Subgroups — D₄×#16864403 #46824377

Order 34432.

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×#16864403	C₂	1	2	yes	maximal, complemented
2	#31771277	C₂	1	1	yes	maximal, characteristic, complemented
3	#31771266	C₂	1	2	yes	maximal, complemented
4	#31771258	C₂	1	3	yes	maximal, complemented
5	#31771246	C₂	1	6	yes	maximal, complemented
6	D₄×D₁₀₇₆	C₂	1	3	yes	maximal, complemented
7	#31771156	C₂	1	3	yes	maximal, complemented
8	#31771118	C₂	1	6	yes	maximal, complemented
9	C₄×#16864403	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₂₆₉	C₂	1	3	yes	maximal, complemented
11	D₄×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16864403	—	2	16	no	—
13	C₂×#16864403	C₂²	1	1	yes	characteristic
14	C₂×#16864403	C₂²	1	6	yes	complemented
15	C₂×#16864403	C₂²	1	6	yes	complemented
16	C₂×#16864403	C₂²	1	2	yes	complemented
17	C₂²×D₁₀₇₆	C₂²	1	6	yes	complemented
18	C₂×D₄×D₂₆₉	C₂²	1	3	yes	—
19	C₂²×C₄×D₂₆₉	C₂²	1	6	yes	complemented
20	#22575450	C₂²	1	3	yes	complemented
21	#22575442	C₂²	1	6	yes	complemented
22	#22575438	C₂²	1	12	yes	complemented
23	#22575429	C₂²	1	6	yes	complemented
24	#22575428	C₂²	1	6	yes	complemented
25	#22575428	C₂²	1	6	yes	complemented
26	#22575428	C₂²	1	6	yes	complemented
27	#22575427	C₂²	1	6	yes	complemented
28	#22575423	C₂²	1	12	yes	complemented
29	Q₈×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₁₀₇₆	C₂²	1	3	yes	complemented
31	C₄×D₁₀₇₆	C₂²	1	3	yes	complemented
32	D₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
33	#22575415	C₂²	1	3	yes	complemented
34	#22575415	C₂²	1	6	yes	complemented
35	#22575415	C₂²	1	3	yes	complemented
36	C₄×C₂²⋊D₂₆₉	C₂²	1	6	yes	complemented
37	#22575413	C₂²	1	6	yes	complemented
38	#22575411	C₂²	1	3	yes	complemented
39	C₄⋊C₄×D₂₆₉	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₂₆₉	C₂²	1	6	yes	complemented
41	C₄²×D₂₆₉	C₂²	1	3	yes	complemented
42	C₂²×Q₈×C₂₆₉	C₂²	1	2	yes	complemented
43	C₄⋊Q₈×C₂₆₉	C₂²	1	3	yes	complemented
44	C₂²⋊Q₈×C₂₆₉	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₂₆₉	C₂²	1	3	yes	—
47	C₂³×D₂₆₉	C₂³	1	6	yes	—
48	#16864403	—	2	24	no	—
49	#16864403	—	2	24	no	—
50	#16864403	—	2	8	no	—
51	#16864403	D₄	1	8	yes	direct product factor
52	C₂×D₁₀₇₆	—	2	24	no	—
53	C₂×D₁₀₇₆	C₂³	1	3	yes	—
