Subgroups — #19753

Order 256.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3185	C₂	1	3	yes	maximal, complemented
2	#3138	C₂	1	3	yes	maximal
3	#1720	C₂	1	6	yes	maximal, complemented
4	#1711	C₂	1	2	yes	maximal, complemented
5	#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	#533	C₂²	1	3	yes	—
7	C₂×#127	C₂²	1	1	yes	characteristic
8	#498	C₂²	1	3	yes	complemented
9	#494	C₂²	1	3	yes	complemented
10	#490	—	2	6	no	—
11	#487	C₂²	1	6	yes	complemented
12	#486	C₂²	1	6	yes	—
13	#486	C₂²	1	6	yes	—
14	#470	—	2	6	no	—
15	#441	—	2	6	no	—
16	#418	—	2	6	no	—
17	C₄×M₁₆	C₂²	1	3	yes	—
18	#322	C₂²	1	3	yes	complemented
19	C₈²	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄	—	2	3	no	—
21	C₂×Q₁₆	—	2	6	no	—
22	C₂×M₁₆	D₄	1	6	yes	—
23	C₄⋊Q₈	C₂³	1	6	yes	—
24	#127	—	2	3	no	—
25	#127	C₂³	1	2	yes	—
26	#125	—	2	3	no	—
27	C₂²⋊Q₈	—	2	6	no	—
28	C₄×Q₈	—	2	6	no	—
29	#117	—	2	3	no	—
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	#106	—	2	6	no	—
32	#103	—	2	12	no	—
33	#102	—	2	12	no	—
34	#102	—	2	12	no	—
35	#97	D₄	1	6	yes	—
36	C₄×C₈	C₂³	1	3	yes	—
37	C₄×C₈	C₂³	1	3	yes	—
38	C₂⁴	—	4	1	no	—
39	C₂×Q₈	—	2	6	no	—
40	C₂×D₄	—	2	6	no	—
41	C₂×D₄	—	4	12	no	—
42	C₂×D₄	—	4	3	no	—
43	C₂²×C₄	C₂×D₄	1	3	yes	—
44	Q₁₆	—	4	12	no	—
45	M₁₆	—	4	6	no	—
46	C₂×C₈	C₂×D₄	1	6	yes	—
47	C₂×C₈	C₂×D₄	1	6	yes	—
48	C₄⋊C₄	—	2	6	no	—
49	C₄⋊C₄	—	2	6	no	—
50	C₄⋊C₄	—	4	6	no	—
51	C₂²⋊C₄	—	4	3	no	—
52	C₄²	—	4	3	no	—
53	C₄²	C₂×D₄	1	3	yes	—
54	C₄²	C₂⁴	1	1	yes	characteristic
55	C₂³	—	4	2	no	—
56	C₂³	—	8	6	no	—
57	C₂³	#127	1	1	yes	characteristic
58	Q₈	—	4	12	no	—
59	D₄	—	4	12	no	—
60	D₄	—	8	6	no	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	—	4	6	no	—
63	C₂×C₄	#127	1	3	yes	—
64	C₂×C₄	C₂²×D₄	1	3	yes	—
65	C₈	—	2	6	no	—
66	C₈	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	8	6	no	—
69	C₂²	—	16	4	no	—
70	C₂²	C₂×#127	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	8	6	no	—
73	C₄	#575	1	6	yes	—
74	C₂	—	4	1	no	—
75	C₂	—	16	2	no	—
76	C₂	#3153	1	3	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3185	C₂	semidirect
3. #1720	C₂	semidirect
4. #1711	C₂	semidirect
5. #1455	C₂	semidirect
8. #498	C₂²	semidirect
9. #494	C₂²	semidirect
11. #487	C₂²	semidirect
18. #322	C₂²	semidirect
19. C₈²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3185	2
#3138	2
#1720	2
#1711	2
#1455	2