Finite Group Databasebeta

← C9×#4749 #13141671

Subgroups — C9×#4749 #13141671

Order 1728.

370 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#932C211yesmaximal, characteristic, complemented
2C2×C9×#213C211yesmaximal, characteristic, complemented
3C9×#908C211yesmaximal, characteristic, complemented
4C9×#907C211yesmaximal, characteristic, complemented
5C3×C9×#100C211yesmaximal, characteristic, complemented
6C9×#823C211yesmaximal, characteristic, complemented
7C9×#822C211yesmaximal, characteristic, complemented
8C3×#4749C311yesmaximal, characteristic
9C9×#472—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C9×#235C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14C3×C9×M16C2211yescharacteristic
15C9×#214—21no—
16C9×#213—21no—
17C9×#213—21no—
18C9×#213—21no—
19C9×#208C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C9×D24—21no—
22C9×#204—21no—
23C9×ΓL(1,𝔽25)—21no—
24D3×C8×C9—21no—
25C3×#932C2×C311yescharacteristic
26C2×C3×#213C2×C311yescharacteristic
27C3×#908C2×C311yescharacteristic
28C3×#907C2×C311yescharacteristic
29C32×#100C2×C311yescharacteristic
30C3×#823C2×C311yescharacteristic
31C3×#822C2×C311yescharacteristic
32C9×AGL(1,ℤ/8)—31no—
33C9×AGL(1,ℤ/8)—31no—
34C2×C9×D8—31no—
35C9×#131—31no—
36C9×#108—31no—
37C9×#100—21no—
38C9×#100—31no—
39C9×#100D311yescharacteristic, complemented
40C23×C3×C9—21no—
41C22×D3×C9—21no—
42C3×D4×C9—21no—
43C3×D4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C9×#67—21no—
46C9×D12—21no—
47C9×D12—21no—
48C9×D12D411yescharacteristic
49C4×D3×C9—21no—
50C9×Dic6D411yescharacteristic
51C3×C8×C9—21no—
52C3×C8×C9—21no—
53C9×C3⋊C8—21no—
54C9×C3⋊C8D411yescharacteristic
55C9×C3⋊C8D411yescharacteristic
56C3×#472—31no—
57#4749C911yescharacteristic, direct product factor
58C2×C32×D4C22×C311yescharacteristic
59C3×#235C22×C311yescharacteristic
60C2×C3×D12C22×C311yescharacteristic
61C32×M16C22×C311yescharacteristic
62C32×M16C22×C311yescharacteristic
63C3×#214—21no—
64C3×#213—21no—
65C3×#213—21no—
66C3×#213—21no—
67C3×#208C22×C311yescharacteristic
68C2×C3×C3⋊C8C22×C311yescharacteristic
69C3×D24—21no—
70C3×#204—21no—
71C3×ΓL(1,𝔽25)—21no—
72C3×D3×C8—21no—
73C9×#41—31no—
74C2×D4×C9—21no—
75C2×D4×C9—31no—
76C2×D4×C9C2×D311yescharacteristic
77C9×SD16—61no—
78C9×SD16—61no—
79C9×D8—61no—
80C9×D8—61no—
81C9×D8—61no—
82C9×D8—61no—
83C9×M16—21no—
84C9×M16—21no—
85C9×M16—31no—
86C9×M16—61no—
87C9×M16C2×D311yescharacteristic
88C9×M16C2×D311yescharacteristic
89C2×C8×C9—31no—
90C2×C8×C9—61no—
91C22×C3×C9—41no—
92C22×C3×C9—81no—
93C22×C3×C9#4111yescharacteristic
94C2×D3×C9—21no—
95C2×D3×C9—41no—
96C2×D3×C9—81no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C9×Dic3—21no—
100C3×AGL(1,ℤ/8)—31no—
101C3×AGL(1,ℤ/8)—31no—
102C2×C3×D8—31no—
103C3×#131—31no—
104#932C2×C911yescharacteristic, complemented
105C2×#213C2×C911yescharacteristic, complemented
106#908C2×C911yescharacteristic, complemented
107#907C2×C911yescharacteristic, complemented
108C3×#108—31no—
109C3×#100—21no—
110C3×#100—31no—
111C3×#100C2×C911yescharacteristic, complemented
112C3×#100C3×D311yescharacteristic
