Finite Group Databasebeta

← 3−1+2×#953 #72220061

Subgroups — 3−1+2×#953 #72220061

Order 2592.

1777 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1777 conjugacy classes of subgroups — too many to draw. Only its 287 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×#67×3−1+2C213yesmaximal, complemented
33−1+2×#217C213yesmaximal, complemented
4C32×#953C311yesmaximal, characteristic
53−1+2×#120—31nomaximal
6C9×#953C313yesmaximal, complemented
7C23×C3×3−1+2—26no—
8C23×C3×3−1+2C2213yes—
9C22×D3×3−1+2C2211yescharacteristic, complemented
10#67×3−1+2—26no—
11C2×Dic3×3−1+2C2213yescomplemented
12C24×C33C2×C311yescharacteristic
13C2×C32×#67C2×C313yes—
14C24×3−1+2—23no—
15C24×3−1+2—21no—
16C24×3−1+2D311yescharacteristic, complemented
17C24×C3×C9C2×C313yescomplemented
18C32×#217C2×C313yes—
19C2×D4×3−1+2—33no—
20C2×C9×#67C2×C319yescomplemented
21C22⋊C4×3−1+2—33no—
22C9×#217C2×C319yescomplemented
23C22×C3×3−1+2—28no—
24C22×C3×3−1+2—26no—
25C22×C3×3−1+2D416yescomplemented
26C22×C3×3−1+2C2311yescharacteristic
27C2×D3×3−1+2—23no—
28Dic3×3−1+2—23no—
29C32×#120—31no—
30C3×#953—31no—
31C3×#953C3211yescharacteristic
32C9×#120—33no—
33C23×C33—26no—
34C23×C33C22×C313yes—
35C22×C32×D3C22×C311yescharacteristic
36C32×#67—26no—
37C2×C32×Dic3C22×C313yes—
38C23×3−1+2—212no—
39C23×3−1+2—236no—
40C23×3−1+2—23no—
41C23×3−1+2—29no—
42C23×3−1+2—26no—
43C23×3−1+2—31no—
44C23×3−1+2C2×D313yes—
45C23×C3×C9—218no—
46C23×C3×C9C22×C319yes—
47C22×D3×C9C22×C313yescomplemented
48D4×3−1+2—66no—
49C2×C4×3−1+2—33no—
50C9×#67—218no—
51C2×C9×Dic3C22×C319yescomplemented
52C2×C3×3−1+2—26no—
53C2×C3×3−1+2C2×D413yes—
54D3×3−1+2—41no—
55C24×C32—21no—
56C24×C32—31no—
57C24×C32—61no—
58C24×C32C3×D311yescharacteristic
59C24×C32C2×C3211yescharacteristic
60C2×C32×D4—33no—
61C2×C3×#67—33no—
62C2×C3×#67C2×C3213yes—
63C24×C9—23no—
64C24×C9C3×D313yescomplemented
65C32×C22⋊C4—33no—
66C3×#217—33no—
67C3×#217C2×C3213yes—
68C2×D4×C9—39no—
69C9×C22⋊C4—39no—
70C22×C33—28no—
71C22×C33—26no—
72C22×C33C3×D416yes—
73C22×C33C23×C311yescharacteristic
74C2×C32×D3—23no—
75C32×Dic3—23no—
76C22×3−1+2—216no—
77C22×3−1+2—248no—
78C22×3−1+2—212no—
79C22×3−1+2—236no—
80C22×3−1+2—28no—
81C22×3−1+2—26no—
82C22×3−1+2—21no—
83C22×3−1+2—218no—
84C22×3−1+2—23no—
85C22×3−1+2—26no—
86C22×3−1+2—63no—
87C22×3−1+2#6716yescomplemented
88C22×3−1+2C22×D311yescharacteristic
89C22×C3×C9—224no—
90C22×C3×C9—218no—
91C22×C3×C9C3×D4118yescomplemented
92C22×C3×C9C23×C313yes—
93C2×D3×C9—29no—
94C4×3−1+2—63no—
95C9×Dic3—29no—
96C3×#120—31no—
97C3×#120—91no—
98#9533−1+211yescharacteristic, direct product factor
99C3×3−1+2#12011yescharacteristic, complemented, Sylow
100C23×C32—26no—
101C23×C32—212no—
102C23×C32—26no—
103C23×C32—23no—
104C23×C32—33no—
105C23×C32—31no—
106C23×C32—63no—
107C23×C32—66no—
108C23×C32—612no—
109C23×C32C2×C3×D313yes—
110C23×C32C22×C3213yes—
111C22×C3×D3—31no—
112C22×C3×D3C22×C3211yescharacteristic
113C32×D4—66no—
114C2×C32×C4—33no—
115C3×#67—26no—
116C3×#67—66no—
117C2×C3×Dic3—33no—
118C2×C3×Dic3C22×C3213yes—
119C23×C9—236no—
120C23×C9—218no—
121C23×C9—29no—
122C23×C9—33no—
123C23×C9C2×C3×D319yes—
124D4×C9—618no—
125C2×C4×C9—39no—
126C2×C33—26no—
127C2×C33C2×C3×D413yes—
128C32×D3—41no—
129C2×3−1+2—212no—
130C2×3−1+2—236no—
131C2×3−1+2—26no—
132C2×3−1+2—23no—
133C2×3−1+2—29no—
134C2×3−1+2—121no—
135C2×3−1+2C2×#6713yes—
136C2×C3×C9—218no—
137C2×C3×C9C2×C3×D419yes—
138D3×C9—43no—
