Finite Group Databasebeta

← #13173059

Subgroups — #13173059

Order 1728.

806 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 207 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×C32×C4×A4C211yesmaximal, characteristic
2C3⋊(C3⋊(A4⋊C8))C212yesmaximal, complemented
3#10593643—312nomaximal
4#10592906—31nomaximal
5C22×C32×A4C411yescharacteristic
6C32×C4×A4C411yescharacteristic
7C32×C4×A4C2211yescharacteristic
8#88855—41nomaximal
9C23×C32×C4D311yescharacteristic
10C2×C3×C4×A4D3112yes—
11C2×#3748—31no—
12C2×#3748—31no—
13C3⋊(A4⋊C8)—324no—
14#63657—32no—
15#63657—32no—
16C2×C32×A4—21no—
17C2×C32×A4C2×C411yescharacteristic
18C2×C33×C4—41no—
19C3⋊(C3⋊(C3⋊C8))—42no—
20#5409—99no—
21#5207—94no—
22C24×C32Dic311yescharacteristic
23C22×C3×A4Dic3112yes—
24C22×C32×C4—31no—
25C22×C32×C4—31no—
26C22×C32×C4—31no—
27C22×C32×C4—31no—
28C22×C32×C4Dic311yescharacteristic
29C22×C32×C4C2×D311yescharacteristic
30C3×C4×A4Dic3112yes—
31C3×C4×A4C2×D3112yes—
32#3748—61no—
33#3748—61no—
34#3748—61no—
35#3748—61no—
36#3748—1212no—
37C2×C3⋊(C3⋊C8)—32no—
38C2×C3⋊(C3⋊C8)—32no—
39C22×C33—41no—
40C32×A4M1611yescharacteristic, complemented
41C33×C4—41no—
42C33×C4—41no—
43C23×C3×C4C3⋊D314yes—
44C2×C4×A4C3⋊D319yes—
45C2×#208—94no—
46C2×#208—94no—
47A4⋊C8—918no—
48#830—98no—
49#830—98no—
50C23×C32—21no—
51C23×C32—31no—
52C23×C32—31no—
53C23×C32—61no—
54C23×C32C2×Dic311yescharacteristic
55C2×C3×A4—212no—
56C2×C3×A4C2×Dic3112yes—
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—412no—
65C2×C32×C4—61no—
66C2×C32×C4S411yescharacteristic
67C3⋊(C3⋊C8)—62no—
68C3⋊(C3⋊C8)—62no—
69C3⋊(C3⋊C8)—1224no—
70#407—271noSylow
71C2×C33—41no—
72C2×C33—81no—
73C24×C3C3⋊Dic314yes—
74C22×A4C3⋊Dic319yes—
75C22×C3×C4—34no—
76C22×C3×C4—34no—
77C22×C3×C4—34no—
78C22×C3×C4—34no—
79C22×C3×C4C3⋊Dic314yes—
80C22×C3×C4C2×C3⋊D314yes—
81C4×A4C3⋊Dic319yes—
82C4×A4C2×C3⋊D319yes—
83#208—184no—
84#208—184no—
85#208—184no—
86#208—184no—
87#208—369no—
88C2×C3⋊C8—98no—
89C2×C3⋊C8—98no—
90C22×C32—31no—
91C22×C32—31no—
92C22×C32—32no—
93C22×C32—31no—
94C22×C32—412no—
95C22×C32—61no—
96C22×C32—61no—
97C22×C32—61no—
98C22×C32#20811yescharacteristic, complemented
99C22×C32SL(2,ℤ/4)11yescharacteristic
100C3×A4#208112yescomplemented
101C32×C4—31no—
102C32×C4—31no—
103C32×C4—412no—
104C32×C4—412no—
105C32×C4SL(2,ℤ/4)11yescharacteristic
106C32×C4C2×S411yescharacteristic
107C23×C4C3⋊(C3⋊D3)11yescharacteristic
108C2×M16—271no—
109C2×M16—271no—
110C22⋊C8—272no—
111C22⋊C8—272no—
112C33—41noSylow
113C23×C3—24no—
114C23×C3—34no—
115C23×C3—34no—