54	C₂×D₁₀₇₆	C₂³	1	12	yes	—
55	C₂×D₁₀₇₆	C₂³	1	6	yes	—
56	C₂×D₁₀₇₆	C₂³	1	6	yes	—
57	C₂×D₁₀₇₆	C₂³	1	3	yes	—
58	C₂×C₂²⋊D₂₆₉	C₂³	1	6	yes	—
59	D₄×D₂₆₉	—	2	12	no	—
60	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
61	C₂×C₄×D₂₆₉	—	2	24	no	—
62	C₂×C₄×D₂₆₉	C₂³	1	3	yes	—
63	C₂×C₄×D₂₆₉	C₂³	1	6	yes	—
64	C₂×C₄×D₂₆₉	C₂³	1	6	yes	—
65	C₂×C₄×D₂₆₉	C₂³	1	6	yes	—
66	C₂×C₄×D₂₆₉	C₂³	1	3	yes	—
67	C₂×C₄×D₂₆₉	C₂³	1	3	yes	—
68	C₄⋊Dic₂₆₉	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₂₆₉	C₂³	1	3	yes	—
70	#16864381	C₂³	1	6	yes	—
71	C₂²⋊Dic₂₆₉	C₂³	1	2	yes	—
72	#16864379	C₂³	1	6	yes	—
73	#16864379	C₂³	1	6	yes	—
74	#16864379	C₂³	1	6	yes	—
75	#16864379	C₂³	1	12	yes	—
76	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
77	C₄×Dic₂₆₉	C₂³	1	3	yes	—
78	C₄×Dic₂₆₉	C₂³	1	3	yes	—
79	C₂×Q₈×C₂₆₉	—	2	8	no	—
80	C₂×Q₈×C₂₆₉	C₂³	1	6	yes	—
81	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₂₆₉	C₂³	1	6	yes	—
84	C₄⋊C₄×C₂₆₉	C₂³	1	3	yes	—
85	C₄⋊C₄×C₂₆₉	C₂³	1	3	yes	—
86	C₄⋊C₄×C₂₆₉	C₂³	1	6	yes	—
87	C₂²⋊C₄×C₂₆₉	C₂³	1	6	yes	—
88	C₄²×C₂₆₉	C₂³	1	3	yes	—
89	C₂²×D₂₆₉	—	2	12	no	—
90	C₂²×D₂₆₉	—	2	12	no	—
91	C₂²×D₂₆₉	—	4	6	no	—
92	C₂²×D₂₆₉	C₂⁴	1	6	yes	—
93	C₂²×D₂₆₉	C₂⁴	1	3	yes	—
94	D₁₀₇₆	—	2	24	no	—
95	D₁₀₇₆	—	2	12	no	—
96	D₁₀₇₆	—	2	6	no	—
97	D₁₀₇₆	—	4	6	no	—
98	D₁₀₇₆	C₂×D₄	1	12	yes	complemented
99	C₂²⋊D₂₆₉	—	2	12	no	—
100	C₂×Dic₂₆₉	—	2	4	no	—
101	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₂₆₉	C₂⁴	1	3	yes	—
103	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
104	C₂×Dic₂₆₉	C₂⁴	1	1	yes	characteristic
105	C₄×D₂₆₉	—	2	12	no	—
106	C₄×D₂₆₉	—	2	12	no	—
107	C₄×D₂₆₉	—	2	6	no	—
108	C₄×D₂₆₉	—	2	12	no	—
109	C₄×D₂₆₉	—	4	3	no	—
110	C₄×D₂₆₉	C₂×D₄	1	12	yes	complemented
111	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
112	Q₈×C₂₆₉	—	2	12	no	—
113	Q₈×C₂₆₉	C₂×D₄	1	4	yes	complemented
114	D₄×C₂₆₉	#41	1	4	yes	complemented
115	C₂×C₄×C₂₆₉	—	2	12	no	—
116	C₂×C₄×C₂₆₉	C₂⁴	1	3	yes	—
117	C₂×C₄×C₂₆₉	C₂⁴	1	6	yes	—
118	C₂×C₄×C₂₆₉	C₂⁴	1	3	yes	—
119	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
120	C₂²×C₂₆₉	—	2	2	no	—
121	C₂²×C₂₆₉	—	2	2	no	—
122	C₂²×C₂₆₉	C₂×#41	1	4	yes	complemented
123	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
124	C₂×D₂₆₉	—	2	3	no	—
125	C₂×D₂₆₉	—	2	3	no	—
