Finite Group Databasebeta

← C9×#5526 #13152500

Subgroups — C9×#5526 #13152500

Order 1728.

886 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic
2C22×C9×Dic6C211yesmaximal, characteristic, complemented
3C3×C9×#117C211yesmaximal, characteristic, complemented
4C9×#897C214yesmaximal, complemented
5C9×#886C214yesmaximal, complemented
6C9×#875C212yesmaximal, complemented
7C9×#869C212yesmaximal, complemented
8C3×#5526C311yesmaximal, characteristic
9C9×#536—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C9×#67C2212yes—
12C22×C9×Dic3C2212yes—
13C9×#235—24no—
14C2×C9×D12C2211yescharacteristic
15C2×C4×D3×C9C2212yes—
16C2×C9×Dic6—24no—
17C2×C9×Dic6C2212yes—
18C2×C9×Dic6C2212yes—
19C2×C9×Dic6C2211yescharacteristic
20C3×C9×C4⋊C4C2212yes—
21C3×C9×C22⋊C4C2212yes—
22C3×C42×C9C2212yes—
23C9×(C2×D3)⋊C4C2218yes—
24C9×C4⋊Dic3C2218yes—
25C2×C3×#235C2×C311yescharacteristic
26C22×C3×Dic6C2×C311yescharacteristic
27C32×#117C2×C311yescharacteristic
28C3×#897C2×C314yes—
29C3×#886C2×C314yes—
30C3×#875C2×C312yes—
31C3×#869C2×C312yes—
32C2×C9×#41—31no—
33C22×Q8×C9—31no—
34C9×C4⋊Q8—32no—
35C9×#124—32no—
36C9×#123—34no—
37C9×C22⋊Q8—34no—
38C9×#117—21no—
39C9×#117D311yescharacteristic, complemented
40C23×C3×C9C2311yescharacteristic
41C22×D3×C9C2312yes—
42C2×C3×C4×C9D414yes—
43C2×C3×C4×C9C2314yes—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C9×#67—24no—
47C2×C9×Dic3—24no—
48C2×C9×Dic3C2314yes—
49C2×C9×Dic3C2312yes—
50C9×D12—22no—
51C4×D3×C9—24no—
52C9×Dic6—24no—
53C9×Dic6—24no—
54C9×Dic6—22no—
55C3×#536—31no—
56#5526C911yescharacteristic, direct product factor
57C22×C32×C4C22×C311yescharacteristic
58C2×C3×#67C22×C312yes—
59C22×C3×Dic3C22×C312yes—
60C3×#235—24no—
61C2×C3×D12C22×C311yescharacteristic
62C2×C3×C4×D3C22×C312yes—
63C2×C3×Dic6—24no—
64C2×C3×Dic6C22×C312yes—
65C2×C3×Dic6C22×C312yes—
66C2×C3×Dic6C22×C311yescharacteristic
67C32×C4⋊C4C22×C312yes—
68C32×C22⋊C4C22×C312yes—
69C32×C42C22×C312yes—
70C3×(C2×D3)⋊C4C22×C318yes—
71C3×C4⋊Dic3C22×C318yes—
72C9×#41—64no—
73C2×Q8×C9—32no—
74C2×Q8×C9—32no—
75C2×Q8×C9—31no—
76C2×Q8×C9—64no—
77C2×D4×C9—32no—
78C2×D4×C9—31no—
79C22×C4×C9—21no—
80C22×C4×C9—32no—
81C22×C4×C9—32no—
82C22×C4×C9C2×D311yescharacteristic
83C9×C4⋊C4—22no—
84C9×C4⋊C4—38no—
85C9×C4⋊C4C2×D312yes—
86C9×C22⋊C4—22no—
87C9×C22⋊C4—38no—
88C9×C22⋊C4C2×D312yes—
89C42×C9—22no—
90C42×C9C2×D312yes—
91C22×C3×C9—22no—
92C22×C3×C9C2×D412yes—
93C22×C3×C9C2411yescharacteristic
94C2×D3×C9—22no—
95C2×D3×C9—24no—
96C3×C4×C9—24no—
97C3×C4×C9C2×D412yes—
98C3×C4×C9C2×D412yes—
99C9×Dic3—24no—
100C9×Dic3—22no—
101C2×C3×#41—31no—
102C22×C3×Q8—31no—
103C2×#235C2×C911yescharacteristic
104C22×Dic6C2×C911yescharacteristic, complemented
105C3×C4⋊Q8—32no—
106C3×#124—32no—
107C3×#123—34no—
108C3×C22⋊Q8—34no—
109C3×#117—21no—
110C3×#117C2×C911yescharacteristic, complemented
111C3×#117C3×D311yescharacteristic
112#897C2×C914yescomplemented
113#886C2×C914yescomplemented
114#875C2×C912yescomplemented
115#869C2×C912yescomplemented
116C23×C32C23×C311yescharacteristic
