Finite Group Databasebeta

← C9×#4733 #13141593

Subgroups — C9×#4733 #13141593

Order 1728.

506 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic, complemented
3C2×C9×Dic12C211yesmaximal, characteristic, complemented
4C2×C9×#204C211yesmaximal, characteristic, complemented
5C3×C9×C22⋊C8C211yesmaximal, characteristic, complemented
6C9×#821C211yesmaximal, characteristic, complemented
7C9×#816C211yesmaximal, characteristic, complemented
8C3×#4733C311yesmaximal, characteristic
9C9×#452—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C9×#235—22no—
13C9×#235—22no—
14C2×C9×D12C2211yescharacteristic
15C2×C4×D3×C9—21no—
16C2×C9×Dic6C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×C3×C8×C9C2211yescharacteristic
19C2×C3×C8×C9C2211yescharacteristic
20C9×#217—21no—
21C9×C4⋊Dic3C2211yescharacteristic
22C9×Dic3⋊C4—21no—
23C9×Dic12—22no—
24C9×#204—22no—
25C2×C3×#235C2×C311yescharacteristic
26C3×#924C2×C311yescharacteristic
27C2×C3×Dic12C2×C311yescharacteristic
28C2×C3×#204C2×C311yescharacteristic
29C2×C9×#41—31no—
30C32×C22⋊C8C2×C311yescharacteristic
31C3×#821C2×C311yescharacteristic
32C3×#816C2×C311yescharacteristic
33C2×C9×Q16—31no—
34C2×C9×SD16—31no—
35C9×C22⋊Q8—31no—
36C9×#103—31no—
37C9×#102—31no—
38C9×C22⋊C8—21no—
39C9×C22⋊C8D311yescharacteristic, complemented
40C23×C3×C9D411yescharacteristic
41C22×D3×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C9×#67—24no—
47C2×C9×Dic3—21no—
48C2×C9×Dic3—21no—
49C2×C9×Dic3—21no—
50C9×D12—21no—
51C9×D12D412yes—
52C4×D3×C9—24no—
53C9×Dic6—21no—
54C9×Dic6—21no—
55C9×Dic6—21no—
56C9×Dic6D412yes—
57C3×C8×C9—21no—
58C3×C8×C9—21no—
59C3×#452—31no—
60#4733C911yescharacteristic, direct product factor
61C22×C32×C4C22×C311yescharacteristic
62C2×C3×#67—21no—
63C3×#235—22no—
64C3×#235—22no—
65C2×C3×D12C22×C311yescharacteristic
66C2×C3×C4×D3—21no—
67C2×C3×Dic6C22×C311yescharacteristic
68C2×C3×Dic6C22×C311yescharacteristic
69C2×C32×C8C22×C311yescharacteristic
70C2×C32×C8C22×C311yescharacteristic
71C3×#217—21no—
72C3×C4⋊Dic3C22×C311yescharacteristic
73C3×Dic3⋊C4—21no—
74C3×Dic12—22no—
75C3×#204—22no—
76C9×#41—62no—
77C9×#41—62no—
78C2×Q8×C9—31no—
79C2×Q8×C9—31no—
80C2×D4×C9—31no—
81C2×D4×C9—61no—
82C22×C4×C9—21no—
83C22×C4×C9—61no—
84C22×C4×C9C2×D311yescharacteristic
85C9×Q16—62no—
86C9×SD16—62no—
87C2×C8×C9—21no—
88C2×C8×C9—21no—
89C2×C8×C9C2×D311yescharacteristic
90C2×C8×C9C2×D311yescharacteristic
91C9×C4⋊C4—31no—
92C9×C4⋊C4—61no—
93C9×C22⋊C4—61no—
94C22×C3×C9—21no—
95C22×C3×C9—21no—
96C22×C3×C9—21no—
97C22×C3×C9C2×D411yescharacteristic
98C2×D3×C9—22no—
99C2×D3×C9—41no—
100C2×D3×C9—41no—
101C3×C4×C9—21no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C9×Dic3—22no—
105C9×Dic3—41no—
106C9×Dic3—41no—
107C2×C3×#41—31no—
108C2×#235C2×C911yescharacteristic
109C2×C3×Q16—31no—
110C2×C3×SD16—31no—
111C3×C22⋊Q8—31no—
112#924C2×C911yescharacteristic, complemented
113C2×Dic12C2×C911yescharacteristic, complemented
114C2×#204C2×C911yescharacteristic, complemented
115C3×#103—31no—
116C3×#102—31no—
117C3×C22⋊C8—21no—
118C3×C22⋊C8C2×C911yescharacteristic, complemented
119C3×C22⋊C8C3×D311yescharacteristic
120#821C2×C911yescharacteristic, complemented
121#816C2×C911yescharacteristic, complemented
122C23×C32C3×D411yescharacteristic
123C22×C3×D3—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4C3×D411yescharacteristic
