Subgroups — C₂³×#544 #10585907

Order 512.

37026 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#544	C₂	1	112	yes	maximal, direct product factor
2	C₂³×C₄⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₂³×#125	C₂	1	4	yes	maximal, complemented
4	C₂³×C₂²⋊Q₈	C₂	1	4	yes	maximal
5	C₂³×#117	C₂	1	2	yes	maximal
6	C₂⁴×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Q₈	C₂²	1	4	yes	—
8	C₂⁵×C₄	C₂²	1	2	yes	—
9	C₂⁵×C₄	C₂²	1	1	yes	characteristic
10	C₂×#544	C₂²	1	1792	yes	direct product factor
11	C₂²×C₄⋊Q₈	C₂²	1	224	yes	complemented
12	C₂²×#125	C₂²	1	224	yes	complemented
13	C₂²×C₂²⋊Q₈	C₂²	1	224	yes	—
14	C₂²×#117	C₂²	1	112	yes	—
15	C₂³×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂³×C₄⋊C₄	C₂²	1	56	yes	—
17	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂³×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂³×C₄⋊C₄	C₂²	1	8	yes	—
21	C₂³×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂³×C₄²	C₂²	1	4	yes	—
23	C₂⁶	C₂³	1	1	yes	characteristic
24	C₂³×Q₈	—	4	32	no	—
25	C₂³×Q₈	C₂³	1	112	yes	—
26	C₂⁴×C₄	—	4	16	no	—
27	C₂⁴×C₄	Q₈	1	32	yes	complemented
28	C₂⁴×C₄	C₂³	1	4	yes	—
29	C₂⁴×C₄	C₂³	1	4	yes	—
30	C₂⁴×C₄	C₂³	1	4	yes	—
31	C₂⁴×C₄	C₂³	1	56	yes	—
32	C₂⁴×C₄	C₂³	1	2	yes	—
33	C₂⁴×C₄	C₂³	1	28	yes	—
34	#544	C₂³	1	4096	yes	direct product factor
35	C₂×C₄⋊Q₈	C₂³	1	1792	yes	complemented
36	C₂×#125	C₂³	1	1792	yes	complemented
37	C₂×C₂²⋊Q₈	C₂³	1	1792	yes	—
38	C₂×#117	C₂³	1	896	yes	—
39	C₂²×C₄⋊C₄	C₂³	1	112	yes	—
40	C₂²×C₄⋊C₄	C₂³	1	448	yes	—
41	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
42	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
43	C₂²×C₄⋊C₄	C₂³	1	112	yes	—
44	C₂²×C₄⋊C₄	C₂³	1	224	yes	—
45	C₂²×C₂²⋊C₄	C₂³	1	112	yes	—
46	C₂²×C₄²	C₂³	1	112	yes	—
47	C₂⁵	—	2	8	no	—
48	C₂⁵	—	2	8	no	—
49	C₂⁵	C₂×Q₈	1	16	yes	—
50	C₂⁵	C₂⁴	1	14	yes	—
51	C₂⁵	C₂⁴	1	1	yes	characteristic
52	C₂²×Q₈	—	4	448	no	—
53	C₂²×Q₈	C₂⁴	1	448	yes	—
54	C₂³×C₄	—	2	32	no	—
55	C₂³×C₄	—	2	32	no	—
56	C₂³×C₄	—	2	32	no	—
57	C₂³×C₄	—	4	224	no	—
58	C₂³×C₄	C₂×Q₈	1	32	yes	—
59	C₂³×C₄	C₂×Q₈	1	448	yes	complemented
60	C₂³×C₄	C₂⁴	1	56	yes	—
61	C₂³×C₄	C₂⁴	1	56	yes	—
62	C₂³×C₄	C₂⁴	1	56	yes	—
63	C₂³×C₄	C₂⁴	1	224	yes	—
64	C₂³×C₄	C₂⁴	1	28	yes	—
65	C₂³×C₄	C₂⁴	1	112	yes	—
66	C₄⋊Q₈	C₂⁴	1	2048	yes	complemented
67	#125	C₂⁴	1	2048	yes	complemented
68	C₂²⋊Q₈	C₂⁴	1	2048	yes	—
69	#117	C₂⁴	1	1024	yes	—
70	C₂×C₄⋊C₄	C₂⁴	1	448	yes	—
71	C₂×C₄⋊C₄	C₂⁴	1	512	yes	—
72	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
73	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
74	C₂×C₄⋊C₄	C₂⁴	1	448	yes	—
75	C₂×C₄⋊C₄	C₂⁴	1	896	yes	—
76	C₂×C₂²⋊C₄	C₂⁴	1	448	yes	—
77	C₂×C₄²	C₂⁴	1	448	yes	—
