Subgroups — C₂×D₄×C₃₁⋊C₈ #16854384

Order 3968.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
2	D₄×C₃₁⋊C₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊(C₃₁⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₃₁	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
9	C₄×D₄×C₃₁	C₄	1	4	yes	—
10	C₄×D₄×C₃₁	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₃₁⋊C₈	—	2	8	no	—
15	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₃₁⋊C₈	C₂²	1	8	yes	complemented
19	C₂²⋊(C₃₁⋊C₈)	C₂²	1	8	yes	complemented
20	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
21	C₄×C₃₁⋊C₈	C₂²	1	4	yes	complemented
22	C₂⁴×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₃₁	C₈	1	8	yes	complemented
24	C₂×D₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₃₁	—	2	4	no	—
26	C₂²×C₄×C₃₁	—	2	4	no	—
27	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
34	C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
35	C₄²×C₃₁	C₂×C₄	1	2	yes	—
36	C₄²×C₃₁	C₂³	1	2	yes	—
37	C₂×C₃₁⋊C₈	—	2	4	no	—
38	C₂×C₃₁⋊C₈	—	2	2	no	—
39	C₂×C₃₁⋊C₈	—	2	16	no	—
40	C₂×C₃₁⋊C₈	D₄	1	4	yes	direct product factor
41	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
42	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
43	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
44	C₂³×C₃₁	—	2	4	no	—
45	C₂³×C₃₁	—	2	4	no	—
46	C₂³×C₃₁	C₂×C₈	1	8	yes	complemented
47	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
48	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	D₄×C₃₁	C₂×C₈	1	16	yes	complemented
50	C₂×C₄×C₃₁	—	2	4	no	—
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	—	2	8	no	—
53	C₂×C₄×C₃₁	—	2	4	no	—
54	C₂×C₄×C₃₁	—	2	4	no	—
55	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	complemented
56	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₃₁	#41	1	2	yes	—
62	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₃₁⋊C₈	—	2	4	no	—
64	C₃₁⋊C₈	—	2	2	no	—
65	C₃₁⋊C₈	C₂×D₄	1	4	yes	direct product factor
66	C₂×C₈×D₄	—	31	1	no	maximal, Sylow
67	C₂²×C₃₁	—	2	4	no	—
68	C₂²×C₃₁	—	2	4	no	—
69	C₂²×C₃₁	—	2	16	no	—
70	C₂²×C₃₁	C₄×D₄	1	2	yes	—
71	C₂²×C₃₁	C₂²×C₈	1	2	yes	—
72	C₂²×C₃₁	C₂²×C₈	1	8	yes	complemented
73	C₂²×C₃₁	#131	1	2	yes	—
74	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
75	C₄×C₃₁	—	2	4	no	—
76	C₄×C₃₁	—	2	2	no	—
77	C₄×C₃₁	C₄×D₄	1	2	yes	—
78	C₄×C₃₁	C₂²×C₈	1	4	yes	complemented
79	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
81	C₂³×C₈	—	31	2	no	—
82	C₂×C₄×D₄	D₃₁	1	1	yes	characteristic
83	C₈×D₄	—	31	8	no	—
84	C₂×C₄⋊C₈	—	31	1	no	—
85	C₂×C₂²⋊C₈	—	31	2	no	—
86	C₂×C₄×C₈	—	31	1	no	—
87	C₂×C₃₁	—	2	8	no	—
88	C₂×C₃₁	C₈×D₄	1	4	yes	complemented
89	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
90	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
91	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
92	C₂²×D₄	Dic₃₁	1	1	yes	characteristic
93	C₂³×C₄	C₂×D₃₁	1	2	yes	—
94	C₂²×C₈	—	31	1	no	—
95	C₂²×C₈	—	31	8	no	—
96	C₂²×C₈	—	31	2	no	—
97	C₂²×C₈	—	31	1	no	—
98	C₂²×C₈	—	62	8	no	—
99	C₄×D₄	Dic₃₁	1	4	yes	—
100	C₄×D₄	C₂×D₃₁	1	4	yes	—
101	C₂×C₄⋊C₄	Dic₃₁	1	1	yes	characteristic
102	C₂×C₂²⋊C₄	Dic₃₁	1	2	yes	—
103	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
104	C₄⋊C₈	—	31	4	no	—
105	C₂²⋊C₈	—	31	8	no	—
106	C₄×C₈	—	31	4	no	—
107	C₃₁	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
108	C₂⁴	C₂×Dic₃₁	1	2	yes	—
