Finite Group Databasebeta

← C9×#5061 #13142916

Subgroups — C9×#5061 #13142916

Order 1728.

532 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#1004C211yesmaximal, characteristic
2C3×C9×#124C211yesmaximal, characteristic, complemented
3C9×#942C212yesmaximal, complemented
4C9×#833C211yesmaximal, characteristic, complemented
5C9×#805C212yesmaximal, complemented
6C3×#5061C311yesmaximal, characteristic
7C9×#455—31nomaximal
8C2×C3×Q8×C9C2211yescharacteristic, complemented
9C2×C3×D4×C9C2211yescharacteristic
10C2×C9×#67—22no—
11C9×#237—22no—
12D3×D4×C9—22no—
13C9×#235C2212yes—
14C3×C9×C22⋊C4—22no—
15C3×C42×C9C2211yescharacteristic, complemented
16C9×#216—22no—
17C9×#215—22no—
18C9×#208C2212yescomplemented
19C3×#1004C2×C311yescharacteristic
20C32×#124C2×C311yescharacteristic
21C3×#942C2×C312yes—
22C9×H(5,𝔽2)—31no—
23C3×#833C2×C311yescharacteristic
24C3×#805C2×C312yes—
25C9×#137—32no—
26C9×#124—21no—
27C9×#124D311yescharacteristic, complemented
28C9×#104—32no—
29C9×#100—31no—
30C23×C3×C9D412yes—
31C22×D3×C9—22no—
32C3×Q8×C9—22no—
33C3×D4×C9—22no—
34C2×C3×C4×C9—21no—
35C2×C3×C4×C9—21no—
36C2×C3×C4×C9C2311yescharacteristic
37C9×#67—24no—
38C9×#67—22no—
39C2×C9×Dic3—22no—
40C9×D12D412yes—
41C4×D3×C9—22no—
42C9×Dic6D412yes—
43C9×C3⋊C8—22no—
44C3×#455—31no—
45#5061C911yescharacteristic, direct product factor
46C2×C32×Q8C22×C311yescharacteristic
47C2×C32×D4C22×C311yescharacteristic
48C2×C3×#67—22no—
49C3×#237—22no—
50C3×D3×D4—22no—
51C3×#235C22×C312yes—
52C32×C22⋊C4—22no—
53C32×C42C22×C311yescharacteristic
54C3×#216—22no—
55C3×#215—22no—
56C3×#208C22×C312yes—
57C9×#41—32no—
58C9×#41—62no—
59C2×Q8×C9—21no—
60C2×Q8×C9C2×D311yescharacteristic, complemented
61C2×D4×C9—21no—
62C2×D4×C9—62no—
63C2×D4×C9—62no—
64C2×D4×C9C2×D311yescharacteristic
65C9×Q16—62no—
66C9×SD16—62no—
67C9×M16—32no—
68C9×C22⋊C4—24no—
69C9×C22⋊C4—22no—
70C42×C9—21no—
71C42×C9C2×D311yescharacteristic, complemented
72C22×C3×C9—22no—
73C22×C3×C9—42no—
74C22×C3×C9C2×D411yescharacteristic
75C2×D3×C9—22no—
76C2×D3×C9—82no—
77C3×C4×C9—41no—
78C3×C4×C9—41no—
79C3×C4×C9C2×D412yes—
80C9×Dic3—22no—
81C3×H(5,𝔽2)—31no—
82#1004C2×C911yescharacteristic
83C3×#137—32no—
84C3×#124—21no—
85C3×#124C2×C911yescharacteristic, complemented
86C3×#124C3×D311yescharacteristic
87#942C2×C912yescomplemented
88C3×#104—32no—
89C3×#100—31no—
90#833C2×C911yescharacteristic, complemented
91#805C2×C912yescomplemented
92C23×C32C3×D412yes—
93C22×C3×D3—22no—
94C32×Q8—22no—
95C32×D4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4C23×C311yescharacteristic
99C3×#67—24no—
100C3×#67—22no—
101C2×C3×Dic3—22no—
102C3×D12C3×D412yes—
103C3×C4×D3—22no—
104C3×Dic6C3×D412yes—
105C23×C9—22no—
106C23×C9—62no—
107C23×C9#6712yes—
108C3×C3⋊C8—22no—
109Q8×C9—22no—
110Q8×C9—32no—
111Q8×C9—42no—
112D4×C9—22no—
113D4×C9—32no—
114D4×C9—42no—
115D4×C9—64no—
116D4×C9—62no—
117C2×C4×C9—22no—
118C2×C4×C9—22no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—62no—
123C2×C4×C9—62no—
124C2×C4×C9C22×D311yescharacteristic
125C8×C9—62no—
126#455—31noSylow
127C2×C3×C9—21no—
128C2×C3×C9—42no—
129C2×C3×C9#12011yescharacteristic
130D3×C9—42no—
131C3×#41—32no—
132C3×#41—62no—
133C2×C3×Q8—21no—
134C2×C3×Q8C22×C911yescharacteristic, complemented
135C2×C3×Q8C2×C3×D311yescharacteristic
136C2×C3×D4—21no—
137C2×C3×D4—62no—
138C2×C3×D4—62no—
139C2×C3×D4C22×C911yescharacteristic
140C2×C3×D4C2×C3×D311yescharacteristic
141C2×#67—22no—
142#237—22no—
143D3×D4—22no—
144#235C22×C912yes—
145C3×Q16—62no—
146C3×SD16—62no—
147C3×M16—32no—
148C3×C22⋊C4—24no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C42—21no—
152C3×C42C22×C911yescharacteristic, complemented
153C3×C42C2×C3×D311yescharacteristic
154#216—22no—
155#215—22no—
156#208C22×C912yescomplemented
157C22×C32—22no—
158C22×C32—42no—
159C22×C32C2×C3×D411yescharacteristic
