Finite Group Databasebeta

← C2×(C22⋊C9)⋊(C22⋊C9) #72244473

Subgroups — C2×(C22⋊C9)⋊(C22⋊C9) #72244473

Order 2592.

540 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 74 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
1(C22⋊C9)⋊(C22⋊C9)C211yesmaximal, characteristic, direct product factor
2C2×C3×C22⋊(C22⋊C9)C312yesmaximal
3C23×C3×C22⋊C9C311yesmaximal, characteristic
4C23×C3×C22⋊C9C311yesmaximal, characteristic
5C2×C9⋊(C22⋊C9)—41nomaximal
6C2×(C22⋊C9)⋊C9—41nomaximal
7C3×C22⋊(C22⋊C9)C2×C312yes—
8C22×C3×C22⋊C9—31no—
9C22×C3×C22⋊C9—31no—
10C22×C3×C22⋊C9—31no—
11C22×C3×C22⋊C9—31no—
12C22×C3×C22⋊C9C2×C311yescharacteristic
13C22×C3×C22⋊C9C2×C311yescharacteristic
14C9⋊(C22⋊C9)—41no—
15(C22⋊C9)⋊C9—41no—
16C25×C32C3211yescharacteristic
17C2×C22⋊(C22⋊C9)—32no—
18C23×C22⋊C9—31no—
19C23×C22⋊C9C913yescomplemented
20C23×C3×C9—41no—
21C23×C3×C9—41no—
22C2×C3×C22⋊C9—31no—
23C2×C3×C22⋊C9—31no—
24C2×C3×C22⋊C9—31no—
25C2×C3×C22⋊C9—31no—
26C2×C3×C22⋊C9—42no—
27C2×C3×C22⋊C9—42no—
28C2×C3×C22⋊C9—122no—
29C2×C3×C22⋊C9A411yescharacteristic
30C2×C3×C22⋊C9A411yescharacteristic
31C2×C9⋊C9—161no—
32C24×C32—31no—
33C24×C32—31no—
34C24×C32—31no—
35C24×C32—31no—
36C24×C32—91no—
37C24×C32—91no—
38C24×C32C2×C3211yescharacteristic
39C22⋊(C22⋊C9)—32no—
40C22×C22⋊C9—31no—
41C22×C22⋊C9—31no—
42C22×C22⋊C9—31no—
43C22×C22⋊C9—31no—
44C22×C22⋊C9—31no—
45C22×C22⋊C9—31no—
46C22×C22⋊C9—31no—
47C22×C22⋊C9—33no—
48C22×C22⋊C9—33no—
49C22×C22⋊C9C2×C913yescomplemented
50C22×C3×C9—41no—
51C22×C3×C9—41no—
52C22×C3×C9—121no—
53C22×C3×C9—121no—
54C22×C3×C9—121no—
55C22×C3×C9—121no—
56C3×C22⋊C9—42no—
57C3×C22⋊C9—42no—
58C3×C22⋊C9—122no—
59C3×C22⋊C9C2×A411yescharacteristic
60C3×C22⋊C9C2×A411yescharacteristic
61C25×C3C3×C911yescharacteristic
62C25×C33−1+211yescharacteristic
63C25×C33−1+212yes—
64C9⋊C9—161noSylow
65C23×C32—32no—
66C23×C32—31no—
67C23×C32—31no—
68C23×C32—31no—
69C23×C32—31no—
70C23×C32—91no—
71C23×C32—91no—
72C23×C32—91no—
73C23×C32—91no—
74C23×C32—91no—
75C23×C32—91no—
76C23×C32—91no—
77C23×C32—91no—
78C23×C32—92no—
79C23×C32—91no—
80C23×C32—92no—
81C23×C32—92no—
82C23×C32C3×A411yescharacteristic
83C23×C32C3×A411yescharacteristic
84C23×C9—43no—
85C23×C9—121no—
86C2×C22⋊C9—31no—
87C2×C22⋊C9—31no—
88C2×C22⋊C9—31no—
89C2×C22⋊C9—31no—
90C2×C22⋊C9—31no—
91C2×C22⋊C9—31no—
92C2×C22⋊C9—33no—
93C2×C22⋊C9—33no—
94C2×C22⋊C9—31no—
95C2×C22⋊C9—122no—
96C2×C22⋊C9—124no—
97C2×C22⋊C9—122no—
98C2×C22⋊C9—122no—
99C2×C22⋊C9C22⋊C913yescomplemented
100C2×C3×C9—41no—
101C2×C3×C9—41no—
102C2×C3×C9—121no—
103C2×C3×C9—121no—
104C2×C3×C9—121no—
105C2×C3×C9—121no—
106C2×C3×C9—162no—
107C24×C3—31no—
108C24×C3—31no—
109C24×C3—31no—
110C24×C3—32no—
111C24×C3—32no—
112C24×C3—31no—
113C24×C3—31no—
114C24×C3—32no—
115C24×C3—32no—
