Finite Group Databasebeta

← C9×#5060 #13142908

Subgroups — C9×#5060 #13142908

Order 1728.

448 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C3×C9×#124C211yesmaximal, characteristic
2C2×C9×#216C212yesmaximal, complemented
3C2×C9×#214C212yesmaximal
4C9×#869C211yesmaximal, characteristic, complemented
5C9×#803C211yesmaximal, characteristic, complemented
6C3×#5060C311yesmaximal, characteristic
7C9×#497—31nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9C2×C3×D4×C9C2211yescharacteristic
10C2×C9×Dic6C2212yes—
11C3×C9×C22⋊C4—22no—
12C3×C42×C9C2211yescharacteristic
13C9×#216—24no—
14C9×#214—24no—
15C9×C4⋊Dic3—22no—
16C2×C9×C3⋊C8C2212yes—
17C32×#124C2×C311yescharacteristic
18C2×C3×#216C2×C312yes—
19C2×C3×#214C2×C312yes—
20C3×#869C2×C311yescharacteristic
21C3×#803C2×C311yescharacteristic
22C2×C9×Q16—32no—
23C2×C9×SD16—32no—
24C9×C4⋊Q8—31no—
25C9×#124—21no—
26C9×#124D311yescharacteristic
27C9×#97—31no—
28C23×C3×C9—21no—
29C3×Q8×C9—22no—
30C3×D4×C9—22no—
31C2×C3×C4×C9—21no—
32C2×C3×C4×C9D412yes—
33C2×C3×C4×C9C2311yescharacteristic
34C2×C9×Dic3—22no—
35C9×Dic6—22no—
36C9×Dic6—22no—
37C9×C3⋊C8D414yes—
38C3×#497—31no—
39#5060C911yescharacteristic, direct product factor
40C2×C32×Q8C22×C311yescharacteristic
41C2×C32×D4C22×C311yescharacteristic
42C2×C3×Dic6C22×C312yes—
43C32×C22⋊C4—22no—
44C32×C42C22×C311yescharacteristic
45C3×#216—24no—
46C3×#214—24no—
47C3×C4⋊Dic3—22no—
48C2×C3×C3⋊C8C22×C312yes—
49C2×Q8×C9—21no—
50C2×Q8×C9—32no—
51C2×Q8×C9C2×D311yescharacteristic
52C2×D4×C9—21no—
53C2×D4×C9C2×D311yescharacteristic
54C9×Q16—64no—
55C9×SD16—64no—
56C2×C8×C9—32no—
57C9×C4⋊C4—62no—
58C9×C22⋊C4—24no—
59C9×C22⋊C4—22no—
60C42×C9—21no—
61C42×C9C2×D311yescharacteristic
62C22×C3×C9—41no—
63C22×C3×C9—42no—
64C22×C3×C9C2×D411yescharacteristic
65C3×C4×C9—22no—
66C3×C4×C9—41no—
67C3×C4×C9C2×D412yes—
68C9×Dic3—42no—
69C2×C3×Q16—32no—
70C2×C3×SD16—32no—
71C3×C4⋊Q8—31no—
72C3×#124—21no—
73C3×#124C2×C911yescharacteristic
74C3×#124C3×D311yescharacteristic
75C2×#216C2×C912yescomplemented
76C2×#214C2×C912yes—
77#869C2×C911yescharacteristic, complemented
78C3×#97—31no—
79#803C2×C911yescharacteristic, complemented
80C23×C32—21no—
81C32×Q8—22no—
82C32×D4—22no—
83C2×C32×C4—21no—
84C2×C32×C4C3×D412yes—
85C2×C32×C4C23×C311yescharacteristic
86C2×C3×Dic3—22no—
87C3×Dic6—22no—
88C3×Dic6—22no—
89C23×C9—22no—
90C23×C9—21no—
91C3×C3⋊C8C3×D414yes—
92Q8×C9—22no—
93Q8×C9—42no—
94Q8×C9—62no—
95Q8×C9—62no—
96D4×C9—22no—
97D4×C9—42no—
98C2×C4×C9—22no—
99C2×C4×C9—21no—
100C2×C4×C9—22no—
101C2×C4×C9—21no—
102C2×C4×C9—62no—
103C2×C4×C9#6712yes—
104C2×C4×C9C22×D311yescharacteristic
105C8×C9—34no—
106#497—31noSylow
107C2×C3×C9—81no—
108C2×C3×C9#12711yescharacteristic
109C2×C3×C9#13712yes—
110C2×C3×Q8—21no—
111C2×C3×Q8—32no—
112C2×C3×Q8C22×C911yescharacteristic
113C2×C3×Q8C2×C3×D311yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4C22×C911yescharacteristic
116C2×C3×D4C2×C3×D311yescharacteristic
117C2×Dic6C22×C912yes—
118C3×Q16—64no—
119C3×SD16—64no—
120C2×C3×C8—32no—
121C3×C4⋊C4—62no—
122C3×C22⋊C4—24no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C42—21no—
126C3×C42C22×C911yescharacteristic
127C3×C42C2×C3×D311yescharacteristic
128#216—24no—
129#214—24no—
