Subgroups — C₂³×#555 #10585930

Order 512.

41746 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×#555	C₂	1	112	yes	maximal, direct product factor
2	C₂³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#124	C₂	1	4	yes	maximal, complemented
4	C₂³×#121	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×D₄	C₂	1	2	yes	maximal, complemented
6	C₂³×#117	C₂	1	4	yes	maximal, complemented
7	C₂⁴×Q₈	C₂²	1	2	yes	—
8	C₂⁴×D₄	C₂²	1	2	yes	—
9	C₂⁴×D₄	C₂²	1	4	yes	—
10	C₂⁵×C₄	C₂²	1	4	yes	—
11	C₂×#555	C₂²	1	1792	yes	direct product factor
12	C₂²×C₄⋊Q₈	C₂²	1	56	yes	complemented
13	C₂²×#124	C₂²	1	224	yes	complemented
14	C₂²×#121	C₂²	1	224	yes	complemented
15	C₂²×C₄×D₄	C₂²	1	112	yes	complemented
16	C₂²×#117	C₂²	1	224	yes	complemented
17	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂³×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂³×C₂²⋊C₄	C₂²	1	8	yes	—
20	C₂³×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂³×C₄²	C₂²	1	4	yes	—
22	C₂³×C₄²	C₂²	1	1	yes	characteristic
23	C₂⁶	C₂³	1	4	yes	—
24	C₂³×Q₈	—	4	16	no	—
25	C₂³×Q₈	C₂³	1	56	yes	—
26	C₂³×D₄	—	2	32	no	—
27	C₂³×D₄	—	4	32	no	—
28	C₂³×D₄	C₂³	1	56	yes	—
29	C₂³×D₄	C₂³	1	112	yes	—
30	C₂⁴×C₄	—	2	64	no	—
31	C₂⁴×C₄	C₂³	1	112	yes	—
32	C₂⁴×C₄	C₂³	1	4	yes	—
33	C₂⁴×C₄	C₂³	1	4	yes	—
34	C₂⁴×C₄	C₂³	1	1	yes	characteristic
35	C₂⁴×C₄	C₂³	1	2	yes	—
36	#555	C₂³	1	4096	yes	direct product factor
37	C₂×C₄⋊Q₈	C₂³	1	448	yes	complemented
38	C₂×#124	C₂³	1	1792	yes	complemented
39	C₂×#121	C₂³	1	1792	yes	complemented
40	C₂×C₄×D₄	C₂³	1	896	yes	complemented
41	C₂×#117	C₂³	1	1792	yes	complemented
42	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
43	C₂²×C₄⋊C₄	C₂³	1	112	yes	—
44	C₂²×C₂²⋊C₄	C₂³	1	224	yes	—
45	C₂²×C₂²⋊C₄	C₂³	1	112	yes	—
46	C₂²×C₄²	C₂³	1	112	yes	—
47	C₂²×C₄²	C₂³	1	28	yes	—
48	C₂⁵	—	2	32	no	—
49	C₂⁵	—	2	32	no	—
50	C₂⁵	—	2	32	no	—
51	C₂⁵	C₂⁴	1	56	yes	—
52	C₂⁵	C₂⁴	1	1	yes	characteristic
53	C₂²×Q₈	—	4	224	no	—
54	C₂²×Q₈	C₂⁴	1	224	yes	—
55	C₂²×D₄	—	2	448	no	—
56	C₂²×D₄	—	4	448	no	—
57	C₂²×D₄	C₂⁴	1	224	yes	—
58	C₂²×D₄	C₂⁴	1	448	yes	—
59	C₂³×C₄	—	2	32	no	—
60	C₂³×C₄	—	2	32	no	—
61	C₂³×C₄	—	2	8	no	—
62	C₂³×C₄	—	2	896	no	—
63	C₂³×C₄	#41	1	32	yes	—
64	C₂³×C₄	C₂⁴	1	14	yes	—
65	C₂³×C₄	C₂⁴	1	56	yes	—
66	C₂³×C₄	C₂⁴	1	56	yes	—
67	C₂³×C₄	C₂⁴	1	448	yes	—
68	C₂³×C₄	C₂⁴	1	28	yes	—
69	C₄⋊Q₈	C₂⁴	1	512	yes	complemented
70	#124	C₂⁴	1	2048	yes	complemented
71	#121	C₂⁴	1	2048	yes	complemented
72	C₄×D₄	C₂⁴	1	1024	yes	complemented
73	#117	C₂⁴	1	2048	yes	complemented
74	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
75	C₂×C₄⋊C₄	C₂⁴	1	448	yes	—
76	C₂×C₂²⋊C₄	C₂⁴	1	896	yes	—
77	C₂×C₂²⋊C₄	C₂⁴	1	448	yes	—
78	C₂×C₄²	C₂⁴	1	448	yes	—
79	C₂×C₄²	C₂⁴	1	112	yes	—
80	C₂⁴	—	2	224	no	—
