Finite Group Databasebeta

← #13158807

Subgroups — #13158807

Order 1728.

736 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C3×#64081C211yesmaximal, characteristic
2#11722173C211yesmaximal, characteristic, complemented
3C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
4#11722051C212yesmaximal, complemented
5#11722023C212yesmaximal
6#10592389—31nomaximal
7#10590264—34nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C33×C42C2211yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
11C3×#3752C412yes—
12C3×#3750C412yes—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15#64081D311yescharacteristic
16C3×#872—34no—
17C32×C4×C8—31no—
18#63647—32no—
19#63641—32no—
20#63630—31no—
21#63630—34no—
22#63630—64no—
23C3×C4×C3⋊C8D314yescomplemented
24Dic3⋊(C3⋊C8)—38no—
25(C2×D3)⋊(C3⋊C8)—38no—
26C22×C3×C3⋊D3C2×C411yescharacteristic
27C2×C33×C4C2×C412yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3C2×C412yes—
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C3×C4×C3⋊D3—22no—
32C3⋊(C3⋊(C3⋊C8))—22no—
33C32×C3⋊C8—22no—
34#5000—91no—
35#4690—94no—
36C2×C4×C3⋊D3C2×D311yescharacteristic
37C2×C3×C4×D3—34no—
38C2×C32×C8—32no—
39C32×C42—24no—
40C32×C42C2×D314yes—
41C32×C42C2×D311yescharacteristic
42#3752Dic312yes—
43#3750Dic312yes—
44C4×C3⋊Dic3C2×D311yescharacteristic
45C2×C3⋊(C3⋊C8)—32no—
46C2×C3⋊(C3⋊C8)—38no—
47C2×C3⋊(C3⋊C8)—68no—
48C3×(C2×D3)⋊C4—38no—
49C3×Dic3⋊C4—38no—
50C3×C4×Dic3—34no—
51C2×C3×C3⋊C8C2×D318yes—
52C22×C33C22×C411yescharacteristic
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—22no—
55C33×C4—22no—
56C33×C4#4112yes—
57C3×C3⋊Dic3—21no—
58C3×C3⋊Dic3—22no—
59C3×#117—91no—
60#872—34no—
61C3×C4×C8—34no—
62#830—92no—
63(C2×D3)⋊C8—98no—
64Dic3⋊C8—98no—
65C4⋊(C3⋊C8)—92no—
66#803—94no—
67#803—91no—
68#803—184no—
69C4×C3⋊C8C3⋊D311yescharacteristic, complemented
70C22×C3⋊D3C2×Dic311yescharacteristic
71C22×C3×D3—34no—
72C2×C32×C4—28no—
73C2×C32×C4—24no—
74C2×C32×C4C4×D318yes—
75C2×C32×C4C2×Dic312yes—
76C2×C32×C4C22×D314yes—
77C2×C32×C4C22×D311yescharacteristic
78C2×C3⋊Dic3C2×Dic312yes—
79C2×C3⋊Dic3C2×Dic311yescharacteristic
80C4×C3⋊D3—22no—
81C2×C3×Dic3—38no—
82C2×C3×Dic3—34no—
83C3×C4×D3—68no—
84C32×C8—62no—
85C3⋊(C3⋊C8)—616no—
86C3⋊(C3⋊C8)—68no—
87C3⋊(C3⋊C8)—62no—
88C3×C3⋊C8—28no—
89#433—271noSylow
90C2×C33#11711yescharacteristic
91C2×C33#13112yes—
92C3×C3⋊D3—41no—
93C22×C3×C4—91no—
94C2×C4×D3—34no—
95C2×C3×C8—38no—
96C3×C4⋊C4—92no—
97C3×C22⋊C4—92no—
98C3×C42—24no—
99C3×C42—91no—
100C3×C42D3214yes—
101C3×C42C2×C3⋊D311yescharacteristic
102(C2×D3)⋊C4—38no—
103Dic3⋊C4—38no—
104C4×Dic3—34no—
