Subgroups — D₄×M₃₂ #33437

Order 256.

367 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	#2169	C₂	1	2	yes	maximal, complemented
4	#2168	C₂	1	4	yes	maximal, complemented
5	D₄×C₁₆	C₂	1	2	yes	maximal, complemented
6	#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	#2112	C₂	1	2	yes	maximal, complemented
8	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈	C₂²	1	2	yes	—
10	C₂×C₄×D₄	C₄	1	1	yes	characteristic
11	C₂×M₃₂	—	2	8	no	—
12	C₂×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×M₃₂	C₂²	1	2	yes	—
14	C₂×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×M₃₂	C₂²	1	4	yes	complemented
16	C₂²×C₁₆	C₂²	1	4	yes	complemented
17	C₈×D₄	C₄	1	2	yes	—
18	C₈×D₄	C₄	1	1	yes	characteristic
19	C₈×D₄	C₄	1	1	yes	characteristic
20	C₈×D₄	C₂²	1	1	yes	characteristic
21	C₈×D₄	C₂²	1	2	yes	—
22	C₈×D₄	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
24	C₂×C₂²⋊C₈	C₄	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆	C₂²	1	2	yes	complemented
27	C₄⋊C₁₆	C₂²	1	2	yes	complemented
28	C₂²⋊C₁₆	C₂²	1	4	yes	complemented
29	C₂²⋊C₁₆	C₂²	1	4	yes	complemented
30	#346	C₂²	1	2	yes	complemented
31	C₄×C₁₆	C₂²	1	2	yes	complemented
32	C₂²×D₄	C₈	1	1	yes	characteristic
33	C₂³×C₄	C₂×C₄	1	2	yes	—
34	C₂²×C₈	—	2	4	no	—
35	C₂²×C₈	—	2	4	no	—
36	C₂²×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	2	yes	—
41	C₂²×C₈	C₂³	1	2	yes	—
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₄×D₄	C₈	1	2	yes	—
44	C₄×D₄	C₈	1	1	yes	characteristic
45	C₄×D₄	C₈	1	1	yes	characteristic
46	C₄×D₄	C₂×C₄	1	2	yes	—
47	C₄×D₄	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
50	C₂×C₂²⋊C₄	C₈	1	2	yes	—
51	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
52	M₃₂	—	2	4	no	—
53	M₃₂	—	2	2	no	—
54	M₃₂	—	2	8	no	—
55	M₃₂	D₄	1	4	yes	direct product factor
56	C₂×C₁₆	—	2	8	no	—
57	C₂×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₆	C₂³	1	4	yes	—
59	C₂×C₁₆	C₂³	1	2	yes	—
60	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈	C₂×C₄	1	2	yes	—
67	C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₈	C₂³	1	1	yes	characteristic
71	C₂⁴	C₂×C₈	1	2	yes	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	C₂×C₈	1	2	yes	—
74	C₂×D₄	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	2	4	no	—
78	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄	C₂×C₈	1	2	yes	—
80	C₂²×C₄	C₂×C₈	1	2	yes	—
81	C₂²×C₄	C₂×C₈	1	2	yes	—
82	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄	C₂²×C₄	1	2	yes	—
84	C₂²×C₄	C₂²×C₄	1	2	yes	—
85	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₈	#41	1	2	yes	—
101	C₂×C₈	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄	C₂×C₈	1	2	yes	—
103	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
104	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
105	C₂²⋊C₄	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄	C₂×C₈	1	2	yes	—
109	C₄²	C₂×C₈	1	1	yes	characteristic
110	C₄²	C₂×C₈	1	1	yes	characteristic
111	C₄²	C₂²×C₄	1	1	yes	characteristic
112	C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₁₆	—	2	4	no	—
114	C₁₆	—	2	2	no	—
115	C₁₆	C₂×D₄	1	4	yes	complemented
116	C₂³	—	2	4	no	—
117	C₂³	—	2	4	no	—
118	C₂³	—	4	2	no	—
119	C₂³	C₂²×C₈	1	2	yes	—
120	C₂³	C₂²×C₈	1	1	yes	characteristic
121	C₂³	C₂²×C₈	1	2	yes	—
122	D₄	—	2	4	no	—
123	D₄	—	2	2	no	—
124	D₄	M₃₂	1	4	yes	direct product factor
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	C₄×D₄	1	2	yes	—
135	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄	C₂²×C₈	1	2	yes	—
139	C₂×C₄	C₂²×C₈	1	2	yes	—
140	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄	#131	1	2	yes	—
143	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	C₄×D₄	1	2	yes	—
149	C₈	C₂²×D₄	1	1	yes	characteristic
150	C₈	C₂×#41	1	1	yes	characteristic
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	4	8	no	—
159	C₂²	C₈×D₄	1	2	yes	—
160	C₂²	C₂×M₃₂	1	4	yes	complemented
161	C₂²	C₂³×C₈	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	C₈×D₄	1	2	yes	—
168	C₄	C₂×M₃₂	1	2	yes	complemented
169	C₄	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄	C₂×#131	1	1	yes	characteristic
171	C₂	—	2	2	no	—
172	C₂	—	2	4	no	—
173	C₂	—	4	2	no	—
174	C₂	C₂×C₈×D₄	1	1	yes	characteristic
175	C₂	C₂²×M₃₂	1	1	yes	characteristic
176	C₂	#3407	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×M₃₂	C₂	semidirect
3. #2169	C₂	semidirect
4. #2168	C₂	semidirect
5. D₄×C₁₆	C₂	semidirect
6. #2150	C₂	semidirect
7. #2112	C₂	semidirect
8. C₄×M₃₂	C₂	semidirect
15. C₂×M₃₂	C₂²	semidirect
16. C₂²×C₁₆	C₂²	semidirect
26. C₄⋊C₁₆	C₂²	semidirect
27. C₄⋊C₁₆	C₂²	semidirect
28. C₂²⋊C₁₆	C₂²	semidirect
29. C₂²⋊C₁₆	C₂²	semidirect
30. #346	C₂²	semidirect
31. C₄×C₁₆	C₂²	semidirect
55. M₃₂	D₄	direct (N × H)
115. C₁₆	C₂×D₄	semidirect
124. D₄	M₃₂	direct (N × H)
160. C₂²	C₂×M₃₂	semidirect
168. C₄	C₂×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂	2
C₂×C₈×D₄	2
#2169	2
#2168	2
D₄×C₁₆	2
#2150	2
#2112	2
C₄×M₃₂	2

Subgroups — D4×M32 #33437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×M₃₂ #33437