Finite Group Databasebeta

← #13141391

Subgroups — #13141391

Order 1728.

1004 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C22×C4⋊AGL(1,ℤ/9)C213yesmaximal, complemented
2C2×(C2×AGL(1,ℤ/9))⋊C4C213yesmaximal, complemented
33−1+2×#95C211yesmaximal, characteristic, complemented
4C3×#4671—31nomaximal
5#10588351C311yesmaximal, characteristic, complemented
6C23×AGL(1,ℤ/9)C2213yes—
7C22×C4×3−1+2C2213yes—
8C22×3−1+2⋊C4C2211yescharacteristic
9C2×C4⋊AGL(1,ℤ/9)—212no—
10C2×C4⋊AGL(1,ℤ/9)—26no—
11(C2×AGL(1,ℤ/9))⋊C4—26no—
12C22×C3×D12—33no—
13C2×C3×(C2×D3)⋊C4—33no—
14C22×D36C2×C313yescomplemented
15C32×#95D311yescharacteristic
16C2×(C2×D9)⋊C4C2×C313yescomplemented
17C9×#95—21no—
18C9×#95C2×C311yescharacteristic, complemented
19C23×3−1+2C2311yescharacteristic
20C22×AGL(1,ℤ/9)—26no—
21C22×AGL(1,ℤ/9)—26no—
22C22×AGL(1,ℤ/9)—23no—
23C22×AGL(1,ℤ/9)—23no—
24C22×AGL(1,ℤ/9)D416yescomplemented
25C2×C4×3−1+2—26no—
26C2×C4×3−1+2D416yescomplemented
27C2×3−1+2⋊C4—23no—
28C4⋊AGL(1,ℤ/9)—224no—
29C3×#392—91no—
30#4671—31no—
31C23×C3×D3—33no—
32C22×C32×C4C2×D313yes—
33C22×C3×Dic3—31no—
34C2×C3×D12—612no—
35C2×C3×D12—66no—
36C23×D9C22×C313yes—
37C3×(C2×D3)⋊C4—66no—
38C22×C4×C9—23no—
39C22×C4×C9C22×C313yes—
40C22×Dic9C22×C311yescharacteristic
41C2×D36—212no—
42C2×D36—26no—
43(C2×D9)⋊C4—26no—
44C22×3−1+2C2×D413yes—
45C22×3−1+2C2×D413yes—
46C22×3−1+2#4111yescharacteristic
47C2×AGL(1,ℤ/9)—212no—
48C2×AGL(1,ℤ/9)—26no—
49C2×AGL(1,ℤ/9)—43no—
50C2×AGL(1,ℤ/9)—46no—
51C2×AGL(1,ℤ/9)—43no—
52C4×3−1+2—26no—
533−1+2⋊C4—41no—
54C22×C3×D4—93no—
55C22×D12—33no—
56C2×C3×C22⋊C4—93no—
57C2×(C2×D3)⋊C4—33no—
58C3×#95—31no—
59C3×#95C3×D311yescharacteristic
60C23×C32C22×D311yescharacteristic
61C22×C3×D3—36no—
62C22×C3×D3—63no—
63C22×C3×D3—66no—
64C22×C3×D3—66no—
65C22×C3×D3—63no—
66C2×C32×C4—26no—
67C2×C32×C4D1216yes—
68C2×C3×Dic3—63no—
69C3×D12—624no—
70C23×C9—21no—
71C23×C9C23×C311yescharacteristic
72C22×D9—23no—
73C22×D9—26no—
74C22×D9—26no—
75C22×D9—23no—
76C22×D9C3×D416yescomplemented
77C2×C4×C9—26no—
78C2×C4×C9—26no—
79C2×C4×C9—46no—
80C2×C4×C9C3×D416yescomplemented
81C2×Dic9—23no—
82D36—224no—
83#392—91noSylow
84C2×3−1+2#12013yes—
85C2×3−1+2#12113yes—
86C2×3−1+2#12711yescharacteristic
87AGL(1,ℤ/9)—46no—
88C24×C3—93no—
89C23×D3—33no—
90C2×C3×D4—186no—
91C2×C3×D4—1812no—
92C22×C3×C4—33no—
93C22×C3×C4—91no—
94C22×C3×C4C2×C3×D313yes—
95C22×Dic3—31no—
96C2×D12—612no—
97C2×D12—66no—
98C3×C22⋊C4—186no—
99(C2×D3)⋊C4—66no—
100C22×C32C2×D1213yes—
101C22×C32D3×D413yes—
102C22×C32#23911yescharacteristic
103C2×C3×D3—612no—
104C2×C3×D3—66no—
105C2×C3×D3—123no—
106C2×C3×D3—123no—
107C2×C3×D3—126no—
108C32×C4—26no—
109C3×Dic3—121no—
110C22×C9—23no—
111C22×C9—23no—
112C22×C9—21no—
113C22×C9C2×C3×D413yes—
114C22×C9C2×C3×D413yes—
115C22×C9C3×#4111yescharacteristic
116C2×D9—212no—
