Subgroups — C₂×D₄×C₃₂ #512739

Order 512.

498 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	D₄×C₃₂	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₁₆	C₂²	1	2	yes	—
8	C₂×C₈×D₄	C₄	1	1	yes	characteristic
9	C₂²×C₃₂	—	2	8	no	—
10	C₂²×C₃₂	C₂²	1	2	yes	—
11	C₂²×C₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₃₂	C₂²	1	1	yes	characteristic
13	C₂²×C₃₂	C₂²	1	8	yes	complemented
14	D₄×C₁₆	C₄	1	4	yes	—
15	D₄×C₁₆	C₂²	1	4	yes	—
16	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₄⋊C₃₂	C₂²	1	4	yes	complemented
20	C₂²⋊C₃₂	C₂²	1	8	yes	complemented
21	C₄×C₃₂	C₂²	1	4	yes	complemented
22	C₂³×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×D₄	C₈	1	1	yes	characteristic
24	C₂²×C₁₆	—	2	4	no	—
25	C₂²×C₁₆	—	2	4	no	—
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₁₆	C₂×C₄	1	4	yes	—
29	C₂²×C₁₆	C₂³	1	4	yes	—
30	C₂²×C₁₆	C₂³	1	1	yes	characteristic
31	C₈×D₄	C₈	1	4	yes	—
32	C₈×D₄	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₈	C₈	1	2	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃₂	—	2	4	no	—
37	C₂×C₃₂	—	2	2	no	—
38	C₂×C₃₂	—	2	16	no	—
39	C₂×C₃₂	D₄	1	4	yes	direct product factor
40	C₂×C₃₂	C₂³	1	2	yes	—
41	C₂×C₃₂	C₂³	1	4	yes	—
42	C₂×C₃₂	C₂³	1	2	yes	—
43	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
44	C₂²⋊C₁₆	C₂×C₄	1	8	yes	—
45	C₄×C₁₆	C₂×C₄	1	2	yes	—
46	C₄×C₁₆	C₂³	1	2	yes	—
47	C₂²×D₄	C₁₆	1	1	yes	characteristic
48	C₂³×C₄	C₂×C₈	1	2	yes	—
49	C₂²×C₈	—	2	4	no	—
50	C₂²×C₈	—	2	4	no	—
51	C₂²×C₈	C₂×C₈	1	2	yes	—
52	C₂²×C₈	C₂×C₈	1	4	yes	—
53	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₈	C₂²×C₄	1	4	yes	—
55	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₄×D₄	C₁₆	1	4	yes	—
57	C₄×D₄	C₂×C₈	1	4	yes	—
58	C₂×C₄⋊C₄	C₁₆	1	1	yes	characteristic
59	C₂×C₂²⋊C₄	C₁₆	1	2	yes	—
60	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
61	C₂×C₁₆	—	2	4	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	8	no	—
64	C₂×C₁₆	—	2	4	no	—
65	C₂×C₁₆	—	2	4	no	—
66	C₂×C₁₆	C₂²×C₄	1	2	yes	—
67	C₂×C₁₆	C₂²×C₄	1	4	yes	—
68	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂×D₄	1	2	yes	—
70	C₂×C₁₆	#41	1	2	yes	—
71	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₈	C₂×C₈	1	4	yes	—
73	C₂²⋊C₈	C₂×C₈	1	8	yes	—
74	C₄×C₈	C₂×C₈	1	2	yes	—
75	C₄×C₈	C₂²×C₄	1	2	yes	—
76	C₃₂	—	2	4	no	—
77	C₃₂	—	2	2	no	—
78	C₃₂	C₂×D₄	1	4	yes	direct product factor
79	C₂⁴	C₂×C₁₆	1	2	yes	—
80	C₂×D₄	C₃₂	1	8	yes	direct product factor
81	C₂×D₄	C₂×C₁₆	1	4	yes	—
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	—	2	4	no	—
84	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
85	C₂²×C₄	C₂×C₁₆	1	2	yes	—
86	C₂²×C₄	C₂×C₁₆	1	4	yes	—
87	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
88	C₂²×C₄	C₂²×C₈	1	4	yes	—
