Subgroups — #16848771

Order 3968.

552 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₃₁	C₂	1	2	yes	maximal
2	#14520956	C₂	1	2	yes	maximal
3	#14520948	C₂	1	2	yes	maximal
4	#14520935	C₂	1	4	yes	maximal
5	C₄⋊C₄×Dic₃₁	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
10	C₃₁×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	Q₈×Dic₃₁	—	2	8	no	—
15	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
23	C₃₁⋊#95	C₂²	1	2	yes	—
24	C₃₁⋊#95	C₂²	1	2	yes	—
25	C₃₁⋊#95	C₂²	1	4	yes	—
26	C₃₁⋊#95	C₂²	1	4	yes	—
27	C₂×Q₈×C₃₁	—	2	4	no	—
28	C₂×Q₈×C₃₁	—	2	4	no	—
29	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×Dic₃₁	C₂³	1	2	yes	—
35	C₂²×Dic₃₁	C₂³	1	2	yes	—
36	C₂²×Dic₃₁	C₂³	1	2	yes	—
37	C₂²×Dic₃₁	C₂³	1	2	yes	—
38	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
39	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
40	C₄²×C₃₁	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₃₁	—	2	2	no	—
42	C₄⋊Dic₃₁	—	2	2	no	—
43	C₄⋊Dic₃₁	—	2	8	no	—
44	C₄⋊Dic₃₁	D₄	1	4	yes	—
45	C₄⋊Dic₃₁	Q₈	1	4	yes	complemented
46	C₄×Dic₃₁	—	2	4	no	—
47	C₄×Dic₃₁	—	2	4	no	—
48	C₄×Dic₃₁	—	2	8	no	—
49	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
50	Q₈×C₃₁	—	2	16	no	—
51	C₂×C₄×C₃₁	—	2	4	no	—
52	C₂×C₄×C₃₁	—	2	4	no	—
53	C₂×C₄×C₃₁	—	2	1	no	—
54	C₂×C₄×C₃₁	—	2	1	no	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₃₁	#41	1	2	yes	—
62	C₂×C₄×C₃₁	#41	1	2	yes	—
63	C₂×Dic₃₁	—	2	2	no	—
64	C₂×Dic₃₁	—	2	2	no	—
65	C₂×Dic₃₁	—	2	2	no	—
66	C₂×Dic₃₁	—	2	2	no	—
67	C₂×Dic₃₁	—	2	2	no	—
68	C₂×Dic₃₁	—	2	2	no	—
69	C₂×Dic₃₁	—	2	2	no	—
70	C₂×Dic₃₁	—	2	2	no	—
71	C₂×Dic₃₁	—	2	2	no	—
72	C₂×Dic₃₁	—	2	2	no	—
73	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
74	C₂×Dic₃₁	C₂×Q₈	1	4	yes	—
75	#2365	—	31	1	no	maximal, Sylow
76	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
77	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
80	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
81	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
82	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
83	C₄×C₃₁	—	2	8	no	—
84	C₄×C₃₁	—	2	2	no	—
85	C₄×C₃₁	C₄×D₄	1	4	yes	—
86	C₄×C₃₁	C₄×Q₈	1	4	yes	complemented
87	Dic₃₁	—	2	4	no	—
88	Dic₃₁	—	2	4	no	—
89	Dic₃₁	—	4	2	no	—
90	Dic₃₁	—	4	2	no	—
91	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
92	C₂×C₄×Q₈	—	31	2	no	—
93	#391	—	31	2	no	—
94	#389	—	31	2	no	—
95	#387	—	31	4	no	—
96	C₄×C₄⋊C₄	—	31	2	no	—
97	C₄×C₄⋊C₄	—	31	1	no	—
98	C₄×C₄⋊C₄	—	31	1	no	—
99	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₃₁	#519	1	1	yes	characteristic
102	C₂×C₃₁	#547	1	1	yes	characteristic
103	C₂×C₃₁	D₄×Q₈	1	1	yes	characteristic
104	C₂×C₃₁	#556	1	1	yes	characteristic
105	C₂×C₃₁	#557	1	1	yes	characteristic
106	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
107	C₄⋊Q₈	Dic₃₁	1	8	yes	complemented
108	C₄×Q₈	—	62	8	no	—
109	C₂×C₄⋊C₄	—	31	1	no	—
110	C₂×C₄⋊C₄	—	31	1	no	—
111	C₂×C₄⋊C₄	—	31	4	no	—
112	C₂×C₄⋊C₄	—	31	1	no	—
113	C₂×C₄⋊C₄	—	31	1	no	—