160C2×C3×D3—22no—
161C2×C3×D3—82no—
162C32×C4—41no—
163C32×C4—41no—
164C32×C4C2×C3×D412yes—
165C3×Dic3—22no—
166C22×C9—22no—
167C22×C9—21no—
168C22×C9—42no—
169C22×C9—42no—
170C22×C9—44no—
171C22×C9—62no—
172C22×C9—242no—
173C22×C9C2×#6711yescharacteristic
174C4×C9—22no—
175C4×C9—42no—
176C4×C9—41no—
177C4×C9—42no—
178C4×C9—41no—
179C4×C9—62no—
180C4×C9D3×D412yes—
181H(5,𝔽2)—31no—
182#137—32no—
183#124D3×C911yescharacteristic, complemented
184#104—32no—
185#100—31no—
186C3×C9#45511yescharacteristic, complemented, Sylow
187C23×C3—22no—
188C23×C3—62no—
189C23×C3D4×C912yes—
190C23×C3C3×#6712yes—
191C22×D3—22no—
192C3×Q8—22no—
193C3×Q8—22no—
194C3×Q8—32no—
195C3×Q8—42no—
196C3×D4—22no—
197C3×D4—22no—
198C3×D4—32no—
199C3×D4—42no—
200C3×D4—64no—
201C3×D4—62no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4C23×C911yescharacteristic
212C2×C3×C4C22×C3×D311yescharacteristic
213#67—24no—
214#67—22no—
215C2×Dic3—22no—
216D12D4×C912yes—
217C4×D3—22no—
218Dic6D4×C912yes—
219C3×C8—62no—
220C3⋊C8—22no—
221C2×C32—21no—
222C2×C32—42no—
223C2×C32C3×#12011yescharacteristic
224C3×D3—42no—
225C2×C9—22no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9—42no—
229C2×C9—82no—
230C2×C9—122no—
231C2×C9#93711yescharacteristic
232#41—32no—
233#41—62no—
234C2×Q8C2×D3×C911yescharacteristic, complemented
235C2×D4—62no—
236C2×D4—62no—
237C2×D4C2×D3×C911yescharacteristic
238Q16—62no—
239SD16—62no—
240M16—32no—
241C22⋊C4—22no—
242C42C2×D3×C911yescharacteristic, complemented
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—44no—
250C22×C3—62no—
251C22×C3—242no—
252C22×C3C2×D4×C911yescharacteristic
253C22×C3C2×C3×#6711yescharacteristic
254C2×D3—22no—
255C2×D3—82no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—41no—
259C3×C4—42no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—62no—
264C3×C4C2×D4×C912yes—
265C3×C4C3×D3×D412yes—
266Dic3—22no—
267C32C3×#45511yescharacteristic
268C9—21no—
269C9#506111yescharacteristic, direct product factor
270C23—62no—
271C23C9×#6712yes—
272Q8—22no—
273Q8—32no—
274D4—22no—
275D4—32no—
276D4—64no—
277D4—62no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4C22×D3×C911yescharacteristic
283C8—62no—
284C2×C3—22no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3—42no—
290C2×C3—82no—
291C2×C3—122no—
292C2×C3C9×#12011yescharacteristic
293C2×C3C3×#93711yescharacteristic
294D3—42no—
295C22—22no—
296C22—42no—
297C22—62no—
298C22—242no—
299C22C2×C9×#6711yescharacteristic
300C4—41no—
301C4—41no—
302C4—62no—
303C4D3×D4×C912yes—
304C3—21no—
305C3C9×#45511yescharacteristic, complemented
306C3C3×#506111yescharacteristic
307C2—21no—
308C2—42no—
309C2—122no—
310C2C9×#93711yescharacteristic
subgroupG/H
centerC2×C9#937
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#124C2
socleC2×C32C3×#120
prime radicalC3×#1004C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#124D3
hypocenterC3C9×#455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#124

Complements

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

NH ≅ G/Ntype
2. C3×C9×#124C2semidirect
3. C9×#942C2semidirect
4. C9×#833C2semidirect
5. C9×#805C2semidirect
8. C2×C3×Q8×C9C22semidirect
15. C3×C42×C9C22semidirect
18. C9×#208C22semidirect
27. C9×#124D3semidirect
45. #5061C9direct (N × H)
60. C2×Q8×C9C2×D3semidirect
71. C42×C9C2×D3semidirect
85. C3×#124C2×C9semidirect
87. #942C2×C9semidirect
90. #833C2×C9semidirect
91. #805C2×C9semidirect
134. C2×C3×Q8C22×C9semidirect
152. C3×C42C22×C9semidirect
156. #208C22×C9semidirect
183. #124D3×C9semidirect
186. C3×C9#455semidirect
234. C2×Q8C2×D3×C9semidirect
242. C42C2×D3×C9semidirect
269. C9#5061direct (N × H)
305. C3C9×#455semidirect

Maximal subgroups

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

subgroupindex
C9×#10042
C3×C9×#1242
C9×#9422
C9×#8332
C9×#8052
C3×#50613
C9×#4553