Subgroups — Q₈×#88075 #15879526

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×#1055	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13115798	C₂	1	3	yes	maximal
4	#13115785	C₂	1	3	yes	maximal
5	#13114841	C₂	1	3	yes	maximal
6	C₄×#88075	C₂	1	3	yes	maximal
7	C₂×Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₁₃	—	2	4	no	—
9	Q₈²×C₁₃	C₄	1	8	yes	complemented
10	C₂×C₁₃×C₄⋊Q₈	—	2	3	no	—
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
13	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×Dic₁₃	—	2	1	no	—
16	C₂×#88075	C₂²	1	3	yes	—
17	C₂×#88075	C₂²	1	1	yes	characteristic
18	Q₈×#1055	—	2	4	no	—
19	#11718646	—	2	3	no	—
20	#11718596	C₂²	1	3	yes	—
21	C₄×C₄⋊Dic₁₃	C₂²	1	3	yes	—
22	C₂×C₄⋊#1055	C₂²	1	3	yes	—
23	C₂×C₄×#1055	C₂²	1	3	yes	—
24	#11718121	C₂²	1	3	yes	—
25	#11718121	C₂²	1	3	yes	—
26	#11718121	C₂²	1	6	yes	—
27	C₂²×Q₈×C₁₃	—	2	3	no	—
28	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×C₄⋊Q₈	—	2	24	no	—
31	C₁₃×C₄⋊Q₈	—	2	6	no	—
32	C₁₃×C₄⋊Q₈	C₂×C₄	1	12	yes	—
33	C₄×Q₈×C₁₃	—	2	8	no	—
34	C₄×Q₈×C₁₃	—	2	6	no	—
35	C₄×Q₈×C₁₃	C₂×C₄	1	12	yes	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₁₃	D₄	1	4	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
40	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
42	C₂×C₄²×C₁₃	—	2	3	no	—
43	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
44	Q₈×Dic₁₃	—	4	4	no	—
45	C₂×C₄⋊Dic₁₃	—	2	3	no	—
46	C₂×C₄⋊Dic₁₃	—	2	3	no	—
47	C₂×C₄⋊Dic₁₃	C₂³	1	3	yes	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₁₃	—	2	3	no	—
50	C₂²×#1055	C₂³	1	3	yes	—
51	C₂²×#1055	C₂³	1	1	yes	characteristic
52	#88075	—	2	6	no	—
53	#88075	Q₈	1	4	yes	direct product factor
54	#88071	—	2	3	no	—
55	C₄⋊#1055	—	2	6	no	—
56	C₄×#1055	—	2	6	no	—
57	Q₈×#103	—	13	1	no	maximal, Sylow
58	C₂×Q₈×C₁₃	—	2	12	no	—
59	C₂×Q₈×C₁₃	—	2	1	no	—
60	C₂×Q₈×C₁₃	—	2	1	no	—
61	C₂×Q₈×C₁₃	—	2	1	no	—
62	C₂×Q₈×C₁₃	—	8	3	no	—
63	C₂×Q₈×C₁₃	—	8	3	no	—
64	C₂×Q₈×C₁₃	C₂²⋊C₄	1	4	yes	—
65	C₂×Q₈×C₁₃	SD₁₆	1	4	yes	—
66	C₂×Q₈×C₁₃	Q₁₆	1	4	yes	—
67	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×Q₈×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂×Q₈×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	3	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₁₃	—	2	1	no	—
76	C₂²×Dic₁₃	—	2	3	no	—
77	C₁₃×C₄⋊C₄	—	2	12	no	—
78	C₁₃×C₄⋊C₄	—	2	12	no	—
79	C₁₃×C₄⋊C₄	—	2	3	no	—
80	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
81	C₁₃×C₄⋊C₄	C₂²×C₄	1	6	yes	—
82	C₄²×C₁₃	—	2	12	no	—
83	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
84	C₄²×C₁₃	C₂×D₄	1	6	yes	—
85	C₄⋊Dic₁₃	—	2	1	no	—
86	C₄⋊Dic₁₃	—	2	3	no	—
87	C₄⋊Dic₁₃	—	2	3	no	—
88	C₄⋊Dic₁₃	—	4	6	no	—
89	C₄⋊Dic₁₃	—	8	3	no	—
90	C₄⋊Dic₁₃	C₂×Q₈	1	2	yes	—
91	C₄×Dic₁₃	—	4	6	no	—
92	C₂×#1055	—	2	1	no	—
93	C₂×#1055	—	2	1	no	—
94	C₂×#1055	—	2	3	no	—
95	C₂×#1055	—	2	3	no	—
96	C₂×#1055	—	2	3	no	—
97	C₂×#1055	C₂×Q₈	1	2	yes	—
98	C₂×Q₈²	D₁₃	1	1	yes	characteristic
99	C₂×C₈×Q₈	—	13	1	no	—
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	—	2	2	no	—
107	Q₈×C₁₃	—	2	2	no	—
108	Q₈×C₁₃	—	2	2	no	—
