Subgroups — D₄×Dic₇₁₂ #36711158

Order 22784.

664 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₁₂	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₃₅₆	C₂	1	2	yes	maximal
3	#26022505	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022343	C₂	1	2	yes	maximal, complemented
5	#26022303	C₂	1	2	yes	maximal, complemented
6	#26022188	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₅₆	C₂²	1	4	yes	—
10	D₄×Dic₁₇₈	—	2	2	no	—
11	C₂×Dic₇₁₂	—	2	8	no	—
12	C₂×Dic₇₁₂	C₂²	1	4	yes	—
13	C₂×Dic₇₁₂	C₂²	1	2	yes	—
14	C₂×Dic₇₁₂	C₂²	1	2	yes	—
15	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
16	#19151978	C₂²	1	2	yes	—
17	#19151839	C₂²	1	4	yes	—
18	#19151809	—	2	2	no	—
19	#19151756	—	2	4	no	—
20	C₄×Dic₃₅₆	C₂²	1	2	yes	—
21	#19151424	C₂²	1	1	yes	characteristic
22	#19151403	C₂²	1	2	yes	—
23	#19151403	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
25	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
28	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₇₈	—	2	4	no	—
30	C₂×Dic₃₅₆	—	2	8	no	—
31	C₂×Dic₃₅₆	—	2	4	no	—
32	C₂×Dic₃₅₆	—	2	2	no	—
33	C₂×Dic₃₅₆	—	4	4	no	—
34	C₂×Dic₃₅₆	C₂³	1	4	yes	—
35	C₂×Dic₃₅₆	C₂³	1	2	yes	—
36	#15668012	—	2	2	no	—
37	#15668011	—	4	2	no	—
38	#15667991	—	2	4	no	—
39	#15667989	—	4	4	no	—
40	C₄×Dic₁₇₈	—	2	2	no	—
41	D₄×Dic₈₉	—	4	2	no	—
42	Dic₇₁₂	—	2	8	no	—
43	Dic₇₁₂	—	2	4	no	—
44	Dic₇₁₂	—	4	2	no	—
45	Dic₇₁₂	D₄	1	4	yes	direct product factor
46	#15667946	C₂³	1	2	yes	—
47	#15667937	—	2	2	no	—
48	#15667937	—	2	4	no	—
49	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
50	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₈₉	—	2	4	no	—
52	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
53	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
57	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₇₈	—	2	2	no	—
59	C₂×Dic₁₇₈	—	2	4	no	—
60	C₂×Dic₁₇₈	—	4	8	no	—
61	C₂×Dic₁₇₈	—	4	4	no	—
62	C₂×Dic₁₇₈	—	8	4	no	—
63	C₂²×Dic₈₉	—	4	4	no	—
64	Dic₃₅₆	—	2	4	no	—
65	Dic₃₅₆	—	2	2	no	—
66	Dic₃₅₆	—	2	4	no	—
67	Dic₃₅₆	—	4	8	no	—
68	Dic₃₅₆	—	8	2	no	—
69	Dic₃₅₆	C₂×D₄	1	4	yes	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	—	4	2	no	—
72	C₄⋊Dic₈₉	—	4	2	no	—
73	#13083553	—	4	2	no	—
74	#13083553	—	4	4	no	—
75	C₂²⋊Dic₈₉	—	4	4	no	—
76	C₄×Dic₈₉	—	4	2	no	—
77	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
85	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
86	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₈₉	—	2	2	no	—
88	C₁₆×C₈₉	—	2	1	no	—
89	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
90	C₂³×C₈₉	C₂×D₈	1	2	yes	—
91	D₄×C₈₉	Q₃₂	1	4	yes	complemented
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₈₉	—	4	2	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	4	4	no	—
101	C₂×Dic₈₉	—	8	8	no	—
102	Dic₁₇₈	—	2	4	no	—
103	Dic₁₇₈	—	4	2	no	—
104	Dic₁₇₈	—	4	4	no	—
105	Dic₁₇₈	—	4	4	no	—
106	Dic₁₇₈	—	8	8	no	—
107	C₈×C₈₉	—	2	2	no	—
108	C₈×C₈₉	—	2	1	no	—
109	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₈₉	—	2	2	no	—
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	C₂×Q₃₂	1	4	yes	complemented
114	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	2	2	no	—
117	C₄×C₈₉	C₂×Q₃₂	1	2	yes	complemented
118	C₄×C₈₉	D₄²	1	1	yes	characteristic
119	C₄×C₈₉	#576	1	1	yes	characteristic
120	Dic₈₉	—	4	4	no	—
121	Dic₈₉	—	8	2	no	—
122	Dic₈₉	—	8	4	no	—
123	D₄×Q₃₂	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	—	2	4	no	—
125	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
126	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
127	C₂×C₈₉	#3417	1	1	yes	characteristic
128	C₂²×Q₃₂	—	89	2	no	—
129	D₄×Q₁₆	—	89	2	no	—
130	#2247	—	89	1	no	—
131	#2216	—	89	2	no	—
132	#2209	—	89	2	no	—
133	#2188	—	89	4	no	—
134	C₄×Q₃₂	—	89	1	no	—
135	D₄×C₁₆	D₈₉	1	1	yes	characteristic
136	C₈₉	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₁₆	—	89	4	no	—
138	D₄×Q₈	—	178	2	no	—
139	C₂×Q₃₂	—	89	1	no	—
140	C₂×Q₃₂	—	89	2	no	—
141	C₂×Q₃₂	—	89	4	no	—
142	C₂×Q₃₂	—	89	2	no	—
143	C₂×Q₃₂	—	178	8	no	—
144	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