89	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	2	8	no	—
95	C₂×C₈	C₄×D₄	1	2	yes	—
96	C₂×C₈	C₂²×C₈	1	2	yes	—
97	C₂×C₈	C₂²×C₈	1	4	yes	—
98	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₈	#131	1	2	yes	—
100	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
101	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
102	C₂²⋊C₄	C₂×C₁₆	1	8	yes	—
103	C₄²	C₂×C₁₆	1	2	yes	—
104	C₄²	C₂²×C₈	1	2	yes	—
105	C₁₆	—	2	4	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	C₄×D₄	1	2	yes	—
108	C₁₆	C₂²×D₄	1	1	yes	characteristic
109	C₁₆	C₂×#41	1	1	yes	characteristic
110	C₂³	—	2	4	no	—
111	C₂³	—	2	4	no	—
112	C₂³	C₂×C₃₂	1	8	yes	complemented
113	C₂³	C₂²×C₁₆	1	4	yes	—
114	C₂³	C₂²×C₁₆	1	1	yes	characteristic
115	D₄	C₂×C₃₂	1	16	yes	direct product factor
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	C₂×C₃₂	1	4	yes	complemented
122	C₂×C₄	C₈×D₄	1	2	yes	—
123	C₂×C₄	C₂²×C₁₆	1	2	yes	—
124	C₂×C₄	C₂²×C₁₆	1	4	yes	—
125	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
126	C₂×C₄	C₂²×C₁₆	1	2	yes	—
127	C₂×C₄	#504	1	2	yes	—
128	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	2	4	no	—
131	C₈	C₈×D₄	1	2	yes	—
132	C₈	C₂×C₄×D₄	1	1	yes	characteristic
133	C₈	C₂×#131	1	1	yes	characteristic
134	C₂²	—	2	16	no	—
135	C₂²	—	2	4	no	—
136	C₂²	—	2	4	no	—
137	C₂²	D₄×C₁₆	1	2	yes	—
138	C₂²	C₂²×C₃₂	1	2	yes	—
139	C₂²	C₂²×C₃₂	1	8	yes	complemented
140	C₂²	#2258	1	2	yes	—
141	C₂²	C₂³×C₁₆	1	1	yes	characteristic
142	C₄	—	2	4	no	—
143	C₄	—	2	2	no	—
144	C₄	D₄×C₁₆	1	2	yes	—
145	C₄	C₂²×C₃₂	1	4	yes	complemented
146	C₄	C₂×C₈×D₄	1	1	yes	characteristic
147	C₄	C₂×#504	1	1	yes	characteristic
148	C₂	—	2	8	no	—
149	C₂	D₄×C₃₂	1	4	yes	direct product factor
150	C₂	C₂×D₄×C₁₆	1	1	yes	characteristic
151	C₂	C₂³×C₃₂	1	1	yes	characteristic
152	C₂	C₂×#2258	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃₂	C₂²
commutator	C₂	C₂³×C₃₂
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂³	C₂²×C₁₆
prime radical	C₂²×D₄	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃₂	C₂	semidirect
3. D₄×C₃₂	C₂	direct (N × H)
4. C₂×C₄⋊C₃₂	C₂	semidirect
5. C₂×C₂²⋊C₃₂	C₂	semidirect
6. C₂×C₄×C₃₂	C₂	semidirect
13. C₂²×C₃₂	C₂²	semidirect
19. C₄⋊C₃₂	C₂²	semidirect
20. C₂²⋊C₃₂	C₂²	semidirect
21. C₄×C₃₂	C₂²	semidirect
39. C₂×C₃₂	D₄	direct (N × H)
78. C₃₂	C₂×D₄	direct (N × H)
80. C₂×D₄	C₃₂	direct (N × H)
112. C₂³	C₂×C₃₂	semidirect
115. D₄	C₂×C₃₂	direct (N × H)
121. C₂×C₄	C₂×C₃₂	semidirect
139. C₂²	C₂²×C₃₂	semidirect
145. C₄	C₂²×C₃₂	semidirect
149. C₂	D₄×C₃₂	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×C₃₂	2
C₂×D₄×C₁₆	2
D₄×C₃₂	2
C₂×C₄⋊C₃₂	2
C₂×C₂²⋊C₃₂	2
C₂×C₄×C₃₂	2

Subgroups — C2×D4×C32 #512739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×C₃₂ #512739