Subgroups — #68075

Order 384.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal
2	C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
3	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	#4454	C₂	1	2	yes	maximal, complemented
5	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#1455	—	3	1	no	maximal, Sylow
7	C₂×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
8	C₂×ΓL(1,𝔽₂₅)	C₄	1	4	yes	—
9	C₄²×D₃	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₂²	1	2	yes	—
12	#815	C₄	1	2	yes	—
13	#815	C₄	1	4	yes	—
14	C₄×C₃⋊C₈	C₂²	1	2	yes	—
15	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×M₁₆	—	3	2	no	—
17	C₄×M₁₆	—	3	1	no	—
18	#322	—	3	2	no	—
19	#322	D₃	1	1	yes	characteristic, complemented
20	C₈²	—	3	1	no	—
21	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
25	C₃×C₄²	C₂³	1	1	yes	characteristic
26	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₃	C₂×C₄	1	2	yes	—
28	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
30	ΓL(1,𝔽₂₅)	—	2	4	no	—
31	ΓL(1,𝔽₂₅)	—	2	8	no	—
32	C₂×M₁₆	—	3	4	no	—
33	C₂×M₁₆	—	3	2	no	—
34	C₂×C₄²	—	3	1	no	—
35	#97	—	3	4	no	—
36	#97	—	3	2	no	—
37	C₄×C₈	—	3	2	no	—
38	C₄×C₈	—	3	1	no	—
39	C₄×C₈	C₂×D₃	1	1	yes	characteristic
40	C₄×C₈	C₂×D₃	1	2	yes	—
41	C₂²×D₃	C₄²	1	1	yes	characteristic
42	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
44	C₂×Dic₃	C₄²	1	1	yes	characteristic
45	C₂×Dic₃	C₄²	1	2	yes	—
46	C₄×D₃	—	2	2	no	—
47	C₄×D₃	—	2	4	no	—
48	C₃×C₈	—	2	4	no	—
49	C₃×C₈	M₁₆	1	4	yes	complemented
50	C₃⋊C₈	—	2	2	no	—
51	C₃⋊C₈	M₁₆	1	8	yes	complemented
52	C₂²×C₄	—	3	1	no	—
53	C₂²×C₄	—	3	2	no	—
54	M₁₆	—	6	8	no	—
55	M₁₆	—	6	4	no	—
56	C₂×C₈	—	3	4	no	—
57	C₂×C₈	—	3	2	no	—
58	C₂×C₈	C₄×D₃	1	4	yes	—
59	C₂×C₈	C₄×D₃	1	2	yes	—
60	C₄²	—	3	1	no	—
61	C₄²	—	3	2	no	—
62	C₄²	C₂²×D₃	1	1	yes	characteristic
63	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×D₃	—	2	1	no	—
65	C₂×D₃	—	2	2	no	—
66	C₃×C₄	C₂×M₁₆	1	2	yes	—
67	C₃×C₄	C₂×M₁₆	1	4	yes	—
68	Dic₃	—	2	1	no	—
69	Dic₃	—	2	2	no	—
70	C₂³	—	3	1	no	—
71	C₂×C₄	—	3	1	no	—
72	C₂×C₄	—	3	2	no	—
73	C₂×C₄	—	6	2	no	—
74	C₂×C₄	—	6	4	no	—
75	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	—	3	8	no	—
79	C₈	—	6	2	no	—
80	C₈	ΓL(1,𝔽₂₅)	1	4	yes	complemented
81	C₂×C₃	C₄×M₁₆	1	1	yes	characteristic
82	C₂×C₃	C₄×M₁₆	1	2	yes	—
83	D₃	—	4	1	no	—
84	C₂²	—	6	1	no	—
85	C₂²	—	6	2	no	—
86	C₂²	C₄²×D₃	1	1	yes	characteristic
87	C₄	—	6	1	no	—
88	C₄	—	6	2	no	—
89	C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
90	C₄	D₃×M₁₆	1	4	yes	—
91	C₃	#1455	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	12	1	no	—
93	C₂	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
94	C₂	#4707	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator, socle	C₂²×C₃	C₂×C₄²
Fitting	C₃×#322	C₂
prime radical	C₂²×D₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	#322	D₃
hypocenter	C₃	#1455
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. C₃×#322	C₂	semidirect
4. #4454	C₂	semidirect
5. C₈×C₃⋊C₈	C₂	semidirect
19. #322	D₃	semidirect
49. C₃×C₈	M₁₆	semidirect
51. C₃⋊C₈	M₁₆	semidirect
80. C₈	ΓL(1,𝔽₂₅)	semidirect
91. C₃	#1455	semidirect

Maximal subgroups

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

subgroup	index
C₄×ΓL(1,𝔽₂₅)	2
C₃×#322	2
#4454	2
C₈×C₃⋊C₈	2
#1455	3