Subgroups — Q₈×#88059 #15881812

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 208 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	2	yes	maximal
2	#13115612	C₂	1	6	yes	maximal
3	#13115008	C₂	1	3	yes	maximal
4	C₄×#88059	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₃	C₄	1	6	yes	—
8	C₈×Q₈×C₁₃	C₄	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
11	C₂×#88059	C₂²	1	3	yes	—
12	C₂×#88059	C₂²	1	6	yes	—
13	C₂×#88059	C₂²	1	1	yes	characteristic
14	#11718685	—	2	6	no	—
15	Q₈×#1055	—	2	2	no	—
16	#11718585	C₂²	1	6	yes	—
17	C₄×#6099	C₂²	1	6	yes	—
18	#11718144	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	D₄	1	3	yes	—
26	C₂×C₁₃×C₄⋊C₄	Q₈	1	3	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
28	C₂×#6099	C₂³	1	6	yes	—
29	C₂×#6099	C₂³	1	2	yes	—
30	C₁₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
31	C₁₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
33	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
34	#88059	—	2	6	no	—
35	#88059	—	4	6	no	—
36	#88059	Q₈	1	4	yes	direct product factor
37	C₄⋊#1055	—	2	6	no	—
38	C₄⋊#1055	—	2	6	no	—
39	#88043	—	2	6	no	—
40	C₄×#1055	—	2	6	no	—
41	Q₈×#108	—	13	1	no	maximal, Sylow
42	C₂×Q₈×C₁₃	—	2	4	no	—
43	C₂×Q₈×C₁₃	C₄⋊C₄	1	3	yes	—
44	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₁₃	—	2	3	no	—
49	C₂×C₈×C₁₃	—	2	3	no	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₃	#41	1	2	yes	—
54	C₁₃×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
55	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
56	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
57	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
58	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
59	C₄²×C₁₃	C₂×D₄	1	3	yes	—
60	C₄²×C₁₃	C₂×D₄	1	3	yes	—
61	#6099	—	2	2	no	—
62	#6099	—	2	6	no	—
63	#6099	—	2	6	no	—
64	#6099	C₂×Q₈	1	4	yes	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	6	no	—
67	Q₈×M₁₆	—	13	2	no	—
68	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
69	#1950	—	13	6	no	—
70	#1848	—	13	3	no	—
71	C₄×#108	—	13	3	no	—
72	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
73	Q₈×C₁₃	—	2	6	no	—
74	Q₈×C₁₃	#108	1	4	yes	complemented
75	C₂×C₄×C₁₃	—	2	6	no	—
76	C₂×C₄×C₁₃	—	4	3	no	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
86	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
87	C₈×C₁₃	—	2	6	no	—
88	C₈×C₁₃	C₄×Q₈	1	4	yes	—
89	#1055	—	2	4	no	—
90	#1055	—	4	6	no	—
91	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
92	#446	—	26	6	no	—
93	C₈×Q₈	—	26	2	no	—
94	C₈×Q₈	Dic₁₃	1	6	yes	—
95	C₈×Q₈	Dic₁₃	1	2	yes	—
96	C₂×#108	—	13	1	no	—
97	C₂×#108	—	13	3	no	—
98	C₂×#108	—	13	6	no	—
99	#423	—	13	6	no	—
100	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
101	C₄×M₁₆	—	13	6	no	—
102	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
103	#341	—	13	6	no	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	—	2	1	no	—
106	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₃	#557	1	1	yes	characteristic
108	C₂²×C₁₃	Q₈²	1	1	yes	characteristic
109	C₄×C₁₃	—	2	3	no	—
110	C₄×C₁₃	—	2	3	no	—
111	C₄×C₁₃	—	2	3	no	—
112	C₄×C₁₃	C₂×#108	1	6	yes	—
113	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₄×C₁₃	#520	1	1	yes	characteristic
