Finite Group Databasebeta

← C23×#157⋊D3 #13181849

Subgroups — C23×#157⋊D3 #13181849

Order 1728.

4488 conjugacy classes of subgroups; only the 575 normal subgroups are shown.

normal subgroups only

This group has 4488 conjugacy classes of subgroups — too many to draw. Only its 575 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×#1113C212yesmaximal, complemented
2C23×C3×#157C211yesmaximal, characteristic, complemented
3C22×#157⋊D3C2128yesmaximal, direct product factor
4C23×GO+(4,𝔽2)—31nomaximal
5C23×C3×C3⋊D3—22no—
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×#1113—214no—
8C22×#1113C22128yescomplemented
9C22×C3×#157C22114yescomplemented
10C2×#157⋊D3C221112yesdirect product factor
11C23×D32—32no—
12C23×D32—62no—
13C23×#157D311yescharacteristic, complemented
14C22×GO+(4,𝔽2)—328no—
15C23×C33D411yescharacteristic, complemented
16C22×C3×C3⋊D3—228no—
17C22×C3×C3⋊D3D417yescomplemented
18C22×C3×C3⋊D3C2317yes—
19C2×#1113—284no—
20C2×#1113C23156yescomplemented
21C2×C3×#157C23128yescomplemented
22#157⋊D3C23164yesdirect product factor
23C23×D12—91nomaximal
24C23×C3⋊D3—22no—
25C23×C3⋊D3C2×D311yescharacteristic
26C23×C3×D3—62no—
27C23×C3×D3—62no—
28C23×C3×D3—62no—
29C23×C3×D3—122no—
30C22×D32—328no—
31C22×D32—628no—
32C22×D32—628no—
33C22×D32—614no—
34C22×#157C2×D3114yescomplemented
35C2×GO+(4,𝔽2)—3112no—
36C22×C33C2×D417yescomplemented
37C2×C3×C3⋊D3—256no—
38C2×C3×C3⋊D3C2×D4121yescomplemented
39C2×C3×C3⋊D3C2417yes—
40#1113—256no—
41#1113C24116yescomplemented
42C3×#157C2418yescomplemented
43C24×D3—92no—
44C24×D3—182no—
45C23×C3×C4—91no—
46C22×D12—928no—
47C23×C32—22no—
48C23×C32—42no—
49C23×C32D1211yescharacteristic, complemented
50C22×C3⋊D3—228no—
51C22×C3⋊D3D1217yescomplemented
52C22×C3⋊D3C22×D317yes—
53C22×C3×D3—614no—
54C22×C3×D3—614no—
55C22×C3×D3—628no—
56C22×C3×D3—628no—
57C22×C3×D3—1228no—
58C2×D32—356no—
59C2×D32—656no—
60C2×D32—6168no—
61C2×D32—684no—
62C2×#157C22×D3128yescomplemented
63GO+(4,𝔽2)—364no—
64C23×D4—271noSylow
65C2×C33C22×D417yescomplemented
66C3×C3⋊D3—216no—
67C3×C3⋊D3C22×D417yescomplemented
68C3×C3⋊D3C2511yescharacteristic
69C24×C3—91no—
70C24×C3—182no—
71C23×D3—62no—
72C23×D3—62no—
73C23×D3—62no—
74C23×D3—928no—
75C23×D3—122no—
76C23×D3—1828no—
77C23×D3—1814no—
78C23×D3—1828no—
79C22×C3×C4—914no—
80C2×D12—9112no—
81C22×C32—214no—
82C22×C32—414no—
83C22×C32C2×D1217yescomplemented
84C2×C3⋊D3—256no—
85C2×C3⋊D3C2×D12121yescomplemented
86C2×C3⋊D3C23×D317yes—
87C2×C3×D3—642no—
88C2×C3×D3—614no—
89C2×C3×D3—656no—
90C2×C3×D3—656no—
91C2×C3×D3—1256no—
92D32—316no—
93D32—616no—
94D32—6112no—
95D32—656no—
96#157C23×D318yescomplemented
97C25—272no—
98C22×D4—2728no—
99C23×C4—91no—
100C33C23×D411yescharacteristic, complemented, Sylow
101C23×C3—22no—
102C23×C3—42no—
103C23×C3—97no—
104C23×C3—97no—
105C23×C3—1828no—
106C23×C3GO+(4,𝔽2)11yescharacteristic, complemented
