Subgroups — #16854663

Order 3968.

388 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521180	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521178	C₂	1	2	yes	maximal, complemented
4	#14521177	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521173	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521071	C₂	1	1	yes	maximal, characteristic
8	#14521071	C₂	1	2	yes	maximal
9	#14521001	C₂	1	2	yes	maximal
10	C₄×#92469	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₃₁	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	Q₈×Dic₃₁	C₄	1	1	yes	characteristic
15	C₂×#92469	C₂²	1	1	yes	characteristic
16	C₂×#92469	C₂²	1	2	yes	—
17	C₂×#92469	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×D₃₁	C₄	1	1	yes	characteristic
19	C₄⋊C₄×D₃₁	C₄	1	2	yes	—
20	#11758464	C₄	1	2	yes	—
21	#11758464	C₄	1	1	yes	characteristic
22	C₄²×D₃₁	C₂²	1	1	yes	characteristic
23	C₄²×D₃₁	C₂²	1	2	yes	—
24	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
25	C₄×Dic₆₂	C₂²	1	2	yes	—
26	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
28	C₃₁×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	#11758403	C₂²	1	1	yes	characteristic
31	#11758403	C₂²	1	1	yes	characteristic
32	Dic₃₁⋊C₈	C₂²	1	2	yes	—
33	Dic₃₁⋊C₈	C₂²	1	2	yes	—
34	Dic₃₁⋊C₈	C₂²	1	2	yes	—
35	C₈×Dic₃₁	C₂²	1	2	yes	—
36	C₈×Dic₃₁	C₂²	1	2	yes	—
37	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
39	#11758391	C₂²	1	2	yes	—
40	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	Q₈×D₃₁	—	2	4	no	—
43	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
49	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₄²×C₃₁	C₂³	1	2	yes	—
57	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
59	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
62	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
64	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
65	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₃₁	C₂³	1	2	yes	—
67	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
70	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	#92469	—	2	2	no	—
72	#92469	—	2	4	no	—
73	#92469	Q₈	1	4	yes	complemented
74	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
86	C₄×D₃₁	—	2	2	no	—
87	C₄×D₃₁	—	2	4	no	—
88	C₄×D₃₁	—	4	1	no	—
89	C₄×D₃₁	—	4	2	no	—
90	C₄×D₃₁	C₂×Q₈	1	2	yes	—
91	C₄×D₃₁	#41	1	2	yes	—
92	Dic₆₂	—	2	4	no	—
93	Dic₆₂	M₁₆	1	4	yes	complemented
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	—	2	1	no	—
96	C₈×C₃₁	C₂×Q₈	1	2	yes	complemented
97	C₃₁⋊C₈	—	2	1	no	—
98	C₃₁⋊C₈	—	2	2	no	—
99	C₃₁⋊C₈	C₂×Q₈	1	2	yes	complemented
100	Q₈×M₁₆	—	31	1	no	maximal, Sylow
101	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₃₁	—	2	1	no	—
103	C₂×D₃₁	—	2	1	no	—
104	C₂×D₃₁	C₄×Q₈	1	2	yes	—
105	C₄×C₃₁	—	2	2	no	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
109	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
110	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
111	Dic₃₁	—	2	1	no	—
112	Dic₃₁	—	2	2	no	—
113	Dic₃₁	—	2	1	no	—
114	Dic₃₁	C₄×Q₈	1	2	yes	—
115	Dic₃₁	C₂×M₁₆	1	4	yes	—
116	C₂×C₄×Q₈	—	31	1	no	—
117	#446	—	31	1	no	—
118	#446	—	31	2	no	—
119	#446	—	31	2	no	—
120	#446	D₃₁	1	1	yes	characteristic, complemented
121	C₈×Q₈	—	31	1	no	—
122	C₈×Q₈	—	31	1	no	—
123	#423	—	31	2	no	—
124	#423	—	31	1	no	—
125	C₄×M₁₆	—	31	2	no	—
126	C₄×M₁₆	—	31	1	no	—
127	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
128	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
