Finite Group Databasebeta

← C23×D3×C22⋊C9 #13180094

Subgroups — C23×D3×C22⋊C9 #13180094

Order 1728.

4570 conjugacy classes of subgroups; only the 415 normal subgroups are shown.

normal subgroups only

This group has 4570 conjugacy classes of subgroups — too many to draw. Only its 415 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×C3×C22⋊C9C211yesmaximal, characteristic, complemented
2C22×D3×C22⋊C9C2114yesmaximal, direct product factor
3C25×C3×D3C311yesmaximal, characteristic
4C24×C22⋊C9—31nomaximal
5C23×D3×C9—41nomaximal
6C22×C3×C22⋊C9C2217yescomplemented
7C2×D3×C22⋊C9C22128yesdirect product factor
8C25×C32C2×C311yescharacteristic
9C24×C3×D3—31no—
10C24×C3×D3—31no—
11C24×C3×D3—314no—
12C24×C3×D3C2×C3114yes—
13C23×C22⋊C9—21no—
14C23×C22⋊C9—314no—
15C23×C22⋊C9D311yescharacteristic, direct product factor
16C23×C3×C9—41no—
17C22×D3×C9—414no—
18C2×C3×C22⋊C9C2317yescomplemented
19D3×C22⋊C9C2318yesdirect product factor
20C26×C3—31no—
21C25×D3C911yescharacteristic, complemented
22C24×C32—31no—
23C24×C32—37no—
24C24×C32C22×C317yes—
25C23×C3×D3—314no—
26C23×C3×D3—356no—
27C23×C3×D3—31no—
28C23×C3×D3—314no—
29C23×C3×D3—314no—
30C23×C3×D3—314no—
31C23×C3×D3—384no—
32C23×C3×D3A411yescharacteristic
33C23×C3×D3C22×C3128yes—
34C24×C9—121no—
35C22×C22⋊C9—27no—
36C22×C22⋊C9—328no—
37C22×C22⋊C9C2×D317yesdirect product factor
38C22×C3×C9—47no—
39C2×D3×C9—428no—
40C3×C22⋊C9C2411yescharacteristic, complemented
41C25×C3—21no—
42C25×C3—314no—
43C25×C3—91no—
44C25×C3—914no—
45C25×C3—91no—
46C25×C3C2×C911yescharacteristic, complemented
47C25×C3C3×D311yescharacteristic
48C24×D3—31no—
49C24×D3—31no—
50C24×D3—314no—
51C24×D3C2×C9114yescomplemented
52C23×C32—321no—
53C23×C32—314no—
54C23×C32—37no—
55C23×C32—37no—
56C23×C32C2×A411yescharacteristic
57C23×C32C23×C317yes—
58C22×C3×D3—384no—
59C22×C3×D3—3112no—
60C22×C3×D3—314no—
61C22×C3×D3—384no—
62C22×C3×D3—328no—
63C22×C3×D3—328no—
64C22×C3×D3—356no—
65C22×C3×D3C2×A4114yes—
66C22×C3×D3C23×C318yes—
67C23×C9—41no—
68C23×C9—81no—
69C23×C9—1214no—
70C2×C22⋊C9—27no—
71C2×C22⋊C9—38no—
72C2×C22⋊C9C22×D317yesdirect product factor
73C26—31noSylow
74C2×C3×C9—47no—
75D3×C9—48no—
76C24×C3—27no—
77C24×C3—37no—
78C24×C3—37no—
79C24×C3—31no—
80C24×C3—31no—
81C24×C3—31no—
82C24×C3—328no—
83C24×C3—61no—
84C24×C3—67no—
85C24×C3—914no—
86C24×C3—914no—
87C24×C3—984no—
88C24×C3—914no—
89C24×C3—956no—
90C24×C3—914no—
91C24×C3—91no—
92C24×C3C22×C917yescomplemented
93C24×C3C2×C3×D317yes—
94C23×D3—314no—
95C23×D3—356no—
96C23×D3—31no—
97C23×D3—314no—
98C23×D3—314no—
99C23×D3—314no—
100C23×D3—384no—
101C23×D3C22⋊C911yescharacteristic, direct product factor
102C23×D3C22×C9128yescomplemented
103C22×C32—37no—
104C22×C32—314no—
105C22×C32—321no—
106C22×C32—37no—
107C22×C32C22×A417yes—
108C22×C32C24×C311yescharacteristic
109C2×C3×D3—356no—
110C2×C3×D3—328no—
111C2×C3×D3—356no—
112C2×C3×D3—38no—
113C2×C3×D3—38no—
114C2×C3×D3C22×A4128yes—
115C22×C9—47no—
116C22×C9—87no—
117C22×C9—1228no—
118C22⋊C9—21no—
119C22⋊C9C23×D311yescharacteristic, direct product factor
120C25—314no—
121C25—91no—
122C25—914no—
123C25—91no—
124C25D3×C911yescharacteristic, complemented
125C3×C9—41noSylow
126C23×C3—21no—
127C23×C3—27no—
128C23×C3—321no—
129C23×C3—321no—
130C23×C3—314no—
131C23×C3—314no—
132C23×C3—314no—
133C23×C3—37no—
134C23×C3—37no—
135C23×C3—37no—
136C23×C3—37no—
137C23×C3—38no—
138C23×C3—67no—
139C23×C3—67no—
140C23×C3—621no—
141C23×C3—614no—
142C23×C3—928no—
143C23×C3—984no—
144C23×C3—956no—
145C23×C3—928no—
146C23×C3—9112no—
147C23×C3—984no—
148C23×C3—914no—
149C23×C3C2×C22⋊C911yescharacteristic, complemented
