Finite Group Databasebeta

← #13180648

Subgroups — #13180648

Order 1728.

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

normal subgroups only

This group has 334 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
1C4×A4×C3⋊D3C211yesmaximal, characteristic
2A4⋊#622C212yesmaximal, complemented
3#10593800—31nomaximal
4C2×A4×C3⋊D3C411yescharacteristic
5C32×C4×A4C411yescharacteristic
6#88988—41nomaximal
7A4×C3⋊Dic3C2211yescharacteristic
8C22×C4×C3⋊D3D311yescharacteristic
9C2×#3787—31no—
10C2×#3787—31no—
11C4×D3×A4—62no—
12C22⋊#622—32no—
13#63778—32no—
14C2×C32×A4C2×C411yescharacteristic
15A4×C3⋊D3C812yescomplemented
16C3×C4×C3⋊D3—41no—
17C3⋊#622—42no—
18C2×A4⋊C8—91nomaximal
19C23×C3⋊D3Dic311yescharacteristic
20C2×D3×A4—62no—
21C22×C32×C4Dic311yescharacteristic
22C22×C3⋊Dic3C2×D311yescharacteristic
23C2×C4×C3⋊D3—31no—
24C2×C4×C3⋊D3—31no—
25C2×C4×C3⋊D3—31no—
26C2×C4×C3⋊D3—31no—
27C3×C4×A4—22no—
28C3×C4×A4—42no—
29C2×#622—32no—
30C2×#622—32no—
31#3787—61no—
32#3787—61no—
33#3787—61no—
34#3787—61no—
35Dic3×A4—62no—
36C2×C3×C3⋊D3—41no—
37C32×A4C2×C811yescharacteristic, complemented
38C33×C4—41no—
39C3×C3⋊Dic3—41no—
40C22×C4×D3—62no—
41C2×C4×A4—91no—
42A4⋊C8—92no—
43C23×C32C2×Dic311yescharacteristic
44C22×C3⋊D3—31no—
45C22×C3⋊D3—32no—
46C22×C3⋊D3—31no—
47C22×C3⋊D3C3⋊C812yescomplemented
48C2×C3×A4—22no—
49C2×C3×A4—42no—
50D3×A4—64no—
51C2×C32×C4—31no—
52C2×C32×C4—31no—
53C2×C3⋊Dic3—31no—
54C2×C3⋊Dic3—31no—
55C4×C3⋊D3—31no—
56C4×C3⋊D3—31no—
57C4×C3⋊D3—31no—
58C4×C3⋊D3—61no—
59C4×C3⋊D3S411yescharacteristic
60C3×C4×D3—242no—
61#622—62no—
62#622—62no—
63C2×C22⋊C8—271noSylow
64C2×C33—41no—
65C3×C3⋊D3—42no—
66C23×D3—62no—
67C22×A4—91no—
68C22×C3×C4—22no—
69C22×Dic3—62no—
70C2×C4×D3—182no—
71C2×C4×D3—182no—
72C2×C4×D3—182no—
73C2×C4×D3—182no—
74C4×A4—42no—
75C4×A4—91no—
76C4×A4#15711yescharacteristic
77C2×C3⋊C8—361no—
78C22×C32—31no—
79C22×C32—31no—
80C22×C32C2×C3⋊C811yescharacteristic, complemented
81C2×C3⋊D3—31no—
82C2×C3⋊D3—32no—
83C2×C3⋊D3—32no—
84C2×C3⋊D3—32no—
85C2×C3⋊D3—61no—
86C2×C3⋊D3SL(2,ℤ/4)11yescharacteristic
87C2×C3×D3—242no—
88C3×A4—22no—
89C3×A4—42no—
90C32×C4—31no—
91C32×C4—82no—
92C32×C4—162no—
93C32×C4SL(2,ℤ/4)11yescharacteristic
94C3⋊Dic3—31no—
95C3⋊Dic3C2×S411yescharacteristic
96C3×Dic3—242no—
97C23×C4—91no—
98C22×C8—271no—
99C22×C8—271no—
100C22⋊C8—272no—
101C22⋊C8—272no—
102C33—41noSylow
103C23×C3—22no—
104C22×D3—64no—
105C22×D3—182no—
106C22×D3—182no—
