Finite Group Databasebeta

← C9×#5838 #13153754

Subgroups — C9×#5838 #13153754

Order 1728.

890 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×2−1+4C211yesmaximal, characteristic, complemented
2C9×#1010C211yesmaximal, characteristic, complemented
3C9×#951C213yesmaximal, complemented
4C9×#950C213yesmaximal, complemented
5C9×#948C211yesmaximal, characteristic, complemented
6C2×C9×#216C213yesmaximal, complemented
7C9×#942C213yesmaximal, complemented
8C3×#5838C311yesmaximal, characteristic
9C9×#578—31nomaximal
10C3×C9×#41—23no—
11C3×C9×#41C2213yes—
12C3×C9×#41C2211yescharacteristic
13C2×C3×Q8×C9—21no—
14C2×C3×Q8×C9C2213yes—
15D3×Q8×C9—21no—
16C9×#237—23no—
17C9×#235C2213yes—
18C2×C9×Dic6C2213yes—
19C9×#216C2216yes—
20C9×#216C2213yes—
21C9×#215C2213yes—
22C9×#214C2213yes—
23C9×#213C2211yescharacteristic
24C9×#208C2213yes—
25C2×C9×C3⋊C8C2213yes—
26C32×2−1+4C2×C311yescharacteristic
27C3×#1010C2×C311yescharacteristic
28C3×#951C2×C313yes—
29C3×#950C2×C313yes—
30C3×#948C2×C311yescharacteristic
31C2×C3×#216C2×C313yes—
32C3×#942C2×C313yes—
33C9×2−1+4—21no—
34C9×2−1+4—31no—
35C9×2−1+4D311yescharacteristic, complemented
36C9×#137—33no—
37C9×#137—33no—
38C9×#135—33no—
39C2×C9×Q16—33no—
40C9×#131—31no—
41C3×Q8×C9—23no—
42C3×Q8×C9D411yescharacteristic
43C3×Q8×C9C2313yes—
44C3×D4×C9—23no—
45C3×D4×C9D413yes—
46C3×D4×C9C2311yescharacteristic
47C2×C3×C4×C9—23no—
48C2×C3×C4×C9—23no—
49C2×C3×C4×C9C2313yes—
50C9×#67—23no—
51C2×C9×Dic3—23no—
52C9×D12C2311yescharacteristic
53C4×D3×C9—23no—
54C9×Dic6—23no—
55C9×Dic6C2313yes—
56C9×C3⋊C8C2311yescharacteristic
57C9×C3⋊C8C2313yes—
58C3×#578—31no—
59#5838C911yescharacteristic, direct product factor
60C32×#41—23no—
61C32×#41C22×C313yes—
62C32×#41C22×C311yescharacteristic
63C2×C32×Q8—21no—
64C2×C32×Q8C22×C313yes—
65C3×D3×Q8—21no—
66C3×#237—23no—
67C3×#235C22×C313yes—
68C2×C3×Dic6C22×C313yes—
69C3×#216C22×C316yes—
70C3×#216C22×C313yes—
71C3×#215C22×C313yes—
72C3×#214C22×C313yes—
73C3×#213C22×C311yescharacteristic
74C3×#208C22×C313yes—
75C2×C3×C3⋊C8C22×C313yes—
76C9×#41—26no—
77C9×#41—23no—
78C9×#41—21no—
79C9×#41—23no—
80C9×#41—33no—
81C9×#41—63no—
82C9×#41C2×D313yes—
83C9×#41C2×D311yescharacteristic
84C2×Q8×C9—22no—
85C2×Q8×C9—23no—
86C2×Q8×C9—21no—
87C2×Q8×C9—33no—
88C2×Q8×C9—61no—
89C2×Q8×C9C2×D313yes—
90C9×Q16—33no—
91C9×Q16—36no—
92C9×SD16—33no—
93C9×SD16—33no—
94C9×D8—31no—
95C9×M16—33no—
96C2×C8×C9—33no—
97C22×C3×C9—21no—
98C22×C3×C9C2×D413yes—
99C2×D3×C9—21no—
100C3×C4×C9—23no—
101C3×C4×C9C2×D413yes—
102C3×C4×C9C2411yescharacteristic
103C9×Dic3—23no—
104C3×2−1+4—21no—
105C3×2−1+4—31no—
106C3×2−1+4C2×C911yescharacteristic, complemented
107C3×2−1+4C3×D311yescharacteristic
108#1010C2×C911yescharacteristic, complemented
109C3×#137—33no—
110C3×#137—33no—
111C3×#135—33no—
112C2×C3×Q16—33no—
113C3×#131—31no—
114#951C2×C913yescomplemented
115#950C2×C913yescomplemented
116#948C2×C911yescharacteristic, complemented
117C2×#216C2×C913yescomplemented
