Subgroups — #19604

Order 256.

353 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3170	C₂	1	4	yes	maximal
2	#3160	C₂	1	2	yes	maximal
3	#1713	C₂	1	2	yes	maximal, complemented
4	#1710	C₂	1	2	yes	maximal, complemented
5	#1707	C₂	1	4	yes	maximal, complemented
6	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
7	#533	C₂²	1	4	yes	—
8	#500	C₂²	1	4	yes	—
9	#500	C₂²	1	4	yes	—
10	#499	—	2	4	no	—
11	#498	C₂²	1	4	yes	—
12	#494	C₂²	1	4	yes	—
13	#487	C₂²	1	8	yes	—
14	#467	—	2	8	no	—
15	C₄×Q₁₆	—	2	8	no	—
16	#427	—	2	4	no	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	C₈²	C₂²	1	2	yes	—
20	C₈²	C₂²	1	1	yes	characteristic
21	C₈²	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆	—	2	8	no	—
23	C₂²×C₈	D₄	1	4	yes	—
24	C₂²×C₈	D₄	1	2	yes	—
25	C₄⋊Q₈	C₂³	1	8	yes	—
26	#125	—	2	4	no	—
27	C₂²⋊Q₈	—	2	8	no	—
28	C₄×Q₈	—	2	8	no	—
29	#117	—	2	4	no	—
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	#107	—	2	8	no	—
32	#107	—	2	8	no	—
33	#106	—	2	8	no	—
34	#103	—	2	16	no	—
35	C₄×C₈	D₄	1	4	yes	—
36	C₄×C₈	D₄	1	2	yes	—
37	C₄×C₈	C₂³	1	2	yes	—
38	C₄×C₈	C₂³	1	2	yes	—
39	C₄×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₈	C₂³	1	1	yes	characteristic
41	C₂×Q₈	—	2	8	no	—
42	C₂²×C₄	C₂×D₄	1	2	yes	—
43	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
44	Q₁₆	—	4	16	no	—
45	C₂×C₈	—	2	8	no	—
46	C₂×C₈	Q₁₆	1	8	yes	complemented
47	C₂×C₈	C₂×D₄	1	4	yes	—
48	C₂×C₈	C₂×D₄	1	2	yes	—
49	C₂×C₈	C₂×D₄	1	2	yes	—
50	C₂×C₈	C₂×D₄	1	4	yes	—
51	C₄⋊C₄	—	2	8	no	—
52	C₄⋊C₄	—	2	8	no	—
53	C₄⋊C₄	—	4	8	no	—
54	C₂²⋊C₄	—	4	4	no	—
55	C₄²	—	4	4	no	—
56	C₄²	C₂×D₄	1	1	yes	characteristic
57	C₄²	C₂×D₄	1	2	yes	—
58	C₄²	C₂⁴	1	1	yes	characteristic
59	C₂³	#127	1	1	yes	characteristic
60	Q₈	—	4	16	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	4	8	no	—
63	C₂×C₄	#127	1	2	yes	—
64	C₂×C₄	#127	1	1	yes	characteristic
65	C₂×C₄	C₂×Q₁₆	1	4	yes	—
66	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₄	1	2	yes	—
68	C₈	—	2	4	no	—
69	C₈	C₂×Q₁₆	1	4	yes	—
70	C₈	C₂×Q₁₆	1	4	yes	—
71	C₈	#135	1	8	yes	—
72	C₂²	—	2	2	no	—
73	C₂²	#494	1	2	yes	—
74	C₂²	C₂×#127	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	8	8	no	—
77	C₄	#494	1	2	yes	—
78	C₄	C₂²×Q₁₆	1	2	yes	—
79	C₄	C₂×#135	1	4	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂×#494	1	1	yes	characteristic
82	C₂	#3147	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#127
commutator, Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×#127
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1713	C₂	semidirect
4. #1710	C₂	semidirect
5. #1707	C₂	semidirect
46. C₂×C₈	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3170	2
#3160	2
#1713	2
#1710	2
#1707	2
C₂×C₈²	2