Finite Group Databasebeta

← C22⋊C4×C9⋊A4 #13152105

Subgroups — C22⋊C4×C9⋊A4 #13152105

Order 1728.

575 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 110 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×C9⋊A4C211yesmaximal, characteristic
2C2×C4×C9⋊A4C212yesmaximal, complemented
3C3×A4×C22⋊C4C311yesmaximal, characteristic
4C22×C9×C22⋊C4C311yesmaximal, characteristic, complemented
5C22⋊C4×C22⋊C9C312yesmaximal, complemented
6C22×C9⋊A4—22no—
7C22×C9⋊A4C412yescomplemented
8C22×C9⋊A4C2211yescharacteristic
9C4×C9⋊A4—22no—
10C22⋊C4×3−1+2—41nomaximal
11C23×C3×A4C2×C311yescharacteristic
12C2×C3×C4×A4C2×C312yes—
13C25×C9C2×C311yescharacteristic
14C23×C22⋊C9C2×C312yes—
15C23×C4×C9C2×C312yescomplemented
16C2×C4×C22⋊C9C2×C314yescomplemented
17C2×C9×C22⋊C4—31no—
18C2×C9×C22⋊C4—31no—
19C2×C9×C22⋊C4—32no—
20C23×3−1+2—41no—
21C2×C9⋊A4—22no—
22C2×C9⋊A4C2×C411yescharacteristic
23C2×C9⋊A4D412yes—
24C2×C4×3−1+2—42no—
25C22×C3×C22⋊C4C3211yescharacteristic
26A4×C22⋊C4—31no—
27C22×C3×A4—22no—
28C22×C3×A4C3×C412yes—
29C22×C3×A4C22×C311yescharacteristic
30C3×C4×A4—22no—
31C24×C9—22no—
32C24×C9—31no—
33C24×C9—31no—
34C24×C9—32no—
35C24×C9—62no—
36C24×C9C3×C412yescomplemented
37C24×C9C22×C311yescharacteristic
38C22×C22⋊C9—24no—
39C22×C22⋊C9C3×C414yescomplemented
40C22×C22⋊C9C22×C312yes—
41C32×C22⋊C4—41no—
42C22×C4×C9—22no—
43C22×C4×C9—32no—
44C22×C4×C9—32no—
45C22×C4×C9—62no—
46C4×C22⋊C9—24no—
47C9×C22⋊C4—32no—
48C9×C22⋊C4—31no—
49C9×C22⋊C4—32no—
50C9×C22⋊C4—42no—
51C9×C22⋊C4A411yescharacteristic, complemented
52C22×3−1+2—41no—
53C22×3−1+2—42no—
54C22×3−1+2—82no—
55C9⋊A4C22⋊C411yescharacteristic, direct product factor
56C4×3−1+2—82no—
57C25×C3C2×C3211yescharacteristic
58C23×A4—31no—
59C23×C3×C4C2×C3212yes—
60C2×C4×A4—32no—
61C2×C3×C22⋊C4—31no—
62C2×C3×C22⋊C4—31no—
63C2×C3×C22⋊C4—32no—
64C23×C32—41no—
65C2×C3×A4—22no—
66C2×C3×A4C2×C3×C411yescharacteristic
67C2×C3×A4C3×D412yes—
68C2×C32×C4—42no—
69C23×C9—22no—
70C23×C9—32no—
71C23×C9—31no—
72C23×C9—31no—
73C23×C9—32no—
74C23×C9—32no—
75C23×C9—32no—
76C23×C9—32no—
77C23×C9—31no—
78C23×C9—42no—
79C23×C9—62no—
80C23×C9—62no—
81C23×C9—62no—
82C23×C9—64no—
83C23×C9—62no—
84C23×C9—62no—
85C23×C9—64no—
86C23×C9C2×C3×C411yescharacteristic
87C23×C9C3×D412yes—
88C23×C9C2×A411yescharacteristic
89C2×C22⋊C9—24no—
90C2×C22⋊C9C2×C3×C412yes—
91C2×C22⋊C9C3×D414yes—
92C2×C4×C9—32no—
93C2×C4×C9—44no—
94C2×C4×C9—62no—
95C2×C4×C9—62no—
96C2×C4×C9—62no—
97C2×C4×C9—62no—
98C2×C4×C9C2×A412yescomplemented
99C22×C22⋊C43−1+211yescharacteristic, complemented, Sylow
