Finite Group Databasebeta

← C22×C3×C4×C22⋊C9 #13164839

Subgroups — C22×C3×C4×C22⋊C9 #13164839

Order 1728.

1794 conjugacy classes of subgroups; only the 432 normal subgroups are shown.

normal subgroups only

This group has 1794 conjugacy classes of subgroups — too many to draw. Only its 432 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
2C2×C3×C4×C22⋊C9C216yesmaximal, direct product factor
3C24×C32×C4C311yesmaximal, characteristic
4C22×C4×C22⋊C9C313yesmaximal, direct product factor
5C22×C3×C22⋊C9C414yesdirect product factor
6C22×C3×C22⋊C9C2213yes—
7C22×C3×C4×C9—41nomaximal
8C3×C4×C22⋊C9C2214yesdirect product factor
9C25×C32C2×C311yescharacteristic
10C23×C32×C4—31no—
11C23×C32×C4—36no—
12C23×C32×C4—31no—
13C23×C32×C4C2×C316yes—
14C23×C22⋊C9C2×C313yes—
15C2×C4×C22⋊C9C2×C3118yesdirect product factor
16C23×C3×C9—41no—
17C2×C3×C22⋊C9C2×C416yesdirect product factor
18C2×C3×C22⋊C9C2311yescharacteristic
19C2×C3×C4×C9—46no—
20C24×C3×C4C913yescomplemented
21C24×C3×C4C3211yescharacteristic
22C24×C32—31no—
23C24×C32—33no—
24C24×C32—34no—
25C24×C32C3×C414yes—
26C24×C32C22×C313yes—
27C22×C32×C4—36no—
28C22×C32×C4—38no—
29C22×C32×C4—31no—
30C22×C32×C4—36no—
31C22×C32×C4—36no—
32C22×C32×C4—36no—
33C22×C32×C4—312no—
34C22×C32×C4A411yescharacteristic
35C22×C32×C4C22×C314yes—
36C22×C22⋊C9C3×C4112yesdirect product factor
37C22×C22⋊C9C22×C319yes—
38C22×C4×C9—43no—
39C4×C22⋊C9C22×C3112yesdirect product factor
40C22×C3×C9—44no—
41C22×C3×C9—43no—
42C3×C22⋊C9C22×C411yescharacteristic, direct product factor
43C3×C4×C9—44no—
44C25×C3C2×C913yes—
45C25×C3C2×C3211yescharacteristic
46C23×C3×C4—33no—
47C23×C3×C4—31no—
48C23×C3×C4—33no—
49C23×C3×C4—318no—
50C23×C3×C4—31no—
51C23×C3×C4—36no—
52C23×C3×C4C2×C9118yescomplemented
53C23×C3×C4C2×C3216yes—
54C23×C32—312no—
55C23×C32—36no—
56C23×C32—33no—
57C23×C32—312no—
58C23×C32—32no—
59C23×C32—34no—
60C23×C32—34no—
61C23×C32—33no—
62C23×C32—33no—
63C23×C32C2×C3×C416yes—
64C23×C32C2×A411yescharacteristic
65C23×C32C23×C311yescharacteristic
66C2×C32×C4—312no—
67C2×C32×C4—36no—
68C2×C32×C4—312no—
69C2×C32×C4—34no—
70C2×C32×C4—34no—
71C2×C32×C4C2×A416yes—
72C23×C9—43no—
73C2×C22⋊C9C2×C3×C4118yesdirect product factor
74C2×C22⋊C9C23×C313yes—
75C2×C4×C9—418no—
76C24×C4C3×C911yescharacteristic, complemented, Sylow
77C2×C3×C9—46no—
78C2×C3×C9—41no—
79C24×C3—312no—
80C24×C3—39no—
81C24×C3—34no—
82C24×C3—33no—
83C24×C3—33no—
84C24×C3—31no—
85C24×C3C4×C9112yescomplemented
86C24×C3C22×C919yes—
87C24×C3C32×C414yes—
88C24×C3C22×C3213yes—
89C22×C3×C4—36no—
90C22×C3×C4—318no—
91C22×C3×C4—38no—
92C22×C3×C4—324no—
93C22×C3×C4—33no—
