Subgroups — #72225097

Order 2048.

696 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72216263	C₂	1	6	yes	maximal, complemented
2	C₃₂²	C₂	1	1	yes	maximal, characteristic, complemented
3	#151724	—	2	12	no	—
4	#123403	C₂²	1	4	yes	—
5	C₁₆×C₃₂	C₂²	1	3	yes	—
6	#13725	—	4	24	no	—
7	#12515	—	2	12	no	—
8	C₈×C₃₂	D₄	1	6	yes	—
9	C₁₆²	C₂³	1	1	yes	characteristic
10	C₂×D₃₂	—	8	48	no	—
11	#2246	—	4	24	no	—
12	#1712	—	4	4	no	—
13	C₄×C₃₂	D₈	1	12	yes	—
14	C₈×C₁₆	C₂×D₄	1	3	yes	—
15	C₂×D₁₆	—	8	48	no	—
16	#493	—	8	12	no	—
17	D₃₂	—	16	96	no	—
18	C₂×C₃₂	D₁₆	1	24	yes	—
19	C₄×C₁₆	C₂×D₈	1	6	yes	—
20	C₈²	#127	1	1	yes	characteristic
21	C₂×D₈	—	16	24	no	—
22	#127	—	16	4	no	—
23	D₁₆	—	16	96	no	—
24	C₂×C₁₆	C₂×D₁₆	1	12	yes	—
25	C₄×C₈	#493	1	3	yes	—
26	C₃₂	D₃₂	1	48	yes	complemented
27	C₂×D₄	—	32	12	no	—
28	D₈	—	32	48	no	—
29	C₂×C₈	#2246	1	6	yes	—
30	C₄²	#1712	1	1	yes	characteristic
31	C₁₆	C₂×D₃₂	1	24	yes	—
32	C₂³	—	64	4	no	—
33	D₄	—	64	24	no	—
34	C₂×C₄	#12515	1	3	yes	—
35	C₈	#13725	1	12	yes	—
36	C₂²	—	128	12	no	—
37	C₂²	#123403	1	1	yes	characteristic
38	C₄	#151724	1	6	yes	—
39	C₂	—	256	4	no	—
40	C₂	#72216263	1	3	yes	—

	subgroup	G/H
center, socle	C₂²	#123403
commutator, Frattini, cosocle	C₁₆²	C₂³
LCS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₁₆² ◁ G
LECS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

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

subgroup	index
#72216263	2
C₃₂²	2