Finite Group Databasebeta

← C9×#5835 #13153745

Subgroups — C9×#5835 #13153745

Order 1728.

970 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C9×#1009C211yesmaximal, characteristic, complemented
3C9×#950C213yesmaximal, complemented
4C9×#949C213yesmaximal, complemented
5C9×#948C211yesmaximal, characteristic, complemented
6C9×#932C213yesmaximal, complemented
7C2×C9×#213C213yesmaximal, complemented
8C3×#5835C311yesmaximal, characteristic
9C9×#576—31nomaximal
10C3×C9×#41—21no—
11C3×C9×#41C2211yescharacteristic
12C3×C9×#41C2213yescomplemented
13C2×C3×D4×C9—23no—
14C2×C3×D4×C9C2213yescomplemented
15C9×#239—21no—
16D3×D4×C9—23no—
17C9×#235C2213yescomplemented
18C2×C9×D12C2213yescomplemented
19C9×#216C2211yescharacteristic, complemented
20C9×#215C2213yescomplemented
21C9×#214C2213yescomplemented
22C9×#213C2213yescomplemented
23C9×#213C2216yescomplemented
24C9×#208C2213yescomplemented
25C2×C9×C3⋊C8C2213yescomplemented
26C32×H(5,𝔽2)C2×C311yescharacteristic
27C3×#1009C2×C311yescharacteristic
28C3×#950C2×C313yes—
29C3×#949C2×C313yes—
30C3×#948C2×C311yescharacteristic
31C3×#932C2×C313yes—
32C2×C3×#213C2×C313yes—
33C9×H(5,𝔽2)—21no—
34C9×H(5,𝔽2)—31no—
35C9×H(5,𝔽2)D311yescharacteristic, complemented
36C9×AGL(1,ℤ/8)—33no—
37C9×AGL(1,ℤ/8)—33no—
38C9×#135—33no—
39C2×C9×D8—33no—
40C9×#131—31no—
41C23×C3×C9—23no—
42C22×D3×C9—23no—
43C3×Q8×C9D411yescharacteristic
44C3×Q8×C9C2311yescharacteristic
45C3×D4×C9—23no—
46C3×D4×C9—23no—
47C3×D4×C9D413yes—
48C3×D4×C9C2313yes—
49C2×C3×C4×C9—23no—
50C2×C3×C4×C9C2313yes—
51C9×#67—23no—
52C9×D12—23no—
53C9×D12C2313yes—
54C4×D3×C9—23no—
55C9×Dic6C2311yescharacteristic
56C9×C3⋊C8C2311yescharacteristic
57C9×C3⋊C8C2313yes—
58C3×#576—31no—
59#5835C911yescharacteristic, direct product factor
60C32×#41—21no—
61C32×#41C22×C313yes—
62C32×#41C22×C311yescharacteristic
63C2×C32×D4—23no—
64C2×C32×D4C22×C313yes—
65C3×#239—21no—
66C3×D3×D4—23no—
67C3×#235C22×C313yes—
68C2×C3×D12C22×C313yes—
69C3×#216C22×C311yescharacteristic
70C3×#215C22×C313yes—
71C3×#214C22×C313yes—
72C3×#213C22×C313yes—
73C3×#213C22×C316yes—
74C3×#208C22×C313yes—
75C2×C3×C3⋊C8C22×C313yes—
76C9×#41—22no—
77C9×#41—21no—
78C9×#41—23no—
79C9×#41—21no—
80C9×#41—33no—
81C9×#41—61no—
82C9×#41C2×D311yescharacteristic
83C9×#41C2×D313yescomplemented
84C2×D4×C9—26no—
85C2×D4×C9—23no—
86C2×D4×C9—23no—
87C2×D4×C9—33no—
88C2×D4×C9—63no—
89C2×D4×C9C2×D313yescomplemented
90C9×Q16—31no—
91C9×SD16—33no—
92C9×SD16—33no—
93C9×D8—33no—
94C9×D8—36no—
95C9×M16—33no—
96C2×C8×C9—33no—
