Finite Group Databasebeta

← C9×#5576 #13152697

Subgroups — C9×#5576 #13152697

Order 1728.

940 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#239C211yesmaximal, characteristic, complemented
2C2×D3×Q8×C9C211yesmaximal, characteristic, complemented
3C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
4C9×#897C213yesmaximal, complemented
5C9×#895C213yesmaximal, complemented
6C9×#875C213yesmaximal, complemented
7C4×C9×D12C213yesmaximal, complemented
8C3×#5576C311yesmaximal, characteristic
9C9×#548—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C9×#239—24no—
12D3×Q8×C9—24no—
13C2×C9×D12C2213yes—
14C2×C9×D12C2213yes—
15C2×C4×D3×C9C2213yes—
16C2×C4×D3×C9C2213yes—
17C2×C9×Dic6C2213yes—
18C3×C9×C4⋊C4C2213yes—
19C3×C42×C9C2213yes—
20C9×(C2×D3)⋊C4C2211yescharacteristic
21C9×(C2×D3)⋊C4C2213yes—
22C9×(C2×D3)⋊C4C2216yes—
23C9×C4⋊Dic3C2213yes—
24C2×C3×#239C2×C311yescharacteristic
25C2×C3×D3×Q8C2×C311yescharacteristic
26C32×C4×Q8C2×C311yescharacteristic
27C3×#897C2×C313yes—
28C3×#895C2×C313yes—
29C3×#875C2×C313yes—
30C3×C4×D12C2×C313yes—
31C2×C9×#41—31no—
32C22×Q8×C9—31no—
33C9×#124—33no—
34C9×C22⋊Q8—33no—
35C9×#121—33no—
36C4×Q8×C9—21no—
37C4×Q8×C9D311yescharacteristic, complemented
38C4×D4×C9—33no—
39C22×D3×C9C2313yes—
40C22×D3×C9C2311yescharacteristic
41C3×Q8×C9D414yescomplemented
42C2×C3×C4×C9C2313yes—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3×C4×C9C2313yes—
45C2×C9×Dic3C2313yes—
46C2×C9×Dic3C2311yescharacteristic
47C9×D12—26no—
48C9×D12—26no—
49C4×D3×C9—26no—
50C4×D3×C9—26no—
51C9×Dic6—26no—
52C3×#548—31no—
53#5576C911yescharacteristic, direct product factor
54C2×C32×Q8C22×C311yescharacteristic
55C3×#239—24no—
56C3×D3×Q8—24no—
57C2×C3×D12C22×C313yes—
58C2×C3×D12C22×C313yes—
59C2×C3×C4×D3C22×C313yes—
60C2×C3×C4×D3C22×C313yes—
61C2×C3×Dic6C22×C313yes—
62C32×C4⋊C4C22×C313yes—
63C32×C42C22×C313yes—
64C3×(C2×D3)⋊C4C22×C316yes—
65C3×(C2×D3)⋊C4C22×C311yescharacteristic
66C3×(C2×D3)⋊C4C22×C313yes—
67C3×C4⋊Dic3C22×C313yes—
68C9×#41—64no—
69C2×Q8×C9—21no—
70C2×Q8×C9—33no—
71C2×Q8×C9—64no—
72C2×Q8×C9C2×D311yescharacteristic
73C2×D4×C9—33no—
74C2×D4×C9—33no—
75C22×C4×C9—33no—
76C22×C4×C9—33no—
77C9×C4⋊C4—23no—
78C9×C4⋊C4—33no—
79C9×C4⋊C4C2×D313yes—
80C9×C22⋊C4—31no—
81C9×C22⋊C4—33no—
82C9×C22⋊C4—36no—
83C42×C9—23no—
84C42×C9C2×D313yes—
85C22×C3×C9C2411yescharacteristic
86C2×D3×C9—21no—
87C2×D3×C9—21no—
88C2×D3×C9—23no—
89C2×D3×C9—23no—
90C2×D3×C9—23no—
91C2×D3×C9#4112yescomplemented
92C3×C4×C9—23no—
93C3×C4×C9—21no—
94C3×C4×C9C2×D416yes—
95C9×Dic3—23no—
96C9×Dic3—21no—
97C2×C3×#41—31no—
98C22×C3×Q8—31no—
99C2×#239C2×C911yescharacteristic, complemented
100C2×D3×Q8C2×C911yescharacteristic, complemented
101C3×#124—33no—
102C3×C22⋊Q8—33no—
103C3×#121—33no—
104C3×C4×Q8—21no—
105C3×C4×Q8C2×C911yescharacteristic, complemented
106C3×C4×Q8C3×D311yescharacteristic
107C3×C4×D4—33no—
108#897C2×C913yescomplemented
109#895C2×C913yescomplemented
110#875C2×C913yescomplemented
111C4×D12C2×C913yescomplemented
112C22×C3×D3C23×C313yes—
113C22×C3×D3C23×C311yescharacteristic
114C32×Q8C3×D414yes—
115C2×C32×C4C23×C313yes—
116C2×C32×C4C23×C311yescharacteristic
117C2×C32×C4C23×C313yes—
118C2×C3×Dic3C23×C311yescharacteristic
119C2×C3×Dic3C23×C313yes—
120C3×D12—26no—
121C3×D12—26no—
122C3×C4×D3—26no—
123C3×C4×D3—26no—
124C3×Dic6—26no—
125C23×C9—31no—
126C23×C9—33no—
127Q8×C9—24no—
128Q8×C9—66no—
129Q8×C9D1214yescomplemented
130D4×C9—66no—
131D4×C9—66no—
132C2×C4×C9—21no—
133C2×C4×C9—23no—
134C2×C4×C9—23no—
135C2×C4×C9—33no—
136C2×C4×C9—31no—
137C2×C4×C9—66no—
138C2×C4×C9—66no—
139C2×C4×C9C22×D311yescharacteristic
140C2×C4×C9C22×D313yes—
141C2×C4×C9C22×D313yes—
142#548—31noSylow
143C2×C3×C9C22×D411yescharacteristic
144C2×C3×C9C2×#4111yescharacteristic
145C2×C3×C92−1+411yescharacteristic
146D3×C9—22no—
147D3×C9—43no—
148C3×#41—64no—
149C2×C3×Q8—21no—
150C2×C3×Q8—33no—
151C2×C3×Q8—64no—
152C2×C3×Q8C22×C911yescharacteristic
153C2×C3×Q8C2×C3×D311yescharacteristic
154C2×C3×D4—33no—
155C2×C3×D4—33no—
156C22×C3×C4—33no—
157C22×C3×C4—33no—
158#239—24no—
159D3×Q8—24no—
160C2×D12C22×C913yes—
161C2×D12C22×C913yes—
162C2×C4×D3C22×C913yes—
163C2×C4×D3C22×C913yes—
