Finite Group Databasebeta

← #100×AGL(1,ℤ/9) #13141678

Subgroups — #100×AGL(1,ℤ/9) #13141678

Order 1728.

581 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 118 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×D4×AGL(1,ℤ/9)C211yesmaximal, characteristic
2M16×AGL(1,ℤ/9)C212yesmaximal, complemented
33−1+2×#100C211yesmaximal, characteristic, complemented
4#63391⋊C3C211yesmaximal, characteristic, complemented
5#63383⋊C3C212yesmaximal, complemented
6C3×D3×#100—31nomaximal
7D9×#100C311yesmaximal, characteristic, complemented
8C23×AGL(1,ℤ/9)C412yes—
9C2×D4×3−1+2C2211yescharacteristic
10C2×#611⋊C3C412yes—
11D4×AGL(1,ℤ/9)—22no—
12D4×AGL(1,ℤ/9)—22no—
13C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
14C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
15M16×3−1+2C2212yescomplemented
16#3667⋊C3C2212yescomplemented
17#3663⋊C3—22no—
18C8×AGL(1,ℤ/9)—22no—
19C2×C3×D3×D4—31no—
20C3×D3×M16—32no—
21C2×D4×D9C2×C311yescharacteristic
22C32×#100D311yescharacteristic
23C3×#833—31no—
24C3×#823—32no—
25M16×D9C2×C312yescomplemented
26C9×#100—21no—
27C9×#100C2×C311yescharacteristic, complemented
28#63391C2×C311yescharacteristic, complemented
29#63383C2×C312yescomplemented
30C23×3−1+2C2×C412yes—
31C22×AGL(1,ℤ/9)—22no—
32C22×AGL(1,ℤ/9)—22no—
33C22×AGL(1,ℤ/9)—44no—
34C22×AGL(1,ℤ/9)C2×C411yescharacteristic
35C22×AGL(1,ℤ/9)C2×C412yes—
36D4×3−1+2—22no—
37C2×C4×3−1+2C2311yescharacteristic
38#611⋊C3—24no—
39C2×3−1+2⋊C4C2×C411yescharacteristic
40C4⋊AGL(1,ℤ/9)—22no—
41C4×AGL(1,ℤ/9)D412yes—
42C4×AGL(1,ℤ/9)D412yes—
43C8×3−1+2—22no—
443−1+2⋊C8—22no—
45C2×C3×#100—91no—
46D3×#100—31no—
47C23×C3×D3—32no—
48C2×C32×D4C2×D311yescharacteristic
49C2×C3×#67—32no—
50C3×D3×D4—62no—
51C3×D3×D4—62no—
52C2×C3×D12—31no—
53C2×C3×C4×D3—31no—
54C23×D9C3×C412yes—
55C32×M16C2×D312yes—
56C3×#208—32no—
57C3×ΓL(1,𝔽25)—62no—
58C3×D3×C8—62no—
59C2×D4×C9—21no—
60C2×D4×C9C22×C311yescharacteristic
61C2×#611C3×C412yes—
62D4×D9—22no—
63D4×D9—22no—
64C2×D36C22×C311yescharacteristic
65C2×C4×D9C22×C311yescharacteristic
66C9×M16—22no—
67C9×M16C22×C312yescomplemented
68#3667C22×C312yescomplemented
69#3663—22no—
70C8×D9—22no—
71C22×3−1+2—22no—
72C22×3−1+2—42no—
73C22×3−1+2C22×C411yescharacteristic
74C2×AGL(1,ℤ/9)—22no—
75C2×AGL(1,ℤ/9)—22no—
76C2×AGL(1,ℤ/9)—44no—
77C2×AGL(1,ℤ/9)—44no—
78C2×AGL(1,ℤ/9)—42no—
79C2×AGL(1,ℤ/9)C22⋊C411yescharacteristic
80C2×AGL(1,ℤ/9)C22⋊C411yescharacteristic
81C4×3−1+2C2×D412yes—
823−1+2⋊C4C22⋊C412yes—
83C22×C3×D4—91no—
84C2×D3×D4—31no—
85C2×C3×M16—92no—
86D3×M16—32no—
87C3×#100—31no—
88C3×#100—92no—
89C3×#100—91no—
90C3×#100C3×D311yescharacteristic
91#833—31no—
92#823—32no—
93C23×C32C4×D312yes—
94C22×C3×D3—31no—
