Subgroups — Q₈×#88052 #16016433

Order 3328.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116745	C₂	1	6	yes	maximal, complemented
2	Q₈×#6090	C₂	1	2	yes	maximal, complemented
3	#13116592	C₂	1	3	yes	maximal
4	C₄×#88052	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₁₃	C₄	1	3	yes	—
8	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₁₃	C₂²	1	3	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	C₂×#88052	C₂²	1	3	yes	—
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	C₄⋊#6090	C₂²	1	6	yes	—
16	C₄⋊#6090	C₂²	1	6	yes	—
17	#11718053	C₂²	1	6	yes	—
18	C₄×#6090	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₃	C₈	1	3	yes	—
23	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×C₄⋊C₄	C₈	1	3	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
28	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
29	C₁₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
30	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₁₃	C₂³	1	3	yes	—
34	C₄×C₈×C₁₃	C₂³	1	3	yes	—
35	#88052	—	2	6	no	—
36	#88052	Q₈	1	4	yes	direct product factor
37	C₂×#6090	C₂³	1	6	yes	—
38	C₂×#6090	C₂³	1	2	yes	—
39	Q₈×M₃₂	—	13	1	no	maximal, Sylow
40	C₂×Q₈×C₁₃	—	2	4	no	—
41	C₂×Q₈×C₁₃	C₂×C₈	1	3	yes	—
42	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	—	2	6	no	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₃	#41	1	2	yes	—
52	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	C₂×C₈	1	6	yes	—
54	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
55	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
56	C₄²×C₁₃	C₂×C₈	1	3	yes	—
57	C₄²×C₁₃	C₂×C₈	1	3	yes	—
58	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
59	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
60	#6090	—	2	6	no	—
61	#6090	C₂×Q₈	1	4	yes	complemented
62	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
63	#2183	—	13	6	no	—
64	Q₈×C₁₆	—	13	2	no	—
65	#2150	—	13	3	no	—
66	C₄×M₃₂	—	13	3	no	—
67	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₁₃	—	2	6	no	—
69	Q₈×C₁₃	M₃₂	1	4	yes	complemented
70	C₂×C₄×C₁₃	—	2	6	no	—
71	C₂×C₄×C₁₃	—	4	3	no	—
72	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₁₃	#131	1	2	yes	—
79	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₁₃	—	2	3	no	—
81	C₈×C₁₃	—	2	3	no	—
82	C₈×C₁₃	C₄×Q₈	1	2	yes	—
83	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
86	C₂×M₃₂	—	13	1	no	—
87	C₂×M₃₂	—	13	3	no	—
88	C₈×Q₈	Dic₁₃	1	3	yes	—
89	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
90	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
91	C₈×Q₈	C₂×D₁₃	1	3	yes	—
92	C₂×C₄⋊C₈	Dic₁₃	1	3	yes	—
93	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
94	C₄⋊C₁₆	—	13	6	no	—
95	C₄⋊C₁₆	—	13	6	no	—
96	#346	—	13	6	no	—
97	C₄×C₁₆	—	13	6	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
101	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
102	C₄×C₁₃	—	2	3	no	—
103	C₄×C₁₃	—	2	3	no	—
104	C₄×C₁₃	—	2	3	no	—
105	C₄×C₁₃	C₈×Q₈	1	2	yes	—
106	C₄×C₁₃	C₂×M₃₂	1	6	yes	—
107	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
109	C₂²×Q₈	#1055	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
111	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
112	C₄×Q₈	#1055	1	3	yes	—
113	C₄×Q₈	#1055	1	1	yes	characteristic
114	C₄×Q₈	C₂×Dic₁₃	1	3	yes	—
115	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	#1055	1	3	yes	—
117	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
118	M₃₂	—	13	4	no	—
119	M₃₂	—	26	6	no	—
120	C₂×C₁₆	—	13	2	no	—
121	C₂×C₁₆	—	13	6	no	—
122	C₄⋊C₈	C₂×Dic₁₃	1	3	yes	—
123	C₄⋊C₈	C₂×Dic₁₃	1	6	yes	—
124	C₄⋊C₈	C₂×Dic₁₃	1	3	yes	—
125	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
126	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
127	C₄×C₈	C₂²×D₁₃	1	3	yes	—
128	C₄×C₈	C₂²×D₁₃	1	3	yes	—
129	C₂×C₁₃	—	2	2	no	—
130	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
131	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
132	C₂×C₁₃	#3407	1	1	yes	characteristic
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	C₂×#1055	1	3	yes	—
135	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
136	C₂²×C₄	C₂×#1055	1	3	yes	—
137	C₂²×C₄	C₂×#1055	1	3	yes	—
138	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
139	C₂×C₈	—	2	6	no	—
140	C₂×C₈	Q₈×D₁₃	1	2	yes	—
141	C₂×C₈	#6131	1	2	yes	—
142	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
143	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
144	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
145	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
146	C₄⋊C₄	C₂×#1055	1	6	yes	—
147	C₄⋊C₄	C₂×#1055	1	3	yes	—
148	C₄⋊C₄	C₂×#1055	1	3	yes	—
149	C₄²	C₂×#1055	1	3	yes	—
150	C₄²	C₂×#1055	1	3	yes	—
151	C₄²	C₂²×Dic₁₃	1	3	yes	—
152	C₄²	C₂²×Dic₁₃	1	3	yes	—
153	C₁₆	—	13	4	no	—
154	C₁₆	—	26	6	no	—
155	C₁₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
156	C₂³	C₂²×#1055	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	#88052	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	C₂²×#1055	1	3	yes	—
162	C₂×C₄	C₂²×#1055	1	3	yes	—
163	C₂×C₄	C₂²×#1055	1	3	yes	—
164	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
165	C₂×C₄	C₂²×#1055	1	3	yes	—
166	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
167	C₂×C₄	#88202	1	2	yes	—
168	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	2	3	no	—
171	C₈	Q₈×Dic₁₃	1	2	yes	—
172	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
173	C₈	C₂×#6131	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	Q₈×#1055	1	2	yes	—
177	C₂²	C₂³×#1055	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	Q₈×#1055	1	2	yes	—
182	C₄	C₂×#88052	1	6	yes	—
183	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
184	C₄	C₂×#88202	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×Q₈×#1055	1	1	yes	characteristic
187	C₂	C₂²×#88052	1	1	yes	characteristic
188	C₂	#13127939	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₁₃	C₂
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #13116745	C₂	semidirect
2. Q₈×#6090	C₂	semidirect
36. #88052	Q₈	direct (N × H)
61. #6090	C₂×Q₈	semidirect
69. Q₈×C₁₃	M₃₂	semidirect
155. C₁₃	Q₈×M₃₂	semidirect
158. Q₈	#88052	direct (N × H)

Maximal subgroups

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

subgroup	index
#13116745	2
Q₈×#6090	2
#13116592	2
C₄×#88052	2
C₂×C₈×Q₈×C₁₃	2
Q₈×M₃₂	13

Subgroups — Q8×#88052 #16016433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88052 #16016433