Subgroups — #69388

Order 384.

296 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×C₄×C₃⋊C₈	C₂	1	2	yes	maximal
3	#4524	C₂	1	4	yes	maximal
4	#1744	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₄²	C₄	1	2	yes	—
6	C₂×C₃×C₄²	C₄	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
10	C₄×C₃⋊C₈	C₄	1	8	yes	complemented
11	C₂×C₄×C₈	—	3	2	no	—
12	C₄³	D₃	1	1	yes	characteristic
13	#336	—	3	4	no	—
14	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
15	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
18	C₃×C₄²	—	2	4	no	—
19	C₃×C₄²	—	2	4	no	—
20	C₃×C₄²	C₈	1	8	yes	complemented
21	C₃×C₄²	C₂×C₄	1	4	yes	—
22	C₂×C₃⋊C₈	—	2	8	no	—
23	C₂×C₃⋊C₈	C₂×C₄	1	8	yes	—
24	C₂²×C₈	—	3	4	no	—
25	C₂×C₄²	Dic₃	1	2	yes	—
26	C₂×C₄²	Dic₃	1	2	yes	—
27	C₂×C₄²	C₂×D₃	1	2	yes	—
28	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
29	C₄×C₈	—	3	8	no	—
30	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄	—	2	4	no	—
32	C₂×C₃×C₄	—	2	4	no	—
33	C₂×C₃×C₄	C₄²	1	2	yes	—
34	C₂×C₃×C₄	C₄²	1	2	yes	—
35	C₂×C₃×C₄	C₂×C₈	1	8	yes	—
36	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
37	C₂×C₃×C₄	M₁₆	1	4	yes	—
38	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
39	C₂×C₃×C₄	#41	1	4	yes	—
40	C₃⋊C₈	—	2	8	no	—
41	C₂²×C₄	C₂×Dic₃	1	4	yes	—
42	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
43	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
44	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
45	C₂×C₈	—	3	8	no	—
46	C₂×C₈	—	6	8	no	—
47	C₄²	—	2	4	no	—
48	C₄²	—	2	4	no	—
49	C₄²	C₃⋊C₈	1	8	yes	complemented
50	C₄²	C₄×D₃	1	4	yes	—
51	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₃	#117	1	2	yes	—
53	C₂²×C₃	C₂²×C₈	1	2	yes	—
54	C₂²×C₃	C₂×M₁₆	1	2	yes	—
55	C₃×C₄	—	2	8	no	—
56	C₃×C₄	C₄×C₈	1	4	yes	—
57	C₃×C₄	#97	1	4	yes	—
58	C₃×C₄	#117	1	4	yes	—
59	C₂³	C₂²×Dic₃	1	1	yes	characteristic
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	C₂×C₃⋊C₈	1	8	yes	—
63	C₂×C₄	C₂×C₃⋊C₈	1	4	yes	—
64	C₂×C₄	#208	1	4	yes	—
65	C₂×C₄	C₄×Dic₃	1	2	yes	—
66	C₂×C₄	C₄×Dic₃	1	2	yes	—
67	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
68	C₂×C₄	#235	1	4	yes	—
69	C₈	—	6	8	no	—
70	C₂×C₃	#376	1	1	yes	characteristic
71	C₂×C₃	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂×C₃	C₂×#97	1	1	yes	characteristic
73	C₂×C₃	#431	1	4	yes	—
74	C₂²	C₂²×C₃⋊C₈	1	2	yes	—
75	C₂²	C₂×#208	1	2	yes	—
76	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
77	C₂²	#926	1	2	yes	—
78	C₄	—	2	8	no	—
79	C₄	C₄×C₃⋊C₈	1	4	yes	—
80	C₄	#803	1	4	yes	—
81	C₄	#872	1	4	yes	—
82	C₃	#1744	1	1	yes	characteristic, complemented, Sylow
83	C₂	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
84	C₂	C₂×#803	1	1	yes	characteristic
85	C₂	#4925	1	4	yes	—
86	C₂	#4932	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×C₄³	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄³	D₃
hypocenter	C₃	#1744
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
10. C₄×C₃⋊C₈	C₄	semidirect
20. C₃×C₄²	C₈	semidirect
49. C₄²	C₃⋊C₈	semidirect
82. C₃	#1744	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×C₄×C₃⋊C₈	2
#4524	2
#1744	3