164C2×Dic6C22×C913yes—
165C3×C4⋊C4—23no—
166C3×C4⋊C4—33no—
167C3×C4⋊C4C22×C913yes—
168C3×C4⋊C4C2×C3×D313yes—
169C3×C22⋊C4—31no—
170C3×C22⋊C4—33no—
171C3×C22⋊C4—36no—
172C3×C42—23no—
173C3×C42C22×C913yes—
174C3×C42C2×C3×D313yes—
175(C2×D3)⋊C4C22×C911yescharacteristic
176(C2×D3)⋊C4C22×C913yes—
177(C2×D3)⋊C4C22×C916yes—
178C4⋊Dic3C22×C913yes—
179C22×C32C24×C311yescharacteristic
180C2×C3×D3—21no—
181C2×C3×D3—21no—
182C2×C3×D3—23no—
183C2×C3×D3—23no—
184C2×C3×D3—23no—
185C2×C3×D3C3×#4112yes—
186C32×C4—21no—
187C32×C4—23no—
188C32×C4C2×C3×D416yes—
189C3×Dic3—23no—
190C3×Dic3—21no—
191C22×C9—21no—
192C22×C9—32no—
193C22×C9—61no—
194C22×C9—61no—
195C22×C9—63no—
196C22×C9—63no—
197C22×C9—63no—
198C22×C9C23×D311yescharacteristic
199C4×C9—22no—
200C4×C9—21no—
201C4×C9—23no—
202C4×C9—26no—
203C4×C9—43no—
204C4×C9—61no—
205C4×C9—63no—
206C4×C9C2×D1216yes—
207C2×#41—31no—
208C22×Q8—31no—
209#124—33no—
210C22⋊Q8—33no—
211#121—33no—
212C4×Q8D3×C911yescharacteristic, complemented
213C4×D4—33no—
214C3×C9#54811yescharacteristic, complemented, Sylow
215C23×C3—31no—
216C23×C3—33no—
217C22×D3C23×C913yes—
218C22×D3C23×C911yescharacteristic
219C3×Q8—24no—
220C3×Q8—66no—
221C3×Q8D4×C914yescomplemented
222C3×Q8C3×D1214yes—
223C3×D4—66no—
224C3×D4—66no—
225C2×C3×C4—21no—
226C2×C3×C4—23no—
227C2×C3×C4—23no—
228C2×C3×C4—33no—
229C2×C3×C4—31no—
230C2×C3×C4—66no—
231C2×C3×C4—66no—
232C2×C3×C4C23×C913yes—
233C2×C3×C4C23×C911yescharacteristic
234C2×C3×C4C23×C913yes—
235C2×C3×C4C22×C3×D311yescharacteristic
236C2×C3×C4C22×C3×D313yes—
237C2×C3×C4C22×C3×D313yes—
238C2×Dic3C23×C911yescharacteristic
239C2×Dic3C23×C913yes—
240D12—26no—
241D12—26no—
242C4×D3—26no—
243C4×D3—26no—
244Dic6—26no—
245C2×C32C22×C3×D411yescharacteristic
246C2×C32C2×C3×#4111yescharacteristic
247C2×C32C3×2−1+411yescharacteristic
248C3×D3—22no—
249C3×D3—43no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—21no—
253C2×C9—62no—
254C2×C9—123no—
255C2×C9C22×D1211yescharacteristic
256C2×C9#100711yescharacteristic
257C2×C9D3×#4111yescharacteristic
258#41—64no—
259C2×Q8—33no—
260C2×Q8—64no—
261C2×Q8C2×D3×C911yescharacteristic
262C2×D4—33no—
263C2×D4—33no—
264C22×C4—33no—
265C22×C4—33no—
266C4⋊C4—33no—
267C4⋊C4C2×D3×C913yes—
268C22⋊C4—31no—
269C22⋊C4—33no—
270C22⋊C4—36no—
271C42C2×D3×C913yes—
272C22×C3—21no—
273C22×C3—32no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—63no—
277C22×C3—63no—
278C22×C3—63no—
279C22×C3C24×C911yescharacteristic
280C22×C3C23×C3×D311yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—23no—
284C2×D3—23no—
285C2×D3—23no—
286C2×D3C9×#4112yescomplemented
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—23no—
290C3×C4—23no—
291C3×C4—21no—
292C3×C4—26no—
293C3×C4—43no—
294C3×C4—61no—
295C3×C4—63no—
296C3×C4C2×D4×C916yes—
297C3×C4C2×C3×D1216yes—
298Dic3—23no—
299Dic3—21no—
300C32C3×#54811yescharacteristic
301C9—21no—
302C9#557611yescharacteristic, direct product factor
303C23—31no—
304C23—33no—
305Q8—66no—
306Q8C9×D1214yescomplemented
307D4—66no—
308D4—66no—
309C2×C4—33no—
310C2×C4—31no—
311C2×C4—66no—
312C2×C4—66no—
313C2×C4C22×D3×C911yescharacteristic
314C2×C4C22×D3×C913yes—
315C2×C4C22×D3×C913yes—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—62no—
320C2×C3—123no—
321C2×C3C22×D4×C911yescharacteristic
322C2×C3C2×C9×#4111yescharacteristic
323C2×C3C9×2−1+411yescharacteristic
324C2×C3C22×C3×D1211yescharacteristic
325C2×C3C3×#100711yescharacteristic
326C2×C3C3×D3×#4111yescharacteristic
327D3—22no—
328D3—43no—
329C22—32no—
330C22—61no—
331C22—61no—
332C22—63no—
333C22—63no—
334C22—63no—
335C22C23×D3×C911yescharacteristic
336C4—21no—
337C4—23no—
338C4—61no—
339C4—63no—
340C4C2×C9×D1216yes—
341C3—21no—
342C3C9×#54811yescharacteristic, complemented
343C3C3×#557611yescharacteristic
344C2—62no—
345C2—123no—
346C2C22×C9×D1211yescharacteristic
347C2C9×#100711yescharacteristic
348C2D3×C9×#4111yescharacteristic
subgroupG/H
centerC22×C9C23×D3
commutatorC22×C3C24×C9
FrattiniC22×C3C23×C3×D3
FittingC3×C4×Q8×C9C2
socle, cosocleC22×C32C24×C3
prime radicalC3×#5576C3
hypercenterC4×Q8×C9D3
hypocenterC3C9×#548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C4×Q8×C9

