Subgroups — D₄×Q₈×D₆₇ #22573679

Order 8576.

1568 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₆₇	C₂	1	2	yes	maximal, complemented
2	Q₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₁₃₄	C₂	1	3	yes	maximal, complemented
4	#16864061	C₂	1	3	yes	maximal, complemented
5	Q₈×C₂²⋊D₆₇	C₂	1	2	yes	maximal, complemented
6	#16864059	C₂	1	6	yes	maximal, complemented
7	C₄⋊Q₈×D₆₇	C₂	1	3	yes	maximal, complemented
8	C₂²⋊Q₈×D₆₇	C₂	1	6	yes	maximal, complemented
9	C₄×Q₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₆₇	C₂	1	3	yes	maximal
11	D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₁₃₄	C₂²	1	6	yes	complemented
13	C₂×Q₈×D₆₇	—	2	16	no	—
14	C₂×Q₈×D₆₇	C₂²	1	6	yes	—
15	C₂×Q₈×D₆₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×D₆₇	C₂²	1	6	yes	complemented
17	C₂×Q₈×D₆₇	C₂²	1	2	yes	complemented
18	C₂×D₄×D₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₆₇	C₂²	1	6	yes	—
20	#14524043	C₂²	1	3	yes	complemented
21	#14524042	C₂²	1	6	yes	complemented
22	#14524042	C₂²	1	6	yes	complemented
23	#14524041	C₂²	1	6	yes	complemented
24	#14524023	C₂²	1	6	yes	complemented
25	#14524022	C₂²	1	6	yes	complemented
26	#14524021	C₂²	1	12	yes	complemented
27	#14524020	C₂²	1	12	yes	complemented
28	#14524019	C₂²	1	6	yes	complemented
29	Q₈×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₄×Dic₁₃₄	C₂²	1	3	yes	complemented
31	#14524008	C₂²	1	3	yes	complemented
32	C₄×D₂₆₈	C₂²	1	3	yes	—
33	D₄×Dic₆₇	C₂²	1	3	yes	—
34	#14524005	C₂²	1	3	yes	—
35	C₄×C₂²⋊D₆₇	C₂²	1	6	yes	—
36	(C₂²⋊D₆₇)⋊C₄	C₂²	1	6	yes	—
37	C₄⋊C₄×D₆₇	C₂²	1	3	yes	—
38	C₄⋊C₄×D₆₇	C₂²	1	3	yes	—
39	C₄⋊C₄×D₆₇	C₂²	1	6	yes	—
40	C₂²⋊C₄×D₆₇	C₂²	1	6	yes	—
41	C₄²×D₆₇	C₂²	1	3	yes	—
42	C₂²×Q₈×C₆₇	C₂²	1	2	yes	complemented
43	C₄⋊Q₈×C₆₇	C₂²	1	3	yes	complemented
44	C₂²⋊Q₈×C₆₇	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₆₇	C₂²	1	3	yes	—
47	C₂³×D₆₇	C₂³	1	2	yes	—
48	C₂×Dic₁₃₄	—	2	24	no	—
49	C₂×Dic₁₃₄	C₂³	1	12	yes	—
50	C₂×Dic₁₃₄	C₂³	1	6	yes	—
51	C₂×Dic₁₃₄	C₂³	1	3	yes	—
52	Q₈×D₆₇	—	2	8	no	—
53	Q₈×D₆₇	—	2	24	no	—
54	Q₈×D₆₇	—	2	24	no	—
55	Q₈×D₆₇	D₄	1	8	yes	direct product factor
56	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
