Finite Group Databasebeta

← C3×#10590202 #13157126

Subgroups — C3×#10590202 #13157126

Order 1728.

448 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 60 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×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C3×C22⋊#622C212yesmaximal
3#10590202C311yesmaximal, characteristic, direct product factor
4C22×C33×C4C411yescharacteristic
5C22×C3×C3⋊Dic3C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C3×#622—22no—
8C3×C4⋊(C3⋊Dic3)—22no—
9C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
10C2×C3×C4⋊Dic3—62no—
11C22⋊#622C2×C312yes—
12C23×C33C2×C411yescharacteristic
13C2×C33×C4—21no—
14C2×C33×C4—21no—
15C2×C33×C4C812yescomplemented
16C2×C3×C3⋊Dic3—21no—
17C2×C3×C3⋊Dic3—41no—
18C2×C3×C3⋊Dic3—41no—
19C2×C3×C3⋊Dic3D412yes—
20C3×#622—42no—
21C3×#324—91nomaximal
22C22×C32×C4—22no—
23C22×C32×C4—42no—
24C22×C32×C4C3×C411yescharacteristic
25C22×C3⋊Dic3C3×C411yescharacteristic
26C22×C3⋊Dic3C22×C311yescharacteristic
27C22×C3×Dic3—62no—
28C22×C3×Dic3—62no—
29C2×#622—22no—
30C4⋊(C3⋊Dic3)—22no—
31C3×C4⋊Dic3—68no—
32C22×C33—21no—
33C22×C33—21no—
34C22×C33C22⋊C411yescharacteristic
35C22×C33C2×C811yescharacteristic
36C22×C33M1611yescharacteristic
37C33×C4—22no—
38C3×C3⋊Dic3—22no—
39C3×C3⋊Dic3—41no—
40C2×C3×C4⋊C4—91no—
41C2×C4⋊Dic3—62no—
42C3×C22⋊C8—92no—
43C23×C32—22no—
44C23×C32—42no—
45C23×C32C2×C3×C411yescharacteristic
46C2×C32×C4—21no—
47C2×C32×C4—21no—
48C2×C32×C4—24no—
49C2×C32×C4—24no—
50C2×C32×C4—24no—
51C2×C32×C4—44no—
52C2×C32×C4—44no—
53C2×C32×C4—44no—
54C2×C32×C4C3×C812yescomplemented
55C2×C3⋊Dic3—21no—
56C2×C3⋊Dic3—41no—
57C2×C3⋊Dic3—41no—
58C2×C3⋊Dic3C3×D412yes—
59C2×C3×Dic3—64no—
60C2×C3×Dic3—64no—
61C2×C3×Dic3—124no—
62C2×C3×Dic3—124no—
63#622—42no—
64#324—91noSylow
65C2×C33—21no—
66C2×C33—21no—
67C2×C33C22⋊C811yescharacteristic
68C2×C33#9911yescharacteristic
69C2×C33#10111yescharacteristic
70C22×C3×C4—22no—
71C22×C3×C4—42no—
72C22×C3×C4—91no—
73C22×C3×C4—91no—
74C22×C3×C4#15711yescharacteristic
75C22×Dic3—62no—
76C22×Dic3—62no—
77C2×C3×C8—182no—
78C3×C4⋊C4—182no—
79C4⋊Dic3—68no—
80C22×C32—21no—
81C22×C32—21no—
82C22×C32—24no—
83C22×C32—24no—
84C22×C32—22no—
85C22×C32—22no—
86C22×C32—22no—
87C22×C32—44no—
88C22×C32—44no—
89C22×C32—42no—
90C22×C32—42no—
91C22×C32—42no—
92C22×C32C3×C22⋊C411yescharacteristic
93C22×C32C2×C3×C811yescharacteristic
94C22×C32C3×M1611yescharacteristic
95C32×C4—22no—
96C32×C4—28no—
97C32×C4—48no—
98C3⋊Dic3—22no—
99C3⋊Dic3—41no—
100C3×Dic3—124no—
101C3×Dic3—124no—
102C2×C4⋊C4—91no—
103C22⋊C8—92no—
104C33#32411yescharacteristic, complemented, Sylow
105C23×C3—22no—
106C23×C3—42no—
107C23×C3C2×#15711yescharacteristic
108C2×C3×C4—24no—