107C22×D3—184no—
108C2×A4—42no—
109C2×A4—92no—
110C2×A4C2×#15711yescharacteristic
111C2×C3×C4—62no—
112C2×C3×C4—62no—
113C2×C3×C4—361no—
114C2×Dic3—182no—
115C2×Dic3—182no—
116C4×D3—62no—
117C4×D3—182no—
118C4×D3—182no—
119C4×D3—182no—
120C4×D3—184no—
121C3⋊C8—362no—
122C2×C32—31no—
123C2×C32—31no—
124C2×C32—82no—
125C2×C32—162no—
126C2×C32C2×SL(2,ℤ/4)11yescharacteristic
127C3⋊D3—32no—
128C3⋊D3A4⋊C812yescomplemented
129C3×D3—244no—
130C24—91no—
131C22×C4—91no—
132C22×C4—271no—
133C22×C4—271no—
134C22×C4—271no—
135C22×C4—271no—
136C22×C4C3⋊#15711yescharacteristic
137C2×C8—272no—
138C2×C8—272no—
139C2×C8—541no—
140C2×C8—541no—
141C2×C8—541no—
142C2×C8—541no—
143C22×C3—22no—
144C22×C3—62no—
145C22×C3—62no—
146C22×C3—361no—
147C2×D3—62no—
148C2×D3—184no—
149C2×D3—184no—
150C2×D3—182no—
151C2×D3—184no—
152C2×D3—184no—
153A4—42no—
154A4#378711yescharacteristic, complemented
155C3×C4—22no—
156C3×C4—41no—
157C3×C4—62no—
158C3×C4—162no—
159C3×C4—361no—
160Dic3—62no—
161Dic3—182no—
162C32—82no—
163C32—162no—
164C32C2×A4⋊C811yescharacteristic, complemented
165C23—92no—
166C23—271no—
167C23—271no—
168C23—272no—
169C23C2×C3⋊#15711yescharacteristic
170C2×C4—31no—
171C2×C4—31no—
172C2×C4—91no—
173C2×C4—271no—
174C2×C4—271no—
175C2×C4—271no—
176C2×C4—271no—
177C2×C4—271no—
178C2×C4—541no—
179C8—542no—
180C8—542no—
181C2×C3—22no—
182C2×C3—41no—
183C2×C3—62no—
184C2×C3—62no—
185C2×C3—162no—
186C2×C3—362no—
187D3—64no—
188D3—184no—
189C22—31no—
190C22—31no—
191C22—91no—
192C22—272no—
193C22—272no—
194C22—272no—
195C22—271no—
196C22—541no—
197C22#8898811yescharacteristic, complemented
198C4—31no—
199C4—91no—
200C4—271no—
201C4A4⋊#15711yescharacteristic
202C3—22no—
203C3—41no—
204C3—162no—
205C2—31no—
206C2—31no—
207C2—92no—
208C2—272no—
209C2C2×A4⋊#15711yescharacteristic
subgroupG/H
center, hypercenterC4A4⋊#157
commutator, hypocenterC32×A4C2×C8
FrattiniC2C2×A4⋊#157
FittingC22×C32×C4Dic3
socleC23×C32C2×Dic3
prime radicalC2×A4×C3⋊D3C4
cosocleA4×C3⋊Dic3C22
LCSG ▷ C32×A4
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. A4⋊#622C2semidirect
15. A4×C3⋊D3C8semidirect
37. C32×A4C2×C8semidirect
47. C22×C3⋊D3C3⋊C8semidirect
80. C22×C32C2×C3⋊C8semidirect
128. C3⋊D3A4⋊C8semidirect
154. A4#3787semidirect
164. C32C2×A4⋊C8semidirect
197. C22#88988semidirect

Maximal subgroups

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

subgroupindex
C4×A4×C3⋊D32
A4⋊#6222
#105938003
#889884
C2×A4⋊C89