113#823C2×C911yescharacteristic, complemented
114#822C2×C911yescharacteristic, complemented
115C23×C32—21no—
116C22×C3×D3—21no—
117C32×D4—21no—
118C32×D4—21no—
119C2×C32×C4C23×C311yescharacteristic
120C3×#67—21no—
121C3×D12—21no—
122C3×D12—21no—
123C3×D12C3×D411yescharacteristic
124C3×C4×D3—21no—
125C3×Dic6C3×D411yescharacteristic
126C23×C9—22no—
127C23×C9—21no—
128C23×C9—61no—
129C32×C8—21no—
130C32×C8—21no—
131C3×C3⋊C8—21no—
132C3×C3⋊C8C3×D411yescharacteristic
133C3×C3⋊C8C3×D411yescharacteristic
134Q8×C9—31no—
135D4×C9—21no—
136D4×C9—21no—
137D4×C9—31no—
138D4×C9—41no—
139D4×C9—41no—
140D4×C9—61no—
141D4×C9—61no—
142D4×C9—61no—
143C2×C4×C9—21no—
144C2×C4×C9—61no—
145C2×C4×C9C22×D311yescharacteristic
146C8×C9—21no—
147C8×C9—21no—
148C8×C9—31no—
149C8×C9—31no—
150C8×C9—41no—
151C8×C9—41no—
152C8×C9—61no—
153#472—31noSylow
154C2×C3×C9—21no—
155C2×C3×C9—81no—
156C2×C3×C9#12111yescharacteristic
157D3×C9—41no—
158D3×C9—81no—
159C3×#41—31no—
160C2×C3×D4—21no—
161C2×C3×D4—31no—
162C2×C3×D4C22×C911yescharacteristic
163C2×C3×D4C2×C3×D311yescharacteristic
164#235C22×C911yescharacteristic
165C2×D12C22×C911yescharacteristic
166C3×SD16—61no—
167C3×SD16—61no—
168C3×D8—61no—
169C3×D8—61no—
170C3×D8—61no—
171C3×D8—61no—
172C3×M16—21no—
173C3×M16—21no—
174C3×M16—31no—
175C3×M16—61no—
176C3×M16C22×C911yescharacteristic
177C3×M16C22×C911yescharacteristic
178C3×M16C2×C3×D311yescharacteristic
179C3×M16C2×C3×D311yescharacteristic
180C2×C3×C8—31no—
181C2×C3×C8—61no—
182#214—21no—
183#213—21no—
184#213—21no—
185#213—21no—
186#208C22×C911yescharacteristic
187C2×C3⋊C8C22×C911yescharacteristic
188D24—21no—
189#204—21no—
190ΓL(1,𝔽25)—21no—
191D3×C8—21no—
192C22×C32—41no—
193C22×C32—81no—
194C22×C32C3×#4111yescharacteristic
195C2×C3×D3—21no—
196C2×C3×D3—41no—
197C2×C3×D3—81no—
198C32×C4C2×C3×D411yescharacteristic
199C32×C4C2×C3×D411yescharacteristic
200C3×Dic3—21no—
201C22×C9—21no—
202C22×C9—42no—
203C22×C9—41no—
204C22×C9—61no—
205C22×C9—81no—
206C22×C9—82no—
207C22×C9—121no—
208C22×C9—241no—
209C22×C9#23511yescharacteristic
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—61no—
213C4×C9D3×D411yescharacteristic
214C4×C9D3×D411yescharacteristic
215AGL(1,ℤ/8)—31no—
216AGL(1,ℤ/8)—31no—
217C2×D8—31no—
218#131—31no—
219#108—31no—
220#100—31no—
221#100D3×C911yescharacteristic, complemented
222C3×C9#47211yescharacteristic, complemented, Sylow
223C23×C3—22no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—61no—
227C22×D3—21no—
228C3×Q8—31no—
229C3×D4—21no—
230C3×D4—21no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—31no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—61no—
237C3×D4—61no—
238C3×D4—61no—
239C2×C3×C4—21no—
240C2×C3×C4—61no—
241C2×C3×C4C23×C911yescharacteristic
242C2×C3×C4C22×C3×D311yescharacteristic
243#67—21no—
244D12—21no—
245D12—21no—
246D12D4×C911yescharacteristic
247C4×D3—21no—
248Dic6D4×C911yescharacteristic
249C3×C8—21no—
250C3×C8—21no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8—31no—
254C3×C8—31no—
255C3×C8—41no—
256C3×C8—41no—
257C3×C8—61no—
258C3⋊C8—21no—
