Subgroups — D₄×#14523249 #30652222

Order 16256.

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×#14523249	C₂	1	2	yes	maximal, complemented
2	#22550548	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550537	C₂	1	2	yes	maximal, complemented
4	#22550529	C₂	1	3	yes	maximal, complemented
5	#22550517	C₂	1	6	yes	maximal, complemented
6	D₄×D₅₀₈	C₂	1	3	yes	maximal, complemented
7	#22550427	C₂	1	3	yes	maximal, complemented
8	#22550389	C₂	1	6	yes	maximal, complemented
9	C₄×#14523249	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₁₂₇	C₂	1	3	yes	maximal, complemented
11	D₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523249	—	2	16	no	—
13	C₂×#14523249	C₂²	1	1	yes	characteristic
14	C₂×#14523249	C₂²	1	6	yes	complemented
15	C₂×#14523249	C₂²	1	2	yes	complemented
16	C₂×#14523249	C₂²	1	6	yes	complemented
17	C₂²×D₅₀₈	C₂²	1	6	yes	complemented
18	C₂×D₄×D₁₂₇	C₂²	1	3	yes	—
19	C₂²×C₄×D₁₂₇	C₂²	1	6	yes	complemented
20	#16861206	C₂²	1	3	yes	complemented
21	#16861198	C₂²	1	6	yes	complemented
22	#16861194	C₂²	1	12	yes	complemented
23	#16861185	C₂²	1	6	yes	complemented
24	#16861184	C₂²	1	6	yes	complemented
25	#16861184	C₂²	1	6	yes	complemented
26	#16861184	C₂²	1	6	yes	complemented
27	#16861183	C₂²	1	6	yes	complemented
28	#16861179	C₂²	1	12	yes	complemented
29	Q₈×Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₅₀₈	C₂²	1	3	yes	complemented
31	C₄×D₅₀₈	C₂²	1	3	yes	complemented
32	D₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
33	#16861171	C₂²	1	3	yes	complemented
34	#16861171	C₂²	1	3	yes	complemented
35	#16861171	C₂²	1	6	yes	complemented
36	C₄×C₂²⋊D₁₂₇	C₂²	1	6	yes	complemented
37	(C₂²⋊D₁₂₇)⋊C₄	C₂²	1	6	yes	complemented
38	#16861167	C₂²	1	3	yes	complemented
39	C₄⋊C₄×D₁₂₇	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₁₂₇	C₂²	1	6	yes	complemented
41	C₄²×D₁₂₇	C₂²	1	3	yes	complemented
42	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	complemented
43	C₄⋊Q₈×C₁₂₇	C₂²	1	3	yes	complemented
44	C₂²⋊Q₈×C₁₂₇	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₁₂₇	C₂²	1	3	yes	—
47	C₂³×D₁₂₇	C₂³	1	6	yes	—
48	#14523249	—	2	24	no	—
49	#14523249	—	2	24	no	—
50	#14523249	—	2	8	no	—
51	#14523249	D₄	1	8	yes	direct product factor
52	C₂×D₅₀₈	—	2	24	no	—
53	C₂×D₅₀₈	C₂³	1	12	yes	—
