Finite Group Databasebeta

← C23×D3×#157 #13181846

Subgroups — C23×D3×#157 #13181846

Order 1728.

4488 conjugacy classes of subgroups; only the 882 normal subgroups are shown.

normal subgroups only

This group has 4488 conjugacy classes of subgroups — too many to draw. Only its 882 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×D3×C3⋊D3C211yesmaximal, characteristic
2C23×C3⋊#157C211yesmaximal, characteristic, complemented
3C23×C3×#157C211yesmaximal, characteristic, complemented
4C22×D3×#157C2128yesmaximal, direct product factor
5C24×#157—31nomaximal
6C23×C3⋊(C3⋊D3)C411yescharacteristic, complemented
7C23×C3×C3⋊D3C2211yescharacteristic
8C23×C32×D3C411yescharacteristic, complemented
9C22×D3×C3⋊D3C4114yescomplemented
10C22×D3×C3⋊D3C22114yes—
11C22×C3⋊#157C22114yescomplemented
12C22×C3×#157C22114yescomplemented
13C2×D3×#157C221112yesdirect product factor
14C24×C3⋊D3—31no—
15C23×D32—62no—
16C23×#157—31no—
17C23×#157—328no—
18C23×#157D311yescharacteristic, direct product factor
19C23×C33C2×C411yescharacteristic, complemented
20C22×C3⋊(C3⋊D3)C2×C4114yescomplemented
21C22×C3×C3⋊D3C2×C417yescomplemented
22C22×C3×C3⋊D3C2317yes—
23C22×C32×D3C2×C4114yescomplemented
24C2×D3×C3⋊D3C2×C4184yescomplemented
25C2×D3×C3⋊D3C23128yes—
26C2×C3⋊#157C23128yescomplemented
27C2×C3×#157C23128yescomplemented
28D3×#157C23164yesdirect product factor
29C23×C4×D3—91nomaximal
30C24×C32—31no—
31C23×C3⋊D3—31no—
32C23×C3⋊D3—314no—
33C23×C3⋊D3—314no—
34C23×C3⋊D3—62no—
35C23×C3⋊D3—122no—
36C23×C3⋊D3C2×D311yescharacteristic
37C23×C3×D3—22no—
38C23×C3×D3—62no—
39C22×D32—628no—
40C22×D32—628no—
41C22×#157—314no—
42C22×#157—3112no—
43C22×#157C2×D3114yesdirect product factor
44C22×C33C22×C417yescomplemented
45C2×C3⋊(C3⋊D3)C22×C4128yescomplemented
46C2×C3×C3⋊D3C22×C4121yescomplemented
47C2×C3×C3⋊D3C2417yes—
48C2×C32×D3C22×C4128yescomplemented
49D3×C3⋊D3C22×C4156yescomplemented
50D3×C3⋊D3C2418yes—
51C3⋊#157C2418yescomplemented
52C3×#157C2418yescomplemented
53C24×D3—91no—
54C24×D3—182no—
55C23×C3×C4—91no—
56C23×Dic3—91no—
57C22×C4×D3—928no—
58C23×C32—22no—
59C23×C32—314no—
60C23×C32—42no—
61C23×C32C4×D311yescharacteristic, complemented
62C22×C3⋊D3—314no—
63C22×C3⋊D3—384no—
64C22×C3⋊D3—328no—
65C22×C3⋊D3—628no—
66C22×C3⋊D3—1228no—
67C22×C3⋊D3C4×D317yescomplemented
68C22×C3⋊D3C22×D317yes—
69C22×C3×D3—228no—
70C22×C3×D3—614no—
71C22×C3×D3—614no—
72C2×D32—6168no—
73C2×D32—656no—
74C2×#157—328no—
75C2×#157—364no—
76C2×#157C22×D3128yesdirect product factor
77C24×C4—271noSylow
78C2×C33C23×C417yescomplemented
79C3⋊(C3⋊D3)C23×C418yescomplemented
80C3×C3⋊D3C23×C417yescomplemented
81C3×C3⋊D3C2511yescharacteristic
82C32×D3C23×C418yescomplemented
83C24×C3—62no—
84C24×C3—91no—
85C23×D3—62no—
86C23×D3—91no—
87C23×D3—914no—
88C23×D3—914no—
89C23×D3—1828no—
90C23×D3—1828no—
91C23×D3—182no—
92C23×D3—362no—
93C23×D3#15711yescharacteristic, direct product factor
94C22×C3×C4—914no—
95C22×Dic3—914no—
96C2×C4×D3—9112no—
97C22×C32—214no—
98C22×C32—328no—
99C22×C32—414no—
100C22×C32C2×C4×D317yescomplemented
101C2×C3⋊D3—328no—
102C2×C3⋊D3—356no—
103C2×C3⋊D3—38no—
104C2×C3⋊D3—656no—
105C2×C3⋊D3—1256no—
106C2×C3⋊D3C2×C4×D3121yescomplemented
107C2×C3⋊D3C23×D317yes—
108C2×C3×D3—256no—
109C2×C3×D3—642no—
110C2×C3×D3—614no—
111D32—6112no—
112D32—616no—
113#157—38no—
114#157C23×D318yesdirect product factor
115C25—271no—
116C23×C4—91no—
117C23×C4—2728no—
118C23×C4—271no—
119C33C24×C411yescharacteristic, complemented, Sylow
120C23×C3—22no—
121C23×C3—42no—
122C23×C3—628no—
123C23×C3—97no—
124C23×C3—97no—
125C23×C3C2×#15711yescharacteristic, complemented
126C22×D3—614no—
