Finite Group Databasebeta

← C9×#5174 #13143359

Subgroups — C9×#5174 #13143359

Order 1728.

428 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×SD16C211yesmaximal, characteristic, complemented
2C9×#946C211yesmaximal, characteristic, complemented
3C9×#938C211yesmaximal, characteristic
4C2×C9×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6C9×#832C211yesmaximal, characteristic, complemented
7C9×#818C211yesmaximal, characteristic, complemented
8C3×#5174C311yesmaximal, characteristic
9C9×#468—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C2×C9×#67—21no—
13C2×C4×D3×C9C2211yescharacteristic
14C3×C9×SD16—22no—
15C2×C3×C8×C9C2211yescharacteristic
16C9×#217—21no—
17C9×(C2×D3)⋊C4—21no—
18C9×C4⋊Dic3C2211yescharacteristic
19C9×C4⋊Dic3C2211yescharacteristic
20C9×Dic3⋊C4—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C9×ΓL(1,𝔽25)—22no—
23C2×C32×SD16C2×C311yescharacteristic
24C3×#946C2×C311yescharacteristic
25C3×#938C2×C311yescharacteristic
26C2×C3×ΓL(1,𝔽25)C2×C311yescharacteristic
27C3×#835C2×C311yescharacteristic
28C3×#832C2×C311yescharacteristic
29C3×#818C2×C311yescharacteristic
30C2×C9×SD16—21no—
31C2×C9×SD16D311yescharacteristic, complemented
32C2×C9×M16—31no—
33C9×C22⋊Q8—31no—
34C9×#121—31no—
35C9×#107—31no—
36C9×#103—31no—
37C9×#102—31no—
38C23×C3×C9—21no—
39C22×D3×C9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C3×D4×C9—21no—
43C3×D4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C9×#67—42no—
47C2×C9×Dic3—21no—
48C2×C9×Dic3—21no—
49C2×C9×Dic3D411yescharacteristic
50C4×D3×C9—21no—
51C4×D3×C9—21no—
52C3×C8×C9D412yescomplemented
53C9×C3⋊C8—21no—
54C3×#468—31no—
55#5174C911yescharacteristic, direct product factor
56C2×C32×Q8C22×C311yescharacteristic
57C2×C32×D4C22×C311yescharacteristic
58C2×C3×#67—21no—
59C2×C3×C4×D3C22×C311yescharacteristic
60C32×SD16—22no—
61C2×C32×C8C22×C311yescharacteristic
62C3×#217—21no—
63C3×(C2×D3)⋊C4—21no—
64C3×C4⋊Dic3C22×C311yescharacteristic
65C3×C4⋊Dic3C22×C311yescharacteristic
66C3×Dic3⋊C4—21no—
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C3×ΓL(1,𝔽25)—22no—
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D4×C9—21no—
72C2×D4×C9—61no—
73C2×D4×C9C2×D311yescharacteristic
74C22×C4×C9—31no—
75C9×SD16—24no—
76C9×SD16—22no—
77C9×M16—62no—
78C2×C8×C9—21no—
79C2×C8×C9—31no—
80C2×C8×C9C2×D311yescharacteristic
81C9×C4⋊C4—31no—
82C9×C4⋊C4—31no—
83C9×C4⋊C4—61no—
84C9×C22⋊C4—61no—
85C9×C22⋊C4—61no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9—41no—
89C22×C3×C9C2×D411yescharacteristic
90C2×D3×C9—21no—
91C2×D3×C9—21no—
92C2×D3×C9—21no—
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C9×Dic3—21no—
97C9×Dic3—41no—
98C9×Dic3—41no—
99C2×C3×SD16—21no—
100C2×C3×SD16C2×C911yescharacteristic, complemented
101C2×C3×SD16C3×D311yescharacteristic
102C2×C3×M16—31no—
103C3×C22⋊Q8—31no—
104C3×#121—31no—
105#946C2×C911yescharacteristic, complemented
106#938C2×C911yescharacteristic
107C2×ΓL(1,𝔽25)C2×C911yescharacteristic, complemented
108C3×#107—31no—
109C3×#103—31no—
110C3×#102—31no—
111#835C2×C911yescharacteristic, complemented
112#832C2×C911yescharacteristic, complemented
113#818C2×C911yescharacteristic, complemented
114C23×C32—21no—
115C22×C3×D3C3×D411yescharacteristic
116C32×Q8—21no—
