Finite Group Databasebeta

← C9×#4715 #13141519

Subgroups — C9×#4715 #13141519

Order 1728.

412 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×C9×Dic12C212yesmaximal, complemented
2C2×C9×#204C212yesmaximal
3C9×#875C211yesmaximal, characteristic
4C9×#869C211yesmaximal, characteristic, complemented
5C3×C9×#97C211yesmaximal, characteristic, complemented
6C3×#4715C311yesmaximal, characteristic
7C9×#497—31nomaximal
8C2×C9×D12C2211yescharacteristic
9C2×C9×Dic6C2212yes—
10C2×C9×Dic6C2211yescharacteristic
11C2×C3×C8×C9C2212yes—
12C3×C42×C9C2211yescharacteristic
13C9×(C2×D3)⋊C4—22no—
14C9×C4⋊Dic3—22no—
15C9×Dic12—24no—
16C9×#204—24no—
17C2×C3×Dic12C2×C312yes—
18C2×C3×#204C2×C312yes—
19C3×#875C2×C311yescharacteristic
20C3×#869C2×C311yescharacteristic
21C32×#97C2×C311yescharacteristic
22C2×C9×Q16—32no—
23C2×C9×SD16—32no—
24C9×C4⋊Q8—31no—
25C9×#124—31no—
26C9×#97—21no—
27C9×#97D311yescharacteristic, complemented
28C22×D3×C9—21no—
29C2×C3×C4×C9D412yes—
30C2×C3×C4×C9C2311yescharacteristic
31C2×C9×Dic3—22no—
32C2×C9×Dic3—21no—
33C9×D12—22no—
34C9×Dic6—22no—
35C9×Dic6—22no—
36C9×Dic6—22no—
37C3×C8×C9D414yes—
38C3×#497—31no—
39#4715C911yescharacteristic, direct product factor
40C2×C3×D12C22×C311yescharacteristic
41C2×C3×Dic6C22×C312yes—
42C2×C3×Dic6C22×C311yescharacteristic
43C2×C32×C8C22×C312yes—
44C32×C42C22×C311yescharacteristic
45C3×(C2×D3)⋊C4—22no—
46C3×C4⋊Dic3—22no—
47C3×Dic12—24no—
48C3×#204—24no—
49C2×Q8×C9—31no—
50C2×Q8×C9—32no—
51C2×D4×C9—31no—
52C9×Q16—64no—
53C9×SD16—64no—
54C2×C8×C9—22no—
55C2×C8×C9C2×D312yes—
56C9×C4⋊C4—62no—
57C9×C22⋊C4—62no—
58C42×C9—21no—
59C42×C9C2×D311yescharacteristic
60C22×C3×C9C2×D411yescharacteristic
61C2×D3×C9—41no—
62C2×D3×C9—42no—
63C3×C4×C9—22no—
64C3×C4×C9C2×D412yes—
65C9×Dic3—42no—
66C9×Dic3—41no—
67C2×C3×Q16—32no—
68C2×C3×SD16—32no—
69C3×C4⋊Q8—31no—
70C3×#124—31no—
71C2×Dic12C2×C912yescomplemented
72C2×#204C2×C912yes—
73#875C2×C911yescharacteristic
74#869C2×C911yescharacteristic, complemented
75C3×#97—21no—
76C3×#97C2×C911yescharacteristic, complemented
77C3×#97C3×D311yescharacteristic
78C22×C3×D3—21no—
79C2×C32×C4C3×D412yes—
80C2×C32×C4C23×C311yescharacteristic
81C2×C3×Dic3—22no—
82C2×C3×Dic3—21no—
83C3×D12—22no—
84C3×Dic6—22no—
85C3×Dic6—22no—
86C3×Dic6—22no—
87C23×C9—61no—
88C32×C8C3×D414yes—
89Q8×C9—62no—
90Q8×C9—62no—
91Q8×C9—62no—
92D4×C9—62no—
93C2×C4×C9—22no—
94C2×C4×C9—21no—
95C2×C4×C9—61no—
96C2×C4×C9—62no—
97C2×C4×C9D1212yes—
98C2×C4×C9C22×D311yescharacteristic
99C8×C9—24no—
100C8×C9D1214yes—
101#497—31noSylow
102C2×C3×C9#12711yescharacteristic
103C2×C3×C9#13712yes—
104D3×C9—81no—
105C2×C3×Q8—31no—
106C2×C3×Q8—32no—
107C2×C3×D4—31no—
108C2×D12C22×C911yescharacteristic
109C2×Dic6C22×C912yes—
110C2×Dic6C22×C911yescharacteristic
111C3×Q16—64no—
112C3×SD16—64no—
113C2×C3×C8—22no—
114C2×C3×C8C22×C912yes—
115C2×C3×C8C2×C3×D312yes—
116C3×C4⋊C4—62no—
117C3×C22⋊C4—62no—
118C3×C42—21no—
119C3×C42C22×C911yescharacteristic
120C3×C42C2×C3×D311yescharacteristic
121(C2×D3)⋊C4—22no—
122C4⋊Dic3—22no—
123Dic12—24no—
124#204—24no—
125C22×C32C2×C3×D411yescharacteristic
126C2×C3×D3—41no—
127C2×C3×D3—42no—
