Finite Group Databasebeta

← C9×#5032 #13142813

Subgroups — C9×#5032 #13142813

Order 1728.

724 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C9×#1004C211yesmaximal, characteristic, complemented
2C3×C9×#120C211yesmaximal, characteristic, complemented
3C9×#953C212yesmaximal, complemented
4C9×#834C211yesmaximal, characteristic, complemented
5C9×#806C212yesmaximal, complemented
6C3×#5032C311yesmaximal, characteristic
7C9×B(4,𝔽2)—31nomaximal
8C24×C3×C9C2211yescharacteristic, complemented
9C2×C3×D4×C9—21no—
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C2×C9×#67—22no—
13C2×C9×#67C2212yes—
14C9×#237—22no—
15D3×D4×C9—22no—
16C9×#235—21no—
17C3×C9×C22⋊C4—21no—
18C3×C9×C22⋊C4C2211yescharacteristic, complemented
19C9×#217—22no—
20C9×#217C2212yescomplemented
21C3×#1004C2×C311yescharacteristic
22C32×#120C2×C311yescharacteristic
23C3×#953C2×C312yes—
24C9×H(5,𝔽2)—31no—
25C3×#834C2×C311yescharacteristic
26C3×#806C2×C312yes—
27C9×#120—21no—
28C9×#120—32no—
29C9×#120D311yescharacteristic, complemented
30C9×#99—31no—
31C9×#99—32no—
32C23×C3×C9—21no—
33C23×C3×C9—22no—
34C23×C3×C9—42no—
35C23×C3×C9D411yescharacteristic, complemented
36C23×C3×C9C2311yescharacteristic
37C22×D3×C9—21no—
38C22×D3×C9D412yescomplemented
39C3×D4×C9—22no—
40C3×D4×C9—42no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9D411yescharacteristic, complemented
43C9×#67—22no—
44C9×#67—22no—
45C9×#67—22no—
46C9×#67—44no—
47C2×C9×Dic3—21no—
48C2×C9×Dic3—22no—
49C2×C9×Dic3D412yescomplemented
50C9×D12—21no—
51C4×D3×C9—22no—
52C9×Dic6—21no—
53C3×B(4,𝔽2)—31no—
54#5032C911yescharacteristic, direct product factor
55C24×C32C22×C311yescharacteristic
56C2×C32×D4—21no—
57C2×C32×D4C22×C311yescharacteristic
58C2×C3×#67—21no—
59C2×C3×#67—22no—
60C2×C3×#67C22×C312yes—
61C3×#237—22no—
62C3×D3×D4—22no—
63C3×#235—21no—
64C24×C9—21no—
65C24×C9C2×D311yescharacteristic, complemented
66C32×C22⋊C4—21no—
67C32×C22⋊C4C22×C311yescharacteristic
68C3×#217—22no—
69C3×#217C22×C312yes—
70C9×#41—62no—
71C9×#41—61no—
72C2×D4×C9—22no—
73C2×D4×C9—21no—
74C2×D4×C9—21no—
75C2×D4×C9—32no—
76C2×D4×C9—62no—
77C2×D4×C9—61no—
78C2×D4×C9—62no—
79C2×D4×C9C2×D311yescharacteristic
80C9×C22⋊C4—22no—
81C9×C22⋊C4—21no—
82C9×C22⋊C4—21no—
83C9×C22⋊C4—32no—
84C9×C22⋊C4—62no—
85C9×C22⋊C4C2×D311yescharacteristic, complemented
86C22×C3×C9—22no—
87C22×C3×C9—24no—
88C22×C3×C9—21no—
89C22×C3×C9—41no—
90C22×C3×C9—41no—
91C22×C3×C9—42no—
92C22×C3×C9—41no—
93C22×C3×C9—41no—
94C22×C3×C9C2×D412yes—
95C22×C3×C9C2×D411yescharacteristic
96C2×D3×C9—22no—
97C2×D3×C9—42no—
98C2×D3×C9—81no—
99C3×C4×C9—21no—
100C3×C4×C9—41no—
101C9×Dic3—22no—
102C9×Dic3—42no—
103C3×H(5,𝔽2)—31no—
104#1004C2×C911yescharacteristic, complemented
105C3×#120—21no—
106C3×#120—32no—
107C3×#120C2×C911yescharacteristic, complemented
108C3×#120C3×D311yescharacteristic
109#953C2×C912yescomplemented
110C3×#99—31no—
111C3×#99—32no—
112#834C2×C911yescharacteristic, complemented
113#806C2×C912yescomplemented
114C23×C32—21no—
115C23×C32—22no—
116C23×C32—42no—
117C23×C32C3×D411yescharacteristic
118C23×C32C23×C311yescharacteristic
119C22×C3×D3—21no—
120C22×C3×D3C3×D412yes—
121C32×D4—22no—
122C32×D4—42no—
123C2×C32×C4—21no—
124C2×C32×C4C3×D411yescharacteristic
125C3×#67—22no—
126C3×#67—22no—
127C3×#67—22no—
128C3×#67—44no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3C3×D412yes—
132C3×D12—21no—
133C3×C4×D3—22no—
134C3×Dic6—21no—
135C23×C9—22no—
136C23×C9—22no—
137C23×C9—21no—
138C23×C9—21no—
139C23×C9—21no—
140C23×C9—32no—
141C23×C9—42no—
142C23×C9—42no—
143C23×C9—44no—
144C23×C9—61no—
145C23×C9#6711yescharacteristic, complemented
146C23×C9C22×D311yescharacteristic
147Q8×C9—61no—
148D4×C9—22no—
149D4×C9—42no—
150D4×C9—42no—
151D4×C9—44no—
152D4×C9—61no—
153D4×C9—62no—
154D4×C9—62no—
155D4×C9—62no—
156D4×C9—124no—
157C2×C4×C9—22no—
158C2×C4×C9—21no—
159C2×C4×C9—21no—
160C2×C4×C9—32no—
161C2×C4×C9—62no—
162C2×C4×C9—61no—
163C2×C4×C9—62no—
164C2×C4×C9#6711yescharacteristic, complemented
165B(4,𝔽2)—31noSylow
166C2×C3×C9—21no—
167C2×C3×C9—22no—
168C2×C3×C9—42no—
169C2×C3×C9—41no—
170C2×C3×C9#12011yescharacteristic
171D3×C9—42no—
172C24×C3—21no—
173C24×C3C22×C911yescharacteristic, complemented
174C24×C3C2×C3×D311yescharacteristic
175C3×#41—62no—
176C3×#41—61no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—32no—
182C2×C3×D4—62no—
183C2×C3×D4—61no—
184C2×C3×D4—62no—
185C2×C3×D4C22×C911yescharacteristic
186C2×C3×D4C2×C3×D311yescharacteristic
187C2×#67—21no—
188C2×#67—22no—
189C2×#67C22×C912yes—
190#237—22no—
191D3×D4—22no—
192#235—21no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—21no—
195C3×C22⋊C4—21no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—32no—
198C3×C22⋊C4—62no—
199C3×C22⋊C4C22×C911yescharacteristic, complemented