58	D₄×D₆₇	Q₈	1	8	yes	direct product factor
59	C₂²×Dic₆₇	C₂³	1	6	yes	—
60	C₂×C₄×D₆₇	—	2	24	no	—
61	C₂×C₄×D₆₇	C₂³	1	3	yes	—
62	C₂×C₄×D₆₇	C₂³	1	6	yes	—
63	C₂×C₄×D₆₇	C₂³	1	6	yes	—
64	C₂×C₄×D₆₇	C₂³	1	6	yes	—
65	C₂×C₄×D₆₇	C₂³	1	3	yes	—
66	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₆₇	C₂³	1	3	yes	—
68	C₄⋊Dic₆₇	C₂³	1	3	yes	—
69	C₄⋊Dic₆₇	C₂³	1	6	yes	—
70	Dic₆₇⋊C₄	C₂³	1	6	yes	—
71	Dic₆₇⋊C₄	C₂³	1	6	yes	—
72	Dic₆₇⋊C₄	C₂³	1	12	yes	—
73	C₂²⋊Dic₆₇	C₂³	1	6	yes	—
74	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
75	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
76	C₄×Dic₆₇	C₂³	1	3	yes	—
77	C₄×Dic₆₇	C₂³	1	3	yes	—
78	C₄×Dic₆₇	C₂³	1	3	yes	—
79	C₂×Q₈×C₆₇	—	2	8	no	—
80	C₂×Q₈×C₆₇	C₂³	1	6	yes	—
81	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂³	1	6	yes	—
84	C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
85	C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
86	C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
87	C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
88	C₄²×C₆₇	C₂³	1	3	yes	—
89	C₂³×C₆₇	C₂⁴	1	2	yes	—
90	C₂²×D₆₇	—	2	4	no	—
91	C₂²×D₆₇	—	2	4	no	—
92	C₂²×D₆₇	C₂×Q₈	1	8	yes	complemented
93	C₂²×D₆₇	C₂⁴	1	2	yes	—
94	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
95	Q₈×C₆₇	—	2	12	no	—
96	Q₈×C₆₇	C₂×D₄	1	4	yes	complemented
97	D₄×C₆₇	C₂×Q₈	1	4	yes	complemented
98	C₂×C₄×C₆₇	—	2	12	no	—
99	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
100	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
101	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
102	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₆₇	C₂×Q₈	1	8	yes	complemented
104	C₂×Dic₆₇	—	2	12	no	—
105	C₂×Dic₆₇	C₂⁴	1	3	yes	—
106	C₂×Dic₆₇	C₂⁴	1	6	yes	—
107	C₂×Dic₆₇	C₂⁴	1	3	yes	—
108	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
109	D₂₆₈	C₂×Q₈	1	4	yes	complemented
110	C₄×D₆₇	—	2	12	no	—
111	C₄×D₆₇	—	2	6	no	—
112	C₄×D₆₇	—	2	12	no	—
113	C₄×D₆₇	—	2	12	no	—
114	C₄×D₆₇	C₂×D₄	1	12	yes	—
115	C₄×D₆₇	C₂×Q₈	1	4	yes	complemented
116	Dic₁₃₄	—	2	12	no	—
117	Dic₁₃₄	—	2	24	no	—
118	Dic₁₃₄	C₂×D₄	1	12	yes	complemented
