Finite Group Databasebeta

← #13173444

Subgroups — #13173444

Order 1728.

2776 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 2776 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C24×C3×C3⋊D3C211yesmaximal, characteristic
2C2×#88994C216yesmaximal, complemented
3#10593847—31nomaximal
4C24×C33C411yescharacteristic, complemented
5C23×C3×C3⋊D3—212no—
6C23×C3×C3⋊D3C413yescomplemented
7C23×C3×C3⋊D3C2213yes—
8C22×C3⋊#157C2214yescomplemented
9#88994—212no—
10C24×C3⋊D3D311yescharacteristic
11C24×C3×D3—62no—
12C2×C22⋊#157—36no—
13C23×C33—26no—
14C23×C33C2×C413yes—
15C22×C3×C3⋊D3—232no—
16C22×C3×C3⋊D3—26no—
17C22×C3×C3⋊D3—212no—
18C22×C3×C3⋊D3—212no—
19C22×C3×C3⋊D3C2×C413yes—
20C22×C3×C3⋊D3D4112yes—
21C22×C3×C3⋊D3C2311yescharacteristic
22C2×C3⋊#157—212no—
23#5247—91nomaximal
24C24×C32—22no—
25C24×C32—42no—
26C24×C32Dic311yescharacteristic, complemented
27C23×C3⋊D3—212no—
28C23×C3⋊D3Dic313yescomplemented
29C23×C3⋊D3C2×D313yes—
30C23×C3×D3—66no—
31C23×C3×D3—66no—
32C23×C3×D3—648no—
33C22×#157—34no—
34C22⋊#157—612no—
35C22×C33—28no—
36C22×C33—26no—
37C22×C33C22⋊C416yescomplemented
38C22×C33C22×C411yescharacteristic
39C2×C3×C3⋊D3—224no—
40C2×C3×C3⋊D3—26no—
41C2×C3×C3⋊D3—224no—
42C2×C3×C3⋊D3C22⋊C416yescomplemented
43C2×C3×C3⋊D3C2×D416yes—
44C3⋊#157—44no—
45C25×C3—91no—
46C24×D3—62no—
47C2×#217—96no—
48C23×C32—26no—
49C23×C32—224no—
50C23×C32—26no—
51C23×C32—424no—
52C23×C32—46no—
53C23×C32C2×Dic313yes—
54C22×C3⋊D3—232no—
55C22×C3⋊D3—26no—
56C22×C3⋊D3—212no—
57C22×C3⋊D3—212no—
58C22×C3⋊D3C2×Dic313yes—
59C22×C3⋊D3#67112yes—
60C22×C3⋊D3C22×D311yescharacteristic
61C22×C3×D3—66no—
62C22×C3×D3—624no—
63C22×C3×D3—62no—
64C22×C3×D3—648no—
65C22×C3×D3—648no—
66C22×C3×D3—6128no—
67C22×C3×D3—624no—
68C2×#157—612no—
69#379—271noSylow
70C2×C33—26no—
71C2×C33C2×C22⋊C413yes—
72C3×C3⋊D3—26no—
73C3×C3⋊D3#12014yes—
74C24×C3—22no—
75C24×C3—42no—
76C24×C3—93no—
77C24×C3—93no—
78C24×C3—1812no—
79C24×C3#15711yescharacteristic, complemented
80C23×D3—66no—
81C23×D3—66no—
82C23×D3—648no—
83C22×Dic3—94no—
84#217—1812no—
85C22×C32—28no—
86C22×C32—26no—
87C22×C32—232no—
88C22×C32—224no—
89C22×C32—22no—
90C22×C32—212no—
91C22×C32—432no—
92C22×C32—424no—
93C22×C32—412no—
94C22×C32—42no—
95C22×C32#21716yescomplemented
96C22×C32C22×Dic311yescharacteristic
97C2×C3⋊D3—224no—
98C2×C3⋊D3—26no—
99C2×C3⋊D3—224no—
100C2×C3⋊D3#21716yescomplemented
101C2×C3⋊D3C2×#6716yes—
102C2×C3×D3—612no—
103C2×C3×D3—612no—
104C2×C3×D3—696no—
105C2×C3×D3—696no—
106C2×C3×D3—624no—
107#157—124no—
108C25—91no—
109C2×C22⋊C4—276no—
110C33#37911yescharacteristic, complemented, Sylow