54	C₂×D₅₀₈	C₂³	1	3	yes	—
55	C₂×D₅₀₈	C₂³	1	6	yes	—
56	C₂×D₅₀₈	C₂³	1	6	yes	—
57	C₂×D₅₀₈	C₂³	1	3	yes	—
58	C₂×C₂²⋊D₁₂₇	C₂³	1	6	yes	—
59	D₄×D₁₂₇	—	2	12	no	—
60	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
61	C₂×C₄×D₁₂₇	—	2	24	no	—
62	C₂×C₄×D₁₂₇	C₂³	1	3	yes	—
63	C₂×C₄×D₁₂₇	C₂³	1	6	yes	—
64	C₂×C₄×D₁₂₇	C₂³	1	6	yes	—
65	C₂×C₄×D₁₂₇	C₂³	1	6	yes	—
66	C₂×C₄×D₁₂₇	C₂³	1	3	yes	—
67	C₂×C₄×D₁₂₇	C₂³	1	3	yes	—
68	C₄⋊Dic₁₂₇	C₂³	1	3	yes	—
69	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
70	Dic₁₂₇⋊C₄	C₂³	1	6	yes	—
71	C₂²⋊Dic₁₂₇	C₂³	1	2	yes	—
72	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
73	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
74	(C₂×D₁₂₇)⋊C₄	C₂³	1	12	yes	—
75	(C₂×D₁₂₇)⋊C₄	C₂³	1	6	yes	—
76	C₄×Dic₁₂₇	C₂³	1	3	yes	—
77	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
78	C₄×Dic₁₂₇	C₂³	1	3	yes	—
79	C₂×Q₈×C₁₂₇	—	2	8	no	—
80	C₂×Q₈×C₁₂₇	C₂³	1	6	yes	—
81	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₂₇	C₂³	1	6	yes	—
84	C₄⋊C₄×C₁₂₇	C₂³	1	3	yes	—
85	C₄⋊C₄×C₁₂₇	C₂³	1	3	yes	—
86	C₄⋊C₄×C₁₂₇	C₂³	1	6	yes	—
87	C₂²⋊C₄×C₁₂₇	C₂³	1	6	yes	—
88	C₄²×C₁₂₇	C₂³	1	3	yes	—
89	C₂²×D₁₂₇	—	2	12	no	—
90	C₂²×D₁₂₇	—	2	12	no	—
91	C₂²×D₁₂₇	—	4	6	no	—
92	C₂²×D₁₂₇	C₂⁴	1	6	yes	—
93	C₂²×D₁₂₇	C₂⁴	1	3	yes	—
94	D₅₀₈	—	2	24	no	—
95	D₅₀₈	—	2	12	no	—
96	D₅₀₈	—	2	6	no	—
97	D₅₀₈	—	4	6	no	—
98	D₅₀₈	C₂×D₄	1	12	yes	complemented
99	C₂²⋊D₁₂₇	—	2	12	no	—
100	C₂×Dic₁₂₇	—	2	4	no	—
101	C₂×Dic₁₂₇	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₁₂₇	C₂⁴	1	3	yes	—
103	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
104	C₂×Dic₁₂₇	C₂⁴	1	1	yes	characteristic
105	C₄×D₁₂₇	—	2	12	no	—
106	C₄×D₁₂₇	—	2	12	no	—
107	C₄×D₁₂₇	—	2	6	no	—
108	C₄×D₁₂₇	—	2	12	no	—
109	C₄×D₁₂₇	—	4	3	no	—
110	C₄×D₁₂₇	C₂×D₄	1	12	yes	complemented
111	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
112	Q₈×C₁₂₇	—	2	12	no	—
113	Q₈×C₁₂₇	C₂×D₄	1	4	yes	complemented
114	D₄×C₁₂₇	#41	1	4	yes	complemented
115	C₂×C₄×C₁₂₇	—	2	12	no	—
116	C₂×C₄×C₁₂₇	C₂⁴	1	3	yes	—
117	C₂×C₄×C₁₂₇	C₂⁴	1	6	yes	—
118	C₂×C₄×C₁₂₇	C₂⁴	1	3	yes	—
119	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
120	C₂²×C₁₂₇	—	2	2	no	—
121	C₂²×C₁₂₇	—	2	2	no	—
122	C₂²×C₁₂₇	C₂×#41	1	4	yes	complemented
123	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₂₇	—	2	3	no	—