97C22×C3×C9—23no—
98C22×C3×C9—83no—
99C22×C3×C9C2×D413yes—
100C2×D3×C9—23no—
101C2×D3×C9—83no—
102C3×C4×C9—21no—
103C3×C4×C9C2×D413yes—
104C3×C4×C9C2411yescharacteristic
105C9×Dic3—21no—
106C3×H(5,𝔽2)—21no—
107C3×H(5,𝔽2)—31no—
108C3×H(5,𝔽2)C2×C911yescharacteristic, complemented
109C3×H(5,𝔽2)C3×D311yescharacteristic
110#1009C2×C911yescharacteristic, complemented
111C3×AGL(1,ℤ/8)—33no—
112C3×AGL(1,ℤ/8)—33no—
113C3×#135—33no—
114C2×C3×D8—33no—
115C3×#131—31no—
116#950C2×C913yescomplemented
117#949C2×C913yescomplemented
118#948C2×C911yescharacteristic, complemented
119#932C2×C913yescomplemented
120C2×#213C2×C913yescomplemented
121C23×C32—23no—
122C22×C3×D3—23no—
123C32×Q8C3×D411yescharacteristic
124C32×Q8C23×C311yescharacteristic
125C32×D4—23no—
126C32×D4—23no—
127C32×D4C3×D413yes—
128C32×D4C23×C313yes—
129C2×C32×C4—23no—
130C2×C32×C4C23×C313yes—
131C3×#67—23no—
132C3×D12—23no—
133C3×D12C23×C313yes—
134C3×C4×D3—23no—
135C3×Dic6C23×C311yescharacteristic
136C23×C9—26no—
137C23×C9—23no—
138C23×C9—63no—
139C3×C3⋊C8C23×C311yescharacteristic
140C3×C3⋊C8C23×C313yes—
141Q8×C9—21no—
142Q8×C9—21no—
143Q8×C9—31no—
144Q8×C9#6711yescharacteristic
145Q8×C9C22×D311yescharacteristic
146D4×C9—26no—
147D4×C9—26no—
148D4×C9—23no—
149D4×C9—23no—
150D4×C9—23no—
151D4×C9—23no—
152D4×C9—33no—
153D4×C9—63no—
154D4×C9—63no—
155D4×C9#6713yes—
156D4×C9C22×D313yes—
157C2×C4×C9—26no—
158C2×C4×C9—23no—
159C2×C4×C9—23no—
160C2×C4×C9—63no—
161C2×C4×C9C22×D313yes—
162C8×C9—31no—
163C8×C9—33no—
164#576—31noSylow
165C2×C3×C9—23no—
166C2×C3×C9—43no—
167C2×C3×C9C22×D411yescharacteristic
168D3×C9—43no—
169C3×#41—22no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C3×#41—23no—
174C3×#41—33no—
175C3×#41—61no—
176C3×#41C22×C911yescharacteristic
177C3×#41C22×C913yescomplemented
178C3×#41C2×C3×D311yescharacteristic
179C3×#41C2×C3×D313yes—
180C2×C3×D4—26no—
181C2×C3×D4—23no—
182C2×C3×D4—23no—
183C2×C3×D4—23no—
184C2×C3×D4—33no—
185C2×C3×D4—63no—
186C2×C3×D4C22×C913yescomplemented
187C2×C3×D4C2×C3×D313yes—
188#239—21no—
189D3×D4—23no—
190#235C22×C913yescomplemented
191C2×D12C22×C913yescomplemented
192C3×Q16—31no—
193C3×SD16—33no—
194C3×SD16—33no—
195C3×D8—33no—
196C3×D8—36no—
197C3×M16—33no—
198C2×C3×C8—33no—
199#216C22×C911yescharacteristic, complemented
200#215C22×C913yescomplemented
201#214C22×C913yescomplemented
202#213C22×C913yescomplemented
203#213C22×C916yescomplemented
204#208C22×C913yescomplemented
205C2×C3⋊C8C22×C913yescomplemented
206C22×C32—23no—
207C22×C32—83no—
208C22×C32C2×C3×D413yes—
