Finite Group Databasebeta

← C2×#11721704 #13153766

Subgroups — C2×#11721704 #13153766

Order 1728.

2360 conjugacy classes of subgroups; only the 372 normal subgroups are shown.

normal subgroups only

This group has 2360 conjugacy classes of subgroups — too many to draw. Only its 372 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
1C25×3−1+2C211yesmaximal, characteristic, complemented
2C22×#611⋊C3C213yesmaximal, complemented
3#11721704C218yesmaximal, direct product factor
4C2×#3676⋊C3C213yesmaximal, complemented
5C2×C3×#953—31nomaximal
6C2×#63515C311yesmaximal, characteristic, complemented
7C24×3−1+2—212no—
8C24×3−1+2C2213yes—
9C24×3−1+2C2214yescomplemented
10C23×AGL(1,ℤ/9)C2211yescharacteristic, complemented
11C2×#611⋊C3—212no—
12C2×#611⋊C3C22112yescomplemented
13C22×3−1+2⋊C4C2213yescomplemented
14#3676⋊C3C22112yescomplemented
15C25×C32D311yescharacteristic
16C22×C3×#67—33no—
17C25×C9—21no—
18C25×C9C2×C311yescharacteristic, complemented
19C3×#953—38no—
20C2×C3×#217—33no—
21C22×#611C2×C313yescomplemented
22#63515C2×C318yescomplemented
23C2×#3676C2×C313yescomplemented
24C23×3−1+2—232no—
25C23×3−1+2—212no—
26C23×3−1+2—224no—
27C23×3−1+2D4112yescomplemented
28C23×3−1+2C2316yes—
29C23×3−1+2C2311yescharacteristic
30C22×AGL(1,ℤ/9)—26no—
31C22×AGL(1,ℤ/9)C2312yescomplemented
32#611⋊C3—224no—
33C2×3−1+2⋊C4—26no—
34C2×3−1+2⋊C4C2316yescomplemented
35C2×C3×#120—91no—
36C2×#953—31no—
37C24×C32—212no—
38C24×C32C2×D314yes—
39C24×C32C2×D313yes—
40C23×C3×D3—31no—
41C2×C3×#67—312no—
42C2×C3×#67—612no—
43C22×C3×Dic3—33no—
44C24×C9—224no—
45C24×C9—23no—
46C24×C9—24no—
47C24×C9—212no—
48C24×C9C22×C313yes—
49C24×C9C22×C314yescomplemented
50C23×D9C22×C311yescharacteristic, complemented
51C3×#217—312no—
52C2×#611—212no—
53C2×#611C22×C3112yescomplemented
54C22×Dic9C22×C313yescomplemented
55#3676C22×C3112yescomplemented
56C22×3−1+2—232no—
57C22×3−1+2—224no—
58C22×3−1+2—212no—
59C22×3−1+2C2×D416yes—
60C22×3−1+2C2×D4112yescomplemented
61C22×3−1+2C2411yescharacteristic
62C2×AGL(1,ℤ/9)—26no—
63C2×AGL(1,ℤ/9)—44no—
643−1+2⋊C4—26no—
65C25×C3—31no—
66C25×C3C3×D311yescharacteristic
67C22×C3×D4—93no—
68C22×#67—33no—
69C3×#120—98no—
70C2×C3×C22⋊C4—93no—
71#953—38no—
72C2×#217—33no—
73C23×C32—232no—
74C23×C32—224no—
75C23×C32—212no—
76C23×C32#67112yes—
77C23×C32C22×D316yes—
78C23×C32C22×D311yescharacteristic
79C22×C3×D3—32no—
80C22×C3×D3—66no—
81C3×#67—624no—
82C2×C3×Dic3—36no—
83C2×C3×Dic3—66no—
84C23×C9—264no—
85C23×C9—232no—
86C23×C9—224no—
87C23×C9—248no—
88C23×C9—212no—
89C23×C9—224no—
90C23×C9—212no—
91C23×C9—26no—
92C23×C9—21no—
93C23×C9C3×D4112yescomplemented
94C23×C9C23×C316yes—
95C23×C9C23×C311yescharacteristic
96C22×D9—26no—
97C22×D9C23×C312yescomplemented
98#611—224no—
99C2×Dic9—26no—
100C2×Dic9C23×C316yescomplemented
101C2×#120—91noSylow
102C2×3−1+2—212no—
103C2×3−1+2#12014yescomplemented
104C2×3−1+2C22×D413yes—
105AGL(1,ℤ/9)—42no—
106C24×C3—212no—
107C24×C3—33no—
108C24×C3—34no—
109C24×C3—612no—
110C24×C3—91no—
111C24×C3C2×C3×D313yes—
112C24×C3C2×C3×D314yes—
113C23×D3—31no—
114C2×C3×D4—912no—
115C2×C3×D4—1812no—
116C22×C3×C4—93no—
117C2×#67—312no—
118C2×#67—612no—
119C22×Dic3—33no—
120C3×C22⋊C4—912no—
121#217—312no—
122C22×C32—232no—
123C22×C32—224no—
124C22×C32—212no—
125C22×C32C2×#67112yes—
126C22×C32C2×#6716yes—
127C22×C32C23×D311yescharacteristic
128C2×C3×D3—66no—
129C2×C3×D3—124no—
130C3×Dic3—66no—
131C22×C9—264no—
132C22×C9—232no—
133C22×C9—248no—
134C22×C9—224no—
135C22×C9—224no—
136C22×C9—212no—
137C22×C9—21no—
138C22×C9—26no—
139C22×C9—212no—
