Finite Group Databasebeta

← C22⋊C9×#217 #13147304

Subgroups — C22⋊C9×#217 #13147304

Order 1728.

786 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 95 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×Dic3×C22⋊C9C212yesmaximal, complemented
3C22×C3×#217C311yesmaximal, characteristic
4C22⋊C4×C22⋊C9—31nomaximal
5C22×C3×C22⋊C9—22no—
6C22×C3×C22⋊C9C412yescomplemented
7C22×C3×C22⋊C9C2211yescharacteristic
8Dic3×C22⋊C9—22no—
9C9×#217—41nomaximal
10C25×C32C2×C311yescharacteristic
11C23×C3×Dic3C2×C312yes—
12C23×C22⋊C9—21no—
13C23×C22⋊C9D311yescharacteristic
14C2×C3×#217—31no—
15C2×C3×#217—31no—
16C2×C3×#217—32no—
17C2×C4×C22⋊C9—32no—
18C23×C3×C9—41no—
19C2×C3×C22⋊C9—22no—
20C2×C3×C22⋊C9C2×C411yescharacteristic
21C2×C3×C22⋊C9D412yes—
22C2×C9×Dic3—42no—
23C22×C3×C22⋊C4—31no—
24C22×#217C911yescharacteristic, complemented
25C24×C32—22no—
26C24×C32—31no—
27C24×C32—31no—
28C24×C32—32no—
29C24×C32—62no—
30C24×C32C3×C412yes—
31C24×C32C22×C311yescharacteristic
32C22×C3×Dic3—22no—
33C22×C3×Dic3—32no—
34C22×C3×Dic3—32no—
35C22×C3×Dic3—62no—
36C22×C22⋊C9—24no—
37C22×C22⋊C9—22no—
38C22×C22⋊C9—21no—
39C22×C22⋊C9—22no—
40C22×C22⋊C9Dic312yescomplemented
41C22×C22⋊C9C2×D311yescharacteristic
42C3×#217—32no—
43C3×#217—31no—
44C3×#217—32no—
45C3×#217A411yescharacteristic
46C4×C22⋊C9—62no—
47C9×C22⋊C4—121no—
48C22×C3×C9—41no—
49C22×C3×C9—42no—
50C22×C3×C9—82no—
51C3×C22⋊C9C22⋊C411yescharacteristic, complemented
52C9×Dic3—82no—
53C25×C3—21no—
54C25×C3C2×C911yescharacteristic
55C25×C3C3×D311yescharacteristic
56C23×C3×C4—32no—
57C23×Dic3C2×C912yescomplemented
58C2×C3×C22⋊C4—91no—
59C2×C3×C22⋊C4—92no—
60C2×C3×C22⋊C4—91no—
61C2×#217—31no—
62C2×#217—31no—
63C2×#217—32no—
64C23×C32—22no—
65C23×C32—31no—
66C23×C32—32no—
67C23×C32—31no—
68C23×C32—32no—
69C23×C32—32no—
70C23×C32—32no—
71C23×C32—32no—
72C23×C32—31no—
73C23×C32—62no—
74C23×C32—62no—
75C23×C32—62no—
76C23×C32—64no—
77C23×C32—62no—
78C23×C32—62no—
79C23×C32—64no—
80C23×C32C2×C3×C411yescharacteristic
81C23×C32C3×D412yes—
82C23×C32C2×A411yescharacteristic
83C2×C3×Dic3—32no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3C2×A412yes—
89C23×C9—41no—
90C23×C9—81no—
91C2×C22⋊C9—24no—
92C2×C22⋊C9—22no—
93C2×C22⋊C9—21no—
94C2×C22⋊C9—22no—
95C2×C22⋊C9C2×Dic311yescharacteristic
96C2×C22⋊C9#6712yes—
97C2×C4×C9—122no—
98C22×C22⋊C4—31noSylow
99C2×C3×C9—41no—
100C2×C3×C9—42no—
101C2×C3×C9—82no—
102C24×C3—24no—
103C24×C3—22no—
104C24×C3—22no—
105C24×C3—22no—
106C24×C3—21no—
107C24×C3—32no—
108C24×C3—31no—
109C24×C3—32no—
110C24×C3—31no—
111C24×C3—31no—
112C24×C3—31no—
113C24×C3—61no—
114C24×C3—61no—
115C24×C3—62no—
116C24×C3—62no—
117C24×C3—62no—
118C24×C3—64no—
119C24×C3C4×C912yescomplemented
120C24×C3C22×C911yescharacteristic
121C24×C3C3×Dic312yes—
122C24×C3C2×C3×D311yescharacteristic
123C22×C3×C4—62no—
124C22×C3×C4—92no—
125C22×C3×C4—92no—
126C22×C3×C4—182no—
127C22×Dic3—22no—
128C22×Dic3—32no—
129C22×Dic3—32no—
130C22×Dic3—62no—
131C3×C22⋊C4—31no—
132C3×C22⋊C4—92no—
