Finite Group Databasebeta

← C9×#5028 #13142784

Subgroups — C9×#5028 #13142784

Order 1728.

406 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
2C2×C9×#216C211yesmaximal, characteristic, complemented
3C2×C9×#215C211yesmaximal, characteristic
4C9×#875C211yesmaximal, characteristic
5C9×#810C211yesmaximal, characteristic, complemented
6C9×#809C211yesmaximal, characteristic
7C9×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5028C311yesmaximal, characteristic
9C9×#464—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×C4⋊C4—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C42×C9—21no—
16C3×C42×C9C2211yescharacteristic
17C9×#216—22no—
18C9×#215—22no—
19C9×(C2×D3)⋊C4—21no—
20C9×(C2×D3)⋊C4—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23C32×C4×Q8C2×C311yescharacteristic
24C2×C3×#216C2×C311yescharacteristic
25C2×C3×#215C2×C311yescharacteristic
26C3×#875C2×C311yescharacteristic
27C3×#810C2×C311yescharacteristic
28C3×#809C2×C311yescharacteristic
29C3×C4⋊(C3⋊C8)C2×C311yescharacteristic
30C2×C9×Q16—31no—
31C2×C9×SD16—31no—
32C9×#124—31no—
33C4×Q8×C9—21no—
34C4×Q8×C9D311yescharacteristic, complemented
35C9×C4⋊C8—31no—
36C9×#103—31no—
37C9×#102—31no—
38C22×D3×C9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9D412yes—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9D411yescharacteristic
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C2×C9×Dic3—21no—
47C9×D12—21no—
48C9×D12—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C9×C3⋊C8—21no—
52C9×C3⋊C8—21no—
53C3×#464—31no—
54#5028C911yescharacteristic, direct product factor
55C2×C32×Q8C22×C311yescharacteristic
56C2×C3×D12C22×C311yescharacteristic
57C2×C3×Dic6C22×C311yescharacteristic
58C32×C4⋊C4—21no—
59C32×C4⋊C4C22×C311yescharacteristic
60C32×C42—21no—
61C32×C42C22×C311yescharacteristic
62C3×#216—22no—
63C3×#215—22no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C2×C3×C3⋊C8C22×C311yescharacteristic
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C2×Q8×C9—21no—
69C2×Q8×C9—31no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D4×C9—31no—
72C9×Q16—62no—
73C9×SD16—62no—
74C2×C8×C9—31no—
75C2×C8×C9—31no—
76C9×C4⋊C4—22no—
77C9×C4⋊C4—21no—
78C9×C4⋊C4—21no—
79C9×C4⋊C4C2×D311yescharacteristic
80C9×C22⋊C4—61no—
81C9×C22⋊C4—61no—
82C42×C9—22no—
83C42×C9—21no—
84C42×C9—21no—
85C42×C9C2×D311yescharacteristic
86C22×C3×C9C2×D411yescharacteristic
87C2×D3×C9—41no—
88C2×D3×C9—41no—
89C2×D3×C9—41no—
90C3×C4×C9—22no—
91C3×C4×C9—21no—
92C3×C4×C9—21no—
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C9×Dic3—41no—
97C2×C3×Q16—31no—
98C2×C3×SD16—31no—
99C3×#124—31no—
100C3×C4×Q8—21no—
101C3×C4×Q8C2×C911yescharacteristic, complemented
102C3×C4×Q8C3×D311yescharacteristic
103C2×#216C2×C911yescharacteristic, complemented
104C2×#215C2×C911yescharacteristic
105#875C2×C911yescharacteristic
106C3×C4⋊C8—31no—
107C3×#103—31no—
108C3×#102—31no—
109#810C2×C911yescharacteristic, complemented
110#809C2×C911yescharacteristic
111C4⋊(C3⋊C8)C2×C911yescharacteristic, complemented
