Subgroups — Q₈×#13050589 #24921040

Order 10624.

1068 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128426	C₂	1	3	yes	maximal, complemented
3	#19128409	C₂	1	6	yes	maximal, complemented
4	#19128408	C₂	1	3	yes	maximal, complemented
5	Q₈×C₂²⋊D₈₃	C₂	1	2	yes	maximal, complemented
6	#19128256	C₂	1	3	yes	maximal, complemented
7	#19128253	C₂	1	6	yes	maximal, complemented
8	C₄×#13050589	C₂	1	3	yes	maximal
9	C₂×Q₈×Dic₈₃	C₂	1	2	yes	maximal, complemented
10	C₄×Q₈×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13050589	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
14	#14581792	C₂²	1	3	yes	—
15	#14581791	C₂²	1	6	yes	—
16	#14581790	C₂²	1	6	yes	—
17	#14581783	C₂²	1	6	yes	—
18	#14581782	C₂²	1	12	yes	—
19	#14581772	C₂²	1	6	yes	—
20	#14581771	C₂²	1	6	yes	—
21	#14581769	C₂²	1	12	yes	—
22	#14581768	C₂²	1	6	yes	—
23	Q₈×Dic₈₃	C₂²	1	2	yes	—
24	Q₈×Dic₈₃	C₂²	1	6	yes	—
25	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
26	Q₈×Dic₈₃	C₂²	1	6	yes	—
27	C₄×Dic₁₆₆	C₂²	1	3	yes	—
28	C₄×Dic₁₆₆	C₂²	1	3	yes	—
29	#14581757	C₂²	1	3	yes	—
30	#14581757	C₂²	1	3	yes	—
31	#14581757	C₂²	1	6	yes	—
32	D₄×Dic₈₃	C₂²	1	3	yes	—
33	C₄×C₂²⋊D₈₃	C₂²	1	6	yes	—
34	(C₂²⋊D₈₃)⋊C₄	C₂²	1	6	yes	—
35	#14581750	C₂²	1	3	yes	—
36	#14581748	C₂²	1	6	yes	—
37	C₂×C₄⋊Dic₈₃	C₂²	1	6	yes	—
38	C₄⋊C₄×D₈₃	C₂²	1	3	yes	—
39	C₂×C₄×Dic₈₃	C₂²	1	6	yes	—
40	C₄²×D₈₃	C₂²	1	3	yes	—
41	C₂²×Q₈×C₈₃	C₂²	1	2	yes	—
42	C₄⋊Q₈×C₈₃	C₂²	1	3	yes	—
43	C₂²⋊Q₈×C₈₃	C₂²	1	6	yes	—
44	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
45	C₄×D₄×C₈₃	C₂²	1	3	yes	—
46	#13050589	Q₈	1	8	yes	direct product factor
47	C₂×Dic₁₆₆	C₂³	1	3	yes	—
48	C₂×Dic₁₆₆	C₂³	1	6	yes	—
49	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
50	Q₈×D₈₃	—	2	4	no	—
51	C₂×C₂²⋊D₈₃	C₂³	1	2	yes	—
52	C₂²×Dic₈₃	C₂³	1	6	yes	—
53	C₂²×Dic₈₃	C₂³	1	2	yes	—
54	C₂×C₄×D₈₃	C₂³	1	3	yes	—
55	C₂×C₄×D₈₃	C₂³	1	3	yes	—
56	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₈₃	C₂³	1	6	yes	—
58	C₄⋊Dic₈₃	C₂³	1	12	yes	—
59	C₄⋊Dic₈₃	C₂³	1	3	yes	—
60	C₄⋊Dic₈₃	C₂³	1	3	yes	—
61	Dic₈₃⋊C₄	C₂³	1	12	yes	—
62	Dic₈₃⋊C₄	C₂³	1	6	yes	—
63	Dic₈₃⋊C₄	C₂³	1	6	yes	—