200C3×C22⋊C4C2×C3×D311yescharacteristic
201#217—22no—
202#217C22×C912yescomplemented
203C22×C32—22no—
204C22×C32—24no—
205C22×C32—21no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—42no—
210C22×C32—41no—
211C22×C32C2×C3×D412yes—
212C22×C32C2×C3×D411yescharacteristic
213C2×C3×D3—22no—
214C2×C3×D3—42no—
215C2×C3×D3—81no—
216C32×C4—21no—
217C32×C4—41no—
218C3×Dic3—22no—
219C3×Dic3—42no—
220C22×C9—24no—
221C22×C9—22no—
222C22×C9—24no—
223C22×C9—21no—
224C22×C9—22no—
225C22×C9—21no—
226C22×C9—41no—
227C22×C9—44no—
228C22×C9—41no—
229C22×C9—41no—
230C22×C9—42no—
231C22×C9—42no—
232C22×C9—41no—
233C22×C9—42no—
234C22×C9—42no—
235C22×C9—42no—
236C22×C9—44no—
237C22×C9—41no—
238C22×C9—62no—
239C22×C9—122no—
240C22×C9—241no—
241C22×C9D3×D412yes—
242C22×C9C2×#6711yescharacteristic
243C4×C9—21no—
244C4×C9—41no—
245C4×C9—42no—
246C4×C9—41no—
247C4×C9—62no—
248C4×C9—122no—
249H(5,𝔽2)—31no—
250#120—32no—
251#120D3×C911yescharacteristic, complemented
252#99—31no—
253#99—32no—
254C3×C9B(4,𝔽2)11yescharacteristic, complemented, Sylow
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—21no—
258C23×C3—22no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—32no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—44no—
267C23×C3—61no—
268C23×C3D4×C911yescharacteristic, complemented
269C23×C3C23×C911yescharacteristic
270C23×C3C3×#6711yescharacteristic
271C23×C3C22×C3×D311yescharacteristic
272C22×D3—21no—
273C22×D3D4×C912yescomplemented
274C3×Q8—61no—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—42no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—44no—
281C3×D4—61no—
282C3×D4—62no—
283C3×D4—62no—
284C3×D4—62no—
285C3×D4—124no—
286C2×C3×C4—22no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—32no—
291C2×C3×C4—62no—
292C2×C3×C4—61no—
293C2×C3×C4—62no—
294C2×C3×C4D4×C911yescharacteristic, complemented
295C2×C3×C4C3×#6711yescharacteristic
296#67—22no—
297#67—22no—
298#67—22no—
299#67—44no—
300C2×Dic3—21no—
301C2×Dic3—22no—
302C2×Dic3D4×C912yescomplemented
303D12—21no—
304C4×D3—22no—
305Dic6—21no—
306C2×C32—21no—
307C2×C32—22no—
308C2×C32—42no—
309C2×C32—41no—
310C2×C32C3×#12011yescharacteristic
311C3×D3—42no—
312C2×C9—22no—
313C2×C9—21no—
314C2×C9—22no—
315C2×C9—21no—
316C2×C9—42no—
317C2×C9—44no—
318C2×C9—42no—
319C2×C9—41no—
320C2×C9—81no—
321C2×C9—122no—
322C2×C9#93711yescharacteristic
323C24C2×D3×C911yescharacteristic, complemented
324#41—62no—
325#41—61no—
326C2×D4—21no—
327C2×D4—32no—
328C2×D4—62no—
329C2×D4—61no—
330C2×D4—62no—
331C2×D4C2×D3×C911yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—32no—
334C22⋊C4—62no—
335C22⋊C4C2×D3×C911yescharacteristic, complemented
336C22×C3—24no—
337C22×C3—22no—
338C22×C3—24no—
339C22×C3—22no—
340C22×C3—24no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—41no—
346C22×C3—44no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—41no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—44no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—62no—
363C22×C3—122no—
364C22×C3—241no—
365C22×C3C2×D4×C912yes—
366C22×C3C2×D4×C911yescharacteristic
367C22×C3C3×D3×D412yes—
368C22×C3C2×C3×#6711yescharacteristic
369C2×D3—22no—
370C2×D3—42no—
371C2×D3—81no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—42no—
377C3×C4—41no—
378C3×C4—62no—
379C3×C4—122no—
380Dic3—22no—
381Dic3—42no—
382C32C3×B(4,𝔽2)11yescharacteristic
383C9—21no—
384C9#503211yescharacteristic, direct product factor
385C23—22no—
386C23—21no—
387C23—32no—
388C23—42no—
389C23—61no—
390C23C9×#6711yescharacteristic, complemented
391C23C22×D3×C911yescharacteristic
392Q8—61no—
393D4—22no—
394D4—42no—
395D4—61no—
396D4—62no—
397D4—62no—
398D4—62no—
399D4—124no—
400C2×C4—21no—
401C2×C4—32no—
402C2×C4—62no—
403C2×C4—61no—
404C2×C4—62no—
405C2×C4C9×#6711yescharacteristic, complemented
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—42no—
413C2×C3—44no—
414C2×C3—41no—
415C2×C3—42no—
416C2×C3—42no—
417C2×C3—41no—
418C2×C3—81no—
419C2×C3—122no—
420C2×C3C9×#12011yescharacteristic
421C2×C3C3×#93711yescharacteristic
422D3—42no—
423C22—22no—
424C22—24no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—42no—
430C22—41no—
431C22—62no—
432C22—122no—
433C22—241no—
434C22D3×D4×C912yes—
435C22C2×C9×#6711yescharacteristic
436C4—21no—
437C4—41no—
438C4—62no—
439C4—122no—
440C3—21no—
441C3C9×B(4,𝔽2)11yescharacteristic, complemented
442C3C3×#503211yescharacteristic
443C2—21no—
444C2—22no—
445C2—42no—
446C2—41no—
447C2—122no—
448C2C9×#93711yescharacteristic
subgroupG/H
centerC2×C9#937
commutatorC23×C3C23×C9
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#120C2
socleC2×C32C3×#120
prime radicalC3×#5032C3
cosocleC23×C32C23×C3
hypercenterC9×#120D3
hypocenterC3C9×B(4,𝔽2)
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C23×C9 ◁ C9×#120

