Finite Group Databasebeta

← C9×#5141 #13143251

Subgroups — C9×#5141 #13143251

Order 1728.

374 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
1C3×C9×#131C211yesmaximal, characteristic, complemented
2C9×#951C211yesmaximal, characteristic, complemented
3C9×#949C211yesmaximal, characteristic
4C2×C9×#204C211yesmaximal, characteristic, complemented
5C9×#824C211yesmaximal, characteristic, complemented
6C9×#823C211yesmaximal, characteristic, complemented
7C9×#819C211yesmaximal, characteristic, complemented
8C3×#5141C311yesmaximal, characteristic
9C9×#471—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×M16—21no—
14C3×C9×M16C2211yescharacteristic
15C2×C3×C8×C9—21no—
16C2×C3×C8×C9C2211yescharacteristic
17C9×#216—21no—
18C9×#215—21no—
19C9×#214—21no—
20C9×#213—21no—
21C9×#208C2211yescharacteristic
22C9×#208C2211yescharacteristic
23C9×#204—21no—
24C9×#204—21no—
25C32×#131C2×C311yescharacteristic
26C3×#951C2×C311yescharacteristic
27C3×#949C2×C311yescharacteristic
28C2×C3×#204C2×C311yescharacteristic
29C3×#824C2×C311yescharacteristic
30C3×#823C2×C311yescharacteristic
31C3×#819C2×C311yescharacteristic
32C9×#137—31no—
33C9×AGL(1,ℤ/8)—31no—
34C2×C9×SD16—31no—
35C9×#131—21no—
36C9×#131D311yescharacteristic, complemented
37C9×#108—31no—
38C9×#101—31no—
39C9×#100—31no—
40C22×D3×C9—21no—
41C3×Q8×C9D411yescharacteristic
42C3×D4×C9—21no—
43C3×D4×C9D411yescharacteristic
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C9×Dic3—21no—
47C9×D12—21no—
48C9×D12—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C3×C8×C9—21no—
52C3×C8×C9D411yescharacteristic
53C3×C8×C9D411yescharacteristic
54C9×C3⋊C8—21no—
55C9×C3⋊C8—21no—
56C3×#471—31no—
57#5141C911yescharacteristic, direct product factor
58C32×#41C22×C311yescharacteristic
59C2×C3×D12C22×C311yescharacteristic
60C2×C3×Dic6C22×C311yescharacteristic
61C32×M16—21no—
62C32×M16C22×C311yescharacteristic
63C2×C32×C8—21no—
64C2×C32×C8C22×C311yescharacteristic
65C3×#216—21no—
66C3×#215—21no—
67C3×#214—21no—
68C3×#213—21no—
69C3×#208C22×C311yescharacteristic
70C3×#208C22×C311yescharacteristic
71C3×#204—21no—
72C3×#204—21no—
73C9×#41—21no—
74C9×#41C2×D311yescharacteristic
75C2×Q8×C9—31no—
76C2×D4×C9—31no—
77C9×Q16—61no—
78C9×SD16—61no—
79C9×SD16—61no—
80C9×SD16—61no—
81C9×SD16—61no—
82C9×D8—61no—
83C9×M16—22no—
84C9×M16—21no—
85C9×M16—21no—
86C9×M16—31no—
87C9×M16—31no—
88C9×M16C2×D311yescharacteristic
89C2×C8×C9—22no—
90C2×C8×C9—21no—
91C2×C8×C9—21no—
92C2×C8×C9C2×D311yescharacteristic
93C22×C3×C9—21no—
94C22×C3×C9#4111yescharacteristic
95C2×D3×C9—41no—
96C2×D3×C9—81no—
97C3×C4×C9—21no—
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C9×Dic3—41no—
101C3×#137—31no—
102C3×AGL(1,ℤ/8)—31no—
103C2×C3×SD16—31no—
104C3×#131—21no—
105C3×#131C2×C911yescharacteristic, complemented
106C3×#131C3×D311yescharacteristic
107#951C2×C911yescharacteristic, complemented
108#949C2×C911yescharacteristic
109C2×#204C2×C911yescharacteristic, complemented
110C3×#108—31no—
111C3×#101—31no—
112C3×#100—31no—
