Subgroups — D₄×#92499 #16859203

Order 3968.

1648 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92499	C₂	1	2	yes	maximal, complemented
2	#14521721	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521710	C₂	1	2	yes	maximal, complemented
4	#14521702	C₂	1	3	yes	maximal, complemented
5	#14521690	C₂	1	6	yes	maximal, complemented
6	D₄×D₁₂₄	C₂	1	3	yes	maximal, complemented
7	#14521600	C₂	1	3	yes	maximal, complemented
8	#14521562	C₂	1	6	yes	maximal, complemented
9	C₄×#92499	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₃₁	C₂	1	3	yes	maximal, complemented
11	D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₃₁	C₂²	1	2	yes	complemented
13	C₂×#92499	—	2	16	no	—
14	C₂×#92499	C₂²	1	1	yes	characteristic
15	C₂×#92499	C₂²	1	6	yes	complemented
16	C₂×#92499	C₂²	1	6	yes	complemented
17	C₂×#92499	C₂²	1	2	yes	complemented
18	C₂×D₄×D₃₁	C₂²	1	3	yes	—
19	C₂²×D₁₂₄	C₂²	1	6	yes	complemented
20	C₂²×C₄×D₃₁	C₂²	1	6	yes	complemented
21	C₃₁×C₄⋊Q₈	C₂²	1	3	yes	complemented
22	C₃₁×C₂²⋊Q₈	C₂²	1	6	yes	complemented
23	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
24	C₄×D₄×C₃₁	C₂²	1	3	yes	—
25	#11758524	C₂²	1	6	yes	complemented
26	#11758523	C₂²	1	6	yes	complemented
27	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
28	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
29	#11758507	C₂²	1	6	yes	complemented
30	C₄×C₂²⋊D₃₁	C₂²	1	6	yes	complemented
31	#11758472	C₂²	1	6	yes	complemented
32	#11758472	C₂²	1	6	yes	complemented
33	#11758472	C₂²	1	6	yes	complemented
34	#11758471	C₂²	1	12	yes	complemented
35	#11758470	C₂²	1	3	yes	complemented
36	#11758470	C₂²	1	6	yes	complemented
37	#11758470	C₂²	1	3	yes	complemented
38	#11758469	C₂²	1	3	yes	complemented
39	C₄⋊C₄×D₃₁	C₂²	1	3	yes	complemented
40	#11758459	C₂²	1	12	yes	complemented
41	(C₂²⋊D₃₁)⋊C₄	C₂²	1	6	yes	complemented
42	C₂²⋊C₄×D₃₁	C₂²	1	6	yes	complemented
43	#11758451	C₂²	1	3	yes	complemented
44	C₄×D₁₂₄	C₂²	1	3	yes	complemented
45	C₄×D₁₂₄	C₂²	1	3	yes	complemented
46	C₄²×D₃₁	C₂²	1	3	yes	complemented
47	C₂³×D₃₁	C₂³	1	6	yes	—
48	C₂×Q₈×C₃₁	—	2	8	no	—
49	C₂×Q₈×C₃₁	C₂³	1	6	yes	—
50	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
53	C₂×C₂²⋊D₃₁	C₂³	1	6	yes	—
54	C₂²×Dic₃₁	C₂³	1	2	yes	—
55	#92499	—	2	8	no	—
56	#92499	—	2	24	no	—
57	#92499	—	2	24	no	—
58	#92499	D₄	1	8	yes	direct product factor
59	D₄×D₃₁	—	2	12	no	—
60	C₂×D₁₂₄	—	2	24	no	—
61	C₂×D₁₂₄	C₂³	1	3	yes	—
62	C₂×D₁₂₄	C₂³	1	6	yes	—
63	C₂×D₁₂₄	C₂³	1	12	yes	—
64	C₂×D₁₂₄	C₂³	1	6	yes	—
65	C₂×D₁₂₄	C₂³	1	3	yes	—
66	C₂×C₄×D₃₁	—	2	24	no	—
67	C₂×C₄×D₃₁	C₂³	1	3	yes	—
68	C₂×C₄×D₃₁	C₂³	1	6	yes	—
69	C₂×C₄×D₃₁	C₂³	1	6	yes	—
70	C₂×C₄×D₃₁	C₂³	1	6	yes	—
71	C₂×C₄×D₃₁	C₂³	1	3	yes	—
72	C₂×C₄×D₃₁	C₂³	1	3	yes	—
73	C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
