Subgroups — Q₈×C₄⋊(C₆₇⋊C₈) #30874625

Order 17152.

896 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₆₇⋊C₈	C₂	1	2	yes	maximal
2	#22560928	C₂	1	6	yes	maximal
3	#22560361	C₂	1	3	yes	maximal
4	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₆₇⋊C₈	—	2	8	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	3	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
12	#16863151	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
19	C₄³×C₆₇	C₂²	1	3	yes	—
20	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
21	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	—	2	12	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
24	C₄⋊(C₆₇⋊C₈)	—	4	6	no	—
25	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	direct product factor
26	C₄×C₆₇⋊C₈	—	2	12	no	—
27	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Q₈×C₆₇	—	2	4	no	—
29	C₄×Q₈×C₆₇	C₈	1	8	yes	complemented
30	C₄×Q₈×C₆₇	D₄	1	4	yes	—
31	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#95×C₆₇	C₂×C₄	1	6	yes	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	6	no	—
46	C₂×C₆₇⋊C₈	—	2	6	no	—
47	C₂×C₆₇⋊C₈	—	2	6	no	—
48	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
49	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
50	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
51	C₂×Q₈×C₆₇	M₁₆	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	—	2	6	no	—
60	C₄⋊C₄×C₆₇	C₄⋊C₄	1	12	yes	—
61	C₄⋊C₄×C₆₇	C₂×C₈	1	12	yes	—
62	C₄²×C₆₇	—	2	6	no	—
63	C₄²×C₆₇	—	2	6	no	—
64	C₄²×C₆₇	—	2	3	no	—
65	C₄²×C₆₇	—	2	3	no	—
66	C₄²×C₆₇	—	4	3	no	—
67	C₄²×C₆₇	C₂×C₈	1	12	yes	—
68	C₄²×C₆₇	C₂×D₄	1	6	yes	—
69	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
70	C₄²×C₆₇	C₂×Q₈	1	6	yes	—
71	C₄²×C₆₇	#41	1	2	yes	—
72	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
73	Q₈×C₆₇	C₄⋊C₈	1	16	yes	complemented
74	C₂×C₄×C₆₇	—	2	3	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	3	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	3	no	—
79	C₂×C₄×C₆₇	—	2	3	no	—
80	C₂×C₄×C₆₇	—	2	3	no	—
81	C₂×C₄×C₆₇	—	2	3	no	—
82	C₂×C₄×C₆₇	—	2	3	no	—
83	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
85	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
86	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
88	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
89	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
91	C₂×C₄×C₆₇	#131	1	2	yes	—
92	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
98	C₆₇⋊C₈	—	2	4	no	—
99	C₆₇⋊C₈	—	4	6	no	—
100	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
102	C₂²×C₆₇	#520	1	1	yes	characteristic
103	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
104	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
105	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
106	C₂²×C₆₇	#568	1	1	yes	characteristic
107	C₄×C₆₇	—	2	6	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	—	2	6	no	—
110	C₄×C₆₇	—	4	3	no	—
111	C₄×C₆₇	C₂×C₄⋊C₈	1	12	yes	—
112	C₄×C₆₇	C₈×Q₈	1	4	yes	complemented
113	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
114	C₄×C₆₇	#550	1	1	yes	characteristic
115	C₄×C₆₇	#557	1	1	yes	characteristic
116	C₄×C₆₇	Q₈²	1	1	yes	characteristic
117	Q₈×C₄⋊C₈	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
120	C₂×C₆₇	#2906	1	1	yes	characteristic
121	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
122	C₂×C₆₇	Q₈×M₁₆	1	1	yes	characteristic
123	C₂×C₆₇	#2988	1	1	yes	characteristic
124	C₂×C₆₇	#2991	1	1	yes	characteristic
125	C₂×C₈×Q₈	—	67	2	no	—
126	C₄²×Q₈	D₆₇	1	1	yes	characteristic
127	#1939	—	67	6	no	—
128	#1842	—	67	3	no	—
129	C₄×C₄⋊C₈	—	67	3	no	—
130	C₆₇	Q₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
132	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
133	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
134	C₈×Q₈	—	134	8	no	—
135	C₂×C₄⋊C₈	—	67	6	no	—
136	C₂×C₄⋊C₈	—	67	1	no	—
137	C₂×C₄⋊C₈	—	67	3	no	—
138	C₂×C₄⋊C₈	—	67	6	no	—
139	C₂×C₄×C₈	—	67	6	no	—
140	C₄×C₄⋊C₄	Dic₆₇	1	3	yes	—
141	C₄×C₄⋊C₄	Dic₆₇	1	3	yes	—
142	C₄×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
143	C₄³	C₂×D₆₇	1	3	yes	—
144	#336	—	67	6	no	—
145	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
146	C₂²×C₈	—	67	2	no	—
147	C₂²×C₈	—	67	6	no	—
148	C₄×Q₈	—	2	4	no	—
149	C₄×Q₈	C₆₇⋊C₈	1	8	yes	complemented