113#824C2×C911yescharacteristic, complemented
114#823C2×C911yescharacteristic, complemented
115#819C2×C911yescharacteristic, complemented
116C22×C3×D3—21no—
117C32×Q8C3×D411yescharacteristic
118C32×D4—21no—
119C32×D4C3×D411yescharacteristic
120C2×C32×C4—21no—
121C2×C32×C4C23×C311yescharacteristic
122C2×C3×Dic3—21no—
123C3×D12—21no—
124C3×D12—21no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C23×C9—61no—
128C32×C8—21no—
129C32×C8C3×D411yescharacteristic
130C32×C8C3×D411yescharacteristic
131C3×C3⋊C8—21no—
132C3×C3⋊C8—21no—
133Q8×C9—21no—
134Q8×C9—61no—
135Q8×C9—61no—
136Q8×C9D1211yescharacteristic
137D4×C9—22no—
138D4×C9—21no—
139D4×C9—21no—
140D4×C9—61no—
141D4×C9—61no—
142D4×C9D1211yescharacteristic
143C2×C4×C9—22no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—61no—
147C2×C4×C9C22×D311yescharacteristic
148C8×C9—22no—
149C8×C9—21no—
150C8×C9—21no—
151C8×C9—21no—
152C8×C9—61no—
153C8×C9—61no—
154C8×C9#6711yescharacteristic
155C8×C9#6711yescharacteristic
156#471—31noSylow
157C2×C3×C9—21no—
158C2×C3×C9—41no—
159C2×C3×C9#12111yescharacteristic
160D3×C9—81no—
161C3×#41—21no—
162C3×#41C22×C911yescharacteristic
163C3×#41C2×C3×D311yescharacteristic
164C2×C3×Q8—31no—
165C2×C3×D4—31no—
166C2×D12C22×C911yescharacteristic
167C2×Dic6C22×C911yescharacteristic
168C3×Q16—61no—
169C3×SD16—61no—
170C3×SD16—61no—
171C3×SD16—61no—
172C3×SD16—61no—
173C3×D8—61no—
174C3×M16—22no—
175C3×M16—21no—
176C3×M16—21no—
177C3×M16—21no—
178C3×M16—31no—
179C3×M16—31no—
180C3×M16C22×C911yescharacteristic
181C3×M16C2×C3×D311yescharacteristic
182C2×C3×C8—22no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8C22×C911yescharacteristic
187C2×C3×C8C2×C3×D311yescharacteristic
188#216—21no—
189#215—21no—
190#214—21no—
191#213—21no—
192#208C22×C911yescharacteristic
193#208C22×C911yescharacteristic
194#204—21no—
195#204—21no—
196C22×C32—21no—
197C22×C32C3×#4111yescharacteristic
198C2×C3×D3—41no—
199C2×C3×D3—81no—
200C32×C4—21no—
201C32×C4C2×C3×D411yescharacteristic
202C32×C4C2×C3×D411yescharacteristic
203C3×Dic3—41no—
204C22×C9—22no—
205C22×C9—21no—
206C22×C9—21no—
207C22×C9—121no—
208C22×C9—241no—
209C22×C9#23511yescharacteristic
210C4×C9—22no—
211C4×C9—21no—
212C4×C9—21no—
213C4×C9—21no—
214C4×C9—121no—
215C4×C9C2×D1211yescharacteristic
216C4×C9C2×#6711yescharacteristic
217#137—31no—
218AGL(1,ℤ/8)—31no—
219C2×SD16—31no—
220#131D3×C911yescharacteristic, complemented
221#108—31no—
222#101—31no—
223#100—31no—
224C3×C9#47111yescharacteristic, complemented, Sylow
225C23×C3—61no—
226C22×D3—21no—
227C3×Q8—21no—
228C3×Q8—61no—
229C3×Q8—61no—
230C3×Q8D4×C911yescharacteristic
231C3×Q8C3×D1211yescharacteristic
232C3×D4—22no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—61no—
237C3×D4—61no—
238C3×D4D4×C911yescharacteristic
239C3×D4C3×D1211yescharacteristic
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—61no—
245C2×C3×C4C23×C911yescharacteristic
246C2×C3×C4C22×C3×D311yescharacteristic
247C2×Dic3—21no—
248D12—21no—
249D12—21no—
250Dic6—21no—
251Dic6—21no—
252C3×C8—22no—
253C3×C8—21no—