125	C₂×D₁₂₇	—	2	3	no	—
126	C₂×D₁₂₇	—	2	6	no	—
127	C₂×D₁₂₇	—	2	6	no	—
128	C₂×D₁₂₇	—	2	6	no	—
129	C₂×D₁₂₇	—	4	24	no	—
130	C₂×D₁₂₇	C₂²×D₄	1	6	yes	—
131	C₄×C₁₂₇	—	2	6	no	—
132	C₄×C₁₂₇	—	2	3	no	—
133	C₄×C₁₂₇	C₂²×D₄	1	6	yes	—
134	C₄×C₁₂₇	C₂×#41	1	2	yes	complemented
135	Dic₁₂₇	—	2	2	no	—
136	Dic₁₂₇	—	2	3	no	—
137	Dic₁₂₇	—	2	1	no	—
138	Dic₁₂₇	C₂²×D₄	1	2	yes	—
139	C₂×C₁₂₇	—	2	4	no	—
140	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₁₂₇	C₂²×#41	1	1	yes	characteristic
142	C₂×C₁₂₇	#585	1	1	yes	characteristic
143	D₁₂₇	—	2	6	no	—
144	D₁₂₇	—	4	6	no	—
145	D₄×#41	—	127	1	no	maximal, Sylow
146	C₁₂₇	D₄×#41	1	1	yes	characteristic, complemented, Sylow
147	C₂²×#41	—	127	2	no	—
148	#550	—	127	1	no	—
149	D₄×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
150	#548	—	127	2	no	—
151	#547	—	127	3	no	—
152	#546	—	127	6	no	—
153	D₄²	—	127	3	no	—
154	#532	—	127	3	no	—
155	#525	—	127	6	no	—
156	C₄×#41	—	127	1	no	—
157	C₂×C₄×D₄	—	127	3	no	—
158	C₂×#41	—	127	1	no	—
159	C₂×#41	—	127	6	no	—
160	C₂×#41	—	127	6	no	—
161	C₂×#41	—	127	2	no	—
162	C₂×#41	—	254	16	no	—
163	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	complemented
164	C₂²×D₄	—	127	3	no	—
165	C₂²×D₄	—	127	6	no	—
166	C₂³×C₄	—	127	6	no	—
167	C₄⋊Q₈	C₂×D₁₂₇	1	3	yes	complemented
168	#127	—	127	3	no	—
169	#124	—	127	6	no	—
170	#123	—	127	12	no	—
171	C₂²⋊Q₈	—	127	6	no	—
172	C₂²⋊Q₈	C₂×D₁₂₇	1	6	yes	complemented
173	#121	—	127	6	no	—
174	#121	—	127	6	no	—
175	#121	—	127	6	no	—
176	#121	—	127	6	no	—
177	#120	—	127	12	no	—
178	C₄×Q₈	—	127	1	no	—
179	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
180	C₄×D₄	—	127	1	no	—
181	C₄×D₄	—	127	6	no	—
182	C₄×D₄	—	127	3	no	—
183	C₄×D₄	—	127	3	no	—
184	C₄×D₄	—	127	6	no	—
185	C₄×D₄	—	127	6	no	—
186	C₄×D₄	—	127	3	no	—
187	C₄×D₄	—	127	3	no	—
188	C₄×D₄	C₂×D₁₂₇	1	3	yes	—
189	#117	—	127	3	no	—
190	C₂×C₄⋊C₄	—	127	3	no	—
191	C₂×C₂²⋊C₄	—	127	6	no	—
192	C₂×C₄²	—	127	3	no	—
193	C₂⁴	—	127	6	no	—
194	#41	—	127	8	no	—
195	#41	—	254	24	no	—
196	#41	—	254	24	no	—
197	#41	—	254	8	no	—
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	C₂²×D₁₂₇	1	6	yes	—
200	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
201	C₂×D₄	—	127	6	no	—
202	C₂×D₄	—	127	3	no	—
203	C₂×D₄	—	127	6	no	—
