Finite Group Databasebeta

← C22×C3×#3792 #13181506

Subgroups — C22×C3×#3792 #13181506

Order 1728.

1336 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 312 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
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3792C2112yesmaximal, direct product factor
3C22×C3×#622C212yesmaximal, complemented
4C22×#3792C311yesmaximal, characteristic, direct product factor
5C24×C33C411yescharacteristic
6C22×C3×C3⋊Dic3C416yes—
7C22×C3×C3⋊Dic3C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2216yes—
10C3×#3792C22116yesdirect product factor
11C2×C3×#622C22112yescomplemented
12C23×C3⋊Dic3C2×C311yescharacteristic
13C23×C3×Dic3—62no—
14C2×#3792C2×C3112yesdirect product factor
15C22×#622C2×C312yescomplemented
16C23×C33—24no—
17C23×C33C2×C416yes—
18C23×C33C2×C411yescharacteristic
19C2×C3×C3⋊Dic3C2×C4112yes—
20C2×C3×C3⋊Dic3C2×C413yes—
21C2×C3×C3⋊Dic3C2×C416yes—
22C2×C3×C3⋊Dic3C2313yes—
23C2×C3×C3⋊Dic3C2314yes—
24C3×#622C2318yescomplemented
25C22×C3×M16—91nomaximal
26C24×C32—22no—
27C24×C32—42no—
28C24×C32C3×C411yescharacteristic
29C22×C3⋊Dic3C3×C416yes—
30C22×C3⋊Dic3C3×C411yescharacteristic
31C22×C3⋊Dic3C22×C311yescharacteristic
32C22×C3⋊Dic3C22×C316yes—
33C22×C3×Dic3—612no—
34C22×C3×Dic3—62no—
35C22×C3×Dic3—612no—
36C22×C3×Dic3—62no—
37#3792C22×C3116yesdirect product factor
38C2×#622C22×C3112yescomplemented
39C22×C33—212no—
40C22×C33M1614yescomplemented
41C22×C33C22×C414yes—
42C22×C33C22×C413yes—
43C3×C3⋊Dic3C22×C413yes—
44C3×C3⋊Dic3C22×C414yes—
45C3×C3⋊Dic3C2411yescharacteristic
46C23×C3×C4—91no—
47C23×Dic3—62no—
48C2×C3×M16—912no—
49C22×C3×C8—92no—
50C23×C32—24no—
51C23×C32—216no—
52C23×C32—212no—
53C23×C32—22no—
54C23×C32—416no—
55C23×C32—412no—
56C23×C32—42no—
57C23×C32C2×C3×C416yes—
58C23×C32C2×C3×C411yescharacteristic
59C2×C3⋊Dic3C2×C3×C4112yes—
60C2×C3⋊Dic3C2×C3×C413yes—
61C2×C3⋊Dic3C2×C3×C416yes—
62C2×C3⋊Dic3C23×C313yes—
63C2×C3⋊Dic3C23×C314yes—
64C2×C3×Dic3—624no—
65C2×C3×Dic3—612no—
66C2×C3×Dic3—66no—
67C2×C3×Dic3—66no—
68C2×C3×Dic3—68no—
69#622C23×C318yescomplemented
70C22×M16—91noSylow
71C2×C33—24no—
72C2×C33C2×M1616yescomplemented
73C2×C33C23×C411yescharacteristic
74C24×C3—22no—
75C24×C3—42no—
76C24×C3#15711yescharacteristic
77C22×C3×C4—96no—
78C22×C3×C4—96no—
79C22×C3×C4—91no—
80C22×C3×C4—91no—
81C22×Dic3—612no—
82C22×Dic3—62no—
83C22×Dic3—62no—
84C22×Dic3—612no—
85C3×M16—916no—
86C2×C3×C8—912no—
87C22×C32—212no—
88C22×C32—248no—
89C22×C32—28no—
90C22×C32—26no—
91C22×C32—28no—
92C22×C32—448no—
93C22×C32—48no—
94C22×C32—48no—
95C22×C32—46no—
96C22×C32C3×M1614yescomplemented
97C22×C32C22×C3×C414yes—
98C22×C32C22×C3×C413yes—
99C3⋊Dic3C22×C3×C413yes—
100C3⋊Dic3C22×C3×C414yes—
101C3⋊Dic3C24×C311yescharacteristic
102C3×Dic3—66no—
103C3×Dic3—68no—
104C3×Dic3—62no—
105C23×C4—91no—
106C2×M16—912no—
