Finite Group Databasebeta

← C2×#11721396 #13152321

Subgroups — C2×#11721396 #13152321

Order 1728.

792 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 252 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
1C2×C42×3−1+2C211yesmaximal, characteristic, complemented
2C2×(C2×AGL(1,ℤ/9))⋊C4C213yesmaximal
3C2×(3−1+2⋊C4)⋊C4C213yesmaximal, complemented
4#11721396C218yesmaximal, direct product factor
5C2×C3×#876—31nomaximal
6C2×#63438C311yesmaximal, characteristic, complemented
7C23×AGL(1,ℤ/9)C2211yescharacteristic
8C22×C4×3−1+2C2213yes—
9C22×3−1+2⋊C4C2213yes—
10C42×3−1+2C2214yescomplemented
11(C2×AGL(1,ℤ/9))⋊C4C22112yes—
12(3−1+2⋊C4)⋊C4C22112yescomplemented
13C2×C32×C42D311yescharacteristic
14C2×C3×(C2×D3)⋊C4—33no—
15C2×C3×Dic3⋊C4—33no—
16C3×#876—38no—
17C2×C42×C9—21no—
18C2×C42×C9C2×C311yescharacteristic, complemented
19C2×(C2×D9)⋊C4C2×C313yes—
20C2×Dic9⋊C4C2×C313yescomplemented
21#63438C2×C318yescomplemented
22C23×3−1+2C2311yescharacteristic
23C22×AGL(1,ℤ/9)—26no—
24C22×AGL(1,ℤ/9)C2312yes—
25C2×C4×3−1+2—26no—
26C2×C4×3−1+2C2316yes—
27C2×3−1+2⋊C4—26no—
28C2×3−1+2⋊C4C2316yes—
29C2×C3×#126—91no—
30C2×#876—31no—
31C23×C3×D3—31no—
32C22×C32×C4C2×D313yes—
33C22×C3×Dic3—33no—
34C23×D9C22×C311yescharacteristic
35C32×C42C2×D314yes—
36C3×(C2×D3)⋊C4—312no—
37C3×Dic3⋊C4—312no—
38C22×C4×C9—23no—
39C22×C4×C9C22×C313yes—
40C22×Dic9C22×C313yes—
41C42×C9—24no—
42C42×C9C22×C314yescomplemented
43(C2×D9)⋊C4C22×C3112yes—
44Dic9⋊C4C22×C3112yescomplemented
45C22×3−1+2#4116yes—
46C22×3−1+2C2411yescharacteristic
47C2×AGL(1,ℤ/9)—26no—
48C2×AGL(1,ℤ/9)—44no—
49C4×3−1+2—26no—
503−1+2⋊C4—26no—
51C3×#126—98no—
52C2×C3×C4⋊C4—93no—
53C2×C3×C22⋊C4—93no—
54C2×C3×C42—31no—
55C2×C3×C42C3×D311yescharacteristic
56C2×(C2×D3)⋊C4—33no—
57C2×Dic3⋊C4—33no—
58#876—38no—
59C23×C32C22×D311yescharacteristic
60C22×C3×D3—32no—
61C22×C3×D3—66no—
62C2×C32×C4—26no—
63C2×C32×C4C22×D316yes—
64C2×C3×Dic3—36no—
65C2×C3×Dic3—66no—
66C23×C9—21no—
67C23×C9C23×C311yescharacteristic
68C22×D9—26no—
69C22×D9C23×C312yes—
70C2×C4×C9—212no—
71C2×C4×C9—26no—
72C2×C4×C9—26no—
73C2×C4×C9C23×C316yes—
74C2×Dic9—26no—
75C2×Dic9C23×C316yes—
76C2×#126—91noSylow
77C2×3−1+2#12614yescomplemented
78C2×3−1+2C2×#4113yes—
79AGL(1,ℤ/9)—42no—
80C24×C3—91no—
81C23×D3—31no—
82C22×C3×C4—33no—
83C22×C3×C4—93no—
84C22×C3×C4C2×C3×D313yes—
85C22×Dic3—33no—
86C3×C4⋊C4—912no—
87C3×C22⋊C4—912no—
88C3×C42—34no—
89C3×C42C2×C3×D314yes—
90(C2×D3)⋊C4—312no—
91Dic3⋊C4—312no—
92C22×C32#23516yes—
93C22×C32C23×D311yescharacteristic
94C2×C3×D3—66no—
95C2×C3×D3—124no—
96C32×C4—26no—
97C3×Dic3—66no—
