Finite Group Databasebeta

← #13160232

Subgroups — #13160232

Order 1728.

588 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 100 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×#64128C211yesmaximal, characteristic
2C3⋊D3×Dic12C211yesmaximal, characteristic, complemented
3#11722199C211yesmaximal, characteristic, complemented
4C32×#829C211yesmaximal, characteristic, complemented
5#11722011C211yesmaximal, characteristic, complemented
6#11722008C211yesmaximal, characteristic, complemented
7#11722002C211yesmaximal, characteristic, complemented
8#10592617—31nomaximal
9#10591042—34nomaximal
10C3×#3832—21no—
11C3⋊D3×Dic6—21no—
12C33×Q16C2211yescharacteristic
13#88853C2211yescharacteristic
14C3×#3755—21no—
15C3×#3745C2211yescharacteristic
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×Dic12C2211yescharacteristic
18#88799—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
20C32×C3⋊C16C2211yescharacteristic
21#64128D311yescharacteristic
22Q16×C3⋊D3—31no—
23C3×#919—34no—
24D3×Dic12—34no—
25C32×Q32—31no—
26#63656—31no—
27#63656—34no—
28#63656—64no—
29#63655—31no—
30#63627—31no—
31#63625—31no—
32C3×#829D314yescomplemented
33#63548—34no—
34#63547—34no—
35#63543—34no—
36C33×Q8—21no—
37#6308—21no—
38C3×C4⋊(C3⋊D3)—21no—
39C3×C4⋊(C3⋊D3)—41no—
40C3×C4×C3⋊D3—41no—
41C3×C4×C3⋊D3D411yescharacteristic
42C32×Dic6—21no—
43#6293—41no—
44Dic3×C3⋊D3—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47#5195—91no—
48#4918—94no—
49#3832—21no—
50Q8×C3⋊D3—61no—
51C3×#239—64no—
52D3×Dic6—64no—
53C32×Q16—24no—
54C32×Q16—31no—
55C32×Q16C2×D314yes—
56C32×Q16C2×D311yescharacteristic
57#3756—61no—
58#3755—21no—
59#3746—31no—
60#3746—34no—
61#3746—64no—
62#3745C2×D311yescharacteristic
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#215—64no—
65C3×Dic12C2×D314yes—
66C3×D24—34no—
67C3×D3×C8—34no—
68#3719—64no—
69C32×C16—31no—
70C3⋊(C3⋊C16)—31no—
71C3⋊(C3⋊C16)—34no—
72C3⋊(C3⋊C16)—64no—
73C3×C3⋊C16C2×D314yes—
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3D811yescharacteristic
76C33×C4—41no—
77C33×C4C2×D411yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3D811yescharacteristic
80C32×Dic3—41no—
81C3×#135—91no—
82#919—34no—
83D3×Q16—94no—
84C2×Dic12—91no—
85C3×Q32—34no—
86#829—94no—
87#829—91no—
88#829—184no—
89#829C3⋊D311yescharacteristic, complemented
90#828—94no—
91#827—91no—
92#827—91no—
93#812—91no—
94#800—94no—
95#798—94no—
96C32×Q8—21no—
97C32×Q8—24no—
98C32×Q8—44no—
99C32×Q8—61no—
100C4⋊(C3⋊D3)—21no—
101C4⋊(C3⋊D3)—41no—
102C4×C3⋊D3—41no—
103C4×C3⋊D3—121no—
104C4×C3⋊D3#6711yescharacteristic
105#634—61no—
106#634—64no—
107#634—124no—
108#634—121no—
109C3×D12—64no—
110C3×D12—124no—
111C3×C4×D3—34no—
112C3×C4×D3—124no—
113C3×Dic6—24no—
114#627—124no—
115D3×Dic3—124no—
116C32×C8—24no—
117C32×C8C22×D314yes—
118C32×C8C22×D311yescharacteristic
119C3⋊(C3⋊C8)#6711yescharacteristic
120C3×C3⋊C8—34no—
121#510—271noSylow
122C2×C33C2×D811yescharacteristic
123C3×C3⋊D3—21no—
124C3×C3⋊D3—81no—
125C3×#41—181no—
126#239—64no—
127D3×Q8—184no—
128C2×Dic6—181no—
129C3×Q16—24no—
130C3×Q16—34no—
131C3×Q16D3214yes—
132C3×Q16C2×C3⋊D311yescharacteristic
133C3×SD16—181no—
134C3×D8—91no—
135C2×C3×C8—91no—
