Finite Group Databasebeta

← #13166074

Subgroups — #13166074

Order 1728.

418 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 44 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×C4⋊(C3⋊S4)C212yesmaximal, complemented
2A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3#10593179—32nomaximal
4#10591166—31nomaximal
5C2×C3×C3⋊S4—22no—
6C32×C4×A4C2211yescharacteristic
7#88810—41nomaximal
8C4⋊(C3⋊S4)D312yescomplemented
9C3×C4⋊S4—32no—
10C3×C4⋊S4—62no—
11C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
12C3×#930—32no—
13C2×#3713—31no—
14A4×C3⋊C8—32no—
15A4×C3⋊C8—61no—
16#63576—31no—
17#63562—32no—
18C2×C32×A4D411yescharacteristic
19C3×C3⋊S4—42no—
20C3×C4⋊(C3⋊D3)—42no—
21C3×C3⋊(C3⋊C8)—41no—
22#5402—91no—
23#5038—92no—
24C2×C3⋊S4—22no—
25C2×C3×S4—64no—
26C2×C3×S4—62no—
27C22×C32×C4C2×D311yescharacteristic
28C2×C3×#67—62no—
29C2×C3×D12—32no—
30C3×C4×A4—22no—
31C3×C4×A4—21no—
32C3×C4×A4—41no—
33C3×C4×A4C2×D312yes—
34C2×C3⋊(C3⋊C8)—31no—
35C2×C3⋊(C3⋊C8)—31no—
36C3×(C2×D3)⋊C4—62no—
37C3×C4⋊Dic3—32no—
38#3714—122no—
39#3713—61no—
40#3713—61no—
41C2×C3×C3⋊D3—82no—
42C32×A4D811yescharacteristic, complemented
43C33×C4—41no—
44C4⋊S4—32no—
45C4⋊S4—62no—
46C3×#121—92no—
47C2×#213—92no—
48#930—32no—
49C22×C3⋊C8—32no—
50C22×C3⋊C8—61no—
51C8×A4—91no—
52#832—92no—
53#809—92no—
54#807—92no—
55C23×C32D1211yescharacteristic
56C22×C3×D3—62no—
57C2×C3×A4—22no—
58C2×C3×A4—21no—
59C2×C3×A4—41no—
60C2×C3×A4#6712yes—
61C3⋊S4—42no—
62C3×S4—124no—
63C3×S4—122no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C4⋊(C3⋊D3)—42no—
67C3×#67—122no—
68C3×#67—122no—
69C2×C3×Dic3—62no—
70C3×D12—62no—
71C3×D12—62no—
72C3×D12—122no—
73C3×D12—242no—
74C3⋊(C3⋊C8)—31no—
75C3⋊(C3⋊C8)S411yescharacteristic, complemented
76C3×C3⋊C8—122no—
77C3×C3⋊C8—241no—
78#466—271noSylow
79C2×C33—41no—
80C3×C3⋊D3—162no—
81C2×S4—64no—
82C2×S4—62no—
83C2×C3×D4—92no—
84C2×C3×D4—182no—
85C22×C3×C4—21no—
86C22×C3×C4D3212yes—
87C2×#67—62no—
88C2×D12—32no—
89C4×A4—22no—
90C4×A4—41no—
91C4×A4D3211yescharacteristic
92C3×C4⋊C4—92no—
93C3×C22⋊C4—182no—
94#213—182no—
95#213—182no—
96(C2×D3)⋊C4—62no—
97C4⋊Dic3—32no—
98C2×C3⋊C8—92no—
99C2×C3⋊C8—92no—
100C2×C3⋊C8—181no—
101C2×C3⋊C8—181no—
102D24—361no—
103C22×C32—31no—
104C22×C32—31no—
105C22×C32D2411yescharacteristic, complemented
106C2×C3⋊D3—82no—
107C2×C3×D3—122no—
108C2×C3×D3—122no—
109C2×C3×D3—122no—
110C2×C3×D3—242no—
111C2×C3×D3—244no—
112C3×A4—22no—
113C3×A4—21no—
114C3×A4—41no—
115C3×A4#21312yescomplemented
116C32×C4—31no—
117C32×C4—42no—
118C32×C4—81no—
119C32×C4—82no—
120C32×C4—161no—
121C32×C4C2×S411yescharacteristic
122C3×Dic3—122no—
123C2×D8—271no—
124C22×C8—91no—
125#121—92no—
126#107—271no—
127#102—272no—
128C33—41noSylow
129C23×C3—21no—
130C23×C3—182no—
