Finite Group Databasebeta

← C23×#88102 #16808145

Subgroups — C23×#88102 #16808145

Order 3328.

3614 conjugacy classes of subgroups; only the 1464 normal subgroups are shown.

normal subgroups only

This group has 3614 conjugacy classes of subgroups — too many to draw. Only its 1464 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
1C23×C4⋊#261C212yesmaximal
2C22×#88102C2128yesmaximal, direct product factor
3C23×C8×D13C211yesmaximal, characteristic
4C24×#261—22no—
5C23×C4×D13C2211yescharacteristic
6C22×C4⋊#261—214no—
7C22×C4⋊#261C22128yes—
8C23×C8×C13C411yescharacteristic, complemented
9C23×#1055C411yescharacteristic, complemented
10C22×C8×D13C4114yescomplemented
11C22×C8×D13C22114yes—
12C2×#88102C221112yesdirect product factor
13C24×D13D411yescharacteristic
14C23×#261—228no—
15C23×#261—416no—
16C23×C4×C13C2×C411yescharacteristic
17C23×Dic13Q811yescharacteristic
18C22×C4×D13C2×C417yes—
19C22×C4×D13D417yes—
20C22×C4×D13Q817yes—
21C22×C4×D13C2317yes—
22C2×C4⋊#261—284no—
23C2×C4⋊#261C23156yes—
24C22×C8×C13C2×C4114yescomplemented
25C22×#1055C2×C4114yescomplemented
26C2×C8×D13C2×C4184yescomplemented
27C2×C8×D13C23128yes—
28#88102C23164yesdirect product factor
29C23×#107—131nomaximal, Sylow
30C24×C13C4⋊C411yescharacteristic
31C23×D13C4⋊C417yes—
32C23×D13D818yes—
33C23×D13Q1618yes—
34C23×D13C2×D417yes—
35C22×#261—256no—
36C22×#261—4112no—
37C22×C4×C13C4⋊C417yes—
38C22×C4×C13C22×C417yes—
39C22×Dic13C4⋊C417yes—
40C22×Dic13C2×Q817yes—
41C2×C4×D13C4⋊C4142yes—
42C2×C4×D13C22×C4121yes—
43C2×C4×D13C2×D4121yes—
44C2×C4×D13C2×Q8121yes—
45C2×C4×D13C2417yes—
46C4⋊#261—256no—
47C4⋊#261C24116yes—
48C2×C8×C13C22×C4128yescomplemented
49C2×#1055C22×C4128yescomplemented
50C8×D13C22×C4156yescomplemented
51C8×D13C2418yes—
52C24×C8—131no—
53C23×C4⋊C4—132no—
54C22×#107—1328no—
55C23×C13#10718yescomplemented
56C23×C13C2×C4⋊C417yes—
57C22×D13#107156yescomplemented
58C22×D13C2×C4⋊C4121yes—
59C22×D13C2×D8128yes—
60C22×D13C2×Q16128yes—
61C22×D13C22×D417yes—
62C2×#261—216no—
63C2×#261—4112no—
64C2×C4×C13C2×C4⋊C4121yes—
65C2×C4×C13C23×C417yes—
66C2×Dic13C2×C4⋊C4121yes—
67C2×Dic13C22×Q817yes—
68C4×D13C2×C4⋊C4142yes—
69C4×D13C23×C417yes—
70C4×D13C22×D417yes—
71C4×D13C22×Q817yes—
72C4×D13C2511yescharacteristic
73C8×C13C23×C418yescomplemented
74#1055C23×C418yescomplemented
75C24×C4—131no—
76C24×C4—262no—
77C23×C8—131no—
78C23×C8—1314no—
79C23×C8—1314no—
80C23×C8#26111yescharacteristic, complemented
81C22×C4⋊C4—1328no—
82C22×C4⋊C4—2614no—
83C2×#107—13112no—
84C22×C13C2×#107128yescomplemented
85C22×C13C22×C4⋊C417yes—
86C2×D13C2×#107184yescomplemented
87C2×D13C22×C4⋊C417yes—
88C2×D13C22×D8114yes—
89C2×D13C22×Q16114yes—
90C2×D13C23×D411yescharacteristic
91#261—416no—
92C4×C13C22×C4⋊C417yes—
93C4×C13C24×C411yescharacteristic
94Dic13C22×C4⋊C417yes—
95Dic13C23×Q811yescharacteristic
