Subgroups — Q₈×#88058 #15881759

Order 3328.

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⋊Dic₁₃	C₂	1	2	yes	maximal
2	#13115595	C₂	1	6	yes	maximal
3	#13114996	C₂	1	3	yes	maximal
4	C₄×#88058	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×Q₈×Dic₁₃	—	2	2	no	—
8	C₈×Q₈×C₁₃	C₄	1	8	yes	complemented
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
11	C₂×#88058	C₂²	1	3	yes	—
12	C₂×#88058	C₂²	1	6	yes	—
13	C₂×#88058	C₂²	1	1	yes	characteristic
14	#11718646	—	2	6	no	—
15	#11718596	C₂²	1	6	yes	—
16	C₄×C₄⋊Dic₁₃	C₂²	1	6	yes	—
17	#11718143	C₂²	1	6	yes	—
18	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
22	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
23	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
25	Q₈×Dic₁₃	—	4	8	no	—
26	C₂×C₄⋊Dic₁₃	—	2	6	no	—
27	C₂×C₄⋊Dic₁₃	—	2	6	no	—
28	C₂×C₄⋊Dic₁₃	C₂³	1	6	yes	—
29	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
30	C₂×C₄×Dic₁₃	—	2	6	no	—
31	C₁₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
33	#88071	—	2	6	no	—
34	#88058	—	2	6	no	—
35	#88058	—	4	6	no	—
36	#88058	Q₈	1	4	yes	direct product factor
37	Q₈×#107	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
39	C₂×Q₈×C₁₃	D₈	1	4	yes	—
40	C₂×Q₈×C₁₃	Q₁₆	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	2	no	—
45	C₂²×Dic₁₃	—	2	6	no	—
46	C₂×C₈×C₁₃	—	2	3	no	—
47	C₂×C₈×C₁₃	—	2	3	no	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₃	#41	1	2	yes	—
52	C₁₃×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
53	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
54	C₄²×C₁₃	C₂×Q₈	1	6	yes	—
55	C₄⋊Dic₁₃	—	2	2	no	—
56	C₄⋊Dic₁₃	—	2	6	no	—
57	C₄⋊Dic₁₃	—	2	6	no	—
58	C₄⋊Dic₁₃	—	4	12	no	—
59	C₄⋊Dic₁₃	—	8	6	no	—
60	C₄⋊Dic₁₃	C₂×Q₈	1	4	yes	—
61	C₄×Dic₁₃	—	4	12	no	—
62	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
63	Q₈×C₄⋊C₄	—	13	2	no	—
64	#1947	—	13	6	no	—
65	#1845	—	13	3	no	—
66	C₄×#107	—	13	3	no	—
67	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
68	Q₈×C₁₃	#107	1	16	yes	complemented
69	C₂×C₄×C₁₃	—	2	3	no	—
70	C₂×C₄×C₁₃	—	2	3	no	—
71	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
73	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
74	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
75	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
79	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
81	C₂×Dic₁₃	—	2	4	no	—
82	C₂×Dic₁₃	—	4	2	no	—
83	C₂×Dic₁₃	—	4	2	no	—
84	C₂×Dic₁₃	—	4	6	no	—
85	C₂×Dic₁₃	—	4	6	no	—
86	C₂×Dic₁₃	—	4	6	no	—
87	C₈×C₁₃	—	2	6	no	—
88	C₈×C₁₃	C₄×Q₈	1	4	yes	complemented
89	C₂×C₄×Q₈	—	26	2	no	—
90	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
91	C₈×Q₈	Dic₁₃	1	8	yes	complemented
92	C₂×#107	—	13	1	no	—
93	C₂×#107	—	13	3	no	—
94	C₂×#107	—	13	6	no	—
95	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
96	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
97	#389	—	26	6	no	—
98	#384	—	13	6	no	—
99	C₄×C₄⋊C₄	—	13	6	no	—
100	#340	—	13	6	no	—
101	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
103	C₂²×C₁₃	#550	1	1	yes	characteristic
104	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
105	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
106	C₂²×C₁₃	#576	1	1	yes	characteristic
107	C₂²×C₁₃	#578	1	1	yes	characteristic
108	C₄×C₁₃	—	2	6	no	—
109	C₄×C₁₃	C₂×#107	1	12	yes	—
110	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
111	C₄×C₁₃	#520	1	1	yes	characteristic
112	C₄×C₁₃	#557	1	1	yes	characteristic
113	C₄×C₁₃	Q₈²	1	1	yes	characteristic
114	Dic₁₃	—	4	4	no	—
115	Dic₁₃	—	8	6	no	—
116	C₂²×Q₈	D₅₂	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
118	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
119	C₄×Q₈	—	52	8	no	—
120	C₄×Q₈	Dic₂₆	1	4	yes	—
121	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
