Finite Group Databasebeta

← #13152785

Subgroups — #13152785

Order 1728.

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

normal subgroups only

This group has 845 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×#120C211yesmaximal, characteristic, complemented
2#11721704C212yesmaximal, complemented
3#63499⋊C3C213yesmaximal, complemented
4#63498⋊C3C213yesmaximal, complemented
5#63447⋊C3C216yesmaximal, complemented
6C3×#5594—31nomaximal
7#10589696C311yesmaximal, characteristic, complemented
8C24×3−1+2C2211yescharacteristic, complemented
9C23×AGL(1,ℤ/9)C2211yescharacteristic, complemented
10C2×D4×3−1+2C2213yes—
11C2×#611⋊C3C2216yes—
12C2×3−1+2⋊Q8C2213yescomplemented
13C22⋊C4×3−1+2C2213yescomplemented
14#3676⋊C3C2213yescomplemented
15#3676⋊C3C2216yescomplemented
16(C2×AGL(1,ℤ/9))⋊C4C2216yescomplemented
17C4×3−1+2⋊C4C2213yescomplemented
18C32×#120D311yescharacteristic
19C3×#953—32no—
20C3×#937—33no—
21C3×#936—33no—
22C3×#885—36no—
23C9×#120—21no—
24C9×#120C2×C311yescharacteristic, complemented
25#63515C2×C312yescomplemented
26#63499C2×C313yescomplemented
27#63498C2×C313yescomplemented
28#63447C2×C316yescomplemented
29C23×3−1+2—43no—
30C23×3−1+2C2311yescharacteristic
31C23×3−1+2C2313yes—
32C22×AGL(1,ℤ/9)—43no—
33C22×AGL(1,ℤ/9)C2312yes—
34D4×3−1+2—43no—
35C2×C4×3−1+2C2313yes—
36#611⋊C3—46no—
37C2×3−1+2⋊C4C2316yes—
383−1+2⋊Q8—43no—
39C3×H(3,𝔽4)—91no—
40#5594—31no—
41C24×C32C2×D311yescharacteristic
42C23×C3×D3—31no—
43C2×C32×D4C2×D313yes—
44C2×C3×#67—36no—
45C2×C3×Dic6—33no—
46C24×C9—21no—
47C24×C9C22×C311yescharacteristic, complemented
48C23×D9C22×C311yescharacteristic, complemented
49C32×C22⋊C4C2×D313yes—
50C3×#217—36no—
51C3×#217—33no—
52C3×(C2×D3)⋊C4—36no—
53C3×C4×Dic3—33no—
54C2×D4×C9—23no—
55C2×D4×C9C22×C313yes—
56C2×#611C22×C316yes—
57C2×Dic18C22×C313yescomplemented
58C9×C22⋊C4—23no—
59C9×C22⋊C4C22×C313yescomplemented
60#3676C22×C313yescomplemented
61#3676C22×C316yescomplemented
62(C2×D9)⋊C4C22×C316yescomplemented
63C4×Dic9C22×C313yescomplemented
64C22×3−1+2—23no—
65C22×3−1+2—26no—
66C22×3−1+2—23no—
67C22×3−1+2—44no—
68C22×3−1+2C2411yescharacteristic
69C2×AGL(1,ℤ/9)—26no—
70C2×AGL(1,ℤ/9)—44no—
71C4×3−1+2—23no—
723−1+2⋊C4—26no—
73C3×#124—96no—
74C3×#124—93no—
75C3×#120—31no—
76C3×#120—93no—
77C3×#120—92no—
78C3×#120C3×D311yescharacteristic
79#953—32no—
80#937—33no—
81#936—33no—
82#885—36no—
83C23×C32—43no—
84C23×C32C22×D311yescharacteristic
85C23×C32C22×D313yes—
86C22×C3×D3—32no—
87C22×C3×D3—123no—
88C32×D4—43no—
89C2×C32×C4C22×D313yes—
90C3×#67—126no—
91C2×C3×Dic3—36no—
92C3×Dic6—123no—
93C23×C9—23no—
94C23×C9—21no—
95C23×C9—43no—
96C23×C9—46no—
97C23×C9C23×C311yescharacteristic
98C23×C9C23×C313yes—
99C22×D9—43no—
100C22×D9C23×C312yes—
101D4×C9—43no—
102D4×C9—46no—
103C2×C4×C9—23no—
104C2×C4×C9C23×C313yes—
105#611—46no—
106C2×Dic9C23×C316yes—
107Dic18—43no—
108H(3,𝔽4)—91noSylow
109C2×3−1+2—43no—
110C2×3−1+2—41no—
111C2×3−1+2H(5,𝔽2)13yes—
112AGL(1,ℤ/9)—42no—
113C24×C3—31no—
114C24×C3—91no—
115C24×C3C2×C3×D311yescharacteristic
116C23×D3—31no—
117C2×C3×Q8—93no—
118C2×C3×D4—33no—
119C2×C3×D4—96no—
120C2×C3×D4C2×C3×D313yes—
121C2×#67—36no—
122C2×Dic6—33no—
123C3×C22⋊C4—33no—
124C3×C22⋊C4—96no—
125C3×C22⋊C4—96no—
126C3×C22⋊C4—93no—
127C3×C22⋊C4C2×C3×D313yes—
128C3×C42—93no—
129#217—36no—
130#217—33no—
131(C2×D3)⋊C4—36no—
132C4×Dic3—33no—
133C22×C32—26no—
134C22×C32—23no—
135C22×C32—23no—
136C22×C32—44no—
137C22×C32C23×D311yescharacteristic