130C4⋊Dic3—22no—
131C2×C3⋊C8C22×C912yes—
132C22×C32—41no—
133C22×C32—42no—
134C22×C32C2×C3×D411yescharacteristic
135C32×C4—22no—
136C32×C4—41no—
137C32×C4C2×C3×D412yes—
138C3×Dic3—42no—
139C22×C9—21no—
140C22×C9—42no—
141C22×C9—44no—
142C22×C9—41no—
143C22×C9—42no—
144C22×C9C2×#6711yescharacteristic
145C4×C9—22no—
146C4×C9—22no—
147C4×C9—42no—
148C4×C9—42no—
149C4×C9—41no—
150C4×C9—122no—
151C4×C9D3×D412yes—
152C2×Q16—32no—
153C2×SD16—32no—
154C4⋊Q8—31no—
155#124D3×C911yescharacteristic
156#97—31no—
157C3×C9#49711yescharacteristic, complemented, Sylow
158C23×C3—22no—
159C23×C3—21no—
160C23×C3—21no—
161C3×Q8—22no—
162C3×Q8—22no—
163C3×Q8—42no—
164C3×Q8—62no—
165C3×Q8—62no—
166C3×D4—22no—
167C3×D4—22no—
168C3×D4—42no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4—62no—
175C2×C3×C4D4×C912yes—
176C2×C3×C4C23×C911yescharacteristic
177C2×C3×C4C3×#6712yes—
178C2×C3×C4C22×C3×D311yescharacteristic
179C2×Dic3—22no—
180Dic6—22no—
181Dic6—22no—
182C3×C8—34no—
183C3⋊C8D4×C914yes—
184C2×C32—81no—
185C2×C32C3×#12711yescharacteristic
186C2×C32C3×#13712yes—
187C2×C9—21no—
188C2×C9—22no—
189C2×C9—82no—
190C2×C9—81no—
191C2×C9#94011yescharacteristic
192C2×C9#95112yes—
193C2×Q8—32no—
194C2×Q8C2×D3×C911yescharacteristic
195C2×D4C2×D3×C911yescharacteristic
196Q16—64no—
197SD16—64no—
198C2×C8—32no—
199C4⋊C4—62no—
200C22⋊C4—22no—
201C42C2×D3×C911yescharacteristic
202C22×C3—21no—
203C22×C3—42no—
204C22×C3—44no—
205C22×C3—41no—
206C22×C3—42no—
207C22×C3—41no—
208C22×C3—42no—
209C22×C3C2×D4×C911yescharacteristic
210C22×C3C2×C3×#6711yescharacteristic
211C3×C4—22no—
212C3×C4—22no—
213C3×C4—22no—
214C3×C4—42no—
215C3×C4—42no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4—122no—
219C3×C4C2×D4×C912yes—
220C3×C4C3×D3×D412yes—
221Dic3—42no—
222C32C3×#49711yescharacteristic
223C9—21no—
224C9#506011yescharacteristic, direct product factor
225C23—21no—
226Q8—22no—
227Q8—62no—
228Q8—62no—
229D4—22no—
230C2×C4—21no—
231C2×C4—62no—
232C2×C4C9×#6712yes—
233C2×C4C22×D3×C911yescharacteristic
234C8—34no—
235C2×C3—21no—
236C2×C3—22no—
237C2×C3—82no—
238C2×C3—81no—
239C2×C3—81no—
240C2×C3C9×#12711yescharacteristic
241C2×C3C9×#13712yes—
242C2×C3C3×#94011yescharacteristic
243C2×C3C3×#95112yes—
244C22—41no—
245C22—42no—
246C22C2×C9×#6711yescharacteristic
247C4—22no—
248C4—41no—
249C4—122no—
250C4D3×D4×C912yes—
251C3—21no—
252C3C9×#49711yescharacteristic, complemented
253C3C3×#506011yescharacteristic
254C2—81no—
255C2C9×#94011yescharacteristic
256C2C9×#95112yes—
subgroupG/H
centerC22×C9C2×#67
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#124C2
socleC22×C32C2×C3×D4
prime radicalC2×C32×D4C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#124D3
hypocenterC3C9×#497
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C42×C9 ◁ C9×#124

Complements

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

NH ≅ G/Ntype
2. C2×C9×#216C2semidirect
4. C9×#869C2semidirect
5. C9×#803C2semidirect
39. #5060C9direct (N × H)
75. C2×#216C2×C9semidirect
77. #869C2×C9semidirect
79. #803C2×C9semidirect
157. C3×C9#497semidirect
224. C9#5060direct (N × H)
252. C3C9×#497semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1242
C2×C9×#2162
C2×C9×#2142
C9×#8692
C9×#8032
C3×#50603
C9×#4973