139C24×C3—21no—
140C24×C3—31no—
141C24×C3—61no—
142C24×C3C2×3−1+211yescharacteristic, complemented
143C24×C3C32×D311yescharacteristic
144C2×C3×D4—33no—
145C2×C3×D4—93no—
146C2×#67C2×3−1+213yescomplemented
147C3×C22⋊C4—33no—
148C3×C22⋊C4—93no—
149#217C2×3−1+213yescomplemented
150C22×C32—28no—
151C22×C32—26no—
152C22×C32—216no—
153C22×C32—28no—
154C22×C32—212no—
155C22×C32—26no—
156C22×C32—26no—
157C22×C32—21no—
158C22×C32—36no—
159C22×C32—31no—
160C22×C32—66no—
161C22×C32—61no—
162C22×C32—66no—
163C22×C32—612no—
164C22×C32—68no—
165C22×C32—616no—
166C22×C32—63no—
167C22×C32C3×#6716yes—
168C22×C32C32×D416yes—
169C22×C32C22×C3×D311yescharacteristic
170C22×C32C23×C3211yescharacteristic
171C2×C3×D3—23no—
172C2×C3×D3—63no—
173C32×C4—63no—
174C3×Dic3—23no—
175C3×Dic3—63no—
176C22×C9—248no—
177C22×C9—224no—
178C22×C9—236no—
179C22×C9—218no—
180C22×C9—218no—
181C22×C9—23no—
182C22×C9—69no—
183C22×C9C3×#67118yescomplemented
184C22×C9C22×C3×D313yes—
185C4×C9—69no—
186#120—31noSylow
187C33C3×#12011yescharacteristic
1883−1+2—21no—
1893−1+2—23no—
1903−1+2#95311yescharacteristic, direct product factor
191C3×C9C3×#12013yescomplemented
192C23×C3—212no—
193C23×C3—26no—
194C23×C3—26no—
195C23×C3—23no—
196C23×C3—31no—
197C23×C3—33no—
198C23×C3—66no—
199C23×C3—612no—
200C23×C3—63no—
201C23×C3—91no—
202C23×C3C22×3−1+213yes—
203C23×C3C2×C32×D313yes—
204C22×D3C22×3−1+211yescharacteristic, complemented
205C3×D4—66no—
206C3×D4—186no—
207C2×C3×C4—33no—
208C2×C3×C4—93no—
209#67—26no—
210C2×Dic3C22×3−1+213yescomplemented
211C2×C32—26no—
212C2×C32—212no—
213C2×C32—26no—
214C2×C32—23no—
215C2×C32—33no—
216C2×C32—63no—
217C2×C32—66no—
218C2×C32—612no—
219C2×C32—121no—
220C2×C32C2×C3×#6713yes—
221C2×C32C2×C32×D413yes—
222C3×D3—41no—
223C3×D3—121no—
224C2×C9—236no—
225C2×C9—218no—
226C2×C9—29no—
227C2×C9—123no—
228C2×C9C2×C3×#6719yes—
229C24D3×3−1+211yescharacteristic, complemented
230C2×D4—33no—
231C22⋊C4—33no—
232C22×C3—28no—
233C22×C3—216no—
234C22×C3—28no—
235C22×C3—212no—
236C22×C3—26no—
237C22×C3—26no—
238C22×C3—26no—
239C22×C3—21no—
240C22×C3—36no—
241C22×C3—31no—
242C22×C3—66no—
243C22×C3—68no—
244C22×C3—616no—
245C22×C3—612no—
246C22×C3—63no—
247C22×C3—66no—
248C22×C3—61no—
249C22×C3—183no—
250C22×C3D4×3−1+216yescomplemented
251C22×C3C23×3−1+211yescharacteristic
252C22×C3C32×#6716yes—
253C22×C3C22×C32×D311yescharacteristic
254C2×D3—23no—
255C3×C4—63no—
256C3×C4—183no—
257Dic3—23no—
258C32—21no—
259C32—31no—
260C32—61no—
261C32C3×#95311yescharacteristic
262C32C32×#12011yescharacteristic
263C9—23no—
264C9C3×#95313yescomplemented
265C23—26no—
266C23—31no—
267C23C2×D3×3−1+213yes—
268D4—66no—
269C2×C4—33no—
270C2×C3—212no—
271C2×C3—26no—
272C2×C3—26no—
273C2×C3—23no—
274C2×C3—33no—
275C2×C3—66no—
276C2×C3—612no—
277C2×C3—63no—
278C2×C3—121no—
279C2×C3—361no—
280C2×C3C2×D4×3−1+213yes—
281C2×C3C2×C32×#6713yes—
282D3—41no—
283C22—28no—
284C22—26no—
285C22—63no—
286C22#67×3−1+216yescomplemented
287C22C22×D3×3−1+211yescharacteristic
288C4—63no—
289C3—21no—
290C3—31no—
291C3—61no—
292C33−1+2×#12011yescharacteristic, complemented
293C3C32×#95311yescharacteristic
294C2—26no—
295C2—121no—
296C2C2×#67×3−1+213yes—
subgroupG/H
center, FrattiniC22×C3C22×C32×D3
commutator, socle, cosocleC22×C32C23×C32
FittingC24×C3×3−1+2C2
prime radicalC32×#953C3
hypercenterC24×3−1+2D3
hypocenterC33−1+2×#120
LCSG ▷ C22×C32 ▷ C3
UCSC1 ◁ C22×C3 ◁ C24×3−1+2