259C3⋊C8D4×C911yescharacteristic
260C3⋊C8D4×C911yescharacteristic
261C2×C32—21no—
262C2×C32—81no—
263C2×C32C3×#12111yescharacteristic
264C3×D3—41no—
265C3×D3—81no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9—41no—
269C2×C9—82no—
270C2×C9—81no—
271C2×C9—121no—
272C2×C9—241no—
273C2×C9#88411yescharacteristic
274#41—31no—
275C2×D4—31no—
276C2×D4C2×D3×C911yescharacteristic
277SD16—61no—
278SD16—61no—
279D8—61no—
280D8—61no—
281D8—61no—
282D8—61no—
283M16—31no—
284M16—61no—
285M16C2×D3×C911yescharacteristic
286M16C2×D3×C911yescharacteristic
287C2×C8—31no—
288C2×C8—61no—
289C22×C3—21no—
290C22×C3—42no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—61no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—82no—
297C22×C3—121no—
298C22×C3—241no—
299C22×C3C9×#4111yescharacteristic
300C22×C3C3×#23511yescharacteristic
301C2×D3—21no—
302C2×D3—41no—
303C2×D3—81no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—61no—
307C3×C4C2×D4×C911yescharacteristic
308C3×C4C2×D4×C911yescharacteristic
309C3×C4C3×D3×D411yescharacteristic
310C3×C4C3×D3×D411yescharacteristic
311Dic3—21no—
312C32C3×#47211yescharacteristic
313C9—21no—
314C9#474911yescharacteristic, direct product factor
315C23—21no—
316C23—61no—
317Q8—31no—
318D4—21no—
319D4—21no—
320D4—31no—
321D4—61no—
322D4—61no—
323D4—61no—
324C2×C4—61no—
325C2×C4C22×D3×C911yescharacteristic
326C8—21no—
327C8—21no—
328C8—31no—
329C8—31no—
330C8—61no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—41no—
335C2×C3—82no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3—121no—
339C2×C3—241no—
340C2×C3C9×#12111yescharacteristic
341C2×C3C3×#88411yescharacteristic
342D3—41no—
343D3—81no—
344C22—41no—
345C22—61no—
346C22—81no—
347C22—121no—
348C22—241no—
349C22C9×#23511yescharacteristic
350C4—61no—
351C4D3×D4×C911yescharacteristic
352C4D3×D4×C911yescharacteristic
353C3—21no—
354C3C9×#47211yescharacteristic, complemented
355C3C3×#474911yescharacteristic
356C2—21no—
357C2—81no—
358C2—121no—
359C2—241no—
360C2C9×#88411yescharacteristic
subgroupG/H
centerC2×C9#884
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#100C2
socleC2×C32C3×#121
prime radicalC3×#4749C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#100D3
hypocenterC3C9×#472
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#100

Complements

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

NH ≅ G/Ntype
1. C9×#932C2semidirect
2. C2×C9×#213C2semidirect
3. C9×#908C2semidirect
4. C9×#907C2semidirect
5. C3×C9×#100C2semidirect
6. C9×#823C2semidirect
7. C9×#822C2semidirect
39. C9×#100D3semidirect
57. #4749C9direct (N × H)
104. #932C2×C9semidirect
105. C2×#213C2×C9semidirect
106. #908C2×C9semidirect
107. #907C2×C9semidirect
111. C3×#100C2×C9semidirect
113. #823C2×C9semidirect
114. #822C2×C9semidirect
221. #100D3×C9semidirect
222. C3×C9#472semidirect
314. C9#4749direct (N × H)
354. C3C9×#472semidirect

Maximal subgroups

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

subgroupindex
C9×#9322
C2×C9×#2132
C9×#9082
C9×#9072
C3×C9×#1002
C9×#8232
C9×#8222
C3×#47493
C9×#4723