118#942C2×C913yescomplemented
119C32×Q8—23no—
120C32×Q8C3×D411yescharacteristic
121C32×Q8C23×C313yes—
122C32×D4—23no—
123C32×D4C3×D413yes—
124C32×D4C23×C311yescharacteristic
125C2×C32×C4—23no—
126C2×C32×C4—23no—
127C2×C32×C4C23×C313yes—
128C3×#67—23no—
129C2×C3×Dic3—23no—
130C3×D12C23×C311yescharacteristic
131C3×C4×D3—23no—
132C3×Dic6—23no—
133C3×Dic6C23×C313yes—
134C3×C3⋊C8C23×C311yescharacteristic
135C3×C3⋊C8C23×C313yes—
136Q8×C9—26no—
137Q8×C9—23no—
138Q8×C9—21no—
139Q8×C9—23no—
140Q8×C9—33no—
141Q8×C9—63no—
142Q8×C9#6711yescharacteristic
143Q8×C9C22×D313yes—
144D4×C9—26no—
145D4×C9—23no—
146D4×C9—23no—
147D4×C9—21no—
148D4×C9—31no—
149D4×C9—63no—
150D4×C9#6713yes—
151D4×C9C22×D311yescharacteristic
152C2×C4×C9—26no—
153C2×C4×C9—26no—
154C2×C4×C9—23no—
155C2×C4×C9—23no—
156C2×C4×C9—23no—
157C2×C4×C9—63no—
158C2×C4×C9—63no—
159C2×C4×C9C22×D313yes—
160C8×C9—31no—
161C8×C9—33no—
162#578—31noSylow
163C2×C3×C9—23no—
164C2×C3×C9—41no—
165C2×C3×C9C22×D411yescharacteristic
166D3×C9—41no—
167C3×#41—26no—
168C3×#41—23no—
169C3×#41—23no—
170C3×#41—23no—
171C3×#41—21no—
172C3×#41—33no—
173C3×#41—63no—
174C3×#41C22×C913yes—
175C3×#41C22×C911yescharacteristic
176C3×#41C2×C3×D313yes—
177C3×#41C2×C3×D311yescharacteristic
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—23no—
182C2×C3×Q8—33no—
183C2×C3×Q8—61no—
184C2×C3×Q8C22×C913yes—
185C2×C3×Q8C2×C3×D313yes—
186D3×Q8—21no—
187#237—23no—
188#235C22×C913yes—
189C2×Dic6C22×C913yes—
190C3×Q16—33no—
191C3×Q16—36no—
192C3×SD16—33no—
193C3×SD16—33no—
194C3×D8—31no—
195C3×M16—33no—
196C2×C3×C8—33no—
197#216C22×C916yes—
198#216C22×C913yes—
199#215C22×C913yes—
200#214C22×C913yes—
201#213C22×C911yescharacteristic
202#208C22×C913yes—
203C2×C3⋊C8C22×C913yes—
204C22×C32—21no—
205C22×C32C2×C3×D413yes—
206C2×C3×D3—21no—
207C32×C4—23no—
208C32×C4C2×C3×D413yes—
209C32×C4C24×C311yescharacteristic
210C3×Dic3—23no—
211C22×C9—22no—
212C22×C9—21no—
213C22×C9—23no—
214C22×C9—61no—
215C22×C9C2×#6713yes—
216C4×C9—26no—
217C4×C9—23no—
218C4×C9—21no—
219C4×C9—23no—
220C4×C9—63no—
221C4×C9C2×#6713yes—
222C4×C9C23×D311yescharacteristic
2232−1+4—31no—
2242−1+4D3×C911yescharacteristic, complemented
225#137—33no—
226#137—33no—
227#135—33no—
228C2×Q16—33no—
229#131—31no—
230C3×C9#57811yescharacteristic, complemented, Sylow
231C3×Q8—26no—
232C3×Q8—23no—
233C3×Q8—23no—
234C3×Q8—21no—
235C3×Q8—23no—
236C3×Q8—33no—
237C3×Q8—63no—
238C3×Q8D4×C911yescharacteristic
239C3×Q8C23×C913yes—
240C3×Q8C3×#6711yescharacteristic
241C3×Q8C22×C3×D313yes—
242C3×D4—26no—
243C3×D4—23no—
244C3×D4—23no—
245C3×D4—23no—
246C3×D4—21no—
247C3×D4—31no—
248C3×D4—63no—
249C3×D4D4×C913yes—
250C3×D4C23×C911yescharacteristic
251C3×D4C3×#6713yes—
252C3×D4C22×C3×D311yescharacteristic
253C2×C3×C4—26no—
254C2×C3×C4—26no—
255C2×C3×C4—23no—
256C2×C3×C4—23no—
257C2×C3×C4—23no—
258C2×C3×C4—23no—
259C2×C3×C4—23no—
260C2×C3×C4—63no—
261C2×C3×C4—63no—
262C2×C3×C4C23×C913yes—
263C2×C3×C4C22×C3×D313yes—
