Subgroups — Q₈×C₃₁×#106 #21479483

Order 7936.

838 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal
3	C₃₁×#1948	C₂	1	6	yes	maximal
4	C₃₁×#1844	C₂	1	3	yes	maximal
5	C₄×C₃₁×#106	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
7	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₃₁	C₄	1	8	yes	complemented
9	C₂×C₃₁×#106	C₂²	1	3	yes	—
10	C₂×C₃₁×#106	C₂²	1	6	yes	—
11	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
14	C₃₁×#389	—	2	6	no	—
15	C₃₁×#384	C₂²	1	6	yes	—
16	C₄×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
17	C₃₁×#340	C₂²	1	6	yes	—
18	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
20	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₄×Q₈×C₃₁	—	4	8	no	—
22	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
23	C₄×Q₈×C₃₁	Q₈	1	4	yes	—
24	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
26	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
27	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
29	C₂×C₄²×C₃₁	—	2	6	no	—
30	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
31	C₃₁×#106	—	2	6	no	—
32	C₃₁×#106	—	4	6	no	—
33	C₃₁×#106	Q₈	1	4	yes	direct product factor
34	C₃₁×C₄⋊C₈	C₂×C₄	1	12	yes	—
35	C₄×C₈×C₃₁	C₂×C₄	1	12	yes	—
36	C₃₁×#95	—	2	6	no	—
37	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
38	C₂×Q₈×C₃₁	SD₁₆	1	8	yes	—
39	C₂²×C₄×C₃₁	—	2	2	no	—
40	C₂²×C₄×C₃₁	—	2	6	no	—
41	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
44	C₂×C₈×C₃₁	—	2	3	no	—
45	C₂×C₈×C₃₁	—	2	3	no	—
46	C₂×C₈×C₃₁	C₂²×C₄	1	6	yes	—
47	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₃₁	#41	1	2	yes	—
50	C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₄⋊C₄×C₃₁	—	2	6	no	—
52	C₄⋊C₄×C₃₁	—	2	6	no	—
53	C₄⋊C₄×C₃₁	—	4	12	no	—
54	C₄⋊C₄×C₃₁	—	8	6	no	—
55	C₄⋊C₄×C₃₁	C₄⋊C₄	1	12	yes	—
56	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
57	C₄²×C₃₁	—	4	12	no	—
58	C₄²×C₃₁	C₂²×C₄	1	6	yes	—
59	C₄²×C₃₁	C₂×Q₈	1	6	yes	—
60	Q₈×#106	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₃₁	#106	1	16	yes	direct product factor
63	C₂×C₄×C₃₁	—	2	6	no	—
64	C₂×C₄×C₃₁	—	2	4	no	—
65	C₂×C₄×C₃₁	—	4	2	no	—
66	C₂×C₄×C₃₁	—	4	2	no	—
67	C₂×C₄×C₃₁	—	4	6	no	—
68	C₂×C₄×C₃₁	—	4	6	no	—
69	C₂×C₄×C₃₁	—	4	6	no	—
70	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	6	yes	—
71	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₃₁	C₂×SD₁₆	1	12	yes	—
73	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
77	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
79	C₈×C₃₁	—	2	6	no	—
80	C₈×C₃₁	C₄×Q₈	1	4	yes	complemented
81	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
82	Q₈×C₄⋊C₄	C₂×C₃₁	1	2	yes	—
83	#1948	C₂×C₃₁	1	6	yes	—
84	#1844	C₂×C₃₁	1	3	yes	—
85	C₄×#106	C₂×C₃₁	1	3	yes	—
86	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
88	C₂²×C₃₁	#550	1	1	yes	characteristic
89	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
90	C₂²×C₃₁	#577	1	2	yes	—
91	C₄×C₃₁	—	2	6	no	—
92	C₄×C₃₁	—	4	4	no	—
93	C₄×C₃₁	—	8	6	no	—
94	C₄×C₃₁	C₂×#106	1	12	yes	—
95	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
96	C₄×C₃₁	#520	1	1	yes	characteristic
97	C₄×C₃₁	#557	1	1	yes	characteristic
98	C₄×C₃₁	Q₈²	1	1	yes	characteristic
99	C₂×C₄×Q₈	—	2	2	no	—
100	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
101	C₈×Q₈	C₄×C₃₁	1	8	yes	complemented
102	C₂×#106	C₂²×C₃₁	1	3	yes	—
103	C₂×#106	C₂²×C₃₁	1	6	yes	—
104	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
105	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
106	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
107	#389	—	2	6	no	—
108	#384	C₂²×C₃₁	1	6	yes	—
109	C₄×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
110	#340	C₂²×C₃₁	1	6	yes	—
111	C₂×C₃₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
