Finite Group Databasebeta

← C9×#4752 #13141692

Subgroups — C9×#4752 #13141692

Order 1728.

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

normal subgroups only

This group has 464 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×#1007C211yesmaximal, characteristic
2C9×#944C211yesmaximal, characteristic, complemented
3C9×#909C212yesmaximal, complemented
4C3×C9×#101C211yesmaximal, characteristic, complemented
5C9×#825C212yesmaximal, complemented
6C3×#4752C311yesmaximal, characteristic
7C9×#456—31nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9C9×#239—22no—
10D3×Q8×C9—22no—
11C9×#235—22no—
12C9×#235C2212yes—
13C2×C9×Dic6C2211yescharacteristic
14C3×C9×M16C2212yes—
15C9×Dic3⋊C4—22no—
16C4×C9×Dic3C2211yescharacteristic
17C9×Dic12—22no—
18C9×#204—22no—
19C3×#1007C2×C311yescharacteristic
20C3×#944C2×C311yescharacteristic
21C3×#909C2×C312yes—
22C9×2−1+4—31no—
23C32×#101C2×C311yescharacteristic
24C3×#825C2×C312yes—
25C9×#137—32no—
26C9×C4⋊Q8—31no—
27C9×#104—32no—
28C9×#101—21no—
29C9×#101D311yescharacteristic, complemented
30C3×Q8×C9—22no—
31C2×C3×C4×C9D412yes—
32C2×C3×C4×C9C2311yescharacteristic
33C9×#67—22no—
34C2×C9×Dic3—21no—
35C2×C9×Dic3—21no—
36C9×D12—22no—
37C9×D12D412yes—
38C4×D3×C9—24no—
39C4×D3×C9—22no—
40C9×Dic6—22no—
41C9×Dic6—22no—
42C9×Dic6D412yes—
43C3×C8×C9—22no—
44C3×#456—31no—
45#4752C911yescharacteristic, direct product factor
46C2×C32×Q8C22×C311yescharacteristic
47C3×#239—22no—
48C3×D3×Q8—22no—
49C3×#235—22no—
50C3×#235C22×C312yes—
51C2×C3×Dic6C22×C311yescharacteristic
52C32×M16C22×C312yes—
53C3×Dic3⋊C4—22no—
54C3×C4×Dic3C22×C311yescharacteristic
55C3×Dic12—22no—
56C3×#204—22no—
57C9×#41—32no—
58C9×#41—62no—
59C9×#41—62no—
60C2×Q8×C9—21no—
61C2×Q8×C9—31no—
62C2×Q8×C9—62no—
63C2×Q8×C9C2×D311yescharacteristic
64C9×Q16—62no—
65C9×SD16—62no—
66C9×M16—22no—
67C9×M16C2×D312yes—
68C9×C4⋊C4—62no—
69C42×C9—31no—
70C22×C3×C9C2×D411yescharacteristic
71C2×D3×C9—22no—
72C3×C4×C9—22no—
73C3×C4×C9C2×D412yes—
74C9×Dic3—22no—
75C9×Dic3—22no—
76C9×Dic3—41no—
77C3×2−1+4—31no—
78#1007C2×C911yescharacteristic
79C3×#137—32no—
80C3×C4⋊Q8—31no—
81#944C2×C911yescharacteristic, complemented
82#909C2×C912yescomplemented
83C3×#104—32no—
84C3×#101—21no—
85C3×#101C2×C911yescharacteristic, complemented
86C3×#101C3×D311yescharacteristic
87#825C2×C912yescomplemented
88C32×Q8—22no—
89C2×C32×C4C3×D412yes—
90C2×C32×C4C23×C311yescharacteristic
91C3×#67—22no—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—21no—
94C3×D12—22no—
95C3×D12C3×D412yes—
96C3×C4×D3—24no—
97C3×C4×D3—22no—
98C3×Dic6—22no—
99C3×Dic6—22no—
100C3×Dic6C3×D412yes—
101C32×C8—22no—
102Q8×C9—22no—
103Q8×C9—32no—
104Q8×C9—42no—
105Q8×C9—62no—
106Q8×C9—62no—
107D4×C9—32no—
108D4×C9—62no—
109D4×C9—62no—
110C2×C4×C9—22no—
111C2×C4×C9—21no—
112C2×C4×C9—64no—
113C2×C4×C9—61no—
114C2×C4×C9—62no—
115C2×C4×C9—61no—
116C2×C4×C9D1212yes—
117C2×C4×C9C22×D311yescharacteristic
118C8×C9—22no—
119C8×C9—42no—
120#456—31noSylow
121C2×C3×C9—21no—
122C2×C3×C9#12011yescharacteristic
123D3×C9—42no—
124C3×#41—32no—
125C3×#41—62no—
126C3×#41—62no—
127C2×C3×Q8—21no—
128C2×C3×Q8—31no—
129C2×C3×Q8—62no—
130C2×C3×Q8C22×C911yescharacteristic
131C2×C3×Q8C2×C3×D311yescharacteristic
132#239—22no—
133D3×Q8—22no—
134#235—22no—
135#235C22×C912yes—
136C2×Dic6C22×C911yescharacteristic
137C3×Q16—62no—
138C3×SD16—62no—
139C3×M16—22no—
140C3×M16C22×C912yes—
141C3×M16C2×C3×D312yes—
142C3×C4⋊C4—62no—
143C3×C42—31no—
