Finite Group Databasebeta

← A4×#3744 #13173159

Subgroups — A4×#3744 #13173159

Order 1728.

562 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 81 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
1A4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2A4×#634C211yesmaximal, characteristic, complemented
3C32×C8×A4C211yesmaximal, characteristic, complemented
4C22×#3744C311yesmaximal, characteristic, complemented
5A4×#204—34nomaximal
6C2×A4×C3⋊D3—21no—
7C32×C4×A4C2211yescharacteristic
8A4×C3⋊Dic3—21no—
9C3×#3744—41nomaximal
10C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
11C22×#634C2×C311yescharacteristic, complemented
12A4×D12—34no—
13A4×Dic6—34no—
14C22×C32×C8C2×C311yescharacteristic, complemented
15C2×#3744—31no—
16C2×#3744—31no—
17C2×#3744—31no—
18C2×#3744—31no—
19C3×C8×A4—24no—
20C3×C8×A4D314yescomplemented
21C2×C32×A4D411yescharacteristic
22A4×C3⋊D3—41no—
23C3×C4⋊(C3⋊D3)—41no—
24C3×#634—41no—
25C33×C8—41no—
26C22×#204—34no—
27A4×SD16—91no—
28C23×C3⋊D3—21no—
29C2×D3×A4—64no—
30C22×C32×C4C22×C311yescharacteristic
31C22×C3⋊Dic3—21no—
32C2×C4⋊(C3⋊D3)—31no—
33C2×C4⋊(C3⋊D3)—31no—
34C2×C4⋊(C3⋊D3)—61no—
35C2×#634—31no—
36C2×#634—31no—
37C2×#634—61no—
38C3×C4×A4—24no—
39C3×C4×A4C2×D314yes—
40Dic3×A4—64no—
41C2×C32×C8—31no—
42C2×C32×C8—31no—
43#3744—32no—
44#3744—31no—
45#3744—31no—
46#3744—31no—
47#3744A411yescharacteristic, direct product factor
48C3×#204—124no—
49C2×C3×C3⋊D3—81no—
50C32×A4SD1611yescharacteristic, complemented
51C33×C4—41no—
52C3×C3⋊Dic3—81no—
53C22×D12—34no—
54C22×Dic6—34no—
55Q8×A4—91no—
56D4×A4—91no—
57C22×C3×C8C3×D314yescomplemented
58C2×#204—94no—
59C2×#204—94no—
60C2×#204—94no—
61C2×#204—94no—
62C8×A4—24no—
63C8×A4C3⋊D311yescharacteristic, complemented
64C23×C32C3×D411yescharacteristic
65C22×C3⋊D3—41no—
66C22×C3⋊D3—61no—
67C22×C3⋊D3—61no—
68C22×C3⋊D3—121no—
69C2×C3×A4—24no—
70C2×C3×A4D1214yes—
71D3×A4—124no—
72C2×C32×C4—31no—
73C2×C32×C4—31no—
74C2×C3⋊Dic3—61no—
75C2×C3⋊Dic3—61no—
76C4⋊(C3⋊D3)—31no—
77C4⋊(C3⋊D3)—61no—
78C4⋊(C3⋊D3)—61no—
79C4⋊(C3⋊D3)C2×A411yescharacteristic
80#634—31no—
81#634—61no—
82#634—61no—
83#634C2×A411yescharacteristic, complemented
84C3×D12—124no—
85C3×Dic6—124no—
86C32×C8—31no—
87C32×C8—44no—
88C32×C8—84no—
89C32×C8C2×A411yescharacteristic, complemented
90C22×SD16—91noSylow
91C2×C33—41no—
92C3×C3⋊D3—161no—
93C23×D3—64no—
94C22×A4—181no—
95C22×C3×C4C2×C3×D314yes—
96C22×Dic3—64no—
97C2×D12—94no—
98C2×D12—94no—
99C2×D12—184no—
100C2×Dic6—94no—
101C2×Dic6—94no—
102C2×Dic6—184no—
103C4×A4—24no—
104C4×A4—181no—
105C4×A4C2×C3⋊D311yescharacteristic
106C3×SD16—361no—
107C2×C3×C8—34no—
108C2×C3×C8—34no—
109#204—34no—
110#204—94no—
111#204—94no—
112#204—98no—
113#204—94no—
114C22×C32—31no—
115C22×C32—31no—
116C22×C32C3×SD1611yescharacteristic, complemented
117C2×C3⋊D3—21no—
118C2×C3⋊D3—61no—
119C2×C3⋊D3—121no—
120C2×C3⋊D3—121no—
121C2×C3⋊D3—121no—
122C2×C3⋊D3—121no—
123C2×C3×D3—244no—
124C3×A4—24no—
125C3×A4#20414yescomplemented
126C32×C4—31no—
127C32×C4—44no—
128C32×C4—84no—
129C32×C4C22×A411yescharacteristic
130C3⋊Dic3—21no—
131C3⋊Dic3—61no—
132C3×Dic3—244no—
133C22×Q8—91no—
134C22×D4—91no—
135C2×SD16—271no—
136C2×SD16—271no—
137C2×SD16—271no—
138C2×SD16—271no—