107C22×D3—614no—
108C22×D3—614no—
109C22×D3—628no—
110C22×D3—628no—
111C22×D3—956no—
112C22×D3—1228no—
113C22×D3—1856no—
114C22×D3—1884no—
115C22×D3—18168no—
116C2×C3×C4—928no—
117D12—964no—
118C2×C32—214no—
119C2×C32—414no—
120C2×C32C22×D1217yescomplemented
121C3⋊D3—216no—
122C3⋊D3C22×D1217yescomplemented
123C3⋊D3C24×D311yescharacteristic
124C3×D3—614no—
125C3×D3—62no—
126C3×D3—616no—
127C3×D3—616no—
128C3×D3—1216no—
129C24—91no—
130C24—182no—
131C24—2728no—
132C24—5414no—
133C2×D4—27112no—
134C22×C4—914no—
135C22×C3—214no—
136C22×C3—414no—
137C22×C3—97no—
138C22×C3—921no—
139C22×C3—1856no—
140C22×C3C2×GO+(4,𝔽2)17yescomplemented
141C2×D3—642no—
142C2×D3—614no—
143C2×D3—656no—
144C2×D3—656no—
145C2×D3—916no—
146C2×D3—1256no—
147C2×D3—1816no—
148C2×D3—1856no—
149C2×D3—18112no—
150C3×C4—98no—
151C32—22no—
152C32—42no—
153C32C23×D1211yescharacteristic, complemented
154C23—97no—
155C23—97no—
156C23—1828no—
157C23—2756no—
158C23—5484no—
159C23#157⋊D311yescharacteristic, direct product factor
160D4—2764no—
161C2×C4—928no—
162C2×C3—214no—
163C2×C3—414no—
164C2×C3—91no—
165C2×C3—97no—
166C2×C3—1816no—
167C2×C3C22×GO+(4,𝔽2)17yescomplemented
168D3—614no—
169D3—62no—
170D3—616no—
171D3—616no—
172D3—1216no—
173C22—97no—
174C22—921no—
175C22—1856no—
176C22—2716no—
177C22—5456no—
178C22C2×#157⋊D317yesdirect product factor
179C4—98no—
180C3—22no—
181C3—42no—
182C3C23×GO+(4,𝔽2)11yescharacteristic, complemented
183C2—91no—
184C2—97no—
185C2—1816no—
186C2C22×#157⋊D317yesdirect product factor
subgroupG/H
center, hypercenterC23#157⋊D3
commutator, cosocleC3×C3⋊D3C25
FrattiniC1G
Fitting, socleC23×C33D4
hypocenterC33C23×D4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×#1113C2semidirect
2. C23×C3×#157C2semidirect
3. C22×#157⋊D3C2direct (N × H)
8. C22×#1113C22semidirect
9. C22×C3×#157C22semidirect
10. C2×#157⋊D3C22direct (N × H)
13. C23×#157D3semidirect
15. C23×C33D4semidirect
17. C22×C3×C3⋊D3D4semidirect
20. C2×#1113C23semidirect
21. C2×C3×#157C23semidirect
22. #157⋊D3C23direct (N × H)
34. C22×#157C2×D3semidirect
36. C22×C33C2×D4semidirect
38. C2×C3×C3⋊D3C2×D4semidirect
41. #1113C24semidirect
42. C3×#157C24semidirect
49. C23×C32D12semidirect
51. C22×C3⋊D3D12semidirect
62. C2×#157C22×D3semidirect
65. C2×C33C22×D4semidirect
67. C3×C3⋊D3C22×D4semidirect
83. C22×C32C2×D12semidirect
85. C2×C3⋊D3C2×D12semidirect
96. #157C23×D3semidirect
100. C33C23×D4semidirect
106. C23×C3GO+(4,𝔽2)semidirect
120. C2×C32C22×D12semidirect
122. C3⋊D3C22×D12semidirect
140. C22×C3C2×GO+(4,𝔽2)semidirect
153. C32C23×D12semidirect
159. C23#157⋊D3direct (N × H)
167. C2×C3C22×GO+(4,𝔽2)semidirect
178. C22C2×#157⋊D3direct (N × H)
182. C3C23×GO+(4,𝔽2)semidirect
186. C2C22×#157⋊D3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#11132
C23×C3×#1572
C22×#157⋊D32
C23×GO+(4,𝔽2)3
C23×D129