Subgroups — Q₈×#88106 #16807265

Order 3328.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131251	C₂	1	6	yes	maximal, complemented
2	Q₈×#6092	C₂	1	2	yes	maximal, complemented
3	#13131168	C₂	1	3	yes	maximal
4	C₄×#88106	C₂	1	3	yes	maximal
5	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×#88106	C₂²	1	3	yes	—
8	C₂×#88106	C₂²	1	1	yes	characteristic
9	Q₈×#1055	C₄	1	3	yes	—
10	Q₈×#1055	C₄	1	1	yes	characteristic
11	Q₈×#1055	C₂²	1	1	yes	characteristic
12	Q₈×#1055	C₂²	1	3	yes	—
13	C₂×C₄⋊#1055	C₄	1	3	yes	—
14	C₂×C₄×#1055	C₂²	1	3	yes	—
15	#11718231	C₂²	1	6	yes	—
16	C₄⋊#6092	C₂²	1	6	yes	—
17	C₄⋊#6092	C₂²	1	6	yes	—
18	C₄×#6092	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₃	C₈	1	3	yes	—
21	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
22	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×C₄⋊C₄	C₈	1	3	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
26	C₂²×#1055	C₂×C₄	1	3	yes	—
27	C₂²×#1055	C₂³	1	1	yes	characteristic
28	#88106	—	2	6	no	—
29	#88106	Q₈	1	4	yes	direct product factor
30	C₂×#6092	C₂³	1	6	yes	—
31	C₂×#6092	C₂³	1	2	yes	—
32	C₄⋊#1055	C₂×C₄	1	3	yes	—
33	C₄⋊#1055	C₂×C₄	1	6	yes	—
34	C₄⋊#1055	C₂×C₄	1	3	yes	—
35	C₄×#1055	C₂×C₄	1	3	yes	—
36	C₄×#1055	C₂×C₄	1	3	yes	—
37	C₄×#1055	C₂³	1	3	yes	—
38	C₄×#1055	C₂³	1	3	yes	—
39	Q₈×M₃₂	—	13	1	no	maximal, Sylow
40	C₂×Q₈×C₁₃	—	2	4	no	—
41	C₂×Q₈×C₁₃	C₂×C₈	1	3	yes	—
42	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₃×C₄⋊C₄	C₂×C₈	1	6	yes	—
47	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
48	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
49	C₄²×C₁₃	C₂×C₈	1	3	yes	—
50	C₄²×C₁₃	C₂×C₈	1	3	yes	—
51	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
52	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
53	C₂×#1055	—	2	6	no	—
54	C₂×#1055	C₂²×C₄	1	3	yes	—
55	C₂×#1055	C₂²×C₄	1	3	yes	—
56	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
57	C₂×#1055	C₂×Q₈	1	2	yes	—
58	C₂×#1055	#41	1	2	yes	—
59	C₂×#1055	C₂⁴	1	1	yes	characteristic
60	#6092	—	2	6	no	—
61	#6092	C₂×Q₈	1	4	yes	complemented
62	C₂×C₈×Q₈	—	13	1	no	—
63	#2183	—	13	6	no	—
64	Q₈×C₁₆	—	13	2	no	—
65	#2150	—	13	3	no	—
66	C₄×M₃₂	—	13	3	no	—
67	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₁₃	—	2	6	no	—
69	Q₈×C₁₃	M₃₂	1	4	yes	complemented
70	C₂×C₄×C₁₃	—	2	6	no	—
71	C₂×C₄×C₁₃	—	4	3	no	—
72	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₁₃	#131	1	2	yes	—
79	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
80	#1055	—	2	3	no	—
81	#1055	—	2	3	no	—
82	#1055	C₄×Q₈	1	2	yes	—
83	#1055	C₂²×Q₈	1	1	yes	characteristic
84	#1055	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×Q₈	#261	1	1	yes	characteristic
86	C₂×M₃₂	—	13	1	no	—
87	C₂×M₃₂	—	13	3	no	—
88	C₈×Q₈	—	13	3	no	—
89	C₈×Q₈	—	13	1	no	—
90	C₈×Q₈	—	13	1	no	—
91	C₈×Q₈	—	13	3	no	—
92	C₂×C₄⋊C₈	—	13	3	no	—
93	C₂×C₄×C₈	—	13	3	no	—
94	C₄⋊C₁₆	—	13	6	no	—
95	C₄⋊C₁₆	—	13	6	no	—
96	#346	—	13	6	no	—
97	C₄×C₁₆	—	13	6	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
101	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
102	C₄×C₁₃	—	2	3	no	—
