Subgroups — #16854732

Order 3968.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₃₁	C₂	1	1	yes	maximal, characteristic
2	C₈×D₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521114	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	#14521101	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521100	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×(C₂×D₃₁)⋊C₈	C₂	1	2	yes	maximal, complemented
10	C₂²⋊C₈×D₃₁	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₃₁	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₃₁	C₂²	1	2	yes	complemented
15	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	D₄×Dic₃₁	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
18	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	complemented
19	C₂×C₈×D₃₁	C₂²	1	2	yes	complemented
20	C₂×C₈×D₃₁	C₂²	1	2	yes	complemented
21	#11758470	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₃₁	C₄	1	1	yes	characteristic
23	(C₂²⋊D₃₁)⋊C₄	C₄	1	2	yes	—
24	C₂²⋊C₄×D₃₁	C₄	1	2	yes	—
25	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
26	C₄²×D₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	complemented
31	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	complemented
35	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	complemented
36	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	C₈×Dic₃₁	C₂²	1	2	yes	complemented
39	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₃₁	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
45	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
46	D₄×D₃₁	C₈	1	8	yes	complemented
47	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₃₁	—	4	2	no	—
49	C₂×C₄×D₃₁	—	4	2	no	—
50	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₃₁	C₂³	1	2	yes	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	C₂³	1	2	yes	—
58	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
64	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
65	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
67	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₃₁⋊C₈	—	4	2	no	—
72	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
75	C₈×D₃₁	—	4	2	no	—
76	C₈×D₃₁	—	4	2	no	—
77	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂²×D₃₁	—	4	2	no	—
79	C₂²×D₃₁	—	4	2	no	—
80	C₂²×D₃₁	C₂×C₈	1	8	yes	complemented
81	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
82	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₃₁	C₂×C₈	1	4	yes	complemented
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₃₁	C₂×C₈	1	8	yes	complemented
91	C₂×Dic₃₁	—	4	2	no	—
92	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
95	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
96	D₁₂₄	C₂×C₈	1	4	yes	complemented
97	C₄×D₃₁	—	2	2	no	—
98	C₄×D₃₁	—	2	2	no	—
99	C₄×D₃₁	—	2	1	no	—
100	C₄×D₃₁	—	2	1	no	—
101	C₄×D₃₁	—	4	2	no	—
102	C₄×D₃₁	—	4	2	no	—
103	C₄×D₃₁	—	4	1	no	—
104	C₄×D₃₁	C₂×C₈	1	4	yes	complemented
