Finite Group Databasebeta

← C23×#88111 #16808032

Subgroups — C23×#88111 #16808032

Order 3328.

7646 conjugacy classes of subgroups; only the 1976 normal subgroups are shown.

normal subgroups only

This group has 7646 conjugacy classes of subgroups — too many to draw. Only its 1976 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×#1057C212yesmaximal, complemented
2C22×#88111C2128yesmaximal, direct product factor
3C24×C4×D13C211yesmaximal, characteristic
4C25×D13C411yescharacteristic
5C24×C4×C13C411yescharacteristic
6C24×Dic13C2211yescharacteristic
7C23×C4×D13—216no—
8C23×C4×D13C4114yes—
9C23×C4×D13C22114yes—
10C23×#1057—216no—
11C23×#1057C22128yescomplemented
12C2×#88111C221112yesdirect product factor
13C25×C13C2×C411yescharacteristic
14C24×D13—28no—
15C24×D13—28no—
16C24×D13C8116yescomplemented
17C24×D13C2×C4114yes—
18C23×C4×C13—28no—
19C23×C4×C13C2×C4114yes—
20C23×Dic13C2×C417yes—
21C23×Dic13D4116yes—
22C23×Dic13C2317yes—
23C22×C4×D13—2112no—
24C22×C4×D13—2112no—
25C22×C4×D13C2×C4184yes—
26C22×C4×D13C23128yes—
27C22×#1057—2112no—
28C22×#1057C23156yescomplemented
29#88111C23164yesdirect product factor
30C23×C22⋊C8—131nomaximal, Sylow
31C24×C13C22⋊C418yes—
32C24×C13C2×C818yes—
33C24×C13M1618yes—
34C24×C13C22×C417yes—
35C23×D13—2128no—
36C23×D13—256no—
37C23×D13—256no—
38C23×D13C2×C81112yescomplemented
39C23×D13C22×C4128yes—
40C22×C4×C13—256no—
41C22×C4×C13C22×C4128yes—
42C22×Dic13C22⋊C4156yes—
43C22×Dic13C22×C4121yes—
44C22×Dic13C2×D4156yes—
45C22×Dic13C2417yes—
46C2×C4×D13—2336no—
47C2×C4×D13—2112no—
48C2×C4×D13C22×C4156yes—
49C2×C4×D13C2418yes—
50C2×#1057—2112no—
51C2×#1057C24116yescomplemented
52C25×C4—131no—
53C24×C8—132no—
54C22×C22⋊C8—1328no—
55C23×C13C22⋊C8164yescomplemented
56C23×C13C2×C22⋊C4128yes—
57C23×C13C22×C8128yes—
58C23×C13C2×M16128yes—
59C23×C13C23×C417yes—
60C22×D13—256no—
61C22×D13—256no—
62C22×D13—2448no—
63C22×D13C22×C81112yescomplemented
64C22×D13C23×C418yes—
65C2×C4×C13—256no—
66C2×C4×C13C23×C418yes—
67C2×Dic13C2×C22⋊C4184yes—
68C2×Dic13C23×C417yes—
69C2×Dic13C22×D4128yes—
70C2×Dic13C2511yescharacteristic
71C4×D13—2112no—
72C4×D13—216no—
73#1057—216no—
74C26—131no—
75C24×C4—131no—
76C24×C4—1314no—
77C24×C4—1314no—
78C24×C4—2616no—
79C24×C4#26111yescharacteristic
80C23×C8—1328no—
81C23×C8—2616no—
82C2×C22⋊C8—13112no—
83C22×C13C2×C22⋊C81112yescomplemented
84C22×C13C22×C22⋊C4114yes—
85C22×C13C23×C8114yes—
86C22×C13C22×M16114yes—
87C22×C13C24×C411yescharacteristic
88C2×D13—2224no—
89C2×D13—28no—
90C2×D13—28no—
91C2×D13C23×C8116yescomplemented
92C4×C13—28no—
93Dic13C22×C22⋊C4114yes—
94Dic13C23×D412yes—
