Subgroups — Q₈×#88076 #15879494

Order 3328.

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×#6099	C₂	1	2	yes	maximal
2	#13115826	C₂	1	6	yes	maximal
3	#13114832	C₂	1	3	yes	maximal
4	C₄×#88076	C₂	1	3	yes	maximal
5	C₂×Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
6	Q₈²×C₁₃	—	2	2	no	—
7	Q₈²×C₁₃	—	2	6	no	—
8	C₂×C₁₃×C₄⋊Q₈	C₄	1	6	yes	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
10	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
11	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88076	C₂²	1	3	yes	—
14	C₂×#88076	C₂²	1	1	yes	characteristic
15	#11718685	—	2	6	no	—
16	Q₈×#1055	—	2	2	no	—
17	#11718585	C₂²	1	6	yes	—
18	C₄×#6099	C₂²	1	6	yes	—
19	#11718122	C₂²	1	6	yes	—
20	#11718122	C₂²	1	6	yes	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×Q₈×C₁₃	C₂×C₄	1	6	yes	—
23	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₁₃×C₄⋊Q₈	—	2	12	no	—
25	C₁₃×C₄⋊Q₈	—	2	12	no	—
26	C₁₃×C₄⋊Q₈	—	2	6	no	—
27	C₁₃×C₄⋊Q₈	—	2	6	no	—
28	C₄×Q₈×C₁₃	—	2	6	no	—
29	C₄×Q₈×C₁₃	—	2	2	no	—
30	C₄×Q₈×C₁₃	—	2	6	no	—
31	C₄×Q₈×C₁₃	—	2	6	no	—
32	C₄×Q₈×C₁₃	D₄	1	2	yes	—
33	C₄×Q₈×C₁₃	D₄	1	6	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
40	C₂×#6099	C₂³	1	6	yes	—
41	C₂×#6099	C₂³	1	2	yes	—
42	#88076	—	2	6	no	—
43	#88076	Q₈	1	4	yes	direct product factor
44	C₄⋊#1055	—	2	6	no	—
45	C₄⋊#1055	—	2	6	no	—
46	#88043	—	2	6	no	—
47	C₄×#1055	—	2	6	no	—
48	Q₈×#101	—	13	1	no	maximal, Sylow
49	C₂×Q₈×C₁₃	—	2	4	no	—
50	C₂×Q₈×C₁₃	—	2	12	no	—
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	—	2	2	no	—
53	C₂×Q₈×C₁₃	—	8	6	no	—
54	C₂×Q₈×C₁₃	C₂²⋊C₄	1	3	yes	—
55	C₂×Q₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂×Q₈×C₁₃	#41	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	—	2	12	no	—
64	C₁₃×C₄⋊C₄	—	2	6	no	—
65	C₁₃×C₄⋊C₄	—	2	6	no	—
66	C₁₃×C₄⋊C₄	—	2	6	no	—
67	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
68	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
69	C₄²×C₁₃	—	2	6	no	—
70	C₄²×C₁₃	—	2	6	no	—
71	C₄²×C₁₃	C₂×D₄	1	6	yes	—
72	C₄²×C₁₃	C₂×D₄	1	6	yes	—
73	#6099	—	2	2	no	—
74	#6099	—	2	6	no	—
75	#6099	—	2	6	no	—
76	#6099	C₂×Q₈	1	4	yes	—
77	C₂×#1055	—	2	2	no	—
78	C₂×#1055	—	2	6	no	—
79	C₂×Q₈²	D₁₃	1	1	yes	characteristic
80	Q₈×M₁₆	—	13	2	no	—
81	#1997	—	13	6	no	—
82	#1801	—	13	3	no	—
83	C₄×#101	—	13	3	no	—
84	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₁₃	—	2	4	no	—
86	Q₈×C₁₃	—	2	4	no	—
87	Q₈×C₁₃	—	2	6	no	—
88	Q₈×C₁₃	—	8	12	no	—
89	Q₈×C₁₃	#101	1	4	yes	complemented
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	6	no	—
93	C₂×C₄×C₁₃	—	2	6	no	—
94	C₂×C₄×C₁₃	—	4	3	no	—
95	C₂×C₄×C₁₃	—	4	6	no	—
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
97	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
98	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
99	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
100	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
104	#1055	—	2	4	no	—
105	#1055	—	4	6	no	—
106	Q₈²	—	2	2	no	—
107	Q₈²	—	2	6	no	—
108	C₂×C₄⋊Q₈	Dic₁₃	1	6	yes	—
109	C₂×C₄⋊Q₈	C₂×D₁₃	1	3	yes	—
110	C₂×C₄×Q₈	Dic₁₃	1	2	yes	—
111	C₂×C₄×Q₈	C₂×D₁₃	1	3	yes	—
112	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
113	#446	—	26	6	no	—
114	C₈×Q₈	—	26	2	no	—
115	#423	—	13	6	no	—
116	C₂×#101	—	13	1	no	—
117	C₂×#101	—	13	3	no	—
118	C₄×M₁₆	—	13	6	no	—
119	#343	—	13	6	no	—
120	#343	—	13	6	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₂²×C₁₃	#519	1	1	yes	characteristic
126	C₄×C₁₃	—	2	4	no	—
127	C₄×C₁₃	—	2	3	no	—
128	C₄×C₁₃	—	2	6	no	—
129	C₄×C₁₃	—	4	6	no	—
130	C₄×C₁₃	C₂×#101	1	6	yes	—
131	C₄×C₁₃	#548	1	2	yes	—