127C2×C32×C4C23×C311yescharacteristic
128C3×#67—24no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C3×D12—21no—
133C3×D12C3×D412yes—
134C3×C4×D3—24no—
135C3×Dic6—21no—
136C3×Dic6—21no—
137C3×Dic6—21no—
138C3×Dic6C3×D412yes—
139C23×C9—21no—
140C23×C9—61no—
141C23×C9D1211yescharacteristic
142C32×C8—21no—
143C32×C8—21no—
144Q8×C9—32no—
145Q8×C9—61no—
146Q8×C9—61no—
147Q8×C9—61no—
148D4×C9—32no—
149D4×C9—64no—
150D4×C9—61no—
151C2×C4×C9—21no—
152C2×C4×C9—21no—
153C2×C4×C9—21no—
154C2×C4×C9—21no—
155C2×C4×C9—41no—
156C2×C4×C9—41no—
157C2×C4×C9—64no—
158C2×C4×C9—61no—
159C2×C4×C9—61no—
160C2×C4×C9—61no—
161C2×C4×C9D1211yescharacteristic
162C2×C4×C9C22×D311yescharacteristic
163C8×C9—21no—
164C8×C9—21no—
165C8×C9—41no—
166C8×C9—41no—
167#452—31noSylow
168C2×C3×C9—41no—
169C2×C3×C9#12011yescharacteristic
170C2×C3×C9#13511yescharacteristic
171C2×C3×C9#13711yescharacteristic
172D3×C9—42no—
173C3×#41—62no—
174C3×#41—62no—
175C2×C3×Q8—31no—
176C2×C3×Q8—31no—
177C2×C3×D4—31no—
178C2×C3×D4—61no—
179C22×C3×C4—21no—
180C22×C3×C4—61no—
181C22×C3×C4C22×C911yescharacteristic
182C22×C3×C4C2×C3×D311yescharacteristic
183C2×#67—21no—
184#235—22no—
185#235—22no—
186C2×D12C22×C911yescharacteristic
187C2×C4×D3—21no—
188C2×Dic6C22×C911yescharacteristic
189C2×Dic6C22×C911yescharacteristic
190C3×Q16—62no—
191C3×SD16—62no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8C22×C911yescharacteristic
195C2×C3×C8C22×C911yescharacteristic
196C2×C3×C8C2×C3×D311yescharacteristic
197C2×C3×C8C2×C3×D311yescharacteristic
198C3×C4⋊C4—31no—
199C3×C4⋊C4—61no—
200C3×C22⋊C4—61no—
201#217—21no—
202C4⋊Dic3C22×C911yescharacteristic
203Dic3⋊C4—21no—
204Dic12—22no—
205#204—22no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32C2×C3×D411yescharacteristic
210C2×C3×D3—22no—
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C32×C4—21no—
214C32×C4C2×C3×D411yescharacteristic
215C32×C4C2×C3×D411yescharacteristic
216C3×Dic3—22no—
217C3×Dic3—41no—
218C3×Dic3—41no—
219C22×C9—22no—
220C22×C9—21no—
221C22×C9—21no—
222C22×C9—21no—
223C22×C9—21no—
224C22×C9—41no—
225C22×C9—41no—
226C22×C9—62no—
227C22×C9—121no—
228C22×C9—121no—
229C22×C9C2×D1211yescharacteristic
230C4×C9—21no—
231C4×C9—21no—
232C4×C9—21no—
233C4×C9—41no—
234C4×C9—62no—
235C4×C9—121no—
236C4×C9—121no—
237C4×C9D3×D411yescharacteristic
238C4×C9D3×D411yescharacteristic
239C2×#41—31no—
240C2×Q16—31no—
241C2×SD16—31no—
242C22⋊Q8—31no—
243#103—31no—
244#102—31no—
245C22⋊C8D3×C911yescharacteristic, complemented
246C3×C9#45211yescharacteristic, complemented, Sylow
247C23×C3—21no—
248C23×C3—61no—
249C23×C3D4×C911yescharacteristic
250C23×C3C3×D1211yescharacteristic
251C22×D3—21no—
252C3×Q8—32no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×D4—32no—
257C3×D4—64no—
258C3×D4—61no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—64no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4D4×C911yescharacteristic
272C2×C3×C4C23×C911yescharacteristic
273C2×C3×C4C3×D1211yescharacteristic
274C2×C3×C4C22×C3×D311yescharacteristic
275#67—24no—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—21no—
279D12—21no—
280D12D4×C912yes—
281C4×D3—24no—
282Dic6—21no—
283Dic6—21no—
284Dic6—21no—
285Dic6D4×C912yes—
286C3×C8—21no—
287C3×C8—21no—
288C3×C8—21no—
289C3×C8—21no—
290C3×C8—41no—
291C3×C8—41no—