127C22×D3—614no—
128C22×D3—914no—
129C22×D3—928no—
130C22×D3—984no—
131C22×D3—1856no—
132C22×D3—18168no—
133C22×D3—1828no—
134C22×D3—3628no—
135C22×D3C2×#157114yesdirect product factor
136C2×C3×C4—928no—
137C2×Dic3—928no—
138C4×D3—964no—
139C2×C32—214no—
140C2×C32—38no—
141C2×C32—414no—
142C2×C32C22×C4×D317yescomplemented
143C3⋊D3—38no—
144C3⋊D3—616no—
145C3⋊D3—1216no—
146C3⋊D3C22×C4×D317yescomplemented
147C3⋊D3C24×D311yescharacteristic
148C3×D3—216no—
149C3×D3—614no—
150C3×D3—62no—
151C24—31no—
152C24—91no—
153C24—2714no—
154C24—2714no—
155C24—271no—
156C22×C4—914no—
157C22×C4—27112no—
158C22×C4—2714no—
159C22×C3—214no—
160C22×C3—414no—
161C22×C3—656no—
162C22×C3—97no—
163C22×C3—921no—
164C22×C3C22×#15717yescomplemented
165C2×D3—642no—
166C2×D3—614no—
167C2×D3—928no—
168C2×D3—98no—
169C2×D3—956no—
170C2×D3—1816no—
171C2×D3—18112no—
172C2×D3—1856no—
173C2×D3—3656no—
174C2×D3C22×#157128yesdirect product factor
175C3×C4—98no—
176Dic3—98no—
177C32—22no—
178C32—42no—
179C32C23×C4×D311yescharacteristic, complemented
180C23—314no—
181C23—97no—
182C23—97no—
183C23—2728no—
184C23—2784no—
185C23—2714no—
186C23D3×#15711yescharacteristic, direct product factor
187C2×C4—928no—
188C2×C4—2764no—
189C2×C4—2728no—
190C2×C3—214no—
191C2×C3—414no—
192C2×C3—616no—
193C2×C3—91no—
194C2×C3—97no—
195C2×C3C23×#15717yescomplemented
196D3—614no—
197D3—62no—
198D3—98no—
199D3—1816no—
200D3—3616no—
201D3C23×#15718yesdirect product factor
202C22—328no—
203C22—97no—
204C22—921no—
205C22—278no—
206C22—2756no—
207C22—2728no—
208C22C2×D3×#15717yesdirect product factor
209C4—98no—
210C4—278no—
211C3—22no—
212C3—42no—
213C3C24×#15711yescharacteristic, complemented
214C2—38no—
215C2—91no—
216C2—97no—
217C2—278no—
218C2C22×D3×#15717yesdirect product factor
subgroupG/H
center, hypercenterC23D3×#157
commutator, hypocenterC33C24×C4
FrattiniC1G
Fitting, socleC23×C33C2×C4
prime radicalC23×D3×C3⋊D3C2
cosocleC3×C3⋊D3C25
LCSG ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C23×C3⋊#157C2semidirect
3. C23×C3×#157C2semidirect
4. C22×D3×#157C2direct (N × H)
6. C23×C3⋊(C3⋊D3)C4semidirect
8. C23×C32×D3C4semidirect
9. C22×D3×C3⋊D3C4semidirect
11. C22×C3⋊#157C22semidirect
12. C22×C3×#157C22semidirect
13. C2×D3×#157C22direct (N × H)
18. C23×#157D3direct (N × H)
19. C23×C33C2×C4semidirect
20. C22×C3⋊(C3⋊D3)C2×C4semidirect
21. C22×C3×C3⋊D3C2×C4semidirect
23. C22×C32×D3C2×C4semidirect
24. C2×D3×C3⋊D3C2×C4semidirect
26. C2×C3⋊#157C23semidirect
27. C2×C3×#157C23semidirect
28. D3×#157C23direct (N × H)
43. C22×#157C2×D3direct (N × H)
44. C22×C33C22×C4semidirect
45. C2×C3⋊(C3⋊D3)C22×C4semidirect
46. C2×C3×C3⋊D3C22×C4semidirect
48. C2×C32×D3C22×C4semidirect
49. D3×C3⋊D3C22×C4semidirect
51. C3⋊#157C24semidirect
52. C3×#157C24semidirect
61. C23×C32C4×D3semidirect
67. C22×C3⋊D3C4×D3semidirect
76. C2×#157C22×D3direct (N × H)
78. C2×C33C23×C4semidirect
79. C3⋊(C3⋊D3)C23×C4semidirect
80. C3×C3⋊D3C23×C4semidirect
82. C32×D3C23×C4semidirect
93. C23×D3#157direct (N × H)
100. C22×C32C2×C4×D3semidirect
106. C2×C3⋊D3C2×C4×D3semidirect
114. #157C23×D3direct (N × H)
119. C33C24×C4semidirect
125. C23×C3C2×#157semidirect
135. C22×D3C2×#157direct (N × H)
142. C2×C32C22×C4×D3semidirect
146. C3⋊D3C22×C4×D3semidirect
164. C22×C3C22×#157semidirect
174. C2×D3C22×#157direct (N × H)
179. C32C23×C4×D3semidirect
186. C23D3×#157direct (N × H)
195. C2×C3C23×#157semidirect
201. D3C23×#157direct (N × H)
208. C22C2×D3×#157direct (N × H)
213. C3C24×#157semidirect
218. C2C22×D3×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3×C3⋊D32
C23×C3⋊#1572
C23×C3×#1572
C22×D3×#1572
C24×#1573
C23×C4×D39