109	Q₈×C₁₃	—	8	12	no	—
110	Q₈×C₁₃	#103	1	16	yes	complemented
111	Q₈×C₁₃	C₄×Q₈	1	4	yes	complemented
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	3	no	—
116	C₂×C₄×C₁₃	—	2	3	no	—
117	C₂×C₄×C₁₃	—	2	6	no	—
118	C₂×C₄×C₁₃	—	4	3	no	—
119	C₂×C₄×C₁₃	—	4	3	no	—
120	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
121	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
122	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
123	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×Dic₁₃	—	2	2	no	—
131	C₂×Dic₁₃	—	4	1	no	—
132	C₂×Dic₁₃	—	4	1	no	—
133	C₂×Dic₁₃	—	4	3	no	—
134	C₂×Dic₁₃	—	4	3	no	—
135	C₂×Dic₁₃	—	4	3	no	—
136	#1055	—	2	2	no	—
137	#1055	—	4	3	no	—
138	Q₈²	—	2	4	no	—
139	Q₈²	Dic₁₃	1	8	yes	complemented
140	C₂×C₄⋊Q₈	—	2	3	no	—
141	C₂×C₄⋊Q₈	C₂×D₁₃	1	3	yes	—
142	C₂×C₄×Q₈	—	2	1	no	—
143	C₂×C₄×Q₈	—	26	1	no	—
144	C₂×C₄×Q₈	C₂×D₁₃	1	3	yes	—
145	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
146	C₈×Q₈	—	26	4	no	—
147	C₂×C₄⋊C₈	—	13	3	no	—
148	C₂×#103	—	13	1	no	—
149	C₂×#103	—	13	3	no	—
150	C₂×C₄×C₈	—	13	3	no	—
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	4	no	—
165	C₄×C₁₃	—	2	6	no	—
166	C₄×C₁₃	—	4	6	no	—
167	C₄×C₁₃	C₂×#103	1	12	yes	—
168	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₄×C₁₃	#519	1	1	yes	characteristic
170	C₄×C₁₃	#548	1	1	yes	characteristic
171	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
172	Dic₁₃	—	4	2	no	—
173	Dic₁₃	—	8	3	no	—
174	C₂²×Q₈	—	2	3	no	—
175	C₂²×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
176	C₂²×Q₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	—	13	1	no	—
178	C₂²×C₈	—	13	3	no	—
179	C₄⋊Q₈	—	2	24	no	—
180	C₄⋊Q₈	—	2	6	no	—
181	C₄⋊Q₈	C₂×Dic₁₃	1	12	yes	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	6	no	—
184	C₄×Q₈	—	52	4	no	—
185	C₄×Q₈	C₂×Dic₁₃	1	12	yes	—
186	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
187	C₄×Q₈	C₂²⋊D₁₃	1	4	yes	—
188	C₂×C₄⋊C₄	—	2	3	no	—
189	C₂×C₄⋊C₄	—	2	3	no	—
190	C₂×C₄⋊C₄	—	13	1	no	—
191	C₂×C₄⋊C₄	—	13	3	no	—
192	C₂×C₄⋊C₄	—	26	3	no	—
193	C₂×C₄⋊C₄	—	26	3	no	—
194	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
195	C₂×C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
196	C₂×C₄²	—	2	3	no	—
197	C₂×C₄²	—	26	3	no	—
198	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
199	C₄⋊C₈	—	26	6	no	—
200	#103	—	13	4	no	—
201	#103	—	26	6	no	—
202	C₄×C₈	—	26	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	—	2	12	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	8	3	no	—
216	C₂×Q₈	—	8	3	no	—
217	C₂×Q₈	#6106	1	4	yes	—
218	C₂×Q₈	#6107	1	4	yes	—
219	C₂×Q₈	C₂²⋊Dic₁₃	1	4	yes	—
220	C₂×Q₈	Q₈×D₁₃	1	2	yes	—
221	C₂×Q₈	#6131	1	2	yes	—
222	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	3	no	—
225	C₂²×C₄	—	26	1	no	—
226	C₂²×C₄	—	26	3	no	—
227	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
228	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
229	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
230	C₂×C₈	—	13	2	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	26	1	no	—
