Finite Group Databasebeta

← C9×#4764 #13141727

Subgroups — C9×#4764 #13141727

Order 1728.

380 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×#940C211yesmaximal, characteristic
2C9×#890C211yesmaximal, characteristic, complemented
3C3×C9×#102C211yesmaximal, characteristic, complemented
4C9×#832C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic, complemented
6C9×#815C211yesmaximal, characteristic, complemented
7C9×#809C211yesmaximal, characteristic, complemented
8C3×#4764C311yesmaximal, characteristic
9C9×#490—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C2×C9×#67—21no—
13C2×C9×D12C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C9×C4⋊Dic3—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C4×C9×Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C3×#940C2×C311yescharacteristic
22C3×#890C2×C311yescharacteristic
23C32×#102C2×C311yescharacteristic
24C3×#832C2×C311yescharacteristic
25C3×#821C2×C311yescharacteristic
26C3×#815C2×C311yescharacteristic
27C3×#809C2×C311yescharacteristic
28C9×#127—31no—
29C9×#125—31no—
30C9×#102—21no—
31C9×#102—31no—
32C9×#102—31no—
33C9×#102—31no—
34C9×#102D311yescharacteristic, complemented
35C9×#97—31no—
36C23×C3×C9—21no—
37C22×D3×C9—21no—
38C3×D4×C9—21no—
39C3×D4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C9×#67—42no—
43C9×#67—42no—
44C2×C9×Dic3—21no—
45C2×C9×Dic3D411yescharacteristic
46C2×C9×Dic3D411yescharacteristic
47C9×D12—21no—
48C9×D12—21no—
49C3×C8×C9—21no—
50C9×C3⋊C8—21no—
51C3×#490—31no—
52#4764C911yescharacteristic, direct product factor
53C2×C32×D4C22×C311yescharacteristic
54C2×C3×#67—21no—
55C2×C3×#67—21no—
56C2×C3×D12C22×C311yescharacteristic
57C2×C32×C8C22×C311yescharacteristic
58C32×C4⋊C4C22×C311yescharacteristic
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3C22×C311yescharacteristic
61C3×Dic3⋊C4—21no—
62C3×C4×Dic3C22×C311yescharacteristic
63C2×C3×C3⋊C8C22×C311yescharacteristic
64C2×D4×C9—21no—
65C2×D4×C9—31no—
66C2×D4×C9—61no—
67C2×D4×C9—61no—
68C2×D4×C9C2×D311yescharacteristic
69C2×C8×C9—21no—
70C2×C8×C9—31no—
71C2×C8×C9C2×D311yescharacteristic
72C9×C4⋊C4—21no—
73C9×C4⋊C4—31no—
74C9×C4⋊C4—61no—
75C9×C4⋊C4—61no—
76C9×C4⋊C4C2×D311yescharacteristic
77C42×C9—31no—
78C22×C3×C9—41no—
79C22×C3×C9—41no—
80C22×C3×C9—41no—
81C22×C3×C9C2×D411yescharacteristic
82C2×D3×C9—41no—
83C2×D3×C9—41no—
84C2×D3×C9—41no—
85C3×C4×C9—41no—
86C3×C4×C9#4111yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C9×Dic3—21no—
89C9×Dic3—21no—
90C9×Dic3—41no—
91C3×#127—31no—
92C3×#125—31no—
93#940C2×C911yescharacteristic
94#890C2×C911yescharacteristic, complemented
95C3×#102—21no—
96C3×#102—31no—
97C3×#102—31no—
98C3×#102—31no—
99C3×#102C2×C911yescharacteristic, complemented
100C3×#102C3×D311yescharacteristic
101C3×#97—31no—
102#832C2×C911yescharacteristic, complemented
103#821C2×C911yescharacteristic, complemented
104#815C2×C911yescharacteristic, complemented
105#809C2×C911yescharacteristic, complemented
106C23×C32—21no—
107C22×C3×D3—21no—
108C32×D4—21no—
109C32×D4—21no—
110C2×C32×C4—21no—
111C2×C32×C4C23×C311yescharacteristic
112C3×#67—42no—
113C3×#67—42no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3C3×D411yescharacteristic
116C2×C3×Dic3C3×D411yescharacteristic
117C3×D12—21no—
118C3×D12—21no—
119C23×C9—21no—
120C23×C9—41no—
121C23×C9—61no—
122C32×C8—21no—
123C3×C3⋊C8—21no—
124D4×C9—22no—
125D4×C9—21no—
126D4×C9—21no—
127D4×C9—41no—
128D4×C9—61no—
129D4×C9—61no—
130D4×C9—122no—
131D4×C9—122no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—31no—
135C2×C4×C9—31no—
136C2×C4×C9—41no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139C8×C9—21no—
140C8×C9—41no—
141C8×C9—61no—
142#490—31noSylow
143C2×C3×C9—81no—
144C2×C3×C9#12411yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
147D3×C9—81no—
148C2×C3×D4—21no—
149C2×C3×D4—31no—
150C2×C3×D4—61no—
151C2×C3×D4—61no—
152C2×C3×D4C22×C911yescharacteristic
153C2×C3×D4C2×C3×D311yescharacteristic
154C2×#67—21no—
155C2×#67—21no—
156C2×D12C22×C911yescharacteristic
157C2×C3×C8—21no—
158C2×C3×C8—31no—
159C2×C3×C8C22×C911yescharacteristic
160C2×C3×C8C2×C3×D311yescharacteristic
161C3×C4⋊C4—21no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4C22×C911yescharacteristic
166C3×C4⋊C4C2×C3×D311yescharacteristic
167C3×C42—31no—
168C4⋊Dic3—21no—
169C4⋊Dic3C22×C911yescharacteristic
170Dic3⋊C4—21no—
171C4×Dic3C22×C911yescharacteristic
172C2×C3⋊C8C22×C911yescharacteristic
173C22×C32—41no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32C2×C3×D411yescharacteristic
177C2×C3×D3—41no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C32×C4—41no—
