Subgroups — Q₈×#88057 #15881746

Order 3328.

838 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#13115600	C₂	1	6	yes	maximal
3	#13114993	C₂	1	3	yes	maximal
4	C₄×#88057	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×#88057	C₂²	1	3	yes	—
12	C₂×#88057	C₂²	1	6	yes	—
13	C₂×#88057	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	#88057	—	2	6	no	—
35	#88057	—	4	6	no	—
36	#88057	Q₈	1	4	yes	direct product factor
37	Q₈×#106	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
39	C₂×Q₈×C₁₃	SD₁₆	1	8	yes	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	—	2	6	no	—
45	C₂×C₈×C₁₃	—	2	3	no	—
46	C₂×C₈×C₁₃	—	2	3	no	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₁₃	#41	1	2	yes	—
51	C₁₃×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
52	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
53	C₄²×C₁₃	C₂×Q₈	1	6	yes	—
54	C₄⋊Dic₁₃	—	2	2	no	—
55	C₄⋊Dic₁₃	—	2	6	no	—
56	C₄⋊Dic₁₃	—	2	6	no	—
57	C₄⋊Dic₁₃	—	4	12	no	—
58	C₄⋊Dic₁₃	—	8	6	no	—
59	C₄⋊Dic₁₃	C₂×Q₈	1	4	yes	—
60	C₄×Dic₁₃	—	4	12	no	—
61	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
62	Q₈×C₄⋊C₄	—	13	2	no	—
63	#1948	—	13	6	no	—
64	#1844	—	13	3	no	—
65	C₄×#106	—	13	3	no	—
66	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
67	Q₈×C₁₃	#106	1	16	yes	complemented
68	C₂×C₄×C₁₃	—	2	6	no	—
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
71	C₂×C₄×C₁₃	C₂×SD₁₆	1	12	yes	—
72	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
76	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
78	C₂×Dic₁₃	—	2	4	no	—
79	C₂×Dic₁₃	—	4	2	no	—
80	C₂×Dic₁₃	—	4	2	no	—
81	C₂×Dic₁₃	—	4	6	no	—
82	C₂×Dic₁₃	—	4	6	no	—
83	C₂×Dic₁₃	—	4	6	no	—
84	C₈×C₁₃	—	2	6	no	—
85	C₈×C₁₃	C₄×Q₈	1	4	yes	complemented
86	C₂×C₄×Q₈	—	26	2	no	—
87	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
88	C₈×Q₈	Dic₁₃	1	8	yes	complemented
89	C₂×#106	—	13	1	no	—
90	C₂×#106	—	13	3	no	—
91	C₂×#106	—	13	6	no	—
92	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
93	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
94	#389	—	26	6	no	—
95	#384	—	13	6	no	—
96	C₄×C₄⋊C₄	—	13	6	no	—
97	#340	—	13	6	no	—
98	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
100	C₂²×C₁₃	#550	1	1	yes	characteristic
101	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
102	C₂²×C₁₃	#577	1	2	yes	—
103	C₄×C₁₃	—	2	6	no	—
104	C₄×C₁₃	C₂×#106	1	12	yes	—
105	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
106	C₄×C₁₃	#520	1	1	yes	characteristic
107	C₄×C₁₃	#557	1	1	yes	characteristic
108	C₄×C₁₃	Q₈²	1	1	yes	characteristic
109	Dic₁₃	—	4	4	no	—
110	Dic₁₃	—	8	6	no	—
111	C₂²×Q₈	D₅₂	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
113	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₄×Q₈	—	52	8	no	—
115	C₄×Q₈	Dic₂₆	1	4	yes	—
116	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
117	C₂×C₄⋊C₄	—	13	2	no	—
118	C₂×C₄⋊C₄	—	13	6	no	—
119	C₂×C₄⋊C₄	—	26	6	no	—
120	C₂×C₄⋊C₄	—	26	6	no	—
121	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
122	C₂×C₄²	—	26	6	no	—
123	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
