Finite Group Databasebeta

← #13141187

Subgroups — #13141187

Order 1728.

344 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 36 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×C9⋊(C22⋊A4)C211yesmaximal, characteristic
2C3×#379⋊C3C311yesmaximal, characteristic
3C9×#379C311yesmaximal, characteristic, complemented
4#379⋊C9C312yesmaximal, complemented
5C9⋊(C22⋊A4)C411yescharacteristic, complemented
6C4×C9⋊A4—41nomaximal
7C2×C3×C22⋊A4C2×C311yescharacteristic
8C25×C9C2×C311yescharacteristic
9C2×C22⋊(C22⋊C9)C2×C312yes—
10C2×C9×C22⋊C4—31no—
11C2×C9×C22⋊C4—31no—
12C2×C9⋊A4—41no—
13C2×C9⋊A4—82no—
14C3×#379C3211yescharacteristic
15#379⋊C3—31no—
16C3×C22⋊A4C3×C411yescharacteristic
17C3×C4×A4—41no—
18C24×C9—31no—
19C24×C9—31no—
20C24×C9—62no—
21C24×C9—62no—
22C24×C9C3×C411yescharacteristic, complemented
23C22⋊(C22⋊C9)C3×C412yescomplemented
24C22×C4×C9—31no—
25C22×C4×C9A411yescharacteristic, complemented
26C4×C22⋊C9—42no—
27C9×C22⋊C4—62no—
28C9×C22⋊C4—62no—
29C9⋊A4—41no—
30C9⋊A4—82no—
31C4×3−1+2—161no—
32C25×C3C2×C3211yescharacteristic
33C2×C22⋊A4—31no—
34C2×C3×C22⋊C4—31no—
35C2×C3×C22⋊C4—31no—
36C2×C3×A4—41no—
37C2×C3×A4—82no—
38C23×C9—22no—
39C23×C9—32no—
40C23×C9—32no—
41C23×C9—31no—
42C23×C9—31no—
43C23×C9—62no—
44C23×C9—62no—
45C23×C9—62no—
46C23×C9—62no—
47C23×C9—62no—
48C23×C9—64no—
49C23×C9—62no—
50C23×C9—62no—
51C23×C9—64no—
52C23×C9C2×A411yescharacteristic
53C2×C22⋊C9—42no—
54C2×C22⋊C9—84no—
55C2×C4×C9—61no—
56C2×C4×C9—62no—
57C2×C4×C9—61no—
58#3793−1+211yescharacteristic, complemented, Sylow
59C2×3−1+2—161no—
60C24×C3—31no—
61C24×C3—31no—
62C24×C3—62no—
63C24×C3—62no—
64C24×C3C32×C411yescharacteristic
65C22⋊A4—31no—
66C22×C3×C4—31no—
67C22×C3×C4C3×A411yescharacteristic
68C4×A4—121no—
69C3×C22⋊C4—62no—
70C3×C22⋊C4—62no—
71C3×A4—41no—
72C3×A4—82no—
73C32×C4—161no—
74C22×C9—22no—
75C22×C9—31no—
76C22×C9—31no—
77C22×C9—32no—
78C22×C9—32no—
79C22×C9—64no—
80C22×C9—62no—
81C22×C9—62no—
82C22×C9—62no—
83C22×C9—62no—
84C22×C9—62no—
85C22×C9—62no—
86C22×C9—64no—
87C22×C9—62no—
88C22×C9C4×A411yescharacteristic
89C22⋊C9—42no—
90C22⋊C9—84no—
91C4×C9—41no—
92C4×C9—121no—
93C4×C9—162no—
94C25C2×3−1+211yescharacteristic
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
973−1+2—161noSylow
98C23×C3—22no—
99C23×C3—32no—
100C23×C3—32no—
101C23×C3—31no—
102C23×C3—31no—
103C23×C3—62no—
104C23×C3—62no—
105C23×C3—62no—
106C23×C3—62no—
107C23×C3—62no—
108C23×C3—64no—
109C23×C3—62no—
110C23×C3—62no—
111C23×C3—64no—
112C23×C3C2×C3×A411yescharacteristic
113C2×A4—121no—
114C2×A4—242no—
115C2×C3×C4—61no—
116C2×C3×C4—62no—
117C2×C3×C4—61no—
118C2×C32—161no—
119C2×C9—31no—
120C2×C9—31no—
121C2×C9—62no—
122C2×C9—62no—
123C2×C9—162no—
124C2×C9#120⋊C311yescharacteristic
125C24—31no—
126C24—31no—
127C24—62no—
128C24—62no—
129C24C4×3−1+211yescharacteristic, complemented
130C22×C4—31no—
131C22×C4C9⋊A411yescharacteristic, complemented
132C22⋊C4—62no—
133C22⋊C4—62no—
134C22×C3—22no—
135C22×C3—31no—
136C22×C3—31no—
137C22×C3—32no—
138C22×C3—32no—
139C22×C3—64no—
140C22×C3—62no—
141C22×C3—62no—
142C22×C3—62no—
143C22×C3—62no—
144C22×C3—62no—
145C22×C3—62no—
146C22×C3—64no—
147C22×C3—62no—
148C22×C3C3×C4×A411yescharacteristic
149A4—121no—
150A4—242no—
151C3×C4—41no—
152C3×C4—121no—
153C3×C4—481no—
154C32—161no—
155C9—162no—
156C9#379⋊C311yescharacteristic, complemented
157C23—22no—
158C23—32no—
159C23—32no—
160C23—31no—
161C23—31no—
162C23—62no—
163C23—62no—
164C23—62no—
165C23—62no—
166C23—62no—
167C23—64no—
168C23—62no—
169C23—62no—
170C23—64no—
171C23C2×C9⋊A411yescharacteristic
172C2×C4—61no—
173C2×C4—62no—
174C2×C4—61no—
175C2×C3—31no—
176C2×C3—31no—
177C2×C3—62no—
178C2×C3—62no—
179C2×C3—481no—
180C2×C3C3×#120⋊C311yescharacteristic
181C22—22no—
182C22—31no—
183C22—31no—
184C22—32no—
185C22—32no—
186C22—64no—
187C22—62no—
188C22—62no—
189C22—62no—
190C22—62no—
191C22—62no—
192C22—62no—
193C22—64no—
194C22—62no—
195C22C4×C9⋊A411yescharacteristic
196C4—41no—
197C4—121no—
198C3—481no—
199C3C3×#379⋊C311yescharacteristic
200C2—31no—
201C2—31no—
202C2—62no—
203C2—62no—
204C2#1172133011yescharacteristic
subgroupG/H
centerC2×C3C3×#120⋊C3
commutatorC24×C3C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC9×#379C3
prime radicalC2×C3×C22⋊A4C2×C3
cosocleC25×C3C2×C32
hypercenterC2×C9#120⋊C3
hypocenterC24C4×3−1+2
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C2×C3 ◁ C2×C9

Complements

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

NH ≅ G/Ntype
3. C9×#379C3semidirect
4. #379⋊C9C3semidirect
5. C9⋊(C22⋊A4)C4semidirect
22. C24×C9C3×C4semidirect
23. C22⋊(C22⋊C9)C3×C4semidirect
25. C22×C4×C9A4semidirect
58. #3793−1+2semidirect
129. C24C4×3−1+2semidirect
131. C22×C4C9⋊A4semidirect
156. C9#379⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C9⋊(C22⋊A4)2
C3×#379⋊C33
C9×#3793
#379⋊C93
C4×C9⋊A44