Finite Group Databasebeta

← C9×#5751 #13153406

Subgroups — C9×#5751 #13153406

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#1010C211yesmaximal, characteristic, complemented
2C9×#1009C211yesmaximal, characteristic, complemented
3C3×C9×#131C211yesmaximal, characteristic, complemented
4C9×#909C213yesmaximal, complemented
5C9×#908C213yesmaximal, complemented
6C9×#904C213yesmaximal, complemented
7C2×C9×#204C213yesmaximal, complemented
8C3×#5751C311yesmaximal, characteristic
9C9×#577—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C9×#239—21no—
12D3×Q8×C9—21no—
13C9×#237—23no—
14D3×D4×C9—23no—
15C9×#235C2213yes—
16C9×#235C2213yescomplemented
17C2×C9×D12C2213yes—
18C2×C9×Dic6C2213yescomplemented
19C3×C9×M16C2213yescomplemented
20C2×C3×C8×C9C2213yescomplemented
21C9×Dic12C2213yescomplemented
22C9×D24C2213yescomplemented
23C9×#204C2213yescomplemented
24C9×#204C2216yescomplemented
25C9×#204C2211yescharacteristic, complemented
26C3×#1010C2×C311yescharacteristic
27C3×#1009C2×C311yescharacteristic
28C32×#131C2×C311yescharacteristic
29C3×#909C2×C313yes—
30C3×#908C2×C313yes—
31C3×#904C2×C313yes—
32C2×C3×#204C2×C313yes—
33C9×2−1+4—31no—
34C9×H(5,𝔽2)—31no—
35C9×#137—33no—
36C9×AGL(1,ℤ/8)—33no—
37C9×#135—33no—
38C2×C9×SD16—33no—
39C9×#131—21no—
40C9×#131D311yescharacteristic, complemented
41C22×D3×C9—23no—
42C3×Q8×C9D411yescharacteristic
43C3×D4×C9D413yes—
44C2×C3×C4×C9C2313yes—
45C9×#67—23no—
46C9×#67—23no—
47C2×C9×Dic3—23no—
48C9×D12—23no—
49C9×D12C2313yes—
50C9×D12C2311yescharacteristic
51C4×D3×C9—23no—
52C4×D3×C9—23no—
53C9×Dic6—23no—
54C9×Dic6C2313yes—
55C9×Dic6C2311yescharacteristic
56C3×C8×C9C2311yescharacteristic
57C3×C8×C9C2313yes—
58C3×#577—31no—
59#5751C911yescharacteristic, direct product factor
60C32×#41C22×C311yescharacteristic
61C3×#239—21no—
62C3×D3×Q8—21no—
63C3×#237—23no—
64C3×D3×D4—23no—
65C3×#235C22×C313yes—
66C3×#235C22×C313yes—
67C2×C3×D12C22×C313yes—
68C2×C3×Dic6C22×C313yes—
69C32×M16C22×C313yes—
70C2×C32×C8C22×C313yes—
71C3×Dic12C22×C313yes—
72C3×D24C22×C313yes—
73C3×#204C22×C313yes—
74C3×#204C22×C316yes—
75C3×#204C22×C311yescharacteristic
76C9×#41—21no—
77C9×#41—33no—
78C9×#41—33no—
79C9×#41—61no—
80C9×#41—63no—
81C9×#41C2×D311yescharacteristic
82C2×Q8×C9—33no—
83C2×Q8×C9—61no—
84C2×D4×C9—33no—
85C2×D4×C9—63no—
86C9×Q16—33no—
87C9×SD16—33no—
88C9×SD16—36no—
89C9×SD16—31no—
90C9×D8—33no—
91C9×M16—23no—
92C9×M16C2×D313yescomplemented
93C2×C8×C9—23no—
94C2×C8×C9C2×D313yescomplemented
95C22×C3×C9C2×D413yes—
96C2×D3×C9—21no—
97C2×D3×C9—23no—
98C2×D3×C9—83no—
99C3×C4×C9C2×D413yes—
100C3×C4×C9C2411yescharacteristic
101C9×Dic3—23no—
102C9×Dic3—21no—
103C3×2−1+4—31no—
104C3×H(5,𝔽2)—31no—
105#1010C2×C911yescharacteristic, complemented
106#1009C2×C911yescharacteristic, complemented
107C3×#137—33no—
108C3×AGL(1,ℤ/8)—33no—
109C3×#135—33no—
110C2×C3×SD16—33no—
111C3×#131—21no—
112C3×#131C2×C911yescharacteristic, complemented
113C3×#131C3×D311yescharacteristic
114#909C2×C913yescomplemented
115#908C2×C913yescomplemented
116#904C2×C913yescomplemented
117C2×#204C2×C913yescomplemented
118C22×C3×D3—23no—
119C32×Q8C3×D411yescharacteristic
120C32×D4C3×D413yes—
121C2×C32×C4C23×C313yes—
122C3×#67—23no—
123C3×#67—23no—
124C2×C3×Dic3—23no—
125C3×D12—23no—
126C3×D12C23×C313yes—
127C3×D12C23×C311yescharacteristic
128C3×C4×D3—23no—
129C3×C4×D3—23no—
130C3×Dic6—23no—
131C3×Dic6C23×C313yes—
132C3×Dic6C23×C311yescharacteristic
133C23×C9—63no—
134C32×C8C23×C311yescharacteristic
135C32×C8C23×C313yes—
136Q8×C9—21no—
137Q8×C9—33no—
138Q8×C9—31no—
139Q8×C9—63no—
140Q8×C9D1211yescharacteristic
141D4×C9—23no—
142D4×C9—31no—
143D4×C9—33no—
144D4×C9—63no—
145D4×C9—63no—
146D4×C9—63no—
147D4×C9D1213yes—
148C2×C4×C9—23no—
149C2×C4×C9—63no—
150C2×C4×C9—63no—
151C2×C4×C9—63no—
152C2×C4×C9C22×D313yes—
153C8×C9—21no—
154C8×C9—23no—
155C8×C9C22×D311yescharacteristic
156C8×C9C22×D313yes—
157#577—31noSylow
158C2×C3×C9—23no—
159C2×C3×C9C22×D411yescharacteristic
160D3×C9—41no—
161D3×C9—43no—
162C3×#41—21no—
163C3×#41—33no—
164C3×#41—33no—
165C3×#41—61no—
166C3×#41—63no—