136#216—184no—
137#215—64no—
138Dic12—94no—
139Dic12—91no—
140Dic12—184no—
141Dic12—181no—
142Dic12C2×C3⋊D311yescharacteristic
143D24—34no—
144D3×C8—34no—
145C3×C16—34no—
146C3⋊C16—94no—
147C3⋊C16—91no—
148C3⋊C16—184no—
149C3⋊C16C2×C3⋊D311yescharacteristic
150C2×C3⋊D3—41no—
151C2×C3⋊D3#21311yescharacteristic
152C2×C3×D3—34no—
153C2×C3×D3—124no—
154C32×C4—24no—
155C32×C4—41no—
156C32×C4—44no—
157C32×C4—84no—
158C32×C4—121no—
159C32×C4D3×D414yes—
160C32×C4C2×#6711yescharacteristic
161C3⋊Dic3—121no—
162C3⋊Dic3—124no—
163C3⋊Dic3—244no—
164C3⋊Dic3#21311yescharacteristic
165C3×Dic3—34no—
166C3×Dic3—44no—
167#135—91no—
168C2×Q16—271no—
169Q32—31no—
170Q32—271no—
171SD32—271no—
172SD32—271no—
173M32—271no—
174C33#51011yescharacteristic, complemented, Sylow
175C3×Q8—24no—
176C3×Q8—21no—
177C3×Q8—44no—
178C3×Q8—64no—
179C3×D4—181no—
180C3×D4—361no—
181C2×C3×C4—91no—
182C2×C3×C4—361no—
183C2×Dic3—361no—
184D12—64no—
185D12—124no—
186C4×D3—34no—
187C4×D3—124no—
188C4×D3—364no—
189Dic6—21no—
190Dic6—184no—
191Dic6—181no—
192Dic6—364no—
193Dic6—364no—
194Dic6—361no—
195C3×C8—24no—
196C3×C8—91no—
197C3×C8C2×D3214yes—
198C3×C8C22×C3⋊D311yescharacteristic
199C3⋊C8—34no—
200C2×C32—24no—
201C2×C32D3×D814yes—
202C2×C32C2×#21311yescharacteristic
203C3⋊D3—21no—
204C3⋊D3—81no—
205C3×D3—64no—
206C3×D3—244no—
207#41—181no—
208C2×Q8—541no—
209Q16—31no—
210Q16—271no—
211Q16—541no—
212Q16D3×C3⋊D311yescharacteristic
213SD16—181no—
214D8—91no—
215C2×C8—91no—
216C16—31no—
217C16—271no—
218C22×C3—91no—
219C22×C3—361no—
220C2×D3—34no—
221C2×D3—124no—
222C3×C4—24no—
223C3×C4—41no—
224C3×C4—44no—
225C3×C4—84no—
226C3×C4—91no—
227C3×C4—124no—
228C3×C4D3×#6714yes—
229C3×C4D4×C3⋊D311yescharacteristic
230Dic3—34no—
231Dic3—41no—
232Dic3—361no—
233Dic3—364no—
234Dic3—724no—
235C32—24no—
236C32#491814yescomplemented
237C32#519511yescharacteristic, complemented
238Q8—21no—
239Q8—61no—
240Q8—541no—
241Q8—1081no—
242D4—181no—
243D4—361no—
244C2×C4—91no—
245C2×C4—361no—
246C2×C4—1081no—
247C8—91no—
248C8C2×D3×C3⋊D311yescharacteristic
249C2×C3—24no—
250C2×C3—181no—
251C2×C3—721no—
252C2×C3D3×#21314yes—
253C2×C3D8×C3⋊D311yescharacteristic
254D3—64no—
255D3—244no—
256C22—91no—
257C22—361no—
258C4—41no—
259C4—91no—
260C4—121no—
261C4—1081no—
262C4C3⋊D3×#6711yescharacteristic
263C3—24no—
264C3#1059104214yescomplemented
265C3#1059261711yescharacteristic, complemented
266C2—181no—
267C2—721no—
268C2C3⋊D3×#21311yescharacteristic
subgroupG/H
centerC2C3⋊D3×#213
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
prime radicalC3×#64128C2
hypercenterQ16D3×C3⋊D3
hypocenterC33#510
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×Dic12C2semidirect
3. #11722199C2semidirect
4. C32×#829C2semidirect
5. #11722011C2semidirect
6. #11722008C2semidirect
7. #11722002C2semidirect
32. C3×#829D3semidirect
89. #829C3⋊D3semidirect
174. C33#510semidirect
236. C32#4918semidirect
237. C32#5195semidirect
264. C3#10591042semidirect
265. C3#10592617semidirect

Maximal subgroups

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

subgroupindex
C3×#641282
C3⋊D3×Dic122
#117221992
C32×#8292
#117220112
#117220082
#117220022
#105926173
#105910423