Subgroups — Q₈×#88056 #15879527

Order 3328.

1086 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13115797	C₂	1	3	yes	maximal
4	#13115783	C₂	1	3	yes	maximal
5	#13114842	C₂	1	3	yes	maximal
6	C₄×#88056	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₃	—	2	1	no	—
10	C₂×Q₈×Dic₁₃	—	2	1	no	—
11	Q₈×Dic₂₆	—	2	4	no	—
12	Q₈×Dic₂₆	C₄	1	8	yes	complemented
13	C₂×#88142	—	2	3	no	—
14	C₂×#88123	C₂²	1	3	yes	—
15	C₂×C₄×Dic₂₆	C₂²	1	3	yes	—
16	C₈×Q₈×C₁₃	—	2	4	no	—
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
19	C₂×#88056	C₂²	1	3	yes	—
20	C₂×#88056	C₂²	1	1	yes	characteristic
21	#11718646	—	2	3	no	—
22	#11718596	C₂²	1	3	yes	—
23	C₄×C₄⋊Dic₁₃	C₂²	1	3	yes	—
24	#11718143	C₂²	1	3	yes	—
25	#11718143	C₂²	1	3	yes	—
26	#11718143	C₂²	1	6	yes	—
27	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×Dic₂₆	—	2	3	no	—
29	C₂²×Dic₂₆	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₄×Q₈×C₁₃	D₄	1	4	yes	—
34	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
36	Q₈×Dic₁₃	—	2	8	no	—
37	Q₈×Dic₁₃	—	4	4	no	—
38	C₂×C₄⋊Dic₁₃	—	2	3	no	—
39	C₂×C₄⋊Dic₁₃	—	2	3	no	—
40	C₂×C₄⋊Dic₁₃	—	2	3	no	—
41	C₂×C₄⋊Dic₁₃	C₂³	1	3	yes	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	3	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
44	C₂×#6101	—	2	3	no	—
45	C₂×C₄×Dic₁₃	—	2	3	no	—
46	C₂×C₄×Dic₁₃	—	2	3	no	—
47	#88142	—	2	24	no	—
48	#88123	—	2	6	no	—
49	#88123	C₂×C₄	1	12	yes	—
50	C₄×Dic₂₆	—	2	6	no	—
51	C₄×Dic₂₆	C₂×C₄	1	12	yes	—
52	C₁₃×C₄⋊C₈	—	2	6	no	—
53	C₄×C₈×C₁₃	—	2	6	no	—
54	#88071	—	2	3	no	—
55	#88056	—	2	6	no	—
56	#88056	Q₈	1	4	yes	direct product factor
57	Q₈×#103	—	13	1	no	maximal, Sylow
58	C₂×Q₈×C₁₃	C₂²⋊C₄	1	4	yes	—
59	C₂×Q₈×C₁₃	SD₁₆	1	4	yes	—
60	C₂×Q₈×C₁₃	Q₁₆	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₁₃	—	2	1	no	—
65	C₂²×Dic₁₃	—	2	1	no	—
66	C₂²×Dic₁₃	—	2	3	no	—
67	C₂²×Dic₁₃	—	2	3	no	—
68	C₂×Dic₂₆	—	2	1	no	—
69	C₂×Dic₂₆	—	2	1	no	—
70	C₂×Dic₂₆	—	2	1	no	—
71	C₂×Dic₂₆	—	2	12	no	—
72	C₂×Dic₂₆	—	8	3	no	—
73	C₂×Dic₂₆	—	8	3	no	—
74	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₆	C₂×Q₈	1	2	yes	—
76	C₂×Dic₂₆	#41	1	2	yes	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	3	no	—
80	C₂×C₈×C₁₃	—	2	3	no	—
81	C₂×C₈×C₁₃	—	2	3	no	—
82	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
83	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
85	C₄²×C₁₃	C₂×D₄	1	6	yes	—
86	C₄⋊Dic₁₃	—	2	12	no	—
87	C₄⋊Dic₁₃	—	2	3	no	—
88	C₄⋊Dic₁₃	—	2	1	no	—
89	C₄⋊Dic₁₃	—	2	3	no	—
90	C₄⋊Dic₁₃	—	2	3	no	—
91	C₄⋊Dic₁₃	—	4	6	no	—
92	C₄⋊Dic₁₃	—	8	3	no	—
93	C₄⋊Dic₁₃	C₂²×C₄	1	6	yes	—
94	C₄⋊Dic₁₃	C₂×Q₈	1	2	yes	—
95	#6101	—	2	12	no	—
96	C₄×Dic₁₃	—	2	12	no	—
97	C₄×Dic₁₃	—	4	6	no	—
98	C₂×Q₈²	—	13	1	no	—
99	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
100	Q₈×C₄⋊C₄	—	13	1	no	—
101	#1990	—	13	3	no	—
102	#1987	—	13	3	no	—
103	#1803	—	13	3	no	—
104	C₄×#103	—	13	3	no	—
105	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₁₃	#103	1	16	yes	complemented
107	C₂×C₄×C₁₃	—	2	3	no	—
108	C₂×C₄×C₁₃	—	2	3	no	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
111	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
