Subgroups — #12880

Order 256.

161 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	#1726	C₂	1	1	yes	maximal, characteristic
3	#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆	C₂²	1	4	yes	—
5	C₂×#97	C₄	1	2	yes	—
6	C₂×#97	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#376	C₄	1	1	yes	characteristic
10	C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈	C₂×C₄	1	2	yes	—
13	C₂²×C₈	C₂³	1	1	yes	characteristic
14	C₂×C₄²	C₂×C₄	1	2	yes	—
15	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆	—	2	4	no	—
17	C₂×C₁₆	—	2	8	no	—
18	#97	—	2	4	no	—
19	#97	—	2	2	no	—
20	C₄×C₈	—	2	4	no	—
21	C₄×C₈	D₄	1	2	yes	—
22	C₄×C₈	Q₈	1	2	yes	—
23	#95	C₈	1	4	yes	—
24	C₂²×C₄	C₂×C₈	1	4	yes	—
25	C₂²×C₄	C₂×C₈	1	2	yes	—
26	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₈	M₁₆	1	4	yes	—
32	C₂×C₈	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈	#41	1	4	yes	—
35	C₄²	—	2	2	no	—
36	C₄²	—	2	2	no	—
37	C₄²	C₄⋊C₄	1	2	yes	—
38	C₄²	M₁₆	1	2	yes	—
39	C₁₆	—	4	4	no	—
40	C₂³	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄	—	2	4	no	—
42	C₂×C₄	—	2	4	no	—
43	C₂×C₄	—	2	8	no	—
44	C₂×C₄	C₄⋊C₈	1	4	yes	—
45	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄	#117	1	2	yes	—
47	C₂×C₄	C₂×M₁₆	1	2	yes	—
48	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	—	2	4	no	—
51	C₈	#124	1	2	yes	—
52	C₈	#125	1	2	yes	—
53	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂²	#431	1	2	yes	—
55	C₂²	#504	1	4	yes	—
56	C₄	—	2	4	no	—
57	C₄	—	4	4	no	—
58	C₄	#382	1	1	yes	characteristic
59	C₄	#423	1	1	yes	characteristic
60	C₄	#432	1	2	yes	—
61	C₂	#1831	1	1	yes	characteristic
62	C₂	#2151	1	2	yes	—
63	C₂	#2164	1	4	yes	—

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

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₁₆	2
#1726	2
#1374	2