109C2×C3×C4—24no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—24no—
113C2×C3×C4—44no—
114C2×C3×C4—44no—
115C2×C3×C4—44no—
116C2×C3×C4—92no—
117C2×C3×C4—181no—
118C2×C3×C4—361no—
119C2×C3×C4—361no—
120C2×C3×C4#62212yescomplemented
121C2×Dic3—64no—
122C2×Dic3—64no—
123C2×Dic3—124no—
124C2×Dic3—124no—
125C3×C8—362no—
126C2×C32—21no—
127C2×C32—21no—
128C2×C32—24no—
129C2×C32—24no—
130C2×C32—22no—
131C2×C32—22no—
132C2×C32—22no—
133C2×C32—44no—
134C2×C32—44no—
135C2×C32—42no—
136C2×C32—42no—
137C2×C32—42no—
138C2×C32C3×C22⋊C811yescharacteristic
139C2×C32C3×#9911yescharacteristic
140C2×C32C3×#10111yescharacteristic
141C22×C4—91no—
142C22×C4—91no—
143C22×C4C3×#15711yescharacteristic
144C2×C8—182no—
145C4⋊C4—182no—
146C22×C3—24no—
147C22×C3—24no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—22no—
151C22×C3—22no—
152C22×C3—22no—
153C22×C3—44no—
154C22×C3—44no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—42no—
158C22×C3C2×#62211yescharacteristic
159C22×C3#379211yescharacteristic
160C22×C3C22⋊#15711yescharacteristic
161C3×C4—28no—
162C3×C4—22no—
163C3×C4—48no—
164C3×C4—182no—
165C3×C4—361no—
166Dic3—124no—
167Dic3—124no—
168C32—22no—
169C32—42no—
170C32C3×#32411yescharacteristic, complemented
171C23C2×C3×#15711yescharacteristic
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—92no—
175C2×C4—181no—
176C2×C4—361no—
177C2×C4—361no—
178C2×C4C3×#62212yescomplemented
179C8—362no—
180C2×C3—24no—
181C2×C3—24no—
182C2×C3—21no—
183C2×C3—21no—
184C2×C3—22no—
185C2×C3—22no—
186C2×C3—22no—
187C2×C3—44no—
188C2×C3—44no—
189C2×C3—42no—
190C2×C3—42no—
191C2×C3—42no—
192C2×C3#6377411yescharacteristic
193C2×C3#6378011yescharacteristic
194C2×C3C22⋊#62211yescharacteristic
195C22—21no—
196C22—21no—
197C22C2×C3×#62211yescharacteristic
198C22C3×#379211yescharacteristic
199C22C3×C22⋊#15711yescharacteristic
200C4—22no—
201C4—182no—
202C4—361no—
203C3—22no—
204C3—42no—
205C3#1059020211yescharacteristic, direct product factor
206C2—21no—
207C2—21no—
208C2C3×#6377411yescharacteristic
209C2C3×#6378011yescharacteristic
210C2C3×C22⋊#62211yescharacteristic
subgroupG/H
centerC22×C3C22⋊#157
commutatorC22×C32C2×C3×C8
FrattiniC23C2×C3×#157
FittingC22×C33×C4C4
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
cosocleC22×C3⋊Dic3C22×C3
hypercenterC22×C3×C4#157
hypocenterC32C3×#324
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. #10590202C3direct (N × H)
15. C2×C33×C4C8semidirect
54. C2×C32×C4C3×C8semidirect
104. C33#324semidirect
120. C2×C3×C4#622semidirect
170. C32C3×#324semidirect
178. C2×C4C3×#622semidirect
205. C3#10590202direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C3×C22⋊#6222
#105902023
C3×#3249