112	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×Dic₁₃	—	2	2	no	—
120	C₂×Dic₁₃	—	2	2	no	—
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	2	6	no	—
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	—	4	1	no	—
125	C₂×Dic₁₃	—	4	3	no	—
126	C₂×Dic₁₃	—	4	3	no	—
127	C₂×Dic₁₃	—	4	1	no	—
128	C₂×Dic₁₃	—	4	3	no	—
129	C₂×Dic₁₃	—	4	3	no	—
130	C₂×Dic₁₃	—	4	3	no	—
131	Dic₂₆	—	2	2	no	—
132	Dic₂₆	—	2	2	no	—
133	Dic₂₆	—	2	2	no	—
134	Dic₂₆	—	8	12	no	—
135	Dic₂₆	C₄×Q₈	1	4	yes	complemented
136	C₈×C₁₃	—	2	2	no	—
137	C₈×C₁₃	—	4	3	no	—
138	Q₈²	—	13	8	no	—
139	Q₈²	—	26	4	no	—
140	C₂×C₄⋊Q₈	—	13	3	no	—
141	C₂×C₄⋊Q₈	—	26	3	no	—
142	C₂×C₄×Q₈	—	13	3	no	—
143	C₂×C₄×Q₈	—	26	1	no	—
144	C₂×C₄×Q₈	—	26	1	no	—
145	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
146	C₈×Q₈	—	2	4	no	—
147	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
148	C₂×#103	—	13	1	no	—
149	C₂×#103	—	13	3	no	—
150	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
151	#389	—	26	3	no	—
152	#384	—	13	3	no	—
153	C₄×C₄⋊C₄	—	13	3	no	—
154	#340	—	13	6	no	—
155	#340	—	13	3	no	—
156	#340	—	13	3	no	—
157	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	#548	1	1	yes	characteristic
159	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
160	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
161	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
162	C₂²×C₁₃	#576	1	1	yes	characteristic
163	C₂²×C₁₃	#577	1	1	yes	characteristic
164	C₄×C₁₃	—	2	6	no	—
165	C₄×C₁₃	C₂×#103	1	12	yes	—
166	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₁₃	#519	1	1	yes	characteristic
168	C₄×C₁₃	#548	1	1	yes	characteristic
169	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
170	Dic₁₃	—	2	4	no	—
171	Dic₁₃	—	4	2	no	—
172	Dic₁₃	—	4	6	no	—
173	Dic₁₃	—	8	3	no	—
174	C₂²×Q₈	—	13	1	no	—
175	C₂²×Q₈	—	26	3	no	—
176	C₂²×Q₈	D₅₂	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
178	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	13	12	no	—
180	C₄⋊Q₈	—	26	24	no	—
181	C₄⋊Q₈	—	26	6	no	—
182	C₄×Q₈	—	13	12	no	—
183	C₄×Q₈	—	26	8	no	—
184	C₄×Q₈	—	26	6	no	—
185	C₄×Q₈	—	52	4	no	—
186	C₄×Q₈	C₄×D₁₃	1	4	yes	—
187	C₄×Q₈	C₂²⋊D₁₃	1	4	yes	—
188	C₂×C₄⋊C₄	—	13	3	no	—
189	C₂×C₄⋊C₄	—	13	1	no	—
190	C₂×C₄⋊C₄	—	13	3	no	—
191	C₂×C₄⋊C₄	—	26	3	no	—
192	C₂×C₄⋊C₄	—	26	3	no	—
193	C₂×C₄⋊C₄	—	26	3	no	—
194	C₂×C₄⋊C₄	—	26	3	no	—
195	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
196	C₂×C₄²	—	26	3	no	—
197	C₂×C₄²	—	26	3	no	—
198	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
199	C₄⋊C₈	—	2	6	no	—
200	#103	—	13	4	no	—
201	#103	—	26	6	no	—
202	C₄×C₈	—	2	6	no	—
203	#95	—	26	3	no	—
204	C₂×C₁₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₁₃	C₂²×#103	1	1	yes	characteristic
206	C₂×C₁₃	#2897	1	1	yes	characteristic
207	C₂×C₁₃	#3362	1	1	yes	characteristic
208	C₂×C₁₃	#3363	1	1	yes	characteristic
209	C₂×C₁₃	Q₈×SD₁₆	1	1	yes	characteristic
210	C₂×C₁₃	Q₈×Q₁₆	1	1	yes	characteristic
211	C₂×Q₈	—	13	2	no	—
212	C₂×Q₈	—	13	2	no	—
213	C₂×Q₈	—	13	2	no	—
214	C₂×Q₈	—	26	12	no	—
215	C₂×Q₈	—	26	1	no	—
216	C₂×Q₈	—	26	1	no	—
217	C₂×Q₈	—	26	1	no	—
218	C₂×Q₈	—	104	3	no	—
219	C₂×Q₈	—	104	3	no	—