109	C₂×D₄	C₃₁⋊C₈	1	8	yes	direct product factor
110	C₂×D₄	C₂×Dic₃₁	1	4	yes	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
115	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
116	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
118	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
119	C₂×C₈	—	31	4	no	—
120	C₂×C₈	—	31	2	no	—
121	C₂×C₈	—	31	2	no	—
122	C₂×C₈	—	31	4	no	—
123	C₂×C₈	—	62	16	no	—
124	C₂×C₈	—	62	4	no	—
125	C₂×C₈	—	62	2	no	—
126	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
127	C₂²⋊C₄	C₂×Dic₃₁	1	8	yes	—
128	C₄²	C₂×Dic₃₁	1	2	yes	—
129	C₄²	C₂²×D₃₁	1	2	yes	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	4	no	—
132	C₂³	C₂×C₃₁⋊C₈	1	8	yes	complemented
133	C₂³	C₂²×Dic₃₁	1	4	yes	—
134	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
135	D₄	C₂×C₃₁⋊C₈	1	16	yes	direct product factor
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	complemented
142	C₂×C₄	D₄×D₃₁	1	2	yes	—
143	C₂×C₄	#92497	1	2	yes	—
144	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
145	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
146	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
147	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
148	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
149	C₈	—	31	4	no	—
150	C₈	—	62	4	no	—
151	C₈	—	62	2	no	—
152	C₂²	—	2	16	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	C₂²×C₃₁⋊C₈	1	2	yes	—
156	C₂²	C₂²×C₃₁⋊C₈	1	8	yes	complemented
157	C₂²	D₄×Dic₃₁	1	2	yes	—
158	C₂²	#11758525	1	2	yes	—
159	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	2	2	no	—
162	C₄	C₂²×C₃₁⋊C₈	1	4	yes	complemented
163	C₄	D₄×Dic₃₁	1	2	yes	—
164	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
165	C₄	C₂×#92497	1	1	yes	characteristic
166	C₂	—	2	8	no	—
167	C₂	D₄×C₃₁⋊C₈	1	4	yes	direct product factor
168	C₂	C₂×D₄×Dic₃₁	1	1	yes	characteristic
169	C₂	C₂³×C₃₁⋊C₈	1	1	yes	characteristic
170	C₂	C₂×#11758525	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×D₄×C₃₁	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂²×D₄×C₃₁	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₃₁
hypocenter	C₃₁	C₂×C₈×D₄
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃₁⋊C₈	C₂	semidirect
2. D₄×C₃₁⋊C₈	C₂	direct (N × H)
3. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
4. C₂×C₂²⋊(C₃₁⋊C₈)	C₂	semidirect
5. C₂×C₄×C₃₁⋊C₈	C₂	semidirect
18. C₂²×C₃₁⋊C₈	C₂²	semidirect
19. C₂²⋊(C₃₁⋊C₈)	C₂²	semidirect
20. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
21. C₄×C₃₁⋊C₈	C₂²	semidirect
23. C₂×D₄×C₃₁	C₈	semidirect
40. C₂×C₃₁⋊C₈	D₄	direct (N × H)
46. C₂³×C₃₁	C₂×C₈	semidirect
49. D₄×C₃₁	C₂×C₈	semidirect
55. C₂×C₄×C₃₁	C₂×C₈	semidirect
65. C₃₁⋊C₈	C₂×D₄	direct (N × H)
72. C₂²×C₃₁	C₂²×C₈	semidirect
78. C₄×C₃₁	C₂²×C₈	semidirect
88. C₂×C₃₁	C₈×D₄	semidirect
107. C₃₁	C₂×C₈×D₄	semidirect
109. C₂×D₄	C₃₁⋊C₈	direct (N × H)
132. C₂³	C₂×C₃₁⋊C₈	semidirect
135. D₄	C₂×C₃₁⋊C₈	direct (N × H)
141. C₂×C₄	C₂×C₃₁⋊C₈	semidirect
156. C₂²	C₂²×C₃₁⋊C₈	semidirect
162. C₄	C₂²×C₃₁⋊C₈	semidirect
167. C₂	D₄×C₃₁⋊C₈	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×C₃₁⋊C₈	2
D₄×C₃₁⋊C₈	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×C₂²⋊(C₃₁⋊C₈)	2
C₂×C₄×C₃₁⋊C₈	2
C₂×C₄×D₄×C₃₁	2
C₂×C₈×D₄	31

Subgroups — C2×D4×C31⋊C8 #16854384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×C₃₁⋊C₈ #16854384