115	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
116	C₄×C₁₃	#550	1	1	yes	characteristic
117	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
118	C₂×M₁₆	—	13	2	no	—
119	C₂×M₁₆	—	13	6	no	—
120	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
121	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
122	C₄×Q₈	D₅₂	1	1	yes	characteristic
123	C₄×Q₈	D₅₂	1	3	yes	—
124	C₄×Q₈	C₂×Dic₁₃	1	3	yes	—
125	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	Dic₂₆	1	3	yes	—
127	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
128	#108	—	13	4	no	—
129	#108	—	26	6	no	—
130	#108	—	52	6	no	—
131	C₄⋊C₈	—	26	6	no	—
132	C₄⋊C₈	—	26	6	no	—
133	C₄⋊C₈	C₂×Dic₁₃	1	6	yes	—
134	C₄⋊C₈	C₂×Dic₁₃	1	6	yes	—
135	#97	—	26	6	no	—
136	C₄×C₈	—	26	6	no	—
137	C₄×C₈	C₂×Dic₁₃	1	6	yes	—
138	C₄×C₈	C₂×Dic₁₃	1	6	yes	—
139	C₂×C₁₃	—	2	1	no	—
140	C₂×C₁₃	—	2	1	no	—
141	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
143	C₂×C₁₃	#2916	1	1	yes	characteristic
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	C₄⋊Dic₁₃	1	3	yes	—
146	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
148	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
149	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
150	M₁₆	—	13	4	no	—
151	M₁₆	—	26	2	no	—
152	M₁₆	—	26	6	no	—
153	M₁₆	—	26	6	no	—
154	C₂×C₈	—	2	3	no	—
155	C₂×C₈	—	2	3	no	—
156	C₂×C₈	—	26	2	no	—
157	C₂×C₈	—	26	6	no	—
158	C₂×C₈	Q₈×D₁₃	1	2	yes	—
159	C₂×C₈	#6131	1	2	yes	—
160	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
161	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
162	C₄⋊C₄	C₄⋊Dic₁₃	1	6	yes	—
163	C₄⋊C₄	C₄⋊Dic₁₃	1	3	yes	—
164	C₄⋊C₄	C₄⋊Dic₁₃	1	3	yes	—
165	C₄²	C₂×D₅₂	1	3	yes	—
166	C₄²	C₂×D₅₂	1	3	yes	—
167	C₄²	C₂²×Dic₁₃	1	3	yes	—
168	C₄²	C₂²×Dic₁₃	1	3	yes	—
169	C₁₃	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
171	Q₈	—	2	6	no	—
172	Q₈	#88059	1	4	yes	direct product factor
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
179	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
181	C₂×C₄	C₂×#6131	1	1	yes	characteristic
182	C₂×C₄	#88256	1	1	yes	characteristic
183	C₂×C₄	#88257	1	1	yes	characteristic
184	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
185	C₈	—	2	6	no	—
186	C₈	—	26	4	no	—
187	C₈	—	52	6	no	—
188	C₈	Q₈×Dic₁₃	1	4	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	Q₈×Dic₂₆	1	1	yes	characteristic
192	C₂²	#11719275	1	1	yes	characteristic
193	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×#88059	1	6	yes	—
198	C₄	Q₈×D₅₂	1	1	yes	characteristic
199	C₄	#11719282	1	1	yes	characteristic
200	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
201	C₄	#11719532	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	Q₈×C₄⋊Dic₁₃	1	1	yes	characteristic
205	C₂	C₂²×#88059	1	1	yes	characteristic
206	C₂	#13123447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	Q₈×#108
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
36. #88059	Q₈	direct (N × H)
74. Q₈×C₁₃	#108	semidirect
169. C₁₃	Q₈×#108	semidirect
172. Q₈	#88059	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
#13115612	2
#13115008	2
C₄×#88059	2
C₂×C₈×Q₈×C₁₃	2
Q₈×#108	13

Subgroups — Q8×#88059 #15881812

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88059 #15881812