140C22×C9C2×C3×D416yes—
141C22×C9C2×C3×D4112yescomplemented
142C22×C9C24×C311yescharacteristic
143C2×D9—26no—
144C2×D9—44no—
145Dic9—26no—
146C25AGL(1,ℤ/9)11yescharacteristic, complemented
147C22×D4—93no—
148#120—98no—
149C2×C22⋊C4—93no—
1503−1+2C2×#12011yescharacteristic, complemented, Sylow
151C23×C3—232no—
152C23×C3—212no—
153C23×C3—224no—
154C23×C3—312no—
155C23×C3—36no—
156C23×C3—31no—
157C23×C3—612no—
158C23×C3—624no—
159C23×C3—632no—
160C23×C3—92no—
161C23×C3—186no—
162C23×C3C3×#67112yes—
163C23×C3C22×C3×D316yes—
164C23×C3C22×C3×D311yescharacteristic
165C22×D3—32no—
166C22×D3—66no—
167C3×D4—1824no—
168C2×C3×C4—96no—
169C2×C3×C4—186no—
170#67—624no—
171C2×Dic3—36no—
172C2×Dic3—66no—
173C2×C32—212no—
174C2×C32#95314yes—
175C2×C32C22×#6713yes—
176C3×D3—122no—
177C2×C9—224no—
178C2×C9—212no—
179C2×C9—24no—
180C2×C9—23no—
181C2×C9C3×#12014yescomplemented
182C2×C9C22×C3×D413yes—
183D9—42no—
184C24—212no—
185C24—91no—
186C24C2×AGL(1,ℤ/9)13yes—
187C24C2×AGL(1,ℤ/9)14yescomplemented
188C2×D4—912no—
189C2×D4—1812no—
190C22×C4—93no—
191C22⋊C4—912no—
192C22×C3—232no—
193C22×C3—224no—
194C22×C3—212no—
195C22×C3—31no—
196C22×C3—36no—
197C22×C3—312no—
198C22×C3—612no—
199C22×C3—624no—
200C22×C3—632no—
201C22×C3—186no—
202C22×C3—364no—
203C22×C3C2×C3×#6716yes—
204C22×C3C2×C3×#67112yes—
205C22×C3C23×C3×D311yescharacteristic
206C2×D3—66no—
207C2×D3—124no—
208C3×C4—186no—
209Dic3—66no—
210C32C2×#95311yescharacteristic
211C9—21no—
212C9C2×C3×#12011yescharacteristic, complemented
213C23—232no—
214C23—212no—
215C23—224no—
216C23—92no—
217C23—186no—
218C23#611⋊C3112yescomplemented
219C23C22×AGL(1,ℤ/9)16yes—
220C23C22×AGL(1,ℤ/9)11yescharacteristic
221D4—1824no—
222C2×C4—96no—
223C2×C4—186no—
224C2×C3—212no—
225C2×C3—33no—
226C2×C3—34no—
227C2×C3—612no—
228C2×C3—362no—
229C2×C3C3×#95314yes—
230C2×C3C22×C3×#6713yes—
231D3—122no—
232C22—232no—
233C22—212no—
234C22—224no—
235C22—186no—
236C22—364no—
237C22C2×#611⋊C316yes—
238C22C2×#611⋊C3112yescomplemented
239C22C23×AGL(1,ℤ/9)11yescharacteristic
240C4—186no—
241C3—31no—
242C3C2×C3×#95311yescharacteristic
243C2—212no—
244C2—362no—
245C2#1172170414yesdirect product factor
246C2C22×#611⋊C313yes—
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC25×3−1+2C2
socleC23×C3C22×C3×D3
hypercenterC25AGL(1,ℤ/9)
hypocenterC9C2×C3×#120
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C25×3−1+2C2semidirect
2. C22×#611⋊C3C2semidirect
3. #11721704C2direct (N × H)
4. C2×#3676⋊C3C2semidirect
6. C2×#63515C3semidirect
9. C24×3−1+2C22semidirect
10. C23×AGL(1,ℤ/9)C22semidirect
12. C2×#611⋊C3C22semidirect
13. C22×3−1+2⋊C4C22semidirect
14. #3676⋊C3C22semidirect
18. C25×C9C2×C3semidirect
21. C22×#611C2×C3semidirect
22. #63515C2×C3semidirect
23. C2×#3676C2×C3semidirect
27. C23×3−1+2D4semidirect
31. C22×AGL(1,ℤ/9)C23semidirect
34. C2×3−1+2⋊C4C23semidirect
49. C24×C9C22×C3semidirect
50. C23×D9C22×C3semidirect
53. C2×#611C22×C3semidirect
54. C22×Dic9C22×C3semidirect
55. #3676C22×C3semidirect
60. C22×3−1+2C2×D4semidirect
93. C23×C9C3×D4semidirect
97. C22×D9C23×C3semidirect
100. C2×Dic9C23×C3semidirect
103. C2×3−1+2#120semidirect
141. C22×C9C2×C3×D4semidirect
146. C25AGL(1,ℤ/9)semidirect
150. 3−1+2C2×#120semidirect
181. C2×C9C3×#120semidirect
187. C24C2×AGL(1,ℤ/9)semidirect
212. C9C2×C3×#120semidirect
218. C23#611⋊C3semidirect
238. C22C2×#611⋊C3semidirect
245. C2#11721704direct (N × H)

Maximal subgroups

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

subgroupindex
C25×3−1+22
C22×#611⋊C32
#117217042
C2×#3676⋊C32
C2×C3×#9533
C2×#635153