Subgroups — #12676

Order 256.

153 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#368	C₂	1	2	yes	maximal
2	C₂×#346	C₂	1	1	yes	maximal, characteristic
3	C₄×#108	C₂	1	2	yes	maximal
4	#1381	C₂	1	2	yes	maximal
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#108	C₂²	1	4	yes	—
8	C₄×M₁₆	—	2	2	no	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#368	C₄	1	8	yes	complemented
11	#346	C₄	1	4	yes	—
12	#341	—	2	2	no	—
13	C₂×M₁₆	—	2	4	no	—
14	C₂²×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₈	C₂³	1	1	yes	characteristic
16	C₂×C₄²	D₄	1	1	yes	characteristic
17	C₂×C₁₆	—	2	2	no	—
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₆	C₂×C₄	1	4	yes	—
21	#108	—	2	4	no	—
22	#108	C₂×C₄	1	8	yes	—
23	#97	—	4	2	no	—
24	C₄×C₈	—	4	2	no	—
25	C₄×C₈	C₂×C₄	1	2	yes	—
26	C₄×C₈	Q₈	1	2	yes	—
27	C₂²×C₄	Q₁₆	1	2	yes	—
28	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
29	M₁₆	—	2	8	no	—
30	M₁₆	—	4	4	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	4	4	no	—
33	C₂×C₈	C₄⋊C₄	1	2	yes	—
34	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈	C₂²×C₄	1	2	yes	—
36	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈	#41	1	2	yes	—
38	C₄²	C₄⋊C₄	1	2	yes	—
39	C₄²	D₈	1	2	yes	—
40	C₁₆	—	2	2	no	—
41	C₁₆	C₄²	1	4	yes	—
42	C₂³	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₄	—	4	2	no	—
44	C₂×C₄	#107	1	4	yes	—
45	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄	C₄×D₄	1	2	yes	—
47	C₂×C₄	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄	#135	1	2	yes	—
49	C₈	—	2	2	no	—
50	C₈	—	4	8	no	—
51	C₈	C₂×C₄²	1	1	yes	characteristic
52	C₈	#117	1	1	yes	characteristic
53	C₈	C₄×Q₈	1	2	yes	—
54	C₂²	—	2	2	no	—
55	C₂²	C₂×#107	1	1	yes	characteristic
56	C₂²	C₄×Q₁₆	1	2	yes	—
57	C₄	—	2	4	no	—
58	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
59	C₄	#427	1	1	yes	characteristic
60	C₄	C₄×D₈	1	2	yes	—
61	C₂	—	2	2	no	—
62	C₂	C₄×#107	1	1	yes	characteristic
63	C₂	#2161	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂³	C₂×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
10. #368	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#368	2
C₂×#346	2
C₄×#108	2
#1381	2