Finite Group Databasebeta

← C9×#5066 #13142946

Subgroups — C9×#5066 #13142946

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#125C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic
3C9×#809C211yesmaximal, characteristic
4C9×#809C211yesmaximal, characteristic
5C9×#808C211yesmaximal, characteristic, complemented
6C9×#807C211yesmaximal, characteristic, complemented
7C9×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5066C311yesmaximal, characteristic
9C9×#478—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C4×D3×C9—21no—
12C3×C9×C4⋊C4—21no—
13C3×C9×C4⋊C4—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C42×C9C2211yescharacteristic
17C9×(C2×D3)⋊C4—21no—
18C9×C4⋊Dic3C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C32×#125C2×C311yescharacteristic
22C3×C4×D12C2×C311yescharacteristic
23C3×#809C2×C311yescharacteristic
24C3×#809C2×C311yescharacteristic
25C3×#808C2×C311yescharacteristic
26C3×#807C2×C311yescharacteristic
27C3×C4⋊(C3⋊C8)C2×C311yescharacteristic
28C9×#125—21no—
29C9×#125D311yescharacteristic, complemented
30C4×D4×C9—31no—
31C9×#107—31no—
32C9×#106—31no—
33C9×C4⋊C8—31no—
34C9×#102—31no—
35C9×#102—31no—
36C22×D3×C9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9D411yescharacteristic
40C2×C3×C4×C9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C9×Dic3—21no—
43C9×D12—21no—
44C9×D12Q812yes—
45C4×D3×C9—42no—
46C9×C3⋊C8—21no—
47C9×C3⋊C8—21no—
48C3×#478—31no—
49#5066C911yescharacteristic, direct product factor
50C2×C3×D12C22×C311yescharacteristic
51C2×C3×C4×D3—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4C22×C311yescharacteristic
55C32×C4⋊C4C22×C311yescharacteristic
56C32×C42C22×C311yescharacteristic
57C3×(C2×D3)⋊C4—21no—
58C3×C4⋊Dic3C22×C311yescharacteristic
59C2×C3×C3⋊C8C22×C311yescharacteristic
60C2×C3×C3⋊C8C22×C311yescharacteristic
61C2×D4×C9—31no—
62C22×C4×C9—61no—
63C2×C8×C9—31no—
64C2×C8×C9—31no—
65C9×C4⋊C4—22no—
66C9×C4⋊C4—22no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4—21no—
71C9×C4⋊C4—31no—
72C9×C4⋊C4C2×D311yescharacteristic
73C9×C4⋊C4C2×D311yescharacteristic
74C9×C22⋊C4—61no—
75C42×C9—21no—
76C42×C9C2×D311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C2×D3×C9—22no—
79C2×D3×C9—41no—
80C2×D3×C9—41no—
81C3×C4×C9—21no—
82C3×C4×C9—21no—
83C3×C4×C9—41no—
84C3×C4×C9—41no—
85C3×C4×C9C2×Q811yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C9×Dic3—41no—
88C3×#125—21no—
89C3×#125C2×C911yescharacteristic, complemented
90C3×#125C3×D311yescharacteristic
91C3×C4×D4—31no—
92C4×D12C2×C911yescharacteristic
93C3×#107—31no—
94C3×#106—31no—
95C3×C4⋊C8—31no—
96C3×#102—31no—
97C3×#102—31no—
98#809C2×C911yescharacteristic
99#809C2×C911yescharacteristic
100#808C2×C911yescharacteristic, complemented
101#807C2×C911yescharacteristic, complemented
102C4⋊(C3⋊C8)C2×C911yescharacteristic, complemented
103C22×C3×D3—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C3×D411yescharacteristic
108C2×C32×C4C23×C311yescharacteristic
109C2×C3×Dic3—21no—
110C3×D12—21no—
111C3×D12C3×Q812yes—
112C3×C4×D3—42no—
113C23×C9—61no—
114C3×C3⋊C8—21no—
115C3×C3⋊C8—21no—
116D4×C9—32no—
117D4×C9—61no—
118C2×C4×C9—22no—
119C2×C4×C9—22no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—61no—
126C2×C4×C9—122no—
127C2×C4×C9#6711yescharacteristic
128C2×C4×C9#6711yescharacteristic
129C2×C4×C9C22×D311yescharacteristic
130C8×C9—61no—
131C8×C9—61no—
132#478—31noSylow
133C2×C3×C9C22⋊Q811yescharacteristic
134C2×C3×C9#13511yescharacteristic
135C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
136D3×C9—42no—
137C2×C3×D4—31no—
138C22×C3×C4—61no—
139C2×D12C22×C911yescharacteristic
140C2×C4×D3—21no—
141C2×C3×C8—31no—
142C2×C3×C8—31no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4C22×C911yescharacteristic
153C3×C4⋊C4C22×C911yescharacteristic
154C3×C4⋊C4C2×C3×D311yescharacteristic
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C22⋊C4—61no—
157C3×C42—21no—
158C3×C42C22×C911yescharacteristic
159C3×C42C2×C3×D311yescharacteristic
160(C2×D3)⋊C4—21no—
161C4⋊Dic3C22×C911yescharacteristic
162C2×C3⋊C8C22×C911yescharacteristic
163C2×C3⋊C8C22×C911yescharacteristic
164C22×C32C2×C3×D411yescharacteristic