150C23×C3C23×C917yescomplemented
151C23×C3D3×A411yescharacteristic
152C23×C3C22×C3×D317yes—
153C22×D3—384no—
154C22×D3—3112no—
155C22×D3—314no—
156C22×D3—384no—
157C22×D3—328no—
158C22×D3—328no—
159C22×D3—356no—
160C22×D3C2×C22⋊C9114yesdirect product factor
161C22×D3C23×C918yescomplemented
162C2×C32—37no—
163C2×C32—31no—
164C2×C32C23×A417yes—
165C3×D3—38no—
166C3×D3C23×A418yes—
167C2×C9—47no—
168C2×C9—87no—
169C2×C9—128no—
170C24—37no—
171C24—31no—
172C24—31no—
173C24—328no—
174C24—914no—
175C24—914no—
176C24—984no—
177C24—914no—
178C24—956no—
179C24—914no—
180C24—91no—
181C24C2×D3×C917yescomplemented
182C22×C3—27no—
183C22×C3—21no—
184C22×C3—37no—
185C22×C3—37no—
186C22×C3—314no—
187C22×C3—314no—
188C22×C3—37no—
189C22×C3—321no—
190C22×C3—328no—
191C22×C3—37no—
192C22×C3—321no—
193C22×C3—67no—
194C22×C3—621no—
195C22×C3—67no—
196C22×C3—614no—
197C22×C3—98no—
198C22×C3—956no—
199C22×C3—98no—
200C22×C3—956no—
201C22×C3—928no—
202C22×C3C22×C22⋊C917yescomplemented
203C22×C3C24×C911yescharacteristic, complemented
204C22×C3C2×D3×A417yes—
205C22×C3C23×C3×D311yescharacteristic
206C2×D3—356no—
207C2×D3—328no—
208C2×D3—38no—
209C2×D3—356no—
210C2×D3—38no—
211C2×D3C22×C22⋊C9128yesdirect product factor
212C32C24×A411yescharacteristic
213C9—41no—
214C9—81no—
215C23—321no—
216C23—314no—
217C23—37no—
218C23—37no—
219C23—314no—
220C23—38no—
221C23—928no—
222C23—984no—
223C23—956no—
224C23—928no—
225C23—9112no—
226C23—984no—
227C23—914no—
228C23D3×C22⋊C911yescharacteristic, direct product factor
229C23C22×D3×C917yescomplemented
230C2×C3—27no—
231C2×C3—31no—
232C2×C3—37no—
233C2×C3—31no—
234C2×C3—37no—
235C2×C3—38no—
236C2×C3—61no—
237C2×C3—67no—
238C2×C3—98no—
239C2×C3C23×C22⋊C917yescomplemented
240C2×C3C22×D3×A417yes—
241D3—38no—
242D3C23×C22⋊C918yesdirect product factor
243C22—37no—
244C22—314no—
245C22—37no—
246C22—321no—
247C22—328no—
248C22—98no—
249C22—956no—
250C22—98no—
251C22—956no—
252C22—928no—
253C22C2×D3×C22⋊C917yesdirect product factor
254C22C23×D3×C911yescharacteristic, complemented
255C3—21no—
256C3C24×C22⋊C911yescharacteristic, complemented
257C3C23×D3×A411yescharacteristic
258C2—31no—
259C2—37no—
260C2—38no—
261C2—98no—
262C2C22×D3×C22⋊C917yesdirect product factor
subgroupG/H
center, hypercenterC23×C3D3×A4
commutator, hypocenterC22×C3C24×C9
FrattiniC3C23×D3×A4
Fitting, socleC25×C32C2×C3
prime radicalC25×C3×D3C3
cosocleC22×C32C24×C3
LCSG ▷ C22×C3
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×C22⋊C9C2semidirect
2. C22×D3×C22⋊C9C2direct (N × H)
6. C22×C3×C22⋊C9C22semidirect
7. C2×D3×C22⋊C9C22direct (N × H)
15. C23×C22⋊C9D3direct (N × H)
18. C2×C3×C22⋊C9C23semidirect
19. D3×C22⋊C9C23direct (N × H)
21. C25×D3C9semidirect
37. C22×C22⋊C9C2×D3direct (N × H)
40. C3×C22⋊C9C24semidirect
46. C25×C3C2×C9semidirect
51. C24×D3C2×C9semidirect
72. C2×C22⋊C9C22×D3direct (N × H)
92. C24×C3C22×C9semidirect
101. C23×D3C22⋊C9direct (N × H)
102. C23×D3C22×C9semidirect
119. C22⋊C9C23×D3direct (N × H)
124. C25D3×C9semidirect
149. C23×C3C2×C22⋊C9semidirect
150. C23×C3C23×C9semidirect
160. C22×D3C2×C22⋊C9direct (N × H)
161. C22×D3C23×C9semidirect
181. C24C2×D3×C9semidirect
202. C22×C3C22×C22⋊C9semidirect
203. C22×C3C24×C9semidirect
211. C2×D3C22×C22⋊C9direct (N × H)
228. C23D3×C22⋊C9direct (N × H)
229. C23C22×D3×C9semidirect
239. C2×C3C23×C22⋊C9semidirect
242. D3C23×C22⋊C9direct (N × H)
253. C22C2×D3×C22⋊C9direct (N × H)
254. C22C23×D3×C9semidirect
256. C3C24×C22⋊C9semidirect
262. C2C22×D3×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C92
C22×D3×C22⋊C92
C25×C3×D33
C24×C22⋊C93
C23×D3×C94