Subgroups — Q₈×#13083574 #26033833

Order 11392.

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×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152326	C₂	1	3	yes	maximal, complemented
3	#19152308	C₂	1	3	yes	maximal, complemented
4	#19152303	C₂	1	6	yes	maximal, complemented
5	Q₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₂²⋊D₈₉	C₂	1	2	yes	maximal, complemented
7	#19152155	C₂	1	3	yes	maximal, complemented
8	#19152152	C₂	1	6	yes	maximal, complemented
9	C₄×#13083574	C₂	1	3	yes	maximal
10	C₄×Q₈×D₈₉	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083574	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₈₉	C₂²	1	2	yes	—
14	#15668012	C₂²	1	3	yes	—
15	#15668012	C₂²	1	3	yes	—
16	#15668011	C₂²	1	6	yes	—
17	#15668010	C₂²	1	6	yes	—
18	#15668004	C₂²	1	6	yes	—
19	#15668002	C₂²	1	12	yes	—
20	#15667992	C₂²	1	6	yes	—
21	#15667991	C₂²	1	6	yes	—
22	#15667991	C₂²	1	6	yes	—
23	#15667990	C₂²	1	12	yes	—
24	#15667988	C₂²	1	6	yes	—
25	Q₈×Dic₈₉	C₂²	1	2	yes	—
26	C₄×Dic₁₇₈	C₂²	1	6	yes	—
27	C₄×Dic₁₇₈	C₂²	1	6	yes	—
28	C₄×Dic₁₇₈	C₂²	1	3	yes	—
29	C₄×Dic₁₇₈	C₂²	1	3	yes	—
30	#15667977	C₂²	1	6	yes	—
31	C₄×D₃₅₆	C₂²	1	3	yes	—
32	C₄×D₃₅₆	C₂²	1	3	yes	—
33	C₄×C₂²⋊D₈₉	C₂²	1	6	yes	—
34	C₄×C₂²⋊D₈₉	C₂²	1	6	yes	—
35	#15667971	C₂²	1	3	yes	—
36	#15667969	C₂²	1	6	yes	—
37	C₄⋊C₄×D₈₉	C₂²	1	6	yes	—
38	C₄²×D₈₉	C₂²	1	6	yes	—
39	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₈₉	C₂²	1	3	yes	—
41	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₈₉	C₂²	1	3	yes	—
43	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
45	C₂×C₄²×C₈₉	C₂²	1	3	yes	—
46	#13083574	Q₈	1	8	yes	direct product factor
47	C₂×Dic₁₇₈	C₂³	1	6	yes	—
48	C₂×Dic₁₇₈	C₂³	1	6	yes	—
49	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
50	Q₈×D₈₉	—	2	8	no	—
51	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₈₉	C₂³	1	2	yes	—
53	C₂×C₄×D₈₉	C₂³	1	6	yes	—
54	C₂×C₄×D₈₉	C₂³	1	6	yes	—
55	C₂×C₄×D₈₉	C₂³	1	2	yes	—
56	C₄⋊Dic₈₉	C₂³	1	3	yes	—
57	C₄⋊Dic₈₉	C₂³	1	3	yes	—
58	C₄⋊Dic₈₉	C₂³	1	6	yes	—
59	C₄⋊Dic₈₉	C₂³	1	6	yes	—
60	#13083553	C₂³	1	6	yes	—
61	#13083553	C₂³	1	6	yes	—
62	#13083553	C₂³	1	12	yes	—
63	#13083553	C₂³	1	12	yes	—
64	C₂²⋊Dic₈₉	C₂³	1	6	yes	—
65	#13083551	C₂³	1	6	yes	—
66	#13083551	C₂³	1	6	yes	—
67	C₄×Dic₈₉	C₂³	1	6	yes	—
68	C₄×Dic₈₉	C₂³	1	6	yes	—
69	C₄×Dic₈₉	C₂³	1	6	yes	—
70	C₂×Q₈×C₈₉	—	2	4	no	—
71	C₂×Q₈×C₈₉	C₂³	1	3	yes	—
72	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
74	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
75	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
77	C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
78	C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
79	C₄²×C₈₉	C₂³	1	3	yes	—
80	C₄²×C₈₉	C₂³	1	3	yes	—
81	C₄²×C₈₉	C₂³	1	3	yes	—
82	C₄²×C₈₉	C₂³	1	3	yes	—
83	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₂²×D₈₉	C₂⁴	1	2	yes	—
85	Q₈×C₈₉	—	2	6	no	—
86	Q₈×C₈₉	#41	1	4	yes	complemented
87	C₂×C₄×C₈₉	—	2	6	no	—
88	C₂×C₄×C₈₉	—	4	3	no	—