112C22×C3×D3—21no—
113C32×Q8—21no—
114C32×Q8C3×D412yes—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4C3×D411yescharacteristic
118C2×C32×C4C3×D411yescharacteristic
119C2×C32×C4C23×C311yescharacteristic
120C2×C3×Dic3—21no—
121C3×D12—21no—
122C3×D12—21no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125C23×C9—61no—
126C3×C3⋊C8—21no—
127C3×C3⋊C8—21no—
128Q8×C9—22no—
129Q8×C9—21no—
130Q8×C9—22no—
131Q8×C9—61no—
132Q8×C9—61no—
133Q8×C9D1212yes—
134D4×C9—61no—
135D4×C9—61no—
136C2×C4×C9—22no—
137C2×C4×C9—21no—
138C2×C4×C9—22no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—61no—
144C2×C4×C9#6711yescharacteristic
145C2×C4×C9#6711yescharacteristic
146C2×C4×C9C22×D311yescharacteristic
147C8×C9—61no—
148C8×C9—61no—
149#464—31noSylow
150C2×C3×C9#12111yescharacteristic
151C2×C3×C9#13511yescharacteristic
152C2×C3×C9#13711yescharacteristic
153D3×C9—81no—
154C2×C3×Q8—21no—
155C2×C3×Q8—31no—
156C2×C3×Q8C22×C911yescharacteristic
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—31no—
159C2×D12C22×C911yescharacteristic
160C2×Dic6C22×C911yescharacteristic
161C3×Q16—62no—
162C3×SD16—62no—
163C2×C3×C8—31no—
164C2×C3×C8—31no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4C22×C911yescharacteristic
170C3×C4⋊C4C2×C3×D311yescharacteristic
171C3×C22⋊C4—61no—
172C3×C22⋊C4—61no—
173C3×C42—22no—
174C3×C42—21no—
175C3×C42—21no—
176C3×C42—21no—
177C3×C42C22×C911yescharacteristic
178C3×C42C2×C3×D311yescharacteristic
179#216—22no—
180#215—22no—
181(C2×D3)⋊C4—21no—
182(C2×D3)⋊C4—21no—
183C2×C3⋊C8C22×C911yescharacteristic
184C2×C3⋊C8C22×C911yescharacteristic
185C22×C32C2×C3×D411yescharacteristic
186C2×C3×D3—41no—
187C2×C3×D3—41no—
188C2×C3×D3—41no—
189C32×C4—22no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4C2×C3×D411yescharacteristic
194C32×C4C3×#4111yescharacteristic
195C3×Dic3—41no—
196C22×C9—21no—
197C22×C9—121no—
198C22×C9—121no—
199C22×C9—121no—
200C22×C9C2×#6711yescharacteristic
201C4×C9—22no—
202C4×C9—24no—
203C4×C9—22no—
204C4×C9—21no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—41no—
209C4×C9—42no—
210C4×C9—41no—
211C4×C9—121no—
212C4×C9C2×D1211yescharacteristic
213C4×C9#23511yescharacteristic
214C2×Q16—31no—
215C2×SD16—31no—
216#124—31no—
217C4×Q8D3×C911yescharacteristic, complemented
218C4⋊C8—31no—
219#103—31no—
220#102—31no—
221C3×C9#46411yescharacteristic, complemented, Sylow
222C23×C3—61no—
223C22×D3—21no—
224C3×Q8—22no—
225C3×Q8—21no—
226C3×Q8—21no—
227C3×Q8—22no—
228C3×Q8—61no—
229C3×Q8—61no—
230C3×Q8D4×C912yes—
231C3×Q8C3×D1212yes—
232C3×D4—61no—
233C3×D4—61no—
234C2×C3×C4—22no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—61no—
244C2×C3×C4D4×C911yescharacteristic
245C2×C3×C4D4×C911yescharacteristic
246C2×C3×C4C23×C911yescharacteristic
247C2×C3×C4C3×#6711yescharacteristic
248C2×C3×C4C3×#6711yescharacteristic
249C2×C3×C4C22×C3×D311yescharacteristic
250C2×Dic3—21no—
251D12—21no—
252D12—21no—
253Dic6—21no—
254Dic6—21no—
255C3×C8—61no—
256C3×C8—61no—
257C3⋊C8—21no—
258C3⋊C8—21no—
259C2×C32C3×#12111yescharacteristic
