Finite Group Databasebeta

← C9×#5831 #13153704

Subgroups — C9×#5831 #13153704

Order 1728.

1034 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
1C2×C3×C9×#41C211yesmaximal, characteristic, complemented
2C2×D3×Q8×C9C211yesmaximal, characteristic, complemented
3Q8×C9×Dic3C211yesmaximal, characteristic, complemented
4C9×#938C213yesmaximal, complemented
5C9×#936C213yesmaximal, complemented
6C4×C9×#67C213yesmaximal, complemented
7C9×#924C213yesmaximal, complemented
8C3×#5831C311yesmaximal, characteristic
9C9×#548—31nomaximal
10C3×C9×#41—24no—
11C2×C3×Q8×C9C2211yescharacteristic
12C2×C3×D4×C9C2213yes—
13C22×C3×C4×C9C2213yes—
14C2×C9×#67C2213yes—
15D3×Q8×C9—24no—
16C2×C4×D3×C9C2213yes—
17C2×C9×Dic6C2213yes—
18C9×#217C2216yes—
19C9×#217C2213yes—
20C9×(C2×D3)⋊C4C2211yescharacteristic
21C9×C4⋊Dic3C2213yes—
22C9×Dic3⋊C4C2213yes—
23C4×C9×Dic3C2213yes—
24C2×C32×#41C2×C311yescharacteristic
25C2×C3×D3×Q8C2×C311yescharacteristic
26C3×Q8×Dic3C2×C311yescharacteristic
27C3×#938C2×C313yes—
28C3×#936C2×C313yes—
29C3×C4×#67C2×C313yes—
30C3×#924C2×C313yes—
31C2×C9×#41—21no—
32C2×C9×#41D311yescharacteristic, complemented
33C22×Q8×C9—31no—
34C9×#124—33no—
35C9×C22⋊Q8—33no—
36C9×#121—33no—
37C4×Q8×C9—31no—
38C4×D4×C9—33no—
39C23×C3×C9C2313yes—
40C22×D3×C9C2311yescharacteristic
41C3×Q8×C9D414yescomplemented
42C3×D4×C9—26no—
43C2×C3×C4×C9—26no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3×C4×C9C2313yes—
46C9×#67—26no—
47C2×C9×Dic3C2313yes—
48C2×C9×Dic3C2313yes—
49C2×C9×Dic3C2311yescharacteristic
50C4×D3×C9—26no—
51C9×Dic6—26no—
52C3×#548—31no—
53#5831C911yescharacteristic, direct product factor
54C32×#41—24no—
55C2×C32×Q8C22×C311yescharacteristic
56C2×C32×D4C22×C313yes—
57C22×C32×C4C22×C313yes—
58C2×C3×#67C22×C313yes—
59C3×D3×Q8—24no—
60C2×C3×C4×D3C22×C313yes—
61C2×C3×Dic6C22×C313yes—
62C3×#217C22×C316yes—
63C3×#217C22×C313yes—
64C3×(C2×D3)⋊C4C22×C311yescharacteristic
65C3×C4⋊Dic3C22×C313yes—
66C3×Dic3⋊C4C22×C313yes—
67C3×C4×Dic3C22×C313yes—
68C9×#41—28no—
69C9×#41—24no—
70C2×Q8×C9—21no—
71C2×Q8×C9—33no—
72C2×Q8×C9—64no—
73C2×Q8×C9C2×D311yescharacteristic
74C2×D4×C9—23no—
75C2×D4×C9—33no—
76C2×D4×C9C2×D313yes—
77C22×C4×C9—23no—
78C22×C4×C9—33no—
79C22×C4×C9C2×D313yes—
80C9×C4⋊C4—33no—
81C9×C4⋊C4—33no—
82C9×C22⋊C4—33no—
83C9×C22⋊C4—36no—
84C9×C22⋊C4—31no—
85C42×C9—33no—
86C22×C3×C9—23no—
87C22×C3×C9—23no—
88C22×C3×C9—23no—
89C22×C3×C9C2411yescharacteristic
90C2×D3×C9—21no—
91C2×D3×C9—21no—
92C2×D3×C9#4112yescomplemented
93C3×C4×C9—21no—
94C3×C4×C9C2×D416yes—
95C9×Dic3—23no—
96C9×Dic3—23no—
97C9×Dic3—21no—
98C2×C3×#41—21no—
99C2×C3×#41C2×C911yescharacteristic, complemented
100C2×C3×#41C3×D311yescharacteristic
101C22×C3×Q8—31no—
102C2×D3×Q8C2×C911yescharacteristic, complemented
103C3×#124—33no—
104C3×C22⋊Q8—33no—
105C3×#121—33no—
106C3×C4×Q8—31no—
107C3×C4×D4—33no—
108Q8×Dic3C2×C911yescharacteristic, complemented
109#938C2×C913yescomplemented
110#936C2×C913yescomplemented
111C4×#67C2×C913yescomplemented
112#924C2×C913yescomplemented
113C23×C32C23×C313yes—
114C22×C3×D3C23×C311yescharacteristic
115C32×Q8C3×D414yes—
116C32×D4—26no—
117C2×C32×C4—26no—