64	Dic₈₃⋊C₄	C₂³	1	6	yes	—
65	C₂²⋊Dic₈₃	C₂³	1	6	yes	—
66	(C₂×D₈₃)⋊C₄	C₂³	1	6	yes	—
67	C₄×Dic₈₃	C₂³	1	6	yes	—
68	C₄×Dic₈₃	C₂³	1	3	yes	—
69	C₄×Dic₈₃	C₂³	1	6	yes	—
70	C₄×Dic₈₃	C₂³	1	3	yes	—
71	C₄×Dic₈₃	C₂³	1	3	yes	—
72	C₄×Dic₈₃	C₂³	1	6	yes	—
73	C₂×Q₈×C₈₃	—	2	8	no	—
74	C₂×Q₈×C₈₃	C₂³	1	6	yes	—
75	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₈₃	C₂³	1	6	yes	—
78	C₄⋊C₄×C₈₃	C₂³	1	6	yes	—
79	C₄⋊C₄×C₈₃	C₂³	1	3	yes	—
80	C₄⋊C₄×C₈₃	C₂³	1	3	yes	—
81	C₂²⋊C₄×C₈₃	C₂³	1	6	yes	—
82	C₄²×C₈₃	C₂³	1	3	yes	—
83	C₂³×C₈₃	C₂⁴	1	2	yes	—
84	C₂²×D₈₃	C₂⁴	1	1	yes	characteristic
85	Q₈×C₈₃	—	2	12	no	—
86	Q₈×C₈₃	#41	1	4	yes	complemented
87	D₄×C₈₃	C₂×Q₈	1	4	yes	complemented
88	C₂×C₄×C₈₃	—	2	12	no	—
89	C₂×C₄×C₈₃	C₂⁴	1	3	yes	—
90	C₂×C₄×C₈₃	C₂⁴	1	6	yes	—
91	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂⁴	1	3	yes	—
93	C₂²⋊D₈₃	C₂×Q₈	1	8	yes	complemented
94	C₂×Dic₈₃	—	4	6	no	—
95	C₂×Dic₈₃	C₂×Q₈	1	8	yes	complemented
96	C₂×Dic₈₃	C₂⁴	1	3	yes	—
97	C₂×Dic₈₃	C₂⁴	1	6	yes	—
98	C₂×Dic₈₃	C₂⁴	1	3	yes	—
99	C₂×Dic₈₃	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₈₃	C₂⁴	1	2	yes	—
101	C₄×D₈₃	—	2	6	no	—
102	C₄×D₈₃	—	4	3	no	—
103	C₄×D₈₃	C₂×Q₈	1	4	yes	complemented
104	Dic₁₆₆	—	2	6	no	—
105	Dic₁₆₆	—	4	6	no	—
106	Dic₁₆₆	C₂×Q₈	1	4	yes	complemented
107	C₂²×C₈₃	—	2	2	no	—
108	C₂²×C₈₃	—	2	2	no	—
109	C₂²×C₈₃	C₂²×Q₈	1	4	yes	—
110	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
111	C₂×D₈₃	—	2	1	no	—
112	C₂×D₈₃	—	2	1	no	—
113	C₂×D₈₃	C₂²×Q₈	1	2	yes	—
114	C₄×C₈₃	—	2	6	no	—
115	C₄×C₈₃	—	2	3	no	—
116	C₄×C₈₃	C₂²×Q₈	1	2	yes	—
117	C₄×C₈₃	C₂×#41	1	6	yes	—
118	Dic₈₃	—	2	3	no	—
119	Dic₈₃	—	2	6	no	—
120	Dic₈₃	—	2	3	no	—
121	Dic₈₃	C₂²×Q₈	1	4	yes	—
122	Dic₈₃	C₂²×Q₈	1	2	yes	—
123	C₂×C₈₃	—	2	4	no	—
124	C₂×C₈₃	C₂³×Q₈	1	1	yes	characteristic
125	C₂×C₈₃	C₂²×#41	1	1	yes	characteristic
126	C₂×C₈₃	#585	1	1	yes	characteristic
127	D₈₃	—	2	2	no	—
128	Q₈×#41	—	83	1	no	maximal, Sylow
129	C₈₃	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
130	Q₈²	—	83	1	no	—
131	#557	—	83	3	no	—
132	#554	—	83	6	no	—
133	#554	—	83	3	no	—
134	D₄×Q₈	—	83	2	no	—
135	D₄×Q₈	D₈₃	1	1	yes	characteristic, complemented
