Finite Group Databasebeta

← #72220038

Subgroups — #72220038

Order 2592.

1903 conjugacy classes of subgroups; only the 262 normal subgroups are shown.

normal subgroups only

This group has 1903 conjugacy classes of subgroups — too many to draw. Only its 262 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
1C24×C3×3−1+2C211yesmaximal, characteristic, complemented
2C2×#10619980C213yesmaximal, complemented
3#13047994C213yesmaximal, complemented
4C3×#64162—31nomaximal
5#11721874C311yesmaximal, characteristic, complemented
6#11721704—33nomaximal
7C23×C3×3−1+2—26no—
8C23×C3×3−1+2C2213yes—
9C22×C3⋊AGL(1,ℤ/9)C2211yescharacteristic, complemented
10#10619980—26no—
11C2×C3⋊(3−1+2⋊C4)C2213yescomplemented
12C24×C33D311yescharacteristic
13C2×C3×#638—33no—
14C24×3−1+2—23no—
15C24×3−1+2D313yescomplemented
16C24×C3×C9—21no—
17C24×C3×C9C2×C311yescharacteristic, complemented
18C3×#3757—33no—
19C2×#6233C2×C313yescomplemented
20C2×#611⋊C3—39no—
21#88551C2×C313yescomplemented
22#3676⋊C3—39no—
23C22×C3×3−1+2—28no—
24C22×C3×3−1+2—26no—
25C22×C3×3−1+2D416yescomplemented
26C22×C3×3−1+2C2311yescharacteristic
27C2×C3⋊AGL(1,ℤ/9)—23no—
28C3⋊(3−1+2⋊C4)—23no—
29#64162—31no—
30C3×#953—93no—
31C3×#953—91no—
32#63515—33no—
33C23×C33—26no—
34C23×C33C2×D313yes—
35C22×C3×C3⋊D3—31no—
36C3×#638—66no—
37C2×C3×C3⋊Dic3—33no—
38C23×3−1+2—236no—
39C23×3−1+2—29no—
40C23×3−1+2—218no—
41C23×3−1+2C2×D319yes—
42C23×C3×C9—212no—
43C23×C3×C9—23no—
44C23×C3×C9—26no—
45C23×C3×C9C22×C313yes—
46C22×C3⋊D9C22×C311yescharacteristic, complemented
47C22×AGL(1,ℤ/9)—33no—
48#6233—26no—
49C2×C3⋊Dic9C22×C313yescomplemented
50#611⋊C3—618no—
51C2×3−1+2⋊C4—39no—
52C2×C3×3−1+2—26no—
53C2×C3×3−1+2C2×D413yes—
54C3⋊AGL(1,ℤ/9)—41no—
55C24×C32—21no—
56C24×C32—33no—
57C24×C32C3×D311yescharacteristic
58C24×C32C3⋊D311yescharacteristic
59C2×#638—33no—
60C2×C3×#67—99no—
61C2×C3×#67—93no—
62C24×C9—23no—
63C24×C9C3×D313yescomplemented
64#3757—33no—
65C3×#217—99no—
66C3×#217—93no—
67C2×#611—39no—
68#3676—39no—
69C22×C33—28no—
70C22×C33—26no—
71C22×C33#6716yes—
72C22×C33C22×D311yescharacteristic
73C2×C3×C3⋊D3—63no—
74C3×C3⋊Dic3—63no—
75C22×3−1+2—248no—
76C22×3−1+2—236no—
77C22×3−1+2—224no—
78C22×3−1+2—218no—
79C22×3−1+2—23no—
80C22×3−1+2—218no—
81C22×3−1+2#67118yescomplemented
82C22×3−1+2C22×D313yes—
83C22×C3×C9—216no—
84C22×C3×C9—212no—
85C22×C3×C9—28no—
86C22×C3×C9—26no—
87C22×C3×C9—21no—
88C22×C3×C9—26no—
89C22×C3×C9C3×D416yescomplemented