138C2×C3×D3—66no—
139C2×C3×D3—124no—
140C32×C4—23no—
141C3×Dic3—66no—
142C22×C9—26no—
143C22×C9—26no—
144C22×C9—23no—
145C22×C9—23no—
146C22×C9—21no—
147C22×C9—46no—
148C22×C9—43no—
149C22×C9—48no—
150C22×C9—44no—
151C22×C9C24×C311yescharacteristic
152C2×D9—26no—
153C2×D9—44no—
154C4×C9—23no—
155C4×C9—43no—
156Dic9—26no—
157#124—93no—
158#124—96no—
159#120—92no—
160#120—93no—
161#120AGL(1,ℤ/9)11yescharacteristic, complemented
1623−1+2H(3,𝔽4)11yescharacteristic, complemented, Sylow
163C23×C3—33no—
164C23×C3—31no—
165C23×C3—43no—
166C23×C3—92no—
167C23×C3—123no—
168C23×C3—363no—
169C23×C3C22×C3×D311yescharacteristic
170C23×C3C22×C3×D313yes—
171C22×D3—32no—
172C22×D3—123no—
173C3×Q8—363no—
174C3×D4—43no—
175C3×D4—123no—
176C3×D4—366no—
177C2×C3×C4—33no—
178C2×C3×C4—96no—
179C2×C3×C4C22×C3×D313yes—
180#67—126no—
181C2×Dic3—36no—
182Dic6—123no—
183C2×C32—41no—
184C2×C32—43no—
185C2×C32#100413yes—
186C3×D3—122no—
187C2×C9—23no—
188C2×C9—46no—
189C2×C9—41no—
190C2×C9—43no—
191C2×C9—81no—
192C2×C9C3×H(5,𝔽2)13yes—
193D9—42no—
194C24—91no—
195C24C2×AGL(1,ℤ/9)11yescharacteristic, complemented
196C2×Q8—93no—
197C2×D4—96no—
198C2×D4C2×AGL(1,ℤ/9)13yes—
199C22⋊C4—93no—
200C22⋊C4—96no—
201C22⋊C4—96no—
202C22⋊C4C2×AGL(1,ℤ/9)13yescomplemented
203C42—93no—
204C22×C3—26no—
205C22×C3—23no—
206C22×C3—23no—
207C22×C3—31no—
208C22×C3—44no—
209C22×C3—63no—
210C22×C3—63no—
211C22×C3—66no—
212C22×C3—124no—
213C22×C3—186no—
214C22×C3—364no—
215C22×C3C23×C3×D311yescharacteristic
216C2×D3—66no—
217C2×D3—124no—
218C3×C4—23no—
219C3×C4—63no—
220C3×C4—186no—
221Dic3—66no—
222C32#559411yescharacteristic
223C9—21no—
224C9C3×H(3,𝔽4)11yescharacteristic, complemented
225C23—43no—
226C23—92no—
227C23—363no—
228C23C22×AGL(1,ℤ/9)11yescharacteristic
229C23C22×AGL(1,ℤ/9)13yes—
230Q8—363no—
231D4—43no—
232D4—366no—
233C2×C4—96no—
234C2×C4C22×AGL(1,ℤ/9)13yes—
235C2×C3—33no—
236C2×C3—41no—
237C2×C3—43no—
238C2×C3—123no—
239C2×C3—121no—
240C2×C3—362no—
241C2×C3C3×#100413yes—
242D3—122no—
243C22—23no—
244C22—23no—
245C22—26no—
246C22—44no—
247C22—186no—
248C22—364no—
249C22C23×AGL(1,ℤ/9)11yescharacteristic
250C4—23no—
251C4—186no—
252C3—31no—
253C3C3×#559411yescharacteristic
254C2—43no—
255C2—41no—
256C2—362no—
257C2#63710⋊C313yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#120C2
hypercenter#120AGL(1,ℤ/9)
hypocenterC9C3×H(3,𝔽4)
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#120C2semidirect
2. #11721704C2semidirect
3. #63499⋊C3C2semidirect
4. #63498⋊C3C2semidirect
5. #63447⋊C3C2semidirect
7. #10589696C3semidirect
8. C24×3−1+2C22semidirect
9. C23×AGL(1,ℤ/9)C22semidirect
12. C2×3−1+2⋊Q8C22semidirect
13. C22⋊C4×3−1+2C22semidirect
14. #3676⋊C3C22semidirect
15. #3676⋊C3C22semidirect
16. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
17. C4×3−1+2⋊C4C22semidirect
24. C9×#120C2×C3semidirect
25. #63515C2×C3semidirect
26. #63499C2×C3semidirect
27. #63498C2×C3semidirect
28. #63447C2×C3semidirect
47. C24×C9C22×C3semidirect
48. C23×D9C22×C3semidirect
57. C2×Dic18C22×C3semidirect
59. C9×C22⋊C4C22×C3semidirect
60. #3676C22×C3semidirect
61. #3676C22×C3semidirect
62. (C2×D9)⋊C4C22×C3semidirect
63. C4×Dic9C22×C3semidirect
161. #120AGL(1,ℤ/9)semidirect
162. 3−1+2H(3,𝔽4)semidirect
195. C24C2×AGL(1,ℤ/9)semidirect
202. C22⋊C4C2×AGL(1,ℤ/9)semidirect
224. C9C3×H(3,𝔽4)semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1202
#117217042
#63499⋊C32
#63498⋊C32
#63447⋊C32
C3×#55943
#105896963