Subgroups — #16854818

Order 3968.

432 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#92497	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521173	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
6	(C₂²⋊D₃₁)⋊C₈	C₂	1	2	yes	maximal, complemented
7	#14521100	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521099	C₂	1	2	yes	maximal, complemented
9	C₂×Dic₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#92497	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
14	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	D₄×Dic₃₁	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
18	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
19	C₂×C₈×D₃₁	C₂²	1	2	yes	—
20	#11758469	C₄	1	1	yes	characteristic
21	#11758464	C₄	1	1	yes	characteristic
22	(C₂²⋊D₃₁)⋊C₄	C₄	1	2	yes	—
23	#11758454	C₄	1	2	yes	—
24	C₄²×D₃₁	C₂²	1	1	yes	characteristic
25	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
30	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
32	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
33	Dic₃₁⋊C₈	C₂²	1	2	yes	—
34	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
35	Dic₃₁⋊C₈	C₂²	1	2	yes	—
36	Dic₃₁⋊C₈	C₂²	1	2	yes	—
37	C₈×Dic₃₁	C₂²	1	2	yes	—
38	C₈×Dic₃₁	C₂²	1	2	yes	—
39	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
40	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
44	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
45	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
46	#92497	C₈	1	8	yes	complemented
47	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
50	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
59	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
61	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
66	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
67	C₄×Dic₃₁	C₂³	1	2	yes	—
68	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₃₁⋊C₈	—	4	2	no	—
70	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
73	C₈×D₃₁	—	4	2	no	—
74	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
76	D₄×C₃₁	C₂×C₈	1	4	yes	complemented
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₃₁	C₂×C₈	1	8	yes	complemented
84	C₂×Dic₃₁	—	4	2	no	—
85	C₂×Dic₃₁	C₂×C₈	1	8	yes	complemented
86	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
89	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
91	C₄×D₃₁	—	2	1	no	—
92	C₄×D₃₁	—	2	1	no	—
93	C₄×D₃₁	—	4	1	no	—
94	C₄×D₃₁	C₂×C₈	1	4	yes	complemented
95	Dic₆₂	C₂×C₈	1	4	yes	complemented
96	C₈×C₃₁	—	2	2	no	—
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₃₁⋊C₈	—	2	2	no	—
100	C₃₁⋊C₈	—	2	1	no	—
101	C₃₁⋊C₈	—	2	1	no	—
102	#2988	—	31	1	no	maximal, Sylow
103	C₂²×C₃₁	—	2	2	no	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	C₂²×C₈	1	4	yes	—
106	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₃₁	—	2	1	no	—
108	C₂×D₃₁	—	2	1	no	—
109	C₂×D₃₁	C₂²×C₈	1	2	yes	—
110	C₄×C₃₁	—	2	2	no	—
111	C₄×C₃₁	—	2	1	no	—
112	C₄×C₃₁	C₂²×C₈	1	2	yes	—
113	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
114	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
115	Dic₃₁	—	2	1	no	—
116	Dic₃₁	—	2	1	no	—
117	Dic₃₁	—	2	2	no	—
118	Dic₃₁	C₂²×C₈	1	4	yes	—
119	Dic₃₁	C₂²×C₈	1	2	yes	—
120	C₄×#41	—	31	1	no	—
121	C₈×Q₈	—	31	1	no	—
122	C₈×Q₈	—	31	1	no	—
123	C₈×D₄	—	31	2	no	—
124	C₈×D₄	—	31	1	no	—
125	C₈×D₄	—	31	2	no	—
126	C₈×D₄	D₃₁	1	1	yes	characteristic, complemented
127	#431	—	31	2	no	—
128	#431	—	31	1	no	—
129	C₂×C₄⋊C₈	—	31	2	no	—
130	C₂×C₄⋊C₈	—	31	1	no	—
131	C₂×C₃₁	—	2	4	no	—
132	C₂×C₃₁	#520	1	1	yes	characteristic
133	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
134	C₂×C₃₁	#568	1	1	yes	characteristic
135	D₃₁	—	2	2	no	—
136	C₂×#41	—	31	1	no	—
137	C₂²×C₈	—	31	2	no	—
138	C₂²×C₈	—	31	2	no	—
139	C₂²×C₈	C₂×D₃₁	1	2	yes	—
