Subgroups — C₂×C₄×Q₈×D₃₁ #16858884

Order 3968.

1664 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×Dic₆₂	C₂	1	3	yes	maximal, complemented
4	C₂×#11758464	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×D₃₁	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₄⋊C₄×D₃₁	C₂	1	3	yes	maximal
7	C₂×C₄²×D₃₁	C₂	1	3	yes	maximal
8	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×Q₈×D₃₁	C₄	1	16	yes	direct product factor
11	C₂×Q₈×D₃₁	C₂²	1	8	yes	—
12	C₂²×C₄×D₃₁	C₂²	1	3	yes	—
13	C₂²×C₄×D₃₁	C₂²	1	3	yes	—
14	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₆₂	C₂²	1	3	yes	—
16	C₄×Q₈×C₃₁	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
18	C₂×C₄²×C₃₁	C₂²	1	3	yes	—
19	Q₈×Dic₃₁	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Dic₃₁	C₂²	1	3	yes	—
21	C₂×Dic₃₁⋊C₄	C₂²	1	6	yes	—
22	C₂×C₄×Dic₃₁	C₂²	1	3	yes	—
23	C₂×C₄×Dic₃₁	C₂²	1	3	yes	—
24	C₂×C₄×Dic₃₁	C₂²	1	3	yes	—
25	C₄⋊C₄×D₃₁	C₂²	1	24	yes	—
26	#11758464	C₂²	1	24	yes	complemented
27	C₄²×D₃₁	C₂²	1	24	yes	—
28	C₄×Dic₆₂	C₂²	1	24	yes	complemented
29	C₂³×D₃₁	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
33	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃₁	C₂³	1	3	yes	—
36	C₂²×Dic₃₁	C₂³	1	3	yes	—
37	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
38	Q₈×D₃₁	C₂×C₄	1	64	yes	direct product factor
39	C₂×C₄×D₃₁	—	2	12	no	—
40	C₂×C₄×D₃₁	C₂×C₄	1	24	yes	—
41	C₂×C₄×D₃₁	Q₈	1	8	yes	direct product factor
42	C₂×C₄×D₃₁	C₂³	1	12	yes	—
43	C₂×C₄×D₃₁	C₂³	1	12	yes	—
44	C₂×C₄×D₃₁	C₂³	1	4	yes	—
45	C₂×Dic₆₂	C₂×C₄	1	24	yes	complemented
46	C₂×Dic₆₂	C₂³	1	12	yes	—
47	C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
48	C₄²×C₃₁	C₂³	1	12	yes	—
49	C₄⋊Dic₃₁	C₂³	1	12	yes	—
50	Dic₃₁⋊C₄	C₂³	1	24	yes	—
51	C₄×Dic₃₁	C₂³	1	12	yes	—
52	C₄×Dic₃₁	C₂³	1	12	yes	—
53	C₄×Dic₃₁	C₂³	1	12	yes	—
54	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
55	C₂²×D₃₁	C₂²×C₄	1	4	yes	—
56	C₂²×D₃₁	C₂×Q₈	1	4	yes	—
57	C₂²×D₃₁	#41	1	4	yes	—
58	C₂²×D₃₁	C₂⁴	1	2	yes	—
59	Q₈×C₃₁	C₂²×C₄	1	16	yes	complemented
60	C₂×C₄×C₃₁	—	2	6	no	—
61	C₂×C₄×C₃₁	C₂²×C₄	1	12	yes	—
62	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	complemented
63	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
64	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
65	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
66	C₂×Dic₃₁	—	2	6	no	—
67	C₂×Dic₃₁	C₂²×C₄	1	12	yes	—
68	C₂×Dic₃₁	C₂×Q₈	1	4	yes	complemented
69	C₂×Dic₃₁	C₂⁴	1	6	yes	—
70	C₂×Dic₃₁	C₂⁴	1	6	yes	—
71	C₂×Dic₃₁	C₂⁴	1	2	yes	—
72	C₄×D₃₁	—	2	24	no	—
73	C₄×D₃₁	C₂²×C₄	1	48	yes	—
74	C₄×D₃₁	C₂×Q₈	1	16	yes	direct product factor