117C32×Q8—21no—
118C32×D4—21no—
119C32×D4—21no—
120C2×C32×C4—21no—
121C2×C32×C4C23×C311yescharacteristic
122C3×#67—42no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3C3×D411yescharacteristic
126C3×C4×D3—21no—
127C3×C4×D3—21no—
128C23×C9—21no—
129C23×C9—31no—
130C23×C9—41no—
131C32×C8C3×D412yes—
132C3×C3⋊C8—21no—
133Q8×C9—22no—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137D4×C9—22no—
138D4×C9—21no—
139D4×C9—21no—
140D4×C9—41no—
141D4×C9—122no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—31no—
145C2×C4×C9—41no—
146C2×C4×C9—61no—
147C2×C4×C9—61no—
148C2×C4×C9—61no—
149C2×C4×C9—61no—
150C2×C4×C9C22×D311yescharacteristic
151C8×C9—22no—
152C8×C9—61no—
153C8×C9#6712yescomplemented
154#468—31noSylow
155C2×C3×C9—81no—
156C2×C3×C9#12111yescharacteristic
157C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
158C2×C3×C9#13711yescharacteristic
159D3×C9—41no—
160C2×C3×Q8—21no—
161C2×C3×Q8C22×C911yescharacteristic
162C2×C3×Q8C2×C3×D311yescharacteristic
163C2×C3×D4—21no—
164C2×C3×D4—61no—
165C2×C3×D4C22×C911yescharacteristic
166C2×C3×D4C2×C3×D311yescharacteristic
167C22×C3×C4—31no—
168C2×#67—21no—
169C2×C4×D3C22×C911yescharacteristic
170C3×SD16—24no—
171C3×SD16—22no—
172C3×SD16—22no—
173C3×M16—62no—
174C2×C3×C8—21no—
175C2×C3×C8—31no—
176C2×C3×C8C22×C911yescharacteristic
177C2×C3×C8C2×C3×D311yescharacteristic
178C3×C4⋊C4—31no—
179C3×C4⋊C4—31no—
180C3×C4⋊C4—61no—
181C3×C22⋊C4—61no—
182C3×C22⋊C4—61no—
183#217—21no—
184(C2×D3)⋊C4—21no—
185C4⋊Dic3C22×C911yescharacteristic
186C4⋊Dic3C22×C911yescharacteristic
187Dic3⋊C4—21no—
188C2×C3⋊C8C22×C911yescharacteristic
189ΓL(1,𝔽25)—22no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32C2×C3×D411yescharacteristic
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C2×C3×D3—21no—
197C32×C4—41no—
198C32×C4C2×C3×D411yescharacteristic
199C32×C4C3×#4111yescharacteristic
200C3×Dic3—21no—
201C3×Dic3—41no—
202C3×Dic3—41no—
203C22×C9—21no—
204C22×C9—42no—
205C22×C9—41no—
206C22×C9—41no—
207C22×C9—41no—
208C22×C9—61no—
209C22×C9—61no—
210C22×C9—61no—
211C22×C9—81no—
212C22×C9—81no—
213C22×C9D3×D411yescharacteristic
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—42no—
217C4×C9—41no—
218C4×C9—61no—
219C4×C9—121no—
220C4×C9—121no—
221C4×C9#23711yescharacteristic
222C4×C9C2×#6711yescharacteristic
223C2×SD16D3×C911yescharacteristic, complemented
224C2×M16—31no—
225C22⋊Q8—31no—
226#121—31no—
227#107—31no—
228#103—31no—
229#102—31no—
230C3×C9#46811yescharacteristic, complemented, Sylow
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—31no—
234C23×C3—41no—
235C22×D3D4×C911yescharacteristic
236C3×Q8—22no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—41no—
242C3×D4—22no—
243C3×D4—21no—
244C3×D4—21no—
245C3×D4—21no—
246C3×D4—21no—
247C3×D4—41no—
248C3×D4—122no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—31no—
253C2×C3×C4—41no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4C23×C911yescharacteristic
259C2×C3×C4C22×C3×D311yescharacteristic
260#67—42no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3D4×C911yescharacteristic
264C4×D3—21no—
265C4×D3—21no—
266C3×C8—22no—
267C3×C8—61no—
268C3×C8D4×C912yescomplemented
269C3×C8C3×#6712yes—
270C3⋊C8—21no—
271C2×C32—81no—