74	C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
75	C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
76	C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
77	C₄²×C₃₁	C₂³	1	3	yes	—
78	C₂²⋊Dic₃₁	C₂³	1	2	yes	—
79	(C₂×D₃₁)⋊C₄	C₂³	1	6	yes	—
80	(C₂×D₃₁)⋊C₄	C₂³	1	6	yes	—
81	(C₂×D₃₁)⋊C₄	C₂³	1	6	yes	—
82	(C₂×D₃₁)⋊C₄	C₂³	1	12	yes	—
83	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₃₁	C₂³	1	3	yes	—
85	Dic₃₁⋊C₄	C₂³	1	6	yes	—
86	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
87	C₄×Dic₃₁	C₂³	1	3	yes	—
88	C₄×Dic₃₁	C₂³	1	3	yes	—
89	C₂³×C₃₁	C₂⁴	1	2	yes	—
90	C₂²×D₃₁	—	2	12	no	—
91	C₂²×D₃₁	—	2	12	no	—
92	C₂²×D₃₁	—	4	6	no	—
93	C₂²×D₃₁	C₂⁴	1	6	yes	—
94	C₂²×D₃₁	C₂⁴	1	3	yes	—
95	Q₈×C₃₁	—	2	12	no	—
96	Q₈×C₃₁	C₂×D₄	1	4	yes	complemented
97	D₄×C₃₁	#41	1	4	yes	complemented
98	C₂×C₄×C₃₁	—	2	12	no	—
99	C₂×C₄×C₃₁	C₂⁴	1	3	yes	—
100	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
101	C₂×C₄×C₃₁	C₂⁴	1	3	yes	—
102	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₃₁	—	2	12	no	—
104	C₂×Dic₃₁	—	2	4	no	—
105	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
106	C₂×Dic₃₁	C₂⁴	1	3	yes	—
107	C₂×Dic₃₁	C₂⁴	1	2	yes	—
108	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
109	D₁₂₄	—	2	24	no	—
110	D₁₂₄	—	2	12	no	—
111	D₁₂₄	—	2	6	no	—
112	D₁₂₄	—	4	6	no	—
113	D₁₂₄	C₂×D₄	1	12	yes	complemented
114	C₄×D₃₁	—	2	12	no	—
115	C₄×D₃₁	—	2	12	no	—
116	C₄×D₃₁	—	2	6	no	—
117	C₄×D₃₁	—	2	12	no	—
118	C₄×D₃₁	—	4	3	no	—
119	C₄×D₃₁	C₂×D₄	1	12	yes	complemented
120	D₄×#41	—	31	1	no	maximal, Sylow
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	2	2	no	—
123	C₂²×C₃₁	C₂×#41	1	4	yes	complemented
124	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
125	C₂×D₃₁	—	2	3	no	—
126	C₂×D₃₁	—	2	3	no	—
127	C₂×D₃₁	—	2	6	no	—
128	C₂×D₃₁	—	2	6	no	—
129	C₂×D₃₁	—	2	6	no	—
130	C₂×D₃₁	—	4	24	no	—
131	C₂×D₃₁	C₂²×D₄	1	6	yes	—
132	C₄×C₃₁	—	2	6	no	—
133	C₄×C₃₁	—	2	3	no	—
134	C₄×C₃₁	C₂²×D₄	1	6	yes	—
135	C₄×C₃₁	C₂×#41	1	2	yes	complemented
136	Dic₃₁	—	2	2	no	—
137	Dic₃₁	—	2	3	no	—
138	Dic₃₁	—	2	1	no	—
139	Dic₃₁	C₂²×D₄	1	2	yes	—
140	C₂²×#41	—	31	2	no	—
141	#550	—	31	1	no	—
142	D₄×Q₈	D₃₁	1	1	yes	characteristic, complemented
143	#548	—	31	2	no	—
144	#547	—	31	3	no	—
145	#546	—	31	6	no	—
146	D₄²	—	31	3	no	—
147	#532	—	31	3	no	—
148	#525	—	31	6	no	—
149	C₄×#41	—	31	1	no	—
150	C₂×C₄×D₄	—	31	3	no	—
151	C₂×C₃₁	—	2	4	no	—
152	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
153	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
154	C₂×C₃₁	#585	1	1	yes	characteristic
155	D₃₁	—	2	6	no	—
156	D₃₁	—	4	6	no	—
157	C₂×#41	—	31	1	no	—
158	C₂×#41	—	31	6	no	—
159	C₂×#41	—	31	6	no	—
160	C₂×#41	—	31	2	no	—
161	C₂×#41	—	62	16	no	—
162	C₂²×Q₈	C₂×D₃₁	1	2	yes	complemented
