Subgroups — #69393

Order 384.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
2	#4524	C₂	1	6	yes	maximal
3	#1839	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₄²	C₄	1	3	yes	—
5	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄²	C₂²	1	3	yes	—
7	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
8	C₄³	D₃	1	1	yes	characteristic
9	#336	—	3	6	no	—
10	C₂²×C₃×C₄	C₂×C₄	1	3	yes	—
11	C₂²×C₃×C₄	C₂×C₄	1	3	yes	—
12	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
13	C₃×C₄²	—	2	6	no	—
14	C₃×C₄²	—	2	6	no	—
15	C₃×C₄²	C₈	1	4	yes	complemented
16	C₂×C₃⋊C₈	—	2	12	no	—
17	C₂²×C₈	—	3	4	no	—
18	C₂×C₄²	Dic₃	1	3	yes	—
19	C₂×C₄²	Dic₃	1	1	yes	characteristic
20	C₂×C₄²	C₂×D₃	1	3	yes	—
21	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	—	2	3	no	—
23	C₂×C₃×C₄	—	2	3	no	—
24	C₂×C₃×C₄	—	2	3	no	—
25	C₂×C₃×C₄	—	2	3	no	—
26	C₂×C₃×C₄	C₂×C₈	1	6	yes	—
27	C₂×C₃×C₄	M₁₆	1	6	yes	—
28	C₂×C₃×C₄	#41	1	6	yes	—
29	C₃⋊C₈	—	4	4	no	—
30	C₂²×C₄	C₂×Dic₃	1	3	yes	—
31	C₂²×C₄	C₂×Dic₃	1	3	yes	—
32	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
33	C₂×C₈	—	6	12	no	—
34	C₄²	—	2	6	no	—
35	C₄²	—	2	6	no	—
36	C₄²	C₃⋊C₈	1	4	yes	complemented
37	C₂²×C₃	#117	1	3	yes	—
38	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
39	C₂²×C₃	C₂×M₁₆	1	3	yes	—
40	C₃×C₄	—	2	6	no	—
41	C₃×C₄	—	2	6	no	—
42	C₃×C₄	#126	1	4	yes	—
43	C₂³	C₂²×Dic₃	1	1	yes	characteristic
44	C₂×C₄	—	2	3	no	—
45	C₂×C₄	—	2	3	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	2	3	no	—
48	C₂×C₄	C₂×C₃⋊C₈	1	6	yes	—
49	C₂×C₄	#208	1	6	yes	—
50	C₂×C₄	#235	1	6	yes	—
51	C₈	—	12	4	no	—
52	C₂×C₃	#383	1	1	yes	characteristic
53	C₂×C₃	#431	1	3	yes	—
54	C₂×C₃	#432	1	3	yes	—
55	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
56	C₂²	C₂×#208	1	3	yes	—
57	C₂²	#926	1	3	yes	—
58	C₄	—	2	6	no	—
59	C₄	—	2	6	no	—
60	C₄	#876	1	4	yes	—
61	C₃	#1839	1	1	yes	characteristic, complemented, Sylow
62	C₂	#4925	1	3	yes	—
63	C₂	#4926	1	3	yes	—
64	C₂	#4937	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₈
Fitting	C₃×C₄³	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄³	D₃
hypocenter	C₃	#1839
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
15. C₃×C₄²	C₈	semidirect
36. C₄²	C₃⋊C₈	semidirect
61. C₃	#1839	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄³	2
#4524	2
#1839	3