89	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	complemented
90	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
91	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
92	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
93	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
95	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₈₉	C₂×Q₈	1	8	yes	complemented
97	C₂×Dic₈₉	C₂⁴	1	6	yes	—
98	C₂×Dic₈₉	C₂⁴	1	6	yes	—
99	C₂×Dic₈₉	C₂⁴	1	2	yes	—
100	D₃₅₆	C₂×Q₈	1	4	yes	complemented
101	C₄×D₈₉	—	2	12	no	—
102	C₄×D₈₉	—	4	6	no	—
103	C₄×D₈₉	C₂×Q₈	1	8	yes	complemented
104	Dic₁₇₈	—	2	12	no	—
105	Dic₁₇₈	—	4	6	no	—
106	Dic₁₇₈	C₂×Q₈	1	4	yes	complemented
107	C₂²×C₈₉	—	2	1	no	—
108	C₂²×C₈₉	—	2	1	no	—
109	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
110	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
111	C₂×D₈₉	—	2	2	no	—
112	C₂×D₈₉	—	2	2	no	—
113	C₂×D₈₉	C₂²×Q₈	1	4	yes	—
114	C₄×C₈₉	—	2	3	no	—
115	C₄×C₈₉	—	2	3	no	—
116	C₄×C₈₉	—	2	3	no	—
117	C₄×C₈₉	C₂²×Q₈	1	2	yes	—
118	C₄×C₈₉	C₂²×Q₈	1	2	yes	—
119	C₄×C₈₉	C₂×#41	1	6	yes	—
120	Dic₈₉	—	2	6	no	—
121	Dic₈₉	—	2	6	no	—
122	Dic₈₉	C₂²×Q₈	1	4	yes	—
123	C₂×C₈₉	—	2	2	no	—
124	C₂×C₈₉	C₂³×Q₈	1	1	yes	characteristic
125	C₂×C₈₉	C₂²×#41	1	1	yes	characteristic
126	C₂×C₈₉	#585	1	1	yes	characteristic
127	D₈₉	—	2	4	no	—
128	Q₈×#41	—	89	1	no	maximal, Sylow
129	C₈₉	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
130	Q₈²	—	89	1	no	—
131	#557	—	89	3	no	—
132	#554	—	89	3	no	—
133	#554	—	89	6	no	—
134	D₄×Q₈	—	89	2	no	—
135	D₄×Q₈	—	89	1	no	—
136	#533	—	89	6	no	—
137	#533	—	89	3	no	—
138	C₄×#41	—	89	3	no	—
139	C₂×C₄×Q₈	—	89	2	no	—
140	C₂×C₄×Q₈	D₈₉	1	1	yes	characteristic, complemented
141	C₂×#41	—	89	1	no	—
142	C₂²×Q₈	—	89	2	no	—
143	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
144	C₄⋊Q₈	—	89	6	no	—
145	C₄⋊Q₈	—	89	6	no	—
146	C₄⋊Q₈	—	89	3	no	—
147	C₄⋊Q₈	—	89	3	no	—
148	#125	—	89	12	no	—
149	#125	—	89	6	no	—
150	C₂²⋊Q₈	—	89	6	no	—
151	C₂²⋊Q₈	—	89	6	no	—
152	C₂²⋊Q₈	—	89	6	no	—
153	C₂²⋊Q₈	—	89	12	no	—
154	C₂²⋊Q₈	—	89	6	no	—
155	C₄×Q₈	—	89	2	no	—
156	C₄×Q₈	—	89	6	no	—
157	C₄×Q₈	—	89	6	no	—
158	C₄×Q₈	—	89	3	no	—
159	C₄×Q₈	—	89	3	no	—
160	C₄×Q₈	—	89	6	no	—
161	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
162	C₄×Q₈	C₂×D₈₉	1	3	yes	—
163	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
164	C₄×Q₈	C₂×D₈₉	1	3	yes	—
165	C₄×D₄	—	89	6	no	—
166	C₄×D₄	—	89	6	no	—
167	C₄×D₄	—	89	3	no	—
168	C₄×D₄	—	89	3	no	—
169	#117	—	89	3	no	—
170	#117	—	89	6	no	—
171	C₂×C₄⋊C₄	—	89	6	no	—
172	C₂×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
173	C₂×C₄²	—	89	6	no	—
174	C₂×C₄²	C₂×D₈₉	1	3	yes	—
175	#41	—	89	8	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	89	1	no	—
178	C₂×Q₈	—	89	6	no	—
179	C₂×Q₈	—	89	6	no	—
180	C₂×Q₈	—	178	8	no	—
181	C₂×Q₈	C₂²×D₈₉	1	3	yes	—
182	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
183	C₂×D₄	—	89	2	no	—
184	C₂×D₄	—	89	1	no	—
185	C₂²×C₄	—	89	2	no	—
186	C₂²×C₄	—	89	6	no	—
187	C₂²×C₄	—	89	6	no	—
