Subgroups — C₂³×#446 #10574211

Order 512.

8242 conjugacy classes of subgroups.

The subgroup lattice has 151 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#446	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄⋊C₈	C₂	1	2	yes	maximal
5	C₂³×#97	C₂	1	2	yes	maximal
6	C₂³×C₄×C₈	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Q₈	C₄	1	1	yes	characteristic
8	C₂⁴×C₈	C₂²	1	1	yes	characteristic
9	C₂⁴×C₈	C₂²	1	2	yes	—
10	C₂⁴×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Q₈	C₄	1	28	yes	—
12	C₂²×C₄×Q₈	C₂²	1	28	yes	—
13	C₂³×C₄⋊C₄	C₄	1	2	yes	—
14	C₂³×C₄⋊C₄	C₄	1	1	yes	characteristic
15	C₂³×C₄²	C₂²	1	1	yes	characteristic
16	C₂³×C₄²	C₂²	1	2	yes	—
17	C₂×#446	C₂²	1	448	yes	direct product factor
18	C₂²×C₄⋊C₈	C₂²	1	56	yes	—
19	C₂²×C₄⋊C₈	C₂²	1	28	yes	—
20	C₂²×#97	C₂²	1	56	yes	—
21	C₂²×C₄×C₈	C₂²	1	28	yes	—
22	C₂³×Q₈	—	2	16	no	—
23	C₂³×Q₈	C₂×C₄	1	28	yes	—
24	C₂⁴×C₄	C₂×C₄	1	2	yes	—
25	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂⁴×C₄	C₂×C₄	1	2	yes	—
27	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂⁴×C₄	C₂³	1	1	yes	characteristic
29	C₂³×C₈	—	2	8	no	—
30	C₂³×C₈	—	2	16	no	—
31	C₂³×C₈	Q₈	1	16	yes	complemented
32	C₂³×C₈	C₂³	1	14	yes	—
33	C₂³×C₈	C₂³	1	28	yes	—
34	C₂³×C₈	C₂³	1	14	yes	—
35	C₂×C₄×Q₈	C₂×C₄	1	336	yes	—
36	C₂×C₄×Q₈	C₂³	1	112	yes	—
37	C₂²×C₄⋊C₄	C₂×C₄	1	56	yes	—
38	C₂²×C₄⋊C₄	C₂×C₄	1	28	yes	—
39	C₂²×C₄²	C₂×C₄	1	28	yes	—
40	C₂²×C₄²	C₂×C₄	1	14	yes	—
41	C₂²×C₄²	C₂³	1	14	yes	—
42	C₂²×C₄²	C₂³	1	28	yes	—
43	#446	C₂³	1	512	yes	direct product factor
44	C₂×C₄⋊C₈	C₂³	1	112	yes	—
45	C₂×C₄⋊C₈	C₂³	1	224	yes	—
46	C₂×#97	C₂³	1	224	yes	—
47	C₂×C₄×C₈	C₂³	1	112	yes	—
48	C₂⁵	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Q₈	—	2	224	no	—
50	C₂²×Q₈	C₂²×C₄	1	112	yes	—
51	C₂³×C₄	—	2	16	no	—
52	C₂³×C₄	—	2	8	no	—
53	C₂³×C₄	—	2	16	no	—
54	C₂³×C₄	M₁₆	1	16	yes	—
55	C₂³×C₄	C₂²×C₄	1	14	yes	—
56	C₂³×C₄	C₂²×C₄	1	28	yes	—
57	C₂³×C₄	C₂²×C₄	1	28	yes	—
58	C₂³×C₄	C₂²×C₄	1	7	yes	—
59	C₂³×C₄	C₂²×C₄	1	14	yes	—
60	C₂³×C₄	C₂×Q₈	1	8	yes	—
61	C₂³×C₄	#41	1	8	yes	—
62	C₂³×C₄	C₂⁴	1	7	yes	—
63	C₂²×C₈	—	2	56	no	—
64	C₂²×C₈	—	2	112	no	—
65	C₂²×C₈	C₂×Q₈	1	112	yes	complemented
66	C₂²×C₈	C₂⁴	1	28	yes	—
67	C₂²×C₈	C₂⁴	1	56	yes	—
68	C₂²×C₈	C₂⁴	1	28	yes	—
69	C₄×Q₈	C₂²×C₄	1	448	yes	—
70	C₄×Q₈	C₂⁴	1	64	yes	—
71	C₂×C₄⋊C₄	C₂²×C₄	1	224	yes	—
72	C₂×C₄⋊C₄	C₂²×C₄	1	112	yes	—
73	C₂×C₄²	C₂²×C₄	1	168	yes	—
74	C₂×C₄²	C₂²×C₄	1	84	yes	—
75	C₂×C₄²	C₂⁴	1	28	yes	—
76	C₂×C₄²	C₂⁴	1	56	yes	—
77	C₄⋊C₈	C₂⁴	1	64	yes	—
78	C₄⋊C₈	C₂⁴	1	128	yes	—
79	#97	C₂⁴	1	128	yes	—
80	C₄×C₈	C₂⁴	1	64	yes	—
81	C₂⁴	C₄×Q₈	1	8	yes	—