95C22×C3×D3—32no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C22×C3×D3—124no—
99C32×D4—22no—
100C2×C32×C4C22×D311yescharacteristic
101C3×#67—64no—
102C2×C3×Dic3—31no—
103C3×D12—62no—
104C3×C4×D3—32no—
105C3×C4×D3—32no—
106C23×C9—22no—
107C23×C9C2×C3×C412yes—
108C22×D9—22no—
109C22×D9—22no—
110C22×D9—44no—
111C22×D9C2×C3×C411yescharacteristic
112C22×D9C2×C3×C412yes—
113C32×C8—22no—
114C3×C3⋊C8—62no—
115D4×C9—22no—
116D4×C9—42no—
117C2×C4×C9—21no—
118C2×C4×C9C23×C311yescharacteristic
119#611—24no—
120C2×Dic9C2×C3×C411yescharacteristic
121D36—22no—
122C4×D9C3×D412yes—
123C4×D9C3×D412yes—
124C8×C9—22no—
125C8×C9—42no—
126C9⋊C8—22no—
127C2×#100—91noSylow
128C2×3−1+2—21no—
129C2×3−1+2—42no—
130C2×3−1+2C2×C22⋊C411yescharacteristic
131AGL(1,ℤ/9)—21no—
132AGL(1,ℤ/9)—42no—
133AGL(1,ℤ/9)#10012yesdirect product factor
134C24×C3—92no—
135C23×D3—32no—
136C2×C3×D4—31no—
137C2×C3×D4—91no—
138C2×C3×D4—92no—
139C2×C3×D4—182no—
140C2×C3×D4—182no—
141C2×C3×D4C2×C3×D311yescharacteristic
142C22×C3×C4—91no—
143C2×#67—32no—
144D3×D4—62no—
145D3×D4—62no—
146C2×D12—31no—
147C2×C4×D3—31no—
148C3×M16—32no—
149C3×M16—92no—
150C3×M16—182no—
151C3×M16C2×C3×D312yes—
152C2×C3×C8—182no—
153#208—32no—
154ΓL(1,𝔽25)—62no—
155D3×C8—62no—
156C22×C32—22no—
157C22×C32—42no—
158C22×C32C2×C4×D311yescharacteristic
159C2×C3×D3—31no—
160C2×C3×D3—31no—
161C2×C3×D3—62no—
162C2×C3×D3—62no—
163C2×C3×D3—124no—
164C2×C3×D3—124no—
165C2×C3×D3—122no—
166C32×C4D3×D412yes—
167C3×Dic3—32no—
168C22×C9—22no—
169C22×C9—21no—
170C22×C9—42no—
171C22×C9—42no—
172C22×C9—82no—
173C22×C9C22×C3×C411yescharacteristic
174C2×D9—22no—
175C2×D9—22no—
176C2×D9—44no—
177C2×D9—44no—
178C2×D9—42no—
179C2×D9C3×C22⋊C411yescharacteristic
180C2×D9C3×C22⋊C411yescharacteristic
181C4×C9—22no—
182C4×C9C2×C3×D412yes—
183Dic9C3×C22⋊C412yes—
184C22×D4—91no—
185C2×M16—92no—
186#100—91no—
187#100—92no—
188#100AGL(1,ℤ/9)11yescharacteristic, direct product factor
1893−1+2C2×#10011yescharacteristic, complemented, Sylow
190C23×C3—32no—
191C23×C3—91no—
192C23×C3—92no—
193C23×C3—182no—
194C23×C3—182no—
195C23×C3—364no—
196C23×C3C3×C4×D312yes—
197C22×D3—31no—
198C22×D3—32no—
199C22×D3—62no—
200C22×D3—62no—
201C22×D3—124no—
202C3×D4—22no—
203C3×D4—62no—
204C3×D4—182no—
205C3×D4—184no—
206C2×C3×C4—31no—
207C2×C3×C4—92no—
208C2×C3×C4—92no—
209C2×C3×C4—91no—
210C2×C3×C4C22×C3×D311yescharacteristic
211#67—64no—
212C2×Dic3—31no—
213D12—62no—
214C4×D3—32no—
215C4×D3—32no—
216C3×C8—22no—
217C3×C8—62no—
218C3×C8—182no—
219C3⋊C8—62no—
220C2×C32—21no—
221C2×C32—42no—
222C2×C32D3×C22⋊C411yescharacteristic
223C3×D3—32no—
224C3×D3—61no—
225C3×D3—122no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9—41no—