188	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
189	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
190	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
191	C₄⋊C₄	—	89	12	no	—
192	C₄⋊C₄	—	89	6	no	—
193	C₄⋊C₄	—	89	6	no	—
194	C₄⋊C₄	—	89	6	no	—
195	C₄⋊C₄	—	89	3	no	—
196	C₄⋊C₄	—	89	3	no	—
197	C₄⋊C₄	—	89	12	no	—
198	C₄⋊C₄	—	89	6	no	—
199	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
200	C₄⋊C₄	C₂²×D₈₉	1	6	yes	—
201	C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
202	C₂²⋊C₄	—	89	6	no	—
203	C₂²⋊C₄	—	89	6	no	—
204	C₂²⋊C₄	—	89	6	no	—
205	C₄²	—	89	6	no	—
206	C₄²	—	89	6	no	—
207	C₄²	—	89	6	no	—
208	C₄²	C₂²×D₈₉	1	3	yes	—
209	C₄²	C₂²×D₈₉	1	3	yes	—
210	C₄²	C₂²×D₈₉	1	3	yes	—
211	C₄²	C₂²×D₈₉	1	3	yes	—
212	C₂³	—	89	2	no	—
213	C₂³	C₂³×D₈₉	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	89	4	no	—
216	Q₈	—	356	6	no	—
217	Q₈	—	178	12	no	—
218	Q₈	#13083574	1	4	yes	direct product factor
219	D₄	—	89	8	no	—
220	D₄	—	89	4	no	—
221	C₂×C₄	—	2	6	no	—
222	C₂×C₄	—	4	3	no	—
223	C₂×C₄	—	89	2	no	—
224	C₂×C₄	—	89	8	no	—
225	C₂×C₄	—	89	6	no	—
226	C₂×C₄	—	89	6	no	—
227	C₂×C₄	—	356	6	no	—
228	C₂×C₄	—	178	12	no	—
229	C₂×C₄	Q₈×D₈₉	1	4	yes	complemented
230	C₂×C₄	C₂³×D₈₉	1	3	yes	—
231	C₂×C₄	C₂³×D₈₉	1	3	yes	—
232	C₂×C₄	C₂³×D₈₉	1	3	yes	—
233	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
234	C₂×C₄	C₂³×D₈₉	1	3	yes	—
235	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	89	4	no	—
239	C₂²	—	178	2	no	—
240	C₂²	—	178	2	no	—
241	C₂²	C₂×Q₈×D₈₉	1	2	yes	—
242	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	89	4	no	—
247	C₄	—	178	6	no	—
248	C₄	—	178	6	no	—
249	C₄	C₂×Q₈×D₈₉	1	2	yes	—
250	C₄	C₂×Q₈×D₈₉	1	2	yes	—
251	C₄	C₂×#13083574	1	6	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	178	4	no	—
254	C₂	C₂²×Q₈×D₈₉	1	1	yes	characteristic
255	C₂	C₂²×#13083574	1	1	yes	characteristic
256	C₂	#19152434	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Frattini	C₂²	C₂⁴×D₈₉
Fitting	C₂×C₄×Q₈×C₈₉	C₂
prime radical	C₂×#13083574	C₂²
hypercenter	C₂×C₄×Q₈	D₈₉
hypocenter	C₈₉	Q₈×#41
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₇₈	C₂	semidirect
2. #19152326	C₂	semidirect
3. #19152308	C₂	semidirect
4. #19152303	C₂	semidirect
5. Q₈×D₃₅₆	C₂	semidirect
6. Q₈×C₂²⋊D₈₉	C₂	semidirect
7. #19152155	C₂	semidirect
8. #19152152	C₂	semidirect
10. C₄×Q₈×D₈₉	C₂	semidirect
11. C₂×C₄×Q₈×C₈₉	C₂	semidirect
46. #13083574	Q₈	direct (N × H)
86. Q₈×C₈₉	#41	semidirect
89. C₂×C₄×C₈₉	C₂×Q₈	semidirect
96. C₂²⋊D₈₉	C₂×Q₈	semidirect
100. D₃₅₆	C₂×Q₈	semidirect
103. C₄×D₈₉	C₂×Q₈	semidirect
106. Dic₁₇₈	C₂×Q₈	semidirect
129. C₈₉	Q₈×#41	semidirect
140. C₂×C₄×Q₈	D₈₉	semidirect
218. Q₈	#13083574	direct (N × H)
229. C₂×C₄	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₁₇₈	2
#19152326	2
#19152308	2
#19152303	2
Q₈×D₃₅₆	2
Q₈×C₂²⋊D₈₉	2
#19152155	2
#19152152	2
C₄×#13083574	2
C₄×Q₈×D₈₉	2
C₂×C₄×Q₈×C₈₉	2
Q₈×#41	89

Subgroups — Q8×#13083574 #26033833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13083574 #26033833