Complements

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

NH ≅ G/Ntype
1. C24×C3×3−1+2C2semidirect
2. C2×#67×3−1+2C2semidirect
3. 3−1+2×#217C2semidirect
6. C9×#953C3semidirect
9. C22×D3×3−1+2C22semidirect
11. C2×Dic3×3−1+2C22semidirect
16. C24×3−1+2D3semidirect
17. C24×C3×C9C2×C3semidirect
20. C2×C9×#67C2×C3semidirect
22. C9×#217C2×C3semidirect
25. C22×C3×3−1+2D4semidirect
47. C22×D3×C9C22×C3semidirect
51. C2×C9×Dic3C22×C3semidirect
64. C24×C9C3×D3semidirect
87. C22×3−1+2#67semidirect
91. C22×C3×C9C3×D4semidirect
98. #9533−1+2direct (N × H)
99. C3×3−1+2#120semidirect
142. C24×C3C2×3−1+2semidirect
146. C2×#67C2×3−1+2semidirect
149. #217C2×3−1+2semidirect
183. C22×C9C3×#67semidirect
190. 3−1+2#953direct (N × H)
191. C3×C9C3×#120semidirect
204. C22×D3C22×3−1+2semidirect
210. C2×Dic3C22×3−1+2semidirect
229. C24D3×3−1+2semidirect
250. C22×C3D4×3−1+2semidirect
264. C9C3×#953semidirect
286. C22#67×3−1+2semidirect
292. C33−1+2×#120semidirect

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×#67×3−1+22
3−1+2×#2172
C32×#9533
3−1+2×#1203
C9×#9533