Subgroups — #16848675

Order 3968.

464 conjugacy classes of subgroups.

The subgroup lattice has 246 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	1	yes	maximal, characteristic
2	#14520956	C₂	1	1	yes	maximal, characteristic
3	#14520934	C₂	1	2	yes	maximal
4	#14520932	C₂	1	2	yes	maximal
5	#14520916	C₂	1	2	yes	maximal
6	#14520914	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₃₁⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
10	#14520894	C₂	1	2	yes	maximal
11	C₃₁×#391	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	Q₈×Dic₃₁	C₄	1	8	yes	complemented
16	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
19	C₂×Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#95	C₂²	1	2	yes	—
27	C₃₁×#95	C₂²	1	2	yes	—
28	C₃₁⋊#95	C₂²	1	1	yes	characteristic
29	C₃₁⋊#95	C₂²	1	2	yes	—
30	C₃₁⋊#95	C₂²	1	2	yes	—
31	C₃₁⋊#95	C₂²	1	2	yes	—
32	C₃₁⋊#95	C₂²	1	1	yes	characteristic
33	C₃₁⋊#95	C₂²	1	2	yes	—
34	C₃₁⋊#95	C₂²	1	2	yes	—
35	C₃₁⋊#95	C₂²	1	2	yes	—
36	C₂×Q₈×C₃₁	—	2	2	no	—
37	C₂×Q₈×C₃₁	—	2	2	no	—
38	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₃₁	C₂³	1	2	yes	—
46	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
47	C₂²×Dic₃₁	C₂³	1	2	yes	—
48	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₄²×C₃₁	—	2	2	no	—
52	C₄⋊Dic₃₁	—	2	2	no	—
53	C₄⋊Dic₃₁	C₂×C₄	1	8	yes	—
54	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
55	Dic₃₁⋊C₄	—	2	2	no	—
56	Dic₃₁⋊C₄	—	2	2	no	—
57	C₄×Dic₃₁	—	2	2	no	—
58	C₄×Dic₃₁	—	2	4	no	—
59	C₄×Dic₃₁	—	2	4	no	—
60	C₄×Dic₃₁	C₂×C₄	1	8	yes	—
61	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
62	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
63	Q₈×C₃₁	—	2	8	no	—
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	1	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	2	1	no	—
70	C₂×C₄×C₃₁	—	2	1	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	2	1	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₁	#41	1	2	yes	—
77	C₂×C₄×C₃₁	#41	1	2	yes	—
78	C₂×C₄×C₃₁	#41	1	2	yes	—
79	C₂×Dic₃₁	—	2	1	no	—
80	C₂×Dic₃₁	—	2	2	no	—
81	C₂×Dic₃₁	—	2	2	no	—
82	C₂×Dic₃₁	—	2	1	no	—
83	C₂×Dic₃₁	—	2	1	no	—
84	C₂×Dic₃₁	—	2	1	no	—
85	C₂×Dic₃₁	—	2	1	no	—
86	C₂×Dic₃₁	—	2	1	no	—
87	C₂×Dic₃₁	—	2	1	no	—
88	C₂×Dic₃₁	—	2	2	no	—
89	C₂×Dic₃₁	—	2	1	no	—
90	C₂×Dic₃₁	—	2	2	no	—
91	C₂×Dic₃₁	—	2	2	no	—
92	C₂×Dic₃₁	—	2	1	no	—
93	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
94	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
95	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
96	C₂×Dic₃₁	#41	1	2	yes	—
97	#2356	—	31	1	no	maximal, Sylow
98	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
100	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
101	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
102	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
103	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
105	C₄×C₃₁	—	2	4	no	—
106	C₄×C₃₁	—	2	2	no	—
107	C₄×C₃₁	—	4	1	no	—
108	C₄×C₃₁	—	4	2	no	—
109	C₄×C₃₁	#117	1	4	yes	—
110	Dic₃₁	—	2	4	no	—
111	Dic₃₁	—	2	2	no	—
112	Dic₃₁	—	2	2	no	—
113	Dic₃₁	—	2	2	no	—
114	Dic₃₁	—	4	1	no	—
115	Dic₃₁	—	4	2	no	—
116	C₂×C₄×Q₈	—	31	1	no	—
117	#391	—	31	1	no	—
118	#391	D₃₁	1	1	yes	characteristic
119	#387	—	31	2	no	—
120	#387	—	31	2	no	—
121	#383	—	31	2	no	—
122	#382	—	31	1	no	—
123	C₄×C₄⋊C₄	—	31	1	no	—
124	C₄×C₄⋊C₄	—	31	1	no	—
125	C₄×C₄⋊C₄	—	31	1	no	—
126	#376	—	31	2	no	—
127	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