114	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
115	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
116	C₂×C₄²	—	31	4	no	—
117	C₂×C₄²	—	31	2	no	—
118	C₂×C₄²	—	31	2	no	—
119	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
120	#95	—	31	2	no	—
121	#95	—	31	2	no	—
122	#95	—	31	4	no	—
123	#95	—	31	4	no	—
124	C₃₁	#2365	1	1	yes	characteristic, complemented, Sylow
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	2	4	no	—
127	C₂×Q₈	C₂×Dic₃₁	1	8	yes	—
128	C₂²×C₄	—	31	2	no	—
129	C₂²×C₄	—	31	2	no	—
130	C₂²×C₄	—	31	2	no	—
131	C₂²×C₄	—	31	2	no	—
132	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
134	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
136	C₄⋊C₄	—	31	4	no	—
137	C₄⋊C₄	—	31	4	no	—
138	C₄⋊C₄	—	62	8	no	—
139	C₄⋊C₄	—	62	2	no	—
140	C₄⋊C₄	—	62	2	no	—
141	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
142	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
143	C₄²	—	62	8	no	—
144	C₄²	—	62	4	no	—
145	C₄²	—	62	4	no	—
146	C₄²	C₂×Dic₃₁	1	4	yes	—
147	C₂³	C₂³×D₃₁	1	1	yes	characteristic
148	Q₈	—	2	16	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	31	4	no	—
154	C₂×C₄	—	31	4	no	—
155	C₂×C₄	—	62	2	no	—
156	C₂×C₄	—	62	2	no	—
157	C₂×C₄	—	62	2	no	—
158	C₂×C₄	—	62	2	no	—
159	C₂×C₄	—	62	2	no	—
160	C₂×C₄	—	62	2	no	—
161	C₂×C₄	—	62	2	no	—
162	C₂×C₄	—	62	2	no	—
163	C₂×C₄	—	62	2	no	—
164	C₂×C₄	—	62	2	no	—
165	C₂×C₄	D₄×D₃₁	1	2	yes	—
166	C₂×C₄	#92497	1	2	yes	—
167	C₂×C₄	Q₈×D₃₁	1	2	yes	—
168	C₂×C₄	#92499	1	2	yes	—
169	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
170	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
171	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
172	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
173	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
174	C₂²	C₂×#92497	1	1	yes	characteristic
175	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
176	C₂²	C₂×#92499	1	1	yes	characteristic
177	C₂²	#11758564	1	1	yes	characteristic
178	C₂²	#11758564	1	1	yes	characteristic
179	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
180	C₄	—	2	8	no	—
181	C₄	—	2	2	no	—
182	C₄	—	62	4	no	—
183	C₄	—	62	4	no	—
184	C₄	—	124	2	no	—
185	C₄	—	124	2	no	—
186	C₄	D₄×Dic₃₁	1	4	yes	—
187	C₄	Q₈×Dic₃₁	1	4	yes	complemented
188	C₂	C₂×D₄×Dic₃₁	1	1	yes	characteristic
189	C₂	C₂×Q₈×Dic₃₁	1	1	yes	characteristic
190	C₂	#14521527	1	1	yes	characteristic
191	C₂	#14521702	1	1	yes	characteristic
192	C₂	#14521714	1	1	yes	characteristic
193	C₂	#14521773	1	1	yes	characteristic
194	C₂	#14521774	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#2365
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
10. C₃₁×C₄⋊Q₈	C₄	semidirect
45. C₄⋊Dic₃₁	Q₈	semidirect
86. C₄×C₃₁	C₄×Q₈	semidirect
107. C₄⋊Q₈	Dic₃₁	semidirect
124. C₃₁	#2365	semidirect
187. C₄	Q₈×Dic₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₃₁	2
#14520956	2
#14520948	2
#14520935	2
C₄⋊C₄×Dic₃₁	2
C₄×C₄⋊Dic₃₁	2
C₂×C₃₁×C₄⋊Q₈	2
#2365	31