163	C₂²×D₄	—	31	3	no	—
164	C₂²×D₄	—	31	6	no	—
165	C₂³×C₄	—	31	6	no	—
166	C₄⋊Q₈	C₂×D₃₁	1	3	yes	complemented
167	#127	—	31	3	no	—
168	#124	—	31	6	no	—
169	#123	—	31	12	no	—
170	C₂²⋊Q₈	—	31	6	no	—
171	C₂²⋊Q₈	C₂×D₃₁	1	6	yes	complemented
172	#121	—	31	6	no	—
173	#121	—	31	6	no	—
174	#121	—	31	6	no	—
175	#121	—	31	6	no	—
176	#120	—	31	12	no	—
177	C₄×Q₈	—	31	1	no	—
178	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
179	C₄×D₄	—	31	1	no	—
180	C₄×D₄	—	31	6	no	—
181	C₄×D₄	—	31	3	no	—
182	C₄×D₄	—	31	3	no	—
183	C₄×D₄	—	31	6	no	—
184	C₄×D₄	—	31	6	no	—
185	C₄×D₄	—	31	3	no	—
186	C₄×D₄	—	31	3	no	—
187	C₄×D₄	C₂×D₃₁	1	3	yes	—
188	#117	—	31	3	no	—
189	C₂×C₄⋊C₄	—	31	3	no	—
190	C₂×C₂²⋊C₄	—	31	6	no	—
191	C₂×C₄²	—	31	3	no	—
192	C₃₁	D₄×#41	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	31	6	no	—
194	#41	—	31	8	no	—
195	#41	—	62	24	no	—
196	#41	—	62	24	no	—
197	#41	—	62	8	no	—
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	C₂²×D₃₁	1	6	yes	—
200	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
201	C₂×D₄	—	31	6	no	—
202	C₂×D₄	—	31	3	no	—
203	C₂×D₄	—	31	3	no	—
204	C₂×D₄	—	31	12	no	—
205	C₂×D₄	—	31	6	no	—
206	C₂×D₄	—	31	6	no	—
207	C₂×D₄	—	62	24	no	—
208	C₂×D₄	—	62	12	no	—
209	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
210	C₂²×C₄	—	31	3	no	—
211	C₂²×C₄	—	31	2	no	—
212	C₂²×C₄	—	31	3	no	—
213	C₂²×C₄	—	31	6	no	—
214	C₂²×C₄	—	31	6	no	—
215	C₂²×C₄	—	31	6	no	—
216	C₂²×C₄	—	31	3	no	—
217	C₂²×C₄	—	62	24	no	—
218	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
219	C₄⋊C₄	—	31	6	no	—
220	C₄⋊C₄	—	31	1	no	—
221	C₄⋊C₄	—	31	3	no	—
222	C₄⋊C₄	C₂²×D₃₁	1	3	yes	—
223	C₄⋊C₄	C₂²×D₃₁	1	3	yes	—
224	C₄⋊C₄	C₂²×D₃₁	1	6	yes	—
225	C₂²⋊C₄	—	31	2	no	—
226	C₂²⋊C₄	—	31	6	no	—
227	C₂²⋊C₄	—	31	6	no	—
228	C₂²⋊C₄	—	31	6	no	—
229	C₂²⋊C₄	—	31	12	no	—
230	C₂²⋊C₄	C₂²×D₃₁	1	6	yes	—
231	C₄²	—	31	1	no	—
232	C₄²	—	31	3	no	—
233	C₄²	—	31	3	no	—
234	C₄²	C₂²×D₃₁	1	3	yes	—
235	C₂³	—	31	3	no	—
236	C₂³	—	31	6	no	—
237	C₂³	—	62	12	no	—
238	C₂³	—	62	12	no	—
239	C₂³	—	124	6	no	—
240	C₂³	C₂³×D₃₁	1	2	yes	—
241	Q₈	—	2	12	no	—
242	Q₈	D₄×D₃₁	1	4	yes	complemented
243	D₄	—	31	12	no	—
244	D₄	—	62	24	no	—
245	D₄	—	62	12	no	—
246	D₄	—	62	12	no	—
247	D₄	—	62	6	no	—
248	D₄	—	124	6	no	—
249	D₄	#92499	1	4	yes	direct product factor
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	31	1	no	—
252	C₂×C₄	—	31	12	no	—
253	C₂×C₄	—	31	1	no	—
254	C₂×C₄	—	31	3	no	—
255	C₂×C₄	—	31	2	no	—
256	C₂×C₄	—	62	12	no	—
257	C₂×C₄	—	62	12	no	—
258	C₂×C₄	—	62	4	no	—
259	C₂×C₄	—	62	12	no	—
260	C₂×C₄	—	62	6	no	—
261	C₂×C₄	—	124	3	no	—
262	C₂×C₄	C₂³×D₃₁	1	3	yes	—
263	C₂×C₄	C₂³×D₃₁	1	6	yes	—
264	C₂×C₄	C₂³×D₃₁	1	3	yes	—
265	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	31	6	no	—