111C23×C3—224no—
112C23×C3—26no—
113C23×C3—26no—
114C23×C3—424no—
115C23×C3—46no—
116C23×C3—91no—
117C23×C3—93no—
118C23×C3—912no—
119C23×C3—1812no—
120C23×C3—1812no—
121C23×C3—1832no—
122C23×C3—186no—
123C23×C3C2×#15713yes—
124C22×D3—66no—
125C22×D3—624no—
126C22×D3—62no—
127C22×D3—648no—
128C22×D3—648no—
129C22×D3—6128no—
130C22×D3—624no—
131C2×Dic3—1812no—
132C2×C32—26no—
133C2×C32—224no—
134C2×C32—26no—
135C2×C32—424no—
136C2×C32—46no—
137C2×C32C2×#21713yes—
138C3⋊D3—26no—
139C3⋊D3#95314yes—
140C3×D3—68no—
141C3×D3—624no—
142C24—93no—
143C24—93no—
144C24—1812no—
145C24C3⋊#15711yescharacteristic, complemented
146C22×C4—274no—
147C22⋊C4—5412no—
148C22×C3—232no—
149C22×C3—224no—
150C22×C3—28no—
151C22×C3—26no—
152C22×C3—22no—
153C22×C3—212no—
154C22×C3—432no—
155C22×C3—424no—
156C22×C3—42no—
157C22×C3—412no—
158C22×C3—96no—
159C22×C3—96no—
160C22×C3—1824no—
161C22×C3—1824no—
162C22×C3—186no—
163C22×C3C22⋊#15716yescomplemented
164C22×C3C22×#15711yescharacteristic
165C2×D3—612no—
166C2×D3—612no—
167C2×D3—696no—
168C2×D3—696no—
169C2×D3—624no—
170Dic3—364no—
171C32—22no—
172C32—42no—
173C32#524711yescharacteristic, complemented
174C23—26no—
175C23—91no—
176C23—93no—
177C23—912no—
178C23—1812no—
179C23—1812no—
180C23—1832no—
181C23—186no—
182C23C2×C3⋊#15713yes—
183C2×C4—5412no—
184C2×C3—224no—
185C2×C3—26no—
186C2×C3—26no—
187C2×C3—424no—
188C2×C3—46no—
189C2×C3—94no—
190C2×C3—186no—
191C2×C3C2×C22⋊#15713yes—
192D3—68no—
193D3—624no—
194C22—28no—
195C22—26no—
196C22—96no—
197C22—96no—
198C22—1824no—
199C22—1824no—
200C22—186no—
201C22#8899416yescomplemented
202C22C22×C3⋊#15711yescharacteristic
203C4—1084no—
204C3—22no—
205C3—42no—
206C3#1059384711yescharacteristic, complemented
207C2—26no—
208C2—94no—
209C2—186no—
210C2C2×#8899413yes—
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC24×C33C4
prime radicalC24×C3×C3⋊D3C2
cosocleC22×C3×C3⋊D3C23
hypercenterC24C3⋊#157
hypocenterC33#379
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88994C2semidirect
4. C24×C33C4semidirect
6. C23×C3×C3⋊D3C4semidirect
8. C22×C3⋊#157C22semidirect
26. C24×C32Dic3semidirect
28. C23×C3⋊D3Dic3semidirect
37. C22×C33C22⋊C4semidirect
42. C2×C3×C3⋊D3C22⋊C4semidirect
79. C24×C3#157semidirect
95. C22×C32#217semidirect
100. C2×C3⋊D3#217semidirect
110. C33#379semidirect
145. C24C3⋊#157semidirect
163. C22×C3C22⋊#157semidirect
173. C32#5247semidirect
201. C22#88994semidirect
206. C3#10593847semidirect

Maximal subgroups

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

subgroupindex
C24×C3×C3⋊D32
C2×#889942
#105938473
#52479