Subgroups — #16848623

Order 3968.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758464	C₂	1	2	yes	maximal
3	#14520945	C₂	1	2	yes	maximal
4	#14520944	C₂	1	4	yes	maximal
5	#14520918	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₃₁	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#384	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
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	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
18	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₃₁⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
22	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
23	#11758464	—	2	8	no	—
24	C₄×Dic₆₂	C₄	1	8	yes	complemented
25	C₃₁⋊#95	C₂²	1	2	yes	—
26	C₃₁⋊#95	C₂²	1	4	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
28	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃₁	C₂³	1	2	yes	—
32	C₂²×Dic₃₁	C₂³	1	2	yes	—
33	C₂²×Dic₃₁	C₂³	1	2	yes	—
34	C₂²×Dic₃₁	C₂³	1	2	yes	—
35	C₂×Dic₆₂	C₂×C₄	1	4	yes	—
36	C₂×Dic₆₂	D₄	1	4	yes	—
37	C₂×Dic₆₂	Q₈	1	4	yes	—
38	C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₄⋊C₄×C₃₁	—	4	2	no	—
41	C₄²×C₃₁	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₃₁	—	2	2	no	—
43	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₃₁	Q₈	1	4	yes	complemented
45	Dic₃₁⋊C₄	—	2	8	no	—
46	Dic₃₁⋊C₄	—	4	4	no	—
47	Dic₃₁⋊C₄	C₂×C₄	1	8	yes	—
48	C₄×Dic₃₁	—	2	4	no	—
49	C₄×Dic₃₁	—	2	8	no	—
50	C₄×Dic₃₁	C₂×C₄	1	8	yes	—
51	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₃₁	—	2	1	no	—
53	C₂×C₄×C₃₁	—	2	1	no	—
54	C₂×C₄×C₃₁	—	2	4	no	—
55	C₂×C₄×C₃₁	—	2	4	no	—
56	C₂×C₄×C₃₁	—	2	4	no	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₃₁	#41	1	2	yes	—
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	C₂²×C₄	1	4	yes	—
72	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
73	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
74	C₂×Dic₃₁	C₂×Q₈	1	4	yes	—
75	C₂×Dic₃₁	C₂×Q₈	1	4	yes	—
76	Dic₆₂	C₄⋊C₄	1	16	yes	complemented
77	Q₈×C₄⋊C₄	—	31	1	no	maximal, Sylow
78	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
83	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
85	C₄×C₃₁	—	2	2	no	—
86	C₄×C₃₁	—	4	4	no	—
87	C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
88	C₄×C₃₁	C₄×Q₈	1	4	yes	complemented
89	Dic₃₁	—	2	4	no	—
90	Dic₃₁	—	2	4	no	—
91	Dic₃₁	—	4	2	no	—
92	Dic₃₁	C₂×C₄⋊C₄	1	8	yes	—
93	C₂×C₄×Q₈	—	31	2	no	—
94	C₂×C₄×Q₈	—	31	1	no	—
95	#389	—	31	4	no	—
96	#389	—	31	2	no	—
97	#384	—	31	2	no	—
98	#384	D₃₁	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	31	2	no	—
100	C₄×C₄⋊C₄	—	31	1	no	—
101	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
103	C₂×C₃₁	#520	1	1	yes	characteristic
104	C₂×C₃₁	D₄×Q₈	1	1	yes	characteristic
105	C₂×C₃₁	#550	1	1	yes	characteristic
106	C₂×C₃₁	#557	1	1	yes	characteristic
107	C₂×C₃₁	Q₈²	1	1	yes	characteristic
108	C₂²×Q₈	—	31	1	no	—
109	C₄×Q₈	—	31	8	no	—