269	C₂²	—	62	3	no	—
270	C₂²	—	62	3	no	—
271	C₂²	—	62	6	no	—
272	C₂²	—	62	6	no	—
273	C₂²	—	62	6	no	—
274	C₂²	—	124	24	no	—
275	C₂²	C₂×#92499	1	4	yes	complemented
276	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
277	C₄	—	2	6	no	—
278	C₄	—	2	3	no	—
279	C₄	—	31	2	no	—
280	C₄	—	62	2	no	—
281	C₄	—	62	3	no	—
282	C₄	—	62	1	no	—
283	C₄	C₂×D₄×D₃₁	1	6	yes	—
284	C₄	C₂×#92499	1	2	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	—	62	6	no	—
287	C₂	—	124	6	no	—
288	C₂	C₂²×D₄×D₃₁	1	1	yes	characteristic
289	C₂	C₂²×#92499	1	1	yes	characteristic
290	C₂	#14521886	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁵
Fitting	D₄×Q₈×C₃₁	C₂
hypercenter	D₄×Q₈	D₃₁
hypocenter	C₃₁	D₄×#41
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#92499	C₂	semidirect
2. #14521721	C₂	semidirect
3. #14521710	C₂	semidirect
4. #14521702	C₂	semidirect
5. #14521690	C₂	semidirect
6. D₄×D₁₂₄	C₂	semidirect
7. #14521600	C₂	semidirect
8. #14521562	C₂	semidirect
9. C₄×#92499	C₂	semidirect
10. C₄×D₄×D₃₁	C₂	semidirect
11. D₄×Q₈×C₃₁	C₂	semidirect
12. C₂²×Q₈×C₃₁	C₂²	semidirect
15. C₂×#92499	C₂²	semidirect
16. C₂×#92499	C₂²	semidirect
17. C₂×#92499	C₂²	semidirect
19. C₂²×D₁₂₄	C₂²	semidirect
20. C₂²×C₄×D₃₁	C₂²	semidirect
21. C₃₁×C₄⋊Q₈	C₂²	semidirect
22. C₃₁×C₂²⋊Q₈	C₂²	semidirect
23. C₄×Q₈×C₃₁	C₂²	semidirect
25. #11758524	C₂²	semidirect
26. #11758523	C₂²	semidirect
27. Q₈×Dic₃₁	C₂²	semidirect
29. #11758507	C₂²	semidirect
30. C₄×C₂²⋊D₃₁	C₂²	semidirect
31. #11758472	C₂²	semidirect
32. #11758472	C₂²	semidirect
33. #11758472	C₂²	semidirect
34. #11758471	C₂²	semidirect
35. #11758470	C₂²	semidirect
36. #11758470	C₂²	semidirect
37. #11758470	C₂²	semidirect
38. #11758469	C₂²	semidirect
39. C₄⋊C₄×D₃₁	C₂²	semidirect
40. #11758459	C₂²	semidirect
41. (C₂²⋊D₃₁)⋊C₄	C₂²	semidirect
42. C₂²⋊C₄×D₃₁	C₂²	semidirect
43. #11758451	C₂²	semidirect
44. C₄×D₁₂₄	C₂²	semidirect
45. C₄×D₁₂₄	C₂²	semidirect
46. C₄²×D₃₁	C₂²	semidirect
58. #92499	D₄	direct (N × H)
96. Q₈×C₃₁	C₂×D₄	semidirect
97. D₄×C₃₁	#41	semidirect
113. D₁₂₄	C₂×D₄	semidirect
119. C₄×D₃₁	C₂×D₄	semidirect
123. C₂²×C₃₁	C₂×#41	semidirect
135. C₄×C₃₁	C₂×#41	semidirect
142. D₄×Q₈	D₃₁	semidirect
162. C₂²×Q₈	C₂×D₃₁	semidirect
166. C₄⋊Q₈	C₂×D₃₁	semidirect
171. C₂²⋊Q₈	C₂×D₃₁	semidirect
178. C₄×Q₈	C₂×D₃₁	semidirect
192. C₃₁	D₄×#41	semidirect
242. Q₈	D₄×D₃₁	semidirect
249. D₄	#92499	direct (N × H)
275. C₂²	C₂×#92499	semidirect
284. C₄	C₂×#92499	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#92499	2
#14521721	2
#14521710	2
#14521702	2
#14521690	2
D₄×D₁₂₄	2
#14521600	2
#14521562	2
C₄×#92499	2
C₄×D₄×D₃₁	2
D₄×Q₈×C₃₁	2
D₄×#41	31

Subgroups — D4×#92499 #16859203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#92499 #16859203