116C24×C3—31no—
117C24×C3—31no—
118C24×C3—31no—
119C24×C3—91no—
120C24×C3—92no—
121C24×C3—91no—
122C24×C3—91no—
123C24×C3—91no—
124C24×C3—92no—
125C24×C3C2×C3×C911yescharacteristic
126C24×C3C2×3−1+211yescharacteristic
127C24×C3C2×3−1+212yes—
128C22×C32—31no—
129C22×C32—31no—
130C22×C32—31no—
131C22×C32—31no—
132C22×C32—32no—
133C22×C32—91no—
134C22×C32—91no—
135C22×C32—91no—
136C22×C32—91no—
137C22×C32—91no—
138C22×C32—91no—
139C22×C32—91no—
140C22×C32—91no—
141C22×C32—92no—
142C22×C32—92no—
143C22×C32—92no—
144C22×C32—91no—
145C22×C32C2×C3×A411yescharacteristic
146C22×C32C2×C3×A411yescharacteristic
147C22×C9—43no—
148C22×C9—123no—
149C22×C9—123no—
150C22×C9—121no—
151C22×C9—121no—
152C22×C9—121no—
153C22×C9—121no—
154C22×C9—121no—
155C22×C9—121no—
156C22×C9—121no—
157C22⋊C9—31no—
158C22⋊C9—122no—
159C22⋊C9—124no—
160C22⋊C9—122no—
161C22⋊C9—122no—
162C22⋊C9C2×C22⋊C913yescomplemented
163C25C9⋊C911yescharacteristic, complemented, Sylow
164C3×C9—41no—
165C3×C9—41no—
166C3×C9—162no—
167C23×C3—32no—
168C23×C3—32no—
169C23×C3—32no—
170C23×C3—32no—
171C23×C3—31no—
172C23×C3—31no—
173C23×C3—32no—
174C23×C3—32no—
175C23×C3—31no—
176C23×C3—32no—
177C23×C3—32no—
178C23×C3—31no—
179C23×C3—31no—
180C23×C3—31no—
181C23×C3—31no—
182C23×C3—31no—
183C23×C3—91no—
184C23×C3—92no—
185C23×C3—91no—
186C23×C3—91no—
187C23×C3—91no—
188C23×C3—92no—
189C23×C3—91no—
190C23×C3—91no—
191C23×C3—92no—
192C23×C3—92no—
193C23×C3—91no—
194C23×C3—91no—
195C23×C3—91no—
196C23×C3—91no—
197C23×C3—92no—
198C23×C3—92no—
199C23×C3—91no—
200C23×C3—91no—
201C23×C3—91no—
202C23×C3—91no—
203C23×C3—91no—
204C23×C3—91no—
205C23×C3—92no—
206C23×C3—92no—
207C23×C3—92no—
208C23×C3—92no—
209C23×C3—92no—
210C23×C3—92no—
211C23×C3—91no—
212C23×C3—92no—
213C23×C3—91no—
214C23×C3—92no—
215C23×C3—92no—
216C23×C3—92no—
217C23×C3—94no—
218C23×C3—94no—
219C23×C3—92no—
220C23×C3C9×A411yescharacteristic
221C23×C3C9⋊A412yes—
222C23×C3C9⋊A412yes—
223C23×C3C9⋊A411yescharacteristic
224C23×C3C3×C22⋊C911yescharacteristic
225C23×C3#109411yescharacteristic
226C2×C32—31no—
227C2×C32—31no—
228C2×C32—31no—
229C2×C32—31no—
230C2×C32—91no—
231C2×C32—91no—
232C2×C32A4211yescharacteristic
233C2×C9—43no—
234C2×C9—123no—
235C2×C9—123no—
236C2×C9—121no—
237C2×C9—121no—
238C2×C9—121no—
239C2×C9—121no—
240C2×C9—121no—
241C2×C9—121no—
242C2×C9—121no—
243C2×C9—482no—
244C24—31no—
245C24—31no—
246C24—31no—
247C24—31no—
248C24—91no—
249C24—91no—
250C24C2×C9⋊C911yescharacteristic, complemented
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—32no—
254C22×C3—32no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—32no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—32no—
265C22×C3—32no—
266C22×C3—32no—
267C22×C3—91no—
268C22×C3—91no—
269C22×C3—92no—
