Finite Group Databasebeta

← C9×#4737 #13141596

Subgroups — C9×#4737 #13141596

Order 1728.

368 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
1C9×#930C211yesmaximal, characteristic
2C9×#926C211yesmaximal, characteristic, complemented
3C3×C9×C22⋊C8C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic, complemented
6C9×#818C211yesmaximal, characteristic, complemented
7C9×#817C211yesmaximal, characteristic, complemented
8C3×#4737C311yesmaximal, characteristic
9C9×#482—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C2×C9×D12C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C9×#217—21no—
16C9×(C2×D3)⋊C4—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×C4⋊Dic3C2211yescharacteristic
19C9×C4⋊Dic3C2211yescharacteristic
20C4×C9×Dic3—21no—
21C3×#930C2×C311yescharacteristic
22C3×#926C2×C311yescharacteristic
23C32×C22⋊C8C2×C311yescharacteristic
24C3×#821C2×C311yescharacteristic
25C3×#821C2×C311yescharacteristic
26C3×#818C2×C311yescharacteristic
27C3×#817C2×C311yescharacteristic
28C9×#121—31no—
29C9×#117—31no—
30C9×#107—31no—
31C9×#106—31no—
32C9×#102—31no—
33C9×#102—31no—
34C9×C22⋊C8—21no—
35C9×C22⋊C8D311yescharacteristic, complemented
36C23×C3×C9D411yescharacteristic
37C22×D3×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C9×#67—42no—
43C2×C9×Dic3—21no—
44C2×C9×Dic3—21no—
45C2×C9×Dic3—21no—
46C9×D12—21no—
47C9×D12—21no—
48C3×C8×C9—21no—
49C3×C8×C9—21no—
50C3×#482—31no—
51#4737C911yescharacteristic, direct product factor
52C22×C32×C4C22×C311yescharacteristic
53C2×C3×#67—21no—
54C2×C3×D12C22×C311yescharacteristic
55C2×C32×C8C22×C311yescharacteristic
56C2×C32×C8C22×C311yescharacteristic
57C3×#217—21no—
58C3×(C2×D3)⋊C4—21no—
59C3×C4⋊Dic3C22×C311yescharacteristic
60C3×C4⋊Dic3C22×C311yescharacteristic
61C3×C4⋊Dic3C22×C311yescharacteristic
62C3×C4×Dic3—21no—
63C2×D4×C9—31no—
64C2×D4×C9—61no—
65C22×C4×C9—21no—
66C22×C4×C9C2×D311yescharacteristic
67C2×C8×C9—21no—
68C2×C8×C9—21no—
69C2×C8×C9C2×D311yescharacteristic
70C2×C8×C9C2×D311yescharacteristic
71C9×C4⋊C4—31no—
72C9×C4⋊C4—31no—
73C9×C4⋊C4—31no—
74C9×C22⋊C4—61no—
75C9×C22⋊C4—61no—
76C42×C9—61no—
77C22×C3×C9—21no—
78C22×C3×C9—21no—
79C22×C3×C9—21no—
80C22×C3×C9C2×D411yescharacteristic
81C2×D3×C9—41no—
82C2×D3×C9—41no—
83C2×D3×C9—41no—
84C3×C4×C9—21no—
85C3×C4×C9#4111yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C9×Dic3—41no—
88C9×Dic3—41no—
89C9×Dic3—41no—
90C3×#121—31no—
91C3×#117—31no—
92#930C2×C911yescharacteristic
93#926C2×C911yescharacteristic, complemented
94C3×#107—31no—
95C3×#106—31no—
96C3×#102—31no—
97C3×#102—31no—
98C3×C22⋊C8—21no—
99C3×C22⋊C8C2×C911yescharacteristic, complemented
100C3×C22⋊C8C3×D311yescharacteristic
101#821C2×C911yescharacteristic, complemented
102#821C2×C911yescharacteristic, complemented
103#818C2×C911yescharacteristic, complemented
104#817C2×C911yescharacteristic, complemented
105C23×C32C3×D411yescharacteristic
106C22×C3×D3—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4C3×D411yescharacteristic
110C2×C32×C4C23×C311yescharacteristic
111C3×#67—42no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C3×D12—21no—
116C3×D12—21no—
117C23×C9—21no—
118C23×C9—61no—
119C23×C9D1211yescharacteristic
120C32×C8—21no—
121C32×C8—21no—
122D4×C9—61no—
123D4×C9—61no—
124D4×C9—122no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—41no—
130C2×C4×C9—41no—
131C2×C4×C9—61no—
132C2×C4×C9—61no—
133C2×C4×C9—61no—
134C2×C4×C9D1211yescharacteristic
135C2×C4×C9C22×D311yescharacteristic
136C8×C9—21no—
137C8×C9—21no—
138C8×C9—41no—
139C8×C9—41no—
140#482—31noSylow
141C2×C3×C9—41no—
142C2×C3×C9#12311yescharacteristic
143C2×C3×C9#13511yescharacteristic
144C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
145D3×C9—81no—
146C2×C3×D4—31no—
147C2×C3×D4—61no—
148C22×C3×C4—21no—
149C22×C3×C4C22×C911yescharacteristic
150C22×C3×C4C2×C3×D311yescharacteristic
151C2×#67—21no—
152C2×D12C22×C911yescharacteristic
153C2×C3×C8—21no—
154C2×C3×C8—21no—
155C2×C3×C8C22×C911yescharacteristic
156C2×C3×C8C22×C911yescharacteristic
157C2×C3×C8C2×C3×D311yescharacteristic
158C2×C3×C8C2×C3×D311yescharacteristic
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—31no—
162C3×C22⋊C4—61no—
163C3×C22⋊C4—61no—
164C3×C42—61no—
165#217—21no—
166(C2×D3)⋊C4—21no—
167C4⋊Dic3C22×C911yescharacteristic
168C4⋊Dic3C22×C911yescharacteristic
169C4⋊Dic3C22×C911yescharacteristic
170C4×Dic3—21no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32C2×C3×D411yescharacteristic
175C2×C3×D3—41no—
