Subgroups — #11772661

Order 1152.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₃⋊C₃₂)	C₂	1	2	yes	maximal
3	#67508	—	3	4	no	maximal
4	C₂×C₃²×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
6	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
7	C₃⋊(C₃⋊C₃₂)	C₄	1	4	yes	complemented
8	C₃×C₄×C₁₆	D₃	1	4	yes	—
9	C₂×C₃⋊C₃₂	—	3	8	no	—
10	C₂×C₃²×C₈	C₈	1	1	yes	characteristic
11	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃²×C₄²	C₈	1	1	yes	characteristic
13	C₃²×C₁₆	C₈	1	2	yes	—
14	C₃²×C₁₆	C₂×C₄	1	2	yes	—
15	#1397	—	9	1	no	Sylow
16	C₂×C₃×C₁₆	Dic₃	1	4	yes	—
17	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
18	C₃×C₄×C₈	Dic₃	1	4	yes	—
19	C₃⋊C₃₂	—	3	16	no	—
20	C₂×C₃²×C₄	C₁₆	1	2	yes	—
21	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
22	C₃²×C₈	C₁₆	1	2	yes	—
23	C₃²×C₈	C₄²	1	1	yes	characteristic
24	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
25	C₂×C₃₂	—	9	2	no	—
26	C₄×C₁₆	C₃⋊D₃	1	1	yes	characteristic
27	C₂×C₃×C₈	C₃⋊C₈	1	4	yes	—
28	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
29	C₃×C₄²	C₃⋊C₈	1	4	yes	—
30	C₃×C₁₆	C₃⋊C₈	1	8	yes	—
31	C₃×C₁₆	C₄×D₃	1	8	yes	—
32	C₂²×C₃²	C₂×C₁₆	1	1	yes	characteristic
33	C₃²×C₄	—	2	2	no	—
34	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
35	C₃²×C₄	C₂×C₁₆	1	1	yes	characteristic
36	C₂×C₁₆	C₃⋊Dic₃	1	1	yes	characteristic
37	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
38	C₄×C₈	C₃⋊Dic₃	1	1	yes	characteristic
39	C₃₂	—	9	4	no	—
40	C₂×C₃×C₄	C₃⋊C₁₆	1	8	yes	—
41	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
42	C₃×C₈	C₃⋊C₁₆	1	8	yes	—
43	C₃×C₈	C₂×C₃⋊C₈	1	4	yes	—
44	C₃×C₈	C₄×Dic₃	1	4	yes	—
45	C₂×C₃²	C₄×C₁₆	1	1	yes	characteristic
46	C₂×C₃²	M₆₄	1	2	yes	—
47	C₂×C₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
48	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
49	C₄²	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
50	C₁₆	C₃⋊(C₃⋊C₈)	1	2	yes	—
51	C₁₆	C₄×C₃⋊D₃	1	2	yes	—
52	C₂²×C₃	C₂×C₃⋊C₁₆	1	4	yes	—
53	C₃×C₄	—	2	8	no	—
54	C₃×C₄	C₄×C₃⋊C₈	1	4	yes	—
55	C₃×C₄	C₂×C₃⋊C₁₆	1	4	yes	—
56	C₃²	#1397	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄	C₃⋊(C₃⋊C₁₆)	1	2	yes	—
58	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
59	C₈	C₃⋊(C₃⋊C₁₆)	1	2	yes	—
60	C₈	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
61	C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
62	C₂×C₃	C₄×C₃⋊C₁₆	1	4	yes	—
63	C₂×C₃	#4499	1	8	yes	—
64	C₂²	C₂×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
67	C₄	C₂×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
68	C₃	#67508	1	4	yes	complemented
69	C₂	C₄×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
70	C₂	#10589365	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
7. C₃⋊(C₃⋊C₃₂)	C₄	semidirect
56. C₃²	#1397	semidirect
68. C₃	#67508	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₁₆	2
C₂×C₃⋊(C₃⋊C₃₂)	2
#67508	3