Subgroups — Q₈×#88064 #15879491

Order 3328.

970 conjugacy classes of subgroups.

The subgroup lattice has 330 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×#6099	C₂	1	1	yes	maximal, characteristic
3	#13115825	C₂	1	3	yes	maximal
4	#13115824	C₂	1	3	yes	maximal
5	#13114829	C₂	1	3	yes	maximal
6	C₄×#88064	C₂	1	3	yes	maximal
7	Q₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₃	C₄	1	2	yes	—
10	Q₈×Dic₂₆	—	2	1	no	—
11	Q₈×Dic₂₆	—	2	3	no	—
12	Q₈×Dic₂₆	—	2	1	no	—
13	Q₈×Dic₂₆	—	2	3	no	—
14	C₂×#88142	C₄	1	6	yes	—
15	C₂×#88123	C₂²	1	3	yes	—
16	C₂×C₄×Dic₂₆	C₂²	1	3	yes	—
17	C₁₃×#446	—	2	3	no	—
18	C₈×Q₈×C₁₃	—	2	1	no	—
19	C₁₃×#423	C₂²	1	3	yes	—
20	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
21	C₂×#88064	C₂²	1	3	yes	—
22	C₂×#88064	C₂²	1	1	yes	characteristic
23	#11718685	—	2	3	no	—
24	Q₈×#1055	—	2	1	no	—
25	#11718585	C₂²	1	3	yes	—
26	C₄×#6099	C₂²	1	3	yes	—
27	#11718146	C₂²	1	3	yes	—
28	#11718146	C₂²	1	3	yes	—
29	#11718146	C₂²	1	6	yes	—
30	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₂₆	C₂×C₄	1	6	yes	—
32	C₂²×Dic₂₆	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	C₂³	1	3	yes	—
34	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₄×Q₈×C₁₃	D₄	1	3	yes	—
37	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₁₃	D₄	1	3	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
41	Q₈×Dic₁₃	—	2	6	no	—
42	Q₈×Dic₁₃	—	2	2	no	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	6	yes	—
44	C₂×C₄⋊Dic₁₃	C₂³	1	3	yes	—
45	C₂×#6101	C₂×C₄	1	6	yes	—
46	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
47	C₂×#6099	C₂³	1	3	yes	—
48	C₂×#6099	C₂³	1	1	yes	characteristic
49	#88142	—	2	12	no	—
50	#88142	—	2	6	no	—
51	#88142	—	2	6	no	—
52	#88123	—	2	3	no	—
53	#88123	—	2	3	no	—
54	#88123	—	2	3	no	—
55	#88123	—	2	3	no	—
56	C₄×Dic₂₆	—	2	3	no	—
57	C₄×Dic₂₆	—	2	3	no	—
58	C₄×Dic₂₆	—	2	3	no	—
59	C₄×Dic₂₆	—	2	3	no	—
60	C₁₃×C₄⋊C₈	—	2	3	no	—
61	C₁₃×C₄⋊C₈	—	2	3	no	—
62	C₁₃×#97	—	2	3	no	—
63	C₄×C₈×C₁₃	—	2	3	no	—
64	#88064	—	2	6	no	—
65	#88064	Q₈	1	4	yes	direct product factor
66	C₄⋊#1055	—	2	3	no	—
67	C₄⋊#1055	—	2	3	no	—
68	#88043	—	2	3	no	—
69	C₄×#1055	—	2	3	no	—
70	Q₈×#101	—	13	1	no	maximal, Sylow
71	C₂×Q₈×C₁₃	—	2	4	no	—
72	C₂×Q₈×C₁₃	C₂²⋊C₄	1	3	yes	—
73	C₂×Q₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
79	C₂×Dic₂₆	—	2	1	no	—
80	C₂×Dic₂₆	—	2	1	no	—
81	C₂×Dic₂₆	—	2	1	no	—
82	C₂×Dic₂₆	—	2	1	no	—
83	C₂×Dic₂₆	—	2	6	no	—
84	C₂×Dic₂₆	—	2	6	no	—
85	C₂×Dic₂₆	—	8	6	no	—
86	C₂×Dic₂₆	C₂×Q₈	1	2	yes	—
87	C₂×Dic₂₆	#41	1	2	yes	—
88	C₁₃×M₁₆	—	2	1	no	—
89	C₁₃×M₁₆	—	2	3	no	—
90	C₁₃×M₁₆	—	2	3	no	—
91	C₁₃×M₁₆	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	3	no	—
94	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
95	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	3	yes	—
96	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	3	yes	—
97	C₄²×C₁₃	C₂×D₄	1	3	yes	—
98	C₄²×C₁₃	C₂×D₄	1	3	yes	—
99	C₄²×C₁₃	C₂×D₄	1	3	yes	—
100	C₄²×C₁₃	C₂×D₄	1	3	yes	—
101	C₄⋊Dic₁₃	—	2	6	no	—
102	C₄⋊Dic₁₃	—	2	6	no	—
103	C₄⋊Dic₁₃	—	2	3	no	—
104	C₄⋊Dic₁₃	—	2	3	no	—
105	#6101	—	2	6	no	—
106	#6101	—	2	6	no	—
107	C₄×Dic₁₃	—	2	6	no	—
108	C₄×Dic₁₃	—	2	6	no	—
109	#6099	—	2	1	no	—
110	#6099	—	2	3	no	—
