Finite Group Databasebeta

← #10588682⋊C3 #13142707

Subgroups — #10588682⋊C3 #13142707

Order 1728.

608 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C22×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic
23−1+2×#117C211yesmaximal, characteristic, complemented
3C2×#3667⋊C3C211yesmaximal, characteristic, complemented
4#63369⋊C3C214yesmaximal, complemented
5C3×#5001—31nomaximal
6#10588682C311yesmaximal, characteristic, complemented
7C23×AGL(1,ℤ/9)—21no—
8C22×C4×3−1+2C2211yescharacteristic
9C2×C4⋊AGL(1,ℤ/9)—21no—
10C2×C4⋊AGL(1,ℤ/9)—22no—
11C2×C4⋊AGL(1,ℤ/9)C414yescomplemented
12C2×C4⋊AGL(1,ℤ/9)C2212yes—
13C4⋊C4×3−1+2C2212yescomplemented
14C22⋊C4×3−1+2—21no—
15C42×3−1+2—21no—
16#3667⋊C3—22no—
17C2×3−1+2⋊C8C2212yescomplemented
18C22×C3×D12—31no—
19C32×#117D311yescharacteristic
20C2×C3×#208—31no—
21C22×D36C2×C311yescharacteristic
22C3×#809—34no—
23C9×#117—21no—
24C9×#117C2×C311yescharacteristic, complemented
25C2×#3667C2×C311yescharacteristic, complemented
26#63369C2×C314yescomplemented
27C23×3−1+2D411yescharacteristic
28C22×AGL(1,ℤ/9)—24no—
29C22×AGL(1,ℤ/9)—22no—
30C22×AGL(1,ℤ/9)—44no—
31C2×C4×3−1+2—22no—
32C2×C4×3−1+2C2×C412yes—
33C2×C4×3−1+2D411yescharacteristic
34C2×C4×3−1+2D412yes—
35C2×C4×3−1+2C2311yescharacteristic
36C4⋊AGL(1,ℤ/9)—24no—
37C4⋊AGL(1,ℤ/9)—22no—
38C4⋊AGL(1,ℤ/9)C2×C414yes—
393−1+2⋊C8—22no—
40C3×#418—91no—
41#5001—31no—
42C23×C3×D3—61no—
43C22×C32×C4C2×D311yescharacteristic
44C2×C3×D12—34no—
45C2×C3×D12—32no—
46C2×C3×D12—62no—
47C2×C3×D12—61no—
48C23×D9—21no—
49C32×C4⋊C4C2×D312yes—
50C32×C22⋊C4—21no—
51C32×C42—21no—
52C3×#208—62no—
53C2×C3×C3⋊C8—32no—
54C22×C4×C9—21no—
55C22×C4×C9C22×C311yescharacteristic
56C2×D36—21no—
57C2×D36—22no—
58C2×D36C3×C414yescomplemented
59C2×D36C22×C312yes—
60C9×C4⋊C4—22no—
61C9×C4⋊C4C22×C312yescomplemented
62C9×C22⋊C4—22no—
63C9×C22⋊C4—21no—
64C42×C9—22no—
65C42×C9—21no—
66#3667—22no—
67C2×C9⋊C8C22×C312yescomplemented
68C22×3−1+2—22no—
69C22×3−1+2C22⋊C412yes—
70C22×3−1+2C2×D411yescharacteristic
71C2×AGL(1,ℤ/9)—24no—
72C2×AGL(1,ℤ/9)—48no—
73C2×AGL(1,ℤ/9)—44no—
74C4×3−1+2—42no—
75C4×3−1+2C22⋊C412yes—
76C4×3−1+2C22×C411yescharacteristic
77C4×3−1+2C2×D411yescharacteristic
78C22×C3×D4—91no—
79C22×D12—31no—
80C2×C3×M16—91no—
81C3×#117—31no—
82C3×#117C3×D311yescharacteristic
83C2×#208—31no—
84C3×#102—94no—
85#809—34no—
86C23×C32#6711yescharacteristic
87C22×C3×D3—64no—
88C22×C3×D3—62no—
89C22×C3×D3—124no—
90C2×C32×C4—22no—
91C2×C32×C4C4×D312yes—
92C2×C32×C4D1212yes—