78	C₂⁴	—	2	128	no	—
79	C₂⁴	—	2	56	no	—
80	C₂⁴	—	2	56	no	—
81	C₂⁴	C₂²×Q₈	1	112	yes	—
82	C₂⁴	C₂²×Q₈	1	8	yes	—
83	C₂⁴	H(5,𝔽₂)	1	8	yes	—
84	C₂⁴	2₋¹⁺⁴	1	8	yes	—
85	C₂⁴	C₂⁵	1	28	yes	—
86	C₂⁴	C₂⁵	1	7	yes	—
87	C₂×Q₈	—	4	896	no	—
88	C₂×Q₈	C₂⁵	1	256	yes	—
89	C₂²×C₄	—	2	224	no	—
90	C₂²×C₄	—	2	224	no	—
91	C₂²×C₄	—	2	224	no	—
92	C₂²×C₄	—	4	448	no	—
93	C₂²×C₄	C₂²×Q₈	1	224	yes	—
94	C₂²×C₄	C₂²×Q₈	1	896	yes	complemented
95	C₂²×C₄	C₂⁵	1	112	yes	—
96	C₂²×C₄	C₂⁵	1	112	yes	—
97	C₂²×C₄	C₂⁵	1	112	yes	—
98	C₂²×C₄	C₂⁵	1	128	yes	—
99	C₂²×C₄	C₂⁵	1	56	yes	—
100	C₂²×C₄	C₂⁵	1	64	yes	—
101	C₄⋊C₄	C₂⁵	1	256	yes	—
102	C₄⋊C₄	C₂⁵	1	128	yes	—
103	C₄⋊C₄	C₂⁵	1	128	yes	—
104	C₄⋊C₄	C₂⁵	1	256	yes	—
105	C₄⋊C₄	C₂⁵	1	512	yes	—
106	C₂²⋊C₄	C₂⁵	1	256	yes	—
107	C₄²	C₂⁵	1	256	yes	—
108	C₂³	—	2	56	no	—
109	C₂³	—	2	56	no	—
110	C₂³	—	2	448	no	—
111	C₂³	#544	1	64	yes	direct product factor
112	C₂³	C₂³×Q₈	1	112	yes	—
113	C₂³	C₂³×Q₈	1	28	yes	—
114	C₂³	C₂×H(5,𝔽₂)	1	28	yes	—
115	C₂³	C₂×2₋¹⁺⁴	1	28	yes	—
116	C₂³	C₂⁶	1	8	yes	—
117	C₂³	C₂⁶	1	7	yes	—
118	Q₈	—	4	256	no	—
119	C₂×C₄	—	2	224	no	—
120	C₂×C₄	—	2	224	no	—
121	C₂×C₄	—	2	224	no	—
122	C₂×C₄	—	4	128	no	—
123	C₂×C₄	C₂³×Q₈	1	224	yes	—
124	C₂×C₄	C₂³×Q₈	1	256	yes	complemented
125	C₂×C₄	C₂⁶	1	32	yes	—
126	C₂×C₄	C₂⁶	1	32	yes	—
127	C₂×C₄	C₂⁶	1	32	yes	—
128	C₂×C₄	C₂⁶	1	16	yes	—
129	C₂²	—	2	8	no	—
130	C₂²	—	2	8	no	—
131	C₂²	—	2	224	no	—
132	C₂²	C₂×#544	1	112	yes	direct product factor
133	C₂²	C₂⁴×Q₈	1	16	yes	—
134	C₂²	C₂⁴×Q₈	1	14	yes	—
135	C₂²	C₂²×H(5,𝔽₂)	1	14	yes	—
136	C₂²	C₂²×2₋¹⁺⁴	1	14	yes	—
137	C₂²	C₂⁷	1	1	yes	characteristic
138	C₄	—	2	32	no	—
139	C₄	—	2	32	no	—
140	C₄	—	2	32	no	—
141	C₄	C₂⁴×Q₈	1	32	yes	—
142	C₂	—	2	16	no	—
143	C₂	C₂²×#544	1	28	yes	direct product factor
144	C₂	C₂⁵×Q₈	1	1	yes	characteristic
145	C₂	C₂³×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂	C₂³×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#544	C₂	direct (N × H)
2. C₂³×C₄⋊Q₈	C₂	semidirect
3. C₂³×#125	C₂	semidirect
10. C₂×#544	C₂²	direct (N × H)
11. C₂²×C₄⋊Q₈	C₂²	semidirect
12. C₂²×#125	C₂²	semidirect
27. C₂⁴×C₄	Q₈	semidirect
34. #544	C₂³	direct (N × H)
35. C₂×C₄⋊Q₈	C₂³	semidirect
36. C₂×#125	C₂³	semidirect
59. C₂³×C₄	C₂×Q₈	semidirect
66. C₄⋊Q₈	C₂⁴	semidirect
67. #125	C₂⁴	semidirect
94. C₂²×C₄	C₂²×Q₈	semidirect
111. C₂³	#544	direct (N × H)
124. C₂×C₄	C₂³×Q₈	semidirect
132. C₂²	C₂×#544	direct (N × H)
143. C₂	C₂²×#544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#544	2
C₂³×C₄⋊Q₈	2
C₂³×#125	2
C₂³×C₂²⋊Q₈	2
C₂³×#117	2
C₂⁴×C₄⋊C₄	2

Subgroups — C23×#544 #10585907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#544 #10585907