140	C₄×Q₈	—	31	1	no	—
141	C₄×Q₈	—	31	1	no	—
142	C₄×D₄	—	31	1	no	—
143	C₄×D₄	—	31	2	no	—
144	C₄×D₄	—	31	2	no	—
145	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
146	#117	—	31	1	no	—
147	#117	—	31	2	no	—
148	C₂×C₄²	—	31	2	no	—
149	C₂×C₄²	—	31	1	no	—
150	C₄⋊C₈	—	31	2	no	—
151	C₄⋊C₈	—	31	1	no	—
152	C₄⋊C₈	—	31	2	no	—
153	C₄⋊C₈	—	31	2	no	—
154	C₄⋊C₈	—	31	1	no	—
155	C₄⋊C₈	—	31	1	no	—
156	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
157	C₂²⋊C₈	—	31	2	no	—
158	C₂²⋊C₈	—	31	1	no	—
159	C₂²⋊C₈	—	31	1	no	—
160	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
161	C₄×C₈	—	31	2	no	—
162	C₄×C₈	—	31	2	no	—
163	C₄×C₈	—	31	1	no	—
164	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
165	C₃₁	#2988	1	1	yes	characteristic, complemented, Sylow
166	#41	—	31	8	no	—
167	C₂×Q₈	—	31	1	no	—
168	C₂×D₄	—	31	2	no	—
169	C₂×D₄	C₄×D₃₁	1	1	yes	characteristic
170	C₂²×C₄	—	31	2	no	—
171	C₂²×C₄	—	31	1	no	—
172	C₂²×C₄	—	31	2	no	—
173	C₂²×C₄	—	31	1	no	—
174	C₂²×C₄	—	31	1	no	—
175	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	31	1	no	—
178	C₂×C₈	—	31	1	no	—
179	C₂×C₈	—	31	2	no	—
180	C₂×C₈	—	124	2	no	—
181	C₂×C₈	—	124	2	no	—
182	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₃₁	1	2	yes	—
184	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
185	C₄⋊C₄	—	31	2	no	—
186	C₄⋊C₄	—	31	2	no	—
187	C₄⋊C₄	—	31	1	no	—
188	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
189	C₂²⋊C₄	—	31	2	no	—
190	C₂²⋊C₄	—	31	2	no	—
191	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
192	C₄²	—	31	2	no	—
193	C₄²	—	31	1	no	—
194	C₄²	—	31	2	no	—
195	C₄²	—	31	1	no	—
196	C₄²	—	31	2	no	—
197	C₄²	—	31	1	no	—
198	C₄²	C₂²×D₃₁	1	1	yes	characteristic
199	C₂³	—	31	1	no	—
200	C₂³	C₂×C₄×D₃₁	1	2	yes	—
201	Q₈	—	31	4	no	—
202	D₄	—	31	8	no	—
203	D₄	C₈×D₃₁	1	4	yes	complemented
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	31	8	no	—
207	C₂×C₄	—	31	1	no	—
208	C₂×C₄	—	31	2	no	—
209	C₂×C₄	—	31	4	no	—
210	C₂×C₄	—	31	1	no	—
211	C₂×C₄	—	31	1	no	—
212	C₂×C₄	—	31	2	no	—
213	C₂×C₄	—	62	1	no	—
214	C₂×C₄	—	62	1	no	—
215	C₂×C₄	—	124	1	no	—
216	C₂×C₄	—	124	2	no	—
217	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	62	2	no	—
225	C₈	—	62	1	no	—
226	C₈	—	62	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	31	2	no	—
230	C₂²	—	62	1	no	—
231	C₂²	—	62	1	no	—
232	C₂²	C₂×C₈×D₃₁	1	4	yes	—
233	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	31	2	no	—
237	C₄	—	31	4	no	—
238	C₄	—	62	1	no	—
239	C₄	—	62	1	no	—
240	C₄	—	62	2	no	—
241	C₄	C₂×C₈×D₃₁	1	2	yes	—
242	C₄	#11758561	1	1	yes	characteristic
243	C₄	#11758567	1	1	yes	characteristic
244	C₂	—	2	4	no	—
245	C₂	—	62	2	no	—
246	C₂	#14521528	1	1	yes	characteristic
247	C₂	C₂²×C₈×D₃₁	1	1	yes	characteristic
248	C₂	#14521814	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator	C₂×C₃₁	C₂³×C₈
Fitting	C₈×D₄×C₃₁	C₂
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₂×#92497	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₈×D₄	D₃₁
hypocenter	C₃₁	#2988
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₆₂	C₂	semidirect
3. #14521173	C₂	semidirect
4. D₄×C₃₁⋊C₈	C₂	semidirect
5. C₈×C₂²⋊D₃₁	C₂	semidirect
6. (C₂²⋊D₃₁)⋊C₈	C₂	semidirect
7. #14521100	C₂	semidirect
8. #14521099	C₂	semidirect
9. C₂×Dic₃₁⋊C₈	C₂	semidirect
10. C₄⋊C₈×D₃₁	C₂	semidirect
11. C₈×D₄×C₃₁	C₂	semidirect
46. #92497	C₈	semidirect
76. D₄×C₃₁	C₂×C₈	semidirect
83. C₂²⋊D₃₁	C₂×C₈	semidirect
85. C₂×Dic₃₁	C₂×C₈	semidirect
94. C₄×D₃₁	C₂×C₈	semidirect
95. Dic₆₂	C₂×C₈	semidirect
126. C₈×D₄	D₃₁	semidirect
165. C₃₁	#2988	semidirect
203. D₄	C₈×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#92497	2
C₈×Dic₆₂	2
#14521173	2
D₄×C₃₁⋊C₈	2
C₈×C₂²⋊D₃₁	2
(C₂²⋊D₃₁)⋊C₈	2
#14521100	2
#14521099	2
C₂×Dic₃₁⋊C₈	2
C₄⋊C₈×D₃₁	2
C₈×D₄×C₃₁	2
#2988	31