139C22×C8C3×C3⋊D311yescharacteristic, complemented
140C33—41noSylow
141C23×C3C3×D1214yes—
142C22×D3—124no—
143C22×D3—184no—
144C22×D3—184no—
145C22×D3—364no—
146C2×A4—24no—
147C2×A4—361no—
148C2×A4C4⋊(C3⋊D3)11yescharacteristic
149C3×Q8—361no—
150C3×D4—361no—
151C2×C3×C4—34no—
152C2×C3×C4—34no—
153C2×Dic3—184no—
154C2×Dic3—184no—
155D12—34no—
156D12—94no—
157D12—184no—
158D12—184no—
159Dic6—34no—
160Dic6—94no—
161Dic6—184no—
162Dic6—184no—
163C3×C8—34no—
164C3×C8—41no—
165C3×C8—84no—
166C3×C8D3×A414yescomplemented
167C2×C32—31no—
168C2×C32—31no—
169C2×C32—44no—
170C2×C32—84no—
171C2×C32D4×A411yescharacteristic
172C3⋊D3—41no—
173C3⋊D3—121no—
174C3×D3—484no—
175C24—181no—
176C2×Q8—271no—
177C2×Q8—271no—
178C2×Q8—541no—
179C2×D4—271no—
180C2×D4—271no—
181C2×D4—541no—
182C22×C4—181no—
183C22×C4C2×C3×C3⋊D311yescharacteristic
184SD16—91no—
185SD16—271no—
186SD16—271no—
187SD16—272no—
188SD16—271no—
189C2×C8—31no—
190C2×C8—31no—
191C22×C3—34no—
192C22×C3—34no—
193C22×C3—721no—
194C22×C3C3×#20414yescomplemented
195C2×D3—64no—
196C2×D3—184no—
197C2×D3—364no—
198C2×D3—364no—
199C2×D3—364no—
200C2×D3—364no—
201A4—24no—
202A4#374411yescharacteristic, direct product factor
203C3×C4—34no—
204C3×C4—41no—
205C3×C4—84no—
206C3×C4—721no—
207C3×C4C2×D3×A414yes—
208Dic3—64no—
209Dic3—184no—
210C32—44no—
211C32—84no—
212C32A4×SD1611yescharacteristic, complemented
213C23—361no—
214C23—541no—
215C23—541no—
216C23—1081no—
217C23C3×C4⋊(C3⋊D3)11yescharacteristic
218Q8—91no—
219Q8—271no—
220Q8—541no—
221Q8—541no—
222D4—91no—
223D4—271no—
224D4—541no—
225D4—541no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—541no—
229C2×C4—541no—
230C8—31no—
231C8A4×C3⋊D311yescharacteristic, complemented
232C2×C3—34no—
233C2×C3—34no—
234C2×C3—41no—
235C2×C3—84no—
236C2×C3—1441no—
237C2×C3A4×D1214yes—
238D3—124no—
239D3—364no—
240C22—31no—
241C22—31no—
242C22—181no—
243C22—541no—
244C22—1081no—
245C22—1081no—
246C22—1081no—
247C22—1081no—
248C22C3×#374411yescharacteristic, complemented
249C4—31no—
250C4—181no—
251C4—541no—
252C4C2×A4×C3⋊D311yescharacteristic
253C3—41no—
254C3—84no—
255C3A4×#20414yescomplemented
256C2—31no—
257C2—31no—
258C2—361no—
259C2—1081no—
260C2A4×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2A4×C4⋊(C3⋊D3)
commutator, cosocleC22×C32×C4C22×C3
FrattiniC4C2×A4×C3⋊D3
FittingC22×C32×C8C2×C3
socleC23×C32C3×D4
prime radicalA4×C4⋊(C3⋊D3)C2
hypercenterC8A4×C3⋊D3
hypocenterC22×C32C3×SD16
LCSG ▷ C22×C32×C4 ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. A4×#634C2semidirect
3. C32×C8×A4C2semidirect
4. C22×#3744C3semidirect
11. C22×#634C2×C3semidirect
14. C22×C32×C8C2×C3semidirect
20. C3×C8×A4D3semidirect
47. #3744A4direct (N × H)
50. C32×A4SD16semidirect
57. C22×C3×C8C3×D3semidirect
63. C8×A4C3⋊D3semidirect
83. #634C2×A4semidirect
89. C32×C8C2×A4semidirect
116. C22×C32C3×SD16semidirect
125. C3×A4#204semidirect
139. C22×C8C3×C3⋊D3semidirect
166. C3×C8D3×A4semidirect
194. C22×C3C3×#204semidirect
202. A4#3744direct (N × H)
212. C32A4×SD16semidirect
231. C8A4×C3⋊D3semidirect
248. C22C3×#3744semidirect
255. C3A4×#204semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊(C3⋊D3)2
A4×#6342
C32×C8×A42
C22×#37443
A4×#2043
C3×#37444