Subgroups — #33410

Order 256.

329 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504	C₂	1	1	yes	maximal, characteristic
2	#2913	C₂	1	2	yes	maximal
3	#2159	C₂	1	6	yes	maximal, complemented
4	#2158	C₂	1	3	yes	maximal, complemented
5	C₂×#366	C₂	1	3	yes	maximal, complemented
6	C₂×#131	C₂²	1	1	yes	characteristic
7	#520	—	2	2	no	—
8	#504	C₄	1	8	yes	complemented
9	C₂×M₃₂	C₂²	1	3	yes	—
10	C₂²×C₁₆	C₂²	1	3	yes	—
11	#428	—	2	6	no	—
12	#427	C₂²	1	6	yes	—
13	C₂×#107	C₂²	1	6	yes	—
14	#367	C₂²	1	6	yes	—
15	#366	C₂²	1	3	yes	—
16	#366	C₂²	1	6	yes	—
17	#366	C₂²	1	1	yes	characteristic
18	C₂×#41	D₄	1	1	yes	characteristic
19	#131	C₂×C₄	1	4	yes	—
20	#131	Q₈	1	4	yes	—
21	C₂×M₁₆	D₄	1	3	yes	—
22	C₂²×C₈	C₂³	1	3	yes	—
23	C₄×Q₈	—	4	2	no	—
24	C₄×D₄	—	4	6	no	—
25	#117	—	2	6	no	—
26	C₂×C₄⋊C₄	—	2	6	no	—
27	M₃₂	C₂×C₄	1	12	yes	complemented
28	C₂×C₁₆	C₂×C₄	1	12	yes	complemented
29	C₂×C₁₆	C₂³	1	1	yes	characteristic
30	C₂×C₁₆	C₂³	1	3	yes	—
31	#107	—	2	6	no	—
32	#107	C₂³	1	6	yes	—
33	#107	C₂³	1	2	yes	—
34	#106	—	2	6	no	—
35	#41	C₄⋊C₄	1	4	yes	—
36	#41	Q₁₆	1	4	yes	—
37	C₂×Q₈	D₈	1	1	yes	characteristic
38	C₂×D₄	D₈	1	3	yes	—
39	C₂²×C₄	—	4	6	no	—
40	C₂²×C₄	C₂×D₄	1	3	yes	—
41	M₁₆	C₄⋊C₄	1	12	yes	—
42	C₂×C₈	C₂²×C₄	1	6	yes	—
43	C₂×C₈	C₂×D₄	1	3	yes	—
44	C₂×C₈	C₂×Q₈	1	6	yes	—
45	C₂×C₈	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	6	no	—
48	C₄⋊C₄	—	4	6	no	—
49	C₂²⋊C₄	—	4	6	no	—
50	C₄²	—	4	6	no	—
51	C₁₆	C₂²×C₄	1	2	yes	—
52	C₁₆	C₂²×C₄	1	6	yes	—
53	C₂³	C₂×D₈	1	3	yes	—
54	Q₈	#107	1	4	yes	—
55	D₄	#107	1	12	yes	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	—	4	6	no	—
58	C₂×C₄	—	16	6	no	—
59	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
60	C₂×C₄	C₂×D₈	1	3	yes	—
61	C₂×C₄	C₂×Q₁₆	1	6	yes	—
62	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
63	C₈	C₂×C₄⋊C₄	1	6	yes	—
64	C₈	C₂³×C₄	1	1	yes	characteristic
65	C₈	C₂²×Q₈	1	1	yes	characteristic
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	C₂×#107	1	6	yes	—
69	C₂²	C₂²×D₈	1	1	yes	characteristic
70	C₄	—	8	6	no	—
71	C₄	—	8	2	no	—
72	C₄	C₂×#107	1	6	yes	—
73	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₄	C₂²×Q₁₆	1	1	yes	characteristic
75	C₂	—	2	6	no	—
76	C₂	C₂²×#107	1	1	yes	characteristic
77	C₂	#3415	1	1	yes	characteristic
78	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2159	C₂	semidirect
4. #2158	C₂	semidirect
5. C₂×#366	C₂	semidirect
8. #504	C₄	semidirect
27. M₃₂	C₂×C₄	semidirect
28. C₂×C₁₆	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504	2
#2913	2
#2159	2
#2158	2
C₂×#366	2