111	#6099	—	2	3	no	—
112	#6099	C₂×Q₈	1	2	yes	—
113	C₂×#1055	—	2	1	no	—
114	C₂×#1055	—	2	3	no	—
115	C₂×Q₈²	—	13	1	no	—
116	Q₈×M₁₆	—	13	1	no	—
117	Q₈×M₁₆	D₁₃	1	1	yes	characteristic
118	#1997	—	13	3	no	—
119	#1997	—	13	3	no	—
120	#1801	—	13	3	no	—
121	C₄×#101	—	13	3	no	—
122	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	6	no	—
124	Q₈×C₁₃	#101	1	4	yes	complemented
125	C₂×C₄×C₁₃	—	2	6	no	—
126	C₂×C₄×C₁₃	—	4	3	no	—
127	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
129	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
130	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	2	6	no	—
140	C₂×Dic₁₃	—	4	6	no	—
141	C₂×Dic₁₃	C₄×Q₈	1	4	yes	—
142	Dic₂₆	—	2	2	no	—
143	Dic₂₆	—	2	2	no	—
144	Dic₂₆	—	2	2	no	—
145	Dic₂₆	—	2	2	no	—
146	Dic₂₆	—	8	6	no	—
147	Dic₂₆	—	8	6	no	—
148	C₈×C₁₃	—	2	2	no	—
149	C₈×C₁₃	—	4	3	no	—
150	#1055	—	2	2	no	—
151	#1055	—	4	3	no	—
152	Q₈²	—	26	1	no	—
153	Q₈²	—	26	3	no	—
154	Q₈²	—	26	1	no	—
155	Q₈²	—	26	3	no	—
156	C₂×C₄⋊Q₈	—	13	6	no	—
157	C₂×C₄⋊Q₈	—	13	3	no	—
158	C₂×C₄×Q₈	—	13	2	no	—
159	C₂×C₄×Q₈	—	13	3	no	—
160	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
161	#446	—	2	3	no	—
162	#446	—	26	3	no	—
163	C₈×Q₈	—	2	1	no	—
164	C₈×Q₈	—	26	1	no	—
165	#423	—	13	3	no	—
166	#423	C₂×D₁₃	1	3	yes	—
167	C₂×#101	—	13	1	no	—
168	C₂×#101	—	13	3	no	—
169	C₄×M₁₆	—	13	3	no	—
170	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
171	#343	—	13	6	no	—
172	#343	—	13	3	no	—
173	#343	—	13	3	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
178	C₂²×C₁₃	#519	1	1	yes	characteristic
179	C₄×C₁₃	—	2	3	no	—
180	C₄×C₁₃	—	2	3	no	—
181	C₄×C₁₃	—	2	3	no	—
182	C₄×C₁₃	C₂×#101	1	6	yes	—
183	C₄×C₁₃	#548	1	1	yes	characteristic
184	C₄×C₁₃	#548	1	1	yes	characteristic
185	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
186	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
187	Dic₁₃	—	2	4	no	—
188	Dic₁₃	—	4	6	no	—
189	C₂²×Q₈	—	13	6	no	—
190	C₂²×Q₈	—	13	1	no	—
191	C₂²×Q₈	C₄×D₁₃	1	1	yes	characteristic
192	C₂×M₁₆	—	13	1	no	—
193	C₂×M₁₆	—	13	3	no	—
194	C₂×M₁₆	C₂²×D₁₃	1	3	yes	—
195	C₂×M₁₆	C₂²×D₁₃	1	1	yes	characteristic
196	C₄⋊Q₈	—	26	12	no	—
197	C₄⋊Q₈	—	26	6	no	—
198	C₄⋊Q₈	—	26	6	no	—
199	C₄⋊Q₈	—	26	3	no	—
200	C₄⋊Q₈	—	26	3	no	—
201	C₄⋊Q₈	—	26	3	no	—
202	C₄⋊Q₈	—	26	3	no	—
203	C₄×Q₈	—	26	6	no	—
204	C₄×Q₈	—	26	2	no	—
205	C₄×Q₈	—	26	3	no	—
206	C₄×Q₈	—	26	3	no	—
207	C₄×Q₈	—	26	3	no	—
208	C₄×Q₈	—	26	3	no	—
209	C₄×Q₈	D₅₂	1	1	yes	characteristic
210	C₄×Q₈	D₅₂	1	3	yes	—
211	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
212	C₄×Q₈	C₂²⋊D₁₃	1	3	yes	—
213	C₂×C₄⋊C₄	—	13	6	no	—
214	C₂×C₄⋊C₄	—	13	6	no	—
215	C₂×C₄⋊C₄	—	13	3	no	—
216	C₂×C₄⋊C₄	C₄×D₁₃	1	3	yes	—
217	C₂×C₄²	—	13	6	no	—
218	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
219	C₄⋊C₈	—	2	3	no	—
220	C₄⋊C₈	—	2	3	no	—
221	C₄⋊C₈	—	26	3	no	—
222	C₄⋊C₈	—	26	3	no	—
223	#101	—	13	4	no	—
224	#101	—	26	6	no	—
225	#97	—	2	3	no	—
226	#97	—	26	3	no	—
227	C₄×C₈	—	2	3	no	—
228	C₄×C₈	—	26	3	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	—	2	1	no	—
231	C₂×C₁₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₁₃	C₂²×#101	1	1	yes	characteristic
233	C₂×C₁₃	#2888	1	1	yes	characteristic
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	13	2	no	—
