Subgroups — #33329

Order 256.

358 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3407	C₂	1	2	yes	maximal
2	#2875	C₂	1	1	yes	maximal, characteristic
3	#2110	C₂	1	6	yes	maximal, complemented
4	C₂×#347	C₂	1	6	yes	maximal, complemented
5	C₂×#131	C₄	1	1	yes	characteristic
6	C₂×#131	C₂²	1	1	yes	characteristic
7	C₄×#41	C₄	1	1	yes	characteristic
8	#504	C₄	1	16	yes	complemented
9	C₂×M₃₂	C₂²	1	12	yes	—
10	#405	C₄	1	3	yes	—
11	#405	C₂²	1	3	yes	—
12	C₄×M₁₆	C₄	1	3	yes	—
13	C₂×#97	C₂²	1	3	yes	—
14	#347	C₂²	1	4	yes	—
15	#347	C₂²	1	12	yes	—
16	C₂×#41	C₂×C₄	1	1	yes	characteristic
17	#131	—	2	4	no	—
18	#131	C₂×C₄	1	8	yes	—
19	C₂×M₁₆	C₂×C₄	1	3	yes	—
20	C₂×M₁₆	C₂×C₄	1	3	yes	—
21	C₂²×C₈	C₂×C₄	1	3	yes	—
22	C₂²×C₈	C₂³	1	3	yes	—
23	C₄×Q₈	—	2	1	no	—
24	C₄×D₄	—	2	3	no	—
25	#117	C₂×C₄	1	3	yes	—
26	C₂×C₄²	C₂×C₄	1	3	yes	—
27	M₃₂	C₂×C₄	1	24	yes	complemented
28	M₃₂	C₂³	1	8	yes	—
29	C₂×C₁₆	C₂×C₄	1	24	yes	complemented
30	#97	C₂×C₄	1	3	yes	—
31	#97	C₂×C₄	1	3	yes	—
32	#97	C₂³	1	1	yes	characteristic
33	#97	C₂³	1	3	yes	—
34	C₄×C₈	C₂×C₄	1	6	yes	—
35	#41	C₄²	1	4	yes	—
36	#41	M₁₆	1	4	yes	—
37	C₂×Q₈	M₁₆	1	1	yes	characteristic
38	C₂×D₄	M₁₆	1	3	yes	—
39	C₂²×C₄	—	2	3	no	—
40	C₂²×C₄	C₂²×C₄	1	3	yes	—
41	M₁₆	—	2	6	no	—
42	M₁₆	C₄²	1	12	yes	—
43	C₂×C₈	—	2	6	no	—
44	C₂×C₈	C₂²×C₄	1	3	yes	—
45	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈	C₂²×C₄	1	3	yes	—
47	C₂×C₈	C₂²×C₄	1	12	yes	—
48	C₂×C₈	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄	—	2	3	no	—
50	C₂²⋊C₄	—	2	3	no	—
51	C₄²	—	2	3	no	—
52	C₄²	C₂²×C₄	1	3	yes	—
53	C₄²	C₂²×C₄	1	1	yes	characteristic
54	C₁₆	C₂²×C₄	1	16	yes	—
55	C₂³	C₂×M₁₆	1	3	yes	—
56	Q₈	#97	1	4	yes	—
57	D₄	#97	1	12	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	3	no	—
60	C₂×C₄	—	8	3	no	—
61	C₂×C₄	C₂×C₄²	1	6	yes	—
62	C₂×C₄	C₂×M₁₆	1	6	yes	—
63	C₂×C₄	C₂×M₁₆	1	3	yes	—
64	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	2	3	no	—
67	C₈	C₂×C₄²	1	6	yes	—
68	C₈	C₂³×C₄	1	2	yes	—
69	C₂²	—	4	3	no	—
70	C₂²	C₂×#97	1	6	yes	—
71	C₂²	C₂²×M₁₆	1	1	yes	characteristic
72	C₄	—	4	3	no	—
73	C₄	—	4	1	no	—
74	C₄	C₂×#97	1	6	yes	—
75	C₄	C₂²×C₄²	1	1	yes	characteristic
76	C₄	C₂²×M₁₆	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	2	6	no	—
79	C₂	C₂²×#97	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂²×#97
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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. #2110	C₂	semidirect
4. C₂×#347	C₂	semidirect
8. #504	C₄	semidirect
27. M₃₂	C₂×C₄	semidirect
29. 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
#3407	2
#2875	2
#2110	2
C₂×#347	2