117C22×C3×D3C23×C312yes—
118C2×C32×C4C3×D414yes—
119C2×C32×C4C23×C314yes—
120C2×C32×C4C23×C311yescharacteristic
121C2×C32×C4C23×C311yescharacteristic
122C3×#67—24no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3C23×C312yes—
125C2×C3×Dic3C23×C314yes—
126C3×D12—22no—
127C3×C4×D3—24no—
128C3×Dic6—24no—
129C3×Dic6—24no—
130C3×Dic6—22no—
131C23×C9—21no—
132C23×C9—32no—
133C23×C9C22×D311yescharacteristic
134Q8×C9—64no—
135Q8×C9—64no—
136Q8×C9—62no—
137D4×C9—64no—
138D4×C9—62no—
139C2×C4×C9—24no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—24no—
143C2×C4×C9—34no—
144C2×C4×C9—32no—
145C2×C4×C9—64no—
146C2×C4×C9—64no—
147C2×C4×C9D1214yes—
148C2×C4×C9C22×D314yes—
149C2×C4×C9C22×D311yescharacteristic
150C2×C4×C9C22×D311yescharacteristic
151#536—31noSylow
152C2×C3×C9—22no—
153C2×C3×C9C22×D411yescharacteristic
154C2×C3×C92−1+412yes—
155D3×C9—42no—
156C3×#41—64no—
157C2×C3×Q8—32no—
158C2×C3×Q8—32no—
159C2×C3×Q8—31no—
160C2×C3×Q8—64no—
161C2×C3×D4—32no—
162C2×C3×D4—31no—
163C22×C3×C4—21no—
164C22×C3×C4—32no—
165C22×C3×C4—32no—
166C22×C3×C4C22×C911yescharacteristic
167C22×C3×C4C2×C3×D311yescharacteristic
168C2×#67C22×C912yes—
169C22×Dic3C22×C912yes—
170#235—24no—
171C2×D12C22×C911yescharacteristic
172C2×C4×D3C22×C912yes—
173C2×Dic6—24no—
174C2×Dic6C22×C911yescharacteristic
175C2×Dic6C22×C912yes—
176C2×Dic6C22×C912yes—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—38no—
179C3×C4⋊C4C22×C912yes—
180C3×C4⋊C4C2×C3×D312yes—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—38no—
183C3×C22⋊C4C22×C912yes—
184C3×C22⋊C4C2×C3×D312yes—
185C3×C42—22no—
186C3×C42C22×C912yes—
187C3×C42C2×C3×D312yes—
188(C2×D3)⋊C4C22×C918yes—
189C4⋊Dic3C22×C918yes—
190C22×C32—22no—
191C22×C32C2×C3×D412yes—
192C22×C32C24×C311yescharacteristic
193C2×C3×D3—22no—
194C2×C3×D3—24no—
195C32×C4—24no—
196C32×C4C2×C3×D412yes—
197C32×C4C2×C3×D412yes—
198C3×Dic3—24no—
199C3×Dic3—22no—
200C22×C9—22no—
201C22×C9—21no—
202C22×C9—22no—
203C22×C9—42no—
204C22×C9—62no—
205C22×C9—64no—
206C22×C9C2×D1212yes—
207C22×C9C23×D311yescharacteristic
208C4×C9—24no—
209C4×C9—22no—
210C4×C9—22no—
211C4×C9—44no—
212C4×C9—62no—
213C4×C9—64no—
214C4×C9C2×D1212yes—
215C4×C9C2×D1212yes—
216C2×#41—31no—
217C22×Q8—31no—
218C4⋊Q8—32no—
219#124—32no—
220#123—34no—
221C22⋊Q8—34no—
222#117D3×C911yescharacteristic, complemented
223C3×C9#53611yescharacteristic, complemented, Sylow
224C23×C3—21no—
225C23×C3—32no—
226C23×C3C23×C911yescharacteristic
227C23×C3C22×C3×D311yescharacteristic
228C22×D3C23×C912yes—
229C3×Q8—64no—
230C3×Q8—64no—
231C3×Q8—62no—
232C3×D4—64no—
233C3×D4—62no—
234C2×C3×C4—24no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—34no—
239C2×C3×C4—32no—
240C2×C3×C4—64no—
241C2×C3×C4—64no—
242C2×C3×C4D4×C914yes—
243C2×C3×C4C23×C914yes—
244C2×C3×C4C23×C911yescharacteristic
245C2×C3×C4C23×C911yescharacteristic
246C2×C3×C4C3×D1214yes—
247C2×C3×C4C22×C3×D314yes—
248C2×C3×C4C22×C3×D311yescharacteristic
249C2×C3×C4C22×C3×D311yescharacteristic
250#67—24no—