131C23×C3#62612yes—
132C22×D3—62no—
133C2×A4—22no—
134C2×A4—41no—
135C2×A4#62511yescharacteristic
136S4—124no—
137S4—122no—
138C3×D4—182no—
139C3×D4—182no—
140C3×D4—362no—
141C3×D4—362no—
142C2×C3×C4—32no—
143C2×C3×C4—32no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146C2×C3×C4—182no—
147#67—122no—
148#67—122no—
149C2×Dic3—62no—
150D12—62no—
151D12—62no—
152D12—122no—
153D12—242no—
154C3×C8—361no—
155C3⋊C8—32no—
156C3⋊C8—61no—
157C3⋊C8—92no—
158C3⋊C8—181no—
159C2×C32—31no—
160C2×C32—31no—
161C2×C32—42no—
162C2×C32—81no—
163C2×C32—82no—
164C2×C32—161no—
165C2×C32C4⋊S411yescharacteristic
166C3⋊D3—162no—
167C3×D3—242no—
168C3×D3—482no—
169C3×D3—484no—
170C2×D4—92no—
171C2×D4—182no—
172C22×C4#111311yescharacteristic
173D8—541no—
174D8—541no—
175C2×C8—271no—
176C2×C8—271no—
177C4⋊C4—92no—
178C22⋊C4—182no—
179C22×C3—21no—
180C22×C3—32no—
181C22×C3—32no—
182C22×C3—61no—
183C22×C3—61no—
184C22×C3—362no—
185C22×C3—362no—
186C22×C3—362no—
187C22×C3#371412yescomplemented
188C2×D3—122no—
189C2×D3—122no—
190C2×D3—122no—
191C2×D3—242no—
192C2×D3—244no—
193A4—22no—
194A4—41no—
195A4#371311yescharacteristic, complemented
196C3×C4—21no—
197C3×C4—32no—
198C3×C4—41no—
199C3×C4—61no—
200C3×C4—82no—
201C3×C4—161no—
202C3×C4—362no—
203C3×C4D3×S412yes—
204Dic3—122no—
205C32—42no—
206C32—81no—
207C32—82no—
208C32—161no—
209C32#540211yescharacteristic, complemented
210C23—182no—
211C23#629511yescharacteristic
212D4—182no—
213D4—182no—
214D4—362no—
215D4—362no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—182no—
219C8—91no—
220C8—271no—
221C2×C3—21no—
222C2×C3—32no—
223C2×C3—32no—
224C2×C3—41no—
225C2×C3—61no—
226C2×C3—61no—
227C2×C3—82no—
228C2×C3—161no—
229C2×C3—722no—
230C2×C3#6420712yes—
231D3—242no—
232D3—482no—
233D3—484no—
234C22—31no—
235C22—31no—
236C22—362no—
237C22—362no—
238C22—362no—
239C22#8881011yescharacteristic, complemented
240C4—31no—
241C4—362no—
242C4#8910111yescharacteristic
243C3—21no—
244C3—41no—
245C3—82no—
246C3—161no—
247C3#1059317912yescomplemented
248C2—31no—
249C2—31no—
250C2—722no—
251C2#1172421411yescharacteristic
subgroupG/H
centerC2#11724214
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4#89101
FittingC22×C32×C4C2×D3
socleC23×C32D12
hypocenterC32×A4D8
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×C4⋊(C3⋊S4)C2semidirect
2. A4×C3⋊(C3⋊C8)C2semidirect
8. C4⋊(C3⋊S4)D3semidirect
11. C22×C3⋊(C3⋊C8)D3semidirect
42. C32×A4D8semidirect
75. C3⋊(C3⋊C8)S4semidirect
105. C22×C32D24semidirect
115. C3×A4#213semidirect
187. C22×C3#3714semidirect
195. A4#3713semidirect
209. C32#5402semidirect
239. C22#88810semidirect
247. C3#10593179semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊(C3⋊S4)2
A4×C3⋊(C3⋊C8)2
#105931793
#105911663
#888104