Subgroups — #12885

Order 256.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2153	C₂	1	1	yes	maximal, characteristic
2	#2151	C₂	1	1	yes	maximal, characteristic
3	#1747	C₂	1	1	yes	maximal, characteristic
4	#1379	C₂	1	2	yes	maximal
5	#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	#405	C₂²	1	2	yes	—
10	#405	C₂²	1	1	yes	characteristic
11	#364	—	2	2	no	—
12	C₄⋊C₁₆	—	2	2	no	—
13	#338	C₄	1	1	yes	characteristic
14	H(3,ℤ/4)	C₄	1	1	yes	characteristic
15	H(3,ℤ/4)	C₄	1	2	yes	—
16	C₂×M₁₆	C₂×C₄	1	2	yes	—
17	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	#117	C₂×C₄	1	2	yes	—
20	#117	C₂×C₄	1	1	yes	characteristic
21	M₃₂	—	2	2	no	—
22	M₃₂	—	2	2	no	—
23	M₃₂	—	2	2	no	—
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	—	2	2	no	—
28	#97	—	2	2	no	—
29	#97	Q₈	1	2	yes	—
30	C₄×C₈	—	2	2	no	—
31	C₄×C₈	D₄	1	2	yes	—
32	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
33	M₁₆	—	2	2	no	—
34	M₁₆	—	2	2	no	—
35	M₁₆	—	2	2	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	C₄⋊C₄	1	2	yes	—
38	C₂×C₈	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈	#41	1	2	yes	—
41	C₂×C₈	#41	1	2	yes	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	2	no	—
44	C₂²⋊C₄	—	2	1	no	—
45	C₂²⋊C₄	—	2	2	no	—
46	C₄²	—	2	2	no	—
47	C₄²	C₄⋊C₄	1	2	yes	—
48	C₁₆	—	4	1	no	—
49	C₁₆	—	4	1	no	—
50	C₁₆	—	4	2	no	—
51	C₂³	#131	1	1	yes	characteristic
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄	#117	1	2	yes	—
56	C₂×C₄	#131	1	2	yes	—
57	C₂×C₄	#131	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	4	2	no	—
60	C₈	#124	1	1	yes	characteristic
61	C₈	#124	1	1	yes	characteristic
62	C₈	#125	1	2	yes	—
63	C₂²	—	4	1	no	—
64	C₂²	#424	1	1	yes	characteristic
65	C₂²	#433	1	2	yes	—
66	C₄	—	4	2	no	—
67	C₄	—	8	2	no	—
68	C₄	#382	1	1	yes	characteristic
69	C₄	#424	1	1	yes	characteristic
70	C₄	#433	1	2	yes	—
71	C₂	—	2	1	no	—
72	C₂	—	2	2	no	—
73	C₂	#1832	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#124
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1832
prime radical	C₂³	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ 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
#2153	2
#2151	2
#1747	2
#1379	2
#1378	2