181C32×C4C3×#4111yescharacteristic
182C32×C4C3×#4111yescharacteristic
183C3×Dic3—21no—
184C3×Dic3—21no—
185C3×Dic3—41no—
186C22×C9—21no—
187C22×C9—42no—
188C22×C9—41no—
189C22×C9—41no—
190C22×C9—41no—
191C22×C9—81no—
192C22×C9—81no—
193C22×C9—121no—
194C22×C9—121no—
195C22×C9—121no—
196C22×C9D3×D411yescharacteristic
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—41no—
200C4×C9—61no—
201C4×C9—61no—
202C4×C9—81no—
203C4×C9—121no—
204C4×C9#23511yescharacteristic
205C4×C9#23711yescharacteristic
206#127—31no—
207#125—31no—
208#102—31no—
209#102—31no—
210#102—31no—
211#102D3×C911yescharacteristic, complemented
212#97—31no—
213C3×C9#49011yescharacteristic, complemented, Sylow
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—41no—
217C23×C3—61no—
218C22×D3—21no—
219C3×D4—22no—
220C3×D4—21no—
221C3×D4—21no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—41no—
225C3×D4—61no—
226C3×D4—61no—
227C3×D4—122no—
228C3×D4—122no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—41no—
235C2×C3×C4—61no—
236C2×C3×C4C23×C911yescharacteristic
237C2×C3×C4C22×C3×D311yescharacteristic
238#67—42no—
239#67—42no—
240C2×Dic3—21no—
241C2×Dic3D4×C911yescharacteristic
242C2×Dic3D4×C911yescharacteristic
243D12—21no—
244D12—21no—
245C3×C8—21no—
246C3×C8—21no—
247C3×C8—41no—
248C3×C8—61no—
249C3⋊C8—21no—
250C2×C32—81no—
251C2×C32C3×#12411yescharacteristic
252C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
253C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
254C3×D3—81no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—82no—
259C2×C9—81no—
260C2×C9—241no—
261C2×C9#88511yescharacteristic
262C2×C9#91111yescharacteristic
263C2×C9#91411yescharacteristic
264C2×D4—31no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4C2×D3×C911yescharacteristic
268C2×C8—31no—
269C2×C8C2×D3×C911yescharacteristic
270C4⋊C4—31no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4C2×D3×C911yescharacteristic
274C42—31no—
275C22×C3—21no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—121no—
286C22×C3—121no—
287C22×C3—121no—
288C22×C3C2×D4×C911yescharacteristic
289C22×C3C3×D3×D411yescharacteristic
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—41no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—61no—
298C3×C4—61no—
299C3×C4—81no—
300C3×C4—121no—
301C3×C4C9×#4111yescharacteristic
302C3×C4C9×#4111yescharacteristic
303C3×C4C3×#23511yescharacteristic
304C3×C4C3×#23711yescharacteristic
305Dic3—21no—
306Dic3—21no—
307Dic3—41no—
308C32C3×#49011yescharacteristic
309C9—21no—
310C9#476411yescharacteristic, direct product factor
311C23—21no—
312C23—61no—
313D4—21no—
314D4—21no—
315D4—61no—
316D4—61no—
317D4—122no—
318D4—122no—
319C2×C4—21no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—61no—
323C2×C4C22×D3×C911yescharacteristic
324C8—21no—
325C8—61no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—82no—
330C2×C3—81no—
331C2×C3—81no—
332C2×C3—241no—
333C2×C3C9×#12411yescharacteristic
334C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
335C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
336C2×C3C3×#88511yescharacteristic
337C2×C3C3×#91111yescharacteristic
338C2×C3C3×#91411yescharacteristic
339D3—81no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22D3×D4×C911yescharacteristic
347C4—41no—
348C4—61no—
349C4—61no—
350C4—121no—
351C4C9×#23511yescharacteristic
352C4C9×#23711yescharacteristic
353C3—21no—
354C3C9×#49011yescharacteristic, complemented
355C3C3×#476411yescharacteristic
356C2—81no—
357C2—241no—
358C2C9×#88511yescharacteristic
359C2C9×#91111yescharacteristic
360C2C9×#91411yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#102C2
socleC22×C32C2×C3×D4
prime radicalC3×#940C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#102D3
hypocenterC3C9×#490
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#102

Complements

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

NH ≅ G/Ntype
2. C9×#890C2semidirect
3. C3×C9×#102C2semidirect
4. C9×#832C2semidirect
5. C9×#821C2semidirect
6. C9×#815C2semidirect
7. C9×#809C2semidirect
34. C9×#102D3semidirect
52. #4764C9direct (N × H)
94. #890C2×C9semidirect
99. C3×#102C2×C9semidirect
102. #832C2×C9semidirect
103. #821C2×C9semidirect
104. #815C2×C9semidirect
105. #809C2×C9semidirect
211. #102D3×C9semidirect
213. C3×C9#490semidirect
310. C9#4764direct (N × H)
354. C3C9×#490semidirect

Maximal subgroups

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

subgroupindex
C9×#9402
C9×#8902
C3×C9×#1022
C9×#8322
C9×#8212
C9×#8152
C9×#8092
C3×#47643
C9×#4903