Subgroups — #67806

Order 384.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#818	C₂	1	2	yes	maximal
3	#1380	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊Dic₃	—	2	2	no	—
6	C₂×C₃×C₁₆	—	2	2	no	—
7	C₂×C₃×C₁₆	C₄	1	2	yes	complemented
8	#818	—	2	2	no	—
9	#818	C₄	1	4	yes	complemented
10	C₂²×C₁₆	D₃	1	1	yes	characteristic
11	C₂×#107	—	3	2	no	—
12	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
13	C₂²×Dic₃	—	4	2	no	—
14	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₈	D₄	1	2	yes	—
17	C₂×C₃×C₈	Q₈	1	1	yes	characteristic
18	C₄⋊Dic₃	—	2	4	no	—
19	C₄⋊Dic₃	—	4	2	no	—
20	C₃×C₁₆	—	2	2	no	—
21	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
22	C₂×C₄⋊C₄	—	6	2	no	—
23	C₂×C₁₆	—	2	2	no	—
24	C₂×C₁₆	Dic₃	1	2	yes	complemented
25	#107	—	3	4	no	—
26	#107	—	6	2	no	—
27	C₂³×C₃	D₈	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
29	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₃×C₄	SD₁₆	1	2	yes	—
31	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
32	C₂×Dic₃	—	4	4	no	—
33	C₂×Dic₃	—	8	2	no	—
34	C₂×Dic₃	—	8	2	no	—
35	C₃×C₈	C₄²	1	1	yes	characteristic
36	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₃×C₈	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄	—	12	2	no	—
39	C₂²×C₄	D₁₂	1	1	yes	characteristic
40	C₂×C₈	Dic₆	1	1	yes	characteristic
41	C₂×C₈	C₄×D₃	1	2	yes	—
42	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
43	C₂×C₈	#67	1	2	yes	—
44	C₄⋊C₄	—	6	4	no	—
45	C₄⋊C₄	—	12	2	no	—
46	C₁₆	—	2	2	no	—
47	C₂²×C₃	#102	1	2	yes	—
48	C₂²×C₃	#107	1	1	yes	characteristic
49	C₂²×C₃	D₁₆	1	1	yes	characteristic
50	C₂²×C₃	SD₃₂	1	2	yes	—
51	C₂²×C₃	Q₃₂	1	1	yes	characteristic
52	C₃×C₄	#95	1	1	yes	characteristic
53	C₃×C₄	#103	1	2	yes	—
54	C₃×C₄	#106	1	1	yes	characteristic
55	Dic₃	—	8	4	no	—
56	C₂³	D₂₄	1	1	yes	characteristic
57	C₂×C₄	—	12	4	no	—
58	C₂×C₄	—	24	2	no	—
59	C₂×C₄	—	24	2	no	—
60	C₂×C₄	#204	1	2	yes	—
61	C₂×C₄	Dic₁₂	1	1	yes	characteristic
62	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
63	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
64	C₈	C₄×Dic₃	1	1	yes	characteristic
65	C₈	Dic₃⋊C₄	1	2	yes	—
66	C₈	#217	1	1	yes	characteristic
67	C₂×C₃	#340	1	1	yes	characteristic
68	C₂×C₃	#357	1	2	yes	—
69	C₂×C₃	#358	1	2	yes	—
70	C₂×C₃	#366	1	1	yes	characteristic
71	C₂×C₃	#367	1	1	yes	characteristic
72	C₂²	D₄₈	1	1	yes	characteristic
73	C₂²	#800	1	2	yes	—
74	C₂²	Dic₂₄	1	1	yes	characteristic
75	C₂²	#818	1	1	yes	characteristic
76	C₂²	#821	1	2	yes	—
77	C₄	—	24	4	no	—
78	C₄	#816	1	2	yes	—
79	C₄	#817	1	1	yes	characteristic
80	C₄	C₃⋊#95	1	1	yes	characteristic
81	C₃	#1380	1	1	yes	characteristic, complemented, Sylow
82	C₂	#4503	1	2	yes	—
83	C₂	#4504	1	1	yes	characteristic
84	C₂	#4505	1	1	yes	characteristic
85	C₂	#4509	1	2	yes	—
86	C₂	#4551	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₂₄
commutator	C₃×C₈	C₄²
Frattini	C₂²×C₈	C₂×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle, prime radical	C₂³×C₃	D₈
cosocle	C₂²×C₃×C₈	C₂²
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	#1380
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
7. C₂×C₃×C₁₆	C₄	semidirect
9. #818	C₄	semidirect
24. C₂×C₁₆	Dic₃	semidirect
81. C₃	#1380	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₁₆	2
C₂×#818	2
#1380	3