150	C₄×Q₈	Dic₁₃₄	1	4	yes	—
151	C₄×Q₈	D₂₆₈	1	4	yes	—
152	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
155	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
156	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
157	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
158	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
159	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
160	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
161	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
162	C₄⋊C₈	—	268	6	no	—
163	C₄⋊C₈	—	67	4	no	—
164	C₄⋊C₈	—	134	12	no	—
165	C₄⋊C₈	—	134	6	no	—
166	C₄×C₈	—	134	12	no	—
167	#95	C₂×Dic₆₇	1	6	yes	—
168	C₂×Q₈	C₄⋊Dic₆₇	1	4	yes	—
169	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
170	C₂×Q₈	#11761612	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
172	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
173	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
175	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
176	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
178	C₂×C₈	—	67	4	no	—
179	C₂×C₈	—	134	6	no	—
180	C₂×C₈	—	134	6	no	—
181	C₂×C₈	—	134	6	no	—
182	C₂×C₈	—	134	2	no	—
183	C₂×C₈	—	134	2	no	—
184	C₄⋊C₄	—	2	6	no	—
185	C₄⋊C₄	C₄⋊Dic₆₇	1	12	yes	—
186	C₄⋊C₄	C₂×C₆₇⋊C₈	1	12	yes	—
187	C₄²	—	2	6	no	—
188	C₄²	—	2	6	no	—
189	C₄²	—	2	3	no	—
190	C₄²	—	2	3	no	—
191	C₄²	—	4	3	no	—
192	C₄²	C₂×C₆₇⋊C₈	1	12	yes	—
193	C₄²	C₂×D₂₆₈	1	6	yes	—
194	C₄²	Q₈×D₆₇	1	2	yes	—
195	C₄²	C₂×Dic₁₃₄	1	6	yes	—
196	C₄²	#11761629	1	2	yes	—
197	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
198	Q₈	C₄⋊(C₆₇⋊C₈)	1	16	yes	direct product factor
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	—	2	3	no	—
208	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
210	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
211	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
212	C₂×C₄	C₂²×C₆₇⋊C₈	1	6	yes	—
213	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
214	C₂×C₄	C₂²×C₆₇⋊C₈	1	6	yes	—
215	C₂×C₄	C₂×#11761612	1	6	yes	—
216	C₂×C₄	#14524050	1	2	yes	—
217	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
218	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
219	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
220	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
221	C₂×C₄	#14524089	1	1	yes	characteristic
222	C₂×C₄	#14524090	1	1	yes	characteristic
223	C₈	—	268	6	no	—
224	C₈	—	134	4	no	—
225	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
226	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
227	C₂²	#16863879	1	1	yes	characteristic
228	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
229	C₂²	C₂²×#11761612	1	1	yes	characteristic
230	C₂²	C₂×#14524050	1	1	yes	characteristic
231	C₂²	#16864163	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	2	6	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	3	no	—
236	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	12	yes	—
237	C₄	Q₈×C₆₇⋊C₈	1	4	yes	complemented
238	C₄	Q₈×D₂₆₈	1	1	yes	characteristic
239	C₄	#16864068	1	1	yes	characteristic
240	C₄	#16864125	1	1	yes	characteristic
241	C₄	Q₈×Dic₁₃₄	1	1	yes	characteristic
242	C₂	Q₈×C₄⋊Dic₆₇	1	1	yes	characteristic
243	C₂	C₂²×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
244	C₂	#22568777	1	1	yes	characteristic
245	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
246	C₂	Q₈×#11761612	1	1	yes	characteristic
247	C₂	#22569320	1	1	yes	characteristic
248	C₂	#22569341	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₄²×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂³×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×Q₈	D₆₇
hypocenter	C₆₇	Q₈×C₄⋊C₈
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
25. C₄⋊(C₆₇⋊C₈)	Q₈	direct (N × H)
29. C₄×Q₈×C₆₇	C₈	semidirect
73. Q₈×C₆₇	C₄⋊C₈	semidirect
112. C₄×C₆₇	C₈×Q₈	semidirect
130. C₆₇	Q₈×C₄⋊C₈	semidirect
149. C₄×Q₈	C₆₇⋊C₈	semidirect
198. Q₈	C₄⋊(C₆₇⋊C₈)	direct (N × H)
237. C₄	Q₈×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₆₇⋊C₈	2
#22560928	2
#22560361	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₄²×Q₈×C₆₇	2
Q₈×C₄⋊C₈	67

Subgroups — Q8×C4⋊(C67⋊C8) #30874625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₄⋊(C₆₇⋊C₈) #30874625