82	C₂⁴	C₂×M₁₆	1	8	yes	—
83	C₂⁴	#131	1	8	yes	—
84	C₂⁴	C₂³×C₄	1	7	yes	—
85	C₂×Q₈	—	2	448	no	—
86	C₂×Q₈	C₂³×C₄	1	64	yes	—
87	C₂²×C₄	—	2	112	no	—
88	C₂²×C₄	—	2	56	no	—
89	C₂²×C₄	—	2	112	no	—
90	C₂²×C₄	C₄×Q₈	1	56	yes	—
91	C₂²×C₄	C₂×M₁₆	1	112	yes	—
92	C₂²×C₄	C₂³×C₄	1	28	yes	—
93	C₂²×C₄	C₂³×C₄	1	56	yes	—
94	C₂²×C₄	C₂³×C₄	1	56	yes	—
95	C₂²×C₄	C₂³×C₄	1	21	yes	—
96	C₂²×C₄	C₂³×C₄	1	28	yes	—
97	C₂²×C₄	C₂²×Q₈	1	28	yes	—
98	C₂²×C₄	C₂×#41	1	28	yes	—
99	C₂²×C₄	C₂⁵	1	7	yes	—
100	C₂×C₈	—	2	56	no	—
101	C₂×C₈	—	2	112	no	—
102	C₂×C₈	C₂²×Q₈	1	112	yes	complemented
103	C₂×C₈	C₂⁵	1	8	yes	—
104	C₂×C₈	C₂⁵	1	16	yes	—
105	C₂×C₈	C₂⁵	1	8	yes	—
106	C₄⋊C₄	C₂³×C₄	1	128	yes	—
107	C₄⋊C₄	C₂³×C₄	1	64	yes	—
108	C₄²	C₂³×C₄	1	112	yes	—
109	C₄²	C₂³×C₄	1	56	yes	—
110	C₄²	C₂⁵	1	8	yes	—
111	C₄²	C₂⁵	1	16	yes	—
112	C₂³	#446	1	64	yes	direct product factor
113	C₂³	C₂×C₄×Q₈	1	28	yes	—
114	C₂³	C₂²×M₁₆	1	28	yes	—
115	C₂³	C₂×#131	1	28	yes	—
116	C₂³	C₂⁴×C₄	1	7	yes	—
117	Q₈	—	2	128	no	—
118	C₂×C₄	—	2	112	no	—
119	C₂×C₄	—	2	56	no	—
120	C₂×C₄	—	2	112	no	—
121	C₂×C₄	C₂×C₄×Q₈	1	84	yes	—
122	C₂×C₄	C₂²×M₁₆	1	112	yes	—
123	C₂×C₄	C₂⁴×C₄	1	8	yes	—
124	C₂×C₄	C₂⁴×C₄	1	16	yes	—
125	C₂×C₄	C₂⁴×C₄	1	16	yes	—
126	C₂×C₄	C₂⁴×C₄	1	7	yes	—
127	C₂×C₄	C₂⁴×C₄	1	8	yes	—
128	C₂×C₄	C₂³×Q₈	1	14	yes	—
129	C₂×C₄	C₂²×#41	1	14	yes	—
130	C₂×C₄	C₂⁶	1	1	yes	characteristic
131	C₈	—	2	8	no	—
132	C₈	—	2	16	no	—
133	C₈	C₂³×Q₈	1	16	yes	complemented
134	C₂²	C₂×#446	1	112	yes	direct product factor
135	C₂²	C₂²×C₄×Q₈	1	14	yes	—
136	C₂²	C₂³×M₁₆	1	14	yes	—
137	C₂²	C₂²×#131	1	14	yes	—
138	C₂²	C₂⁵×C₄	1	1	yes	characteristic
139	C₄	—	2	16	no	—
140	C₄	—	2	8	no	—
141	C₄	—	2	16	no	—
142	C₄	C₂²×C₄×Q₈	1	14	yes	—
143	C₄	C₂³×M₁₆	1	16	yes	—
144	C₄	C₂⁴×Q₈	1	1	yes	characteristic
145	C₄	C₂³×#41	1	1	yes	characteristic
146	C₂	C₂²×#446	1	28	yes	direct product factor
147	C₂	C₂³×C₄×Q₈	1	1	yes	characteristic
148	C₂	C₂⁴×M₁₆	1	1	yes	characteristic
149	C₂	C₂³×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂³
commutator	C₂²	C₂⁵×C₄
Frattini, cosocle	C₂×C₄	C₂⁶
socle, prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×#446	C₂	direct (N × H)
17. C₂×#446	C₂²	direct (N × H)
31. C₂³×C₈	Q₈	semidirect
43. #446	C₂³	direct (N × H)
65. C₂²×C₈	C₂×Q₈	semidirect
102. C₂×C₈	C₂²×Q₈	semidirect
112. C₂³	#446	direct (N × H)
133. C₈	C₂³×Q₈	semidirect
134. C₂²	C₂×#446	direct (N × H)
146. C₂	C₂²×#446	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×Q₈	2
C₂²×#446	2
C₂³×C₄⋊C₈	2
C₂³×#97	2
C₂³×C₄×C₈	2

Subgroups — C23×#446 #10574211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#446 #10574211