105	C₈×C₃₁	—	2	1	no	—
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	—	2	1	no	—
108	C₃₁⋊C₈	—	2	1	no	—
109	C₃₁⋊C₈	—	2	2	no	—
110	C₃₁⋊C₈	—	2	1	no	—
111	#2972	—	31	1	no	maximal, Sylow
112	C₂²×C₃₁	—	2	2	no	—
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	C₂²×C₈	1	4	yes	—
115	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₃₁	—	2	1	no	—
117	C₂×D₃₁	—	2	2	no	—
118	C₂×D₃₁	—	2	1	no	—
119	C₂×D₃₁	—	2	2	no	—
120	C₂×D₃₁	—	4	8	no	—
121	C₂×D₃₁	C₂²×C₈	1	2	yes	—
122	C₂×D₃₁	C₂²×C₈	1	4	yes	—
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	C₂²×C₈	1	2	yes	—
126	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
127	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
128	Dic₃₁	—	2	1	no	—
129	Dic₃₁	—	2	2	no	—
130	Dic₃₁	—	2	1	no	—
131	Dic₃₁	C₂²×C₈	1	2	yes	—
132	C₂×C₄×D₄	—	31	1	no	—
133	C₈×D₄	—	31	2	no	—
134	C₈×D₄	—	31	1	no	—
135	C₈×D₄	—	31	1	no	—
136	C₈×D₄	—	31	2	no	—
137	C₈×D₄	—	31	1	no	—
138	C₈×D₄	D₃₁	1	1	yes	characteristic, complemented
139	#431	—	31	1	no	—
140	#431	—	31	1	no	—
141	C₂×C₂²⋊C₈	—	31	2	no	—
142	C₂×C₂²⋊C₈	—	31	2	no	—
143	C₂×C₃₁	—	2	4	no	—
144	C₂×C₃₁	#518	1	1	yes	characteristic
145	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
146	C₂×C₃₁	#568	1	1	yes	characteristic
147	D₃₁	—	2	2	no	—
148	D₃₁	—	2	4	no	—
149	C₂²×D₄	—	31	1	no	—
150	C₂³×C₄	—	31	2	no	—
151	C₂²×C₈	—	31	2	no	—
152	C₂²×C₈	—	31	2	no	—
153	C₂²×C₈	—	31	2	no	—
154	C₂²×C₈	C₂×D₃₁	1	2	yes	complemented
155	C₄×D₄	—	31	1	no	—
156	C₄×D₄	—	31	2	no	—
157	C₄×D₄	—	31	1	no	—
158	C₄×D₄	—	31	2	no	—
159	C₄×D₄	—	31	1	no	—
160	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	31	1	no	—
162	C₂×C₂²⋊C₄	—	31	2	no	—
163	C₂×C₄²	—	31	1	no	—
164	C₄⋊C₈	—	31	1	no	—
165	C₄⋊C₈	—	31	1	no	—
166	C₄⋊C₈	—	31	1	no	—
167	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
168	C₂²⋊C₈	—	31	2	no	—
169	C₂²⋊C₈	—	31	2	no	—
170	C₂²⋊C₈	—	31	1	no	—
171	C₂²⋊C₈	—	31	1	no	—
172	C₂²⋊C₈	—	31	2	no	—
173	C₂²⋊C₈	—	31	2	no	—
174	C₂²⋊C₈	C₂×D₃₁	1	2	yes	complemented
175	C₄×C₈	—	31	2	no	—
176	C₄×C₈	—	31	1	no	—
177	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
178	C₃₁	#2972	1	1	yes	characteristic, complemented, Sylow
179	C₂⁴	—	31	2	no	—
180	C₂×D₄	—	31	2	no	—
181	C₂×D₄	—	31	8	no	—
182	C₂×D₄	—	31	1	no	—
183	C₂×D₄	C₄×D₃₁	1	1	yes	characteristic
184	C₂²×C₄	—	31	1	no	—
185	C₂²×C₄	—	31	2	no	—
186	C₂²×C₄	—	31	1	no	—
187	C₂²×C₄	—	31	2	no	—
188	C₂²×C₄	—	31	1	no	—
189	C₂²×C₄	—	31	2	no	—
190	C₂²×C₄	—	31	2	no	—
191	C₂²×C₄	—	124	2	no	—
192	C₂²×C₄	—	124	2	no	—
193	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	31	1	no	—
196	C₂×C₈	—	31	1	no	—
197	C₂×C₈	—	31	2	no	—
198	C₂×C₈	—	124	2	no	—
199	C₂×C₈	—	124	2	no	—
200	C₂×C₈	—	124	2	no	—
201	C₂×C₈	C₂²×D₃₁	1	2	yes	—
202	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
204	C₄⋊C₄	—	31	2	no	—
205	C₄⋊C₄	—	31	1	no	—
206	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
207	C₂²⋊C₄	—	31	2	no	—
208	C₂²⋊C₄	—	31	2	no	—
209	C₂²⋊C₄	—	31	2	no	—