Complements

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

NH ≅ G/Ntype
1. C9×#1004C2semidirect
2. C3×C9×#120C2semidirect
3. C9×#953C2semidirect
4. C9×#834C2semidirect
5. C9×#806C2semidirect
8. C24×C3×C9C22semidirect
18. C3×C9×C22⋊C4C22semidirect
20. C9×#217C22semidirect
29. C9×#120D3semidirect
35. C23×C3×C9D4semidirect
38. C22×D3×C9D4semidirect
42. C2×C3×C4×C9D4semidirect
49. C2×C9×Dic3D4semidirect
54. #5032C9direct (N × H)
65. C24×C9C2×D3semidirect
85. C9×C22⋊C4C2×D3semidirect
104. #1004C2×C9semidirect
107. C3×#120C2×C9semidirect
109. #953C2×C9semidirect
112. #834C2×C9semidirect
113. #806C2×C9semidirect
145. C23×C9#67semidirect
164. C2×C4×C9#67semidirect
173. C24×C3C22×C9semidirect
199. C3×C22⋊C4C22×C9semidirect
202. #217C22×C9semidirect
251. #120D3×C9semidirect
254. C3×C9B(4,𝔽2)semidirect
268. C23×C3D4×C9semidirect
273. C22×D3D4×C9semidirect
294. C2×C3×C4D4×C9semidirect
302. C2×Dic3D4×C9semidirect
323. C24C2×D3×C9semidirect
335. C22⋊C4C2×D3×C9semidirect
384. C9#5032direct (N × H)
390. C23C9×#67semidirect
405. C2×C4C9×#67semidirect
441. C3C9×B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C9×#10042
C3×C9×#1202
C9×#9532
C9×#8342
C9×#8062
C3×#50323
C9×B(4,𝔽2)3