Finite Group Databasebeta

← A4×#3792 #13180676

Subgroups — A4×#3792 #13180676

Order 1728.

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

normal subgroups only

This group has 338 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×A4×C3⋊Dic3C211yesmaximal, characteristic
2A4×#622C212yesmaximal, complemented
3C22×#3792C311yesmaximal, characteristic, complemented
4C22×C32×A4C411yescharacteristic
5C3×#3792—41nomaximal
6A4×C3⋊Dic3C411yescharacteristic
7A4×C3⋊Dic3C2211yescharacteristic
8C23×C3⋊Dic3C2×C311yescharacteristic
9C2×#3792—31no—
10C2×#3792—32no—
11C2×#3792—31no—
12C22×#622C2×C312yescomplemented
13C2×Dic3×A4—62no—
14C2×C32×A4—21no—
15C2×C32×A4C2×C411yescharacteristic
16C2×C3×C3⋊Dic3—41no—
17C3×#622—42no—
18A4×M16—91nomaximal
19C24×C32C3×C411yescharacteristic
20C22×C3×A4—22no—
21C22×C3×A4—42no—
22C22×C3⋊Dic3—31no—
23C22×C3⋊Dic3—31no—
24C22×C3⋊Dic3—31no—
25C22×C3⋊Dic3—31no—
26C22×C3⋊Dic3C3×C411yescharacteristic
27C22×C3⋊Dic3C22×C311yescharacteristic
28#3792—32no—
29#3792—31no—
30#3792—32no—
31#3792A411yescharacteristic, direct product factor
32C2×#622—32no—
33C2×#622—32no—
34Dic3×A4—62no—
35Dic3×A4—62no—
36C22×C33—41no—
37C32×A4M1611yescharacteristic, complemented
38C3×C3⋊Dic3—41no—
39C3×C3⋊Dic3—41no—
40C23×Dic3—62no—
41C2×C4×A4—91no—
42C8×A4—92no—
43C23×C32—21no—
44C23×C32—31no—
45C23×C32—31no—
46C23×C32—61no—
47C23×C32C2×C3×C411yescharacteristic
48C2×C3×A4—24no—
49C2×C3×A4—22no—
50C2×C3×A4—44no—
51C2×C3×A4—42no—
52C2×C3⋊Dic3—32no—
53C2×C3⋊Dic3—31no—
54C2×C3⋊Dic3—31no—
55C2×C3⋊Dic3—31no—
56C2×C3⋊Dic3—31no—
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—31no—
59C2×C3⋊Dic3—31no—
60C2×C3⋊Dic3C2×A411yescharacteristic
61C2×C3×Dic3—242no—
62#622—32no—
63#622C2×A412yescomplemented
64C22×M16—91noSylow
65C2×C33—41no—
66C2×C33—81no—
67C24×C3—22no—
68C22×A4—42no—
69C22×A4#15711yescharacteristic
70C22×Dic3—62no—
71C22×Dic3—62no—
72C22×Dic3—182no—
73C22×Dic3—182no—
74C22×Dic3—182no—
75C22×Dic3—182no—
76C4×A4—91no—
77C4×A4—91no—
78C3×M16—361no—
79C22×C32—31no—
80C22×C32—31no—
81C22×C32—31no—
82C22×C32—61no—
83C22×C32—61no—
84C22×C32—61no—
85C22×C32—61no—
86C22×C32—82no—
87C22×C32—162no—
88C22×C32C3×M1611yescharacteristic, complemented
89C22×C32C4×A411yescharacteristic
90C3×A4—22no—
91C3×A4—42no—
92C3⋊Dic3—31no—
93C3⋊Dic3—31no—
94C3⋊Dic3C4×A411yescharacteristic
95C3⋊Dic3C22×A411yescharacteristic
96C3×Dic3—242no—
97C3×Dic3—242no—
98C23×C4—91no—
99C2×M16—271no—
100C2×M16—272no—
101C2×M16—271no—
102C22×C8—92no—
103C33—41noSylow
