Finite Group Databasebeta

← C9×#5574 #13152695

Subgroups — C9×#5574 #13152695

Order 1728.

952 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#239C211yesmaximal, characteristic
2C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
3Q8×C9×Dic3C211yesmaximal, characteristic, complemented
4C9×#893C213yesmaximal, complemented
5D3×C9×C4⋊C4C213yesmaximal, complemented
6C4×C9×D12C213yesmaximal, complemented
7C9×#872C213yesmaximal, complemented
8C3×#5574C311yesmaximal, characteristic
9C9×#520—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C9×#239C418yescomplemented
12C2×C9×D12C2213yes—
13C2×C4×D3×C9C2216yes—
14C2×C4×D3×C9C2213yes—
15C3×C9×C4⋊C4C2213yes—
16C3×C42×C9C2213yes—
17C9×(C2×D3)⋊C4C2213yes—
18C9×(C2×D3)⋊C4C2213yes—
19C9×C4⋊Dic3C2213yes—
20C9×Dic3⋊C4C2211yescharacteristic
21C9×Dic3⋊C4C2213yes—
22C4×C9×Dic3C2213yes—
23C2×C3×#239C2×C311yescharacteristic
24C32×C4×Q8C2×C311yescharacteristic
25C3×Q8×Dic3C2×C311yescharacteristic
26C3×#893C2×C313yes—
27C3×D3×C4⋊C4C2×C313yes—
28C3×C4×D12C2×C313yes—
29C3×#872C2×C313yes—
30C2×C9×#41—31no—
31C4×Q8×C9—21no—
32C4×Q8×C9—31no—
33C4×Q8×C9D311yescharacteristic, complemented
34C4×D4×C9—33no—
35C4×D4×C9—33no—
36C9×#117—33no—
37C2×C9×C4⋊C4—33no—
38C22×D3×C9C2313yes—
39C3×Q8×C9C2×C414yescomplemented
40C2×C3×C4×C9C2313yes—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3×C4×C9C2313yes—
43C2×C9×Dic3C2313yes—
44C2×C9×Dic3C2311yescharacteristic
45C2×C9×Dic3C2311yescharacteristic
46C9×D12C2×C4112yescomplemented
47C4×D3×C9—46no—
48C4×D3×C9C2×C4112yescomplemented
49C3×#520—31no—
50#5574C911yescharacteristic, direct product factor
51C2×C32×Q8C22×C311yescharacteristic
52C3×#239C3×C418yes—
53C2×C3×D12C22×C313yes—
54C2×C3×C4×D3C22×C316yes—
55C2×C3×C4×D3C22×C313yes—
56C32×C4⋊C4C22×C313yes—
57C32×C42C22×C313yes—
58C3×(C2×D3)⋊C4C22×C313yes—
59C3×(C2×D3)⋊C4C22×C313yes—
60C3×C4⋊Dic3C22×C313yes—
61C3×Dic3⋊C4C22×C313yes—
62C3×Dic3⋊C4C22×C311yescharacteristic
63C3×C4×Dic3C22×C313yes—
64C9×#41—38no—
65C2×Q8×C9—21no—
66C2×Q8×C9C2×D311yescharacteristic
67C2×D4×C9—33no—
68C22×C4×C9—33no—
69C22×C4×C9—36no—
70C9×C4⋊C4—23no—
71C9×C4⋊C4—33no—
72C9×C4⋊C4—33no—
73C9×C4⋊C4—31no—
74C9×C4⋊C4C2×D313yes—
75C9×C22⋊C4—33no—
76C9×C22⋊C4—33no—
77C42×C9—23no—
78C42×C9—33no—
79C42×C9C2×D313yes—
80C22×C3×C9C2411yescharacteristic
81C2×D3×C9—23no—
82C2×D3×C9—23no—
83C2×D3×C9C22×C416yes—
84C3×C4×C9—23no—
85C3×C4×C9—21no—
86C3×C4×C9C22×C416yes—
87C9×Dic3—23no—
88C9×Dic3—21no—
89C9×Dic3C22×C412yes—
90C2×C3×#41—31no—
91C2×#239C2×C911yescharacteristic
92C3×C4×Q8—21no—
93C3×C4×Q8—31no—
94C3×C4×Q8C2×C911yescharacteristic, complemented
95C3×C4×Q8C3×D311yescharacteristic
96C3×C4×D4—33no—
97C3×C4×D4—33no—
98C3×#117—33no—
99C2×C3×C4⋊C4—33no—
100Q8×Dic3C2×C911yescharacteristic, complemented
101#893C2×C913yescomplemented
102D3×C4⋊C4C2×C913yescomplemented
103C4×D12C2×C913yescomplemented
104#872C2×C913yescomplemented
105C22×C3×D3C23×C313yes—
106C32×Q8C2×C3×C414yes—
107C2×C32×C4C23×C311yescharacteristic
108C2×C32×C4C23×C313yes—
109C2×C32×C4C23×C313yes—
110C2×C3×Dic3C23×C311yescharacteristic
111C2×C3×Dic3C23×C313yes—
112C2×C3×Dic3C23×C311yescharacteristic
113C3×D12C2×C3×C4112yes—
114C3×C4×D3—46no—
115C3×C4×D3C2×C3×C4112yes—
116C23×C9—33no—
117Q8×C9—24no—
118Q8×C9C4×D314yescomplemented
119D4×C9—312no—
120C2×C4×C9—21no—
121C2×C4×C9—23no—
122C2×C4×C9—23no—
123C2×C4×C9—31no—
124C2×C4×C9—312no—
125C2×C4×C9—31no—
126C2×C4×C9—33no—
127C2×C4×C9—126no—
128C2×C4×C9C22×D311yescharacteristic
129C2×C4×C9C22×D313yes—
130C2×C4×C9C22×D313yes—
131#520—31noSylow
132C2×C3×C9C23×C411yescharacteristic
133C2×C3×C9H(5,𝔽2)11yescharacteristic
134C2×C3×C92−1+411yescharacteristic
135D3×C9—26no—
136C3×#41—38no—
137C2×C3×Q8—21no—
138C2×C3×Q8C22×C911yescharacteristic
139C2×C3×Q8C2×C3×D311yescharacteristic
140C2×C3×D4—33no—
141C22×C3×C4—33no—
142C22×C3×C4—36no—
143#239C4×C918yescomplemented
144C2×D12C22×C913yes—
145C2×C4×D3C22×C916yes—
146C2×C4×D3C22×C913yes—
147C3×C4⋊C4—23no—