128C32×C4—22no—
129C32×C4C2×C3×D412yes—
130C3×Dic3—42no—
131C3×Dic3—41no—
132C22×C9—21no—
133C22×C9—121no—
134C22×C9—122no—
135C22×C9C2×D1211yescharacteristic
136C4×C9—22no—
137C4×C9—22no—
138C4×C9—42no—
139C4×C9—121no—
140C4×C9—122no—
141C4×C9C2×D1212yes—
142C2×Q16—32no—
143C2×SD16—32no—
144C4⋊Q8—31no—
145#124—31no—
146#97D3×C911yescharacteristic, complemented
147C3×C9#49711yescharacteristic, complemented, Sylow
148C23×C3—61no—
149C22×D3—21no—
150C3×Q8—62no—
151C3×Q8—62no—
152C3×Q8—62no—
153C3×D4—62no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—61no—
157C2×C3×C4—62no—
158C2×C3×C4D4×C912yes—
159C2×C3×C4C23×C911yescharacteristic
160C2×C3×C4C3×D1212yes—
161C2×C3×C4C22×C3×D311yescharacteristic
162C2×Dic3—22no—
163C2×Dic3—21no—
164D12—22no—
165Dic6—22no—
166Dic6—22no—
167Dic6—22no—
168C3×C8—24no—
169C3×C8D4×C914yes—
170C3×C8C3×D1214yes—
171C2×C32C3×#12711yescharacteristic
172C2×C32C3×#13712yes—
173C3×D3—81no—
174C2×C9—22no—
175C2×C9—21no—
176C2×C9—241no—
177C2×C9#87411yescharacteristic
178C2×C9#90912yes—
179C2×Q8—31no—
180C2×Q8—32no—
181C2×D4—31no—
182Q16—64no—
183SD16—64no—
184C2×C8C2×D3×C912yes—
185C4⋊C4—62no—
186C22⋊C4—62no—
187C42C2×D3×C911yescharacteristic
188C22×C3—21no—
189C22×C3—122no—
190C22×C3—121no—
191C22×C3C2×D4×C911yescharacteristic
192C22×C3C2×C3×D1211yescharacteristic
193C2×D3—41no—
194C2×D3—42no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—22no—
198C3×C4—42no—
199C3×C4—121no—
200C3×C4—122no—
201C3×C4C2×D4×C912yes—
202C3×C4C2×C3×D1212yes—
203Dic3—42no—
204Dic3—41no—
205C32C3×#49711yescharacteristic
206C9—21no—
207C9#471511yescharacteristic, direct product factor
208C23—61no—
209Q8—62no—
210Q8—62no—
211Q8—62no—
212D4—62no—
213C2×C4—61no—
214C2×C4—62no—
215C2×C4C9×D1212yes—
216C2×C4C22×D3×C911yescharacteristic
217C8C9×D1214yes—
218C2×C3—22no—
219C2×C3—21no—
220C2×C3—241no—
221C2×C3C9×#12711yescharacteristic
222C2×C3C9×#13712yes—
223C2×C3C3×#87411yescharacteristic
224C2×C3C3×#90912yes—
225D3—81no—
226C22—122no—
227C22—121no—
228C22C2×C9×D1211yescharacteristic
229C4—22no—
230C4—121no—
231C4—122no—
232C4C2×C9×D1212yes—
233C3—21no—
234C3C9×#49711yescharacteristic, complemented
235C3C3×#471511yescharacteristic
236C2—241no—
237C2C9×#87411yescharacteristic
238C2C9×#90912yes—
subgroupG/H
centerC22×C9C2×D12
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#97C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×D12C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#97D3
hypocenterC3C9×#497
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C42×C9 ◁ C9×#97

Complements

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

NH ≅ G/Ntype
1. C2×C9×Dic12C2semidirect
4. C9×#869C2semidirect
5. C3×C9×#97C2semidirect
27. C9×#97D3semidirect
39. #4715C9direct (N × H)
71. C2×Dic12C2×C9semidirect
74. #869C2×C9semidirect
76. C3×#97C2×C9semidirect
146. #97D3×C9semidirect
147. C3×C9#497semidirect
207. C9#4715direct (N × H)
234. C3C9×#497semidirect

Maximal subgroups

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

subgroupindex
C2×C9×Dic122
C2×C9×#2042
C9×#8752
C9×#8692
C3×C9×#972
C3×#47153
C9×#4973