Finite Group Databasebeta

← #13153253

Subgroups — #13153253

Order 1728.

786 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 202 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
13−1+2×#126C211yesmaximal, characteristic, complemented
2#63457⋊C3C213yesmaximal, complemented
3#63456⋊C3C213yesmaximal, complemented
4#63446⋊C3C213yesmaximal, complemented
5#63444⋊C3C213yesmaximal, complemented
6#11721396C211yesmaximal, characteristic, complemented
7#11721394C211yesmaximal, characteristic, complemented
8C3×#5711—31nomaximal
9#10589813C311yesmaximal, characteristic, complemented
10C23×AGL(1,ℤ/9)C2211yescharacteristic
11C2×#611⋊C3C2213yes—
12C2×C4⋊AGL(1,ℤ/9)C2213yes—
13C2×C4⋊AGL(1,ℤ/9)C2213yes—
14C2×C4⋊AGL(1,ℤ/9)C2213yescomplemented
15C2×C4×AGL(1,ℤ/9)C2213yescomplemented
16C4⋊C4×3−1+2C2213yescomplemented
17C22⋊C4×3−1+2C2213yes—
18C42×3−1+2C2211yescharacteristic, complemented
19(C2×AGL(1,ℤ/9))⋊C4C2213yes—
20(C2×AGL(1,ℤ/9))⋊C4C2213yes—
21(C2×AGL(1,ℤ/9))⋊C4C2213yescomplemented
22(3−1+2⋊C4)⋊C4C2213yescomplemented
23C32×#126D311yescharacteristic
24C3×#895—33no—
25C3×#894—33no—
26C3×#884—33no—
27C3×#882—33no—
28C3×#876—31no—
29C3×#874—31no—
30C9×#126—21no—
31C9×#126C2×C311yescharacteristic, complemented
32#63457C2×C313yescomplemented
33#63456C2×C313yescomplemented
34#63446C2×C313yescomplemented
35#63444C2×C313yescomplemented
36#63438C2×C311yescharacteristic, complemented
37#63436C2×C311yescharacteristic, complemented
38C23×3−1+2C2311yescharacteristic
39C22×AGL(1,ℤ/9)—43no—
40C22×AGL(1,ℤ/9)C2311yescharacteristic
41C22×AGL(1,ℤ/9)C2311yescharacteristic
42C22×AGL(1,ℤ/9)C2313yes—
43C2×C4×3−1+2C2313yes—
44C2×C4×3−1+2C2313yes—
45#611⋊C3—43no—
46C2×3−1+2⋊C4C2313yes—
47C4⋊AGL(1,ℤ/9)—43no—
48C4⋊AGL(1,ℤ/9)—43no—
49C4⋊AGL(1,ℤ/9)—43no—
50C4×AGL(1,ℤ/9)—43no—
51C3×#560—91no—
52#5711—31no—
53C23×C3×D3—31no—
54C2×C3×#67—33no—
55C2×C3×D12—33no—
56C2×C3×D12—33no—
57C2×C3×D12—33no—
58C2×C3×C4×D3—33no—
59C23×D9C22×C311yescharacteristic
60C32×C4⋊C4C2×D313yes—
61C32×C22⋊C4C2×D313yes—
62C32×C42C2×D311yescharacteristic
63C3×(C2×D3)⋊C4—33no—
64C3×(C2×D3)⋊C4—33no—
65C3×(C2×D3)⋊C4—33no—
66C3×Dic3⋊C4—33no—
67C2×#611C22×C313yes—
68C2×D36C22×C313yes—
69C2×D36C22×C313yes—
70C2×D36C22×C313yescomplemented
71C2×C4×D9C22×C313yescomplemented
72C9×C4⋊C4—23no—
73C9×C4⋊C4C22×C313yescomplemented
74C9×C22⋊C4—23no—
75C9×C22⋊C4C22×C313yes—
76C42×C9—21no—
77C42×C9C22×C311yescharacteristic, complemented
78(C2×D9)⋊C4C22×C313yes—
79(C2×D9)⋊C4C22×C313yes—
80(C2×D9)⋊C4C22×C313yescomplemented
81Dic9⋊C4C22×C313yescomplemented
82C22×3−1+2—23no—
83C22×3−1+2C2411yescharacteristic
84C2×AGL(1,ℤ/9)—23no—
85C2×AGL(1,ℤ/9)—23no—
86C2×AGL(1,ℤ/9)—23no—
87C2×AGL(1,ℤ/9)—23no—
88C2×AGL(1,ℤ/9)—23no—
89C2×AGL(1,ℤ/9)—44no—