117C2×D9—26no—
118C2×D9—43no—
119C2×D9—46no—
120C2×D9—43no—
121C4×C9—26no—
122C4×C9—46no—
123Dic9—41no—
124C22×D4—93no—
125C2×C22⋊C4—93no—
126#95AGL(1,ℤ/9)11yescharacteristic, complemented
1273−1+2#39211yescharacteristic, complemented, Sylow
128C23×C3—31no—
129C23×C3—96no—
130C23×C3—183no—
131C23×C3—186no—
132C23×C3—186no—
133C23×C3—183no—
134C23×C3C22×C3×D311yescharacteristic
135C22×D3—36no—
136C22×D3—63no—
137C22×D3—66no—
138C22×D3—66no—
139C22×D3—63no—
140C3×D4—1824no—
141C2×C3×C4—26no—
142C2×C3×C4—36no—
143C2×C3×C4—66no—
144C2×C3×C4—183no—
145C2×C3×C4C3×D1216yes—
146C2×Dic3—63no—
147D12—624no—
148C2×C32#87411yescharacteristic
149C2×C32#88213yes—
150C2×C32#89513yes—
151C3×D3—126no—
152C2×C9—23no—
153C2×C9—23no—
154C2×C9—21no—
155C2×C9C3×#12013yes—
156C2×C9C3×#12113yes—
157C2×C9C3×#12711yescharacteristic
158D9—46no—
159C24—93no—
160C2×D4—1812no—
161C2×D4—186no—
162C22×C4—91no—
163C22×C4C2×AGL(1,ℤ/9)13yes—
164C22⋊C4—186no—
165C22×C3—33no—
166C22×C3—33no—
167C22×C3—31no—
168C22×C3—1812no—
169C22×C3—186no—
170C22×C3—366no—
171C22×C3—363no—
172C22×C3—363no—
173C22×C3C2×C3×D1213yes—
174C22×C3C3×D3×D413yes—
175C22×C3C3×#23911yescharacteristic
176C2×D3—612no—
177C2×D3—66no—
178C2×D3—123no—
179C2×D3—126no—
180C2×D3—123no—
181C3×C4—26no—
182C3×C4—66no—
183C3×C4—361no—
184Dic3—121no—
185C32#467111yescharacteristic
186C9—21no—
187C9C3×#39211yescharacteristic, complemented
188C23—96no—
189C23—183no—
190C23—186no—
191C23—186no—
192C23—183no—
193C23C22×AGL(1,ℤ/9)11yescharacteristic
194D4—1824no—
195C2×C4—26no—
196C2×C4—183no—
197C2×C4C4⋊AGL(1,ℤ/9)16yescomplemented
198C2×C3—33no—
199C2×C3—31no—
200C2×C3—33no—
201C2×C3—366no—
202C2×C3C3×#87411yescharacteristic
203C2×C3C3×#88213yes—
204C2×C3C3×#89513yes—
205D3—126no—
206C22—186no—
207C22—1812no—
208C22—366no—
209C22—363no—
210C22—363no—
211C22C2×C4⋊AGL(1,ℤ/9)13yes—
212C22D4×AGL(1,ℤ/9)13yes—
213C22#8872411yescharacteristic
214C4—26no—
215C4—361no—
216C3—31no—
217C3C3×#467111yescharacteristic
218C2—366no—
219C2#1172139411yescharacteristic
220C2#63444⋊C313yes—
221C2#63457⋊C313yes—
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
Fitting3−1+2×#95C2
hypercenter#95AGL(1,ℤ/9)
hypocenterC9C3×#392
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊AGL(1,ℤ/9)C2semidirect
2. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
3. 3−1+2×#95C2semidirect
5. #10588351C3semidirect
14. C22×D36C2×C3semidirect
16. C2×(C2×D9)⋊C4C2×C3semidirect
18. C9×#95C2×C3semidirect
24. C22×AGL(1,ℤ/9)D4semidirect
26. C2×C4×3−1+2D4semidirect
76. C22×D9C3×D4semidirect
80. C2×C4×C9C3×D4semidirect
126. #95AGL(1,ℤ/9)semidirect
127. 3−1+2#392semidirect
187. C9C3×#392semidirect
197. C2×C4C4⋊AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊AGL(1,ℤ/9)2
C2×(C2×AGL(1,ℤ/9))⋊C42
3−1+2×#952
C3×#46713
#105883513