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	C₂²×Q₈	1	4	yes	complemented
122	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
123	C₂×D₆₇	—	2	2	no	—
124	C₂×D₆₇	—	2	2	no	—
125	C₂×D₆₇	—	2	16	no	—
126	C₂×D₆₇	C₂²×D₄	1	2	yes	—
127	C₂×D₆₇	C₂²×Q₈	1	2	yes	—
128	C₂×D₆₇	C₂²×Q₈	1	4	yes	complemented
129	C₂×D₆₇	2₋¹⁺⁴	1	2	yes	—
130	C₄×C₆₇	—	2	6	no	—
131	C₄×C₆₇	—	2	3	no	—
132	C₄×C₆₇	C₂²×D₄	1	6	yes	—
133	C₄×C₆₇	C₂²×Q₈	1	2	yes	complemented
134	Dic₆₇	—	2	6	no	—
135	Dic₆₇	—	2	3	no	—
136	Dic₆₇	C₂²×D₄	1	6	yes	—
137	Dic₆₇	C₂²×Q₈	1	2	yes	complemented
138	C₂×C₆₇	—	2	4	no	—
139	C₂×C₆₇	C₂³×D₄	1	1	yes	characteristic
140	C₂×C₆₇	C₂³×Q₈	1	1	yes	characteristic
141	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
142	D₆₇	—	2	4	no	—
143	D₆₇	D₄×Q₈	1	4	yes	direct product factor
144	C₂×D₄×Q₈	—	67	1	no	maximal, Sylow
145	C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
146	C₂³×Q₈	—	67	2	no	—
147	D₄×Q₈	—	67	2	no	—
148	D₄×Q₈	—	67	3	no	—
149	D₄×Q₈	—	67	6	no	—
150	D₄×Q₈	—	67	1	no	—
151	D₄×Q₈	—	67	3	no	—
152	D₄×Q₈	D₆₇	1	1	yes	characteristic, direct product factor
153	C₂×C₄⋊Q₈	—	67	3	no	—
154	C₂×C₂²⋊Q₈	—	67	6	no	—
155	C₂×C₄×Q₈	—	67	1	no	—
156	C₂×C₄×D₄	—	67	3	no	—
157	C₂²×Q₈	—	67	1	no	—
158	C₂²×Q₈	—	67	6	no	—
159	C₂²×Q₈	—	67	6	no	—
160	C₂²×Q₈	—	67	2	no	—
161	C₂²×Q₈	—	67	6	no	—
162	C₂²×Q₈	—	134	16	no	—
163	C₂²×Q₈	C₂×D₆₇	1	2	yes	complemented
164	C₂²×D₄	—	67	1	no	—
165	C₂³×C₄	—	67	6	no	—
166	C₄⋊Q₈	—	67	6	no	—
167	C₄⋊Q₈	—	67	6	no	—
168	C₄⋊Q₈	—	67	6	no	—
169	C₄⋊Q₈	—	67	3	no	—
170	C₄⋊Q₈	C₂×D₆₇	1	3	yes	complemented
171	C₂²⋊Q₈	—	67	6	no	—
172	C₂²⋊Q₈	—	67	6	no	—
173	C₂²⋊Q₈	—	67	12	no	—
174	C₂²⋊Q₈	—	67	6	no	—
175	C₂²⋊Q₈	—	67	12	no	—
176	C₂²⋊Q₈	C₂×D₆₇	1	6	yes	complemented
177	C₄×Q₈	—	67	1	no	—
178	C₄×Q₈	—	67	3	no	—
179	C₄×Q₈	—	67	3	no	—
180	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
181	C₄×D₄	—	67	3	no	—
182	C₄×D₄	—	67	6	no	—
183	C₄×D₄	—	67	3	no	—
184	C₄×D₄	—	67	6	no	—
185	C₄×D₄	—	67	3	no	—
186	C₄×D₄	C₂×D₆₇	1	3	yes	—
187	C₂×C₄⋊C₄	—	67	3	no	—
188	C₂×C₄⋊C₄	—	67	3	no	—
189	C₂×C₄⋊C₄	—	67	6	no	—
190	C₂×C₂²⋊C₄	—	67	6	no	—
191	C₂×C₄²	—	67	3	no	—
192	C₂⁴	—	67	2	no	—