118C2×C32×C4C23×C311yescharacteristic
119C2×C32×C4C23×C313yes—
120C3×#67—26no—
121C2×C3×Dic3C23×C311yescharacteristic
122C2×C3×Dic3C23×C313yes—
123C2×C3×Dic3C23×C313yes—
124C3×C4×D3—26no—
125C3×Dic6—26no—
126C23×C9—23no—
127C23×C9—31no—
128C23×C9C22×D313yes—
129Q8×C9—24no—
130Q8×C9—66no—
131Q8×C9#6714yescomplemented
132D4×C9—212no—
133D4×C9—26no—
134D4×C9—66no—
135C2×C4×C9—212no—
136C2×C4×C9—26no—
137C2×C4×C9—21no—
138C2×C4×C9—23no—
139C2×C4×C9—31no—
140C2×C4×C9—33no—
141C2×C4×C9—33no—
142C2×C4×C9—66no—
143C2×C4×C9C22×D311yescharacteristic
144C2×C4×C9C22×D313yes—
145#548—31noSylow
146C2×C3×C9—43no—
147C2×C3×C9C22×D411yescharacteristic
148C2×C3×C9C2×#4111yescharacteristic
149C2×C3×C92−1+411yescharacteristic
150D3×C9—22no—
151C3×#41—28no—
152C3×#41—24no—
153C3×#41—24no—
154C2×C3×Q8—21no—
155C2×C3×Q8—33no—
156C2×C3×Q8—64no—
157C2×C3×Q8C22×C911yescharacteristic
158C2×C3×Q8C2×C3×D311yescharacteristic
159C2×C3×D4—23no—
160C2×C3×D4—33no—
161C2×C3×D4C22×C913yes—
162C2×C3×D4C2×C3×D313yes—
163C22×C3×C4—23no—
164C22×C3×C4—33no—
165C22×C3×C4C22×C913yes—
166C22×C3×C4C2×C3×D313yes—
167C2×#67C22×C913yes—
168D3×Q8—24no—
169C2×C4×D3C22×C913yes—
170C2×Dic6C22×C913yes—
171C3×C4⋊C4—33no—
172C3×C4⋊C4—33no—
173C3×C22⋊C4—33no—
174C3×C22⋊C4—36no—
175C3×C22⋊C4—31no—
176C3×C42—33no—
177#217C22×C916yes—
178#217C22×C913yes—
179(C2×D3)⋊C4C22×C911yescharacteristic
180C4⋊Dic3C22×C913yes—
181Dic3⋊C4C22×C913yes—
182C4×Dic3C22×C913yes—
183C22×C32—23no—
184C22×C32—23no—
185C22×C32—23no—
186C22×C32C24×C311yescharacteristic
187C2×C3×D3—21no—
188C2×C3×D3—21no—
189C2×C3×D3C3×#4112yes—
190C32×C4—21no—
191C32×C4C2×C3×D416yes—
192C3×Dic3—23no—
193C3×Dic3—23no—
194C3×Dic3—21no—
195C22×C9—26no—
196C22×C9—23no—
197C22×C9—23no—
198C22×C9—23no—
199C22×C9—21no—
200C22×C9—32no—
201C22×C9—43no—
202C22×C9—43no—
203C22×C9—61no—
204C22×C9—61no—
205C22×C9C23×D311yescharacteristic
206C4×C9—22no—
207C4×C9—21no—
208C4×C9—26no—
209C4×C9—63no—
210C4×C9—61no—
211C4×C9—63no—
212C4×C9C2×#6716yes—
213C2×#41D3×C911yescharacteristic, complemented
214C22×Q8—31no—
215#124—33no—
216C22⋊Q8—33no—
217#121—33no—
218C4×Q8—31no—
219C4×D4—33no—
220C3×C9#54811yescharacteristic, complemented, Sylow
221C23×C3—23no—
222C23×C3—31no—
223C23×C3C23×C913yes—
224C23×C3C22×C3×D313yes—
225C22×D3C23×C911yescharacteristic
226C3×Q8—24no—
227C3×Q8—66no—
228C3×Q8D4×C914yescomplemented
229C3×Q8C3×#6714yes—
230C3×D4—212no—
231C3×D4—26no—
232C3×D4—26no—
233C3×D4—66no—
234C2×C3×C4—212no—
235C2×C3×C4—26no—
236C2×C3×C4—26no—
237C2×C3×C4—21no—
238C2×C3×C4—23no—
239C2×C3×C4—31no—
240C2×C3×C4—33no—
241C2×C3×C4—33no—
242C2×C3×C4—66no—
243C2×C3×C4C23×C911yescharacteristic
244C2×C3×C4C23×C913yes—
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×C3×C4C22×C3×D313yes—
247#67—26no—
248C2×Dic3C23×C911yescharacteristic
249C2×Dic3C23×C913yes—
250C2×Dic3C23×C913yes—
251C4×D3—26no—
252Dic6—26no—
253C2×C32—43no—
254C2×C32C22×C3×D411yescharacteristic
255C2×C32C2×C3×#4111yescharacteristic
256C2×C32C3×2−1+411yescharacteristic
257C3×D3—22no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9—21no—
261C2×C9—46no—
262C2×C9—43no—
