Subgroups — Q₈×#88038 #16016429

Order 3328.

530 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	C₈×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116744	C₂	1	3	yes	maximal, complemented
3	#13116741	C₂	1	3	yes	maximal, complemented
4	Q₈×#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116590	C₂	1	3	yes	maximal
6	C₄×#88038	C₂	1	3	yes	maximal
7	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₁₃	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	Q₈×#1055	C₄	1	1	yes	characteristic
11	C₂×#88038	C₂²	1	3	yes	—
12	C₂×#88038	C₂²	1	1	yes	characteristic
13	D₁₃×C₄⋊C₈	C₄	1	3	yes	—
14	#11718491	C₄	1	3	yes	—
15	C₄×C₈×D₁₃	C₂²	1	3	yes	—
16	C₈×Dic₂₆	C₂²	1	3	yes	—
17	C₁₃×C₄⋊C₁₆	C₂²	1	3	yes	—
18	C₄×C₁₃×C₁₆	C₂²	1	3	yes	—
19	#11718094	C₂²	1	3	yes	—
20	#11718094	C₂²	1	3	yes	—
21	#11718093	C₂²	1	6	yes	—
22	C₄⋊#6090	C₂²	1	3	yes	—
23	C₄×#6090	C₂²	1	3	yes	—
24	C₂×Q₈×D₁₃	C₈	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
27	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
28	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×D₁₃	C₈	1	3	yes	—
30	#88141	C₈	1	3	yes	—
31	C₄²×D₁₃	C₂×C₄	1	3	yes	—
32	C₄×Dic₂₆	C₂×C₄	1	3	yes	—
33	C₂×C₁₃×C₁₆	C₂³	1	3	yes	—
34	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
35	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₁₃	C₂³	1	3	yes	—
37	#88054	C₂×C₄	1	6	yes	—
38	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
39	C₈×Dic₁₃	C₂³	1	3	yes	—
40	C₂×#6090	C₂³	1	3	yes	—
41	C₂×#6090	C₂³	1	1	yes	characteristic
42	C₄⋊#1055	C₂×C₄	1	3	yes	—
43	C₄×#1055	C₂×C₄	1	3	yes	—
44	#88038	—	2	6	no	—
45	#88038	Q₈	1	4	yes	direct product factor
46	Q₈×M₃₂	—	13	1	no	maximal, Sylow
47	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
48	Q₈×D₁₃	—	2	4	no	—
49	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
50	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
51	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₂₆	C₂×C₈	1	3	yes	—
53	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
56	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
57	C₄⋊Dic₁₃	C₂×C₈	1	3	yes	—
58	#6101	C₂×C₈	1	6	yes	—
59	C₄×Dic₁₃	C₂×C₈	1	3	yes	—
60	C₄×Dic₁₃	C₂×C₈	1	3	yes	—
61	C₄×Dic₁₃	C₂²×C₄	1	3	yes	—
62	C₂×#1055	C₂²×C₄	1	3	yes	—
63	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
64	C₈×D₁₃	—	2	6	no	—
65	C₈×D₁₃	C₂×Q₈	1	2	yes	—
66	C₈×D₁₃	#41	1	2	yes	—
67	C₁₃×C₁₆	—	2	3	no	—
68	C₁₃×C₁₆	C₂×Q₈	1	2	yes	complemented
69	#6090	—	2	3	no	—
70	#6090	C₂×Q₈	1	2	yes	complemented
71	C₂×C₈×Q₈	—	13	1	no	—
72	#2183	—	13	3	no	—
73	#2183	—	13	3	no	—
74	Q₈×C₁₆	—	13	1	no	—
75	Q₈×C₁₆	D₁₃	1	1	yes	characteristic, complemented
76	#2150	—	13	3	no	—
77	C₄×M₃₂	—	13	3	no	—
78	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
79	Q₈×C₁₃	M₃₂	1	4	yes	complemented
80	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
81	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
82	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃	C₂²×C₈	1	3	yes	—
84	C₂×Dic₁₃	C₂²×C₈	1	3	yes	—
85	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
86	C₄×D₁₃	—	2	6	no	—
87	C₄×D₁₃	—	4	3	no	—
88	C₄×D₁₃	C₄×Q₈	1	2	yes	—
89	C₄×D₁₃	#131	1	2	yes	—
90	Dic₂₆	—	2	6	no	—
91	C₈×C₁₃	—	2	3	no	—
92	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
94	#1055	—	2	3	no	—
95	#1055	C₄×Q₈	1	2	yes	—
96	C₂×C₄×Q₈	—	13	1	no	—
97	C₂×M₃₂	—	13	1	no	—
98	C₂×M₃₂	—	13	3	no	—
99	C₈×Q₈	—	13	1	no	—
100	C₈×Q₈	—	13	3	no	—
101	C₈×Q₈	—	13	3	no	—
102	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	13	3	no	—
104	C₂×C₄×C₈	—	13	3	no	—
105	C₄⋊C₁₆	—	13	6	no	—
106	C₄⋊C₁₆	—	13	3	no	—
107	C₄⋊C₁₆	C₂×D₁₃	1	3	yes	—