229C2×C9—42no—
230C2×C9—82no—
231C2×C9C2×C3×C22⋊C411yescharacteristic
232D9—21no—
233D9—42no—
234D9C3×#10012yescomplemented
235C24—92no—
236C2×D4—92no—
237C2×D4—91no—
238C2×D4—182no—
239C2×D4—182no—
240C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
241C22×C4—91no—
242M16—92no—
243M16—182no—
244M16C2×AGL(1,ℤ/9)12yescomplemented
245C2×C8—182no—
246C22×C3—22no—
247C22×C3—31no—
248C22×C3—42no—
249C22×C3—62no—
250C22×C3—91no—
251C22×C3—91no—
252C22×C3—122no—
253C22×C3—182no—
254C22×C3—182no—
255C22×C3—362no—
256C22×C3—364no—
257C22×C3—364no—
258C22×C3C2×C3×C4×D311yescharacteristic
259C2×D3—31no—
260C2×D3—31no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—124no—
264C2×D3—124no—
265C2×D3—122no—
266C3×C4—32no—
267C3×C4—92no—
268C3×C4C3×D3×D412yes—
269Dic3—32no—
270C32D3×#10011yescharacteristic
271C9—21no—
272C9C2×C3×#10011yescharacteristic, complemented
273C23—91no—
274C23—92no—
275C23—182no—
276C23—182no—
277C23—364no—
278C23C4×AGL(1,ℤ/9)12yes—
279D4—22no—
280D4—184no—
281D4—182no—
282C2×C4—91no—
283C2×C4—92no—
284C2×C4—92no—
285C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
286C8—22no—
287C8—182no—
288C2×C3—21no—
289C2×C3—31no—
290C2×C3—42no—
291C2×C3—61no—
292C2×C3—92no—
293C2×C3—122no—
294C2×C3—181no—
295C2×C3—362no—
296C2×C3C3×D3×C22⋊C411yescharacteristic
297D3—32no—
298D3—61no—
299D3—122no—
300C22—22no—
301C22—42no—
302C22—91no—
303C22—91no—
304C22—182no—
305C22—182no—
306C22—362no—
307C22—364no—
308C22—364no—
309C22C2×C4×AGL(1,ℤ/9)11yescharacteristic
310C4—92no—
311C4D4×AGL(1,ℤ/9)12yes—
312C3—31no—
313C3C3×D3×#10011yescharacteristic
314C2—21no—
315C2—42no—
316C2—92no—
317C2—181no—
318C2—362no—
319C2C22⋊C4×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC2C22⋊C4×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#100C2
socleC2×C3C3×D3×C22⋊C4
prime radicalC2×D4×AGL(1,ℤ/9)C2
cosocleC2×C4×C9C23×C3
hypercenter#100AGL(1,ℤ/9)
hypocenterC9C2×C3×#100
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. M16×AGL(1,ℤ/9)C2semidirect
3. 3−1+2×#100C2semidirect
4. #63391⋊C3C2semidirect
5. #63383⋊C3C2semidirect
7. D9×#100C3semidirect
15. M16×3−1+2C22semidirect
16. #3667⋊C3C22semidirect
25. M16×D9C2×C3semidirect
27. C9×#100C2×C3semidirect
28. #63391C2×C3semidirect
29. #63383C2×C3semidirect
67. C9×M16C22×C3semidirect
68. #3667C22×C3semidirect
133. AGL(1,ℤ/9)#100direct (N × H)
188. #100AGL(1,ℤ/9)direct (N × H)
189. 3−1+2C2×#100semidirect
234. D9C3×#100semidirect
244. M16C2×AGL(1,ℤ/9)semidirect
272. C9C2×C3×#100semidirect

Maximal subgroups

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

subgroupindex
C2×D4×AGL(1,ℤ/9)2
M16×AGL(1,ℤ/9)2
3−1+2×#1002
#63391⋊C32
#63383⋊C32
C3×D3×#1003
D9×#1003