167C3×#41C22×C911yescharacteristic
168C3×#41C2×C3×D311yescharacteristic
169C2×C3×Q8—33no—
170C2×C3×Q8—61no—
171C2×C3×D4—33no—
172C2×C3×D4—63no—
173#239—21no—
174D3×Q8—21no—
175#237—23no—
176D3×D4—23no—
177#235C22×C913yes—
178#235C22×C913yescomplemented
179C2×D12C22×C913yes—
180C2×Dic6C22×C913yescomplemented
181C3×Q16—33no—
182C3×SD16—33no—
183C3×SD16—36no—
184C3×SD16—31no—
185C3×D8—33no—
186C3×M16—23no—
187C3×M16C22×C913yescomplemented
188C3×M16C2×C3×D313yes—
189C2×C3×C8—23no—
190C2×C3×C8C22×C913yescomplemented
191C2×C3×C8C2×C3×D313yes—
192Dic12C22×C913yescomplemented
193D24C22×C913yescomplemented
194#204C22×C913yescomplemented
195#204C22×C916yescomplemented
196#204C22×C911yescharacteristic, complemented
197C22×C32C2×C3×D413yes—
198C2×C3×D3—21no—
199C2×C3×D3—23no—
200C2×C3×D3—83no—
201C32×C4C2×C3×D413yes—
202C32×C4C24×C311yescharacteristic
203C3×Dic3—23no—
204C3×Dic3—21no—
205C22×C9—23no—
206C22×C9—61no—
207C22×C9—63no—
208C22×C9—243no—
209C22×C9C2×D1213yes—
210C4×C9—21no—
211C4×C9—23no—
212C4×C9—61no—
213C4×C9—63no—
214C4×C9C2×D1213yes—
215C4×C9C23×D311yescharacteristic
2162−1+4—31no—
217H(5,𝔽2)—31no—
218#137—33no—
219AGL(1,ℤ/8)—33no—
220#135—33no—
221C2×SD16—33no—
222#131D3×C911yescharacteristic, complemented
223C3×C9#57711yescharacteristic, complemented, Sylow
224C23×C3—63no—
225C22×D3—23no—
226C3×Q8—21no—
227C3×Q8—33no—
228C3×Q8—31no—
229C3×Q8—63no—
230C3×Q8D4×C911yescharacteristic
231C3×Q8C3×D1211yescharacteristic
232C3×D4—23no—
233C3×D4—31no—
234C3×D4—33no—
235C3×D4—63no—
236C3×D4—63no—
237C3×D4—63no—
238C3×D4D4×C913yes—
239C3×D4C3×D1213yes—
240C2×C3×C4—23no—
241C2×C3×C4—63no—
242C2×C3×C4—63no—
243C2×C3×C4—63no—
244C2×C3×C4C23×C913yes—
245C2×C3×C4C22×C3×D313yes—
246#67—23no—
247#67—23no—
248C2×Dic3—23no—
249D12—23no—
250D12C23×C913yes—
251D12C23×C911yescharacteristic
252C4×D3—23no—
253C4×D3—23no—
254Dic6—23no—
255Dic6C23×C913yes—
256Dic6C23×C911yescharacteristic
257C3×C8—21no—
258C3×C8—23no—
259C3×C8C23×C911yescharacteristic
260C3×C8C23×C913yes—
261C3×C8C22×C3×D311yescharacteristic
262C3×C8C22×C3×D313yes—
263C2×C32—23no—
264C2×C32C22×C3×D411yescharacteristic
265C3×D3—41no—
266C3×D3—43no—
267C2×C9—23no—
268C2×C9—21no—
269C2×C9—43no—
270C2×C9—121no—
271C2×C9—123no—
272C2×C9C22×D1211yescharacteristic
273#41—33no—
274#41—33no—
275#41—61no—
276#41—63no—
277#41C2×D3×C911yescharacteristic
278C2×Q8—33no—
279C2×Q8—61no—
280C2×D4—33no—
281C2×D4—63no—
282Q16—33no—
283SD16—33no—
284SD16—36no—
285SD16—31no—
286D8—33no—
287M16C2×D3×C913yescomplemented
288C2×C8C2×D3×C913yescomplemented
289C22×C3—23no—
290C22×C3—61no—
291C22×C3—63no—
292C22×C3—243no—
293C22×C3C2×D4×C913yes—
294C22×C3C2×C3×D1213yes—
295C2×D3—21no—
296C2×D3—23no—
297C2×D3—83no—
298C3×C4—21no—
299C3×C4—23no—
300C3×C4—61no—
301C3×C4—63no—
302C3×C4C2×D4×C913yes—
303C3×C4C24×C911yescharacteristic
304C3×C4C2×C3×D1213yes—
305C3×C4C23×C3×D311yescharacteristic
306Dic3—23no—
307Dic3—21no—
308C32C3×#57711yescharacteristic
309C9—21no—
310C9#575111yescharacteristic, direct product factor
311C23—63no—
312Q8—33no—
313Q8—31no—
314Q8—63no—
315Q8C9×D1211yescharacteristic
316D4—31no—
317D4—33no—
318D4—63no—
319D4—63no—
320D4—63no—
321D4C9×D1213yes—
322C2×C4—63no—
323C2×C4—63no—
324C2×C4—63no—
325C2×C4C22×D3×C913yes—
326C8C22×D3×C911yescharacteristic
327C8C22×D3×C913yes—
328C2×C3—23no—
329C2×C3—23no—
330C2×C3—21no—
331C2×C3—43no—
332C2×C3—123no—
333C2×C3—121no—
334C2×C3C22×D4×C911yescharacteristic
335C2×C3C22×C3×D1211yescharacteristic
336D3—43no—
337D3—41no—
338C22—61no—
339C22—63no—
340C22—243no—
341C22C2×C9×D1213yes—
342C4—61no—
343C4—63no—
344C4C2×C9×D1213yes—
345C4C23×D3×C911yescharacteristic
346C3—21no—
347C3C9×#57711yescharacteristic, complemented
348C3C3×#575111yescharacteristic
349C2—23no—
350C2—123no—
351C2—121no—
352C2C22×C9×D1211yescharacteristic
subgroupG/H
centerC2×C9C22×D12
commutatorC3×C4C24×C9
FrattiniC3×C4C23×C3×D3
FittingC3×C9×#131C2
socleC2×C32C22×C3×D4
prime radicalC3×#5751C3
cosocleC32×C4C24×C3
hypercenterC9×#131D3
hypocenterC3C9×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C9×#41 ◁ C9×#131