193	C₂×Q₈	—	2	8	no	—
194	C₂×Q₈	—	67	8	no	—
195	C₂×Q₈	—	67	3	no	—
196	C₂×Q₈	—	67	12	no	—
197	C₂×Q₈	—	67	6	no	—
198	C₂×Q₈	—	134	24	no	—
199	C₂×Q₈	—	134	24	no	—
200	C₂×Q₈	—	134	24	no	—
201	C₂×Q₈	—	134	8	no	—
202	C₂×Q₈	C₂²×D₆₇	1	6	yes	—
203	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
204	C₂×D₄	—	67	2	no	—
205	C₂×D₄	—	67	8	no	—
206	C₂×D₄	—	67	1	no	—
207	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
208	C₂²×C₄	—	67	1	no	—
209	C₂²×C₄	—	67	3	no	—
210	C₂²×C₄	—	67	6	no	—
211	C₂²×C₄	—	67	3	no	—
212	C₂²×C₄	—	67	6	no	—
213	C₂²×C₄	—	67	6	no	—
214	C₂²×C₄	—	67	6	no	—
215	C₂²×C₄	—	134	24	no	—
216	C₂²×C₄	C₂²×D₆₇	1	6	yes	—
217	C₄⋊C₄	—	67	6	no	—
218	C₄⋊C₄	—	67	3	no	—
219	C₄⋊C₄	—	67	6	no	—
220	C₄⋊C₄	—	67	3	no	—
221	C₄⋊C₄	—	67	12	no	—
222	C₄⋊C₄	—	67	6	no	—
223	C₄⋊C₄	C₂²×D₆₇	1	3	yes	—
224	C₄⋊C₄	C₂²×D₆₇	1	3	yes	—
225	C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
226	C₂²⋊C₄	—	67	6	no	—
227	C₂²⋊C₄	—	67	6	no	—
228	C₂²⋊C₄	—	67	6	no	—
229	C₂²⋊C₄	C₂²×D₆₇	1	6	yes	—
230	C₄²	—	67	3	no	—
231	C₄²	—	67	3	no	—
232	C₄²	—	67	3	no	—
233	C₄²	C₂²×D₆₇	1	3	yes	—
234	C₂³	—	67	1	no	—
235	C₂³	—	67	8	no	—
236	C₂³	—	67	2	no	—
237	C₂³	—	134	4	no	—
238	C₂³	—	134	4	no	—
239	C₂³	C₂³×D₆₇	1	2	yes	—
240	Q₈	—	2	12	no	—
241	Q₈	—	67	12	no	—
242	Q₈	—	134	24	no	—
243	Q₈	—	134	12	no	—
244	Q₈	D₄×D₆₇	1	4	yes	direct product factor
245	D₄	—	67	8	no	—
246	D₄	—	67	4	no	—
247	D₄	Q₈×D₆₇	1	4	yes	direct product factor
248	C₂×C₄	—	2	12	no	—
249	C₂×C₄	—	67	1	no	—
250	C₂×C₄	—	67	3	no	—
251	C₂×C₄	—	67	12	no	—
252	C₂×C₄	—	67	4	no	—
253	C₂×C₄	—	67	3	no	—
254	C₂×C₄	—	67	6	no	—
255	C₂×C₄	—	134	12	no	—
256	C₂×C₄	—	134	12	no	—
257	C₂×C₄	—	134	6	no	—
258	C₂×C₄	—	134	12	no	—
259	C₂×C₄	—	134	12	no	—
260	C₂×C₄	C₂³×D₆₇	1	3	yes	—
261	C₂×C₄	C₂³×D₆₇	1	6	yes	—
262	C₂×C₄	C₂³×D₆₇	1	3	yes	—
263	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	67	2	no	—
267	C₂²	—	67	2	no	—
268	C₂²	—	67	2	no	—
269	C₂²	—	67	4	no	—
270	C₂²	—	134	16	no	—
271	C₂²	—	134	2	no	—
272	C₂²	—	134	2	no	—
273	C₂²	C₂×Q₈×D₆₇	1	4	yes	complemented
274	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
275	C₄	—	2	6	no	—