144Dic3⋊C4—22no—
145C4×Dic3C22×C911yescharacteristic
146Dic12—22no—
147#204—22no—
148C22×C32C2×C3×D411yescharacteristic
149C2×C3×D3—22no—
150C32×C4—22no—
151C32×C4C2×C3×D412yes—
152C3×Dic3—22no—
153C3×Dic3—22no—
154C3×Dic3—41no—
155C22×C9—21no—
156C22×C9—62no—
157C22×C9C2×D1211yescharacteristic
158C4×C9—22no—
159C4×C9—22no—
160C4×C9—42no—
161C4×C9—62no—
162C4×C9—62no—
163C4×C9—121no—
164C4×C9D3×D412yes—
1652−1+4—31no—
166#137—32no—
167C4⋊Q8—31no—
168#104—32no—
169#101D3×C911yescharacteristic, complemented
170C3×C9#45611yescharacteristic, complemented, Sylow
171C3×Q8—22no—
172C3×Q8—22no—
173C3×Q8—32no—
174C3×Q8—42no—
175C3×Q8—62no—
176C3×Q8—62no—
177C3×D4—32no—
178C3×D4—62no—
179C3×D4—62no—
180C2×C3×C4—22no—
181C2×C3×C4—21no—
182C2×C3×C4—64no—
183C2×C3×C4—61no—
184C2×C3×C4—62no—
185C2×C3×C4—61no—
186C2×C3×C4D4×C912yes—
187C2×C3×C4C23×C911yescharacteristic
188C2×C3×C4C3×D1212yes—
189C2×C3×C4C22×C3×D311yescharacteristic
190#67—22no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193D12—22no—
194D12D4×C912yes—
195C4×D3—24no—
196C4×D3—22no—
197Dic6—22no—
198Dic6—22no—
199Dic6D4×C912yes—
200C3×C8—22no—
201C3×C8—22no—
202C3×C8—42no—
203C2×C32—21no—
204C2×C32C3×#12011yescharacteristic
205C3×D3—42no—
206C2×C9—21no—
207C2×C9—21no—
208C2×C9—41no—
209C2×C9—122no—
210C2×C9#88211yescharacteristic
211#41—32no—
212#41—62no—
213#41—62no—
214C2×Q8—31no—
215C2×Q8—62no—
216C2×Q8C2×D3×C911yescharacteristic
217Q16—62no—
218SD16—62no—
219M16C2×D3×C912yes—
220C4⋊C4—62no—
221C42—31no—
222C22×C3—21no—
223C22×C3—62no—
224C22×C3C2×D4×C911yescharacteristic
225C22×C3C2×C3×D1211yescharacteristic
226C2×D3—22no—
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—22no—
230C3×C4—42no—
231C3×C4—62no—
232C3×C4—62no—
233C3×C4—121no—
234C3×C4C2×D4×C912yes—
235C3×C4C3×D3×D412yes—
236Dic3—22no—
237Dic3—22no—
238Dic3—41no—
239C32C3×#45611yescharacteristic
240C9—21no—
241C9#475211yescharacteristic, direct product factor
242Q8—22no—
243Q8—32no—
244Q8—62no—
245Q8—62no—
246D4—32no—
247D4—62no—
248D4—62no—
249C2×C4—64no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4C9×D1212yes—
254C2×C4C22×D3×C911yescharacteristic
255C8—22no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—41no—
260C2×C3—122no—
261C2×C3C9×#12011yescharacteristic
262C2×C3C3×#88211yescharacteristic
263D3—42no—
264C22—62no—
265C22C2×C9×D1211yescharacteristic
266C4—22no—
267C4—62no—
268C4—62no—
269C4—121no—
270C4D3×D4×C912yes—
271C3—21no—
272C3C9×#45611yescharacteristic, complemented
273C3C3×#475211yescharacteristic
274C2—21no—
275C2—122no—
276C2C9×#88211yescharacteristic
subgroupG/H
centerC2×C9#882
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#101C2
socleC2×C32C3×#120
prime radicalC3×#1007C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#101D3
hypocenterC3C9×#456
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#101

Complements

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

NH ≅ G/Ntype
2. C9×#944C2semidirect
3. C9×#909C2semidirect
4. C3×C9×#101C2semidirect
5. C9×#825C2semidirect
29. C9×#101D3semidirect
45. #4752C9direct (N × H)
81. #944C2×C9semidirect
82. #909C2×C9semidirect
85. C3×#101C2×C9semidirect
87. #825C2×C9semidirect
169. #101D3×C9semidirect
170. C3×C9#456semidirect
241. C9#4752direct (N × H)
272. C3C9×#456semidirect

Maximal subgroups

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

subgroupindex
C9×#10072
C9×#9442
C9×#9092
C3×C9×#1012
C9×#8252
C3×#47523
C9×#4563