Subgroups — Q₈×#15667946 #36573268

Order 22784.

838 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₈₉	C₂	1	2	yes	maximal
2	#26021011	C₂	1	6	yes	maximal
3	#26020412	C₂	1	3	yes	maximal
4	C₄×#15667946	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₈₉	—	2	2	no	—
7	C₂×#15667946	C₂²	1	3	yes	—
8	C₂×#15667946	C₂²	1	6	yes	—
9	C₂×#15667946	C₂²	1	1	yes	characteristic
10	#19151497	—	2	6	no	—
11	#19151468	C₂²	1	6	yes	—
12	C₄×C₄⋊Dic₈₉	C₂²	1	6	yes	—
13	#19151361	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₈₉	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
18	Q₈×Dic₈₉	—	4	8	no	—
19	C₂×C₄⋊Dic₈₉	—	2	6	no	—
20	C₂×C₄⋊Dic₈₉	—	2	6	no	—
21	C₂×C₄⋊Dic₈₉	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
23	C₂×C₄×Dic₈₉	—	2	6	no	—
24	#15667946	—	2	6	no	—
25	#15667946	—	4	6	no	—
26	#15667946	Q₈	1	4	yes	direct product factor
27	#15667919	—	2	6	no	—
28	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
32	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₈₉	D₄	1	3	yes	—
34	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
35	C₄⋊C₈×C₈₉	C₂×C₄	1	12	yes	—
36	C₄×C₈×C₈₉	C₂×C₄	1	12	yes	—
37	C₂²×Dic₈₉	—	2	2	no	—
38	C₂²×Dic₈₉	—	2	6	no	—
39	C₄⋊Dic₈₉	—	2	2	no	—
40	C₄⋊Dic₈₉	—	2	6	no	—
41	C₄⋊Dic₈₉	—	2	6	no	—
42	C₄⋊Dic₈₉	—	4	12	no	—
43	C₄⋊Dic₈₉	—	8	6	no	—
44	C₄⋊Dic₈₉	C₂×Q₈	1	4	yes	—
45	C₄×Dic₈₉	—	4	12	no	—
46	C₂×Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
47	C₂×Q₈×C₈₉	D₈	1	4	yes	—
48	C₂×Q₈×C₈₉	Q₁₆	1	4	yes	—
49	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	3	no	—
53	C₂×C₈×C₈₉	—	2	3	no	—
54	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
55	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₈₉	#41	1	2	yes	—
58	C₄⋊C₄×C₈₉	C₄⋊C₄	1	12	yes	—
59	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
60	C₄²×C₈₉	C₂×Q₈	1	6	yes	—
61	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₈₉	#107	1	16	yes	complemented
63	C₂×C₄×C₈₉	—	2	3	no	—
64	C₂×C₄×C₈₉	—	2	3	no	—
65	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
66	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
67	C₂×C₄×C₈₉	C₂×D₈	1	6	yes	—
68	C₂×C₄×C₈₉	C₂×Q₁₆	1	6	yes	—
69	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
73	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
74	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
75	C₂×Dic₈₉	—	2	4	no	—
76	C₂×Dic₈₉	—	4	2	no	—
77	C₂×Dic₈₉	—	4	2	no	—
78	C₂×Dic₈₉	—	4	6	no	—
79	C₂×Dic₈₉	—	4	6	no	—
80	C₂×Dic₈₉	—	4	6	no	—
81	C₈×C₈₉	—	2	6	no	—
82	C₈×C₈₉	C₄×Q₈	1	4	yes	complemented
83	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
85	C₂²×C₈₉	#550	1	1	yes	characteristic
86	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
87	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
88	C₂²×C₈₉	#576	1	1	yes	characteristic
89	C₂²×C₈₉	#578	1	1	yes	characteristic
90	C₄×C₈₉	—	2	6	no	—
91	C₄×C₈₉	C₂×#107	1	12	yes	—
92	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
93	C₄×C₈₉	#520	1	1	yes	characteristic
94	C₄×C₈₉	#557	1	1	yes	characteristic
95	C₄×C₈₉	Q₈²	1	1	yes	characteristic
96	Dic₈₉	—	4	4	no	—
97	Dic₈₉	—	8	6	no	—
98	Q₈×#107	—	89	1	no	maximal, Sylow
99	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
101	C₂×C₈₉	#2913	1	1	yes	characteristic
102	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
103	C₂×C₈₉	Q₈×Q₁₆	1	1	yes	characteristic
104	C₂×C₈₉	#3391	1	1	yes	characteristic
105	C₂×C₈₉	#3395	1	1	yes	characteristic
106	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
107	Q₈×C₄⋊C₄	—	89	2	no	—
108	#1947	—	89	6	no	—
109	#1845	—	89	3	no	—
110	C₄×#107	—	89	3	no	—
111	C₈₉	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	—	178	2	no	—
113	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
114	C₈×Q₈	Dic₈₉	1	8	yes	complemented
115	C₂×#107	—	89	1	no	—
116	C₂×#107	—	89	6	no	—