90C22×C3×C9C23×C311yescharacteristic
91C2×C3⋊D9—23no—
92C2×AGL(1,ℤ/9)—69no—
93C3⋊Dic9—23no—
943−1+2⋊C4—69no—
95C3×#120—271no—
96#953—93no—
97#953—91no—
98C3×3−1+2#12011yescharacteristic, complemented, Sylow
99C23×C32—26no—
100C23×C32—212no—
101C23×C32—26no—
102C23×C32—23no—
103C23×C32—39no—
104C23×C32—618no—
105C23×C32C2×C3×D313yes—
106C23×C32C2×C3⋊D313yes—
107C22×C3⋊D3—31no—
108C22×C3×D3—93no—
109C22×C3×D3—91no—
110#638—66no—
111C2×C3⋊Dic3—33no—
112C3×#67—1818no—
113C3×#67—186no—
114C2×C3×Dic3—99no—
115C2×C3×Dic3—93no—
116C23×C9—236no—
117C23×C9—218no—
118C23×C9—29no—
119C23×C9C2×C3×D319yes—
120C22×D9—33no—
121#611—618no—
122C2×Dic9—39no—
123C2×C33—26no—
124C2×C33C2×#6713yes—
125C3×C3⋊D3—121no—
126C2×3−1+2—236no—
127C2×3−1+2—218no—
128C2×3−1+2—29no—
129C2×3−1+2C2×#6719yes—
130C2×C3×C9—212no—
131C2×C3×C9—26no—
132C2×C3×C9—23no—
133C2×C3×C9C2×C3×D413yes—
134C3⋊D9—41no—
135AGL(1,ℤ/9)—123no—
136C24×C3—31no—
137C24×C3—61no—
138C24×C3AGL(1,ℤ/9)13yescomplemented
139C24×C3C3×C3⋊D311yescharacteristic
140C2×C3×D4—273no—
141C2×#67—99no—
142C2×#67—93no—
143C3×C22⋊C4—273no—
144#217—99no—
145#217—93no—
146C22×C32—28no—
147C22×C32—26no—
148C22×C32—216no—
149C22×C32—28no—
150C22×C32—212no—
151C22×C32—26no—
152C22×C32—26no—
153C22×C32—21no—
154C22×C32—318no—
155C22×C32—33no—
156C22×C32—618no—
157C22×C32—624no—
158C22×C32C3×#6716yes—
159C22×C32#63816yes—
160C22×C32C22×C3×D311yescharacteristic
161C22×C32C22×C3⋊D311yescharacteristic
162C2×C3⋊D3—63no—
163C2×C3×D3—189no—
164C2×C3×D3—183no—
165C3⋊Dic3—63no—
166C3×Dic3—189no—
167C3×Dic3—183no—
168C22×C9—248no—
169C22×C9—224no—
170C22×C9—236no—
171C22×C9—218no—
172C22×C9—218no—
173C22×C9—23no—
174C22×C9C3×#67118yescomplemented
175C22×C9C22×C3×D313yes—
176C2×D9—69no—
177Dic9—69no—
178#120—271noSylow
179C33#95311yescharacteristic
1803−1+2—23no—
1813−1+2#95313yescomplemented
182C3×C9—21no—
183C3×C9C3×#12011yescharacteristic, complemented
184C23×C3—218no—
185C23×C3—26no—
186C23×C3—33no—
187C23×C3—66no—
188C23×C3—612no—
189C23×C3—63no—
190C23×C3—271no—
191C23×C3C2×AGL(1,ℤ/9)19yes—
192C23×C3C2×C3×C3⋊D313yes—
193C22×D3—93no—
194C22×D3—91no—
195C3×D4—546no—
196C2×C3×C4—273no—
197#67—1818no—
198#67—186no—
199C2×Dic3—99no—
200C2×Dic3—93no—
201C2×C32—26no—
202C2×C32—212no—
203C2×C32—26no—
204C2×C32—23no—
205C2×C32—39no—
206C2×C32—618no—
207C2×C32C2×C3×#6713yes—
208C2×C32C2×#63813yes—
209C3⋊D3—121no—
210C3×D3—363no—
211C3×D3—361no—