98C22×C9—21no—
99C22×C9—26no—
100C22×C9C3×#4116yes—
101C22×C9C24×C311yescharacteristic
102C2×D9—26no—
103C2×D9—44no—
104C4×C9—212no—
105C4×C9—26no—
106Dic9—26no—
107#126—98no—
108C2×C4⋊C4—93no—
109C2×C22⋊C4—93no—
110C2×C42AGL(1,ℤ/9)11yescharacteristic, complemented
1113−1+2C2×#12611yescharacteristic, complemented, Sylow
112C23×C3—31no—
113C23×C3—92no—
114C23×C3—186no—
115C23×C3C22×C3×D311yescharacteristic
116C22×D3—32no—
117C22×D3—66no—
118C2×C3×C4—26no—
119C2×C3×C4—36no—
120C2×C3×C4—66no—
121C2×C3×C4—96no—
122C2×C3×C4—186no—
123C2×C3×C4C22×C3×D316yes—
124C2×Dic3—36no—
125C2×Dic3—66no—
126C2×C32#87614yes—
127C2×C32C2×#23513yes—
128C3×D3—122no—
129C2×C9—23no—
130C2×C9—24no—
131C2×C9C3×#12614yescomplemented
132C2×C9C2×C3×#4113yes—
133D9—42no—
134C24—91no—
135C22×C4—93no—
136C22×C4C2×AGL(1,ℤ/9)13yes—
137C4⋊C4—912no—
138C22⋊C4—912no—
139C42C2×AGL(1,ℤ/9)14yescomplemented
140C22×C3—31no—
141C22×C3—36no—
142C22×C3—186no—
143C22×C3—364no—
144C22×C3C3×#23516yes—
145C22×C3C23×C3×D311yescharacteristic
146C2×D3—66no—
147C2×D3—124no—
148C3×C4—26no—
149C3×C4—66no—
150C3×C4—186no—
151Dic3—66no—
152C32C2×#87611yescharacteristic
153C9—21no—
154C9C2×C3×#12611yescharacteristic, complemented
155C23—92no—
156C23—186no—
157C23C22×AGL(1,ℤ/9)11yescharacteristic
158C2×C4—26no—
159C2×C4—96no—
160C2×C4—186no—
161C2×C4C22×AGL(1,ℤ/9)16yes—
162C2×C3—33no—
163C2×C3—34no—
164C2×C3—362no—
165C2×C3C3×#87614yes—
166C2×C3C2×C3×#23513yes—
167D3—122no—
168C22—186no—
169C22—364no—
170C22#3697⋊C316yes—
171C22C23×AGL(1,ℤ/9)11yescharacteristic
172C4—26no—
173C4—186no—
174C3—31no—
175C3C2×C3×#87611yescharacteristic
176C2—362no—
177C2#1172139614yesdirect product factor
178C2C2×#3697⋊C313yes—
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C42×3−1+2C2
socleC23×C3C22×C3×D3
prime radicalC23×AGL(1,ℤ/9)C22
hypercenterC2×C42AGL(1,ℤ/9)
hypocenterC9C2×C3×#126
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C42×3−1+2C2semidirect
3. C2×(3−1+2⋊C4)⋊C4C2semidirect
4. #11721396C2direct (N × H)
6. C2×#63438C3semidirect
10. C42×3−1+2C22semidirect
12. (3−1+2⋊C4)⋊C4C22semidirect
18. C2×C42×C9C2×C3semidirect
20. C2×Dic9⋊C4C2×C3semidirect
21. #63438C2×C3semidirect
42. C42×C9C22×C3semidirect
44. Dic9⋊C4C22×C3semidirect
77. C2×3−1+2#126semidirect
110. C2×C42AGL(1,ℤ/9)semidirect
111. 3−1+2C2×#126semidirect
131. C2×C9C3×#126semidirect
139. C42C2×AGL(1,ℤ/9)semidirect
154. C9C2×C3×#126semidirect
177. C2#11721396direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C42×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×(3−1+2⋊C4)⋊C42
#117213962
C2×C3×#8763
C2×#634383