Finite Group Databasebeta

← #13159550

Subgroups — #13159550

Order 1728.

1084 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 188 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×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723503C212yesmaximal
3C3×#63644C211yesmaximal, characteristic, complemented
4C32×#817C211yesmaximal, characteristic, complemented
5#11722053C212yesmaximal, complemented
6C3⋊D3×#106—31nomaximal
7#10590685—34nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×#6289—22no—
10C2×C33×C8C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C8×C3⋊(C3⋊D3)C414yescomplemented
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C32×C4⋊Dic3C2212yes—
15#88806—22no—
16C2×C8×C3⋊D3—31no—
17C2×C8×C3⋊D3—34no—
18C2×C8×C3⋊D3—64no—
19C4⋊C4×C3⋊D3—32no—
20#63900—38no—
21C32×#106—31no—
22#63644D311yescharacteristic, complemented
23#63635—32no—
24C3×#817—34no—
25C3×#817D314yescomplemented
26#63574—38no—
27C22×C3⋊(C3⋊D3)D411yescharacteristic
28C2×C33×C4C2311yescharacteristic
29C2×C3⋊(C3⋊Dic3)D411yescharacteristic
30C4×C3⋊(C3⋊D3)C2×C412yes—
31C4×C3⋊(C3⋊D3)Q812yes—
32C2×C3×C3⋊Dic3—22no—
33C2×C32×Dic3—22no—
34#6289—44no—
35C33×C8C2×C412yescomplemented
36C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
37D3×#106—94no—
38D3×#106—91no—
39C2×C4×C3⋊D3—31no—
40C2×C4×C3⋊D3—34no—
41C2×C4×C3⋊D3—64no—
42C2×C4×C3⋊D3—62no—
43C2×#624—68no—
44C2×C32×C8—24no—
45C2×C32×C8C2×D314yes—
46C2×C32×C8C2×D311yescharacteristic
47C32×C4⋊C4—32no—
48C4⋊(C3⋊Dic3)C2×D312yes—
49#3750—62no—
50C2×C3⋊(C3⋊C8)—31no—
51C2×C3⋊(C3⋊C8)—34no—
52C2×C3⋊(C3⋊C8)—64no—
53C8×C3⋊D3—34no—
54C8×C3⋊D3—316no—
55C8×C3⋊D3—616no—
56C3×C4⋊Dic3—38no—
57C3×C4⋊Dic3C2×D318yes—
58#3724—68no—
59C22×C33C2×D411yescharacteristic
60C2×C3⋊(C3⋊D3)C4⋊C412yes—
61C2×C3⋊(C3⋊D3)SD1614yes—
62C33×C4C22×C411yescharacteristic
63C33×C4C2×Q811yescharacteristic
64C3⋊(C3⋊Dic3)C4⋊C412yes—
65C3×C3⋊Dic3—42no—
66C32×Dic3—42no—
67C2×D3×C8—94no—
68C2×D3×C8—91no—
69C2×D3×C8—184no—
70D3×C4⋊C4—98no—
71D3×C4⋊C4—92no—
72C3×#106—34no—
73C3×#106—91no—
74#817—34no—
75#817C3⋊D311yescharacteristic, complemented
76#808—98no—
77#808—92no—
78C22×C3⋊D3—31no—
79C22×C3⋊D3—34no—
80C22×C3⋊D3—64no—
81C2×C32×C4—24no—
82C2×C32×C4—62no—
83C2×C32×C4C22×D314yes—
84C2×C32×C4C22×D311yescharacteristic
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—31no—
87C2×C3⋊Dic3—34no—
88C2×C3⋊Dic3—64no—
89C4×C3⋊D3—32no—
90C4×C3⋊D3—32no—
91C4×C3⋊D3—38no—
92C4×C3⋊D3—38no—
93C4×C3⋊D3—68no—
94C4×C3⋊D3—68no—
95C4×C3⋊D3—124no—
96C2×C3×Dic3—28no—
97C2×C3×Dic3—68no—
98#624—1216no—
99C32×C8—28no—
100C32×C8C4×D318yescomplemented
101C32×C8C4×D312yescomplemented
102C3⋊(C3⋊C8)—32no—
103C3⋊(C3⋊C8)—38no—
104C3⋊(C3⋊C8)—68no—
105C2×#106—271noSylow
106C2×C33C2×C4⋊C411yescharacteristic
107C2×C33C2×SD1612yes—
108C3⋊(C3⋊D3)#10614yescomplemented
109C2×C4×D3—94no—
110C2×C4×D3—91no—
111C2×C4×D3—184no—
112C2×C4×D3—182no—
113C2×C4×D3—188no—
114C2×C3×C8—24no—
115C2×C3×C8D3214yes—
116C2×C3×C8C2×C3⋊D311yescharacteristic
117C3×C4⋊C4—38no—
118C3×C4⋊C4—92no—
119C4⋊Dic3—38no—
120C4⋊Dic3C2×C3⋊D312yes—
121Dic3⋊C4—182no—
122Dic3⋊C4—188no—
123C2×C3⋊C8—94no—
124C2×C3⋊C8—91no—
125C2×C3⋊C8—184no—
126D3×C8—916no—
127D3×C8—94no—
128D3×C8—1816no—
129C22×C32—24no—
130C22×C32D3×D414yes—
131C22×C32D3×D411yescharacteristic
132C2×C3⋊D3—32no—
133C2×C3⋊D3—34no—