260C2×C32C3×#13511yescharacteristic
261C2×C32C3×#13711yescharacteristic
262C3×D3—81no—
263C2×C9—21no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—241no—
267C2×C9#93011yescharacteristic
268C2×C9#94211yescharacteristic
269C2×C9#95011yescharacteristic
270C2×Q8—31no—
271C2×Q8C2×D3×C911yescharacteristic
272C2×D4—31no—
273Q16—62no—
274SD16—62no—
275C2×C8—31no—
276C2×C8—31no—
277C4⋊C4—21no—
278C4⋊C4C2×D3×C911yescharacteristic
279C22⋊C4—61no—
280C22⋊C4—61no—
281C42—21no—
282C42C2×D3×C911yescharacteristic
283C22×C3—21no—
284C22×C3—121no—
285C22×C3—121no—
286C22×C3—121no—
287C22×C3C2×D4×C911yescharacteristic
288C22×C3C2×C3×#6711yescharacteristic
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C3×C4—24no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—42no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—121no—
307C3×C4C2×D4×C911yescharacteristic
308C3×C4C9×#4111yescharacteristic
309C3×C4C2×C3×D1211yescharacteristic
310C3×C4C3×#23511yescharacteristic
311Dic3—41no—
312C32C3×#46411yescharacteristic
313C9—21no—
314C9#502811yescharacteristic, direct product factor
315C23—61no—
316Q8—21no—
317Q8—61no—
318Q8—61no—
319Q8C9×D1212yes—
320D4—61no—
321D4—61no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—61no—
325C2×C4C9×#6711yescharacteristic
326C2×C4C9×#6711yescharacteristic
327C2×C4C22×D3×C911yescharacteristic
328C8—61no—
329C8—61no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—241no—
334C2×C3C9×#12111yescharacteristic
335C2×C3C9×#13511yescharacteristic
336C2×C3C9×#13711yescharacteristic
337C2×C3C3×#93011yescharacteristic
338C2×C3C3×#94211yescharacteristic
339C2×C3C3×#95011yescharacteristic
340D3—81no—
341C22—121no—
342C22—121no—
343C22—121no—
344C22C2×C9×#6711yescharacteristic
345C4—22no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—121no—
350C4C2×C9×D1211yescharacteristic
351C4C9×#23511yescharacteristic
352C3—21no—
353C3C9×#46411yescharacteristic, complemented
354C3C3×#502811yescharacteristic
355C2—241no—
356C2C9×#93011yescharacteristic
357C2C9×#94211yescharacteristic
358C2C9×#95011yescharacteristic
subgroupG/H
centerC22×C9C2×#67
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C4×Q8×C9C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×D12C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC4×Q8×C9D3
hypocenterC3C9×#464
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C42×C9 ◁ C4×Q8×C9

Complements

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

NH ≅ G/Ntype
1. C3×C4×Q8×C9C2semidirect
2. C2×C9×#216C2semidirect
5. C9×#810C2semidirect
7. C9×C4⋊(C3⋊C8)C2semidirect
34. C4×Q8×C9D3semidirect
54. #5028C9direct (N × H)
101. C3×C4×Q8C2×C9semidirect
103. C2×#216C2×C9semidirect
109. #810C2×C9semidirect
111. C4⋊(C3⋊C8)C2×C9semidirect
217. C4×Q8D3×C9semidirect
221. C3×C9#464semidirect
314. C9#5028direct (N × H)
353. C3C9×#464semidirect

Maximal subgroups

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

subgroupindex
C3×C4×Q8×C92
C2×C9×#2162
C2×C9×#2152
C9×#8752
C9×#8102
C9×#8092
C9×C4⋊(C3⋊C8)2
C3×#50283
C9×#4643