204	C₂×D₄	—	127	3	no	—
205	C₂×D₄	—	127	12	no	—
206	C₂×D₄	—	127	6	no	—
207	C₂×D₄	—	254	24	no	—
208	C₂×D₄	—	254	12	no	—
209	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
210	C₂²×C₄	—	127	3	no	—
211	C₂²×C₄	—	127	2	no	—
212	C₂²×C₄	—	127	3	no	—
213	C₂²×C₄	—	127	3	no	—
214	C₂²×C₄	—	127	6	no	—
215	C₂²×C₄	—	127	6	no	—
216	C₂²×C₄	—	127	6	no	—
217	C₂²×C₄	—	254	24	no	—
218	C₂²×C₄	C₂²×D₁₂₇	1	6	yes	—
219	C₄⋊C₄	—	127	6	no	—
220	C₄⋊C₄	—	127	1	no	—
221	C₄⋊C₄	—	127	3	no	—
222	C₄⋊C₄	C₂²×D₁₂₇	1	3	yes	—
223	C₄⋊C₄	C₂²×D₁₂₇	1	3	yes	—
224	C₄⋊C₄	C₂²×D₁₂₇	1	6	yes	—
225	C₂²⋊C₄	—	127	2	no	—
226	C₂²⋊C₄	—	127	6	no	—
227	C₂²⋊C₄	—	127	6	no	—
228	C₂²⋊C₄	—	127	6	no	—
229	C₂²⋊C₄	—	127	12	no	—
230	C₂²⋊C₄	C₂²×D₁₂₇	1	6	yes	—
231	C₄²	—	127	1	no	—
232	C₄²	—	127	3	no	—
233	C₄²	—	127	3	no	—
234	C₄²	C₂²×D₁₂₇	1	3	yes	—
235	C₂³	—	127	3	no	—
236	C₂³	—	127	6	no	—
237	C₂³	—	508	6	no	—
238	C₂³	—	254	12	no	—
239	C₂³	—	254	12	no	—
240	C₂³	C₂³×D₁₂₇	1	2	yes	—
241	Q₈	—	2	12	no	—
242	Q₈	D₄×D₁₂₇	1	4	yes	complemented
243	D₄	—	127	12	no	—
244	D₄	—	508	6	no	—
245	D₄	—	254	24	no	—
246	D₄	—	254	12	no	—
247	D₄	—	254	12	no	—
248	D₄	—	254	6	no	—
249	D₄	#14523249	1	4	yes	direct product factor
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	127	1	no	—
252	C₂×C₄	—	127	12	no	—
253	C₂×C₄	—	127	1	no	—
254	C₂×C₄	—	127	3	no	—
255	C₂×C₄	—	127	2	no	—
256	C₂×C₄	—	508	3	no	—
257	C₂×C₄	—	254	12	no	—
258	C₂×C₄	—	254	12	no	—
259	C₂×C₄	—	254	4	no	—
260	C₂×C₄	—	254	12	no	—
261	C₂×C₄	—	254	6	no	—
262	C₂×C₄	C₂³×D₁₂₇	1	3	yes	—
263	C₂×C₄	C₂³×D₁₂₇	1	6	yes	—
264	C₂×C₄	C₂³×D₁₂₇	1	3	yes	—
265	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	127	6	no	—
269	C₂²	—	508	24	no	—
270	C₂²	—	254	6	no	—
271	C₂²	—	254	6	no	—
272	C₂²	—	254	6	no	—
273	C₂²	—	254	3	no	—
274	C₂²	—	254	3	no	—
275	C₂²	C₂×#14523249	1	4	yes	complemented
276	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
277	C₄	—	2	6	no	—
278	C₄	—	2	3	no	—
279	C₄	—	127	2	no	—
280	C₄	—	254	2	no	—
281	C₄	—	254	3	no	—
282	C₄	—	254	1	no	—
283	C₄	C₂×D₄×D₁₂₇	1	6	yes	—
284	C₄	C₂×#14523249	1	2	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	—	508	6	no	—
287	C₂	—	254	6	no	—
288	C₂	C₂²×D₄×D₁₂₇	1	1	yes	characteristic
289	C₂	C₂²×#14523249	1	1	yes	characteristic
