Subgroups — #69389

Order 384.

272 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	C₂×#803	C₂	1	2	yes	maximal
3	#4524	C₂	1	4	yes	maximal
4	#1745	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₄²	C₄	1	4	yes	—
6	C₂×C₃×C₄²	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
9	#803	C₄	1	8	yes	complemented
10	C₂×#97	—	3	2	no	—
11	C₄³	D₃	1	1	yes	characteristic
12	#336	—	3	4	no	—
13	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
14	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
15	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
16	C₃×C₄²	—	2	8	no	—
17	C₃×C₄²	—	2	4	no	—
18	C₃×C₄²	C₂×C₄	1	4	yes	—
19	C₂×C₃⋊C₈	—	2	8	no	—
20	C₂×C₃⋊C₈	C₂×C₄	1	8	yes	—
21	C₂²×C₈	—	3	4	no	—
22	C₂×C₄²	Dic₃	1	4	yes	—
23	C₂×C₄²	C₂×D₃	1	2	yes	—
24	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
25	#97	—	3	8	no	—
26	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄	—	2	4	no	—
28	C₂×C₃×C₄	—	2	4	no	—
29	C₂×C₃×C₄	—	2	4	no	—
30	C₂×C₃×C₄	C₄²	1	4	yes	—
31	C₂×C₃×C₄	M₁₆	1	8	yes	—
32	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
33	C₂×C₃×C₄	#41	1	4	yes	—
34	C₃⋊C₈	—	2	8	no	—
35	C₂²×C₄	C₂×Dic₃	1	4	yes	—
36	C₂²×C₄	C₂×Dic₃	1	2	yes	—
37	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
38	C₂×C₈	—	3	8	no	—
39	C₂×C₈	—	6	8	no	—
40	C₄²	—	2	8	no	—
41	C₄²	—	2	4	no	—
42	C₄²	C₄×D₃	1	4	yes	—
43	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₃	#117	1	2	yes	—
45	C₂²×C₃	C₂×M₁₆	1	4	yes	—
46	C₃×C₄	—	2	4	no	—
47	C₃×C₄	—	2	8	no	—
48	C₃×C₄	#117	1	4	yes	—
49	C₂³	C₂²×Dic₃	1	1	yes	characteristic
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	#208	1	8	yes	—
54	C₂×C₄	C₄×Dic₃	1	4	yes	—
55	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
56	C₂×C₄	#235	1	4	yes	—
57	C₈	—	6	8	no	—
58	C₂×C₃	#376	1	1	yes	characteristic
59	C₂×C₃	C₄×M₁₆	1	2	yes	—
60	C₂×C₃	#432	1	4	yes	—
61	C₂²	C₂×#208	1	4	yes	—
62	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
63	C₂²	#926	1	2	yes	—
64	C₄	—	2	4	no	—
65	C₄	—	2	8	no	—
66	C₄	#872	1	4	yes	—
67	C₃	#1745	1	1	yes	characteristic, complemented, Sylow
68	C₂	C₄×#208	1	2	yes	—
69	C₂	#4926	1	4	yes	—
70	C₂	#4932	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₃×C₄³	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄³	D₃
hypocenter	C₃	#1745
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
9. #803	C₄	semidirect
67. C₃	#1745	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
C₂×#803	2
#4524	2
#1745	3