90C4×3−1+2—23no—
91C4×3−1+2—23no—
923−1+2⋊C4—23no—
93C3×#127—91no—
94C3×#126—31no—
95C3×#126—91no—
96C3×#126C3×D311yescharacteristic
97C3×#123—93no—
98C3×#121—93no—
99C3×#121—93no—
100C3×#120—93no—
101#895—33no—
102#894—33no—
103#884—33no—
104#882—33no—
105#876—31no—
106#874—31no—
107C23×C32C22×D311yescharacteristic
108C22×C3×D3—31no—
109C22×C3×D3—31no—
110C22×C3×D3—33no—
111C22×C3×D3—123no—
112C2×C32×C4C22×D313yes—
113C2×C32×C4C22×D313yes—
114C3×#67—123no—
115C2×C3×Dic3—33no—
116C3×D12—123no—
117C3×D12—123no—
118C3×D12—123no—
119C3×C4×D3—123no—
120C23×C9—21no—
121C23×C9C23×C311yescharacteristic
122C22×D9—43no—
123C22×D9C23×C311yescharacteristic
124C22×D9C23×C311yescharacteristic
125C22×D9C23×C313yes—
126C2×C4×C9—23no—
127C2×C4×C9—23no—
128C2×C4×C9C23×C313yes—
129C2×C4×C9C23×C313yes—
130#611—43no—
131C2×Dic9C23×C313yes—
132D36—43no—
133D36—43no—
134D36—43no—
135C4×D9—43no—
136#560—91noSylow
137C2×3−1+2—41no—
138C2×3−1+2H(5,𝔽2)13yes—
139AGL(1,ℤ/9)—43no—
140AGL(1,ℤ/9)—41no—
141AGL(1,ℤ/9)—41no—
142C24×C3—91no—
143C23×D3—31no—
144C2×C3×D4—93no—
145C2×C3×D4—93no—
146C2×C3×D4—93no—
147C2×C3×D4—93no—
148C22×C3×C4—93no—
149C2×#67—33no—
150C2×D12—33no—
151C2×D12—33no—
152C2×D12—33no—
153C2×C4×D3—33no—
154C3×C4⋊C4—33no—
155C3×C4⋊C4—93no—
156C3×C4⋊C4C2×C3×D313yes—
157C3×C22⋊C4—33no—
158C3×C22⋊C4—93no—
159C3×C22⋊C4—93no—
160C3×C22⋊C4—93no—
161C3×C22⋊C4C2×C3×D313yes—
162C3×C42—31no—
163C3×C42C2×C3×D311yescharacteristic
164(C2×D3)⋊C4—33no—
165(C2×D3)⋊C4—33no—
166(C2×D3)⋊C4—33no—
167Dic3⋊C4—33no—
168C22×C32—23no—
169C22×C32C23×D311yescharacteristic
170C2×C3×D3—63no—
171C2×C3×D3—63no—
172C2×C3×D3—63no—
173C2×C3×D3—63no—
174C2×C3×D3—63no—
175C2×C3×D3—124no—
176C32×C4—23no—
177C32×C4—23no—
178C3×Dic3—63no—
179C22×C9—23no—
180C22×C9—21no—
181C22×C9—43no—
182C22×C9C24×C311yescharacteristic
183C2×D9—23no—
184C2×D9—23no—
185C2×D9—23no—
186C2×D9—23no—
187C2×D9—23no—
188C2×D9—44no—
189C4×C9—23no—
190C4×C9—23no—
191C4×C9—43no—
192C4×C9—43no—
193Dic9—23no—
194#127—91no—
195#126—91no—
196#126AGL(1,ℤ/9)11yescharacteristic, complemented
197#123—93no—
198#121—93no—
199#121—93no—
200#120—93no—
2013−1+2#56011yescharacteristic, complemented, Sylow
202C23×C3—31no—
203C23×C3—91no—
204C23×C3—91no—
205C23×C3—93no—
206C23×C3—363no—
207C23×C3C22×C3×D311yescharacteristic
208C22×D3—31no—
209C22×D3—31no—
210C22×D3—33no—
211C22×D3—123no—
212C3×D4—363no—
213C3×D4—363no—
214C3×D4—363no—
215C3×D4—363no—
216C2×C3×C4—33no—
217C2×C3×C4—33no—
218C2×C3×C4—93no—
219C2×C3×C4—363no—
220C2×C3×C4C22×C3×D313yes—
221C2×C3×C4C22×C3×D313yes—
222#67—123no—
223C2×Dic3—33no—
224D12—123no—
225D12—123no—
226D12—123no—
227C4×D3—123no—
228C2×C32—41no—
229C2×C32#100913yes—
230C3×D3—123no—
231C3×D3—121no—
232C3×D3—121no—