128	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
129	C₂×C₃₁	#519	1	1	yes	characteristic
130	C₂×C₃₁	#552	1	1	yes	characteristic
131	C₂×C₃₁	#552	1	1	yes	characteristic
132	C₂×C₃₁	#556	1	1	yes	characteristic
133	C₂×C₃₁	#556	1	1	yes	characteristic
134	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
135	C₄×Q₈	—	31	8	no	—
136	C₂×C₄⋊C₄	—	31	1	no	—
137	C₂×C₄⋊C₄	—	31	2	no	—
138	C₂×C₄⋊C₄	—	31	1	no	—
139	C₂×C₄⋊C₄	—	31	1	no	—
140	C₂×C₄⋊C₄	—	31	1	no	—
141	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
142	C₂×C₄²	—	31	2	no	—
143	C₂×C₄²	—	31	1	no	—
144	C₂×C₄²	—	31	2	no	—
145	C₂×C₄²	—	31	2	no	—
146	C₂×C₄²	—	31	1	no	—
147	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
148	#95	—	31	1	no	—
149	#95	—	31	2	no	—
150	#95	—	31	2	no	—
151	#95	—	31	1	no	—
152	#95	—	31	2	no	—
153	#95	—	31	2	no	—
154	#95	—	31	2	no	—
155	#95	—	31	2	no	—
156	#95	C₂×D₃₁	1	2	yes	—
157	#95	C₂×D₃₁	1	2	yes	—
158	C₃₁	#2356	1	1	yes	characteristic, complemented, Sylow
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	C₄×D₃₁	1	4	yes	—
162	C₂²×C₄	—	31	1	no	—
163	C₂²×C₄	—	31	1	no	—
164	C₂²×C₄	—	31	2	no	—
165	C₂²×C₄	—	31	1	no	—
166	C₂²×C₄	—	31	1	no	—
167	C₂²×C₄	—	31	2	no	—
168	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
169	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
171	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	31	8	no	—
175	C₄⋊C₄	—	31	4	no	—
176	C₄⋊C₄	—	62	2	no	—
177	C₄⋊C₄	—	62	2	no	—
178	C₄⋊C₄	—	62	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	31	8	no	—
181	C₄²	—	31	4	no	—
182	C₄²	—	62	4	no	—
183	C₄²	—	62	2	no	—
184	C₄²	—	62	4	no	—
185	C₂³	C₂³×D₃₁	1	1	yes	characteristic
186	Q₈	—	2	8	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	31	2	no	—
198	C₂×C₄	—	31	4	no	—
199	C₂×C₄	—	31	2	no	—
200	C₂×C₄	—	31	2	no	—
201	C₂×C₄	—	62	1	no	—
202	C₂×C₄	—	62	2	no	—
203	C₂×C₄	—	62	1	no	—
204	C₂×C₄	—	62	1	no	—
205	C₂×C₄	—	62	1	no	—
206	C₂×C₄	—	62	2	no	—
207	C₂×C₄	—	62	1	no	—
208	C₂×C₄	—	62	1	no	—
209	C₂×C₄	—	62	1	no	—
210	C₂×C₄	—	62	2	no	—
211	C₂×C₄	—	62	2	no	—
212	C₂×C₄	—	62	1	no	—
213	C₂×C₄	—	62	2	no	—
214	C₂×C₄	—	62	1	no	—
215	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
216	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
217	C₂×C₄	#92497	1	2	yes	—
218	C₂×C₄	#92497	1	2	yes	—
219	C₂×C₄	#92499	1	2	yes	—
220	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
221	C₂²	C₂×#92497	1	1	yes	characteristic
222	C₂²	C₂×#92497	1	1	yes	characteristic
223	C₂²	C₂×#92499	1	1	yes	characteristic
224	C₂²	#11758564	1	1	yes	characteristic
225	C₂²	#41×D₃₁	1	1	yes	characteristic
226	C₂²	#11758567	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	62	2	no	—
232	C₄	—	62	4	no	—
233	C₄	—	62	2	no	—
234	C₄	—	62	2	no	—
235	C₄	—	124	1	no	—
236	C₄	—	124	2	no	—
237	C₄	#11758454	1	4	yes	—
238	C₂	C₂×#11758454	1	1	yes	characteristic
239	C₂	C₄×#92499	1	1	yes	characteristic
240	C₂	#14521525	1	1	yes	characteristic
241	C₂	#14521733	1	1	yes	characteristic
242	C₂	#14521741	1	1	yes	characteristic
243	C₂	#14521768	1	1	yes	characteristic
244	C₂	#14521773	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#391	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	#391	D₃₁
hypocenter	C₃₁	#2356
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
15. Q₈×Dic₃₁	C₄	semidirect
158. C₃₁	#2356	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
#14520934	2
#14520932	2
#14520916	2
#14520914	2
C₄⋊C₄×Dic₃₁	2
C₄×Dic₃₁⋊C₄	2
C₄×C₄⋊Dic₃₁	2
#14520894	2
C₃₁×#391	2
#2356	31