132	C₄×C₁₃	D₄×Q₈	1	2	yes	—
133	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
134	C₂²×Q₈	C₂×Dic₁₃	1	6	yes	—
135	C₂²×Q₈	C₂²×D₁₃	1	1	yes	characteristic
136	C₂×M₁₆	—	13	2	no	—
137	C₂×M₁₆	—	13	6	no	—
138	C₄⋊Q₈	—	2	12	no	—
139	C₄⋊Q₈	—	2	12	no	—
140	C₄⋊Q₈	—	2	6	no	—
141	C₄⋊Q₈	—	2	6	no	—
142	C₄×Q₈	—	2	6	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	6	no	—
145	C₄×Q₈	—	2	6	no	—
146	C₄×Q₈	C₂²⋊D₁₃	1	2	yes	—
147	C₄×Q₈	C₂²⋊D₁₃	1	6	yes	—
148	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
149	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
150	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
151	C₂×C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
152	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
153	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
154	C₄⋊C₈	—	26	6	no	—
155	C₄⋊C₈	—	26	6	no	—
156	#101	—	13	4	no	—
157	#101	—	26	6	no	—
158	#97	—	26	6	no	—
159	C₄×C₈	—	26	6	no	—
160	C₂×C₁₃	—	2	1	no	—
161	C₂×C₁₃	—	2	1	no	—
162	C₂×C₁₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₁₃	C₂²×#101	1	1	yes	characteristic
164	C₂×C₁₃	#2888	1	1	yes	characteristic
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	2	12	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	8	6	no	—
170	C₂×Q₈	C₂²⋊Dic₁₃	1	3	yes	—
171	C₂×Q₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
172	C₂×Q₈	Q₈×D₁₃	1	2	yes	—
173	C₂×Q₈	#6131	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
176	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
177	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
178	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
179	M₁₆	—	13	4	no	—
180	M₁₆	—	26	2	no	—
181	M₁₆	—	26	6	no	—
182	M₁₆	—	26	6	no	—
183	C₂×C₈	—	26	2	no	—
184	C₂×C₈	—	26	6	no	—
185	C₄⋊C₄	—	2	12	no	—
186	C₄⋊C₄	—	2	6	no	—
187	C₄⋊C₄	—	2	6	no	—
188	C₄⋊C₄	—	2	6	no	—
189	C₄⋊C₄	C₂²⋊Dic₁₃	1	6	yes	—
190	C₄⋊C₄	C₂²⋊Dic₁₃	1	6	yes	—
191	C₄²	—	2	6	no	—
192	C₄²	—	2	6	no	—
193	C₄²	C₂×C₂²⋊D₁₃	1	6	yes	—
194	C₄²	C₂×C₂²⋊D₁₃	1	6	yes	—
195	C₁₃	Q₈×#101	1	1	yes	characteristic, complemented, Sylow
196	C₂³	C₂³×Dic₁₃	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	#88076	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	Q₈×Dic₁₃	1	4	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
211	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
212	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×#6131	1	1	yes	characteristic
214	C₂×C₄	#88254	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₁₃	1	2	yes	—
216	C₈	—	26	4	no	—
217	C₈	—	52	6	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
221	C₂²	#11719517	1	1	yes	characteristic
222	C₂²	C₂²×C₂²⋊Dic₁₃	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×#88076	1	6	yes	—
228	C₄	Q₈×C₂²⋊D₁₃	1	2	yes	—
229	C₄	#11719522	1	2	yes	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	Q₈×C₂²⋊Dic₁₃	1	1	yes	characteristic
233	C₂	C₂²×#88076	1	1	yes	characteristic
234	C₂	#13123315	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₁₃
commutator, prime radical	C₂³×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×Q₈²×C₁₃	C₂
socle	C₂²×C₁₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×Q₈²	D₁₃
hypocenter	C₁₃	Q₈×#101
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
43. #88076	Q₈	direct (N × H)
89. Q₈×C₁₃	#101	semidirect
195. C₁₃	Q₈×#101	semidirect
201. Q₈	#88076	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×#6099	2
#13115826	2
#13114832	2
C₄×#88076	2
C₂×Q₈²×C₁₃	2
Q₈×#101	13

Subgroups — Q8×#88076 #15879494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88076 #15879494