113	C₂×C₃₁	#2912	1	1	yes	characteristic
114	C₂×C₃₁	Q₈×SD₁₆	1	2	yes	—
115	C₂×C₃₁	#3392	1	2	yes	—
116	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
117	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
118	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
119	C₄×Q₈	—	4	8	no	—
120	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
121	C₄×Q₈	Q₈×C₃₁	1	4	yes	—
122	C₂×C₄⋊C₄	—	2	6	no	—
123	C₂×C₄⋊C₄	—	2	6	no	—
124	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
125	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
126	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
127	C₂×C₄²	—	2	6	no	—
128	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
129	#106	—	2	6	no	—
130	#106	—	4	6	no	—
131	#106	Q₈×C₃₁	1	4	yes	direct product factor
132	C₄⋊C₈	C₂×C₄×C₃₁	1	12	yes	—
133	C₄×C₈	C₂×C₄×C₃₁	1	12	yes	—
134	#95	—	2	6	no	—
135	C₃₁	Q₈×#106	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
137	C₂×Q₈	SD₁₆×C₃₁	1	8	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	6	no	—
140	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
141	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
142	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
143	C₂×C₈	—	2	3	no	—
144	C₂×C₈	—	2	3	no	—
145	C₂×C₈	C₂²×C₄×C₃₁	1	6	yes	—
146	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
147	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
148	C₂×C₈	#41×C₃₁	1	2	yes	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	6	no	—
151	C₄⋊C₄	—	2	6	no	—
152	C₄⋊C₄	—	4	12	no	—
153	C₄⋊C₄	—	8	6	no	—
154	C₄⋊C₄	C₄⋊C₄×C₃₁	1	12	yes	—
155	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
156	C₄²	—	4	12	no	—
157	C₄²	C₂²×C₄×C₃₁	1	6	yes	—
158	C₄²	C₂×Q₈×C₃₁	1	6	yes	—
159	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
160	Q₈	C₃₁×#106	1	16	yes	direct product factor
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	6	no	—
166	C₂×C₄	—	4	6	no	—
167	C₂×C₄	—	4	6	no	—
168	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	6	yes	—
169	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	6	yes	—
170	C₂×C₄	C₂×SD₁₆×C₃₁	1	12	yes	—
171	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
173	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
175	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
177	C₈	—	2	6	no	—
178	C₈	C₄×Q₈×C₃₁	1	4	yes	complemented
179	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
180	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
181	C₂²	C₃₁×#550	1	1	yes	characteristic
182	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
183	C₂²	C₃₁×#577	1	2	yes	—
184	C₄	—	2	6	no	—
185	C₄	—	4	4	no	—
186	C₄	—	8	6	no	—
187	C₄	C₂×C₃₁×#106	1	12	yes	—
188	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
189	C₄	C₃₁×#520	1	1	yes	characteristic
190	C₄	C₃₁×#557	1	1	yes	characteristic
191	C₄	Q₈²×C₃₁	1	1	yes	characteristic
192	C₂	Q₈×C₄⋊C₄×C₃₁	1	1	yes	characteristic
193	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic
194	C₂	C₃₁×#2912	1	1	yes	characteristic
195	C₂	Q₈×SD₁₆×C₃₁	1	2	yes	—
196	C₂	C₃₁×#3392	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₃₁	C₄	semidirect
33. C₃₁×#106	Q₈	direct (N × H)
60. Q₈×#106	C₃₁	direct (N × H)
62. Q₈×C₃₁	#106	direct (N × H)
80. C₈×C₃₁	C₄×Q₈	semidirect
101. C₈×Q₈	C₄×C₃₁	semidirect
131. #106	Q₈×C₃₁	direct (N × H)
135. C₃₁	Q₈×#106	direct (N × H)
160. Q₈	C₃₁×#106	direct (N × H)
178. C₈	C₄×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₃₁	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#1948	2
C₃₁×#1844	2
C₄×C₃₁×#106	2
Q₈×#106	31

Subgroups — Q8×C31×#106 #21479483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃₁×#106 #21479483