263C2×C9—62no—
264C2×C9D3×#4111yescharacteristic
265C2×C9#101011yescharacteristic
266C2×C9C22×#6711yescharacteristic
267#41—24no—
268C2×Q8—33no—
269C2×Q8—64no—
270C2×Q8C2×D3×C911yescharacteristic
271C2×D4—33no—
272C2×D4C2×D3×C913yes—
273C22×C4—33no—
274C22×C4C2×D3×C913yes—
275C4⋊C4—33no—
276C4⋊C4—33no—
277C22⋊C4—33no—
278C22⋊C4—36no—
279C22⋊C4—31no—
280C42—33no—
281C22×C3—26no—
282C22×C3—23no—
283C22×C3—23no—
284C22×C3—23no—
285C22×C3—23no—
286C22×C3—23no—
287C22×C3—23no—
288C22×C3—21no—
289C22×C3—32no—
290C22×C3—43no—
291C22×C3—43no—
292C22×C3—61no—
293C22×C3—61no—
294C22×C3C24×C911yescharacteristic
295C22×C3C23×C3×D311yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3C9×#4112yescomplemented
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—26no—
303C3×C4—63no—
304C3×C4—61no—
305C3×C4—63no—
306C3×C4C2×D4×C916yes—
307C3×C4C2×C3×#6716yes—
308Dic3—23no—
309Dic3—23no—
310Dic3—21no—
311C32C3×#54811yescharacteristic
312C9—21no—
313C9#583111yescharacteristic, direct product factor
314C23—31no—
315C23C22×D3×C913yes—
316Q8—66no—
317Q8C9×#6714yescomplemented
318D4—26no—
319D4—66no—
320C2×C4—26no—
321C2×C4—31no—
322C2×C4—33no—
323C2×C4—33no—
324C2×C4—66no—
325C2×C4C22×D3×C911yescharacteristic
326C2×C4C22×D3×C913yes—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—46no—
331C2×C3—43no—
332C2×C3—43no—
333C2×C3—62no—
334C2×C3C22×D4×C911yescharacteristic
335C2×C3C2×C9×#4111yescharacteristic
336C2×C3C9×2−1+411yescharacteristic
337C2×C3C3×D3×#4111yescharacteristic
338C2×C3C3×#101011yescharacteristic
339C2×C3C22×C3×#6711yescharacteristic
340D3—22no—
341C22—23no—
342C22—23no—
343C22—23no—
344C22—32no—
345C22—61no—
346C22—61no—
347C22C23×D3×C911yescharacteristic
348C4—21no—
349C4—63no—
350C4—61no—
351C4—63no—
352C4C2×C9×#6716yes—
353C3—21no—
354C3C9×#54811yescharacteristic, complemented
355C3C3×#583111yescharacteristic
356C2—43no—
357C2—62no—
358C2D3×C9×#4111yescharacteristic
359C2C9×#101011yescharacteristic
360C2C22×C9×#6711yescharacteristic
subgroupG/H
centerC22×C9C23×D3
commutatorC22×C3C24×C9
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×#41C2
socle, cosocleC22×C32C24×C3
prime radicalC3×#5831C3
hypercenterC2×C9×#41D3
hypocenterC3C9×#548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C9×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×#41C2semidirect
2. C2×D3×Q8×C9C2semidirect
3. Q8×C9×Dic3C2semidirect
4. C9×#938C2semidirect
5. C9×#936C2semidirect
6. C4×C9×#67C2semidirect
7. C9×#924C2semidirect
32. C2×C9×#41D3semidirect
41. C3×Q8×C9D4semidirect
53. #5831C9direct (N × H)
92. C2×D3×C9#41semidirect
99. C2×C3×#41C2×C9semidirect
102. C2×D3×Q8C2×C9semidirect
108. Q8×Dic3C2×C9semidirect
109. #938C2×C9semidirect
110. #936C2×C9semidirect
111. C4×#67C2×C9semidirect
112. #924C2×C9semidirect
131. Q8×C9#67semidirect
213. C2×#41D3×C9semidirect
220. C3×C9#548semidirect
228. C3×Q8D4×C9semidirect
298. C2×D3C9×#41semidirect
313. C9#5831direct (N × H)
317. Q8C9×#67semidirect
354. C3C9×#548semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×#412
C2×D3×Q8×C92
Q8×C9×Dic32
C9×#9382
C9×#9362
C4×C9×#672
C9×#9242
C3×#58313
C9×#5483