251C2×Dic3—24no—
252C2×Dic3C23×C912yes—
253C2×Dic3C23×C914yes—
254D12—22no—
255C4×D3—24no—
256Dic6—24no—
257Dic6—24no—
258Dic6—22no—
259C2×C32—22no—
260C2×C32C22×C3×D411yescharacteristic
261C2×C32C3×2−1+412yes—
262C3×D3—42no—
263C2×C9—22no—
264C2×C9—21no—
265C2×C9—22no—
266C2×C9—42no—
267C2×C9—122no—
268C2×C9C22×D1211yescharacteristic
269C2×C9#101012yes—
270#41—64no—
271C2×Q8—32no—
272C2×Q8—32no—
273C2×Q8—31no—
274C2×Q8—64no—
275C2×D4—32no—
276C2×D4—31no—
277C22×C4—32no—
278C22×C4—32no—
279C22×C4C2×D3×C911yescharacteristic
280C4⋊C4—38no—
281C4⋊C4C2×D3×C912yes—
282C22⋊C4—38no—
283C22⋊C4C2×D3×C912yes—
284C42C2×D3×C912yes—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—42no—
290C22×C3—62no—
291C22×C3—64no—
292C22×C3C2×D4×C912yes—
293C22×C3C24×C911yescharacteristic
294C22×C3C2×C3×D1212yes—
295C22×C3C23×C3×D311yescharacteristic
296C2×D3—22no—
297C2×D3—24no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—44no—
303C3×C4—62no—
304C3×C4—64no—
305C3×C4C2×D4×C912yes—
306C3×C4C2×D4×C912yes—
307C3×C4C2×C3×D1212yes—
308C3×C4C2×C3×D1212yes—
309Dic3—24no—
310Dic3—22no—
311C32C3×#53611yescharacteristic
312C9—21no—
313C9#552611yescharacteristic, direct product factor
314C23—32no—
315C23C22×D3×C911yescharacteristic
316Q8—64no—
317Q8—64no—
318Q8—62no—
319D4—64no—
320D4—62no—
321C2×C4—34no—
322C2×C4—32no—
323C2×C4—64no—
324C2×C4—64no—
325C2×C4C9×D1214yes—
326C2×C4C22×D3×C914yes—
327C2×C4C22×D3×C911yescharacteristic
328C2×C4C22×D3×C911yescharacteristic
329C2×C3—22no—
330C2×C3—22no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—42no—
334C2×C3—122no—
335C2×C3C22×D4×C911yescharacteristic
336C2×C3C9×2−1+412yes—
337C2×C3C22×C3×D1211yescharacteristic
338C2×C3C3×#101012yes—
339D3—42no—
340C22—22no—
341C22—62no—
342C22—64no—
343C22C2×C9×D1212yes—
344C22C23×D3×C911yescharacteristic
345C4—24no—
346C4—62no—
347C4—64no—
348C4C2×C9×D1212yes—
349C4C2×C9×D1212yes—
350C3—21no—
351C3C9×#53611yescharacteristic, complemented
352C3C3×#552611yescharacteristic
353C2—22no—
354C2—122no—
355C2C22×C9×D1211yescharacteristic
356C2C9×#101012yes—
subgroupG/H
centerC22×C9C23×D3
commutatorC22×C3C24×C9
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#117C2
socle, cosocleC22×C32C24×C3
prime radicalC2×C3×#235C2×C3
hypercenterC9×#117D3
hypocenterC3C9×#536
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C9×#117

Complements

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

NH ≅ G/Ntype
2. C22×C9×Dic6C2semidirect
3. C3×C9×#117C2semidirect
4. C9×#897C2semidirect
5. C9×#886C2semidirect
6. C9×#875C2semidirect
7. C9×#869C2semidirect
39. C9×#117D3semidirect
56. #5526C9direct (N × H)
104. C22×Dic6C2×C9semidirect
110. C3×#117C2×C9semidirect
112. #897C2×C9semidirect
113. #886C2×C9semidirect
114. #875C2×C9semidirect
115. #869C2×C9semidirect
222. #117D3×C9semidirect
223. C3×C9#536semidirect
313. C9#5526direct (N × H)
351. C3C9×#536semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C22×C9×Dic62
C3×C9×#1172
C9×#8972
C9×#8862
C9×#8752
C9×#8692
C3×#55263
C9×#5363