108	#346	—	13	3	no	—
109	#346	—	13	3	no	—
110	C₄×C₁₆	—	13	3	no	—
111	C₄×C₁₆	C₂×D₁₃	1	3	yes	—
112	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
113	C₂×D₁₃	—	2	1	no	—
114	C₂×D₁₃	—	2	1	no	—
115	C₂×D₁₃	C₈×Q₈	1	2	yes	—
116	C₄×C₁₃	—	2	3	no	—
117	C₄×C₁₃	C₂×M₃₂	1	6	yes	—
118	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
120	Dic₁₃	—	2	3	no	—
121	Dic₁₃	—	2	3	no	—
122	Dic₁₃	C₈×Q₈	1	2	yes	—
123	C₂²×Q₈	—	13	1	no	—
124	C₂²×C₈	—	13	1	no	—
125	C₂²×C₈	—	13	3	no	—
126	C₄×Q₈	—	13	1	no	—
127	C₄×Q₈	—	13	3	no	—
128	C₄×Q₈	—	13	3	no	—
129	C₄×Q₈	C₄×D₁₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	13	3	no	—
131	C₂×C₄²	—	13	3	no	—
132	M₃₂	—	13	4	no	—
133	M₃₂	—	26	6	no	—
134	C₂×C₁₆	—	13	1	no	—
135	C₂×C₁₆	—	13	3	no	—
136	C₂×C₁₆	C₂²×D₁₃	1	3	yes	—
137	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
138	C₄⋊C₈	—	13	6	no	—
139	C₄⋊C₈	—	13	3	no	—
140	C₄⋊C₈	C₄×D₁₃	1	3	yes	—
141	C₄×C₈	—	13	3	no	—
142	C₄×C₈	—	13	3	no	—
143	C₄×C₈	—	13	3	no	—
144	C₄×C₈	C₂²×D₁₃	1	3	yes	—
145	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
146	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
147	C₂×C₁₃	#3407	1	1	yes	characteristic
148	D₁₃	—	2	2	no	—
149	C₂×Q₈	—	13	3	no	—
150	C₂×Q₈	—	26	4	no	—
151	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
152	C₂²×C₄	—	13	1	no	—
153	C₂²×C₄	—	13	3	no	—
154	C₂²×C₄	—	13	3	no	—
155	C₂×C₈	—	13	1	no	—
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	13	2	no	—
158	C₂×C₈	—	13	3	no	—
159	C₂×C₈	—	26	6	no	—
160	C₂×C₈	C₂×C₄×D₁₃	1	3	yes	—
161	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
162	C₄⋊C₄	—	13	6	no	—
163	C₄⋊C₄	—	13	3	no	—
164	C₄⋊C₄	C₈×D₁₃	1	3	yes	—
165	C₄²	—	13	3	no	—
166	C₄²	—	13	3	no	—
167	C₄²	—	13	3	no	—
168	C₄²	C₂×C₄×D₁₃	1	3	yes	—
169	C₁₆	—	2	3	no	—
170	C₁₆	—	13	2	no	—
171	C₁₆	—	26	3	no	—
172	C₁₆	Q₈×D₁₃	1	2	yes	complemented
173	C₁₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	13	1	no	—
175	Q₈	—	26	6	no	—
176	Q₈	#88038	1	4	yes	direct product factor
177	C₂×C₄	—	13	1	no	—
178	C₂×C₄	—	13	2	no	—
179	C₂×C₄	—	13	2	no	—
180	C₂×C₄	—	13	3	no	—
181	C₂×C₄	—	13	3	no	—
182	C₂×C₄	—	26	6	no	—
183	C₂×C₄	—	52	3	no	—
184	C₂×C₄	C₂×C₈×D₁₃	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₁₃	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	13	2	no	—
189	C₈	—	26	3	no	—
190	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
191	C₈	#41×D₁₃	1	1	yes	characteristic
192	C₂²	—	13	2	no	—
193	C₂²	—	26	1	no	—
194	C₂²	—	26	1	no	—
195	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	13	2	no	—
198	C₄	—	26	3	no	—
199	C₄	—	26	3	no	—
200	C₄	C₂×#88038	1	6	yes	—
201	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
202	C₄	D₁₃×#131	1	1	yes	characteristic
203	C₂	—	26	2	no	—
204	C₂	C₈×Q₈×D₁₃	1	1	yes	characteristic
205	C₂	C₂²×#88038	1	1	yes	characteristic
206	C₂	#13127937	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	Q₈×C₁₃×C₁₆	C₂
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	Q₈×C₁₆	D₁₃
hypocenter	C₁₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13116744	C₂	semidirect
3. #13116741	C₂	semidirect
4. Q₈×#6090	C₂	semidirect
7. Q₈×C₁₃×C₁₆	C₂	semidirect
45. #88038	Q₈	direct (N × H)
68. C₁₃×C₁₆	C₂×Q₈	semidirect
70. #6090	C₂×Q₈	semidirect
75. Q₈×C₁₆	D₁₃	semidirect
79. Q₈×C₁₃	M₃₂	semidirect
172. C₁₆	Q₈×D₁₃	semidirect
173. C₁₃	Q₈×M₃₂	semidirect
176. Q₈	#88038	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₁₃	2
#13116744	2
#13116741	2
Q₈×#6090	2
#13116590	2
C₄×#88038	2
Q₈×C₁₃×C₁₆	2
Q₈×M₃₂	13

Subgroups — Q8×#88038 #16016429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88038 #16016429