Subgroups — #11918512

Order 1152.

4272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#585	C₂	1	1	yes	maximal, characteristic
2	#10596290	C₂	1	15	yes	maximal, complemented
3	C₂×#64157	C₂	1	15	yes	maximal, complemented
4	#87296	—	3	4	no	maximal
5	C₃²×2₋¹⁺⁴	C₄	1	6	yes	—
6	C₃²×H(5,𝔽₂)	C₄	1	10	yes	—
7	C₂×C₃²×#41	C₂²	1	15	yes	—
8	#64157	C₂²	1	20	yes	complemented
9	#64157	C₂²	1	60	yes	complemented
10	C₂×#3748	C₂²	1	45	yes	complemented
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	15	yes	complemented
12	C₃×#585	D₃	1	4	yes	—
13	#5819	—	3	60	no	—
14	C₂×#948	—	3	60	no	—
15	C₃²×#41	C₂×C₄	1	60	yes	—
16	C₃²×#41	C₂³	1	20	yes	—
17	C₂×C₃²×Q₈	C₂×C₄	1	15	yes	—
18	C₂×C₃²×D₄	C₂×C₄	1	45	yes	—
19	C₂²×C₃²×C₄	C₂³	1	15	yes	—
20	#3748	C₂³	1	60	yes	—
21	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	15	yes	—
22	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	45	yes	—
23	#3573	—	9	1	no	Sylow
24	C₃×2₋¹⁺⁴	Dic₃	1	24	yes	—
25	C₃×H(5,𝔽₂)	Dic₃	1	40	yes	—
26	C₂×C₃×#41	C₂×D₃	1	60	yes	—
27	#948	—	3	80	no	—
28	#948	—	3	240	no	—
29	C₂×#208	—	3	180	no	—
30	C₂²×C₃⋊C₈	—	3	60	no	—
31	C₂³×C₃²	C₂²×C₄	1	15	yes	—
32	C₃²×Q₈	C₂²×C₄	1	20	yes	—
33	C₃²×D₄	C₂²×C₄	1	60	yes	—
34	C₂×C₃²×C₄	C₂²×C₄	1	45	yes	—
35	C₂×C₃²×C₄	C₂⁴	1	15	yes	—
36	C₃⋊(C₃⋊C₈)	C₂⁴	1	1	yes	characteristic
37	C₃⋊(C₃⋊C₈)	C₂⁴	1	15	yes	—
38	#585	C₃⋊D₃	1	1	yes	characteristic
39	#568	—	9	15	no	—
40	C₂×#131	—	9	15	no	—
41	C₃×#41	C₂×Dic₃	1	240	yes	—
42	C₃×#41	C₂²×D₃	1	80	yes	—
43	C₂×C₃×Q₈	C₂×Dic₃	1	60	yes	—
44	C₂×C₃×D₄	C₂×Dic₃	1	180	yes	—
45	C₂²×C₃×C₄	C₂²×D₃	1	60	yes	—
46	#208	—	3	240	no	—
47	C₂×C₃⋊C₈	—	3	60	no	—
48	C₂×C₃⋊C₈	—	3	180	no	—
49	C₂²×C₃²	—	4	15	no	—
50	C₂²×C₃²	C₂³×C₄	1	15	yes	—
51	C₃²×C₄	C₂³×C₄	1	15	yes	—
52	C₃²×C₄	C₂⁵	1	1	yes	characteristic
53	2₋¹⁺⁴	C₃⋊Dic₃	1	6	yes	—
54	H(5,𝔽₂)	C₃⋊Dic₃	1	10	yes	—
55	C₂×#41	C₂×C₃⋊D₃	1	15	yes	—
56	#131	—	9	20	no	—
57	#131	—	9	60	no	—
58	C₂×M₁₆	—	9	45	no	—
59	C₂²×C₈	—	9	15	no	—
60	C₂³×C₃	C₂²×Dic₃	1	60	yes	—
61	C₃×Q₈	C₂²×Dic₃	1	80	yes	—
62	C₃×D₄	C₂²×Dic₃	1	240	yes	—
63	C₂×C₃×C₄	C₂²×Dic₃	1	180	yes	—
64	C₂×C₃×C₄	C₂³×D₃	1	60	yes	—
65	C₃⋊C₈	—	3	4	no	—
66	C₃⋊C₈	—	3	60	no	—
67	C₂×C₃²	—	2	15	no	—
68	C₂×C₃²	C₂⁴×C₄	1	1	yes	characteristic
69	#41	C₂×C₃⋊Dic₃	1	60	yes	—
70	#41	C₂²×C₃⋊D₃	1	20	yes	—
71	C₂×Q₈	C₂×C₃⋊Dic₃	1	15	yes	—
72	C₂×D₄	C₂×C₃⋊Dic₃	1	45	yes	—
73	C₂²×C₄	C₂²×C₃⋊D₃	1	15	yes	—
74	M₁₆	—	9	60	no	—
75	C₂×C₈	—	9	15	no	—
76	C₂×C₈	—	9	45	no	—
77	C₂²×C₃	—	4	60	no	—
78	C₂²×C₃	C₂³×Dic₃	1	60	yes	—
79	C₃×C₄	C₂³×Dic₃	1	60	yes	—
80	C₃×C₄	C₂⁴×D₃	1	4	yes	—
81	C₃²	#3573	1	1	yes	characteristic, complemented, Sylow
82	C₂³	C₂²×C₃⋊Dic₃	1	15	yes	—
83	Q₈	C₂²×C₃⋊Dic₃	1	20	yes	—
84	D₄	C₂²×C₃⋊Dic₃	1	60	yes	—
85	C₂×C₄	C₂²×C₃⋊Dic₃	1	45	yes	—
86	C₂×C₄	C₂³×C₃⋊D₃	1	15	yes	—
87	C₈	—	9	1	no	—
88	C₈	—	9	15	no	—
89	C₂×C₃	—	2	60	no	—
90	C₂×C₃	C₂⁴×Dic₃	1	4	yes	—
91	C₂²	—	4	15	no	—
92	C₂²	C₂³×C₃⋊Dic₃	1	15	yes	—
93	C₄	C₂³×C₃⋊Dic₃	1	15	yes	—
94	C₄	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
95	C₃	#87296	1	4	yes	complemented
96	C₂	—	2	15	no	—
97	C₂	C₂⁴×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×C₃⋊D₃
commutator, socle	C₂×C₃²	C₂⁴×C₄
Fitting, prime radical	C₃²×#585	C₂
cosocle	C₃²×C₄	C₂⁵
hypercenter	#585	C₃⋊D₃
hypocenter	C₃²	#3573
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
2. #10596290	C₂	semidirect
3. C₂×#64157	C₂	semidirect
8. #64157	C₂²	semidirect
9. #64157	C₂²	semidirect
10. C₂×#3748	C₂²	semidirect
11. C₂²×C₃⋊(C₃⋊C₈)	C₂²	semidirect
81. C₃²	#3573	semidirect
95. C₃	#87296	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#585	2
#10596290	2
C₂×#64157	2
#87296	3