Finite Group Databasebeta

← A4×#3755 #13173178

Subgroups — A4×#3755 #13173178

Order 1728.

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

normal subgroups only

This group has 594 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
1C32×Q8×A4C211yesmaximal, characteristic, complemented
2A4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C22×#3755C311yesmaximal, characteristic, complemented
5A4×#215—34nomaximal
6C2×A4×C3⋊D3—21no—
7C32×C4×A4—21no—
8C32×C4×A4C2211yescharacteristic
9C3×#3755—41nomaximal
10C22×C32×Q8C2×C311yescharacteristic, complemented
11C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
12C3×Q8×A4—24no—
13C3×Q8×A4D314yescomplemented
14A4×D12—34no—
15C2×#3755—31no—
16C2×#3755—31no—
17C2×#3755—31no—
18C2×#3755—31no—
19C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
20A4×C3⋊C8—34no—
21C2×C32×A4D411yescharacteristic
22A4×C3⋊D3—41no—
23C33×Q8—41no—
24C3×C4⋊(C3⋊D3)—41no—
25C3×C3⋊(C3⋊C8)—41no—
26C22×#215—34no—
27A4×SD16—91no—
28C23×C3⋊D3—21no—
29C2×D3×A4—64no—
30C2×C32×Q8—31no—
31C2×C32×Q8—31no—
32C2×C32×Q8—61no—
33C22×C32×C4—21no—
34C22×C32×C4C22×C311yescharacteristic
35C2×C4⋊(C3⋊D3)—31no—
36C2×C4⋊(C3⋊D3)—31no—
37C2×C4⋊(C3⋊D3)—61no—
38C3×C4×A4—28no—
39C3×C4×A4—24no—
40C3×C4×A4—24no—
41C3×C4×A4C2×D314yes—
42#3755—32no—
43#3755—31no—
44#3755—31no—
45#3755—31no—
46#3755A411yescharacteristic, direct product factor
47C2×C3⋊(C3⋊C8)—31no—
48C2×C3⋊(C3⋊C8)—31no—
49C3×#215—124no—
50C2×C3×C3⋊D3—81no—
51C32×A4SD1611yescharacteristic, complemented
52C33×C4—41no—
53C33×C4—81no—
54C22×C3×Q8C3×D314yescomplemented
55C22×D12—34no—
56Q8×A4—24no—
57Q8×A4C3⋊D311yescharacteristic, complemented
58D4×A4—91no—
59C2×#215—94no—
60C2×#215—94no—
61C2×#215—94no—
62C2×#215—94no—
63C22×C3⋊C8—34no—
64C8×A4—91no—
65C23×C32C3×D411yescharacteristic
66C22×C3⋊D3—41no—
67C22×C3⋊D3—61no—
68C22×C3⋊D3—61no—
69C22×C3⋊D3—121no—
70C2×C3×A4—24no—
71C2×C3×A4#6714yes—
72D3×A4—124no—
73C32×Q8—31no—
74C32×Q8—44no—
75C32×Q8—61no—
76C32×Q8—61no—
77C32×Q8—84no—
78C32×Q8C2×A411yescharacteristic, complemented
79C2×C32×C4—31no—
80C2×C32×C4—31no—
81C2×C32×C4—61no—
82C2×C32×C4—61no—
83C4⋊(C3⋊D3)—31no—
84C4⋊(C3⋊D3)—61no—
85C4⋊(C3⋊D3)—61no—
86C4⋊(C3⋊D3)C2×A411yescharacteristic
87C3×D12—124no—
88C3⋊(C3⋊C8)—31no—
89C3⋊(C3⋊C8)C2×A411yescharacteristic, complemented
90C3×C3⋊C8—124no—
91C22×SD16—91noSylow
92C2×C33—41no—
93C3×C3⋊D3—161no—
94C23×D3—64no—
95C22×A4—181no—
96C2×C3×Q8—34no—
97C2×C3×Q8—34no—
98C2×C3×Q8—64no—
99C22×C3×C4—24no—
100C22×C3×C4C2×C3×D314yes—
101C2×D12—94no—
102C2×D12—94no—
103C2×D12—184no—
104C4×A4—28no—
105C4×A4—21no—
106C4×A4—24no—
107C4×A4C2×C3⋊D311yescharacteristic
108C3×SD16—361no—
109#215—34no—
110#215—94no—
111#215—94no—
112#215—98no—
113#215—94no—
114C2×C3⋊C8—94no—
115C2×C3⋊C8—94no—
116C22×C32—31no—
117C22×C32—31no—
118C22×C32C3×SD1611yescharacteristic, complemented
119C2×C3⋊D3—21no—
120C2×C3⋊D3—61no—
121C2×C3⋊D3—121no—
122C2×C3⋊D3—121no—
123C2×C3⋊D3—121no—
124C2×C3⋊D3—121no—
125C2×C3×D3—244no—
126C3×A4—24no—
127C3×A4#21514yescomplemented
128C32×C4—21no—
129C32×C4—31no—
130C32×C4—44no—
131C32×C4—61no—
132C32×C4—84no—
133C32×C4—88no—
134C32×C4—84no—
135C32×C4C22×A411yescharacteristic
136C22×Q8C3×C3⋊D311yescharacteristic, complemented