117	C₂×#107	—	89	3	no	—
118	C₂×C₄⋊C₈	C₂×D₈₉	1	3	yes	—
119	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
120	#389	—	178	6	no	—
121	#384	—	89	6	no	—
122	C₄×C₄⋊C₄	—	89	6	no	—
123	#340	—	89	6	no	—
124	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
126	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
127	C₄×Q₈	—	356	8	no	—
128	C₄×Q₈	Dic₁₇₈	1	4	yes	—
129	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
130	C₂×C₄⋊C₄	—	89	2	no	—
131	C₂×C₄⋊C₄	—	89	6	no	—
132	C₂×C₄⋊C₄	—	178	6	no	—
133	C₂×C₄⋊C₄	—	178	6	no	—
134	C₂×C₄⋊C₄	D₃₅₆	1	3	yes	—
135	C₂×C₄²	—	178	6	no	—
136	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
137	#107	—	89	4	no	—
138	#107	—	356	6	no	—
139	#107	—	178	6	no	—
140	C₄⋊C₈	C₂×Dic₈₉	1	12	yes	—
141	C₄×C₈	C₂×Dic₈₉	1	12	yes	—
142	#95	—	178	6	no	—
143	C₂×Q₈	C₄⋊Dic₈₉	1	4	yes	—
144	C₂×Q₈	D₇₁₂	1	4	yes	—
145	C₂×Q₈	Dic₃₅₆	1	4	yes	—
146	C₂²×C₄	—	178	6	no	—
147	C₂²×C₄	—	178	2	no	—
148	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
149	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
150	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
151	C₂×C₈	—	2	3	no	—
152	C₂×C₈	—	2	3	no	—
153	C₂×C₈	C₂²×Dic₈₉	1	6	yes	—
154	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
155	C₂×C₈	Q₈×D₈₉	1	2	yes	—
156	C₂×C₈	#13083575	1	2	yes	—
157	C₄⋊C₄	—	89	4	no	—
158	C₄⋊C₄	—	356	12	no	—
159	C₄⋊C₄	—	178	6	no	—
160	C₄⋊C₄	—	178	6	no	—
161	C₄⋊C₄	—	178	2	no	—
162	C₄⋊C₄	—	712	6	no	—
163	C₄⋊C₄	C₄⋊Dic₈₉	1	12	yes	—
164	C₄²	—	356	12	no	—
165	C₄²	C₂²×Dic₈₉	1	6	yes	—
166	C₄²	C₂×Dic₁₇₈	1	6	yes	—
167	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
168	Q₈	#15667946	1	16	yes	direct product factor
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	356	2	no	—
172	C₂×C₄	—	356	2	no	—
173	C₂×C₄	—	356	6	no	—
174	C₂×C₄	—	356	6	no	—
175	C₂×C₄	—	356	6	no	—
176	C₂×C₄	—	178	4	no	—
177	C₂×C₄	C₂×C₄⋊Dic₈₉	1	6	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₈₉	1	6	yes	—
179	C₂×C₄	C₂×D₇₁₂	1	6	yes	—
180	C₂×C₄	C₂×Dic₃₅₆	1	6	yes	—
181	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
182	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
184	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
185	C₂×C₄	#15668058	1	1	yes	characteristic
186	C₂×C₄	#15668059	1	1	yes	characteristic
187	C₈	—	2	6	no	—
188	C₈	Q₈×Dic₈₉	1	4	yes	complemented
189	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
190	C₂²	Q₈×D₃₅₆	1	1	yes	characteristic
191	C₂²	#19152270	1	1	yes	characteristic
192	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
193	C₂²	C₂²×Dic₃₅₆	1	1	yes	characteristic
194	C₂²	#19152403	1	1	yes	characteristic
195	C₂²	#19152415	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	356	4	no	—
198	C₄	—	712	6	no	—
199	C₄	C₂×#15667946	1	12	yes	—
200	C₄	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
201	C₄	#19152081	1	1	yes	characteristic
202	C₄	#19152327	1	1	yes	characteristic
203	C₄	Q₈×Dic₁₇₈	1	1	yes	characteristic
204	C₂	Q₈×C₄⋊Dic₈₉	1	1	yes	characteristic
205	C₂	C₂²×#15667946	1	1	yes	characteristic
206	C₂	#26028852	1	1	yes	characteristic
207	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
208	C₂	Q₈×Dic₃₅₆	1	1	yes	characteristic
209	C₂	#26033227	1	1	yes	characteristic
210	C₂	#26033258	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	Q₈×#107
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₈₉	C₄	semidirect
26. #15667946	Q₈	direct (N × H)
62. Q₈×C₈₉	#107	semidirect
82. C₈×C₈₉	C₄×Q₈	semidirect
111. C₈₉	Q₈×#107	semidirect
114. C₈×Q₈	Dic₈₉	semidirect
168. Q₈	#15667946	direct (N × H)
188. C₈	Q₈×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₈₉	2
#26021011	2
#26020412	2
C₄×#15667946	2
C₂×C₈×Q₈×C₈₉	2
Q₈×#107	89

Subgroups — Q8×#15667946 #36573268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#15667946 #36573268