Finite Group Databasebeta

← C22×C3×#3774 #13181048

Subgroups — C22×C3×#3774 #13181048

Order 1728.

1416 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 210 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×C3×#625C212yesmaximal, complemented
2C22×C3×#622C211yesmaximal, characteristic, complemented
3C2×C3×#3774C2112yesmaximal, direct product factor
4C22×#3774C311yesmaximal, characteristic, direct product factor
5C23×C32×D3—22no—
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C3×#625—26no—
8C2×C3×#625C22112yescomplemented
9C2×C3×#622C2216yescomplemented
10C3×#3774C22116yesdirect product factor
11C22×C3×#67—62no—
12C22×#625C2×C312yescomplemented
13C22×#622C2×C311yescharacteristic, complemented
14C2×#3774C2×C3112yesdirect product factor
15C23×C33D411yescharacteristic
16C22×C32×D3—212no—
17C22×C32×D3—48no—
18C2×C3×C3⋊Dic3D413yes—
19C2×C3×C3⋊Dic3C2313yes—
20C3×#625—212no—
21C3×#625C2318yescomplemented
22C3×#622C2314yescomplemented
23C22×C3×D8—91nomaximal
24C24×C32—62no—
25C23×C3×D3—22no—
26C23×C3×D3—22no—
27C23×C3×D3—42no—
28C22×C3⋊Dic3C22×C311yescharacteristic
29C2×C3×#67—612no—
30C2×C3×#67—612no—
31C22×C3×Dic3—62no—
32C2×#625—26no—
33C2×#625C22×C3112yescomplemented
34C2×#622C22×C316yescomplemented
35#3774C22×C3116yesdirect product factor
36C22×C33D814yescomplemented
37C22×C33C2×D413yes—
38C2×C32×D3—28no—
39C2×C32×D3—424no—
40C3×C3⋊Dic3C2×D413yes—
41C3×C3⋊Dic3C2411yescharacteristic
42C22×C3×D4—92no—
43C22×#67—62no—
44C2×C3×D8—912no—
45C22×C3×C8—91no—
46C23×C32—22no—
47C23×C32—42no—
48C23×C32—612no—
49C23×C32—128no—
50C23×C32C3×D411yescharacteristic
51C22×C3×D3—212no—
52C22×C3×D3—212no—
53C22×C3×D3—412no—
54C22×C3×D3—48no—
55C22×C3×D3—416no—
56C22×C3×D3—48no—
57C2×C3⋊Dic3C3×D413yes—
58C2×C3⋊Dic3C23×C313yes—
59C3×#67—68no—
60C3×#67—624no—
61C2×C3×Dic3—66no—
62C2×C3×Dic3—66no—
63#625—212no—
64#625C23×C318yescomplemented
65#622C23×C314yescomplemented
66C22×D8—91noSylow
67C2×C33C2×D816yescomplemented
68C2×C33C22×D411yescharacteristic
69C32×D3—48no—
70C24×C3—62no—
71C24×C3—62no—
72C24×C3—122no—
73C23×D3—22no—
74C2×C3×D4—912no—
75C2×C3×D4—186no—
76C22×C3×C4—91no—
77C2×#67—612no—
78C2×#67—612no—
79C22×Dic3—62no—
80C3×D8—916no—
81C2×C3×C8—96no—
82C22×C32—26no—
83C22×C32—28no—
84C22×C32—48no—
85C22×C32—46no—
86C22×C32—68no—
87C22×C32—1224no—
88C22×C32C3×D814yescomplemented
89C22×C32C2×C3×D413yes—
90C2×C3×D3—28no—
91C2×C3×D3—28no—
92C2×C3×D3—48no—
93C2×C3×D3—424no—
94C2×C3×D3—448no—
95C2×C3×D3—424no—
96C3⋊Dic3C2×C3×D413yes—
97C3⋊Dic3C24×C311yescharacteristic
98C3×Dic3—66no—
99C3×Dic3—62no—
100C22×D4—92no—
101C2×D8—912no—
102C22×C8—91no—
103C33C22×D811yescharacteristic, complemented, Sylow
104C23×C3—22no—