94C22×C3×C4—31no—
95C22×C3×C4—318no—
96C22×C3×C4—318no—
97C22×C3×C4—36no—
98C22×C3×C4—36no—
99C22×C3×C4—318no—
100C22×C3×C4—336no—
101C22×C3×C4—36no—
102C22×C3×C4—312no—
103C22×C3×C4C22⋊C913yesdirect product factor
104C22×C3×C4C22×C9112yescomplemented
105C22×C3×C4C3×A411yescharacteristic
106C22×C3×C4C22×C3214yes—
107C22×C32—36no—
108C22×C32—31no—
109C22×C32—38no—
110C22×C32—36no—
111C22×C32—312no—
112C22×C32—36no—
113C22×C32—36no—
114C22×C32—33no—
115C22×C32—31no—
116C22×C32C4×A414yes—
117C22×C32C22×C3×C411yescharacteristic
118C22×C32C22×A413yes—
119C32×C4—34no—
120C32×C4C22×A414yes—
121C22×C9—49no—
122C22×C9—412no—
123C22⋊C9C22×C3×C413yesdirect product factor
124C4×C9—412no—
125C25C2×C3×C911yescharacteristic
126C23×C4—31no—
127C23×C4—31no—
128C23×C4—36no—
129C23×C4C2×C3×C916yescomplemented
130C3×C9—41noSylow
131C23×C3—39no—
132C23×C3—33no—
133C23×C3—336no—
134C23×C3—318no—
135C23×C3—312no—
136C23×C3—36no—
137C23×C3—32no—
138C23×C3—36no—
139C23×C3—312no—
140C23×C3—336no—
141C23×C3—33no—
142C23×C3—33no—
143C23×C3—312no—
144C23×C3—312no—
145C23×C3—39no—
146C23×C3—39no—
147C23×C3—34no—
148C23×C3—34no—
149C23×C3C2×C4×C9118yescomplemented
150C23×C3C2×C22⋊C913yes—
151C23×C3C23×C913yes—
152C23×C3C2×C32×C416yes—
153C23×C3C2×C3×A411yescharacteristic
154C23×C3C23×C3211yescharacteristic
155C2×C3×C4—312no—
156C2×C3×C4—336no—
157C2×C3×C4—36no—
158C2×C3×C4—318no—
159C2×C3×C4—312no—
160C2×C3×C4—34no—
161C2×C3×C4—336no—
162C2×C3×C4—312no—
163C2×C3×C4—312no—
164C2×C3×C4—34no—
165C2×C3×C4C2×C22⋊C9118yesdirect product factor
166C2×C3×C4C2×C3×A416yes—
167C2×C32—36no—
168C2×C32—31no—
169C2×C32—31no—
170C2×C32C2×C4×A416yes—
171C2×C32C23×A411yescharacteristic
172C2×C9—418no—
173C2×C9—43no—
174C24—33no—
175C24—34no—
176C24—31no—
177C24C3×C4×C914yescomplemented
178C24C22×C3×C913yes—
179C22×C4—36no—
180C22×C4—38no—
181C22×C4—31no—
182C22×C4—36no—
183C22×C4—36no—
184C22×C4—36no—
185C22×C4—312no—
186C22×C4C3×C22⋊C911yescharacteristic, direct product factor
187C22×C4C22×C3×C914yescomplemented
188C22×C3—318no—
189C22×C3—33no—
190C22×C3—36no—
191C22×C3—31no—
192C22×C3—318no—
193C22×C3—324no—
194C22×C3—36no—
195C22×C3—38no—
196C22×C3—36no—
197C22×C3—312no—
198C22×C3—36no—
199C22×C3—31no—
200C22×C3—33no—
201C22×C3—33no—
202C22×C3—39no—
203C22×C3—318no—
204C22×C3—336no—
205C22×C3—318no—
206C22×C3C4×C22⋊C9112yesdirect product factor
207C22×C3C22×C4×C913yescomplemented
208C22×C3C22×C22⋊C919yes—
209C22×C3C3×C4×A414yes—
210C22×C3C22×C32×C411yescharacteristic
211C22×C3C22×C3×A413yes—
212C3×C4—34no—
213C3×C4—312no—
214C3×C4C22×C22⋊C9112yesdirect product factor
215C3×C4C22×C3×A414yes—