148C3×C4⋊C4—33no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—33no—
151C3×C4⋊C4C22×C913yes—
152C3×C4⋊C4C2×C3×D313yes—
153C3×C22⋊C4—33no—
154C3×C22⋊C4—33no—
155C3×C42—23no—
156C3×C42—33no—
157C3×C42C22×C913yes—
158C3×C42C2×C3×D313yes—
159(C2×D3)⋊C4C22×C913yes—
160(C2×D3)⋊C4C22×C913yes—
161C4⋊Dic3C22×C913yes—
162Dic3⋊C4C22×C911yescharacteristic
163Dic3⋊C4C22×C913yes—
164C4×Dic3C22×C913yes—
165C22×C32C24×C311yescharacteristic
166C2×C3×D3—23no—
167C2×C3×D3—23no—
168C2×C3×D3C22×C3×C416yes—
169C32×C4—21no—
170C32×C4—23no—
171C32×C4C22×C3×C416yes—
172C3×Dic3—23no—
173C3×Dic3—21no—
174C3×Dic3C22×C3×C412yes—
175C22×C9—21no—
176C22×C9—36no—
177C22×C9—63no—
178C22×C9—63no—
179C22×C9C23×D311yescharacteristic
180C4×C9—22no—
181C4×C9—21no—
182C4×C9—23no—
183C4×C9—26no—
184C4×C9—32no—
185C4×C9—43no—
186C4×C9—61no—
187C4×C9—63no—
188C4×C9C2×C4×D316yes—
189C2×#41—31no—
190C4×Q8—31no—
191C4×Q8D3×C911yescharacteristic, complemented
192C4×D4—33no—
193C4×D4—33no—
194#117—33no—
195C2×C4⋊C4—33no—
196C3×C9#52011yescharacteristic, complemented, Sylow
197C23×C3—33no—
198C22×D3C23×C913yes—
199C3×Q8—24no—
200C3×Q8C2×C4×C914yescomplemented
201C3×Q8C3×C4×D314yes—
202C3×D4—312no—
203C2×C3×C4—21no—
204C2×C3×C4—23no—
205C2×C3×C4—23no—
206C2×C3×C4—31no—
207C2×C3×C4—312no—
208C2×C3×C4—31no—
209C2×C3×C4—33no—
210C2×C3×C4—126no—
211C2×C3×C4C23×C911yescharacteristic
212C2×C3×C4C23×C913yes—
213C2×C3×C4C23×C913yes—
214C2×C3×C4C22×C3×D311yescharacteristic
215C2×C3×C4C22×C3×D313yes—
216C2×C3×C4C22×C3×D313yes—
217C2×Dic3C23×C911yescharacteristic
218C2×Dic3C23×C913yes—
219C2×Dic3C23×C911yescharacteristic
220D12C2×C4×C9112yescomplemented
221C4×D3—46no—
222C4×D3C2×C4×C9112yescomplemented
223C2×C32C23×C3×C411yescharacteristic
224C2×C32C3×H(5,𝔽2)11yescharacteristic
225C2×C32C3×2−1+411yescharacteristic
226C3×D3—26no—
227C2×C9—21no—
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—66no—
231C2×C9C22×C4×D311yescharacteristic
232C2×C9#100711yescharacteristic
233C2×C9#100911yescharacteristic
234#41—38no—
235C2×Q8C2×D3×C911yescharacteristic
236C2×D4—33no—
237C22×C4—33no—
238C22×C4—36no—
239C4⋊C4—33no—
240C4⋊C4—33no—
241C4⋊C4—31no—
242C4⋊C4C2×D3×C913yes—
243C22⋊C4—33no—
244C22⋊C4—33no—
245C42—33no—
246C42C2×D3×C913yes—
247C22×C3—21no—
248C22×C3—36no—
249C22×C3—63no—
250C22×C3—63no—
251C22×C3C24×C911yescharacteristic
252C22×C3C23×C3×D311yescharacteristic
253C2×D3—23no—
254C2×D3—23no—
255C2×D3C22×C4×C916yes—
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—23no—
259C3×C4—23no—
260C3×C4—21no—
261C3×C4—26no—
262C3×C4—32no—
263C3×C4—43no—
264C3×C4—61no—
265C3×C4—63no—
266C3×C4C22×C4×C916yes—
267C3×C4C2×C3×C4×D316yes—
268Dic3—23no—
269Dic3—21no—
270Dic3C22×C4×C912yes—
271C32C3×#52011yescharacteristic
272C9—21no—
273C9#557411yescharacteristic, direct product factor
274C23—33no—
275Q8C4×D3×C914yescomplemented
276D4—312no—
277C2×C4—31no—
278C2×C4—312no—
279C2×C4—31no—
280C2×C4—33no—
281C2×C4—126no—
282C2×C4C22×D3×C911yescharacteristic
283C2×C4C22×D3×C913yes—
284C2×C4C22×D3×C913yes—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—66no—
289C2×C3C23×C4×C911yescharacteristic
290C2×C3C9×H(5,𝔽2)11yescharacteristic
291C2×C3C9×2−1+411yescharacteristic
292C2×C3C22×C3×C4×D311yescharacteristic
293C2×C3C3×#100711yescharacteristic
294C2×C3C3×#100911yescharacteristic
295D3—26no—
296C22—36no—
297C22—63no—
298C22—63no—
299C22C23×D3×C911yescharacteristic
300C4—21no—
301C4—23no—
302C4—32no—
303C4—61no—
304C4—63no—
305C4C2×C4×D3×C916yes—
306C3—21no—
307C3C9×#52011yescharacteristic, complemented
308C3C3×#557411yescharacteristic
309C2—66no—
310C2C22×C4×D3×C911yescharacteristic
311C2C9×#100711yescharacteristic
312C2C9×#100911yescharacteristic
subgroupG/H
centerC22×C9C23×D3
commutatorC2×C3C23×C4×C9
FrattiniC22×C3C23×C3×D3
FittingC3×C4×Q8×C9C2
socle, cosocleC22×C32C24×C3
prime radicalC2×C3×#239C2×C3
hypercenterC4×Q8×C9D3
hypocenterC3C9×#520
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C4×Q8×C9