233	C₂×C₈	—	26	3	no	—
234	C₂×C₈	—	26	3	no	—
235	C₂×C₈	—	26	3	no	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	2	12	no	—
238	C₄⋊C₄	—	2	3	no	—
239	C₄⋊C₄	—	13	2	no	—
240	C₄⋊C₄	—	26	1	no	—
241	C₄⋊C₄	—	26	3	no	—
242	C₄⋊C₄	—	26	3	no	—
243	C₄⋊C₄	—	52	6	no	—
244	C₄⋊C₄	—	104	3	no	—
245	C₄⋊C₄	C₂²⋊Dic₁₃	1	12	yes	—
246	C₄⋊C₄	C₂²×Dic₁₃	1	6	yes	—
247	C₄²	—	2	12	no	—
248	C₄²	—	52	6	no	—
249	C₄²	C₂²×Dic₁₃	1	6	yes	—
250	C₄²	C₂×C₂²⋊D₁₃	1	6	yes	—
251	C₁₃	Q₈×#103	1	1	yes	characteristic, complemented, Sylow
252	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	2	2	no	—
256	Q₈	—	8	12	no	—
257	Q₈	#88075	1	16	yes	direct product factor
258	Q₈	Q₈×Dic₁₃	1	4	yes	complemented
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	3	no	—
262	C₂×C₄	—	2	3	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	6	no	—
265	C₂×C₄	—	4	3	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	—	52	1	no	—
269	C₂×C₄	—	52	1	no	—
270	C₂×C₄	—	52	3	no	—
271	C₂×C₄	—	52	3	no	—
272	C₂×C₄	—	52	3	no	—
273	C₂×C₄	C₂×#6106	1	6	yes	—
274	C₂×C₄	C₂×#6107	1	6	yes	—
275	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
276	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
277	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#6131	1	1	yes	characteristic
279	C₂×C₄	#88254	1	1	yes	characteristic
280	C₂×C₄	#88254	1	1	yes	characteristic
281	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
283	C₈	—	26	2	no	—
284	C₈	—	52	3	no	—
285	C₂²	C₂²×#6106	1	1	yes	characteristic
286	C₂²	C₂²×#6107	1	1	yes	characteristic
287	C₂²	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₂²	#11719522	1	1	yes	characteristic
289	C₂²	#11719540	1	1	yes	characteristic
290	C₂²	#11719541	1	1	yes	characteristic
291	C₂²	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
292	C₄	—	2	6	no	—
293	C₄	—	2	4	no	—
294	C₄	—	4	6	no	—
295	C₄	—	52	2	no	—
296	C₄	—	104	3	no	—
297	C₄	C₂×#88075	1	12	yes	—
298	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
299	C₄	#11719517	1	1	yes	characteristic
300	C₄	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
301	C₄	#11719522	1	1	yes	characteristic
302	C₂	Q₈×C₂²⋊Dic₁₃	1	1	yes	characteristic
303	C₂	C₂²×#88075	1	1	yes	characteristic
304	C₂	#13123362	1	1	yes	characteristic
305	C₂	#13127507	1	1	yes	characteristic
306	C₂	#13127514	1	1	yes	characteristic
307	C₂	Q₈×#6106	1	1	yes	characteristic
308	C₂	Q₈×#6107	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×Q₈²×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×Q₈²	D₁₃
hypocenter	C₁₃	Q₈×#103
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
9. Q₈²×C₁₃	C₄	semidirect
53. #88075	Q₈	direct (N × H)
110. Q₈×C₁₃	#103	semidirect
111. Q₈×C₁₃	C₄×Q₈	semidirect
139. Q₈²	Dic₁₃	semidirect
251. C₁₃	Q₈×#103	semidirect
257. Q₈	#88075	direct (N × H)
258. Q₈	Q₈×Dic₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#1055	2
Q₈×C₄⋊Dic₁₃	2
#13115798	2
#13115785	2
#13114841	2
C₄×#88075	2
C₂×Q₈²×C₁₃	2
Q₈×#103	13

Subgroups — Q8×#88075 #15879526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88075 #15879526