Finite Group Databasebeta

← A4×#3756 #13173179

Subgroups — A4×#3756 #13173179

Order 1728.

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

normal subgroups only

This group has 534 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
2A4×#634C211yesmaximal, characteristic
3A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
4C22×#3756C311yesmaximal, characteristic, complemented
5A4×#216—34nomaximal
6C32×C4×A4—21no—
7C32×C4×A4C2211yescharacteristic
8A4×C3⋊Dic3—21no—
9C3×#3756—41nomaximal
10C22×C32×Q8C2×C311yescharacteristic
11C22×#634C2×C311yescharacteristic
12C3×Q8×A4—24no—
13C3×Q8×A4D314yes—
14A4×Dic6—34no—
15C2×#3756—31no—
16C2×#3756—31no—
17C2×#3756—31no—
18C2×#3756—31no—
19C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
20A4×C3⋊C8—34no—
21C2×C32×A4D411yescharacteristic
22C33×Q8—41no—
23C3×#634—41no—
24C3×C3⋊(C3⋊C8)—41no—
25C22×#216—34no—
26A4×Q16—91no—
27C2×C32×Q8—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C22×C32×C4—21no—
31C22×C32×C4C22×C311yescharacteristic
32C22×C3⋊Dic3—21no—
33C2×#634—31no—
34C2×#634—31no—
35C2×#634—61no—
36C3×C4×A4—28no—
37C3×C4×A4—24no—
38C3×C4×A4—24no—
39C3×C4×A4C2×D314yes—
40Dic3×A4—64no—
41#3756—32no—
42#3756—31no—
43#3756—31no—
44#3756—31no—
45#3756A411yescharacteristic, direct product factor
46C2×C3⋊(C3⋊C8)—31no—
47C2×C3⋊(C3⋊C8)—31no—
48C3×#216—124no—
49C32×A4Q1611yescharacteristic, complemented
50C33×C4—41no—
51C33×C4—81no—
52C3×C3⋊Dic3—81no—
53C22×C3×Q8C3×D314yes—
54C22×Dic6—34no—
55Q8×A4—24no—
56Q8×A4—91no—
57Q8×A4C3⋊D311yescharacteristic
58C2×#216—94no—
59C2×#216—94no—
60C2×#216—94no—
61C2×#216—94no—
62C22×C3⋊C8—34no—
63C8×A4—91no—
64C23×C32C3×D411yescharacteristic
65C2×C3×A4—24no—
66C2×C3×A4#6714yes—
67C32×Q8—31no—
68C32×Q8—44no—
69C32×Q8—61no—
70C32×Q8—61no—
71C32×Q8—84no—
72C32×Q8C2×A411yescharacteristic
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C2×C32×C4—61no—
77C2×C3⋊Dic3—61no—
78C2×C3⋊Dic3—61no—
79#634—31no—
80#634—61no—
81#634—61no—
82#634C2×A411yescharacteristic
83C3×Dic6—124no—
84C3⋊(C3⋊C8)—31no—
85C3⋊(C3⋊C8)C2×A411yescharacteristic
86C3×C3⋊C8—124no—
87C22×Q16—91noSylow
88C2×C33—41no—
89C2×C3×Q8—34no—
90C2×C3×Q8—34no—
91C2×C3×Q8—64no—
92C22×C3×C4—24no—
93C22×C3×C4C2×C3×D314yes—
94C22×Dic3—64no—
95C2×Dic6—94no—
96C2×Dic6—94no—
97C2×Dic6—184no—
98C4×A4—28no—
99C4×A4—21no—
100C4×A4—24no—
101C4×A4—181no—
102C4×A4C2×C3⋊D311yescharacteristic
103C3×Q16—361no—
104#216—34no—
105#216—94no—
106#216—94no—
107#216—98no—
108#216—94no—
109C2×C3⋊C8—94no—
110C2×C3⋊C8—94no—
111C22×C32—31no—
112C22×C32—31no—
113C22×C32C3×Q1611yescharacteristic, complemented
114C3×A4—24no—
115C3×A4#21614yescomplemented
116C32×C4—21no—
117C32×C4—31no—
118C32×C4—44no—