93C2×C32×C4#6711yescharacteristic
94C2×C32×C4C22×D311yescharacteristic
95C3×D12—34no—
96C3×D12—64no—
97C3×D12—62no—
98C23×C9—21no—
99C23×C9C3×D411yescharacteristic
100C22×D9—24no—
101C22×D9—22no—
102C22×D9—44no—
103C3×C3⋊C8—62no—
104C2×C4×C9—24no—
105C2×C4×C9—22no—
106C2×C4×C9—22no—
107C2×C4×C9—22no—
108C2×C4×C9—21no—
109C2×C4×C9—21no—
110C2×C4×C9C2×C3×C412yes—
111C2×C4×C9C3×D412yes—
112C2×C4×C9C3×D411yescharacteristic
113C2×C4×C9C23×C311yescharacteristic
114D36—24no—
115D36—22no—
116D36C2×C3×C414yes—
117C9⋊C8—22no—
118#418—91noSylow
119C2×3−1+2—22no—
120C2×3−1+2C2×C22⋊C411yescharacteristic
121C2×3−1+2AGL(1,ℤ/8)12yes—
122AGL(1,ℤ/9)—44no—
123C24×C3—181no—
124C23×D3—61no—
125C2×C3×D4—92no—
126C2×C3×D4—94no—
127C2×C3×D4—182no—
128C2×C3×D4—181no—
129C22×C3×C4—31no—
130C22×C3×C4C2×C3×D311yescharacteristic
131C2×D12—34no—
132C2×D12—32no—
133C2×D12—62no—
134C2×D12—61no—
135C3×M16—182no—
136C2×C3×C8—92no—
137C3×C4⋊C4—32no—
138C3×C4⋊C4C2×C3×D312yes—
139C3×C22⋊C4—21no—
140C3×C22⋊C4—61no—
141C3×C42—21no—
142C3×C42—61no—
143#208—62no—
144C2×C3⋊C8—32no—
145C22×C32—22no—
146C22×C32(C2×D3)⋊C412yes—
147C22×C32C2×#6711yescharacteristic
148C2×C3×D3—64no—
149C2×C3×D3—128no—
150C2×C3×D3—124no—
151C32×C4—42no—
152C32×C4(C2×D3)⋊C412yes—
153C32×C4C2×C4×D311yescharacteristic
154C32×C4C2×D1211yescharacteristic
155C22×C9—22no—
156C22×C9—22no—
157C22×C9—21no—
158C22×C9—42no—
159C22×C9C3×C22⋊C412yes—
160C22×C9C2×C3×D411yescharacteristic
161C2×D9—24no—
162C2×D9—48no—
163C2×D9—44no—
164C4×C9—22no—
165C4×C9—21no—
166C4×C9—21no—
167C4×C9—44no—
168C4×C9—42no—
169C4×C9C3×C22⋊C412yes—
170C4×C9C22×C3×C411yescharacteristic
171C4×C9C2×C3×D411yescharacteristic
172C22×D4—91no—
173C2×M16—91no—
174#117AGL(1,ℤ/9)11yescharacteristic, complemented
175#102—94no—
1763−1+2#41811yescharacteristic, complemented, Sylow
177C23×C3—31no—
178C23×C3—182no—
179C23×C3—184no—
180C23×C3—364no—
181C23×C3C3×#6711yescharacteristic
182C22×D3—64no—
183C22×D3—62no—
184C22×D3—124no—
185C3×D4—94no—
186C3×D4—182no—
187C3×D4—184no—
188C2×C3×C4—22no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—62no—
194C2×C3×C4C3×C4×D312yes—
195C2×C3×C4C3×D1212yes—
196C2×C3×C4C3×#6711yescharacteristic
197C2×C3×C4C22×C3×D311yescharacteristic
198D12—34no—
199D12—64no—
200D12—62no—
201C3×C8—182no—
202C3⋊C8—62no—
203C2×C32—22no—
204C2×C32C2×(C2×D3)⋊C411yescharacteristic
205C2×C32#94912yes—
206C3×D3—124no—
207C2×C9—22no—
208C2×C9—21no—
209C2×C9—22no—
210C2×C9—42no—
211C2×C9C2×C3×C22⋊C411yescharacteristic
212C2×C9C3×AGL(1,ℤ/8)12yes—