Complements

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

NH ≅ G/Ntype
1. C9×#1010C2semidirect
2. C9×#1009C2semidirect
3. C3×C9×#131C2semidirect
4. C9×#909C2semidirect
5. C9×#908C2semidirect
6. C9×#904C2semidirect
7. C2×C9×#204C2semidirect
16. C9×#235C22semidirect
18. C2×C9×Dic6C22semidirect
19. C3×C9×M16C22semidirect
20. C2×C3×C8×C9C22semidirect
21. C9×Dic12C22semidirect
22. C9×D24C22semidirect
23. C9×#204C22semidirect
24. C9×#204C22semidirect
25. C9×#204C22semidirect
40. C9×#131D3semidirect
59. #5751C9direct (N × H)
92. C9×M16C2×D3semidirect
94. C2×C8×C9C2×D3semidirect
105. #1010C2×C9semidirect
106. #1009C2×C9semidirect
112. C3×#131C2×C9semidirect
114. #909C2×C9semidirect
115. #908C2×C9semidirect
116. #904C2×C9semidirect
117. C2×#204C2×C9semidirect
178. #235C22×C9semidirect
180. C2×Dic6C22×C9semidirect
187. C3×M16C22×C9semidirect
190. C2×C3×C8C22×C9semidirect
192. Dic12C22×C9semidirect
193. D24C22×C9semidirect
194. #204C22×C9semidirect
195. #204C22×C9semidirect
196. #204C22×C9semidirect
222. #131D3×C9semidirect
223. C3×C9#577semidirect
287. M16C2×D3×C9semidirect
288. C2×C8C2×D3×C9semidirect
310. C9#5751direct (N × H)
347. C3C9×#577semidirect

Maximal subgroups

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

subgroupindex
C9×#10102
C9×#10092
C3×C9×#1312
C9×#9092
C9×#9082
C9×#9042
C2×C9×#2042
C3×#57513
C9×#5773