119C32×C4—61no—
120C32×C4—84no—
121C32×C4—88no—
122C32×C4—84no—
123C32×C4C22×A411yescharacteristic
124C3⋊Dic3—21no—
125C3⋊Dic3—61no—
126C3×Dic3—244no—
127C22×Q8—91no—
128C22×Q8C3×C3⋊D311yescharacteristic
129C2×Q16—271no—
130C2×Q16—271no—
131C2×Q16—271no—
132C2×Q16—271no—
133C22×C8—91no—
134C33—41noSylow
135C23×C3C3×#6714yes—
136C2×A4—24no—
137C2×A4#63811yescharacteristic
138C3×Q8—34no—
139C3×Q8—41no—
140C3×Q8—64no—
141C3×Q8—64no—
142C3×Q8—84no—
143C3×Q8—361no—
144C3×Q8D3×A414yes—
145C2×C3×C4—34no—
146C2×C3×C4—34no—
147C2×C3×C4—64no—
148C2×C3×C4—64no—
149C2×Dic3—184no—
150C2×Dic3—184no—
151Dic6—34no—
152Dic6—94no—
153Dic6—184no—
154Dic6—184no—
155C3×C8—361no—
156C3⋊C8—34no—
157C3⋊C8—94no—
158C2×C32—31no—
159C2×C32—31no—
160C2×C32—44no—
161C2×C32—84no—
162C2×C32D4×A411yescharacteristic
163C2×Q8—31no—
164C2×Q8—31no—
165C2×Q8—61no—
166C2×Q8—271no—
167C2×Q8—271no—
168C2×Q8—541no—
169C22×C4—21no—
170C22×C4—181no—
171C22×C4C2×C3×C3⋊D311yescharacteristic
172Q16—91no—
173Q16—271no—
174Q16—271no—
175Q16—272no—
176Q16—271no—
177C2×C8—271no—
178C2×C8—271no—
179C22×C3—34no—
180C22×C3—34no—
181C22×C3C3×#21614yescomplemented
182A4—24no—
183A4#375611yescharacteristic, direct product factor
184C3×C4—24no—
185C3×C4—34no—
186C3×C4—41no—
187C3×C4—64no—
188C3×C4—81no—
189C3×C4—88no—
190C3×C4—84no—
191C3×C4—721no—
192C3×C4C2×D3×A414yes—
193Dic3—64no—
194Dic3—184no—
195C32—44no—
196C32—84no—
197C32A4×Q1611yescharacteristic, complemented
198C23C3×#63811yescharacteristic
199Q8—31no—
200Q8—61no—
201Q8—61no—
202Q8—91no—
203Q8—271no—
204Q8—541no—
205Q8—541no—
206Q8A4×C3⋊D311yescharacteristic
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—541no—
212C2×C4—541no—
213C8—91no—
214C8—271no—
215C2×C3—34no—
216C2×C3—34no—
217C2×C3—41no—
218C2×C3—84no—
219C2×C3A4×#6714yes—
220C22—31no—
221C22—31no—
222C22C3×#375611yescharacteristic, complemented
223C4—21no—
224C4—31no—
225C4—61no—
226C4—181no—
227C4—541no—
228C4C2×A4×C3⋊D311yescharacteristic
229C3—41no—
230C3—84no—
231C3A4×#21614yescomplemented
232C2—31no—
233C2—31no—
234C2A4×#63811yescharacteristic
subgroupG/H
centerC2A4×#638
commutator, cosocleC22×C32×C4C22×C3
FrattiniC4C2×A4×C3⋊D3
FittingC22×C32×Q8C2×C3
socleC23×C32C3×D4
prime radicalC2×C32×A4D4
hypercenterQ8A4×C3⋊D3
hypocenterC22×C32C3×Q16
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
4. C22×#3756C3semidirect
45. #3756A4direct (N × H)
49. C32×A4Q16semidirect
113. C22×C32C3×Q16semidirect
115. C3×A4#216semidirect
181. C22×C3C3×#216semidirect
183. A4#3756direct (N × H)
197. C32A4×Q16semidirect
222. C22C3×#3756semidirect
231. C3A4×#216semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×A42
A4×#6342
A4×C3⋊(C3⋊C8)2
C22×#37563
A4×#2163
C3×#37564