213D9—44no—
214C24—181no—
215C2×D4—94no—
216C2×D4—92no—
217C2×D4—182no—
218C2×D4—181no—
219C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
220M16—182no—
221C2×C8—92no—
222C4⋊C4C2×AGL(1,ℤ/9)12yescomplemented
223C22⋊C4—21no—
224C42—21no—
225C22×C3—22no—
226C22×C3—31no—
227C22×C3—32no—
228C22×C3—62no—
229C22×C3—184no—
230C22×C3—364no—
231C22×C3—368no—
232C22×C3C3×(C2×D3)⋊C412yes—
233C22×C3C2×C3×#6711yescharacteristic
234C2×D3—64no—
235C2×D3—128no—
236C2×D3—124no—
237C3×C4—31no—
238C3×C4—31no—
239C3×C4—32no—
240C3×C4—42no—
241C3×C4—122no—
242C3×C4C3×(C2×D3)⋊C412yes—
243C3×C4C2×C3×C4×D311yescharacteristic
244C3×C4C2×C3×D1211yescharacteristic
245C32#500111yescharacteristic
246C9—21no—
247C9C3×#41811yescharacteristic, complemented
248C23—182no—
249C23—184no—
250C23—364no—
251C23#611⋊C311yescharacteristic
252D4—94no—
253D4—182no—
254D4—184no—
255C2×C4—22no—
256C2×C4C4×AGL(1,ℤ/9)12yes—
257C2×C4C4⋊AGL(1,ℤ/9)12yes—
258C2×C4#611⋊C311yescharacteristic
259C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
260C8—182no—
261C2×C3—22no—
262C2×C3—31no—
263C2×C3—32no—
264C2×C3—62no—
265C2×C3—364no—
266C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
267C2×C3C3×#94912yes—
268D3—124no—
269C22—22no—
270C22—184no—
271C22—368no—
272C22—364no—
273C22(C2×AGL(1,ℤ/9))⋊C412yes—
274C22C2×#611⋊C311yescharacteristic
275C4—42no—
276C4(C2×AGL(1,ℤ/9))⋊C412yes—
277C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
278C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
279C3—31no—
280C3C3×#500111yescharacteristic
281C2—22no—
282C2—364no—
283C2C2×(C2×AGL(1,ℤ/9))⋊C411yescharacteristic
284C2#63512⋊C312yes—
subgroupG/H
centerC22C2×#611⋊C3
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#117C2
socleC22×C3C2×C3×#67
prime radicalC22×C4⋊AGL(1,ℤ/9)C2
cosocleC2×C4×C9C23×C3
hypercenter#117AGL(1,ℤ/9)
hypocenterC9C3×#418
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#117C2semidirect
3. C2×#3667⋊C3C2semidirect
4. #63369⋊C3C2semidirect
6. #10588682C3semidirect
11. C2×C4⋊AGL(1,ℤ/9)C4semidirect
13. C4⋊C4×3−1+2C22semidirect
17. C2×3−1+2⋊C8C22semidirect
24. C9×#117C2×C3semidirect
25. C2×#3667C2×C3semidirect
26. #63369C2×C3semidirect
58. C2×D36C3×C4semidirect
61. C9×C4⋊C4C22×C3semidirect
67. C2×C9⋊C8C22×C3semidirect
174. #117AGL(1,ℤ/9)semidirect
176. 3−1+2#418semidirect
222. C4⋊C4C2×AGL(1,ℤ/9)semidirect
247. C9C3×#418semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊AGL(1,ℤ/9)2
3−1+2×#1172
C2×#3667⋊C32
#63369⋊C32
C3×#50013
#105886823