165C2×C3×D3—22no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C32×C4—21no—
169C32×C4—21no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4C2×C3×Q811yescharacteristic
173C32×C4C3×#4111yescharacteristic
174C3×Dic3—41no—
175C22×C9—21no—
176C22×C9—62no—
177C22×C9—121no—
178C22×C9—121no—
179C22×C9C2×#6711yescharacteristic
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—21no—
183C4×C9—21no—
184C4×C9—41no—
185C4×C9—41no—
186C4×C9—42no—
187C4×C9—42no—
188C4×C9—41no—
189C4×C9—41no—
190C4×C9—121no—
191C4×C9D3×Q811yescharacteristic
192C4×C9#23911yescharacteristic
193#125D3×C911yescharacteristic, complemented
194C4×D4—31no—
195#107—31no—
196#106—31no—
197C4⋊C8—31no—
198#102—31no—
199#102—31no—
200C3×C9#47811yescharacteristic, complemented, Sylow
201C23×C3—61no—
202C22×D3—21no—
203C3×D4—32no—
204C3×D4—61no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—61no—
215C2×C3×C4—122no—
216C2×C3×C4D4×C911yescharacteristic
217C2×C3×C4D4×C911yescharacteristic
218C2×C3×C4C23×C911yescharacteristic
219C2×C3×C4C3×#6711yescharacteristic
220C2×C3×C4C3×#6711yescharacteristic
221C2×C3×C4C22×C3×D311yescharacteristic
222C2×Dic3—21no—
223D12—21no—
224D12Q8×C912yes—
225C4×D3—42no—
226C3×C8—61no—
227C3×C8—61no—
228C3⋊C8—21no—
229C3⋊C8—21no—
230C2×C32C3×C22⋊Q811yescharacteristic
231C2×C32C3×#13511yescharacteristic
232C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
233C3×D3—42no—
234C2×C9—21no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—122no—
238C2×C9#94611yescharacteristic
239C2×C9#94911yescharacteristic
240C2×C9#95011yescharacteristic
241C2×D4—31no—
242C22×C4—61no—
243C2×C8—31no—
244C2×C8—31no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—31no—
248C4⋊C4C2×D3×C911yescharacteristic
249C4⋊C4C2×D3×C911yescharacteristic
250C22⋊C4—61no—
251C42C2×D3×C911yescharacteristic
252C22×C3—21no—
253C22×C3—62no—
254C22×C3—121no—
255C22×C3—121no—
256C22×C3C2×D4×C911yescharacteristic
257C22×C3C2×C3×#6711yescharacteristic
258C2×D3—22no—
259C2×D3—41no—
260C2×D3—41no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—42no—
270C3×C4—42no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—121no—
276C3×C4C2×Q8×C911yescharacteristic
277C3×C4C9×#4111yescharacteristic
278C3×C4C3×D3×Q811yescharacteristic
279C3×C4C3×#23911yescharacteristic
280Dic3—41no—
281C32C3×#47811yescharacteristic
282C9—21no—
283C9#506611yescharacteristic, direct product factor
284C23—61no—
285D4—32no—
286D4—61no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—61no—
290C2×C4—122no—
291C2×C4C9×#6711yescharacteristic
292C2×C4C9×#6711yescharacteristic
293C2×C4C22×D3×C911yescharacteristic
294C8—61no—
295C8—61no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—122no—
300C2×C3C9×C22⋊Q811yescharacteristic
301C2×C3C9×#13511yescharacteristic
302C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
303C2×C3C3×#94611yescharacteristic
304C2×C3C3×#94911yescharacteristic
305C2×C3C3×#95011yescharacteristic
306D3—42no—
307C22—62no—
308C22—121no—
309C22—121no—
310C22C2×C9×#6711yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—121no—
316C4D3×Q8×C911yescharacteristic
317C4C9×#23911yescharacteristic
318C3—21no—
319C3C9×#47811yescharacteristic, complemented
320C3C3×#506611yescharacteristic
321C2—122no—
322C2C9×#94611yescharacteristic
323C2C9×#94911yescharacteristic
324C2C9×#95011yescharacteristic
subgroupG/H
centerC22×C9C2×#67
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#125C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×D12C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#125D3
hypocenterC3C9×#478
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C42×C9 ◁ C9×#125

Complements

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

NH ≅ G/Ntype
1. C3×C9×#125C2semidirect
5. C9×#808C2semidirect
6. C9×#807C2semidirect
7. C9×C4⋊(C3⋊C8)C2semidirect
29. C9×#125D3semidirect
49. #5066C9direct (N × H)
89. C3×#125C2×C9semidirect
100. #808C2×C9semidirect
101. #807C2×C9semidirect
102. C4⋊(C3⋊C8)C2×C9semidirect
193. #125D3×C9semidirect
200. C3×C9#478semidirect
283. C9#5066direct (N × H)
319. C3C9×#478semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1252
C4×C9×D122
C9×#8092
C9×#8082
C9×#8072
C9×C4⋊(C3⋊C8)2
C3×#50663
C9×#4783