220	C₂×Q₈	#6095	1	4	yes	—
221	C₂×Q₈	Dic₅₂	1	4	yes	—
222	C₂×Q₈	#6103	1	4	yes	—
223	C₂²×C₄	—	26	1	no	—
224	C₂²×C₄	—	26	1	no	—
225	C₂²×C₄	—	26	3	no	—
226	C₂²×C₄	—	26	3	no	—
227	C₂²×C₄	C₂×D₅₂	1	3	yes	—
228	C₂²×C₄	C₂×D₅₂	1	3	yes	—
229	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	3	no	—
233	C₂×C₈	—	2	3	no	—
234	C₂×C₈	—	2	3	no	—
235	C₂×C₈	Q₈×D₁₃	1	2	yes	—
236	C₄⋊C₄	—	13	6	no	—
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	26	12	no	—
239	C₄⋊C₄	—	26	12	no	—
240	C₄⋊C₄	—	26	1	no	—
241	C₄⋊C₄	—	26	3	no	—
242	C₄⋊C₄	—	26	3	no	—
243	C₄⋊C₄	—	26	3	no	—
244	C₄⋊C₄	—	52	6	no	—
245	C₄⋊C₄	—	104	3	no	—
246	C₄⋊C₄	#6103	1	12	yes	—
247	C₄²	—	26	12	no	—
248	C₄²	—	52	6	no	—
249	C₄²	C₂×C₄×D₁₃	1	6	yes	—
250	C₄²	C₂×C₂²⋊D₁₃	1	6	yes	—
251	C₁₃	Q₈×#103	1	1	yes	characteristic, complemented, Sylow
252	C₂³	C₂²×D₅₂	1	1	yes	characteristic
253	Q₈	—	13	4	no	—
254	Q₈	—	26	2	no	—
255	Q₈	—	26	2	no	—
256	Q₈	—	26	2	no	—
257	Q₈	—	104	12	no	—
258	Q₈	#88056	1	16	yes	direct product factor
259	C₂×C₄	—	2	3	no	—
260	C₂×C₄	—	2	3	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	26	2	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	—	26	6	no	—
265	C₂×C₄	—	26	2	no	—
266	C₂×C₄	—	52	1	no	—
267	C₂×C₄	—	52	1	no	—
268	C₂×C₄	—	52	3	no	—
269	C₂×C₄	—	52	3	no	—
270	C₂×C₄	—	52	3	no	—
271	C₂×C₄	—	52	3	no	—
272	C₂×C₄	—	52	3	no	—
273	C₂×C₄	C₂×#6095	1	6	yes	—
274	C₂×C₄	C₂×Dic₅₂	1	6	yes	—
275	C₂×C₄	C₂×#6103	1	6	yes	—
276	C₂×C₄	C₂×#6103	1	6	yes	—
277	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
279	C₂×C₄	#88254	1	1	yes	characteristic
280	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
281	C₂×C₄	#88257	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	3	no	—
285	C₂²	Q₈×D₅₂	1	1	yes	characteristic
286	C₂²	#11719281	1	1	yes	characteristic
287	C₂²	C₂²×#6095	1	1	yes	characteristic
288	C₂²	C₂²×Dic₅₂	1	1	yes	characteristic
289	C₂²	#11719456	1	1	yes	characteristic
290	C₂²	#11719457	1	1	yes	characteristic
291	C₂²	C₂²×#6103	1	1	yes	characteristic
292	C₄	—	2	6	no	—
293	C₄	—	26	4	no	—
294	C₄	—	52	2	no	—
295	C₄	—	52	6	no	—
296	C₄	—	104	3	no	—
297	C₄	C₂×#88056	1	12	yes	—
298	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
299	C₄	#11719277	1	1	yes	characteristic
300	C₄	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
301	C₄	#11719536	1	1	yes	characteristic
302	C₂	Q₈×#6103	1	1	yes	characteristic
303	C₂	C₂²×#88056	1	1	yes	characteristic
304	C₂	#13123363	1	1	yes	characteristic
305	C₂	#13127508	1	1	yes	characteristic
306	C₂	#13127515	1	1	yes	characteristic
307	C₂	Q₈×#6095	1	1	yes	characteristic
308	C₂	Q₈×Dic₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	Q₈×#103
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
12. Q₈×Dic₂₆	C₄	semidirect
56. #88056	Q₈	direct (N × H)
106. Q₈×C₁₃	#103	semidirect
135. Dic₂₆	C₄×Q₈	semidirect
251. C₁₃	Q₈×#103	semidirect
258. Q₈	#88056	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₂₆	2
Q₈×C₄⋊Dic₁₃	2
#13115797	2
#13115783	2
#13114842	2
C₄×#88056	2
C₂×C₈×Q₈×C₁₃	2
Q₈×#103	13

Subgroups — Q8×#88056 #15879527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88056 #15879527