95C25—1316no—
96C25—1314no—
97C25—268no—
98C25—268no—
99C25C2×#26111yescharacteristic
100C23×C4—28no—
101C23×C4—1316no—
102C23×C4—137no—
103C23×C4—137no—
104C23×C4—1328no—
105C23×C4—1384no—
106C23×C4—26112no—
107C23×C4—26112no—
108C23×C4C2×#261114yes—
109C22×C8—1356no—
110C22×C8—26112no—
111C22⋊C8—1364no—
112C2×C13C22×C22⋊C8128yescomplemented
113C2×C13C23×C22⋊C411yescharacteristic
114C2×C13C24×C811yescharacteristic
115C2×C13C23×M1611yescharacteristic
116D13—216no—
117C24—13112no—
118C24—1328no—
119C24—26128no—
120C24—2656no—
121C24—2656no—
122C24D13⋊C818yes—
123C24#611818yes—
124C24C22⋊#26118yes—
125C24C22×#26117yes—
126C22×C4—256no—
127C22×C4—1356no—
128C22×C4—1356no—
129C22×C4—137no—
130C22×C4—1321no—
131C22×C4—138no—
132C22×C4—1356no—
133C22×C4—26112no—
134C22×C4—26336no—
135C22×C4C22×#261128yes—
136C2×C8—1316no—
137C2×C8—26112no—
138C13C23×C22⋊C811yescharacteristic, complemented, Sylow
139C23—13112no—
140C23—138no—
141C23—2656no—
142C23—26448no—
143C23—2656no—
144C23#88111164yesdirect product factor
145C23C2×D13⋊C8128yes—
146C23C2×#6118128yes—
147C23C2×C22⋊#261128yes—
148C23C23×#26117yes—
149C2×C4—256no—
150C2×C4—1328no—
151C2×C4—1384no—
152C2×C4—131no—
153C2×C4—137no—
154C2×C4—26112no—
155C2×C4—2616no—
156C2×C4C23×#26118yes—
157C8—2616no—
158C22—1316no—
159C22—268no—
160C22—268no—
161C22—26224no—
162C22C2×#881111112yesdirect product factor
163C22C22×D13⋊C8114yes—
164C22C22×#6118114yes—
165C22C22×C22⋊#261114yes—
166C22C24×#26111yescharacteristic
167C4—28no—
168C4—1314no—
169C4—132no—
170C2—2616no—
171C2C22×#88111128yesdirect product factor
172C2C23×C22⋊#26111yescharacteristic
173C2C23×D13⋊C811yescharacteristic
174C2C23×#611811yescharacteristic
subgroupG/H
centerC25C2×#261
commutatorC2×C13C24×C8
FrattiniC22C24×#261
FittingC24×C4×C13C4
socleC25×C13C2×C4
prime radicalC25×D13C4
cosocleC2×Dic13C25
hypercenterC24×C4#261
hypocenterC13C23×C22⋊C8
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C25 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. C24×#1057C2semidirect
2. C22×#88111C2direct (N × H)
11. C23×#1057C22semidirect
12. C2×#88111C22direct (N × H)
16. C24×D13C8semidirect
28. C22×#1057C23semidirect
29. #88111C23direct (N × H)
38. C23×D13C2×C8semidirect
51. C2×#1057C24semidirect
55. C23×C13C22⋊C8semidirect
63. C22×D13C22×C8semidirect
83. C22×C13C2×C22⋊C8semidirect
91. C2×D13C23×C8semidirect
112. C2×C13C22×C22⋊C8semidirect
138. C13C23×C22⋊C8semidirect
144. C23#88111direct (N × H)
162. C22C2×#88111direct (N × H)
171. C2C22×#88111direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#10572
C22×#881112
C24×C4×D132
C23×C22⋊C813