212C2×C9—236no—
213C2×C9—218no—
214C2×C9—29no—
215C2×C9C2×C3×#6719yes—
216D9—123no—
217C24C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
218C2×D4—273no—
219C22⋊C4—273no—
220C22×C3—28no—
221C22×C3—224no—
222C22×C3—218no—
223C22×C3—26no—
224C22×C3—36no—
225C22×C3—31no—
226C22×C3—66no—
227C22×C3—68no—
228C22×C3—616no—
229C22×C3—612no—
230C22×C3—66no—
231C22×C3—61no—
232C22×C3—543no—
233C22×C3#611⋊C3118yescomplemented
234C22×C3C22×AGL(1,ℤ/9)13yes—
235C22×C3C3×#63816yes—
236C22×C3C22×C3×C3⋊D311yescharacteristic
237C2×D3—189no—
238C2×D3—183no—
239C3×C4—543no—
240Dic3—189no—
241Dic3—183no—
242C32—21no—
243C32—33no—
244C32C3×#95311yescharacteristic
245C32#6416211yescharacteristic
246C9—23no—
247C9C3×#95313yescomplemented
248C23—26no—
249C23—271no—
250C23C2×C3⋊AGL(1,ℤ/9)13yes—
251D4—546no—
252C2×C4—273no—
253C2×C3—26no—
254C2×C3—218no—
255C2×C3—33no—
256C2×C3—66no—
257C2×C3—612no—
258C2×C3—63no—
259C2×C3—1081no—
260C2×C3C2×#611⋊C319yes—
261C2×C3C2×C3×#63813yes—
262D3—361no—
263D3—363no—
264C22—28no—
265C22—26no—
266C22—543no—
267C22#1061998016yescomplemented
268C22C22×C3⋊AGL(1,ℤ/9)11yescharacteristic
269C4—543no—
270C3—31no—
271C3—61no—
272C3#1172170413yescomplemented
273C3C3×#6416211yescharacteristic
274C2—26no—
275C2—1081no—
276C2C2×#1061998013yes—
subgroupG/H
centerC22C22×C3⋊AGL(1,ℤ/9)
commutator, cosocleC22×C3×C9C23×C3
FrattiniC22×C3C22×C3×C3⋊D3
FittingC24×C3×3−1+2C2
socleC22×C32C22×C3×D3
hypercenterC24C3⋊AGL(1,ℤ/9)
hypocenterC3×C9C3×#120
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C3×3−1+2C2semidirect
2. C2×#10619980C2semidirect
3. #13047994C2semidirect
5. #11721874C3semidirect
9. C22×C3⋊AGL(1,ℤ/9)C22semidirect
11. C2×C3⋊(3−1+2⋊C4)C22semidirect
15. C24×3−1+2D3semidirect
17. C24×C3×C9C2×C3semidirect
19. C2×#6233C2×C3semidirect
21. #88551C2×C3semidirect
25. C22×C3×3−1+2D4semidirect
46. C22×C3⋊D9C22×C3semidirect
49. C2×C3⋊Dic9C22×C3semidirect
63. C24×C9C3×D3semidirect
81. C22×3−1+2#67semidirect
89. C22×C3×C9C3×D4semidirect
98. C3×3−1+2#120semidirect
138. C24×C3AGL(1,ℤ/9)semidirect
174. C22×C9C3×#67semidirect
181. 3−1+2#953semidirect
183. C3×C9C3×#120semidirect
217. C24C3⋊AGL(1,ℤ/9)semidirect
233. C22×C3#611⋊C3semidirect
247. C9C3×#953semidirect
267. C22#10619980semidirect
272. C3#11721704semidirect

Maximal subgroups

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

subgroupindex
C24×C3×3−1+22
C2×#106199802
#130479942
C3×#641623
#117218743
#117217043