104C23×C3—24no—
105C23×C3—22no—
106C23×C3—62no—
107C23×C3—62no—
108C23×C3—64no—
109C2×A4—21no—
110C2×A4—44no—
111C2×A4—42no—
112C2×A4C2×#15711yescharacteristic
113C2×C3×C4—361no—
114C2×Dic3—62no—
115C2×Dic3—182no—
116C2×Dic3—182no—
117C2×Dic3—182no—
118C2×Dic3—182no—
119C2×Dic3—182no—
120C2×Dic3—184no—
121C2×Dic3—182no—
122C2×Dic3—182no—
123C3×C8—362no—
124C2×C32—21no—
125C2×C32—31no—
126C2×C32—31no—
127C2×C32—61no—
128C2×C32—82no—
129C2×C32—84no—
130C2×C32—164no—
131C2×C32—162no—
132C2×C32C2×C4×A411yescharacteristic
133C24C3×#15711yescharacteristic
134C22×C4—91no—
135C22×C4—91no—
136C22×C4—271no—
137C22×C4—271no—
138C22×C4—271no—
139C22×C4—271no—
140M16—91no—
141M16—272no—
142M16—271no—
143M16—272no—
144C2×C8—272no—
145C2×C8—272no—
146C22×C3—22no—
147C22×C3—22no—
148C22×C3—41no—
149C22×C3—62no—
150C22×C3—62no—
151C22×C3—62no—
152C22×C3—64no—
153C22×C3—64no—
154C22×C3—64no—
155C22×C3—64no—
156C22×C3—162no—
157A4—42no—
158A4#379211yescharacteristic, direct product factor
159C3×C4—361no—
160C3×C4—361no—
161Dic3—62no—
162Dic3—62no—
163Dic3—182no—
164Dic3—182no—
165C32—82no—
166C32—162no—
167C32A4×M1611yescharacteristic, complemented
168C23—21no—
169C23—31no—
170C23—31no—
171C23—61no—
172C23C2×C3×#15711yescharacteristic
173C2×C4—91no—
174C2×C4—271no—
175C2×C4—271no—
176C2×C4—271no—
177C2×C4—271no—
178C2×C4—272no—
179C2×C4—271no—
180C2×C4—271no—
181C2×C4—271no—
182C8—92no—
183C8—272no—
184C2×C3—24no—
185C2×C3—22no—
186C2×C3—41no—
187C2×C3—62no—
188C2×C3—62no—
189C2×C3—64no—
190C2×C3—81no—
191C2×C3—164no—
192C2×C3—162no—
193C22—31no—
194C22—31no—
195C22—31no—
196C22—61no—
197C22—61no—
198C22—61no—
199C22—61no—
200C22C3×#379211yescharacteristic, complemented
201C22A4×#15711yescharacteristic
202C4—91no—
203C4—91no—
204C4—271no—
205C4—271no—
206C3—22no—
207C3—41no—
208C3—162no—
209C2—21no—
210C2—31no—
211C2—31no—
212C2—61no—
213C2C2×A4×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×#157
commutator, socleC23×C32C2×C3×C4
FittingC24×C32C3×C4
prime radicalC22×C32×A4C4
cosocleC22×C3⋊Dic3C22×C3
hypercenterC22A4×#157
hypocenterC22×C32C3×M16
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. A4×#622C2semidirect
3. C22×#3792C3semidirect
12. C22×#622C2×C3semidirect
31. #3792A4direct (N × H)
37. C32×A4M16semidirect
63. #622C2×A4semidirect
88. C22×C32C3×M16semidirect
158. A4#3792direct (N × H)
167. C32A4×M16semidirect
200. C22C3×#3792semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C3⋊Dic32
A4×#6222
C22×#37923
C3×#37924
A4×M169