110	C₄×Q₈	—	62	8	no	—
111	C₂×C₄⋊C₄	—	31	2	no	—
112	C₂×C₄⋊C₄	—	31	1	no	—
113	C₂×C₄⋊C₄	—	31	1	no	—
114	C₂×C₄⋊C₄	—	31	4	no	—
115	C₂×C₄⋊C₄	—	31	1	no	—
116	C₂×C₄⋊C₄	—	31	4	no	—
117	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
118	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
120	C₂×C₄²	—	31	2	no	—
121	C₂×C₄²	—	31	4	no	—
122	C₂×C₄²	—	31	2	no	—
123	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
124	#95	—	31	4	no	—
125	#95	—	31	2	no	—
126	C₃₁	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
127	C₂×Q₈	—	31	4	no	—
128	C₂×Q₈	—	31	4	no	—
129	C₂×Q₈	—	31	4	no	—
130	C₂²×C₄	—	31	2	no	—
131	C₂²×C₄	—	31	2	no	—
132	C₂²×C₄	—	31	2	no	—
133	C₂²×C₄	—	31	2	no	—
134	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
135	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	31	8	no	—
142	C₄⋊C₄	—	31	4	no	—
143	C₄⋊C₄	—	31	4	no	—
144	C₄⋊C₄	—	62	8	no	—
145	C₄⋊C₄	—	62	2	no	—
146	C₄⋊C₄	—	124	4	no	—
147	C₄²	—	31	8	no	—
148	C₄²	—	62	8	no	—
149	C₄²	—	62	4	no	—
150	C₄²	C₄×D₃₁	1	4	yes	—
151	C₂³	C₂³×D₃₁	1	1	yes	characteristic
152	Q₈	—	31	16	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	31	4	no	—
159	C₂×C₄	—	31	4	no	—
160	C₂×C₄	—	31	4	no	—
161	C₂×C₄	—	31	4	no	—
162	C₂×C₄	—	31	4	no	—
163	C₂×C₄	—	62	2	no	—
164	C₂×C₄	—	62	2	no	—
165	C₂×C₄	—	62	2	no	—
166	C₂×C₄	—	62	2	no	—
167	C₂×C₄	—	62	2	no	—
168	C₂×C₄	—	62	2	no	—
169	C₂×C₄	—	62	2	no	—
170	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
172	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
173	C₂×C₄	D₄×D₃₁	1	2	yes	—
174	C₂×C₄	#92497	1	2	yes	—
175	C₂×C₄	Q₈×D₃₁	1	2	yes	—
176	C₂×C₄	Q₈×D₃₁	1	2	yes	—
177	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
179	C₂²	C₂×#92497	1	1	yes	characteristic
180	C₂²	#11758561	1	1	yes	characteristic
181	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
182	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
183	C₂²	#11758564	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	31	8	no	—
187	C₄	—	62	4	no	—
188	C₄	—	62	4	no	—
189	C₄	—	124	2	no	—
190	C₄	#11758464	1	4	yes	complemented
191	C₄	C₄⋊C₄×D₃₁	1	4	yes	—
192	C₂	C₂×C₄⋊C₄×D₃₁	1	1	yes	characteristic
193	C₂	C₂×#11758464	1	1	yes	characteristic
194	C₂	#14521530	1	1	yes	characteristic
195	C₂	#14521714	1	1	yes	characteristic
196	C₂	#14521719	1	1	yes	characteristic
197	C₂	#14521774	1	1	yes	characteristic
198	C₂	#14521779	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#384	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	#384	D₃₁
hypocenter	C₃₁	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
24. C₄×Dic₆₂	C₄	semidirect
44. C₄⋊Dic₃₁	Q₈	semidirect
76. Dic₆₂	C₄⋊C₄	semidirect
88. C₄×C₃₁	C₄×Q₈	semidirect
126. C₃₁	Q₈×C₄⋊C₄	semidirect
190. C₄	#11758464	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₆₂	2
C₂×#11758464	2
#14520945	2
#14520944	2
#14520918	2
C₄⋊C₄×Dic₃₁	2
C₄×C₄⋊Dic₃₁	2
C₃₁×#384	2
Q₈×C₄⋊C₄	31