126	C₂×D₂₆₉	—	2	6	no	—
127	C₂×D₂₆₉	—	2	6	no	—
128	C₂×D₂₆₉	—	2	6	no	—
129	C₂×D₂₆₉	—	4	24	no	—
130	C₂×D₂₆₉	C₂²×D₄	1	6	yes	—
131	C₄×C₂₆₉	—	2	6	no	—
132	C₄×C₂₆₉	—	2	3	no	—
133	C₄×C₂₆₉	C₂²×D₄	1	6	yes	—
134	C₄×C₂₆₉	C₂×#41	1	2	yes	complemented
135	Dic₂₆₉	—	2	2	no	—
136	Dic₂₆₉	—	2	3	no	—
137	Dic₂₆₉	—	2	1	no	—
138	Dic₂₆₉	C₂²×D₄	1	2	yes	—
139	C₂×C₂₆₉	—	2	4	no	—
140	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₂₆₉	C₂²×#41	1	1	yes	characteristic
142	C₂×C₂₆₉	#585	1	1	yes	characteristic
143	D₂₆₉	—	2	6	no	—
144	D₂₆₉	—	4	6	no	—
145	C₂₆₉	D₄×#41	1	1	yes	characteristic, complemented, Sylow
146	D₄×#41	—	269	1	no	maximal, Sylow
147	C₂²×#41	—	269	2	no	—
148	#550	—	269	1	no	—
149	D₄×Q₈	D₂₆₉	1	1	yes	characteristic, complemented
150	#548	—	269	2	no	—
151	#547	—	269	3	no	—
152	#546	—	269	6	no	—
153	D₄²	—	269	3	no	—
154	#532	—	269	3	no	—
155	#525	—	269	6	no	—
156	C₄×#41	—	269	1	no	—
157	C₂×C₄×D₄	—	269	3	no	—
158	C₂×#41	—	269	1	no	—
159	C₂×#41	—	269	6	no	—
160	C₂×#41	—	269	6	no	—
161	C₂×#41	—	269	2	no	—
162	C₂×#41	—	538	16	no	—
163	C₂²×Q₈	C₂×D₂₆₉	1	2	yes	complemented
164	C₂²×D₄	—	269	3	no	—
165	C₂²×D₄	—	269	6	no	—
166	C₂³×C₄	—	269	6	no	—
167	C₄⋊Q₈	C₂×D₂₆₉	1	3	yes	complemented
168	#127	—	269	3	no	—
169	#124	—	269	6	no	—
170	#123	—	269	12	no	—
171	C₂²⋊Q₈	—	269	6	no	—
172	C₂²⋊Q₈	C₂×D₂₆₉	1	6	yes	complemented
173	#121	—	269	6	no	—
174	#121	—	269	6	no	—
175	#121	—	269	6	no	—
176	#121	—	269	6	no	—
177	#120	—	269	12	no	—
178	C₄×Q₈	—	269	1	no	—
179	C₄×Q₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
180	C₄×D₄	—	269	1	no	—
181	C₄×D₄	—	269	6	no	—
182	C₄×D₄	—	269	3	no	—
183	C₄×D₄	—	269	3	no	—
184	C₄×D₄	—	269	6	no	—
185	C₄×D₄	—	269	6	no	—
186	C₄×D₄	—	269	3	no	—
187	C₄×D₄	—	269	3	no	—
188	C₄×D₄	C₂×D₂₆₉	1	3	yes	—
189	#117	—	269	3	no	—
190	C₂×C₄⋊C₄	—	269	3	no	—
191	C₂×C₂²⋊C₄	—	269	6	no	—
192	C₂×C₄²	—	269	3	no	—
193	C₂⁴	—	269	6	no	—
194	#41	—	269	8	no	—
195	#41	—	538	24	no	—
196	#41	—	538	24	no	—
197	#41	—	538	8	no	—
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	C₂²×D₂₆₉	1	6	yes	—
200	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
201	C₂×D₄	—	269	6	no	—
202	C₂×D₄	—	269	3	no	—
203	C₂×D₄	—	269	3	no	—
204	C₂×D₄	—	269	12	no	—
205	C₂×D₄	—	269	6	no	—