75	Dic₆₂	C₂²×C₄	1	48	yes	complemented
76	C₂²×C₄×Q₈	—	31	1	no	maximal, Sylow
77	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
78	C₂²×C₃₁	C₂²×Q₈	1	2	yes	—
79	C₂²×C₃₁	C₂×#41	1	2	yes	—
80	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
81	C₂×D₃₁	C₄×Q₈	1	16	yes	direct product factor
82	C₂×D₃₁	C₂³×C₄	1	4	yes	—
83	C₂×D₃₁	C₂²×Q₈	1	4	yes	—
84	C₂×D₃₁	C₂×#41	1	4	yes	—
85	C₄×C₃₁	—	2	6	no	—
86	C₄×C₃₁	C₂³×C₄	1	12	yes	—
87	C₄×C₃₁	C₂²×Q₈	1	4	yes	complemented
88	Dic₃₁	—	2	6	no	—
89	Dic₃₁	C₂³×C₄	1	12	yes	—
90	Dic₃₁	C₂²×Q₈	1	4	yes	complemented
91	C₂³×Q₈	—	31	1	no	—
92	C₂×C₄×Q₈	—	31	1	no	—
93	C₂×C₄×Q₈	—	31	16	no	—
94	C₂×C₄×Q₈	—	31	3	no	—
95	C₂×C₄×Q₈	—	31	3	no	—
96	C₂×C₄×Q₈	D₃₁	1	1	yes	characteristic, direct product factor
97	C₂²×C₄⋊C₄	—	31	3	no	—
98	C₂²×C₄²	—	31	3	no	—
99	C₂×C₃₁	C₂×C₄×Q₈	1	4	yes	complemented
100	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
101	C₂×C₃₁	C₂³×Q₈	1	1	yes	characteristic
102	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
103	D₃₁	C₂×C₄×Q₈	1	8	yes	direct product factor
104	C₂²×Q₈	—	31	16	no	—
105	C₂²×Q₈	—	31	8	no	—
106	C₂²×Q₈	—	31	3	no	—
107	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
108	C₂³×C₄	—	31	3	no	—
109	C₂³×C₄	—	31	1	no	—
110	C₂³×C₄	—	31	3	no	—
111	C₄×Q₈	—	31	8	no	—
112	C₄×Q₈	—	31	24	no	—
113	C₄×Q₈	—	31	24	no	—
114	C₄×Q₈	C₂×D₃₁	1	8	yes	direct product factor
115	C₂×C₄⋊C₄	—	31	6	no	—
116	C₂×C₄⋊C₄	—	31	3	no	—
117	C₂×C₄⋊C₄	—	31	24	no	—
118	C₂×C₄⋊C₄	C₂×D₃₁	1	3	yes	—
119	C₂×C₄²	—	31	3	no	—
120	C₂×C₄²	—	31	3	no	—
121	C₂×C₄²	—	31	3	no	—
122	C₂×C₄²	—	31	24	no	—
123	C₂×C₄²	C₂×D₃₁	1	3	yes	—
124	C₃₁	C₂²×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
125	C₂⁴	—	31	1	no	—
126	C₂×Q₈	—	31	64	no	—
127	C₂×Q₈	—	31	24	no	—
128	C₂×Q₈	—	31	12	no	—
129	C₂×Q₈	C₄×D₃₁	1	8	yes	direct product factor
130	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
131	C₂²×C₄	—	31	1	no	—
132	C₂²×C₄	—	31	3	no	—
133	C₂²×C₄	—	31	24	no	—
134	C₂²×C₄	—	31	8	no	—
135	C₂²×C₄	—	31	4	no	—
136	C₂²×C₄	—	31	12	no	—
137	C₂²×C₄	—	31	3	no	—
138	C₂²×C₄	—	31	12	no	—
139	C₂²×C₄	—	62	12	no	—
140	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
141	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
142	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
143	C₄⋊C₄	—	31	24	no	—
144	C₄⋊C₄	—	31	12	no	—
145	C₄⋊C₄	C₂²×D₃₁	1	12	yes	—
146	C₄²	—	31	12	no	—
147	C₄²	—	31	12	no	—
148	C₄²	—	31	12	no	—
149	C₄²	C₂²×D₃₁	1	12	yes	—
150	C₂³	—	31	4	no	—
151	C₂³	—	31	4	no	—
152	C₂³	—	31	4	no	—
153	C₂³	—	31	2	no	—
154	C₂³	C₂³×D₃₁	1	1	yes	characteristic
155	Q₈	—	31	48	no	—