122	C₂×C₄⋊C₄	—	13	2	no	—
123	C₂×C₄⋊C₄	—	13	6	no	—
124	C₂×C₄⋊C₄	—	26	6	no	—
125	C₂×C₄⋊C₄	—	26	6	no	—
126	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
127	C₂×C₄²	—	26	6	no	—
128	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
129	#107	—	13	4	no	—
130	#107	—	26	6	no	—
131	#107	—	52	6	no	—
132	C₄⋊C₈	C₂×Dic₁₃	1	12	yes	—
133	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
134	#95	—	26	6	no	—
135	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
137	C₂×C₁₃	#2913	1	1	yes	characteristic
138	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
139	C₂×C₁₃	Q₈×Q₁₆	1	1	yes	characteristic
140	C₂×C₁₃	#3391	1	1	yes	characteristic
141	C₂×C₁₃	#3395	1	1	yes	characteristic
142	C₂×Q₈	D₁₀₄	1	4	yes	—
143	C₂×Q₈	Dic₅₂	1	4	yes	—
144	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
145	C₂²×C₄	—	26	2	no	—
146	C₂²×C₄	—	26	6	no	—
147	C₂²×C₄	C₂×D₅₂	1	3	yes	—
148	C₂²×C₄	C₂×D₅₂	1	3	yes	—
149	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
150	C₂×C₈	—	2	3	no	—
151	C₂×C₈	—	2	3	no	—
152	C₂×C₈	Q₈×D₁₃	1	2	yes	—
153	C₂×C₈	#6131	1	2	yes	—
154	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
155	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
156	C₄⋊C₄	—	13	4	no	—
157	C₄⋊C₄	—	26	2	no	—
158	C₄⋊C₄	—	26	6	no	—
159	C₄⋊C₄	—	26	6	no	—
160	C₄⋊C₄	—	52	12	no	—
161	C₄⋊C₄	—	104	6	no	—
162	C₄⋊C₄	C₄⋊Dic₁₃	1	12	yes	—
163	C₄²	—	52	12	no	—
164	C₄²	C₂×Dic₂₆	1	6	yes	—
165	C₄²	C₂²×Dic₁₃	1	6	yes	—
166	C₁₃	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
167	C₂³	C₂²×D₅₂	1	1	yes	characteristic
168	Q₈	#88058	1	16	yes	direct product factor
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	26	4	no	—
172	C₂×C₄	—	52	2	no	—
173	C₂×C₄	—	52	2	no	—
174	C₂×C₄	—	52	6	no	—
175	C₂×C₄	—	52	6	no	—
176	C₂×C₄	—	52	6	no	—
177	C₂×C₄	C₂×D₁₀₄	1	6	yes	—
178	C₂×C₄	C₂×Dic₅₂	1	6	yes	—
179	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
181	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
182	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
183	C₂×C₄	C₂×#6131	1	1	yes	characteristic
184	C₂×C₄	#88256	1	1	yes	characteristic
185	C₂×C₄	#88257	1	1	yes	characteristic
186	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
187	C₈	—	2	6	no	—
188	C₈	Q₈×Dic₁₃	1	4	yes	complemented
189	C₂²	Q₈×D₅₂	1	1	yes	characteristic
190	C₂²	#11719282	1	1	yes	characteristic
191	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
192	C₂²	C₂²×Dic₅₂	1	1	yes	characteristic
193	C₂²	#11719457	1	1	yes	characteristic
194	C₂²	#11719458	1	1	yes	characteristic
195	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	52	4	no	—
198	C₄	—	104	6	no	—
199	C₄	C₂×#88058	1	12	yes	—
200	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
201	C₄	#11719275	1	1	yes	characteristic
202	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
203	C₄	#11719532	1	1	yes	characteristic
204	C₂	Q₈×C₄⋊Dic₁₃	1	1	yes	characteristic
205	C₂	C₂²×#88058	1	1	yes	characteristic
206	C₂	#13123436	1	1	yes	characteristic
207	C₂	Q₈×D₁₀₄	1	1	yes	characteristic
208	C₂	Q₈×Dic₅₂	1	1	yes	characteristic
209	C₂	#13127811	1	1	yes	characteristic
210	C₂	#13127842	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	Q₈×#107
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
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
8. C₈×Q₈×C₁₃	C₄	semidirect
36. #88058	Q₈	direct (N × H)
68. Q₈×C₁₃	#107	semidirect
88. C₈×C₁₃	C₄×Q₈	semidirect
91. C₈×Q₈	Dic₁₃	semidirect
166. C₁₃	Q₈×#107	semidirect
168. Q₈	#88058	direct (N × H)
188. C₈	Q₈×Dic₁₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₁₃	2
#13115595	2
#13114996	2
C₄×#88058	2
C₂×C₈×Q₈×C₁₃	2
Q₈×#107	13

Subgroups — Q8×#88058 #15881759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88058 #15881759