Subgroups — Q₈×C₃⋊(C₃⋊C₁₆) #11797996

Order 1152.

402 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	3	yes	maximal
3	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	3	yes	maximal
4	Q₈×C₃⋊C₁₆	—	3	4	no	maximal
5	C₃²×C₄×Q₈	C₄	1	1	yes	characteristic
6	C₃²×C₄⋊C₈	C₄	1	3	yes	—
7	C₃²×C₄×C₈	C₂²	1	3	yes	—
8	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	3	yes	—
9	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
10	C₃×C₈×Q₈	D₃	1	4	yes	—
11	C₄⋊(C₃⋊C₁₆)	—	3	12	no	—
12	C₄×C₃⋊C₁₆	—	3	12	no	—
13	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
14	C₂×C₃²×C₈	C₂×C₄	1	3	yes	—
15	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
16	C₃²×C₄⋊C₄	C₈	1	3	yes	—
17	C₃²×C₄²	C₂×C₄	1	3	yes	—
18	C₃⋊(C₃⋊C₁₆)	—	2	3	no	—
19	C₃⋊(C₃⋊C₁₆)	Q₈	1	2	yes	direct product factor
20	Q₈×C₁₆	—	9	1	no	Sylow
21	C₃×C₄×Q₈	Dic₃	1	4	yes	—
22	C₃×C₄⋊C₈	Dic₃	1	12	yes	—
23	C₃×C₄×C₈	C₂×D₃	1	12	yes	—
24	C₂×C₃⋊C₁₆	—	3	4	no	—
25	C₂×C₃⋊C₁₆	—	3	12	no	—
26	C₃²×Q₈	C₁₆	1	4	yes	complemented
27	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
28	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
29	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
30	C₃²×C₈	—	2	3	no	—
31	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
32	C₃²×C₈	#41	1	1	yes	characteristic
33	C₈×Q₈	C₃⋊D₃	1	1	yes	characteristic
34	C₄⋊C₁₆	—	9	3	no	—
35	C₄×C₁₆	—	9	3	no	—
36	C₂×C₃×Q₈	C₃⋊C₈	1	4	yes	—
37	C₂×C₃×C₈	C₂×Dic₃	1	12	yes	—
38	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
39	C₃×C₄⋊C₄	C₃⋊C₈	1	12	yes	—
40	C₃×C₄²	C₂×Dic₃	1	12	yes	—
41	C₃⋊C₁₆	—	3	8	no	—
42	C₃⋊C₁₆	—	6	12	no	—
43	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
44	C₃²×C₄	—	2	3	no	—
45	C₃²×C₄	C₂×C₁₆	1	6	yes	—
46	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
47	C₃²×C₄	#131	1	1	yes	characteristic
48	C₄×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
49	C₂×C₁₆	—	9	1	no	—
50	C₂×C₁₆	—	9	3	no	—
51	C₄⋊C₈	C₃⋊Dic₃	1	3	yes	—
52	C₄×C₈	C₂×C₃⋊D₃	1	3	yes	—
53	C₃×Q₈	C₃⋊C₁₆	1	16	yes	complemented
54	C₂×C₃×C₄	C₂×C₃⋊C₈	1	12	yes	—
55	C₂×C₃×C₄	C₂×C₃⋊C₈	1	12	yes	—
56	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
57	C₃×C₈	—	2	12	no	—
58	C₃×C₈	D₃×Q₈	1	4	yes	—
59	C₃×C₈	#239	1	4	yes	—
60	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
61	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
62	C₂×C₃²	#504	1	1	yes	characteristic
63	C₂×Q₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
64	C₂×C₈	C₂×C₃⋊Dic₃	1	3	yes	—
65	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
66	C₄⋊C₄	C₃⋊(C₃⋊C₈)	1	3	yes	—
67	C₄²	C₂×C₃⋊Dic₃	1	3	yes	—
68	C₁₆	—	9	2	no	—
69	C₁₆	—	18	3	no	—
70	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
71	C₃×C₄	—	2	12	no	—
72	C₃×C₄	C₂×C₃⋊C₁₆	1	24	yes	—
73	C₃×C₄	Q₈×Dic₃	1	4	yes	—
74	C₃×C₄	#948	1	4	yes	—
75	C₃²	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
76	Q₈	C₃⋊(C₃⋊C₁₆)	1	4	yes	direct product factor
77	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
78	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
79	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
80	C₈	—	2	3	no	—
81	C₈	Q₈×C₃⋊D₃	1	1	yes	characteristic
82	C₈	#3832	1	1	yes	characteristic
83	C₂×C₃	Q₈×C₃⋊C₈	1	4	yes	—
84	C₂×C₃	C₂²×C₃⋊C₁₆	1	4	yes	—
85	C₂×C₃	#5140	1	4	yes	—
86	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
87	C₄	—	2	3	no	—
88	C₄	C₂×C₃⋊(C₃⋊C₁₆)	1	6	yes	—
89	C₄	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
90	C₄	#64157	1	1	yes	characteristic
91	C₃	Q₈×C₃⋊C₁₆	1	4	yes	complemented
92	C₂	Q₈×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
93	C₂	C₂²×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
94	C₂	#10592839	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₁₆
Fitting	C₃²×C₈×Q₈	C₂
socle, prime radical	C₂²×C₃²	C₂²×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₈×Q₈	C₃⋊D₃
hypocenter	C₃²	Q₈×C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
19. C₃⋊(C₃⋊C₁₆)	Q₈	direct (N × H)
26. C₃²×Q₈	C₁₆	semidirect
53. C₃×Q₈	C₃⋊C₁₆	semidirect
75. C₃²	Q₈×C₁₆	semidirect
76. Q₈	C₃⋊(C₃⋊C₁₆)	direct (N × H)
91. C₃	Q₈×C₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₈×Q₈	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
C₄×C₃⋊(C₃⋊C₁₆)	2
Q₈×C₃⋊C₁₆	3

Subgroups — Q8×C3⋊(C3⋊C16) #11797996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃⋊(C₃⋊C₁₆) #11797996