133C3×C22⋊C4—92no—
134C3×C22⋊C4—91no—
135#217—32no—
136#217—31no—
137#217—32no—
138#217C22⋊C911yescharacteristic, direct product factor
139C22×C32—22no—
140C22×C32—31no—
141C22×C32—32no—
142C22×C32—32no—
143C22×C32—31no—
144C22×C32—31no—
145C22×C32—32no—
146C22×C32—32no—
147C22×C32—32no—
148C22×C32—62no—
149C22×C32—64no—
150C22×C32—62no—
151C22×C32—62no—
152C22×C32—62no—
153C22×C32—64no—
154C22×C32—62no—
155C22×C32C3×C22⋊C411yescharacteristic
156C22×C32C4×A412yes—
157C22×C32C22×A411yescharacteristic
158C3×Dic3—22no—
159C3×Dic3—62no—
160C22×C9—41no—
161C22×C9—42no—
162C22×C9—82no—
163C22×C9—84no—
164C22×C9—81no—
165C22×C9—82no—
166C22⋊C9—21no—
167C22⋊C9#21711yescharacteristic, direct product factor
168C4×C9—242no—
169C25D3×C911yescharacteristic
170C23×C4—32no—
171C2×C22⋊C4—91no—
172C2×C22⋊C4—92no—
173C2×C22⋊C4—91no—
174C3×C9—41noSylow
175C23×C3—24no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—21no—
179C23×C3—22no—
180C23×C3—21no—
181C23×C3—32no—
182C23×C3—32no—
183C23×C3—31no—
184C23×C3—31no—
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—32no—
188C23×C3—32no—
189C23×C3—32no—
190C23×C3—32no—
191C23×C3—31no—
192C23×C3—32no—
193C23×C3—32no—
194C23×C3—31no—
195C23×C3—32no—
196C23×C3—32no—
197C23×C3—61no—
198C23×C3—62no—
199C23×C3—62no—
200C23×C3—61no—
201C23×C3—62no—
202C23×C3—61no—
203C23×C3—62no—
204C23×C3—62no—
205C23×C3—62no—
206C23×C3—62no—
207C23×C3—62no—
208C23×C3—62no—
209C23×C3—64no—
210C23×C3—64no—
211C23×C3—62no—
212C23×C3—62no—
213C23×C3—64no—
214C23×C3—64no—
215C23×C3—62no—
216C23×C3—62no—
217C23×C3—62no—
218C23×C3—62no—
219C23×C3—64no—
220C23×C3—64no—
221C23×C3—64no—
222C23×C3—64no—
223C23×C3—68no—
224C23×C3—64no—
225C23×C3—68no—
226C23×C3C2×C4×C911yescharacteristic
227C23×C3D4×C912yes—
228C23×C3C2×C22⋊C911yescharacteristic
229C23×C3C2×C3×Dic311yescharacteristic
230C23×C3C3×#6712yes—
231C23×C3D3×A411yescharacteristic
232C2×C3×C4—32no—
233C2×C3×C4—92no—
234C2×C3×C4—182no—
235C2×C3×C4—182no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×Dic3—32no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3C2×C22⋊C912yescomplemented
244C2×C32—22no—
245C2×C32—31no—
246C2×C32—32no—
247C2×C32—31no—
248C2×C32—62no—
249C2×C32C2×C4×A411yescharacteristic
250C2×C32D4×A412yes—
251C2×C9—41no—
252C2×C9—42no—
253C2×C9—82no—
254C2×C9—84no—
255C2×C9—81no—
256C2×C9—82no—
257C24—22no—
258C24—31no—
259C24—32no—
260C24—31no—
261C24—62no—
262C24C9×Dic312yescomplemented
263C24C2×D3×C911yescharacteristic
264C22×C4—62no—
265C22×C4—92no—
266C22×C4—92no—
267C22×C4—182no—
268C22⋊C4—31no—
269C22⋊C4—92no—
270C22⋊C4—92no—
271C22⋊C4—91no—
272C22×C3—24no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—31no—
279C22×C3—32no—
280C22×C3—31no—
281C22×C3—32no—
282C22×C3—32no—
283C22×C3—32no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—32no—
287C22×C3—32no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—32no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—32no—
294C22×C3—61no—
295C22×C3—62no—