236	C₂×Q₈	—	13	2	no	—
237	C₂×Q₈	—	26	6	no	—
238	C₂×Q₈	—	26	6	no	—
239	C₂×Q₈	—	26	1	no	—
240	C₂×Q₈	—	26	1	no	—
241	C₂×Q₈	—	26	1	no	—
242	C₂×Q₈	—	26	1	no	—
243	C₂×Q₈	—	104	6	no	—
244	C₂×Q₈	#6103	1	3	yes	—
245	C₂×Q₈	#6103	1	1	yes	characteristic
246	C₂²×C₄	—	13	2	no	—
247	C₂²×C₄	—	13	6	no	—
248	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
249	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
250	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	3	no	—
253	M₁₆	—	2	3	no	—
254	M₁₆	—	13	2	no	—
255	M₁₆	—	26	1	no	—
256	M₁₆	—	26	3	no	—
257	M₁₆	—	26	3	no	—
258	M₁₆	Q₈×D₁₃	1	2	yes	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	3	no	—
261	C₂×C₈	—	26	1	no	—
262	C₂×C₈	—	26	3	no	—
263	C₄⋊C₄	—	26	6	no	—
264	C₄⋊C₄	—	26	6	no	—
265	C₄⋊C₄	—	26	6	no	—
266	C₄⋊C₄	—	26	6	no	—
267	C₄⋊C₄	—	26	3	no	—
268	C₄⋊C₄	—	26	3	no	—
269	C₄⋊C₄	#6103	1	6	yes	—
270	C₄⋊C₄	#6103	1	3	yes	—
271	C₄⋊C₄	#6103	1	3	yes	—
272	C₄²	—	26	6	no	—
273	C₄²	—	26	6	no	—
274	C₄²	C₂×D₅₂	1	3	yes	—
275	C₄²	C₂×D₅₂	1	3	yes	—
276	C₄²	C₂×C₂²⋊D₁₃	1	3	yes	—
277	C₄²	C₂×C₂²⋊D₁₃	1	3	yes	—
278	C₁₃	Q₈×#101	1	1	yes	characteristic, complemented, Sylow
279	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
280	Q₈	—	2	6	no	—
281	Q₈	—	26	2	no	—
282	Q₈	—	26	2	no	—
283	Q₈	—	26	2	no	—
284	Q₈	—	26	2	no	—
285	Q₈	—	104	6	no	—
286	Q₈	—	104	6	no	—
287	Q₈	#88064	1	4	yes	direct product factor
288	C₂×C₄	—	2	6	no	—
289	C₂×C₄	—	4	3	no	—
290	C₂×C₄	—	13	4	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	6	no	—
294	C₂×C₄	—	52	6	no	—
295	C₂×C₄	C₂×#6103	1	3	yes	—
296	C₂×C₄	C₂×#6103	1	3	yes	—
297	C₂×C₄	C₂×#6103	1	3	yes	—
298	C₂×C₄	C₂×#6103	1	3	yes	—
299	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
301	C₂×C₄	#88254	1	1	yes	characteristic
302	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
303	C₂×C₄	#88257	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	3	no	—
307	C₈	—	26	2	no	—
308	C₈	—	52	3	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	C₄×Q₈×D₁₃	1	1	yes	characteristic
312	C₂²	#11719277	1	1	yes	characteristic
313	C₂²	C₂²×#6103	1	1	yes	characteristic
314	C₄	—	2	3	no	—
315	C₄	—	2	3	no	—
316	C₄	—	2	3	no	—
317	C₄	—	26	4	no	—
318	C₄	—	52	6	no	—
319	C₄	C₂×#88064	1	6	yes	—
320	C₄	Q₈×D₅₂	1	1	yes	characteristic
321	C₄	#11719281	1	1	yes	characteristic
322	C₄	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
323	C₄	#11719536	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	Q₈×#6103	1	1	yes	characteristic
327	C₂	C₂²×#88064	1	1	yes	characteristic
328	C₂	#13123314	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6103
commutator, prime radical	C₂³×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	Q₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	Q₈×M₁₆	D₁₃
hypocenter	C₁₃	Q₈×#101
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
65. #88064	Q₈	direct (N × H)
124. Q₈×C₁₃	#101	semidirect
278. C₁₃	Q₈×#101	semidirect
287. Q₈	#88064	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×#6099	2
#13115825	2
#13115824	2
#13114829	2
C₄×#88064	2
Q₈×C₁₃×M₁₆	2
Q₈×#101	13

Subgroups — Q8×#88064 #15879491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88064 #15879491