254C3×C8—21no—
255C3×C8—21no—
256C3×C8—21no—
257C3×C8—61no—
258C3×C8—61no—
259C3×C8D4×C911yescharacteristic
260C3×C8D4×C911yescharacteristic
261C3×C8C3×#6711yescharacteristic
262C3×C8C3×#6711yescharacteristic
263C3⋊C8—21no—
264C3⋊C8—21no—
265C2×C32—21no—
266C2×C32—41no—
267C2×C32C3×#12111yescharacteristic
268C3×D3—81no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—41no—
272C2×C9—42no—
273C2×C9—41no—
274C2×C9—241no—
275C2×C9#93011yescharacteristic
276#41C2×D3×C911yescharacteristic
277C2×Q8—31no—
278C2×D4—31no—
279Q16—61no—
280SD16—61no—
281SD16—61no—
282SD16—61no—
283SD16—61no—
284D8—61no—
285M16—21no—
286M16—31no—
287M16—31no—
288M16C2×D3×C911yescharacteristic
289C2×C8—21no—
290C2×C8C2×D3×C911yescharacteristic
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—121no—
296C22×C3—241no—
297C22×C3C9×#4111yescharacteristic
298C22×C3C3×#23511yescharacteristic
299C2×D3—41no—
300C2×D3—81no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—121no—
307C3×C4C2×D4×C911yescharacteristic
308C3×C4C2×D4×C911yescharacteristic
309C3×C4C2×C3×D1211yescharacteristic
310C3×C4C2×C3×#6711yescharacteristic
311Dic3—41no—
312C32C3×#47111yescharacteristic
313C9—21no—
314C9#514111yescharacteristic, direct product factor
315C23—61no—
316Q8—61no—
317Q8—61no—
318Q8C9×D1211yescharacteristic
319D4—21no—
320D4—61no—
321D4—61no—
322D4C9×D1211yescharacteristic
323C2×C4—21no—
324C2×C4—61no—
325C2×C4C22×D3×C911yescharacteristic
326C8—21no—
327C8—61no—
328C8—61no—
329C8C9×#6711yescharacteristic
330C8C9×#6711yescharacteristic
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—41no—
335C2×C3—42no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—241no—
339C2×C3C9×#12111yescharacteristic
340C2×C3C3×#93011yescharacteristic
341D3—81no—
342C22—21no—
343C22—121no—
344C22—241no—
345C22C9×#23511yescharacteristic
346C4—21no—
347C4—121no—
348C4C2×C9×D1211yescharacteristic
349C4C2×C9×#6711yescharacteristic
350C3—21no—
351C3C9×#47111yescharacteristic, complemented
352C3C3×#514111yescharacteristic
353C2—21no—
354C2—41no—
355C2—241no—
356C2C9×#93011yescharacteristic
subgroupG/H
centerC2×C9#930
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#131C2
socleC2×C32C3×#121
prime radicalC3×#949C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#131D3
hypocenterC3C9×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#131

Complements

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

NH ≅ G/Ntype
1. C3×C9×#131C2semidirect
2. C9×#951C2semidirect
4. C2×C9×#204C2semidirect
5. C9×#824C2semidirect
6. C9×#823C2semidirect
7. C9×#819C2semidirect
36. C9×#131D3semidirect
57. #5141C9direct (N × H)
105. C3×#131C2×C9semidirect
107. #951C2×C9semidirect
109. C2×#204C2×C9semidirect
113. #824C2×C9semidirect
114. #823C2×C9semidirect
115. #819C2×C9semidirect
220. #131D3×C9semidirect
224. C3×C9#471semidirect
314. C9#5141direct (N × H)
351. C3C9×#471semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1312
C9×#9512
C9×#9492
C2×C9×#2042
C9×#8242
C9×#8232
C9×#8192
C3×#51413
C9×#4713