Subgroups — #11772020

Order 1152.

276 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#494	C₂	1	1	yes	maximal, characteristic
2	#10589329	C₂	1	2	yes	maximal
3	#67794	—	3	4	no	maximal
4	C₂×C₃²×Q₁₆	—	2	2	no	—
5	C₃²×C₄⋊Q₈	C₄	1	2	yes	—
6	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
7	#63639	—	2	2	no	—
8	C₃×#494	D₃	1	4	yes	—
9	#4463	—	3	8	no	—
10	C₂×C₃²×Q₈	—	2	2	no	—
11	C₃²×Q₁₆	—	4	2	no	—
12	C₃²×Q₁₆	—	4	2	no	—
13	C₂×C₃²×C₈	D₄	1	2	yes	—
14	C₃²×C₄⋊C₄	—	4	2	no	—
15	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₃⋊(C₃⋊C₁₆)	—	4	2	no	—
17	#1358	—	9	1	no	Sylow
18	C₂×C₃×Q₁₆	—	2	8	no	—
19	C₃×C₄⋊Q₈	Dic₃	1	8	yes	—
20	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
21	#812	—	6	8	no	—
22	C₃²×Q₈	—	4	4	no	—
23	C₂×C₃²×C₄	—	2	1	no	—
24	C₂×C₃²×C₄	—	4	2	no	—
25	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
26	C₃²×C₈	D₈	1	2	yes	—
27	C₃²×C₈	SD₁₆	1	2	yes	—
28	#494	C₃⋊D₃	1	1	yes	characteristic
29	#364	—	9	2	no	—
30	C₂×C₃×Q₈	—	2	8	no	—
31	C₃×Q₁₆	—	4	8	no	—
32	C₃×Q₁₆	—	4	8	no	—
33	C₂×C₃×C₈	#67	1	8	yes	—
34	C₃×C₄⋊C₄	—	4	8	no	—
35	C₃×C₄²	C₂×Dic₃	1	4	yes	—
36	C₃⋊C₁₆	—	12	8	no	—
37	C₂²×C₃²	#100	1	1	yes	characteristic
38	C₃²×C₄	—	2	2	no	—
39	C₃²×C₄	—	8	2	no	—
40	C₃²×C₄	#102	1	2	yes	—
41	C₂×Q₁₆	—	2	2	no	—
42	C₄⋊Q₈	C₃⋊Dic₃	1	2	yes	—
43	M₃₂	—	18	2	no	—
44	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
45	C₃×Q₈	—	4	16	no	—
46	C₂×C₃×C₄	—	2	4	no	—
47	C₂×C₃×C₄	—	4	8	no	—
48	C₂×C₃×C₄	#217	1	4	yes	—
49	C₃×C₈	#213	1	8	yes	—
50	C₃×C₈	#214	1	8	yes	—
51	C₂×C₃²	—	2	1	no	—
52	C₂×C₃²	#331	1	1	yes	characteristic
53	C₂×Q₈	—	2	2	no	—
54	Q₁₆	—	4	2	no	—
55	Q₁₆	—	4	2	no	—
56	C₂×C₈	#638	1	2	yes	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
59	C₁₆	—	36	2	no	—
60	C₂²×C₃	#833	1	4	yes	—
61	C₃×C₄	—	2	8	no	—
62	C₃×C₄	—	8	8	no	—
63	C₃×C₄	#832	1	8	yes	—
64	C₃²	#1358	1	1	yes	characteristic, complemented, Sylow
65	Q₈	—	4	4	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	#3757	1	1	yes	characteristic
69	C₈	#3753	1	2	yes	—
70	C₈	#3754	1	2	yes	—
71	C₂×C₃	—	2	4	no	—
72	C₂×C₃	#4544	1	4	yes	—
73	C₂²	#63660	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	8	2	no	—
76	C₄	#63659	1	2	yes	—
77	C₃	#67794	1	4	yes	complemented
78	C₂	—	2	1	no	—
79	C₂	#10589410	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10589410
commutator	C₃²×C₄²	C₂×C₄
Frattini	C₄×C₈	C₂×C₃⋊D₃
Fitting	C₃²×#494	C₂
socle	C₂×C₃²	#331
prime radical	C₂²×C₃²	#100
cosocle	C₃²×C₄×C₈	C₂²
hypercenter	#494	C₃⋊D₃
hypocenter	C₃²	#1358
LCS	G ▷ C₃²×C₄² ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
64. C₃²	#1358	semidirect
77. C₃	#67794	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#494	2
#10589329	2
#67794	3