Subgroups — Q₈×#88062 #15881809

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6099	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
3	#13115611	C₂	1	3	yes	maximal
4	#13115609	C₂	1	3	yes	maximal
5	#13115007	C₂	1	3	yes	maximal
6	C₄×#88062	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	C₁₃×#446	—	2	3	no	—
10	C₈×Q₈×C₁₃	—	2	1	no	—
11	C₁₃×#423	C₂²	1	3	yes	—
12	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
13	C₂×#88062	C₂²	1	3	yes	—
14	C₂×#88062	C₂²	1	6	yes	—
15	C₂×#88062	C₂²	1	1	yes	characteristic
16	#11718685	—	2	3	no	—
17	Q₈×#1055	—	2	1	no	—
18	Q₈×#1055	C₄	1	6	yes	—
19	Q₈×#1055	C₄	1	2	yes	—
20	#11718585	C₂²	1	3	yes	—
21	C₂×C₄⋊#1055	C₂²	1	3	yes	—
22	C₄×#6099	C₂²	1	3	yes	—
23	C₂×C₄×#1055	C₂²	1	3	yes	—
24	#11718150	C₂²	1	3	yes	—
25	#11718150	C₂²	1	3	yes	—
26	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
27	C₂×C₁₃×M₁₆	C₂³	1	3	yes	—
28	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
30	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	D₄	1	3	yes	—
33	C₂×C₁₃×C₄⋊C₄	Q₈	1	3	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
35	C₂×#6099	C₂³	1	3	yes	—
36	C₂×#6099	C₂³	1	1	yes	characteristic
37	C₂²×#1055	C₂³	1	3	yes	—
38	C₂²×#1055	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊C₈	—	2	3	no	—
40	C₁₃×C₄⋊C₈	—	2	3	no	—
41	C₁₃×#97	—	2	3	no	—
42	C₄×C₈×C₁₃	—	2	3	no	—
43	#88062	—	2	6	no	—
44	#88062	—	4	6	no	—
45	#88062	Q₈	1	4	yes	direct product factor
46	C₄⋊#1055	—	2	3	no	—
47	C₄⋊#1055	—	2	3	no	—
48	C₄⋊#1055	C₂×C₄	1	6	yes	—
49	C₄⋊#1055	C₂×C₄	1	6	yes	—
50	#88043	—	2	3	no	—
51	C₄×#1055	—	2	3	no	—
52	C₄×#1055	C₂×C₄	1	6	yes	—
53	C₄×#1055	C₂×C₄	1	6	yes	—
54	Q₈×#108	—	13	1	no	maximal, Sylow
55	C₂×Q₈×C₁₃	—	2	4	no	—
56	C₂×Q₈×C₁₃	C₄⋊C₄	1	3	yes	—
57	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
59	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₁₃×M₁₆	—	2	1	no	—
62	C₁₃×M₁₆	—	2	3	no	—
63	C₁₃×M₁₆	—	2	3	no	—
64	C₁₃×M₁₆	C₂×Q₈	1	2	yes	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	2	3	no	—
67	C₁₃×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
68	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
69	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
70	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
71	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
72	C₄²×C₁₃	C₂×D₄	1	3	yes	—
73	C₄²×C₁₃	C₂×D₄	1	3	yes	—
74	#6099	—	2	1	no	—
75	#6099	—	2	3	no	—
76	#6099	—	2	3	no	—
77	#6099	C₂×Q₈	1	2	yes	—
78	C₂×#1055	—	2	1	no	—
79	C₂×#1055	—	2	3	no	—
80	C₂×#1055	—	2	3	no	—
81	C₂×#1055	—	2	3	no	—
82	C₂×#1055	C₂²×C₄	1	6	yes	—
83	C₂×#1055	C₂²×C₄	1	2	yes	—
84	C₂×#1055	C₂×Q₈	1	2	yes	—
85	C₂×#1055	#41	1	2	yes	—
86	Q₈×M₁₆	—	13	1	no	—
87	Q₈×M₁₆	D₁₃	1	1	yes	characteristic
88	C₂×C₈×Q₈	—	13	1	no	—
89	#1950	—	13	3	no	—
90	#1950	—	13	3	no	—
91	#1848	—	13	3	no	—
92	C₄×#108	—	13	3	no	—
93	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
94	Q₈×C₁₃	—	2	6	no	—
95	Q₈×C₁₃	#108	1	4	yes	complemented
96	C₂×C₄×C₁₃	—	2	6	no	—
97	C₂×C₄×C₁₃	—	4	3	no	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	—	4	3	no	—
110	#1055	—	2	6	no	—
111	#1055	—	2	2	no	—
112	#1055	—	4	3	no	—
113	#1055	C₄×Q₈	1	4	yes	—
114	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
115	#446	—	2	3	no	—
116	#446	—	26	3	no	—
117	C₈×Q₈	—	2	1	no	—
118	C₈×Q₈	—	13	6	no	—
119	C₈×Q₈	—	13	2	no	—
120	C₈×Q₈	—	26	1	no	—
121	C₂×#108	—	13	1	no	—
122	C₂×#108	—	13	3	no	—
123	C₂×#108	—	13	6	no	—
124	#423	—	13	3	no	—
125	#423	C₂×D₁₃	1	3	yes	—
126	C₂×C₄⋊C₈	—	13	3	no	—
127	C₄×M₁₆	—	13	3	no	—
128	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
129	C₂×C₄×C₈	—	13	3	no	—
130	#341	—	13	3	no	—
131	#341	—	13	3	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₁₃	#557	1	1	yes	characteristic
136	C₂²×C₁₃	Q₈²	1	1	yes	characteristic
137	C₄×C₁₃	—	2	3	no	—
138	C₄×C₁₃	—	2	3	no	—
139	C₄×C₁₃	—	2	3	no	—
140	C₄×C₁₃	C₂×#108	1	6	yes	—