136	#533	—	83	3	no	—
137	#533	—	83	6	no	—
138	C₄×#41	—	83	3	no	—
139	C₂×C₄×Q₈	—	83	2	no	—
140	C₂×C₄×Q₈	—	83	1	no	—
141	C₂×#41	—	83	1	no	—
142	C₂²×Q₈	—	83	1	no	—
143	C₂²×Q₈	C₂×D₈₃	1	2	yes	—
144	C₄⋊Q₈	—	83	6	no	—
145	C₄⋊Q₈	—	83	6	no	—
146	C₄⋊Q₈	—	83	3	no	—
147	C₄⋊Q₈	C₂×D₈₃	1	3	yes	—
148	#125	—	83	12	no	—
149	#125	—	83	6	no	—
150	C₂²⋊Q₈	—	83	6	no	—
151	C₂²⋊Q₈	—	83	6	no	—
152	C₂²⋊Q₈	—	83	6	no	—
153	C₂²⋊Q₈	—	83	12	no	—
154	C₂²⋊Q₈	C₂×D₈₃	1	6	yes	—
155	C₄×Q₈	—	83	1	no	—
156	C₄×Q₈	—	83	6	no	—
157	C₄×Q₈	—	83	2	no	—
158	C₄×Q₈	—	83	6	no	—
159	C₄×Q₈	—	83	3	no	—
160	C₄×Q₈	—	83	3	no	—
161	C₄×Q₈	—	83	6	no	—
162	C₄×Q₈	—	83	3	no	—
163	C₄×Q₈	—	83	3	no	—
164	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
165	C₄×D₄	—	83	3	no	—
166	C₄×D₄	—	83	6	no	—
167	C₄×D₄	—	83	6	no	—
168	C₄×D₄	C₂×D₈₃	1	3	yes	—
169	#117	—	83	3	no	—
170	#117	—	83	6	no	—
171	C₂×C₄⋊C₄	—	83	6	no	—
172	C₂×C₄⋊C₄	—	83	3	no	—
173	C₂×C₄²	—	83	6	no	—
174	C₂×C₄²	—	83	3	no	—
175	#41	—	83	8	no	—
176	C₂×Q₈	—	2	8	no	—
177	C₂×Q₈	—	83	6	no	—
178	C₂×Q₈	—	83	1	no	—
179	C₂×Q₈	—	83	3	no	—
180	C₂×Q₈	—	166	4	no	—
181	C₂×Q₈	C₂²×D₈₃	1	6	yes	—
182	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
183	C₂×D₄	—	83	2	no	—
184	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
185	C₂²×C₄	—	83	6	no	—
186	C₂²×C₄	—	83	3	no	—
187	C₂²×C₄	—	83	2	no	—
188	C₂²×C₄	—	83	1	no	—
189	C₂²×C₄	—	83	3	no	—
190	C₂²×C₄	C₂²×D₈₃	1	6	yes	—
191	C₄⋊C₄	—	83	3	no	—
192	C₄⋊C₄	—	83	6	no	—
193	C₄⋊C₄	—	83	12	no	—
194	C₄⋊C₄	—	83	6	no	—
195	C₄⋊C₄	—	83	6	no	—
196	C₄⋊C₄	—	83	6	no	—
197	C₄⋊C₄	—	83	3	no	—
198	C₄⋊C₄	—	83	12	no	—
199	C₄⋊C₄	C₂²×D₈₃	1	3	yes	—
200	C₄⋊C₄	C₂²×D₈₃	1	3	yes	—
201	C₄⋊C₄	C₂²×D₈₃	1	6	yes	—
202	C₂²⋊C₄	—	83	6	no	—
203	C₂²⋊C₄	—	83	6	no	—
204	C₂²⋊C₄	C₂²×D₈₃	1	6	yes	—
205	C₄²	—	83	3	no	—
206	C₄²	—	83	6	no	—
207	C₄²	—	83	3	no	—
208	C₄²	—	83	6	no	—
209	C₄²	—	83	3	no	—
210	C₄²	—	83	6	no	—
211	C₄²	C₂²×D₈₃	1	3	yes	—
212	C₂³	—	83	1	no	—
213	C₂³	C₂³×D₈₃	1	2	yes	—
214	Q₈	—	2	12	no	—
215	Q₈	—	332	6	no	—
216	Q₈	—	83	4	no	—
217	Q₈	—	166	6	no	—
218	Q₈	#13050589	1	4	yes	direct product factor
219	D₄	—	83	8	no	—
