Subgroups — #11772658

Order 1152.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊(C₃⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#67589	—	3	4	no	maximal
5	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
6	C₈×C₃⋊Dic₃	C₄	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
8	C₃²×C₃₂	C₄	1	2	yes	complemented
9	C₃⋊(C₃⋊C₃₂)	C₄	1	2	yes	complemented
10	C₂×C₃×C₃₂	D₃	1	4	yes	—
11	Dic₃×C₁₆	—	3	4	no	—
12	C₂×C₃⋊C₃₂	—	3	4	no	—
13	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
14	C₄×C₃⋊Dic₃	C₈	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₈)	C₈	1	1	yes	characteristic
16	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₃⋊(C₃⋊C₁₆)	C₈	1	2	yes	—
19	#1397	—	9	1	no	Sylow
20	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
21	C₈×Dic₃	—	3	4	no	—
22	C₂×C₃⋊C₁₆	—	3	4	no	—
23	C₃×C₃₂	Dic₃	1	8	yes	complemented
24	C₃⋊C₃₂	—	3	8	no	—
25	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
26	C₂×C₃⋊Dic₃	C₁₆	1	2	yes	—
27	C₃²×C₈	C₄²	1	1	yes	characteristic
28	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
29	C₃⋊(C₃⋊C₈)	C₁₆	1	2	yes	—
30	C₂×C₃₂	—	9	1	no	—
31	C₂×C₃₂	C₃⋊D₃	1	1	yes	characteristic
32	C₄×C₁₆	—	9	1	no	—
33	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
34	C₄×Dic₃	—	3	4	no	—
35	C₂×C₃⋊C₈	—	3	4	no	—
36	C₃×C₁₆	C₄×D₃	1	4	yes	—
37	C₃×C₁₆	C₂×Dic₃	1	4	yes	—
38	C₃⋊C₁₆	—	3	8	no	—
39	C₂²×C₃²	C₂×C₁₆	1	1	yes	characteristic
40	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
41	C₃²×C₄	C₂×C₁₆	1	1	yes	characteristic
42	C₃⋊Dic₃	—	2	2	no	—
43	C₂×C₁₆	—	9	1	no	—
44	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
45	C₄×C₈	—	9	1	no	—
46	C₃₂	—	9	2	no	—
47	C₃₂	C₃⋊Dic₃	1	2	yes	complemented
48	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
49	C₂×Dic₃	—	3	8	no	—
50	C₃×C₈	D₃×C₈	1	4	yes	—
51	C₃×C₈	C₄×Dic₃	1	4	yes	—
52	C₃⋊C₈	—	3	8	no	—
53	C₂×C₃²	C₄×C₁₆	1	1	yes	characteristic
54	C₂×C₃²	M₆₄	1	2	yes	—
55	C₂×C₈	—	9	1	no	—
56	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
57	C₄²	—	9	1	no	—
58	C₁₆	—	9	2	no	—
59	C₁₆	C₄×C₃⋊D₃	1	1	yes	characteristic
60	C₁₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
61	C₂²×C₃	D₃×C₁₆	1	4	yes	—
62	C₃×C₄	D₃×C₁₆	1	4	yes	—
63	C₃×C₄	C₈×Dic₃	1	4	yes	—
64	Dic₃	—	6	8	no	—
65	C₃²	#1397	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄	—	9	2	no	—
67	C₂×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
68	C₈	—	9	2	no	—
69	C₈	C₈×C₃⋊D₃	1	1	yes	characteristic
70	C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
71	C₂×C₃	#4447	1	8	yes	—
72	C₂×C₃	Dic₃×C₁₆	1	4	yes	—
73	C₂²	C₁₆×C₃⋊D₃	1	1	yes	characteristic
74	C₄	—	18	2	no	—
75	C₄	C₁₆×C₃⋊D₃	1	1	yes	characteristic
76	C₄	C₈×C₃⋊Dic₃	1	1	yes	characteristic
77	C₃	#67589	1	4	yes	complemented
78	C₂	#10589313	1	2	yes	—
79	C₂	C₁₆×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₄×C₁₆
Fitting	C₂×C₃²×C₃₂	C₂
socle, prime radical	C₂²×C₃²	C₂×C₁₆
cosocle	C₂×C₃²×C₁₆	C₂²
hypercenter	C₂×C₃₂	C₃⋊D₃
hypocenter	C₃²	#1397
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
8. C₃²×C₃₂	C₄	semidirect
9. C₃⋊(C₃⋊C₃₂)	C₄	semidirect
23. C₃×C₃₂	Dic₃	semidirect
47. C₃₂	C₃⋊Dic₃	semidirect
65. C₃²	#1397	semidirect
77. C₃	#67589	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₃₂	2
C₁₆×C₃⋊Dic₃	2
C₂×C₃⋊(C₃⋊C₃₂)	2
#67589	3