124	#106	—	13	4	no	—
125	#106	—	26	6	no	—
126	#106	—	52	6	no	—
127	C₄⋊C₈	C₂×Dic₁₃	1	12	yes	—
128	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
129	#95	—	26	6	no	—
130	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
132	C₂×C₁₃	#2912	1	1	yes	characteristic
133	C₂×C₁₃	Q₈×SD₁₆	1	2	yes	—
134	C₂×C₁₃	#3392	1	2	yes	—
135	C₂×Q₈	#6095	1	8	yes	—
136	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
137	C₂²×C₄	—	26	2	no	—
138	C₂²×C₄	—	26	6	no	—
139	C₂²×C₄	C₂×D₅₂	1	3	yes	—
140	C₂²×C₄	C₂×D₅₂	1	3	yes	—
141	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
142	C₂×C₈	—	2	3	no	—
143	C₂×C₈	—	2	3	no	—
144	C₂×C₈	Q₈×D₁₃	1	2	yes	—
145	C₂×C₈	#6131	1	2	yes	—
146	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
147	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
148	C₄⋊C₄	—	13	4	no	—
149	C₄⋊C₄	—	26	2	no	—
150	C₄⋊C₄	—	26	6	no	—
151	C₄⋊C₄	—	26	6	no	—
152	C₄⋊C₄	—	52	12	no	—
153	C₄⋊C₄	—	104	6	no	—
154	C₄⋊C₄	C₄⋊Dic₁₃	1	12	yes	—
155	C₄²	—	52	12	no	—
156	C₄²	C₂×Dic₂₆	1	6	yes	—
157	C₄²	C₂²×Dic₁₃	1	6	yes	—
158	C₁₃	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
159	C₂³	C₂²×D₅₂	1	1	yes	characteristic
160	Q₈	#88057	1	16	yes	direct product factor
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	26	4	no	—
163	C₂×C₄	—	52	2	no	—
164	C₂×C₄	—	52	2	no	—
165	C₂×C₄	—	52	6	no	—
166	C₂×C₄	—	52	6	no	—
167	C₂×C₄	—	52	6	no	—
168	C₂×C₄	C₂×#6095	1	12	yes	—
169	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
171	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
172	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
173	C₂×C₄	C₂×#6131	1	1	yes	characteristic
174	C₂×C₄	#88256	1	1	yes	characteristic
175	C₂×C₄	#88257	1	1	yes	characteristic
176	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
177	C₈	—	2	6	no	—
178	C₈	Q₈×Dic₁₃	1	4	yes	complemented
179	C₂²	Q₈×D₅₂	1	1	yes	characteristic
180	C₂²	#11719282	1	1	yes	characteristic
181	C₂²	C₂²×#6095	1	2	yes	—
182	C₂²	#11719456	1	2	yes	—
183	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
184	C₄	—	2	6	no	—
185	C₄	—	52	4	no	—
186	C₄	—	104	6	no	—
187	C₄	C₂×#88057	1	12	yes	—
188	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
189	C₄	#11719275	1	1	yes	characteristic
190	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
191	C₄	#11719532	1	1	yes	characteristic
192	C₂	Q₈×C₄⋊Dic₁₃	1	1	yes	characteristic
193	C₂	C₂²×#88057	1	1	yes	characteristic
194	C₂	#13123431	1	1	yes	characteristic
195	C₂	Q₈×#6095	1	2	yes	—
196	C₂	#13127818	1	2	yes	—

	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×#106
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. #88057	Q₈	direct (N × H)
67. Q₈×C₁₃	#106	semidirect
85. C₈×C₁₃	C₄×Q₈	semidirect
88. C₈×Q₈	Dic₁₃	semidirect
158. C₁₃	Q₈×#106	semidirect
160. Q₈	#88057	direct (N × H)
178. 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
#13115600	2
#13114993	2
C₄×#88057	2
C₂×C₈×Q₈×C₁₃	2
Q₈×#106	13

Subgroups — Q8×#88057 #15881746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88057 #15881746