137C22×D4—91no—
138C2×SD16—271no—
139C2×SD16—271no—
140C2×SD16—271no—
141C2×SD16—271no—
142C22×C8—91no—
143C33—41noSylow
144C23×C3C3×#6714yes—
145C22×D3—124no—
146C22×D3—184no—
147C22×D3—184no—
148C22×D3—364no—
149C2×A4—24no—
150C2×A4—361no—
151C2×A4#63811yescharacteristic
152C3×Q8—34no—
153C3×Q8—41no—
154C3×Q8—64no—
155C3×Q8—64no—
156C3×Q8—84no—
157C3×Q8D3×A414yescomplemented
158C3×D4—361no—
159C2×C3×C4—34no—
160C2×C3×C4—34no—
161C2×C3×C4—64no—
162C2×C3×C4—64no—
163D12—34no—
164D12—94no—
165D12—184no—
166D12—184no—
167C3×C8—361no—
168C3⋊C8—34no—
169C3⋊C8—94no—
170C2×C32—31no—
171C2×C32—31no—
172C2×C32—44no—
173C2×C32—84no—
174C2×C32D4×A411yescharacteristic
175C3⋊D3—41no—
176C3⋊D3—121no—
177C3×D3—484no—
178C24—181no—
179C2×Q8—31no—
180C2×Q8—31no—
181C2×Q8—61no—
182C2×D4—271no—
183C2×D4—271no—
184C2×D4—541no—
185C22×C4—21no—
186C22×C4C2×C3×C3⋊D311yescharacteristic
187SD16—91no—
188SD16—271no—
189SD16—271no—
190SD16—272no—
191SD16—271no—
192C2×C8—271no—
193C2×C8—271no—
194C22×C3—34no—
195C22×C3—34no—
196C22×C3—721no—
197C22×C3C3×#21514yescomplemented
198C2×D3—64no—
199C2×D3—184no—
200C2×D3—364no—
201C2×D3—364no—
202C2×D3—364no—
203C2×D3—364no—
204A4—24no—
205A4#375511yescharacteristic, direct product factor
206C3×C4—24no—
207C3×C4—34no—
208C3×C4—41no—
209C3×C4—64no—
210C3×C4—81no—
211C3×C4—88no—
212C3×C4—84no—
213C3×C4C2×D3×A414yes—
214C32—44no—
215C32—84no—
216C32A4×SD1611yescharacteristic, complemented
217C23—361no—
218C23—541no—
219C23—541no—
220C23—1081no—
221C23C3×#63811yescharacteristic
222Q8—31no—
223Q8—61no—
224Q8—61no—
225Q8A4×C3⋊D311yescharacteristic, complemented
226D4—91no—
227D4—271no—
228D4—541no—
229D4—541no—
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—61no—
233C2×C4—61no—
234C8—91no—
235C8—271no—
236C2×C3—34no—
237C2×C3—34no—
238C2×C3—41no—
239C2×C3—84no—
240C2×C3—1441no—
241C2×C3A4×#6714yes—
242D3—124no—
243D3—364no—
244C22—31no—
245C22—31no—
246C22—181no—
247C22—541no—
248C22—1081no—
249C22—1081no—
250C22—1081no—
251C22—1081no—
252C22C3×#375511yescharacteristic, complemented
253C4—21no—
254C4—31no—
255C4—61no—
256C4C2×A4×C3⋊D311yescharacteristic
257C3—41no—
258C3—84no—
259C3A4×#21514yescomplemented
260C2—31no—
261C2—31no—
262C2—361no—
263C2—1081no—
264C2A4×#63811yescharacteristic
subgroupG/H
centerC2A4×#638
commutator, cosocleC22×C32×C4C22×C3
FrattiniC4C2×A4×C3⋊D3
FittingC22×C32×Q8C2×C3
socleC23×C32C3×D4
prime radicalA4×C4⋊(C3⋊D3)C2
hypercenterQ8A4×C3⋊D3
hypocenterC22×C32C3×SD16
LCSG ▷ C22×C32×C4 ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
3. A4×C3⋊(C3⋊C8)C2semidirect
4. C22×#3755C3semidirect
10. C22×C32×Q8C2×C3semidirect
13. C3×Q8×A4D3semidirect
19. C22×C3⋊(C3⋊C8)C2×C3semidirect
46. #3755A4direct (N × H)
51. C32×A4SD16semidirect
54. C22×C3×Q8C3×D3semidirect
57. Q8×A4C3⋊D3semidirect
78. C32×Q8C2×A4semidirect
89. C3⋊(C3⋊C8)C2×A4semidirect
118. C22×C32C3×SD16semidirect
127. C3×A4#215semidirect
136. C22×Q8C3×C3⋊D3semidirect
157. C3×Q8D3×A4semidirect
197. C22×C3C3×#215semidirect
205. A4#3755direct (N × H)
216. C32A4×SD16semidirect
225. Q8A4×C3⋊D3semidirect
252. C22C3×#3755semidirect
259. C3A4×#215semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×A42
A4×C4⋊(C3⋊D3)2
A4×C3⋊(C3⋊C8)2
C22×#37553
A4×#2153
C3×#37554