100C2×3−1+2—41no—
101C2×3−1+2—42no—
102C2×3−1+2—82no—
103C24×C3—22no—
104C24×C3—32no—
105C24×C3—31no—
106C24×C3—31no—
107C24×C3—62no—
108C24×C3C32×C412yes—
109C24×C3C22×C3211yescharacteristic
110C22×A4—32no—
111C22×A4—31no—
112C22×A4—62no—
113C22×C3×C4—22no—
114C22×C3×C4—32no—
115C22×C3×C4—32no—
116C22×C3×C4—62no—
117C4×A4—62no—
118C3×C22⋊C4—32no—
119C3×C22⋊C4—31no—
120C3×C22⋊C4—32no—
121C3×C22⋊C4—121no—
122C3×C22⋊C4C3×A411yescharacteristic
123C22×C32—41no—
124C22×C32—42no—
125C22×C32—82no—
126C3×A4C3×C22⋊C411yescharacteristic
127C32×C4—82no—
128C22×C9—22no—
129C22×C9—31no—
130C22×C9—32no—
131C22×C9—32no—
132C22×C9—31no—
133C22×C9—31no—
134C22×C9—32no—
135C22×C9—32no—
136C22×C9—32no—
137C22×C9—42no—
138C22×C9—44no—
139C22×C9—62no—
140C22×C9—64no—
141C22×C9—62no—
142C22×C9—62no—
143C22×C9—62no—
144C22×C9—64no—
145C22×C9—62no—
146C22×C9—84no—
147C22×C9C3×C22⋊C411yescharacteristic, complemented
148C22×C9C4×A412yescomplemented
149C22×C9C22×A411yescharacteristic
150C22⋊C9C3×C22⋊C412yescomplemented
151C4×C9—22no—
152C4×C9—62no—
153C4×C9—84no—
154C25C2×3−1+211yescharacteristic
155C23×C4C2×3−1+212yescomplemented
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—32no—
1593−1+2—41noSylow
160C23×C3—22no—
161C23×C3—32no—
162C23×C3—31no—
163C23×C3—31no—
164C23×C3—32no—
165C23×C3—32no—
166C23×C3—31no—
167C23×C3—32no—
168C23×C3—32no—
169C23×C3—62no—
170C23×C3—62no—
171C23×C3—62no—
172C23×C3—64no—
173C23×C3—62no—
174C23×C3—62no—
175C23×C3—64no—
176C23×C3—121no—
177C23×C3C2×C32×C411yescharacteristic
178C23×C3C32×D412yes—
179C23×C3C2×C3×A411yescharacteristic
180C2×A4—31no—
181C2×A4—32no—
182C2×A4—62no—
183C2×C3×C4—32no—
184C2×C3×C4—62no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—122no—
189C2×C3×C4C2×C3×A412yes—
190C2×C32—41no—
191C2×C32—42no—
192C2×C32—82no—
193C2×C9—22no—
194C2×C9—31no—
195C2×C9—32no—
196C2×C9—31no—
197C2×C9—42no—
198C2×C9—44no—
199C2×C9—62no—
200C2×C9—84no—
201C2×C9C2×C4×A411yescharacteristic
202C2×C9D4×A412yes—
203C24—22no—
204C24—31no—
205C24—32no—
206C24—31no—
207C24—62no—
208C24C4×3−1+212yescomplemented
209C24C22×3−1+211yescharacteristic
210C22×C4—22no—
211C22×C4—32no—
212C22×C4—32no—
213C22×C4—62no—
214C22⋊C4—32no—
215C22⋊C4—31no—
216C22⋊C4—32no—
217C22⋊C4C9⋊A411yescharacteristic, direct product factor
218C22×C3—22no—
219C22×C3—31no—
220C22×C3—32no—
221C22×C3—32no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—32no—
225C22×C3—32no—
226C22×C3—32no—
227C22×C3—62no—
228C22×C3—64no—
229C22×C3—62no—
230C22×C3—62no—
231C22×C3—62no—