134C2×C3⋊D3—38no—
135C2×C3⋊D3—316no—
136C2×C3⋊D3—68no—
137C2×C3⋊D3—616no—
138C32×C4—24no—
139C32×C4—24no—
140C32×C4—122no—
141C32×C4C2×C4×D314yes—
142C32×C4C2×C4×D311yescharacteristic
143C32×C4D3×Q814yes—
144C32×C4D3×Q811yescharacteristic
145C3⋊Dic3—32no—
146C3⋊Dic3—38no—
147C3⋊Dic3—42no—
148C3⋊Dic3—68no—
149C3×Dic3—48no—
150C3×Dic3—128no—
151C22×C8—271no—
152C2×C4⋊C4—272no—
153#106—31no—
154#106—91no—
155#106—272no—
156C33C2×#10611yescharacteristic, complemented, Sylow
157C22×D3—94no—
158C22×D3—91no—
159C22×D3—184no—
160C2×C3×C4—24no—
161C2×C3×C4—68no—
162C2×C3×C4—182no—
163C2×C3×C4C2×D3214yes—
164C2×C3×C4C22×C3⋊D311yescharacteristic
165C2×Dic3—22no—
166C2×Dic3—68no—
167C2×Dic3—94no—
168C2×Dic3—91no—
169C2×Dic3—184no—
170C4×D3—98no—
171C4×D3—98no—
172C4×D3—92no—
173C4×D3—92no—
174C4×D3—188no—
175C4×D3—188no—
176C4×D3—364no—
177C4×D3—3616no—
178C3×C8—28no—
179C3×C8#62418yescomplemented
180C3×C8C4×C3⋊D312yescomplemented
181C3⋊C8—98no—
182C3⋊C8—92no—
183C3⋊C8—188no—
184C2×C32—28no—
185C2×C32—24no—
186C2×C32D3×C4⋊C414yes—
187C2×C32D3×C4⋊C411yescharacteristic
188C2×C32D3×SD1618yes—
189C2×C32D3×SD1612yes—
190C3⋊D3—34no—
191C3⋊D3—316no—
192C3⋊D3—616no—
193C22×C4—271no—
194C22×C4—542no—
195C2×C8—271no—
196C2×C8—274no—
197C2×C8D3×C3⋊D311yescharacteristic
198C4⋊C4—32no—
199C4⋊C4—92no—
200C4⋊C4—542no—
201C22×C3—24no—
202C22×C3#380214yes—
203C22×C3D4×C3⋊D311yescharacteristic
204C2×D3—916no—
205C2×D3—92no—
206C2×D3—94no—
207C2×D3—98no—
208C2×D3—188no—
209C2×D3—1816no—
210C3×C4—24no—
211C3×C4—24no—
212C3×C4—128no—
213C3×C4—362no—
214C3×C4#379714yes—
215C3×C4C2×#62414yes—
216C3×C4C2×C4×C3⋊D311yescharacteristic
217C3×C4Q8×C3⋊D311yescharacteristic
218Dic3—42no—
219Dic3—92no—
220Dic3—98no—
221Dic3—128no—
222Dic3—188no—
223C32—24no—
224C32D3×#10614yescomplemented
225C32D3×#10611yescharacteristic, complemented
226C23—271no—
227C2×C4—62no—
228C2×C4—182no—
229C2×C4—271no—
230C2×C4—272no—
231C2×C4—272no—
232C2×C4—1084no—
233C2×C4C2×D3×C3⋊D311yescharacteristic
234C8—272no—
235C8#628912yescomplemented
236C2×C3—28no—
237C2×C3—24no—
238C2×C3#6379618yes—
239C2×C3#6390014yes—
240C2×C3C4⋊C4×C3⋊D311yescharacteristic
241C2×C3SD16×C3⋊D312yes—
242D3—94no—
243D3—916no—
244D3—1816no—
245C22—274no—
246C22—272no—
247C22#8902611yescharacteristic
248C4—122no—
249C4—272no—
250C4—362no—
251C4#8901911yescharacteristic
252C4C2×#628911yescharacteristic
253C3—24no—
254C3#1059068514yescomplemented
255C3C3⋊D3×#10611yescharacteristic, complemented
256C2—274no—
257C2#1172332112yes—
258C2#1172350311yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#106
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63644C2semidirect
4. C32×#817C2semidirect
5. #11722053C2semidirect
12. C8×C3⋊(C3⋊D3)C4semidirect
22. #63644D3semidirect
25. C3×#817D3semidirect
35. C33×C8C2×C4semidirect
36. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
75. #817C3⋊D3semidirect
100. C32×C8C4×D3semidirect
101. C32×C8C4×D3semidirect
108. C3⋊(C3⋊D3)#106semidirect
156. C33C2×#106semidirect
179. C3×C8#624semidirect
180. C3×C8C4×C3⋊D3semidirect
224. C32D3×#106semidirect
225. C32D3×#106semidirect
235. C8#6289semidirect
254. C3#10590685semidirect
255. C3C3⋊D3×#106semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊(C3⋊D3)2
#117235032
C3×#636442
C32×#8172
#117220532
C3⋊D3×#1063
#105906853