Subgroups — #69358

Order 384.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4904	C₂	1	1	yes	maximal, characteristic
2	#4895	C₂	1	2	yes	maximal, complemented
3	C₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	#4502	C₂	1	1	yes	maximal, characteristic, complemented
5	#4497	C₂	1	2	yes	maximal, complemented
6	#2179	—	3	1	no	maximal, Sylow
7	#907	—	2	2	no	—
8	#904	C₂²	1	2	yes	—
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×#108	C₂²	1	2	yes	—
11	#825	—	2	2	no	—
12	C₈×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	Dic₂₄	C₄	1	1	yes	characteristic, complemented
15	#800	C₄	1	2	yes	complemented
16	D₄₈	C₄	1	1	yes	characteristic, complemented
17	#508	—	3	1	no	—
18	#443	—	3	2	no	—
19	#368	D₃	1	1	yes	characteristic, complemented
20	#359	—	3	2	no	—
21	#346	—	3	1	no	—
22	#235	—	2	2	no	—
23	C₃×M₁₆	—	2	2	no	—
24	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₄×Dic₃	D₄	1	1	yes	characteristic
26	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
27	Dic₁₂	C₂×C₄	1	2	yes	—
28	D₂₄	C₂×C₄	1	2	yes	—
29	#204	—	2	2	no	—
30	ΓL(1,𝔽₂₅)	—	4	2	no	—
31	D₃×C₈	—	4	2	no	—
32	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₃⋊C₁₆	—	2	1	no	—
35	#135	—	3	2	no	—
36	#131	—	6	2	no	—
37	Q₃₂	—	3	1	no	—
38	SD₃₂	—	3	2	no	—
39	D₁₆	—	3	1	no	—
40	C₂×C₁₆	—	3	1	no	—
41	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
42	#108	C₂×D₃	1	2	yes	—
43	#104	—	6	2	no	—
44	C₄×C₈	—	3	1	no	—
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	#67	—	4	2	no	—
47	C₂×Dic₃	—	2	1	no	—
48	D₁₂	—	2	2	no	—
49	C₄×D₃	—	4	2	no	—
50	Dic₆	—	2	2	no	—
51	C₃×C₈	—	4	2	no	—
52	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₃×C₈	#41	1	1	yes	characteristic
54	C₃⋊C₈	D₈	1	2	yes	—
55	#41	—	6	2	no	—
56	Q₁₆	—	3	2	no	—
57	SD₁₆	—	6	2	no	—
58	D₈	—	3	2	no	—
59	M₁₆	—	2	2	no	—
60	M₁₆	—	12	2	no	—
61	C₂×C₈	—	3	1	no	—
62	C₂×C₈	—	12	2	no	—
63	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
64	C₄²	—	3	1	no	—
65	C₁₆	—	6	1	no	—
66	C₁₆	C₄×D₃	1	1	yes	characteristic
67	C₁₆	C₄×D₃	1	1	yes	characteristic
68	C₂²×C₃	#135	1	1	yes	characteristic
69	C₂×D₃	—	4	2	no	—
70	C₃×C₄	C₄×D₄	1	1	yes	characteristic
71	C₃×C₄	C₂×D₈	1	1	yes	characteristic
72	Dic₃	—	4	1	no	—
73	Dic₃	—	4	2	no	—
74	Q₈	—	6	2	no	—
75	D₄	—	6	2	no	—
76	D₄	—	12	2	no	—
77	C₂×C₄	—	6	1	no	—
78	C₂×C₄	—	12	2	no	—
79	C₂×C₄	D₃×D₄	1	1	yes	characteristic
80	C₈	—	3	2	no	—
81	C₈	—	4	2	no	—
82	C₈	C₂×C₄×D₃	1	1	yes	characteristic
83	C₈	#239	1	1	yes	characteristic
84	C₂×C₃	—	2	1	no	—
85	C₂×C₃	C₄×D₈	1	1	yes	characteristic
86	D₃	—	8	2	no	—
87	C₂²	—	12	2	no	—
88	C₂²	#919	1	1	yes	characteristic
89	C₄	—	12	1	no	—
90	C₄	—	12	2	no	—
91	C₄	#893	1	1	yes	characteristic
92	C₄	D₃×D₈	1	1	yes	characteristic
93	C₃	#2179	1	1	yes	characteristic, complemented, Sylow
94	C₂	—	2	1	no	—
95	C₂	—	24	2	no	—
96	C₂	#4872	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#368	C₂
socle	C₂×C₃	C₄×D₈
prime radical	#4904	C₂
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#368	D₃
hypocenter	C₃	#2179
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #4895	C₂	semidirect
3. C₃×#368	C₂	semidirect
4. #4502	C₂	semidirect
5. #4497	C₂	semidirect
14. Dic₂₄	C₄	semidirect
15. #800	C₄	semidirect
16. D₄₈	C₄	semidirect
19. #368	D₃	semidirect
93. C₃	#2179	semidirect

Maximal subgroups

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

subgroup	index
#4904	2
#4895	2
C₃×#368	2
#4502	2
#4497	2
#2179	3