290	C₂	#22550713	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₂₇
commutator, socle, cosocle	C₂²×C₁₂₇	C₂⁵
Fitting	D₄×Q₈×C₁₂₇	C₂
hypercenter	D₄×Q₈	D₁₂₇
hypocenter	C₁₂₇	D₄×#41
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523249	C₂	semidirect
2. #22550548	C₂	semidirect
3. #22550537	C₂	semidirect
4. #22550529	C₂	semidirect
5. #22550517	C₂	semidirect
6. D₄×D₅₀₈	C₂	semidirect
7. #22550427	C₂	semidirect
8. #22550389	C₂	semidirect
9. C₄×#14523249	C₂	semidirect
10. C₄×D₄×D₁₂₇	C₂	semidirect
11. D₄×Q₈×C₁₂₇	C₂	semidirect
14. C₂×#14523249	C₂²	semidirect
15. C₂×#14523249	C₂²	semidirect
16. C₂×#14523249	C₂²	semidirect
17. C₂²×D₅₀₈	C₂²	semidirect
19. C₂²×C₄×D₁₂₇	C₂²	semidirect
20. #16861206	C₂²	semidirect
21. #16861198	C₂²	semidirect
22. #16861194	C₂²	semidirect
23. #16861185	C₂²	semidirect
24. #16861184	C₂²	semidirect
25. #16861184	C₂²	semidirect
26. #16861184	C₂²	semidirect
27. #16861183	C₂²	semidirect
28. #16861179	C₂²	semidirect
29. Q₈×Dic₁₂₇	C₂²	semidirect
30. C₄×D₅₀₈	C₂²	semidirect
31. C₄×D₅₀₈	C₂²	semidirect
33. #16861171	C₂²	semidirect
34. #16861171	C₂²	semidirect
35. #16861171	C₂²	semidirect
36. C₄×C₂²⋊D₁₂₇	C₂²	semidirect
37. (C₂²⋊D₁₂₇)⋊C₄	C₂²	semidirect
38. #16861167	C₂²	semidirect
39. C₄⋊C₄×D₁₂₇	C₂²	semidirect
40. C₂²⋊C₄×D₁₂₇	C₂²	semidirect
41. C₄²×D₁₂₇	C₂²	semidirect
42. C₂²×Q₈×C₁₂₇	C₂²	semidirect
43. C₄⋊Q₈×C₁₂₇	C₂²	semidirect
44. C₂²⋊Q₈×C₁₂₇	C₂²	semidirect
45. C₄×Q₈×C₁₂₇	C₂²	semidirect
51. #14523249	D₄	direct (N × H)
98. D₅₀₈	C₂×D₄	semidirect
110. C₄×D₁₂₇	C₂×D₄	semidirect
113. Q₈×C₁₂₇	C₂×D₄	semidirect
114. D₄×C₁₂₇	#41	semidirect
122. C₂²×C₁₂₇	C₂×#41	semidirect
134. C₄×C₁₂₇	C₂×#41	semidirect
146. C₁₂₇	D₄×#41	semidirect
149. D₄×Q₈	D₁₂₇	semidirect
163. C₂²×Q₈	C₂×D₁₂₇	semidirect
167. C₄⋊Q₈	C₂×D₁₂₇	semidirect
172. C₂²⋊Q₈	C₂×D₁₂₇	semidirect
179. C₄×Q₈	C₂×D₁₂₇	semidirect
242. Q₈	D₄×D₁₂₇	semidirect
249. D₄	#14523249	direct (N × H)
275. C₂²	C₂×#14523249	semidirect
284. C₄	C₂×#14523249	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523249	2
#22550548	2
#22550537	2
#22550529	2
#22550517	2
D₄×D₅₀₈	2
#22550427	2
#22550389	2
C₄×#14523249	2
C₄×D₄×D₁₂₇	2
D₄×Q₈×C₁₂₇	2
D₄×#41	127

Subgroups — D4×#14523249 #30652222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#14523249 #30652222