276	C₄	—	2	3	no	—
277	C₄	—	67	6	no	—
278	C₄	—	67	2	no	—
279	C₄	—	134	6	no	—
280	C₄	—	134	3	no	—
281	C₄	C₂×D₄×D₆₇	1	6	yes	—
282	C₄	C₂×Q₈×D₆₇	1	2	yes	complemented
283	C₂	—	2	4	no	—
284	C₂	—	67	4	no	—
285	C₂	—	134	4	no	—
286	C₂	C₂²×D₄×D₆₇	1	1	yes	characteristic
287	C₂	C₂²×Q₈×D₆₇	1	1	yes	characteristic
288	C₂	2₋¹⁺⁴×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	D₄×Q₈×C₆₇	C₂
hypercenter	D₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₆₇	C₂	semidirect
2. Q₈×D₂₆₈	C₂	semidirect
3. D₄×Dic₁₃₄	C₂	semidirect
4. #16864061	C₂	semidirect
5. Q₈×C₂²⋊D₆₇	C₂	semidirect
6. #16864059	C₂	semidirect
7. C₄⋊Q₈×D₆₇	C₂	semidirect
8. C₂²⋊Q₈×D₆₇	C₂	semidirect
9. C₄×Q₈×D₆₇	C₂	semidirect
11. D₄×Q₈×C₆₇	C₂	semidirect
12. C₂²×Dic₁₃₄	C₂²	semidirect
16. C₂×Q₈×D₆₇	C₂²	semidirect
17. C₂×Q₈×D₆₇	C₂²	semidirect
20. #14524043	C₂²	semidirect
21. #14524042	C₂²	semidirect
22. #14524042	C₂²	semidirect
23. #14524041	C₂²	semidirect
24. #14524023	C₂²	semidirect
25. #14524022	C₂²	semidirect
26. #14524021	C₂²	semidirect
27. #14524020	C₂²	semidirect
28. #14524019	C₂²	semidirect
29. Q₈×Dic₆₇	C₂²	semidirect
30. C₄×Dic₁₃₄	C₂²	semidirect
31. #14524008	C₂²	semidirect
42. C₂²×Q₈×C₆₇	C₂²	semidirect
43. C₄⋊Q₈×C₆₇	C₂²	semidirect
44. C₂²⋊Q₈×C₆₇	C₂²	semidirect
45. C₄×Q₈×C₆₇	C₂²	semidirect
55. Q₈×D₆₇	D₄	direct (N × H)
58. D₄×D₆₇	Q₈	direct (N × H)
92. C₂²×D₆₇	C₂×Q₈	semidirect
96. Q₈×C₆₇	C₂×D₄	semidirect
97. D₄×C₆₇	C₂×Q₈	semidirect
103. C₂²⋊D₆₇	C₂×Q₈	semidirect
109. D₂₆₈	C₂×Q₈	semidirect
115. C₄×D₆₇	C₂×Q₈	semidirect
118. Dic₁₃₄	C₂×D₄	semidirect
121. C₂²×C₆₇	C₂²×Q₈	semidirect
128. C₂×D₆₇	C₂²×Q₈	semidirect
133. C₄×C₆₇	C₂²×Q₈	semidirect
137. Dic₆₇	C₂²×Q₈	semidirect
143. D₆₇	D₄×Q₈	direct (N × H)
145. C₆₇	C₂×D₄×Q₈	semidirect
152. D₄×Q₈	D₆₇	direct (N × H)
163. C₂²×Q₈	C₂×D₆₇	semidirect
170. C₄⋊Q₈	C₂×D₆₇	semidirect
176. C₂²⋊Q₈	C₂×D₆₇	semidirect
180. C₄×Q₈	C₂×D₆₇	semidirect
244. Q₈	D₄×D₆₇	direct (N × H)
247. D₄	Q₈×D₆₇	direct (N × H)
273. C₂²	C₂×Q₈×D₆₇	semidirect
282. C₄	C₂×Q₈×D₆₇	semidirect

Subgroups — D4×Q8×D67 #22573679

Conjugacy classes of subgroups

Complements

Subgroups — D₄×Q₈×D₆₇ #22573679