220	D₄	Q₈×D₈₃	1	4	yes	complemented
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	332	3	no	—
223	C₂×C₄	—	332	6	no	—
224	C₂×C₄	—	83	8	no	—
225	C₂×C₄	—	83	1	no	—
226	C₂×C₄	—	83	2	no	—
227	C₂×C₄	—	83	4	no	—
228	C₂×C₄	—	83	3	no	—
229	C₂×C₄	—	83	3	no	—
230	C₂×C₄	—	83	6	no	—
231	C₂×C₄	—	166	6	no	—
232	C₂×C₄	C₂³×D₈₃	1	6	yes	—
233	C₂×C₄	C₂³×D₈₃	1	3	yes	—
234	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₈₃	1	3	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	83	2	no	—
239	C₂²	—	166	1	no	—
240	C₂²	—	166	1	no	—
241	C₂²	C₂×Q₈×D₈₃	1	4	yes	—
242	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
243	C₄	—	2	6	no	—
244	C₄	—	2	3	no	—
245	C₄	—	83	2	no	—
246	C₄	—	83	4	no	—
247	C₄	—	166	3	no	—
248	C₄	—	166	3	no	—
249	C₄	—	166	6	no	—
250	C₄	C₂×Q₈×D₈₃	1	2	yes	—
251	C₄	C₂×#13050589	1	6	yes	—
252	C₂	—	2	4	no	—
253	C₂	—	166	2	no	—
254	C₂	C₂²×Q₈×D₈₃	1	1	yes	characteristic
255	C₂	C₂²×#13050589	1	1	yes	characteristic
256	C₂	#19128538	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₃
commutator, socle, cosocle	C₂²×C₈₃	C₂⁵
Fitting	D₄×Q₈×C₈₃	C₂
prime radical	C₂×#13050589	C₂²
hypercenter	D₄×Q₈	D₈₃
hypocenter	C₈₃	Q₈×#41
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₆₆	C₂	semidirect
2. #19128426	C₂	semidirect
3. #19128409	C₂	semidirect
4. #19128408	C₂	semidirect
5. Q₈×C₂²⋊D₈₃	C₂	semidirect
6. #19128256	C₂	semidirect
7. #19128253	C₂	semidirect
9. C₂×Q₈×Dic₈₃	C₂	semidirect
10. C₄×Q₈×D₈₃	C₂	semidirect
11. D₄×Q₈×C₈₃	C₂	semidirect
46. #13050589	Q₈	direct (N × H)
86. Q₈×C₈₃	#41	semidirect
87. D₄×C₈₃	C₂×Q₈	semidirect
93. C₂²⋊D₈₃	C₂×Q₈	semidirect
95. C₂×Dic₈₃	C₂×Q₈	semidirect
103. C₄×D₈₃	C₂×Q₈	semidirect
106. Dic₁₆₆	C₂×Q₈	semidirect
129. C₈₃	Q₈×#41	semidirect
135. D₄×Q₈	D₈₃	semidirect
218. Q₈	#13050589	direct (N × H)
220. D₄	Q₈×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₁₆₆	2
#19128426	2
#19128409	2
#19128408	2
Q₈×C₂²⋊D₈₃	2
#19128256	2
#19128253	2
C₄×#13050589	2
C₂×Q₈×Dic₈₃	2
C₄×Q₈×D₈₃	2
D₄×Q₈×C₈₃	2
Q₈×#41	83

Subgroups — Q8×#13050589 #24921040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13050589 #24921040