Complements

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

NH ≅ G/Ntype
2. C3×C4×Q8×C9C2semidirect
3. Q8×C9×Dic3C2semidirect
4. C9×#893C2semidirect
5. D3×C9×C4⋊C4C2semidirect
6. C4×C9×D12C2semidirect
7. C9×#872C2semidirect
11. C9×#239C4semidirect
33. C4×Q8×C9D3semidirect
39. C3×Q8×C9C2×C4semidirect
46. C9×D12C2×C4semidirect
48. C4×D3×C9C2×C4semidirect
50. #5574C9direct (N × H)
94. C3×C4×Q8C2×C9semidirect
100. Q8×Dic3C2×C9semidirect
101. #893C2×C9semidirect
102. D3×C4⋊C4C2×C9semidirect
103. C4×D12C2×C9semidirect
104. #872C2×C9semidirect
118. Q8×C9C4×D3semidirect
143. #239C4×C9semidirect
191. C4×Q8D3×C9semidirect
196. C3×C9#520semidirect
200. C3×Q8C2×C4×C9semidirect
220. D12C2×C4×C9semidirect
222. C4×D3C2×C4×C9semidirect
273. C9#5574direct (N × H)
275. Q8C4×D3×C9semidirect
307. C3C9×#520semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2392
C3×C4×Q8×C92
Q8×C9×Dic32
C9×#8932
D3×C9×C4⋊C42
C4×C9×D122
C9×#8722
C3×#55743
C9×#5203