209C2×C3×D3—23no—
210C2×C3×D3—83no—
211C32×C4—21no—
212C32×C4C2×C3×D413yes—
213C32×C4C24×C311yescharacteristic
214C3×Dic3—21no—
215C22×C9—26no—
216C22×C9—23no—
217C22×C9—23no—
218C22×C9—63no—
219C22×C9—83no—
220C22×C9—86no—
221C22×C9—243no—
222C22×C9C2×#6713yes—
223C4×C9—22no—
224C4×C9—21no—
225C4×C9—21no—
226C4×C9—23no—
227C4×C9—61no—
228C4×C9C2×#6713yes—
229C4×C9C23×D311yescharacteristic
230H(5,𝔽2)—31no—
231H(5,𝔽2)D3×C911yescharacteristic, complemented
232AGL(1,ℤ/8)—33no—
233AGL(1,ℤ/8)—33no—
234#135—33no—
235C2×D8—33no—
236#131—31no—
237C3×C9#57611yescharacteristic, complemented, Sylow
238C23×C3—26no—
239C23×C3—23no—
240C23×C3—23no—
241C23×C3—63no—
242C22×D3—23no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—31no—
246C3×Q8D4×C911yescharacteristic
247C3×Q8C23×C911yescharacteristic
248C3×Q8C3×#6711yescharacteristic
249C3×Q8C22×C3×D311yescharacteristic
250C3×D4—26no—
251C3×D4—26no—
252C3×D4—23no—
253C3×D4—23no—
254C3×D4—23no—
255C3×D4—23no—
256C3×D4—23no—
257C3×D4—23no—
258C3×D4—33no—
259C3×D4—63no—
260C3×D4—63no—
261C3×D4D4×C913yes—
262C3×D4C23×C913yes—
263C3×D4C3×#6713yes—
264C3×D4C22×C3×D313yes—
265C2×C3×C4—26no—
266C2×C3×C4—23no—
267C2×C3×C4—23no—
268C2×C3×C4—23no—
269C2×C3×C4—63no—
270C2×C3×C4C23×C913yes—
271C2×C3×C4C22×C3×D313yes—
272#67—23no—
273D12—23no—
274D12C23×C913yes—
275C4×D3—23no—
276Dic6C23×C911yescharacteristic
277C3×C8—31no—
278C3×C8—33no—
279C3⋊C8C23×C911yescharacteristic
280C3⋊C8C23×C913yes—
281C2×C32—23no—
282C2×C32—43no—
283C2×C32C22×C3×D411yescharacteristic
284C3×D3—43no—
285C2×C9—23no—
286C2×C9—21no—
287C2×C9—43no—
288C2×C9—46no—
289C2×C9—43no—
290C2×C9—123no—
291C2×C9C22×#6711yescharacteristic
292#41—21no—
293#41—33no—
294#41—61no—
295#41C2×D3×C911yescharacteristic
296#41C2×D3×C913yescomplemented
297C2×D4—23no—
298C2×D4—33no—
299C2×D4—63no—
300C2×D4C2×D3×C913yescomplemented
301Q16—31no—
302SD16—33no—
303SD16—33no—
304D8—33no—
305D8—36no—
306M16—33no—
307C2×C8—33no—
308C22×C3—26no—
309C22×C3—23no—
310C22×C3—23no—
311C22×C3—23no—
312C22×C3—63no—
313C22×C3—83no—
314C22×C3—83no—
315C22×C3—86no—
316C22×C3—243no—
317C22×C3C2×D4×C913yes—
318C22×C3C2×C3×#6713yes—
319C2×D3—23no—
320C2×D3—83no—
321C3×C4—22no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—23no—
326C3×C4—61no—
327C3×C4C2×D4×C913yes—
328C3×C4C24×C911yescharacteristic
329C3×C4C2×C3×#6713yes—
330C3×C4C23×C3×D311yescharacteristic
331Dic3—21no—
332C32C3×#57611yescharacteristic
333C9—21no—
334C9#583511yescharacteristic, direct product factor
335C23—23no—