233C2×C9—23no—
234C2×C9—41no—
235C2×C9—81no—
236C2×C9C3×H(5,𝔽2)13yes—
237D9—43no—
238D9—41no—
239D9—41no—
240C24—91no—
241C2×D4—93no—
242C2×D4—93no—
243C2×D4—93no—
244C2×D4—93no—
245C22×C4—93no—
246C4⋊C4—93no—
247C4⋊C4C2×AGL(1,ℤ/9)13yescomplemented
248C22⋊C4—93no—
249C22⋊C4—93no—
250C22⋊C4—93no—
251C22⋊C4C2×AGL(1,ℤ/9)13yes—
252C42C2×AGL(1,ℤ/9)11yescharacteristic, complemented
253C22×C3—23no—
254C22×C3—31no—
255C22×C3—63no—
256C22×C3—183no—
257C22×C3—183no—
258C22×C3—183no—
259C22×C3—183no—
260C22×C3—183no—
261C22×C3—364no—
262C22×C3C23×C3×D311yescharacteristic
263C2×D3—63no—
264C2×D3—63no—
265C2×D3—63no—
266C2×D3—63no—
267C2×D3—63no—
268C2×D3—124no—
269C3×C4—23no—
270C3×C4—23no—
271C3×C4—63no—
272C3×C4—63no—
273C3×C4—183no—
274Dic3—63no—
275C32#571111yescharacteristic
276C9—21no—
277C9C3×#56011yescharacteristic, complemented
278C23—91no—
279C23—91no—
280C23—93no—
281C23—363no—
282C23C22×AGL(1,ℤ/9)11yescharacteristic
283D4—363no—
284D4—363no—
285D4—363no—
286D4—363no—
287C2×C4—93no—
288C2×C4—363no—
289C2×C4C22×AGL(1,ℤ/9)13yes—
290C2×C4C22×AGL(1,ℤ/9)13yes—
291C2×C3—33no—
292C2×C3—41no—
293C2×C3—121no—
294C2×C3—361no—
295C2×C3—361no—
296C2×C3—363no—
297C2×C3C3×#100913yes—
298D3—121no—
299D3—121no—
300D3—123no—
301C22—23no—
302C22—183no—
303C22—183no—
304C22—183no—
305C22—183no—
306C22—183no—
307C22—364no—
308C22C23×AGL(1,ℤ/9)11yescharacteristic
309C4—23no—
310C4—23no—
311C4—183no—
312C3—31no—
313C3C3×#571111yescharacteristic
314C2—41no—
315C2—361no—
316C2—361no—
317C2—363no—
318C2#1172264113yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#126C2
hypercenter#126AGL(1,ℤ/9)
hypocenterC9C3×#560
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#126C2semidirect
2. #63457⋊C3C2semidirect
3. #63456⋊C3C2semidirect
4. #63446⋊C3C2semidirect
5. #63444⋊C3C2semidirect
6. #11721396C2semidirect
7. #11721394C2semidirect
9. #10589813C3semidirect
14. C2×C4⋊AGL(1,ℤ/9)C22semidirect
15. C2×C4×AGL(1,ℤ/9)C22semidirect
16. C4⋊C4×3−1+2C22semidirect
18. C42×3−1+2C22semidirect
21. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
22. (3−1+2⋊C4)⋊C4C22semidirect
31. C9×#126C2×C3semidirect
32. #63457C2×C3semidirect
33. #63456C2×C3semidirect
34. #63446C2×C3semidirect
35. #63444C2×C3semidirect
36. #63438C2×C3semidirect
37. #63436C2×C3semidirect
70. C2×D36C22×C3semidirect
71. C2×C4×D9C22×C3semidirect
73. C9×C4⋊C4C22×C3semidirect
77. C42×C9C22×C3semidirect
80. (C2×D9)⋊C4C22×C3semidirect
81. Dic9⋊C4C22×C3semidirect
196. #126AGL(1,ℤ/9)semidirect
201. 3−1+2#560semidirect
247. C4⋊C4C2×AGL(1,ℤ/9)semidirect
252. C42C2×AGL(1,ℤ/9)semidirect
277. C9C3×#560semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1262
#63457⋊C32
#63456⋊C32
#63446⋊C32
#63444⋊C32
#117213962
#117213942
C3×#57113
#105898133