Subgroups — #69325

Order 384.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#4880	C₂	1	2	yes	maximal
3	C₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
4	#4504	C₂	1	1	yes	maximal, characteristic, complemented
5	#4490	C₂	1	2	yes	maximal, complemented
6	#2158	—	3	1	no	maximal, Sylow
7	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
8	#892	—	2	2	no	—
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×#107	C₂²	1	2	yes	—
11	#818	C₂²	1	2	yes	—
12	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#808	—	2	2	no	—
14	D₃×C₁₆	C₄	1	4	yes	complemented
15	C₂²×C₁₆	—	3	1	no	—
16	#427	—	3	2	no	—
17	#367	—	3	2	no	—
18	#366	—	3	1	no	—
19	#366	D₃	1	1	yes	characteristic, complemented
20	C₂×C₄×D₃	D₄	1	1	yes	characteristic
21	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
22	C₃×C₄⋊C₄	—	2	2	no	—
23	(C₂×D₃)⋊C₄	—	4	2	no	—
24	C₄⋊Dic₃	—	2	2	no	—
25	C₄×Dic₃	—	4	2	no	—
26	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
27	D₃×C₈	C₂×C₄	1	2	yes	—
28	D₃×C₈	Q₈	1	2	yes	—
29	C₃×C₁₆	C₂×C₄	1	2	yes	—
30	C₃⋊C₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈	—	3	1	no	—
32	#117	—	6	2	no	—
33	C₂×C₁₆	—	3	1	no	—
34	C₂×C₁₆	—	3	4	no	—
35	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
36	#107	—	3	2	no	—
37	#107	C₂×D₃	1	2	yes	—
38	#106	—	6	2	no	—
39	C₂²×D₃	D₈	1	1	yes	characteristic
40	C₂×C₃×C₄	—	4	2	no	—
41	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₃	—	4	2	no	—
43	C₂×Dic₃	D₈	1	1	yes	characteristic
44	C₄×D₃	C₄⋊C₄	1	2	yes	—
45	C₄×D₃	Q₁₆	1	2	yes	—
46	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₃⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄	—	3	1	no	—
50	C₂×C₈	—	3	1	no	—
51	C₂×C₈	—	3	2	no	—
52	C₂×C₈	—	3	2	no	—
53	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	6	2	no	—
56	C₂²⋊C₄	—	12	2	no	—
57	C₄²	—	12	2	no	—
58	C₁₆	—	3	2	no	—
59	C₁₆	C₄×D₃	1	2	yes	—
60	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
61	C₂×D₃	—	2	1	no	—
62	C₂×D₃	—	2	1	no	—
63	C₂×D₃	#107	1	2	yes	—
64	C₃×C₄	—	8	2	no	—
65	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
67	Dic₃	—	8	2	no	—
68	Dic₃	#107	1	2	yes	—
69	C₂³	—	3	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	3	2	no	—
72	C₂×C₄	—	3	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	12	2	no	—
75	C₂×C₄	D₃×D₄	1	1	yes	characteristic
76	C₈	—	3	2	no	—
77	C₈	C₂×C₄×D₃	1	1	yes	characteristic
78	C₈	D₃×Q₈	1	1	yes	characteristic
79	C₂×C₃	C₂×#107	1	1	yes	characteristic
80	C₂×C₃	#508	1	1	yes	characteristic
81	C₂×C₃	#508	1	1	yes	characteristic
82	D₃	—	2	2	no	—
83	C₂²	—	3	2	no	—
84	C₂²	—	6	1	no	—
85	C₂²	—	6	1	no	—
86	C₂²	D₃×D₈	1	1	yes	characteristic
87	C₄	—	3	2	no	—
88	C₄	—	8	2	no	—
89	C₄	—	24	2	no	—
90	C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
91	C₄	D₃×Q₁₆	1	1	yes	characteristic
92	C₃	#2158	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	6	2	no	—
94	C₂	D₃×#107	1	1	yes	characteristic
95	C₂	#4912	1	1	yes	characteristic
96	C₂	#4919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#366	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂²×D₃	D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#366	D₃
hypocenter	C₃	#2158
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
3. C₃×#366	C₂	semidirect
4. #4504	C₂	semidirect
5. #4490	C₂	semidirect
14. D₃×C₁₆	C₄	semidirect
19. #366	D₃	semidirect
92. C₃	#2158	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×C₁₆	2
#4880	2
C₃×#366	2
#4504	2
#4490	2
#2158	3