141	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₁₃	#520	1	1	yes	characteristic
143	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
144	C₄×C₁₃	#550	1	1	yes	characteristic
145	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
146	C₂×M₁₆	—	13	1	no	—
147	C₂×M₁₆	—	13	3	no	—
148	C₂×M₁₆	C₂²×D₁₃	1	3	yes	—
149	C₂×M₁₆	C₂²×D₁₃	1	1	yes	characteristic
150	C₂²×C₈	—	13	1	no	—
151	C₂²×C₈	—	13	3	no	—
152	C₄×Q₈	C₄×D₁₃	1	3	yes	—
153	C₄×Q₈	C₄×D₁₃	1	1	yes	characteristic
154	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
155	C₄×Q₈	C₂²⋊D₁₃	1	3	yes	—
156	C₂×C₄⋊C₄	Dic₂₆	1	3	yes	—
157	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
158	#108	—	13	4	no	—
159	#108	—	26	6	no	—
160	#108	—	52	6	no	—
161	C₄⋊C₈	—	2	3	no	—
162	C₄⋊C₈	—	2	3	no	—
163	C₄⋊C₈	—	13	6	no	—
164	C₄⋊C₈	—	13	6	no	—
165	C₄⋊C₈	—	26	3	no	—
166	C₄⋊C₈	—	26	3	no	—
167	#97	—	2	3	no	—
168	#97	—	26	3	no	—
169	C₄×C₈	—	2	3	no	—
170	C₄×C₈	—	13	6	no	—
171	C₄×C₈	—	13	6	no	—
172	C₄×C₈	—	26	3	no	—
173	C₂×C₁₃	—	2	1	no	—
174	C₂×C₁₃	—	2	1	no	—
175	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
177	C₂×C₁₃	#2916	1	1	yes	characteristic
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	#6101	1	3	yes	—
180	C₂×Q₈	#6101	1	1	yes	characteristic
181	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
182	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
183	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
184	M₁₆	—	2	1	no	—
185	M₁₆	—	2	3	no	—
186	M₁₆	—	2	3	no	—
187	M₁₆	—	13	2	no	—
188	M₁₆	—	26	1	no	—
189	M₁₆	—	26	3	no	—
190	M₁₆	—	26	3	no	—
191	M₁₆	Q₈×D₁₃	1	2	yes	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	3	no	—
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	13	6	no	—
198	C₂×C₈	—	26	3	no	—
199	C₂×C₈	—	26	3	no	—
200	C₂×C₈	—	26	1	no	—
201	C₂×C₈	—	26	3	no	—
202	C₄⋊C₄	#6101	1	6	yes	—
203	C₄⋊C₄	#6101	1	3	yes	—
204	C₄⋊C₄	#6101	1	3	yes	—
205	C₄²	C₂×C₄×D₁₃	1	3	yes	—
206	C₄²	C₂×C₄×D₁₃	1	3	yes	—
207	C₄²	C₂×C₂²⋊D₁₃	1	3	yes	—
208	C₄²	C₂×C₂²⋊D₁₃	1	3	yes	—
209	C₁₃	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
210	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
211	Q₈	—	2	6	no	—
212	Q₈	#88062	1	4	yes	direct product factor
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	4	3	no	—
215	C₂×C₄	C₂×#6101	1	3	yes	—
216	C₂×C₄	C₂×#6101	1	3	yes	—
217	C₂×C₄	C₂×#6101	1	3	yes	—
218	C₂×C₄	C₂×#6101	1	3	yes	—
219	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
220	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
221	C₂×C₄	#88254	1	1	yes	characteristic
222	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
223	C₂×C₄	#88256	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	4	3	no	—
227	C₈	—	13	4	no	—
228	C₈	—	26	2	no	—
229	C₈	—	26	6	no	—
230	C₈	—	52	3	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	Q₈×Dic₂₆	1	1	yes	characteristic
234	C₂²	#11719274	1	1	yes	characteristic
235	C₂²	C₂²×#6101	1	1	yes	characteristic
236	C₄	—	2	3	no	—
237	C₄	—	2	3	no	—
238	C₄	—	2	3	no	—
239	C₄	C₂×#88062	1	6	yes	—
240	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
241	C₄	#11719279	1	1	yes	characteristic
242	C₄	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
243	C₄	#11719539	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	Q₈×#6101	1	1	yes	characteristic
247	C₂	C₂²×#88062	1	1	yes	characteristic
248	C₂	#13123445	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	Q₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	Q₈×M₁₆	D₁₃
hypocenter	C₁₃	Q₈×#108
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
45. #88062	Q₈	direct (N × H)
95. Q₈×C₁₃	#108	semidirect
209. C₁₃	Q₈×#108	semidirect
212. Q₈	#88062	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#6099	2
C₂×Q₈×#1055	2
#13115611	2
#13115609	2
#13115007	2
C₄×#88062	2
Q₈×C₁₃×M₁₆	2
Q₈×#108	13

Subgroups — Q8×#88062 #15881809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88062 #15881809