Finite Group Databasebeta

← C9×#4714 #13141496

Subgroups — C9×#4714 #13141496

Order 1728.

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

normal subgroups only

This group has 340 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×#875C211yesmaximal, characteristic
2C9×#869C211yesmaximal, characteristic, complemented
3C3×C9×#97C211yesmaximal, characteristic, complemented
4C9×#821C212yesmaximal
5C9×#816C212yesmaximal, complemented
6C3×#4714C311yesmaximal, characteristic
7C9×#489—31nomaximal
8C2×C9×D12C2211yescharacteristic
9C2×C9×Dic6—21no—
10C2×C9×Dic6C2211yescharacteristic
11C2×C3×C8×C9C2212yes—
12C3×C42×C9C2211yescharacteristic
13C9×(C2×D3)⋊C4—21no—
14C9×(C2×D3)⋊C4—21no—
15C9×C4⋊Dic3—21no—
16C9×C4⋊Dic3C2212yes—
17C3×#875C2×C311yescharacteristic
18C3×#869C2×C311yescharacteristic
19C32×#97C2×C311yescharacteristic
20C3×#821C2×C312yes—
21C3×#816C2×C312yes—
22C9×C4⋊Q8—31no—
23C9×#124—31no—
24C9×#103—32no—
25C9×#102—32no—
26C9×#97—21no—
27C9×#97D311yescharacteristic, complemented
28C22×D3×C9—21no—
29C2×C3×C4×C9D411yescharacteristic
30C2×C3×C4×C9D411yescharacteristic
31C2×C3×C4×C9C2311yescharacteristic
32C2×C9×Dic3—22no—
33C2×C9×Dic3—21no—
34C9×D12—22no—
35C9×Dic6—22no—
36C9×Dic6—42no—
37C3×C8×C9—22no—
38C3×#489—31no—
39#4714C911yescharacteristic, direct product factor
40C2×C3×D12C22×C311yescharacteristic
41C2×C3×Dic6—21no—
42C2×C3×Dic6C22×C311yescharacteristic
43C2×C32×C8C22×C312yes—
44C32×C42C22×C311yescharacteristic
45C3×(C2×D3)⋊C4—21no—
46C3×(C2×D3)⋊C4—21no—
47C3×C4⋊Dic3—21no—
48C3×C4⋊Dic3C22×C312yes—
49C2×Q8×C9—31no—
50C2×Q8×C9—61no—
51C2×D4×C9—31no—
52C2×C8×C9—22no—
53C2×C8×C9C2×D312yes—
54C9×C4⋊C4—32no—
55C9×C4⋊C4—61no—
56C9×C22⋊C4—61no—
57C9×C22⋊C4—61no—
58C42×C9—21no—
59C42×C9C2×D311yescharacteristic
60C22×C3×C9C2×D411yescharacteristic
61C2×D3×C9—41no—
62C2×D3×C9—41no—
63C2×D3×C9—41no—
64C3×C4×C9—21no—
65C3×C4×C9—21no—
66C3×C4×C9#4112yes—
67C9×Dic3—41no—
68C9×Dic3—42no—
69C3×C4⋊Q8—31no—
70C3×#124—31no—
71#875C2×C911yescharacteristic
72#869C2×C911yescharacteristic, complemented
73C3×#103—32no—
74C3×#102—32no—
75C3×#97—21no—
76C3×#97C2×C911yescharacteristic, complemented
77C3×#97C3×D311yescharacteristic
78#821C2×C912yes—
79#816C2×C912yescomplemented
80C22×C3×D3—21no—
81C2×C32×C4C3×D411yescharacteristic
82C2×C32×C4C3×D411yescharacteristic
83C2×C32×C4C23×C311yescharacteristic
84C2×C3×Dic3—22no—
85C2×C3×Dic3—21no—
86C3×D12—22no—
87C3×Dic6—22no—
88C3×Dic6—42no—
89C23×C9—61no—
90C32×C8—22no—
91Q8×C9—62no—
92Q8×C9—122no—
93D4×C9—62no—
94C2×C4×C9—21no—
95C2×C4×C9—21no—
96C2×C4×C9—21no—
97C2×C4×C9—61no—
98C2×C4×C9—62no—
99C2×C4×C9D1211yescharacteristic
100C2×C4×C9D1211yescharacteristic
101C2×C4×C9C22×D311yescharacteristic
102C8×C9—24no—
103C8×C9—22no—
104#489—31noSylow
105C2×C3×C9#12411yescharacteristic
106C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
107C2×C3×C9#13711yescharacteristic
108D3×C9—81no—
109C2×C3×Q8—31no—
110C2×C3×Q8—61no—
111C2×C3×D4—31no—
112C2×D12C22×C911yescharacteristic
113C2×Dic6—21no—
114C2×Dic6C22×C911yescharacteristic
115C2×C3×C8—22no—
116C2×C3×C8C22×C912yes—
117C2×C3×C8C2×C3×D312yes—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—61no—
120C3×C22⋊C4—61no—
121C3×C22⋊C4—61no—
122C3×C42—21no—
123C3×C42C22×C911yescharacteristic
124C3×C42C2×C3×D311yescharacteristic
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3C22×C912yes—
129C22×C32C2×C3×D411yescharacteristic
130C2×C3×D3—41no—