206	C₂×D₄	—	269	6	no	—
207	C₂×D₄	—	538	24	no	—
208	C₂×D₄	—	538	12	no	—
209	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic
210	C₂²×C₄	—	269	3	no	—
211	C₂²×C₄	—	269	2	no	—
212	C₂²×C₄	—	269	3	no	—
213	C₂²×C₄	—	269	6	no	—
214	C₂²×C₄	—	269	6	no	—
215	C₂²×C₄	—	269	6	no	—
216	C₂²×C₄	—	269	3	no	—
217	C₂²×C₄	—	538	24	no	—
218	C₂²×C₄	C₂²×D₂₆₉	1	6	yes	—
219	C₄⋊C₄	—	269	1	no	—
220	C₄⋊C₄	—	269	6	no	—
221	C₄⋊C₄	—	269	3	no	—
222	C₄⋊C₄	C₂²×D₂₆₉	1	3	yes	—
223	C₄⋊C₄	C₂²×D₂₆₉	1	3	yes	—
224	C₄⋊C₄	C₂²×D₂₆₉	1	6	yes	—
225	C₂²⋊C₄	—	269	2	no	—
226	C₂²⋊C₄	—	269	6	no	—
227	C₂²⋊C₄	—	269	6	no	—
228	C₂²⋊C₄	—	269	6	no	—
229	C₂²⋊C₄	—	269	12	no	—
230	C₂²⋊C₄	C₂²×D₂₆₉	1	6	yes	—
231	C₄²	—	269	1	no	—
232	C₄²	—	269	3	no	—
233	C₄²	—	269	3	no	—
234	C₄²	C₂²×D₂₆₉	1	3	yes	—
235	C₂³	—	269	3	no	—
236	C₂³	—	269	6	no	—
237	C₂³	—	538	12	no	—
238	C₂³	—	538	12	no	—
239	C₂³	—	1076	6	no	—
240	C₂³	C₂³×D₂₆₉	1	2	yes	—
241	Q₈	—	2	12	no	—
242	Q₈	D₄×D₂₆₉	1	4	yes	complemented
243	D₄	—	269	12	no	—
244	D₄	—	538	24	no	—
245	D₄	—	538	12	no	—
246	D₄	—	538	12	no	—
247	D₄	—	538	6	no	—
248	D₄	—	1076	6	no	—
249	D₄	#16864403	1	4	yes	direct product factor
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	269	1	no	—
252	C₂×C₄	—	269	12	no	—
253	C₂×C₄	—	269	1	no	—
254	C₂×C₄	—	269	3	no	—
255	C₂×C₄	—	269	2	no	—
256	C₂×C₄	—	538	12	no	—
257	C₂×C₄	—	538	12	no	—
258	C₂×C₄	—	538	4	no	—
259	C₂×C₄	—	538	12	no	—
260	C₂×C₄	—	538	6	no	—
261	C₂×C₄	—	1076	3	no	—
262	C₂×C₄	C₂³×D₂₆₉	1	3	yes	—
263	C₂×C₄	C₂³×D₂₆₉	1	6	yes	—
264	C₂×C₄	C₂³×D₂₆₉	1	3	yes	—
265	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	269	6	no	—
269	C₂²	—	538	3	no	—
270	C₂²	—	538	3	no	—
271	C₂²	—	538	6	no	—
272	C₂²	—	538	6	no	—
273	C₂²	—	538	6	no	—
274	C₂²	—	1076	24	no	—
275	C₂²	C₂×#16864403	1	4	yes	complemented
276	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
277	C₄	—	2	6	no	—
278	C₄	—	2	3	no	—
279	C₄	—	269	2	no	—
280	C₄	—	538	2	no	—
281	C₄	—	538	3	no	—
282	C₄	—	538	1	no	—
283	C₄	C₂×D₄×D₂₆₉	1	6	yes	—
284	C₄	C₂×#16864403	1	2	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	—	538	6	no	—
287	C₂	—	1076	6	no	—
288	C₂	C₂²×D₄×D₂₆₉	1	1	yes	characteristic
289	C₂	C₂²×#16864403	1	1	yes	characteristic
290	C₂	#31771442	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁵
Fitting	D₄×Q₈×C₂₆₉	C₂
hypercenter	D₄×Q₈	D₂₆₉
hypocenter	C₂₆₉	D₄×#41
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#16864403	C₂	semidirect
2. #31771277	C₂	semidirect
3. #31771266	C₂	semidirect
4. #31771258	C₂	semidirect
5. #31771246	C₂	semidirect
6. D₄×D₁₀₇₆	C₂	semidirect
7. #31771156	C₂	semidirect
8. #31771118	C₂	semidirect
9. C₄×#16864403	C₂	semidirect
10. C₄×D₄×D₂₆₉	C₂	semidirect
11. D₄×Q₈×C₂₆₉	C₂	semidirect
14. C₂×#16864403	C₂²	semidirect
15. C₂×#16864403	C₂²	semidirect
16. C₂×#16864403	C₂²	semidirect
17. C₂²×D₁₀₇₆	C₂²	semidirect
19. C₂²×C₄×D₂₆₉	C₂²	semidirect
20. #22575450	C₂²	semidirect
21. #22575442	C₂²	semidirect
22. #22575438	C₂²	semidirect
23. #22575429	C₂²	semidirect
24. #22575428	C₂²	semidirect
25. #22575428	C₂²	semidirect
26. #22575428	C₂²	semidirect
27. #22575427	C₂²	semidirect
28. #22575423	C₂²	semidirect
29. Q₈×Dic₂₆₉	C₂²	semidirect
30. C₄×D₁₀₇₆	C₂²	semidirect
31. C₄×D₁₀₇₆	C₂²	semidirect
33. #22575415	C₂²	semidirect
34. #22575415	C₂²	semidirect
35. #22575415	C₂²	semidirect
36. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
37. #22575413	C₂²	semidirect
38. #22575411	C₂²	semidirect
39. C₄⋊C₄×D₂₆₉	C₂²	semidirect
40. C₂²⋊C₄×D₂₆₉	C₂²	semidirect
41. C₄²×D₂₆₉	C₂²	semidirect
42. C₂²×Q₈×C₂₆₉	C₂²	semidirect
43. C₄⋊Q₈×C₂₆₉	C₂²	semidirect
44. C₂²⋊Q₈×C₂₆₉	C₂²	semidirect
45. C₄×Q₈×C₂₆₉	C₂²	semidirect
51. #16864403	D₄	direct (N × H)
98. D₁₀₇₆	C₂×D₄	semidirect
110. C₄×D₂₆₉	C₂×D₄	semidirect
113. Q₈×C₂₆₉	C₂×D₄	semidirect
114. D₄×C₂₆₉	#41	semidirect
122. C₂²×C₂₆₉	C₂×#41	semidirect
134. C₄×C₂₆₉	C₂×#41	semidirect
145. C₂₆₉	D₄×#41	semidirect
149. D₄×Q₈	D₂₆₉	semidirect
163. C₂²×Q₈	C₂×D₂₆₉	semidirect
167. C₄⋊Q₈	C₂×D₂₆₉	semidirect
172. C₂²⋊Q₈	C₂×D₂₆₉	semidirect
179. C₄×Q₈	C₂×D₂₆₉	semidirect
242. Q₈	D₄×D₂₆₉	semidirect
249. D₄	#16864403	direct (N × H)
275. C₂²	C₂×#16864403	semidirect
284. C₄	C₂×#16864403	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#16864403	2
#31771277	2
#31771266	2
#31771258	2
#31771246	2
D₄×D₁₀₇₆	2
#31771156	2
#31771118	2
C₄×#16864403	2
C₄×D₄×D₂₆₉	2
D₄×Q₈×C₂₆₉	2
D₄×#41	269

Subgroups — D4×#16864403 #46824377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#16864403 #46824377