210	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
211	C₄²	—	31	1	no	—
212	C₄²	—	31	1	no	—
213	C₄²	—	31	1	no	—
214	C₄²	C₂²×D₃₁	1	1	yes	characteristic
215	C₂³	—	31	2	no	—
216	C₂³	—	31	1	no	—
217	C₂³	—	31	8	no	—
218	C₂³	—	124	2	no	—
219	C₂³	—	124	2	no	—
220	C₂³	C₂×C₄×D₃₁	1	2	yes	—
221	D₄	—	31	8	no	—
222	D₄	—	31	4	no	—
223	D₄	C₈×D₃₁	1	4	yes	complemented
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	31	1	no	—
227	C₂×C₄	—	31	4	no	—
228	C₂×C₄	—	31	1	no	—
229	C₂×C₄	—	31	1	no	—
230	C₂×C₄	—	31	2	no	—
231	C₂×C₄	—	62	2	no	—
232	C₂×C₄	—	62	1	no	—
233	C₂×C₄	—	62	2	no	—
234	C₂×C₄	—	62	1	no	—
235	C₂×C₄	—	124	1	no	—
236	C₂×C₄	—	124	2	no	—
237	C₂×C₄	—	124	2	no	—
238	C₂×C₄	—	124	2	no	—
239	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	62	1	no	—
247	C₈	—	62	2	no	—
248	C₈	—	62	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	31	2	no	—
252	C₂²	—	31	4	no	—
253	C₂²	—	62	1	no	—
254	C₂²	—	62	2	no	—
255	C₂²	—	62	1	no	—
256	C₂²	—	62	2	no	—
257	C₂²	—	124	8	no	—
258	C₂²	C₂×C₈×D₃₁	1	4	yes	—
259	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	31	2	no	—
263	C₄	—	62	1	no	—
264	C₄	—	62	2	no	—
265	C₄	—	62	1	no	—
266	C₄	C₂×C₈×D₃₁	1	2	yes	—
267	C₄	#11758561	1	1	yes	characteristic
268	C₄	#11758566	1	1	yes	characteristic
269	C₂	—	2	4	no	—
270	C₂	—	62	2	no	—
271	C₂	—	62	4	no	—
272	C₂	#14521521	1	1	yes	characteristic
273	C₂	C₂²×C₈×D₃₁	1	1	yes	characteristic
274	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×D₄×D₃₁	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₈×D₄	D₃₁
hypocenter	C₃₁	#2972
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×D₁₂₄	C₂	semidirect
3. D₄×C₃₁⋊C₈	C₂	semidirect
4. #14521114	C₂	semidirect
5. C₈×C₂²⋊D₃₁	C₂	semidirect
6. (C₂²⋊D₃₁)⋊C₈	C₂	semidirect
7. #14521101	C₂	semidirect
8. #14521100	C₂	semidirect
9. C₂×(C₂×D₃₁)⋊C₈	C₂	semidirect
10. C₂²⋊C₈×D₃₁	C₂	semidirect
11. C₈×D₄×C₃₁	C₂	semidirect
14. C₂²×C₈×C₃₁	C₂²	semidirect
18. C₂²×C₃₁⋊C₈	C₂²	semidirect
19. C₂×C₈×D₃₁	C₂²	semidirect
20. C₂×C₈×D₃₁	C₂²	semidirect
27. C₃₁×C₄⋊C₈	C₂²	semidirect
28. C₃₁×C₂²⋊C₈	C₂²	semidirect
29. C₄×C₈×C₃₁	C₂²	semidirect
30. C₂²⋊(C₃₁⋊C₈)	C₂²	semidirect
34. (C₂×D₃₁)⋊C₈	C₂²	semidirect
35. (C₂×D₃₁)⋊C₈	C₂²	semidirect
36. Dic₃₁⋊C₈	C₂²	semidirect
37. Dic₃₁⋊C₈	C₂²	semidirect
38. C₈×Dic₃₁	C₂²	semidirect
39. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
40. C₄×C₃₁⋊C₈	C₂²	semidirect
46. D₄×D₃₁	C₈	semidirect
80. C₂²×D₃₁	C₂×C₈	semidirect
83. D₄×C₃₁	C₂×C₈	semidirect
90. C₂²⋊D₃₁	C₂×C₈	semidirect
96. D₁₂₄	C₂×C₈	semidirect
104. C₄×D₃₁	C₂×C₈	semidirect
138. C₈×D₄	D₃₁	semidirect
154. C₂²×C₈	C₂×D₃₁	semidirect
167. C₄⋊C₈	C₂×D₃₁	semidirect
174. C₂²⋊C₈	C₂×D₃₁	semidirect
177. C₄×C₈	C₂×D₃₁	semidirect
178. C₃₁	#2972	semidirect
223. 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×D₄×D₃₁	2
C₈×D₁₂₄	2
D₄×C₃₁⋊C₈	2
#14521114	2
C₈×C₂²⋊D₃₁	2
(C₂²⋊D₃₁)⋊C₈	2
#14521101	2
#14521100	2
C₂×(C₂×D₃₁)⋊C₈	2
C₂²⋊C₈×D₃₁	2
C₈×D₄×C₃₁	2
#2972	31