Subgroups — #19618

Order 256.

345 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	#1723	C₂	1	8	yes	maximal, complemented
4	C₂×#322	C₂	1	1	yes	maximal, characteristic
5	#533	C₂²	1	4	yes	—
6	#500	C₂²	1	4	yes	—
7	#500	C₂²	1	4	yes	—
8	#499	—	2	4	no	—
9	#498	C₂²	1	4	yes	—
10	#494	C₂²	1	4	yes	—
11	#487	C₂²	1	8	yes	—
12	#467	—	2	8	no	—
13	C₄×Q₁₆	—	2	8	no	—
14	#427	—	2	4	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	2	yes	—
17	#322	C₂²	1	4	yes	—
18	C₂×Q₁₆	—	2	8	no	—
19	C₂²×C₈	D₄	1	4	yes	—
20	C₂²×C₈	D₄	1	2	yes	—
21	C₄⋊Q₈	C₂³	1	8	yes	—
22	#125	—	2	4	no	—
23	C₂²⋊Q₈	—	2	8	no	—
24	C₄×Q₈	—	2	8	no	—
25	#117	—	2	4	no	—
26	C₂×C₄²	C₂³	1	1	yes	characteristic
27	#107	—	2	8	no	—
28	#107	—	2	8	no	—
29	#106	—	2	8	no	—
30	#103	—	2	16	no	—
31	C₄×C₈	D₄	1	4	yes	—
32	C₄×C₈	D₄	1	2	yes	—
33	C₄×C₈	C₂³	1	4	yes	—
34	C₄×C₈	C₂³	1	2	yes	—
35	C₂×Q₈	—	2	8	no	—
36	C₂²×C₄	C₂×D₄	1	2	yes	—
37	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
38	Q₁₆	—	4	16	no	—
39	C₂×C₈	—	4	4	no	—
40	C₂×C₈	Q₁₆	1	8	yes	complemented
41	C₂×C₈	C₂×D₄	1	8	yes	—
42	C₂×C₈	C₂×D₄	1	4	yes	—
43	C₄⋊C₄	—	2	8	no	—
44	C₄⋊C₄	—	2	8	no	—
45	C₄⋊C₄	—	4	8	no	—
46	C₂²⋊C₄	—	4	4	no	—
47	C₄²	—	4	4	no	—
48	C₄²	C₂×D₄	1	1	yes	characteristic
49	C₄²	C₂×D₄	1	2	yes	—
50	C₄²	C₂⁴	1	1	yes	characteristic
51	C₂³	#127	1	1	yes	characteristic
52	Q₈	—	4	16	no	—
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	—	4	8	no	—
55	C₂×C₄	#127	1	2	yes	—
56	C₂×C₄	#127	1	1	yes	characteristic
57	C₂×C₄	C₂×Q₁₆	1	4	yes	—
58	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₄	1	2	yes	—
60	C₈	—	2	8	no	—
61	C₈	C₂×Q₁₆	1	8	yes	—
62	C₂²	—	2	2	no	—
63	C₂²	#494	1	2	yes	—
64	C₂²	C₂×#127	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	—	8	8	no	—
67	C₄	#494	1	2	yes	—
68	C₄	C₂²×Q₁₆	1	2	yes	—
69	C₄	#575	1	4	yes	—
70	C₂	—	2	2	no	—
71	C₂	C₂×#494	1	1	yes	characteristic
72	C₂	#3150	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. #1723	C₂	semidirect
40. 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
#1723	2
C₂×#322	2