145	#494	—	89	2	no	—
146	#467	—	89	4	no	—
147	#462	—	178	2	no	—
148	#451	—	178	4	no	—
149	C₄×Q₁₆	—	89	2	no	—
150	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
151	#366	—	89	1	no	—
152	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
153	#358	—	89	2	no	—
154	#358	—	89	4	no	—
155	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
156	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
157	C₂²×Q₈	—	178	4	no	—
158	C₂×Q₁₆	—	89	2	no	—
159	C₂×Q₁₆	—	89	4	no	—
160	C₂×Q₁₆	—	356	4	no	—
161	C₂×Q₁₆	—	178	8	no	—
162	C₂×Q₁₆	—	178	4	no	—
163	C₂×Q₁₆	—	178	2	no	—
164	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
165	C₄⋊Q₈	—	356	2	no	—
166	C₄⋊Q₈	—	178	2	no	—
167	C₂²⋊Q₈	—	356	4	no	—
168	C₂²⋊Q₈	—	178	4	no	—
169	C₄×Q₈	—	178	2	no	—
170	C₄×D₄	—	356	2	no	—
171	C₄×D₄	D₃₅₆	1	1	yes	characteristic
172	Q₃₂	—	89	4	no	—
173	Q₃₂	—	356	2	no	—
174	Q₃₂	—	178	8	no	—
175	Q₃₂	—	178	4	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
178	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
179	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
180	#107	—	89	2	no	—
181	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
182	#103	—	178	2	no	—
183	#103	—	178	4	no	—
184	C₂²⋊C₈	D₃₅₆	1	2	yes	—
185	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
186	C₂×Q₈	—	356	8	no	—
187	C₂×Q₈	—	356	4	no	—
188	C₂×Q₈	—	178	4	no	—
189	C₂×Q₈	—	178	2	no	—
190	C₂×Q₈	—	712	4	no	—
191	C₂×D₄	D₇₁₂	1	1	yes	characteristic
192	C₂²×C₄	—	356	4	no	—
193	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
194	Q₁₆	—	89	4	no	—
195	Q₁₆	—	356	8	no	—
196	Q₁₆	—	178	4	no	—
197	Q₁₆	—	178	4	no	—
198	Q₁₆	—	178	2	no	—
199	Q₁₆	—	712	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
203	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
204	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
205	C₄⋊C₄	—	356	2	no	—
206	C₄⋊C₄	—	356	2	no	—
207	C₄⋊C₄	—	356	2	no	—
208	C₄⋊C₄	—	356	4	no	—
209	C₄⋊C₄	—	178	2	no	—
210	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
211	C₂²⋊C₄	—	356	4	no	—
212	C₂²⋊C₄	D₇₁₂	1	2	yes	—
213	C₄²	—	356	2	no	—
214	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
215	C₁₆	—	2	2	no	—
216	C₁₆	—	2	1	no	—
217	C₁₆	D₄×D₈₉	1	2	yes	—
218	C₂³	C₂×D₇₁₂	1	2	yes	—
219	Q₈	—	356	2	no	—
220	Q₈	—	356	4	no	—
221	Q₈	—	356	4	no	—
222	Q₈	—	178	4	no	—
223	Q₈	—	712	8	no	—
224	D₄	Dic₇₁₂	1	4	yes	direct product factor
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	356	2	no	—
228	C₂×C₄	—	356	2	no	—
229	C₂×C₄	—	356	4	no	—
230	C₂×C₄	—	712	8	no	—
231	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
232	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
233	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	2	no	—
237	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
238	C₈	#15668058	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×Dic₇₁₂	1	4	yes	complemented
242	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	356	4	no	—
246	C₄	—	712	4	no	—
247	C₄	—	712	2	no	—
248	C₄	C₂×Dic₇₁₂	1	2	yes	complemented
249	C₄	D₄×D₃₅₆	1	1	yes	characteristic
250	C₄	#19152403	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	D₄×D₇₁₂	1	1	yes	characteristic
253	C₂	C₂²×Dic₇₁₂	1	1	yes	characteristic
254	C₂	#26033405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	D₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₄×C₈₉	D₈
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₇₁₂	C₂	semidirect
3. #26022505	C₂	semidirect
4. #26022343	C₂	semidirect
5. #26022303	C₂	semidirect
6. #26022188	C₂	semidirect
7. C₄×Dic₇₁₂	C₂	semidirect
45. Dic₇₁₂	D₄	direct (N × H)
91. D₄×C₈₉	Q₃₂	semidirect
113. C₂²×C₈₉	C₂×Q₃₂	semidirect
117. C₄×C₈₉	C₂×Q₃₂	semidirect
136. C₈₉	D₄×Q₃₂	semidirect
224. D₄	Dic₇₁₂	direct (N × H)
241. C₂²	C₂×Dic₇₁₂	semidirect
248. C₄	C₂×Dic₇₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₇₁₂	2
D₄×Dic₃₅₆	2
#26022505	2
#26022343	2
#26022303	2
#26022188	2
C₄×Dic₇₁₂	2
D₄×C₁₆×C₈₉	2
D₄×Q₃₂	89

Subgroups — D4×Dic712 #36711158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₇₁₂ #36711158