Complements

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

NH ≅ G/Ntype
1. C2×C9×#239C2semidirect
2. C2×D3×Q8×C9C2semidirect
3. C3×C4×Q8×C9C2semidirect
4. C9×#897C2semidirect
5. C9×#895C2semidirect
6. C9×#875C2semidirect
7. C4×C9×D12C2semidirect
37. C4×Q8×C9D3semidirect
41. C3×Q8×C9D4semidirect
53. #5576C9direct (N × H)
91. C2×D3×C9#41semidirect
99. C2×#239C2×C9semidirect
100. C2×D3×Q8C2×C9semidirect
105. C3×C4×Q8C2×C9semidirect
108. #897C2×C9semidirect
109. #895C2×C9semidirect
110. #875C2×C9semidirect
111. C4×D12C2×C9semidirect
129. Q8×C9D12semidirect
212. C4×Q8D3×C9semidirect
214. C3×C9#548semidirect
221. C3×Q8D4×C9semidirect
286. C2×D3C9×#41semidirect
302. C9#5576direct (N × H)
306. Q8C9×D12semidirect
342. C3C9×#548semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2392
C2×D3×Q8×C92
C3×C4×Q8×C92
C9×#8972
C9×#8952
C9×#8752
C4×C9×D122
C3×#55763
C9×#5483