336C23—63no—
337Q8—31no—
338Q8C9×#6711yescharacteristic
339Q8C22×D3×C911yescharacteristic
340D4—23no—
341D4—23no—
342D4—33no—
343D4—63no—
344D4—63no—
345D4C9×#6713yes—
346D4C22×D3×C913yes—
347C2×C4—23no—
348C2×C4—63no—
349C2×C4C22×D3×C913yes—
350C8—31no—
351C8—33no—
352C2×C3—23no—
353C2×C3—23no—
354C2×C3—21no—
355C2×C3—43no—
356C2×C3—46no—
357C2×C3—43no—
358C2×C3—43no—
359C2×C3—123no—
360C2×C3C22×D4×C911yescharacteristic
361C2×C3C22×C3×#6711yescharacteristic
362D3—43no—
363C22—23no—
364C22—63no—
365C22—83no—
366C22—243no—
367C22C2×C9×#6713yes—
368C4—21no—
369C4—61no—
370C4C2×C9×#6713yes—
371C4C23×D3×C911yescharacteristic
372C3—21no—
373C3C9×#57611yescharacteristic, complemented
374C3C3×#583511yescharacteristic
375C2—23no—
376C2—43no—
377C2—123no—
378C2C22×C9×#6711yescharacteristic
subgroupG/H
centerC2×C9C22×#67
commutatorC3×C4C24×C9
FrattiniC3×C4C23×C3×D3
FittingC3×C9×H(5,𝔽2)C2
socleC2×C32C22×C3×D4
prime radicalC3×#5835C3
cosocleC32×C4C24×C3
hypercenterC9×H(5,𝔽2)D3
hypocenterC3C9×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C9×#41 ◁ C9×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×C9×H(5,𝔽2)C2semidirect
2. C9×#1009C2semidirect
3. C9×#950C2semidirect
4. C9×#949C2semidirect
5. C9×#948C2semidirect
6. C9×#932C2semidirect
7. C2×C9×#213C2semidirect
12. C3×C9×#41C22semidirect
14. C2×C3×D4×C9C22semidirect
17. C9×#235C22semidirect
18. C2×C9×D12C22semidirect
19. C9×#216C22semidirect
20. C9×#215C22semidirect
21. C9×#214C22semidirect
22. C9×#213C22semidirect
23. C9×#213C22semidirect
24. C9×#208C22semidirect
25. C2×C9×C3⋊C8C22semidirect
35. C9×H(5,𝔽2)D3semidirect
59. #5835C9direct (N × H)
83. C9×#41C2×D3semidirect
89. C2×D4×C9C2×D3semidirect
108. C3×H(5,𝔽2)C2×C9semidirect
110. #1009C2×C9semidirect
116. #950C2×C9semidirect
117. #949C2×C9semidirect
118. #948C2×C9semidirect
119. #932C2×C9semidirect
120. C2×#213C2×C9semidirect
177. C3×#41C22×C9semidirect
186. C2×C3×D4C22×C9semidirect
190. #235C22×C9semidirect
191. C2×D12C22×C9semidirect
199. #216C22×C9semidirect
200. #215C22×C9semidirect
201. #214C22×C9semidirect
202. #213C22×C9semidirect
203. #213C22×C9semidirect
204. #208C22×C9semidirect
205. C2×C3⋊C8C22×C9semidirect
231. H(5,𝔽2)D3×C9semidirect
237. C3×C9#576semidirect
296. #41C2×D3×C9semidirect
300. C2×D4C2×D3×C9semidirect
334. C9#5835direct (N × H)
373. C3C9×#576semidirect

Maximal subgroups

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

subgroupindex
C3×C9×H(5,𝔽2)2
C9×#10092
C9×#9502
C9×#9492
C9×#9482
C9×#9322
C2×C9×#2132
C3×#58353
C9×#5763