292C2×C32—41no—
293C2×C32C3×#12011yescharacteristic
294C2×C32C3×#13511yescharacteristic
295C2×C32C3×#13711yescharacteristic
296C3×D3—42no—
297C2×C9—21no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—42no—
301C2×C9—41no—
302C2×C9—122no—
303C2×C9#88211yescharacteristic
304C2×C9#90411yescharacteristic
305C2×C9#90911yescharacteristic
306#41—62no—
307#41—62no—
308C2×Q8—31no—
309C2×Q8—31no—
310C2×D4—31no—
311C2×D4—61no—
312C22×C4—61no—
313C22×C4C2×D3×C911yescharacteristic
314Q16—62no—
315SD16—62no—
316C2×C8C2×D3×C911yescharacteristic
317C2×C8C2×D3×C911yescharacteristic
318C4⋊C4—31no—
319C4⋊C4—61no—
320C22⋊C4—61no—
321C22×C3—22no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—62no—
332C22×C3—121no—
333C22×C3—121no—
334C22×C3C2×D4×C911yescharacteristic
335C22×C3C2×C3×D1211yescharacteristic
336C2×D3—22no—
337C2×D3—41no—
338C2×D3—41no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—41no—
344C3×C4—62no—
345C3×C4—121no—
346C3×C4—121no—
347C3×C4C2×D4×C911yescharacteristic
348C3×C4C2×D4×C911yescharacteristic
349C3×C4C3×D3×D411yescharacteristic
350C3×C4C3×D3×D411yescharacteristic
351Dic3—22no—
352Dic3—41no—
353Dic3—41no—
354C32C3×#45211yescharacteristic
355C9—21no—
356C9#473311yescharacteristic, direct product factor
357C23—61no—
358C23C9×D1211yescharacteristic
359Q8—32no—
360Q8—61no—
361Q8—61no—
362Q8—61no—
363D4—32no—
364D4—64no—
365D4—61no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—64no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4C9×D1211yescharacteristic
373C2×C4C22×D3×C911yescharacteristic
374C8—21no—
375C8—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—42no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—122no—
383C2×C3C9×#12011yescharacteristic
384C2×C3C9×#13511yescharacteristic
385C2×C3C9×#13711yescharacteristic
386C2×C3C3×#88211yescharacteristic
387C2×C3C3×#90411yescharacteristic
388C2×C3C3×#90911yescharacteristic
389D3—42no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—62no—
394C22—121no—
395C22—121no—
396C22C2×C9×D1211yescharacteristic
397C4—21no—
398C4—62no—
399C4—121no—
400C4—121no—
401C4D3×D4×C911yescharacteristic
402C4D3×D4×C911yescharacteristic
403C3—21no—
404C3C9×#45211yescharacteristic, complemented
405C3C3×#473311yescharacteristic
406C2—41no—
407C2—122no—
408C2C9×#88211yescharacteristic
409C2C9×#90411yescharacteristic
410C2C9×#90911yescharacteristic
subgroupG/H
centerC22×C9C2×D12
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊C8C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×#235C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×C22⋊C8D3
hypocenterC3C9×#452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C22×C4×C9 ◁ C9×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C9×#924C2semidirect
3. C2×C9×Dic12C2semidirect
4. C2×C9×#204C2semidirect
5. C3×C9×C22⋊C8C2semidirect
6. C9×#821C2semidirect
7. C9×#816C2semidirect
39. C9×C22⋊C8D3semidirect
60. #4733C9direct (N × H)
112. #924C2×C9semidirect
113. C2×Dic12C2×C9semidirect
114. C2×#204C2×C9semidirect
118. C3×C22⋊C8C2×C9semidirect
120. #821C2×C9semidirect
121. #816C2×C9semidirect
245. C22⋊C8D3×C9semidirect
246. C3×C9#452semidirect
356. C9#4733direct (N × H)
404. C3C9×#452semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C9×#9242
C2×C9×Dic122
C2×C9×#2042
C3×C9×C22⋊C82
C9×#8212
C9×#8162
C3×#47333
C9×#4523