129	C₂×C₃₁	#568	1	1	yes	characteristic
130	D₃₁	—	2	2	no	—
131	C₂²×Q₈	—	31	1	no	—
132	C₂×M₁₆	—	31	1	no	—
133	C₂×M₁₆	—	31	2	no	—
134	C₂×M₁₆	—	31	1	no	—
135	C₄×Q₈	—	31	1	no	—
136	C₄×Q₈	—	31	1	no	—
137	C₄×Q₈	—	31	2	no	—
138	C₄×Q₈	—	31	1	no	—
139	C₄×Q₈	—	31	2	no	—
140	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	31	2	no	—
142	C₂×C₄⋊C₄	—	31	1	no	—
143	C₂×C₄²	—	31	2	no	—
144	C₂×C₄²	—	31	1	no	—
145	C₄⋊C₈	—	31	2	no	—
146	C₄⋊C₈	—	31	2	no	—
147	C₄⋊C₈	—	31	2	no	—
148	C₄⋊C₈	—	31	1	no	—
149	C₄⋊C₈	—	31	2	no	—
150	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
151	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
152	#97	—	31	1	no	—
153	#97	—	31	1	no	—
154	#97	—	31	2	no	—
155	#97	C₂×D₃₁	1	2	yes	—
156	C₄×C₈	—	31	2	no	—
157	C₄×C₈	—	31	2	no	—
158	C₄×C₈	—	31	1	no	—
159	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
160	C₃₁	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
161	C₂×Q₈	—	31	1	no	—
162	C₂×Q₈	—	31	2	no	—
163	C₂×Q₈	—	62	4	no	—
164	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
165	C₂²×C₄	—	31	1	no	—
166	C₂²×C₄	—	31	2	no	—
167	C₂²×C₄	—	31	1	no	—
168	C₂²×C₄	—	31	2	no	—
169	C₂²×C₄	—	31	1	no	—
170	M₁₆	—	31	4	no	—
171	M₁₆	—	62	4	no	—
172	M₁₆	—	62	2	no	—
173	C₂×C₈	—	31	1	no	—
174	C₂×C₈	—	31	1	no	—
175	C₂×C₈	—	31	2	no	—
176	C₂×C₈	C₂²×D₃₁	1	2	yes	—
177	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
179	C₄⋊C₄	—	31	2	no	—
180	C₄⋊C₄	—	31	2	no	—
181	C₄⋊C₄	—	31	2	no	—
182	C₄⋊C₄	—	31	2	no	—
183	C₄⋊C₄	—	31	1	no	—
184	C₄⋊C₄	C₄×D₃₁	1	2	yes	—
185	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
186	C₄²	—	31	2	no	—
187	C₄²	—	31	2	no	—
188	C₄²	—	31	1	no	—
189	C₄²	—	31	1	no	—
190	C₄²	—	31	2	no	—
191	C₄²	—	31	1	no	—
192	C₄²	C₂²×D₃₁	1	1	yes	characteristic
193	C₄²	C₂²×D₃₁	1	2	yes	—
194	C₂³	—	31	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	31	4	no	—
197	Q₈	—	62	4	no	—
198	C₂×C₄	—	31	1	no	—
199	C₂×C₄	—	31	2	no	—
200	C₂×C₄	—	31	2	no	—
201	C₂×C₄	—	31	2	no	—
202	C₂×C₄	—	31	2	no	—
203	C₂×C₄	—	31	1	no	—
204	C₂×C₄	—	31	1	no	—
205	C₂×C₄	—	62	4	no	—
206	C₂×C₄	—	62	2	no	—
207	C₂×C₄	—	124	1	no	—
208	C₂×C₄	—	124	2	no	—
209	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	1	no	—
216	C₈	—	31	2	no	—
217	C₈	—	62	1	no	—
218	C₈	—	62	2	no	—
219	C₈	Q₈×D₃₁	1	2	yes	complemented
220	C₂²	—	31	2	no	—
221	C₂²	—	62	1	no	—
222	C₂²	—	62	1	no	—
223	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	31	2	no	—
228	C₄	—	31	4	no	—
229	C₄	—	62	1	no	—
230	C₄	—	62	2	no	—
231	C₄	—	62	1	no	—
232	C₄	M₁₆×D₃₁	1	2	yes	—
233	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
234	C₄	#41×D₃₁	1	1	yes	characteristic
235	C₂	—	62	2	no	—
236	C₂	C₄×Q₈×D₃₁	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic
238	C₂	#14521814	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#446	C₂
prime radical	C₂²×D₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#446	D₃₁
hypocenter	C₃₁	Q₈×M₁₆
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #14521180	C₂	semidirect
3. #14521178	C₂	semidirect
4. #14521177	C₂	semidirect
5. C₈×Dic₆₂	C₂	semidirect
6. #14521173	C₂	semidirect
11. C₃₁×#446	C₂	semidirect
73. #92469	Q₈	semidirect
93. Dic₆₂	M₁₆	semidirect
96. C₈×C₃₁	C₂×Q₈	semidirect
99. C₃₁⋊C₈	C₂×Q₈	semidirect
120. #446	D₃₁	semidirect
160. C₃₁	Q₈×M₁₆	semidirect
219. C₈	Q₈×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₃₁	2
#14521180	2
#14521178	2
#14521177	2
C₈×Dic₆₂	2
#14521173	2
#14521071	2
#14521001	2
C₄×#92469	2
C₃₁×#446	2
Q₈×M₁₆	31