264#67—23no—
265C2×Dic3—23no—
266D12C23×C911yescharacteristic
267C4×D3—23no—
268Dic6—23no—
269Dic6C23×C913yes—
270C3×C8—31no—
271C3×C8—33no—
272C3⋊C8C23×C911yescharacteristic
273C3⋊C8C23×C913yes—
274C2×C32—23no—
275C2×C32—41no—
276C2×C32C22×C3×D411yescharacteristic
277C3×D3—41no—
278C2×C9—23no—
279C2×C9—21no—
280C2×C9—43no—
281C2×C9—42no—
282C2×C9—41no—
283C2×C9—121no—
284C2×C9C22×#6711yescharacteristic
285#41—23no—
286#41—33no—
287#41—63no—
288#41C2×D3×C913yes—
289#41C2×D3×C911yescharacteristic
290C2×Q8—21no—
291C2×Q8—33no—
292C2×Q8—61no—
293C2×Q8C2×D3×C913yes—
294Q16—33no—
295Q16—36no—
296SD16—33no—
297SD16—33no—
298D8—31no—
299M16—33no—
300C2×C8—33no—
301C22×C3—22no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—23no—
305C22×C3—61no—
306C22×C3C2×D4×C913yes—
307C22×C3C2×C3×#6713yes—
308C2×D3—21no—
309C3×C4—26no—
310C3×C4—23no—
311C3×C4—23no—
312C3×C4—21no—
313C3×C4—23no—
314C3×C4—63no—
315C3×C4C2×D4×C913yes—
316C3×C4C24×C911yescharacteristic
317C3×C4C2×C3×#6713yes—
318C3×C4C23×C3×D311yescharacteristic
319Dic3—23no—
320C32C3×#57811yescharacteristic
321C9—21no—
322C9#583811yescharacteristic, direct product factor
323Q8—23no—
324Q8—33no—
325Q8—63no—
326Q8C9×#6711yescharacteristic
327Q8C22×D3×C913yes—
328D4—23no—
329D4—31no—
330D4—63no—
331D4C9×#6713yes—
332D4C22×D3×C911yescharacteristic
333C2×C4—23no—
334C2×C4—23no—
335C2×C4—63no—
336C2×C4—63no—
337C2×C4C22×D3×C913yes—
338C8—31no—
339C8—33no—
340C2×C3—23no—
341C2×C3—23no—
342C2×C3—21no—
343C2×C3—43no—
344C2×C3—42no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—121no—
348C2×C3C22×D4×C911yescharacteristic
349C2×C3C22×C3×#6711yescharacteristic
350D3—41no—
351C22—21no—
352C22—61no—
353C22C2×C9×#6713yes—
354C4—23no—
355C4—63no—
356C4C2×C9×#6713yes—
357C4C23×D3×C911yescharacteristic
358C3—21no—
359C3C9×#57811yescharacteristic, complemented
360C3C3×#583811yescharacteristic
361C2—23no—
362C2—41no—
363C2—121no—
364C2C22×C9×#6711yescharacteristic
subgroupG/H
centerC2×C9C22×#67
commutatorC3×C4C24×C9
FrattiniC3×C4C23×C3×D3
FittingC3×C9×2−1+4C2
socleC2×C32C22×C3×D4
prime radicalC3×#5838C3
cosocleC32×C4C24×C3
hypercenterC9×2−1+4D3
hypocenterC3C9×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C9×#41 ◁ C9×2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×C9×2−1+4C2semidirect
2. C9×#1010C2semidirect
3. C9×#951C2semidirect
4. C9×#950C2semidirect
5. C9×#948C2semidirect
6. C2×C9×#216C2semidirect
7. C9×#942C2semidirect
35. C9×2−1+4D3semidirect
59. #5838C9direct (N × H)
106. C3×2−1+4C2×C9semidirect
108. #1010C2×C9semidirect
114. #951C2×C9semidirect
115. #950C2×C9semidirect
116. #948C2×C9semidirect
117. C2×#216C2×C9semidirect
118. #942C2×C9semidirect
224. 2−1+4D3×C9semidirect
230. C3×C9#578semidirect
322. C9#5838direct (N × H)
359. C3C9×#578semidirect

Maximal subgroups

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

subgroupindex
C3×C9×2−1+42
C9×#10102
C9×#9512
C9×#9502
C9×#9482
C2×C9×#2162
C9×#9422
C3×#58383
C9×#5783