176C2×C3×D3—41no—
177C2×C3×D3—41no—
178C32×C4—21no—
179C32×C4C3×#4111yescharacteristic
180C32×C4C3×#4111yescharacteristic
181C3×Dic3—41no—
182C3×Dic3—41no—
183C3×Dic3—41no—
184C22×C9—22no—
185C22×C9—21no—
186C22×C9—21no—
187C22×C9—21no—
188C22×C9—21no—
189C22×C9—41no—
190C22×C9—41no—
191C22×C9—121no—
192C22×C9—121no—
193C22×C9—121no—
194C22×C9C2×D1211yescharacteristic
195C4×C9—21no—
196C4×C9—21no—
197C4×C9—21no—
198C4×C9—41no—
199C4×C9—121no—
200C4×C9—121no—
201C4×C9—121no—
202C4×C9#23711yescharacteristic
203C4×C9#23711yescharacteristic
204#121—31no—
205#117—31no—
206#107—31no—
207#106—31no—
208#102—31no—
209#102—31no—
210C22⋊C8D3×C911yescharacteristic, complemented
211C3×C9#48211yescharacteristic, complemented, Sylow
212C23×C3—21no—
213C23×C3—61no—
214C23×C3D4×C911yescharacteristic
215C23×C3C3×D1211yescharacteristic
216C22×D3—21no—
217C3×D4—61no—
218C3×D4—61no—
219C3×D4—122no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4D4×C911yescharacteristic
232C2×C3×C4C23×C911yescharacteristic
233C2×C3×C4C3×D1211yescharacteristic
234C2×C3×C4C22×C3×D311yescharacteristic
235#67—42no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239D12—21no—
240D12—21no—
241C3×C8—21no—
242C3×C8—21no—
243C3×C8—21no—
244C3×C8—21no—
245C3×C8—41no—
246C3×C8—41no—
247C2×C32—41no—
248C2×C32C3×#12311yescharacteristic
249C2×C32C3×#13511yescharacteristic
250C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
251C3×D3—81no—
252C2×C9—21no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—42no—
256C2×C9—41no—
257C2×C9—241no—
258C2×C9#88611yescharacteristic
259C2×C9#90411yescharacteristic
260C2×C9#90811yescharacteristic
261C2×D4—31no—
262C2×D4—61no—
263C22×C4C2×D3×C911yescharacteristic
264C2×C8C2×D3×C911yescharacteristic
265C2×C8C2×D3×C911yescharacteristic
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—31no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—61no—
272C22×C3—22no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—121no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3C2×D4×C911yescharacteristic
286C22×C3C2×C3×D1211yescharacteristic
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—41no—
295C3×C4—121no—
296C3×C4—121no—
297C3×C4—121no—
298C3×C4C9×#4111yescharacteristic
299C3×C4C9×#4111yescharacteristic
300C3×C4C3×#23711yescharacteristic
301C3×C4C3×#23711yescharacteristic
302Dic3—41no—
303Dic3—41no—
304Dic3—41no—
305C32C3×#48211yescharacteristic
306C9—21no—
307C9#473711yescharacteristic, direct product factor
308C23—61no—
309C23C9×D1211yescharacteristic
310D4—61no—
311D4—61no—
312D4—122no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4C9×D1211yescharacteristic
319C2×C4C22×D3×C911yescharacteristic
320C8—21no—
321C8—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—42no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—241no—
329C2×C3C9×#12311yescharacteristic
330C2×C3C9×#13511yescharacteristic
331C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
332C2×C3C3×#88611yescharacteristic
333C2×C3C3×#90411yescharacteristic
334C2×C3C3×#90811yescharacteristic
335D3—81no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22C2×C9×D1211yescharacteristic
343C4—21no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4C9×#23711yescharacteristic
348C4C9×#23711yescharacteristic
349C3—21no—
350C3C9×#48211yescharacteristic, complemented
351C3C3×#473711yescharacteristic
352C2—41no—
353C2—241no—
354C2C9×#88611yescharacteristic
355C2C9×#90411yescharacteristic
356C2C9×#90811yescharacteristic
subgroupG/H
centerC22×C9C2×D12
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊C8C2
socleC22×C32C2×C3×D4
prime radicalC3×#930C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×C22⋊C8D3
hypocenterC3C9×#482
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C22×C4×C9 ◁ C9×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C9×#926C2semidirect
3. C3×C9×C22⋊C8C2semidirect
4. C9×#821C2semidirect
5. C9×#821C2semidirect
6. C9×#818C2semidirect
7. C9×#817C2semidirect
35. C9×C22⋊C8D3semidirect
51. #4737C9direct (N × H)
93. #926C2×C9semidirect
99. C3×C22⋊C8C2×C9semidirect
101. #821C2×C9semidirect
102. #821C2×C9semidirect
103. #818C2×C9semidirect
104. #817C2×C9semidirect
210. C22⋊C8D3×C9semidirect
211. C3×C9#482semidirect
307. C9#4737direct (N × H)
350. C3C9×#482semidirect

Maximal subgroups

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

subgroupindex
C9×#9302
C9×#9262
C3×C9×C22⋊C82
C9×#8212
C9×#8182
C9×#8172
C3×#47373
C9×#4823