103	C₄×C₁₃	—	2	3	no	—
104	C₄×C₁₃	—	2	3	no	—
105	C₄×C₁₃	C₈×Q₈	1	2	yes	—
106	C₄×C₁₃	C₂×M₃₂	1	6	yes	—
107	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
109	C₂²×Q₈	#1057	1	1	yes	characteristic
110	C₂²×C₈	—	13	1	no	—
111	C₂²×C₈	—	13	3	no	—
112	C₄×Q₈	#1057	1	3	yes	—
113	C₄×Q₈	#1057	1	1	yes	characteristic
114	C₄×Q₈	C₂×#261	1	3	yes	—
115	C₄×Q₈	C₂×#261	1	1	yes	characteristic
116	C₂×C₄⋊C₄	#1057	1	3	yes	—
117	C₂×C₄²	C₂×#261	1	3	yes	—
118	M₃₂	—	13	4	no	—
119	M₃₂	—	26	6	no	—
120	C₂×C₁₆	—	13	2	no	—
121	C₂×C₁₆	—	13	6	no	—
122	C₄⋊C₈	—	13	3	no	—
123	C₄⋊C₈	—	13	6	no	—
124	C₄⋊C₈	—	13	3	no	—
125	C₄×C₈	—	13	3	no	—
126	C₄×C₈	—	13	3	no	—
127	C₄×C₈	—	13	3	no	—
128	C₄×C₈	—	13	3	no	—
129	C₂×C₁₃	—	2	2	no	—
130	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
131	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
132	C₂×C₁₃	#3407	1	1	yes	characteristic
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	C₂×#1057	1	3	yes	—
135	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
136	C₂²×C₄	C₂×#1057	1	3	yes	—
137	C₂²×C₄	C₂×#1057	1	3	yes	—
138	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
139	C₂×C₈	—	13	2	no	—
140	C₂×C₈	—	13	2	no	—
141	C₂×C₈	—	13	1	no	—
142	C₂×C₈	—	13	1	no	—
143	C₂×C₈	—	13	3	no	—
144	C₂×C₈	—	13	3	no	—
145	C₂×C₈	—	26	6	no	—
146	C₄⋊C₄	C₂×#1057	1	6	yes	—
147	C₄⋊C₄	C₂×#1057	1	3	yes	—
148	C₄⋊C₄	C₂×#1057	1	3	yes	—
149	C₄²	C₂×#1057	1	3	yes	—
150	C₄²	C₂×#1057	1	3	yes	—
151	C₄²	C₂²×#261	1	3	yes	—
152	C₄²	C₂²×#261	1	3	yes	—
153	C₁₆	—	13	4	no	—
154	C₁₆	—	26	6	no	—
155	C₁₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
156	C₂³	C₂²×#1057	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	#88106	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	#88240	1	2	yes	—
162	C₂×C₄	Q₈×#261	1	2	yes	—
163	C₂×C₄	C₂²×#1057	1	3	yes	—
164	C₂×C₄	C₂²×#1057	1	3	yes	—
165	C₂×C₄	C₂²×#1057	1	3	yes	—
166	C₂×C₄	C₂²×#1057	1	3	yes	—
167	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
168	C₂×C₄	C₂³×#261	1	1	yes	characteristic
169	C₈	—	13	1	no	—
170	C₈	—	13	1	no	—
171	C₈	—	13	2	no	—
172	C₈	—	26	3	no	—
173	C₈	—	26	3	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	Q₈×#1057	1	2	yes	—
177	C₂²	C₂³×#1057	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₂×#88106	1	6	yes	—
182	C₄	Q₈×#1057	1	2	yes	—
183	C₄	C₂×#88240	1	1	yes	characteristic
184	C₄	C₂×Q₈×#261	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×Q₈×#1057	1	1	yes	characteristic
187	C₂	C₂²×#88106	1	1	yes	characteristic
188	C₂	#13131803	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₁₃	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
2. Q₈×#6092	C₂	semidirect
29. #88106	Q₈	direct (N × H)
61. #6092	C₂×Q₈	semidirect
69. Q₈×C₁₃	M₃₂	semidirect
155. C₁₃	Q₈×M₃₂	semidirect
158. Q₈	#88106	direct (N × H)

Maximal subgroups

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

subgroup	index
#13131251	2
Q₈×#6092	2
#13131168	2
C₄×#88106	2
C₂×Q₈×#1055	2
Q₈×M₃₂	13

Subgroups — Q8×#88106 #16807265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88106 #16807265