Finite Group Databasebeta

← A4×#3788 #13180664

Subgroups — A4×#3788 #13180664

Order 1728.

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

normal subgroups only

This group has 322 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
3C22×#3788C311yesmaximal, characteristic, complemented
4C2×A4×C3⋊D3C411yescharacteristic
5C32×C4×A4C411yescharacteristic
6C3×#3788—41nomaximal
7A4×C3⋊Dic3C2211yescharacteristic
8C22×C4×C3⋊D3C2×C311yescharacteristic
9C22×#622C2×C312yescomplemented
10C2×#3788—31no—
11C2×#3788—31no—
12C2×#3788—32no—
13C4×D3×A4—62no—
14C2×C32×A4C2×C411yescharacteristic
15A4×C3⋊D3—21no—
16C3×C4×C3⋊D3—41no—
17C3×#622—42no—
18A4×M16—91nomaximal
19C23×C3⋊D3C3×C411yescharacteristic
20C2×D3×A4—62no—
21C22×C32×C4C3×C411yescharacteristic
22C22×C3⋊Dic3C22×C311yescharacteristic
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#3788—32no—
32#3788—31no—
33#3788—32no—
34#3788A411yescharacteristic, direct product factor
35Dic3×A4—62no—
36C2×C3×C3⋊D3—41no—
37C32×A4M1611yescharacteristic, complemented
38C33×C4—41no—
39C3×C3⋊Dic3—41no—
40C22×C4×D3—62no—
41C2×C4×A4—91no—
42C8×A4—92no—
43C23×C32C2×C3×C411yescharacteristic
44C22×C3⋊D3—21no—
45C22×C3⋊D3—31no—
46C22×C3⋊D3—31no—
47C22×C3⋊D3—61no—
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—32no—
56C4×C3⋊D3—31no—
57C4×C3⋊D3—31no—
58C4×C3⋊D3—31no—
59C4×C3⋊D3C2×A411yescharacteristic
60C3×C4×D3—242no—
61#622—32no—
62#622C2×A412yescomplemented
63C22×M16—91noSylow
64C2×C33—41no—
65C3×C3⋊D3—81no—
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
77C3×M16—361no—
78C22×C32—31no—
79C22×C32—31no—
80C22×C32C3×M1611yescharacteristic, complemented
81C2×C3⋊D3—31no—
82C2×C3⋊D3—61no—
83C2×C3⋊D3—61no—
84C2×C3⋊D3—61no—
85C2×C3⋊D3—61no—
86C2×C3⋊D3C4×A411yescharacteristic
87C2×C3×D3—242no—
88C3×A4—22no—
89C3×A4—42no—
90C32×C4—31no—
91C32×C4—82no—
92C32×C4—162no—
93C32×C4C4×A411yescharacteristic
94C3⋊Dic3—31no—
95C3⋊Dic3C22×A411yescharacteristic
96C3×Dic3—242no—
97C23×C4—91no—
98C2×M16—271no—
99C2×M16—272no—
100C2×M16—271no—
101C22×C8—92no—
102C33—41noSylow
103C23×C3—22no—
104C22×D3—64no—
105C22×D3—182no—
106C22×D3—182no—
107C22×D3—184no—
108C2×A4—42no—
109C2×A4—181no—
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×C4×A411yescharacteristic
127C3⋊D3—21no—
128C3⋊D3—61no—
129C3×D3—244no—
130C24—91no—
131C22×C4—91no—
132C22×C4—271no—
133C22×C4—271no—
134C22×C4—271no—
135C22×C4—271no—
136C22×C4C3×#15711yescharacteristic
137M16—91no—
138M16—272no—
139M16—271no—
140M16—272no—
141C2×C8—272no—
142C2×C8—272no—
143C22×C3—22no—
144C22×C3—62no—
145C22×C3—62no—
146C22×C3—361no—
147C2×D3—62no—
148C2×D3—182no—
149C2×D3—184no—
150C2×D3—184no—
151C2×D3—184no—
152C2×D3—184no—
153A4—42no—
154A4#378811yescharacteristic, direct product factor
155C3×C4—22no—
156C3×C4—41no—
157C3×C4—62no—
158C3×C4—162no—
159C3×C4—361no—
160Dic3—62no—
161Dic3—182no—
162C32—82no—
163C32—162no—
164C32A4×M1611yescharacteristic, complemented
165C23—181no—
166C23—271no—
167C23—271no—
168C23—541no—
169C23C2×C3×#15711yescharacteristic
170C2×C4—31no—
171C2×C4—31no—
172C2×C4—91no—
173C2×C4—271no—
174C2×C4—271no—
175C2×C4—272no—
176C2×C4—271no—
177C2×C4—271no—
178C2×C4—271no—
179C8—92no—
180C8—272no—
181C2×C3—22no—
182C2×C3—41no—
183C2×C3—62no—
184C2×C3—62no—
185C2×C3—162no—
186C2×C3—721no—
187D3—64no—
188D3—184no—
189C22—31no—
190C22—31no—
191C22—91no—
192C22—271no—
193C22—541no—
194C22—541no—
195C22—541no—
196C22—541no—
197C22C3×#378811yescharacteristic, complemented
198C4—31no—
199C4—91no—
200C4—271no—
201C4A4×#15711yescharacteristic
202C3—22no—
203C3—41no—
204C3—162no—
205C2—31no—
206C2—31no—
207C2—181no—
208C2—541no—
209C2C2×A4×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×#157
commutator, socleC23×C32C2×C3×C4
FittingC22×C32×C4C3×C4
prime radicalC2×A4×C3⋊D3C4
cosocleC22×C3⋊Dic3C22×C3
hypercenterC4A4×#157
hypocenterC22×C32C3×M16
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. A4×#622C2semidirect
3. C22×#3788C3semidirect
9. C22×#622C2×C3semidirect
34. #3788A4direct (N × H)
37. C32×A4M16semidirect
62. #622C2×A4semidirect
80. C22×C32C3×M16semidirect
154. A4#3788direct (N × H)
164. C32A4×M16semidirect
197. C22C3×#3788semidirect

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
C22×#37883
C3×#37884
A4×M169