216C32C22×C4×A411yescharacteristic
217C9—43no—
218C23—33no—
219C23—312no—
220C23—36no—
221C23—32no—
222C23—312no—
223C23—33no—
224C23—33no—
225C23—34no—
226C23—34no—
227C23C2×C3×C4×C916yescomplemented
228C23C2×C3×C22⋊C911yescharacteristic
229C23C23×C3×C911yescharacteristic
230C2×C4—312no—
231C2×C4—36no—
232C2×C4—34no—
233C2×C4—312no—
234C2×C4—34no—
235C2×C4C2×C3×C22⋊C916yesdirect product factor
236C2×C3—33no—
237C2×C3—318no—
238C2×C3—33no—
239C2×C3—31no—
240C2×C3—36no—
241C2×C3—31no—
242C2×C3C2×C4×C22⋊C9118yesdirect product factor
243C2×C3C23×C22⋊C913yes—
244C2×C3C2×C3×C4×A416yes—
245C2×C3C23×C3×A411yescharacteristic
246C22—36no—
247C22—31no—
248C22—36no—
249C22—38no—
250C22—31no—
251C22—33no—
252C22—36no—
253C22—312no—
254C22—36no—
255C22C3×C4×C22⋊C914yesdirect product factor
256C22C22×C3×C4×C911yescharacteristic, complemented
257C22C22×C3×C22⋊C913yes—
258C4—34no—
259C4C22×C3×C22⋊C914yesdirect product factor
260C3C22×C4×C22⋊C913yesdirect product factor
261C3C22×C3×C4×A411yescharacteristic
262C2—31no—
263C2—36no—
264C2—31no—
265C2C2×C3×C4×C22⋊C916yesdirect product factor
266C2C23×C3×C22⋊C911yescharacteristic
subgroupG/H
center, hypercenterC22×C32×C4A4
commutator, hypocenterC22C22×C3×C4×C9
FrattiniC2×C3C23×C3×A4
FittingC24×C32×C4C3
socle, prime radicalC25×C32C2×C3
cosocleC23×C3C23×C32
LCSG ▷ C22
UCSC1 ◁ C22×C32×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C22⋊C9C2direct (N × H)
4. C22×C4×C22⋊C9C3direct (N × H)
5. C22×C3×C22⋊C9C4direct (N × H)
8. C3×C4×C22⋊C9C22direct (N × H)
15. C2×C4×C22⋊C9C2×C3direct (N × H)
17. C2×C3×C22⋊C9C2×C4direct (N × H)
20. C24×C3×C4C9semidirect
36. C22×C22⋊C9C3×C4direct (N × H)
39. C4×C22⋊C9C22×C3direct (N × H)
42. C3×C22⋊C9C22×C4direct (N × H)
52. C23×C3×C4C2×C9semidirect
73. C2×C22⋊C9C2×C3×C4direct (N × H)
76. C24×C4C3×C9semidirect
85. C24×C3C4×C9semidirect
103. C22×C3×C4C22⋊C9direct (N × H)
104. C22×C3×C4C22×C9semidirect
123. C22⋊C9C22×C3×C4direct (N × H)
129. C23×C4C2×C3×C9semidirect
149. C23×C3C2×C4×C9semidirect
165. C2×C3×C4C2×C22⋊C9direct (N × H)
177. C24C3×C4×C9semidirect
186. C22×C4C3×C22⋊C9direct (N × H)
187. C22×C4C22×C3×C9semidirect
206. C22×C3C4×C22⋊C9direct (N × H)
207. C22×C3C22×C4×C9semidirect
214. C3×C4C22×C22⋊C9direct (N × H)
227. C23C2×C3×C4×C9semidirect
235. C2×C4C2×C3×C22⋊C9direct (N × H)
242. C2×C3C2×C4×C22⋊C9direct (N × H)
255. C22C3×C4×C22⋊C9direct (N × H)
256. C22C22×C3×C4×C9semidirect
259. C4C22×C3×C22⋊C9direct (N × H)
260. C3C22×C4×C22⋊C9direct (N × H)
265. C2C2×C3×C4×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
C2×C3×C4×C22⋊C92
C24×C32×C43
C22×C4×C22⋊C93
C22×C3×C4×C94