96C25—131no—
97C23×C4—131no—
98C23×C4—137no—
99C23×C4—137no—
100C23×C4—137no—
101C23×C4—137no—
102C23×C4—2628no—
103C23×C4—5216no—
104C23×C4C2×#26111yescharacteristic
105C22×C8—1314no—
106C22×C8—1384no—
107C22×C8—1328no—
108C22×C8C2×#261114yescomplemented
109C2×C4⋊C4—1356no—
110C2×C4⋊C4—2684no—
111#107—1364no—
112C2×C13C22×#107114yescomplemented
113C2×C13C23×C4⋊C411yescharacteristic
114D13C22×#107114yescomplemented
115D13C23×D811yescharacteristic
116D13C23×Q1611yescharacteristic
117C24—138no—
118C24—138no—
119C24—137no—
120C24—137no—
121C24C4⋊#26111yescharacteristic
122C22×C4—137no—
123C22×C4—137no—
124C22×C4—1321no—
125C22×C4—1342no—
126C22×C4—1321no—
127C22×C4—137no—
128C22×C4—1321no—
129C22×C4—2656no—
130C22×C4—52112no—
131C22×C4C4⋊#26117yes—
132C22×C4C22×#26117yes—
133C2×C8—1328no—
134C2×C8—138no—
135C2×C8—1356no—
136C2×C8C22×#261128yescomplemented
137C4⋊C4—1316no—
138C4⋊C4—2656no—
139C13C23×#10711yescharacteristic, complemented, Sylow
140C23—1328no—
141C23—1356no—
142C23—1328no—
143C23—137no—
144C23—1321no—
145C23#8810218yesdirect product factor
146C23C2×C4⋊#26117yes—
147C2×C4—1321no—
148C2×C4—137no—
149C2×C4—137no—
150C2×C4—1342no—
151C2×C4—137no—
152C2×C4—131no—
153C2×C4—137no—
154C2×C4—2616no—
155C2×C4—52112no—
156C2×C4C2×C4⋊#261121yes—
157C2×C4C23×#26117yes—
158C8—138no—
159C8C23×#26118yescomplemented
160C22—137no—
161C22—131no—
162C22—1314no—
163C22—1314no—
164C22—1384no—
165C22C2×#88102128yesdirect product factor
166C22C22×C4⋊#26117yes—
167C4—137no—
168C4—131no—
169C4—5216no—
170C4C22×C4⋊#26117yes—
171C4C24×#26111yescharacteristic
172C2—131no—
173C2—1314no—
174C2—131no—
175C2C22×#88102114yesdirect product factor
176C2C23×C4⋊#26111yescharacteristic
subgroupG/H
centerC24C4⋊#261
commutatorC4×C13C24×C4
FrattiniC4C24×#261
FittingC23×C8×C13C4
socleC24×C13C4⋊C4
prime radicalC24×D13D4
cosocleC4×D13C25
hypercenterC23×C8#261
hypocenterC13C23×#107
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. C22×#88102C2direct (N × H)
8. C23×C8×C13C4semidirect
9. C23×#1055C4semidirect
10. C22×C8×D13C4semidirect
12. C2×#88102C22direct (N × H)
24. C22×C8×C13C2×C4semidirect
25. C22×#1055C2×C4semidirect
26. C2×C8×D13C2×C4semidirect
28. #88102C23direct (N × H)
48. C2×C8×C13C22×C4semidirect
49. C2×#1055C22×C4semidirect
50. C8×D13C22×C4semidirect
55. C23×C13#107semidirect
57. C22×D13#107semidirect
73. C8×C13C23×C4semidirect
74. #1055C23×C4semidirect
80. C23×C8#261semidirect
84. C22×C13C2×#107semidirect
86. C2×D13C2×#107semidirect
108. C22×C8C2×#261semidirect
112. C2×C13C22×#107semidirect
114. D13C22×#107semidirect
136. C2×C8C22×#261semidirect
139. C13C23×#107semidirect
145. C23#88102direct (N × H)
159. C8C23×#261semidirect
165. C22C2×#88102direct (N × H)
175. C2C22×#88102direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊#2612
C22×#881022
C23×C8×D132
C23×#10713