107C22×C8—92no—
108C33C22×M1611yescharacteristic, complemented, Sylow
109C23×C3—216no—
110C23×C3—24no—
111C23×C3—212no—
112C23×C3—22no—
113C23×C3—416no—
114C23×C3—412no—
115C23×C3—42no—
116C23×C3C2×#15716yes—
117C23×C3C2×#15711yescharacteristic
118C2×C3×C4—93no—
119C2×C3×C4—94no—
120C2×C3×C4—912no—
121C2×C3×C4—96no—
122C2×C3×C4—93no—
123C2×Dic3—624no—
124C2×Dic3—66no—
125C2×Dic3—612no—
126C2×Dic3—66no—
127C2×Dic3—68no—
128C3×C8—98no—
129C2×C32—24no—
130C2×C32—216no—
131C2×C32—212no—
132C2×C32—22no—
133C2×C32—416no—
134C2×C32—412no—
135C2×C32—42no—
136C2×C32C2×C3×M1616yescomplemented
137C2×C32C23×C3×C411yescharacteristic
138C24C3×#15711yescharacteristic
139C22×C4—91no—
140C22×C4—96no—
141C22×C4—96no—
142C22×C4—91no—
143M16—916no—
144C2×C8—912no—
145C22×C3—248no—
146C22×C3—212no—
147C22×C3—28no—
148C22×C3—26no—
149C22×C3—28no—
150C22×C3—448no—
151C22×C3—48no—
152C22×C3—46no—
153C22×C3—48no—
154C22×C3#379214yesdirect product factor
155C22×C3C22×#15714yes—
156C22×C3C22×#15713yes—
157C3×C4—91no—
158C3×C4—93no—
159C3×C4—94no—
160Dic3—66no—
161Dic3—68no—
162Dic3—62no—
163C32—22no—
164C32—42no—
165C32C22×C3×M1611yescharacteristic, complemented
166C23—24no—
167C23C2×C3×#15716yes—
168C23C2×C3×#15711yescharacteristic
169C2×C4—93no—
170C2×C4—94no—
171C2×C4—912no—
172C2×C4—93no—
173C2×C4—96no—
174C8—98no—
175C2×C3—216no—
176C2×C3—24no—
177C2×C3—22no—
178C2×C3—212no—
179C2×C3—416no—
180C2×C3—42no—
181C2×C3—412no—
182C2×C3C2×#379216yesdirect product factor
183C2×C3C23×#15711yescharacteristic
184C22—212no—
185C22C3×#379214yesdirect product factor
186C22C22×C3×#15714yes—
187C22C22×C3×#15713yes—
188C4—91no—
189C4—93no—
190C4—94no—
191C3—22no—
192C3—42no—
193C3C22×#379211yescharacteristic, direct product factor
194C2—24no—
195C2C2×C3×#379216yesdirect product factor
196C2C23×C3×#15711yescharacteristic
subgroupG/H
centerC23×C3C2×#157
commutatorC2×C32C23×C3×C4
FrattiniC2C23×C3×#157
Fitting, prime radicalC24×C33C4
socleC23×C33C2×C4
cosocleC3⋊Dic3C24×C3
hypercenterC24×C3#157
hypocenterC32C22×C3×M16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3792C2direct (N × H)
3. C22×C3×#622C2semidirect
4. C22×#3792C3direct (N × H)
10. C3×#3792C22direct (N × H)
11. C2×C3×#622C22semidirect
14. C2×#3792C2×C3direct (N × H)
15. C22×#622C2×C3semidirect
24. C3×#622C23semidirect
37. #3792C22×C3direct (N × H)
38. C2×#622C22×C3semidirect
40. C22×C33M16semidirect
69. #622C23×C3semidirect
72. C2×C33C2×M16semidirect
96. C22×C32C3×M16semidirect
108. C33C22×M16semidirect
136. C2×C32C2×C3×M16semidirect
154. C22×C3#3792direct (N × H)
165. C32C22×C3×M16semidirect
182. C2×C3C2×#3792direct (N × H)
185. C22C3×#3792direct (N × H)
193. C3C22×#3792direct (N × H)
195. C2C2×C3×#3792direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×C3×#37922
C22×C3×#6222
C22×#37923
C22×C3×M169