C₄×#41	C₂×D₆₇	1	3	yes	—
147	C₂×C₄×Q₈	C₂×D₆₇	1	3	yes	—
148	#446	—	67	18	no	—
149	#446	—	67	18	no	—
150	C₈×Q₈	—	67	1	no	—
151	C₈×Q₈	—	67	9	no	—
152	C₈×Q₈	—	67	9	no	—
153	C₈×Q₈	—	67	3	no	—
154	C₈×Q₈	—	67	6	no	—
155	#424	—	67	18	no	—
156	#423	—	67	9	no	—
157	C₂×C₄⋊C₈	—	67	9	no	—
158	#405	—	67	9	no	—
159	#405	—	67	9	no	—
160	C₄×M₁₆	—	67	9	no	—
161	C₂×C₄×C₈	—	67	9	no	—
162	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
163	C₂²×Q₈	C₂×Dic₆₇	1	3	yes	—
164	#131	—	67	8	no	—
165	#131	—	134	12	no	—
166	C₂×M₁₆	—	67	3	no	—
167	C₂×M₁₆	—	67	9	no	—
168	C₂²×C₈	—	67	3	no	—
169	C₂²×C₈	—	67	9	no	—
170	C₄⋊Q₈	C₂×Dic₆₇	1	9	yes	—
171	C₄⋊Q₈	C₂×Dic₆₇	1	9	yes	—
172	#125	C₂×Dic₆₇	1	18	yes	—
173	C₂²⋊Q₈	C₂×Dic₆₇	1	18	yes	—
174	C₂²⋊Q₈	C₂×Dic₆₇	1	18	yes	—
175	C₄×Q₈	C₂×Dic₆₇	1	3	yes	—
176	C₄×Q₈	C₂×Dic₆₇	1	9	yes	—
177	C₄×Q₈	C₂×Dic₆₇	1	3	yes	—
178	C₄×Q₈	C₂×Dic₆₇	1	3	yes	—
179	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
180	C₄×Q₈	C₂²×D₆₇	1	9	yes	—
181	C₄×Q₈	C₂²×D₆₇	1	6	yes	—
182	C₄×D₄	C₂×Dic₆₇	1	9	yes	—
183	C₄×D₄	C₂×Dic₆₇	1	9	yes	—
184	#117	C₂²×D₆₇	1	9	yes	—
185	C₂×C₄⋊C₄	C₂×Dic₆₇	1	9	yes	—
186	C₂×C₄²	C₂²×D₆₇	1	9	yes	—
187	C₄⋊C₈	—	67	3	no	—
188	C₄⋊C₈	—	67	9	no	—
189	C₄⋊C₈	—	67	18	no	—
190	C₄⋊C₈	—	67	18	no	—
191	#97	—	67	18	no	—
192	C₄×C₈	—	67	3	no	—
193	C₄×C₈	—	67	9	no	—
194	C₄×C₈	—	67	9	no	—
195	C₄×C₈	—	67	9	no	—
196	#41	Q₈×D₆₇	1	4	yes	—
197	#41	#11761629	1	4	yes	—
198	C₂×Q₈	—	2	12	no	—
199	C₂×Q₈	C₂²×Dic₆₇	1	9	yes	—
200	C₂×Q₈	C₂²×Dic₆₇	1	6	yes	—
201	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
202	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
203	C₂×D₄	C₂²×Dic₆₇	1	3	yes	—
204	C₂²×C₄	C₂²×Dic₆₇	1	9	yes	—
205	C₂²×C₄	C₂²×Dic₆₇	1	9	yes	—
206	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
207	M₁₆	—	67	12	no	—
208	M₁₆	—	134	18	no	—
209	C₂×C₈	—	67	12	no	—
210	C₂×C₈	—	67	1	no	—
211	C₂×C₈	—	67	3	no	—
212	C₂×C₈	—	67	3	no	—
213	C₂×C₈	—	67	9	no	—
214	C₂×C₈	—	134	18	no	—
215	C₄⋊C₄	C₂²×Dic₆₇	1	3	yes	—
216	C₄⋊C₄	C₂²×Dic₆₇	1	18	yes	—
217	C₄⋊C₄	C₂²×Dic₆₇	1	9	yes	—
218	C₄⋊C₄	C₂²×Dic₆₇	1	9	yes	—
219	C₄⋊C₄	C₂²×Dic₆₇	1	9	yes	—
220	C₄⋊C₄	C₂²×Dic₆₇	1	18	yes	—
221	C₂²⋊C₄	C₂²×Dic₆₇	1	18	yes	—
222	C₄²	C₂²×Dic₆₇	1	9	yes	—
223	C₄²	C₂²×Dic₆₇	1	9	yes	—
224	C₄²	C₂³×D₆₇	1	3	yes	—
225	C₄²	C₂³×D₆₇	1	9	yes	—
226	C₂³	C₂³×Dic₆₇	1	3	yes	—
227	Q₈	—	2	18	no	—
228	Q₈	—	4	6	no	—
229	Q₈	Q₈×Dic₆₇	1	4	yes	—
230	Q₈	#14524050	1	4	yes	direct product factor
231	D₄	Q₈×Dic₆₇	1	12	yes	—
232	C₂×C₄	—	2	18	no	—
233	C₂×C₄	—	4	9	no	—
234	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
235	C₂×C₄	C₂³×Dic₆₇	1	9	yes	—
236	C₂×C₄	C₂³×Dic₆₇	1	9	yes	—
237	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
238	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
239	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
240	C₂×C₄	C₂×#11761629	1	6	yes	—
241	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
242	C₈	—	67	2	no	—
243	C₈	—	67	6	no	—
244	C₈	—	134	3	no	—
245	C₈	—	134	9	no	—
246	C₂²	—	2	3	no	—
247	C₂²	—	2	3	no	—
248	C₂²	C₂×Q₈×Dic₆₇	1	6	yes	—
249	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
250	C₄	—	2	9	no	—
251	C₄	—	2	3	no	—
252	C₄	—	2	9	no	—
253	C₄	C₂×Q₈×Dic₆₇	1	6	yes	—
254	C₄	C₂×#14524050	1	6	yes	—
255	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
256	C₄	C₂²×#11761629	1	1	yes	characteristic
257	C₂	—	2	6	no	—
258	C₂	C₂²×Q₈×Dic₆₇	1	1	yes	characteristic
259	C₂	C₂²×#14524050	1	1	yes	characteristic
260	C₂	#22574921	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁴×C₄
Fitting	Q₈×#41×C₆₇	C₂
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	Q₈×#41	D₆₇
hypocenter	C₆₇	Q₈×#131
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ Q₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#11761612	C₂	semidirect
2. #22569158	C₂	semidirect
3. #22569144	C₂	semidirect
4. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
32. #14524050	Q₈	direct (N × H)
66. #11761612	C₂×Q₈	semidirect
68. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
99. Q₈×C₆₇	#131	semidirect
138. C₆₇	Q₈×#131	semidirect
230. Q₈	#14524050	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
#22569158	2
#22569144	2
C₂×Q₈×C₆₇⋊C₈	2
#22568777	2
C₄×#14524050	2
Q₈×#41×C₆₇	2
Q₈×#131	67

Subgroups — Q8×#14524050 #31720247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14524050 #31720247