116C23×C3—64no—
117C23×C3C2×C3⋊Dic314yes—
118C2×A4—29no—
119C2×A4C2×C3⋊Dic319yes—
120C2×C3×C4—34no—
121C2×C3×C4—34no—
122C2×C3×C4—34no—
123C2×C3×C4—34no—
124C2×C3×C4—34no—
125C2×C3×C4—34no—
126C2×C3×C4—34no—
127C2×C3×C4—49no—
128C2×C3×C4—64no—
129C2×C3×C4C3⋊S414yes—
130C3⋊C8—188no—
131C3⋊C8—188no—
132C3⋊C8—3618no—
133C2×C32—21no—
134C2×C32—31no—
135C2×C32—31no—
136C2×C32—412no—
137C2×C32—61no—
138C2×C32—812no—
139C2×C32C2×SL(2,ℤ/4)11yescharacteristic
140C24C3⋊(C3⋊Dic3)11yescharacteristic
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4C3⋊(C3⋊Dic3)11yescharacteristic
146C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
147M16—541no—
148M16—541no—
149M16—541no—
150M16—541no—
151C2×C8—272no—
152C2×C8—272no—
153C22×C3—34no—
154C22×C3—34no—
155C22×C3—38no—
156C22×C3—34no—
157C22×C3—49no—
158C22×C3—64no—
159C22×C3—64no—
160C22×C3—64no—
161C22×C3#374814yescomplemented
162C22×C3C3⋊SL(2,ℤ/4)14yes—
163A4#374819yescomplemented
164C3×C4—34no—
165C3×C4—34no—
166C3×C4—49no—
167C3×C4—49no—
168C3×C4C3⋊SL(2,ℤ/4)14yes—
169C3×C4C2×C3⋊S414yes—
170C32—412no—
171C32#540911yescharacteristic, complemented
172C23—21no—
173C23—31no—
174C23—31no—
175C23—61no—
176C23C2×C3⋊(C3⋊Dic3)11yescharacteristic
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—61no—
185C2×C4C3⋊(C3⋊S4)11yescharacteristic
186C8—542no—
187C8—542no—
188C2×C3—24no—
189C2×C3—34no—
190C2×C3—34no—
191C2×C3—49no—
192C2×C3—64no—
193C2×C3—89no—
194C2×C3C2×C3⋊SL(2,ℤ/4)14yes—
195C22—31no—
196C22—31no—
197C22—32no—
198C22—31no—
199C22—61no—
200C22—61no—
201C22—61no—
202C22#8885511yescharacteristic, complemented
203C22C3⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
204C4—31no—
205C4—31no—
206C4C3⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
207C4C2×C3⋊(C3⋊S4)11yescharacteristic
208C3—49no—
209C3#1059364314yescomplemented
210C2—21no—
211C2—31no—
212C2—31no—
213C2—61no—
214C2C2×C3⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
subgroupG/H
center, FrattiniC4C2×C3⋊(C3⋊S4)
commutatorC2×C32×A4C2×C4
FittingC23×C32×C4D3
socleC23×C32C2×Dic3
prime radicalC22×C32×A4C4
cosocleC32×C4×A4C22
hypercenterC2×C4C3⋊(C3⋊S4)
hypocenterC32×A4M16
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊(A4⋊C8))C2semidirect
40. C32×A4M16semidirect
98. C22×C32#208semidirect
100. C3×A4#208semidirect
161. C22×C3#3748semidirect
163. A4#3748semidirect
171. C32#5409semidirect
202. C22#88855semidirect
209. C3#10593643semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
C3⋊(C3⋊(A4⋊C8))2
#105936433
#105929063
#888554