270C22×C3—91no—
271C22×C3—91no—
272C22×C3—91no—
273C22×C3—91no—
274C22×C3—92no—
275C22×C3—92no—
276C22×C3—91no—
277C22×C3—91no—
278C22×C3—92no—
279C22×C3—91no—
280C22×C3—91no—
281C22×C3—91no—
282C22×C3—91no—
283C22×C3—91no—
284C22×C3—91no—
285C22×C3—91no—
286C22×C3—91no—
287C22×C3—92no—
288C22×C3—92no—
289C22×C3—92no—
290C22×C3—92no—
291C22×C3—92no—
292C22×C3—92no—
293C22×C3—92no—
294C22×C3—92no—
295C22×C3—94no—
296C22×C3—94no—
297C22×C3—92no—
298C22×C3—92no—
299C22×C3—92no—
300C22×C3—92no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—92no—
304C22×C3C2×C9×A411yescharacteristic
305C22×C3C2×C9⋊A412yes—
306C22×C3C2×C9⋊A412yes—
307C22×C3C2×C9⋊A411yescharacteristic
308C22×C3C2×C3×C22⋊C911yescharacteristic
309C22×C3C2×#109411yescharacteristic
310C32C2×A4211yescharacteristic
311C9—43no—
312C9—121no—
313C9—482no—
314C23—32no—
315C23—31no—
316C23—31no—
317C23—31no—
318C23—31no—
319C23—91no—
320C23—91no—
321C23—91no—
322C23—91no—
323C23—91no—
324C23—91no—
325C23—91no—
326C23—91no—
327C23—92no—
328C23—91no—
329C23—92no—
330C23—92no—
331C23(C22⋊C9)⋊C911yescharacteristic, complemented
332C23C9⋊(C22⋊C9)11yescharacteristic, complemented
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—32no—
342C2×C3—32no—
343C2×C3—32no—
344C2×C3—32no—
345C2×C3—91no—
346C2×C3—91no—
347C2×C3—91no—
348C2×C3—91no—
349C2×C3—92no—
350C2×C3—92no—
351C2×C3A4×C22⋊C911yescharacteristic
352C2×C3#8888011yescharacteristic
353C2×C3#8888112yes—
354C22—31no—
355C22—31no—
356C22—31no—
357C22—31no—
358C22—32no—
359C22—91no—
360C22—91no—
361C22—91no—
362C22—91no—
363C22—91no—
364C22—91no—
365C22—91no—
366C22—91no—
367C22—92no—
368C22—92no—
369C22—92no—
370C22—91no—
371C22C2×(C22⋊C9)⋊C911yescharacteristic, complemented
372C22C2×C9⋊(C22⋊C9)11yescharacteristic, complemented
373C3C2×A4×C22⋊C911yescharacteristic
374C3C2×#8888011yescharacteristic
375C3C2×#8888112yes—
376C2—31no—
377C2—31no—
378C2—31no—
379C2—31no—
380C2—91no—
381C2—91no—
382C2(C22⋊C9)⋊(C22⋊C9)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C32A42
commutatorC24×C3C2×C3×C9
FrattiniC32C2×A42
Fitting, socle, prime radicalC25×C32C32
cosocleC24×C32C2×C32
hypocenterC24C2×C9⋊C9
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. (C22⋊C9)⋊(C22⋊C9)C2direct (N × H)
19. C23×C22⋊C9C9semidirect
49. C22×C22⋊C9C2×C9semidirect
99. C2×C22⋊C9C22⋊C9semidirect
162. C22⋊C9C2×C22⋊C9semidirect
163. C25C9⋊C9semidirect
250. C24C2×C9⋊C9semidirect
331. C23(C22⋊C9)⋊C9semidirect
332. C23C9⋊(C22⋊C9)semidirect
371. C22C2×(C22⋊C9)⋊C9semidirect
372. C22C2×C9⋊(C22⋊C9)semidirect
382. C2(C22⋊C9)⋊(C22⋊C9)direct (N × H)

Maximal subgroups

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

subgroupindex
(C22⋊C9)⋊(C22⋊C9)2
C2×C3×C22⋊(C22⋊C9)3
C23×C3×C22⋊C93
C2×C9⋊(C22⋊C9)4
C2×(C22⋊C9)⋊C94