105C23×C3—42no—
106C23×C3—612no—
107C23×C3—612no—
108C23×C3—1212no—
109C23×C3—128no—
110C23×C3—1216no—
111C23×C3—128no—
112C23×C3GO+(4,𝔽2)11yescharacteristic
113C22×D3—212no—
114C22×D3—48no—
115C3×D4—98no—
116C3×D4—1812no—
117C2×C3×C4—93no—
118C2×C3×C4—93no—
119#67—68no—
120#67—624no—
121C2×Dic3—66no—
122C2×Dic3—66no—
123C3×C8—94no—
124C2×C32—22no—
125C2×C32—212no—
126C2×C32—412no—
127C2×C32—42no—
128C2×C32—128no—
129C2×C32C2×C3×D816yescomplemented
130C2×C32C22×C3×D411yescharacteristic
131C3×D3—48no—
132C3×D3—416no—
133C3×D3—48no—
134C24—62no—
135C2×D4—912no—
136C2×D4—186no—
137C22×C4—91no—
138D8—916no—
139C2×C8—96no—
140C22×C3—26no—
141C22×C3—28no—
142C22×C3—46no—
143C22×C3—48no—
144C22×C3—68no—
145C22×C3—68no—
146C22×C3—128no—
147C22×C3—1224no—
148C22×C3—1248no—
149C22×C3—1224no—
150C22×C3#377414yesdirect product factor
151C22×C3C2×GO+(4,𝔽2)13yes—
152C2×D3—28no—
153C2×D3—424no—
154C3×C4—91no—
155C3×C4—93no—
156Dic3—66no—
157Dic3—62no—
158C32—22no—
159C32—42no—
160C32C22×C3×D811yescharacteristic, complemented
161C23—612no—
162C23—128no—
163C23C3×GO+(4,𝔽2)11yescharacteristic
164D4—98no—
165D4—1812no—
166C2×C4—93no—
167C2×C4—93no—
168C8—94no—
169C2×C3—212no—
170C2×C3—22no—
171C2×C3—412no—
172C2×C3—42no—
173C2×C3—128no—
174C2×C3—1216no—
175C2×C3—128no—
176C2×C3C2×#377416yesdirect product factor
177C2×C3C22×GO+(4,𝔽2)11yescharacteristic
178D3—48no—
179C22—68no—
180C22—1224no—
181C22C3×#377414yesdirect product factor
182C22C2×C3×GO+(4,𝔽2)13yes—
183C4—91no—
184C4—93no—
185C3—22no—
186C3—42no—
187C3C22×#377411yescharacteristic, direct product factor
188C2—128no—
189C2C2×C3×#377416yesdirect product factor
190C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
Fitting, socleC23×C33D4
hypocenterC32C22×C3×D8
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#625C2semidirect
2. C22×C3×#622C2semidirect
3. C2×C3×#3774C2direct (N × H)
4. C22×#3774C3direct (N × H)
8. C2×C3×#625C22semidirect
9. C2×C3×#622C22semidirect
10. C3×#3774C22direct (N × H)
12. C22×#625C2×C3semidirect
13. C22×#622C2×C3semidirect
14. C2×#3774C2×C3direct (N × H)
21. C3×#625C23semidirect
22. C3×#622C23semidirect
33. C2×#625C22×C3semidirect
34. C2×#622C22×C3semidirect
35. #3774C22×C3direct (N × H)
36. C22×C33D8semidirect
64. #625C23×C3semidirect
65. #622C23×C3semidirect
67. C2×C33C2×D8semidirect
88. C22×C32C3×D8semidirect
103. C33C22×D8semidirect
129. C2×C32C2×C3×D8semidirect
150. C22×C3#3774direct (N × H)
160. C32C22×C3×D8semidirect
176. C2×C3C2×#3774direct (N × H)
181. C22C3×#3774direct (N × H)
187. C3C22×#3774direct (N × H)
189. C2C2×C3×#3774direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C22×C3×#6222
C2×C3×#37742
C22×#37743
C22×C3×D89