81	C₂⁴	—	2	224	no	—
82	C₂⁴	—	2	224	no	—
83	C₂⁴	—	4	256	no	—
84	C₂⁴	C₂×#41	1	16	yes	—
85	C₂⁴	H(5,𝔽₂)	1	8	yes	—
86	C₂⁴	C₂⁵	1	112	yes	—
87	C₂⁴	C₂⁵	1	7	yes	—
88	C₂×Q₈	—	4	448	no	—
89	C₂×Q₈	C₂⁵	1	128	yes	—
90	C₂×D₄	—	2	896	no	—
91	C₂×D₄	—	4	896	no	—
92	C₂×D₄	C₂⁵	1	128	yes	—
93	C₂×D₄	C₂⁵	1	256	yes	—
94	C₂²×C₄	—	2	224	no	—
95	C₂²×C₄	—	2	224	no	—
96	C₂²×C₄	—	2	56	no	—
97	C₂²×C₄	—	2	1792	no	—
98	C₂²×C₄	C₂×#41	1	224	yes	—
99	C₂²×C₄	C₂⁵	1	112	yes	—
100	C₂²×C₄	C₂⁵	1	28	yes	—
101	C₂²×C₄	C₂⁵	1	112	yes	—
102	C₂²×C₄	C₂⁵	1	256	yes	—
103	C₂²×C₄	C₂⁵	1	56	yes	—
104	C₄⋊C₄	C₂⁵	1	128	yes	—
105	C₄⋊C₄	C₂⁵	1	256	yes	—
106	C₂²⋊C₄	C₂⁵	1	512	yes	—
107	C₂²⋊C₄	C₂⁵	1	256	yes	—
108	C₄²	C₂⁵	1	256	yes	—
109	C₄²	C₂⁵	1	64	yes	—
110	C₂³	—	2	224	no	—
111	C₂³	—	2	224	no	—
112	C₂³	—	2	224	no	—
113	C₂³	—	4	896	no	—
114	C₂³	#555	1	64	yes	direct product factor
115	C₂³	C₂²×#41	1	56	yes	—
116	C₂³	C₂×H(5,𝔽₂)	1	28	yes	—
117	C₂³	C₂⁶	1	32	yes	—
118	C₂³	C₂⁶	1	7	yes	—
119	Q₈	—	4	128	no	—
120	D₄	—	2	256	no	—
121	D₄	—	4	256	no	—
122	C₂×C₄	—	2	224	no	—
123	C₂×C₄	—	2	224	no	—
124	C₂×C₄	—	2	56	no	—
125	C₂×C₄	—	2	512	no	—
126	C₂×C₄	C₂²×#41	1	224	yes	—
127	C₂×C₄	C₂⁶	1	32	yes	—
128	C₂×C₄	C₂⁶	1	8	yes	—
129	C₂×C₄	C₂⁶	1	32	yes	—
130	C₂×C₄	C₂⁶	1	16	yes	—
131	C₂²	—	2	32	no	—
132	C₂²	—	2	32	no	—
133	C₂²	—	2	32	no	—
134	C₂²	—	4	448	no	—
135	C₂²	C₂×#555	1	112	yes	direct product factor
136	C₂²	C₂³×#41	1	28	yes	—
137	C₂²	C₂²×H(5,𝔽₂)	1	14	yes	—
138	C₂²	C₂⁷	1	1	yes	characteristic
139	C₄	—	2	32	no	—
140	C₄	—	2	32	no	—
141	C₄	—	2	8	no	—
142	C₄	C₂³×#41	1	32	yes	—
143	C₂	—	4	32	no	—
144	C₂	C₂²×#555	1	28	yes	direct product factor
145	C₂	C₂⁴×#41	1	2	yes	—
146	C₂	C₂³×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
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×#555	C₂	direct (N × H)
2. C₂³×C₄⋊Q₈	C₂	semidirect
3. C₂³×#124	C₂	semidirect
4. C₂³×#121	C₂	semidirect
5. C₂³×C₄×D₄	C₂	semidirect
6. C₂³×#117	C₂	semidirect
11. C₂×#555	C₂²	direct (N × H)
12. C₂²×C₄⋊Q₈	C₂²	semidirect
13. C₂²×#124	C₂²	semidirect
14. C₂²×#121	C₂²	semidirect
15. C₂²×C₄×D₄	C₂²	semidirect
16. C₂²×#117	C₂²	semidirect
36. #555	C₂³	direct (N × H)
37. C₂×C₄⋊Q₈	C₂³	semidirect
38. C₂×#124	C₂³	semidirect
39. C₂×#121	C₂³	semidirect
40. C₂×C₄×D₄	C₂³	semidirect
41. C₂×#117	C₂³	semidirect
69. C₄⋊Q₈	C₂⁴	semidirect
70. #124	C₂⁴	semidirect
71. #121	C₂⁴	semidirect
72. C₄×D₄	C₂⁴	semidirect
73. #117	C₂⁴	semidirect
114. C₂³	#555	direct (N × H)
135. C₂²	C₂×#555	direct (N × H)
144. C₂	C₂²×#555	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×#555	2
C₂³×C₄⋊Q₈	2
C₂³×#124	2
C₂³×#121	2
C₂³×C₄×D₄	2
C₂³×#117	2

Subgroups — C23×#555 #10585930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#555 #10585930