156	Q₈	C₂×C₄×D₃₁	1	16	yes	direct product factor
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	31	12	no	—
159	C₂×C₄	—	31	4	no	—
160	C₂×C₄	—	31	2	no	—
161	C₂×C₄	—	31	6	no	—
162	C₂×C₄	—	31	48	no	—
163	C₂×C₄	—	31	16	no	—
164	C₂×C₄	—	31	6	no	—
165	C₂×C₄	—	62	24	no	—
166	C₂×C₄	—	62	6	no	—
167	C₂×C₄	C₂×C₄×D₃₁	1	12	yes	—
168	C₂×C₄	Q₈×D₃₁	1	4	yes	direct product factor
169	C₂×C₄	C₂³×D₃₁	1	6	yes	—
170	C₂×C₄	C₂³×D₃₁	1	6	yes	—
171	C₂×C₄	C₂³×D₃₁	1	2	yes	—
172	C₂²	—	31	4	no	—
173	C₂²	—	31	4	no	—
174	C₂²	—	31	4	no	—
175	C₂²	—	31	16	no	—
176	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
177	C₂²	C₂×Q₈×D₃₁	1	2	yes	—
178	C₂²	#41×D₃₁	1	2	yes	—
179	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
180	C₄	—	2	6	no	—
181	C₄	—	31	12	no	—
182	C₄	—	31	4	no	—
183	C₄	—	62	6	no	—
184	C₄	C₂²×C₄×D₃₁	1	12	yes	—
185	C₄	C₂×Q₈×D₃₁	1	4	yes	direct product factor
186	C₂	—	31	8	no	—
187	C₂	C₄×Q₈×D₃₁	1	4	yes	direct product factor
188	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
189	C₂	C₂²×Q₈×D₃₁	1	1	yes	characteristic
190	C₂	C₂×#41×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₄×Q₈×C₃₁	C₂
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂³×D₃₁	C₂³
cosocle	C₂²×C₃₁	C₂⁵
hypercenter	C₂×C₄×Q₈	D₃₁
hypocenter	C₃₁	C₂²×C₄×Q₈
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₃₁	C₂	semidirect
3. C₂×C₄×Dic₆₂	C₂	semidirect
4. C₂×#11758464	C₂	semidirect
5. C₄×Q₈×D₃₁	C₂	direct (N × H)
8. C₂×C₄×Q₈×C₃₁	C₂	semidirect
10. C₂×Q₈×D₃₁	C₄	direct (N × H)
16. C₄×Q₈×C₃₁	C₂²	semidirect
19. Q₈×Dic₃₁	C₂²	semidirect
26. #11758464	C₂²	semidirect
28. C₄×Dic₆₂	C₂²	semidirect
30. C₂×Q₈×C₃₁	C₂×C₄	semidirect
38. Q₈×D₃₁	C₂×C₄	direct (N × H)
41. C₂×C₄×D₃₁	Q₈	direct (N × H)
45. C₂×Dic₆₂	C₂×C₄	semidirect
59. Q₈×C₃₁	C₂²×C₄	semidirect
62. C₂×C₄×C₃₁	C₂×Q₈	semidirect
68. C₂×Dic₃₁	C₂×Q₈	semidirect
74. C₄×D₃₁	C₂×Q₈	direct (N × H)
75. Dic₆₂	C₂²×C₄	semidirect
81. C₂×D₃₁	C₄×Q₈	direct (N × H)
87. C₄×C₃₁	C₂²×Q₈	semidirect
90. Dic₃₁	C₂²×Q₈	semidirect
96. C₂×C₄×Q₈	D₃₁	direct (N × H)
99. C₂×C₃₁	C₂×C₄×Q₈	semidirect
103. D₃₁	C₂×C₄×Q₈	direct (N × H)
114. C₄×Q₈	C₂×D₃₁	direct (N × H)
124. C₃₁	C₂²×C₄×Q₈	semidirect
129. C₂×Q₈	C₄×D₃₁	direct (N × H)
156. Q₈	C₂×C₄×D₃₁	direct (N × H)
168. C₂×C₄	Q₈×D₃₁	direct (N × H)
185. C₄	C₂×Q₈×D₃₁	direct (N × H)
187. C₂	C₄×Q₈×D₃₁	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×Q₈×D₃₁	2
C₂×Q₈×Dic₃₁	2
C₂×C₄×Dic₆₂	2
C₂×#11758464	2
C₄×Q₈×D₃₁	2
C₂×C₄⋊C₄×D₃₁	2
C₂×C₄²×D₃₁	2
C₂×C₄×Q₈×C₃₁	2
C₂²×C₄×Q₈	31

Subgroups — C2×C4×Q8×D31 #16858884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Q₈×D₃₁ #16858884