105C2×C3⋊C8—98no—
106C2×C3⋊C8—92no—
107C2×C3⋊C8—188no—
108C2×C3⋊C8C2×C3⋊D312yes—
109C22×C32—24no—
110C22×C32C2×C4×D314yes—
111C22×C32C22×Dic311yescharacteristic
112C2×C3⋊D3—21no—
113C2×C3⋊D3—22no—
114C2×C3×D3—64no—
115C2×C3×D3—68no—
116C32×C4—22no—
117C32×C4—28no—
118C32×C4—28no—
119C32×C4—48no—
120C32×C4#23518yes—
121C32×C4#23512yes—
122C3⋊Dic3—21no—
123C3⋊Dic3—22no—
124C3×Dic3—64no—
125C3×Dic3—68no—
126#117—91no—
127C4⋊C8—272no—
128C22⋊C8—272no—
129#97—271no—
130C4×C8—31no—
131C33#43311yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—34no—
134C2×C3×C4—28no—
135C2×C3×C4—24no—
136C2×C3×C4—92no—
137C2×C3×C4—91no—
138C2×C3×C4—182no—
139C2×C3×C4D3×Dic318yes—
140C2×C3×C4C4×C3⋊D312yes—
141C2×C3×C4C2×D3214yes—
142C2×C3×C4C22×C3⋊D311yescharacteristic
143C2×Dic3—38no—
144C2×Dic3—34no—
145C4×D3—68no—
146C3×C8—68no—
147C3⋊C8—22no—
148C3⋊C8—1816no—
149C3⋊C8—188no—
150C3⋊C8—182no—
151C2×C32—28no—
152C2×C32—24no—
153C2×C32#87214yes—
154C2×C32#90118yes—
155C2×C32#92611yescharacteristic
156C2×C32#94812yes—
157C3⋊D3—41no—
158C3×D3—124no—
159C22×C4—91no—
160C2×C8—32no—
161C2×C8—272no—
162C4⋊C4—92no—
163C22⋊C4—92no—
164C42—91no—
165C42D3×C3⋊D311yescharacteristic
166C22×C3—24no—
167C22×C3—181no—
168C22×C3—182no—
169C22×C3C2×D3×Dic314yes—
170C22×C3C2×C4×C3⋊D311yescharacteristic
171C2×D3—68no—
172C2×D3—64no—
173C3×C4—28no—
174C3×C4—22no—
175C3×C4—28no—
176C3×C4—48no—
177C3×C4—181no—
178C3×C4—182no—
179C3×C4#379818yes—
180C3×C4#382812yes—
181Dic3—68no—
182Dic3—64no—
183C32—24no—
184C32#469014yescomplemented
185C32#500011yescharacteristic, complemented
186C23—91no—
187C2×C4—92no—
188C2×C4—91no—
189C2×C4—182no—
190C2×C4Dic3×C3⋊D312yes—
191C2×C4C2×D3×C3⋊D311yescharacteristic
192C8—62no—
193C8—542no—
194C2×C3—28no—
195C2×C3—24no—
196C2×C3—361no—
197C2×C3#6381418yes—
198C2×C3#6385514yes—
199C2×C3#6408111yescharacteristic
200C2×C3#6411012yes—
201D3—124no—
202C22—182no—
203C22—181no—
204C22C2×Dic3×C3⋊D311yescharacteristic
205C4—22no—
206C4—182no—
207C4—181no—
208C4#8902012yes—
209C3—24no—
210C3#1059026414yescomplemented
211C3#1059238911yescharacteristic, complemented
212C2—361no—
213C2#1172335212yes—
214C2#1172343711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11722173C2semidirect
3. C32×C4×C3⋊C8C2semidirect
4. #11722051C2semidirect
23. C3×C4×C3⋊C8D3semidirect
69. C4×C3⋊C8C3⋊D3semidirect
131. C33#433semidirect
184. C32#4690semidirect
185. C32#5000semidirect
210. C3#10590264semidirect
211. C3#10592389semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
#117221732
C32×C4×C3⋊C82
#117220512
#117220232
#105923893
#105902643