272C2×C32C3×#12111yescharacteristic
273C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
274C2×C32C3×#13711yescharacteristic
275C3×D3—41no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9—82no—
280C2×C9—81no—
281C2×C9—121no—
282C2×C9#91411yescharacteristic
283C2×C9#91511yescharacteristic
284C2×C9#93811yescharacteristic
285C2×Q8C2×D3×C911yescharacteristic
286C2×D4—61no—
287C2×D4C2×D3×C911yescharacteristic
288C22×C4—31no—
289SD16—22no—
290M16—62no—
291C2×C8—31no—
292C2×C8C2×D3×C911yescharacteristic
293C4⋊C4—31no—
294C4⋊C4—31no—
295C4⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22×C3—21no—
299C22×C3—42no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3C2×D4×C911yescharacteristic
312C22×C3C3×D3×D411yescharacteristic
313C2×D3—21no—
314C2×D3—21no—
315C2×D3—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—41no—
320C3×C4—41no—
321C3×C4—61no—
322C3×C4—121no—
323C3×C4—121no—
324C3×C4C2×D4×C911yescharacteristic
325C3×C4C9×#4111yescharacteristic
326C3×C4C3×#23711yescharacteristic
327C3×C4C2×C3×#6711yescharacteristic
328Dic3—21no—
329Dic3—41no—
330Dic3—41no—
331C32C3×#46811yescharacteristic
332C9—21no—
333C9#517411yescharacteristic, direct product factor
334C23—21no—
335C23—31no—
336Q8—21no—
337Q8—21no—
338D4—21no—
339D4—21no—
340D4—122no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4C22×D3×C911yescharacteristic
348C8—61no—
349C8C9×#6712yescomplemented
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—82no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3—121no—
357C2×C3C9×#12111yescharacteristic
358C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
359C2×C3C9×#13711yescharacteristic
360C2×C3C3×#91411yescharacteristic
361C2×C3C3×#91511yescharacteristic
362C2×C3C3×#93811yescharacteristic
363D3—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—61no—
368C22—61no—
369C22—61no—
370C22D3×D4×C911yescharacteristic
371C4—41no—
372C4—61no—
373C4—121no—
374C4—121no—
375C4C9×#23711yescharacteristic
376C4C2×C9×#6711yescharacteristic
377C3—21no—
378C3C9×#46811yescharacteristic, complemented
379C3C3×#517411yescharacteristic
380C2—81no—
381C2—121no—
382C2C9×#91411yescharacteristic
383C2C9×#91511yescharacteristic
384C2C9×#93811yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C9×SD16C2
socleC22×C32C2×C3×D4
prime radicalC3×#938C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC2×C9×SD16D3
hypocenterC3C9×#468
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C2×C9×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×SD16C2semidirect
2. C9×#946C2semidirect
4. C2×C9×ΓL(1,𝔽25)C2semidirect
5. C9×#835C2semidirect
6. C9×#832C2semidirect
7. C9×#818C2semidirect
31. C2×C9×SD16D3semidirect
52. C3×C8×C9D4semidirect
55. #5174C9direct (N × H)
100. C2×C3×SD16C2×C9semidirect
105. #946C2×C9semidirect
107. C2×ΓL(1,𝔽25)C2×C9semidirect
111. #835C2×C9semidirect
112. #832C2×C9semidirect
113. #818C2×C9semidirect
153. C8×C9#67semidirect
223. C2×SD16D3×C9semidirect
230. C3×C9#468semidirect
268. C3×C8D4×C9semidirect
333. C9#5174direct (N × H)
349. C8C9×#67semidirect
378. C3C9×#468semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×SD162
C9×#9462
C9×#9382
C2×C9×ΓL(1,𝔽25)2
C9×#8352
C9×#8322
C9×#8182
C3×#51743
C9×#4683