232C22×C3—64no—
233C22×C3—62no—
234C22×C3—121no—
235C22×C3—122no—
236C22×C3—242no—
237C22×C3C32×C22⋊C411yescharacteristic
238C22×C3C3×C4×A412yes—
239C22×C3C22×C3×A411yescharacteristic
240A4—31no—
241C3×C4—22no—
242C3×C4—62no—
243C3×C4—242no—
244C32—41no—
245C9—42no—
246C9A4×C22⋊C411yescharacteristic, complemented
247C23—22no—
248C23—32no—
249C23—31no—
250C23—31no—
251C23—32no—
252C23—32no—
253C23—31no—
254C23—32no—
255C23—32no—
256C23—62no—
257C23—62no—
258C23—62no—
259C23—64no—
260C23—62no—
261C23—62no—
262C23—64no—
263C23C2×C4×3−1+211yescharacteristic
264C23D4×3−1+212yes—
265C23C2×C9⋊A411yescharacteristic
266C2×C4—32no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4C2×C9⋊A412yescomplemented
272C2×C3—22no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—32no—
276C2×C3—62no—
277C2×C3—121no—
278C2×C3—122no—
279C2×C3—242no—
280C2×C3C2×C3×C4×A411yescharacteristic
281C2×C3C3×D4×A412yes—
282C22—22no—
283C22—31no—
284C22—32no—
285C22—32no—
286C22—31no—
287C22—31no—
288C22—32no—
289C22—32no—
290C22—32no—
291C22—62no—
292C22—62no—
293C22—64no—
294C22—62no—
295C22—62no—
296C22—64no—
297C22—62no—
298C22C22⋊C4×3−1+211yescharacteristic, complemented
299C22C4×C9⋊A412yescomplemented
300C22C22×C9⋊A411yescharacteristic
301C4—22no—
302C4—62no—
303C3—121no—
304C3C3×A4×C22⋊C411yescharacteristic
305C2—22no—
306C2—31no—
307C2—31no—
308C2—32no—
309C2—62no—
310C2C2×C4×C9⋊A411yescharacteristic
311C2D4×C9⋊A412yes—
subgroupG/H
center, FrattiniC22×C3C22×C3×A4
commutatorC23×C3C2×C32×C4
FittingC22×C9×C22⋊C4C3
socle, cosocleC24×C3C22×C32
prime radicalC23×C3×A4C2×C3
hypercenterC9×C22⋊C4A4
hypocenterC22C22⋊C4×3−1+2
LCSG ▷ C23×C3 ▷ C22
UCSC1 ◁ C22×C3 ◁ C9×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C9⋊A4C2semidirect
4. C22×C9×C22⋊C4C3semidirect
5. C22⋊C4×C22⋊C9C3semidirect
7. C22×C9⋊A4C4semidirect
15. C23×C4×C9C2×C3semidirect
16. C2×C4×C22⋊C9C2×C3semidirect
36. C24×C9C3×C4semidirect
39. C22×C22⋊C9C3×C4semidirect
51. C9×C22⋊C4A4semidirect
55. C9⋊A4C22⋊C4direct (N × H)
98. C2×C4×C9C2×A4semidirect
99. C22×C22⋊C43−1+2semidirect
147. C22×C9C3×C22⋊C4semidirect
148. C22×C9C4×A4semidirect
150. C22⋊C9C3×C22⋊C4semidirect
155. C23×C4C2×3−1+2semidirect
208. C24C4×3−1+2semidirect
217. C22⋊C4C9⋊A4direct (N × H)
246. C9A4×C22⋊C4semidirect
271. C2×C4C2×C9⋊A4semidirect
298. C22C22⋊C4×3−1+2semidirect
299. C22C4×C9⋊A4semidirect

Maximal subgroups

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

subgroupindex
C23×C9⋊A42
C2×C4×C9⋊A42
C3×A4×C22⋊C43
C22×C9×C22⋊C43
C22⋊C4×C22⋊C93
C22⋊C4×3−1+24