131C2×C3×D3—41no—
132C2×C3×D3—41no—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4C3×#4112yes—
136C3×Dic3—41no—
137C3×Dic3—42no—
138C22×C9—21no—
139C22×C9—121no—
140C22×C9—121no—
141C22×C9—121no—
142C22×C9C2×D1211yescharacteristic
143C4×C9—21no—
144C4×C9—21no—
145C4×C9—22no—
146C4×C9—41no—
147C4×C9—41no—
148C4×C9—121no—
149C4×C9—122no—
150C4×C9#23512yes—
151C4⋊Q8—31no—
152#124—31no—
153#103—32no—
154#102—32no—
155#97D3×C911yescharacteristic, complemented
156C3×C9#48911yescharacteristic, complemented, Sylow
157C23×C3—61no—
158C22×D3—21no—
159C3×Q8—62no—
160C3×Q8—122no—
161C3×D4—62no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—61no—
166C2×C3×C4—62no—
167C2×C3×C4D4×C911yescharacteristic
168C2×C3×C4D4×C911yescharacteristic
169C2×C3×C4C23×C911yescharacteristic
170C2×C3×C4C3×D1211yescharacteristic
171C2×C3×C4C3×D1211yescharacteristic
172C2×C3×C4C22×C3×D311yescharacteristic
173C2×Dic3—22no—
174C2×Dic3—21no—
175D12—22no—
176Dic6—22no—
177Dic6—42no—
178C3×C8—24no—
179C3×C8—22no—
180C3×C8—22no—
181C2×C32C3×#12411yescharacteristic
182C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
183C2×C32C3×#13711yescharacteristic
184C3×D3—81no—
185C2×C9—21no—
186C2×C9—21no—
187C2×C9—21no—
188C2×C9—241no—
189C2×C9#87511yescharacteristic
190C2×C9#90811yescharacteristic
191C2×C9#90911yescharacteristic
192C2×Q8—31no—
193C2×Q8—61no—
194C2×D4—31no—
195C2×C8C2×D3×C912yes—
196C4⋊C4—32no—
197C4⋊C4—61no—
198C22⋊C4—61no—
199C22⋊C4—61no—
200C42C2×D3×C911yescharacteristic
201C22×C3—21no—
202C22×C3—121no—
203C22×C3—121no—
204C22×C3—121no—
205C22×C3C2×D4×C911yescharacteristic
206C22×C3C2×C3×D1211yescharacteristic
207C2×D3—41no—
208C2×D3—41no—
209C2×D3—41no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—21no—
214C3×C4—22no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—122no—
218C3×C4—121no—
219C3×C4C9×#4112yes—
220C3×C4C3×#23512yes—
221Dic3—41no—
222Dic3—42no—
223C32C3×#48911yescharacteristic
224C9—21no—
225C9#471411yescharacteristic, direct product factor
226C23—61no—
227Q8—62no—
228Q8—122no—
229D4—62no—
230C2×C4—61no—
231C2×C4—62no—
232C2×C4C9×D1211yescharacteristic
233C2×C4C9×D1211yescharacteristic
234C2×C4C22×D3×C911yescharacteristic
235C8—22no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3—241no—
240C2×C3C9×#12411yescharacteristic
241C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
242C2×C3C9×#13711yescharacteristic
243C2×C3C3×#87511yescharacteristic
244C2×C3C3×#90811yescharacteristic
245C2×C3C3×#90911yescharacteristic
246D3—81no—
247C22—121no—
248C22—121no—
249C22—121no—
250C22C2×C9×D1211yescharacteristic
251C4—21no—
252C4—21no—
253C4—122no—
254C4—121no—
255C4C9×#23512yes—
256C3—21no—
257C3C9×#48911yescharacteristic, complemented
258C3C3×#471411yescharacteristic
259C2—241no—
260C2C9×#87511yescharacteristic
261C2C9×#90811yescharacteristic
262C2C9×#90911yescharacteristic
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×#489
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
2. C9×#869C2semidirect
3. C3×C9×#97C2semidirect
5. C9×#816C2semidirect
27. C9×#97D3semidirect
39. #4714C9direct (N × H)
72. #869C2×C9semidirect
76. C3×#97C2×C9semidirect
79. #816C2×C9semidirect
155. #97D3×C9semidirect
156. C3×C9#489semidirect
225. C9#4714direct (N × H)
257. C3C9×#489semidirect

Maximal subgroups

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

subgroupindex
C9×#8752
C9×#8692
C3×C9×#972
C9×#8212
C9×#8162
C3×#47143
C9×#4893