296C22×C3—61no—
297C22×C3—62no—
298C22×C3—62no—
299C22×C3—61no—
300C22×C3—62no—
301C22×C3—62no—
302C22×C3—62no—
303C22×C3—62no—
304C22×C3—64no—
305C22×C3—62no—
306C22×C3—64no—
307C22×C3—62no—
308C22×C3—64no—
309C22×C3—64no—
310C22×C3—68no—
311C22×C3—64no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—62no—
315C22×C3—62no—
316C22×C3—64no—
317C22×C3—64no—
318C22×C3—64no—
319C22×C3—64no—
320C22×C3—68no—
321C22×C3—62no—
322C22×C3—62no—
323C22×C3C9×C22⋊C411yescharacteristic, complemented
324C22×C3C4×C22⋊C912yescomplemented
325C22×C3C3×#21711yescharacteristic
326C22×C3C22×C22⋊C911yescharacteristic
327C22×C3Dic3×A412yes—
328C22×C3C2×D3×A411yescharacteristic
329C3×C4—62no—
330C3×C4—182no—
331Dic3—22no—
332Dic3—62no—
333C32A4×C22⋊C411yescharacteristic
334C9—41no—
335C9—81no—
336C23—22no—
337C23—32no—
338C23—31no—
339C23—31no—
340C23—32no—
341C23—32no—
342C23—32no—
343C23—32no—
344C23—31no—
345C23—62no—
346C23—62no—
347C23—62no—
348C23—64no—
349C23—62no—
350C23—62no—
351C23—64no—
352C23C2×C9×Dic311yescharacteristic
353C23C9×#6712yes—
354C23D3×C22⋊C911yescharacteristic
355C2×C4—32no—
356C2×C4—92no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4—182no—
361C2×C3—24no—
362C2×C3—22no—
363C2×C3—22no—
364C2×C3—21no—
365C2×C3—22no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—32no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—32no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—62no—
375C2×C3—64no—
376C2×C3—62no—
377C2×C3—62no—
378C2×C3C2×C4×C22⋊C911yescharacteristic
379C2×C3D4×C22⋊C912yes—
380C2×C3C2×Dic3×A411yescharacteristic
381C2×C3A4×#6712yes—
382C22—22no—
383C22—31no—
384C22—32no—
385C22—32no—
386C22—31no—
387C22—31no—
388C22—32no—
389C22—32no—
390C22—32no—
391C22—62no—
392C22—62no—
393C22—64no—
394C22—62no—
395C22—62no—
396C22—64no—
397C22—62no—
398C22C9×#21711yescharacteristic, complemented
399C22Dic3×C22⋊C912yescomplemented
400C22C2×D3×C22⋊C911yescharacteristic
401C4—62no—
402C4—182no—
403C3—21no—
404C3C22⋊C4×C22⋊C911yescharacteristic, complemented
405C3A4×#21711yescharacteristic
406C2—22no—
407C2—31no—
408C2—31no—
409C2—32no—
410C2—62no—
411C2C2×Dic3×C22⋊C911yescharacteristic
412C2#67×C22⋊C912yes—
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutatorC23×C3C2×C4×C9
Fitting, prime radicalC25×C32C2×C3
socle, cosocleC24×C32C22×C3
hypercenterC23×C3D3×A4
hypocenterC22×C3C9×C22⋊C4
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×C22⋊C9C2semidirect
6. C22×C3×C22⋊C9C4semidirect
24. C22×#217C9semidirect
40. C22×C22⋊C9Dic3semidirect
51. C3×C22⋊C9C22⋊C4semidirect
57. C23×Dic3C2×C9semidirect
119. C24×C3C4×C9semidirect
138. #217C22⋊C9direct (N × H)
167. C22⋊C9#217direct (N × H)
243. C2×Dic3C2×C22⋊C9semidirect
262. C24C9×Dic3semidirect
323. C22×C3C9×C22⋊C4semidirect
324. C22×C3C4×C22⋊C9semidirect
398. C22C9×#217semidirect
399. C22Dic3×C22⋊C9semidirect
404. C3C22⋊C4×C22⋊C9semidirect

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×Dic3×C22⋊C92
C22×C3×#2173
C22⋊C4×C22⋊C93
C9×#2174