Finite Group Databasebeta

← #13160239

Subgroups — #13160239

Order 1728.

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

normal subgroups only

This group has 596 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×Q16×C3⋊D3C211yesmaximal, characteristic, complemented
2#11723341C211yesmaximal, characteristic
3#11722198C211yesmaximal, characteristic, complemented
4C32×#829C211yesmaximal, characteristic, complemented
5#11722013C211yesmaximal, characteristic, complemented
6#11722012C211yesmaximal, characteristic, complemented
7#11722002C211yesmaximal, characteristic, complemented
8#10592617—31nomaximal
9#10591047—34nomaximal
10C3×Q8×C3⋊D3—21no—
11#89018—21no—
12C33×Q16C2211yescharacteristic
13#88852C2211yescharacteristic
14C3×#3756—21no—
15C3×#3746C2211yescharacteristic
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×Dic12C2211yescharacteristic
18#88795—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
20C32×C3⋊C16C2211yescharacteristic
21#64128—31no—
22Q16×C3⋊D3D311yescharacteristic, complemented
23C3×D3×Q16—34no—
24#63808—34no—
25C32×Q32—31no—
26#63656—31no—
27#63655—31no—
28#63655—34no—
29#63655—64no—
30#63627—31no—
31#63625—31no—
32C3×#829D314yescomplemented
33#63550—34no—
34#63549—34no—
35#63543—34no—
36C33×Q8—21no—
37C4⋊(C3⋊(C3⋊D3))—21no—
38C3×C4×C3⋊D3—41no—
39C3×C4×C3⋊D3D411yescharacteristic
40C3×#634—21no—
41C3×#634—41no—
42C32×Dic6—21no—
43#6292—41no—
44#6289—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47#5179—91no—
48#4918—94no—
49#3832—61no—
50Q8×C3⋊D3—21no—
51C3×D3×Q8—64no—
52#3799—64no—
53C32×Q16—24no—
54C32×Q16—31no—
55C32×Q16C2×D314yes—
56C32×Q16C2×D311yescharacteristic
57#3756—21no—
58#3755—61no—
59#3746C2×D311yescharacteristic
60#3745—31no—
61#3745—34no—
62#3745—64no—
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#216—64no—
65C3×Dic12—34no—
66C3×Dic12C2×D314yes—
67C3×D3×C8—34no—
68#3717—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—
81C2×C3×Q16—91no—
82#919—94no—
83D3×Q16—34no—
84#904—91no—
85C3×Q32—34no—
86#829—94no—
87#829—91no—
88#829C3⋊D311yescharacteristic, complemented
89#828—94no—
90#828—91no—
91#828—91no—
92#828—184no—
93#812—91no—
94#800—94no—
95#798—94no—
96C32×Q8—21no—
97C32×Q8—24no—
98C32×Q8—44no—
99C32×Q8—61no—
100C4⋊(C3⋊D3)—61no—
101C4⋊(C3⋊D3)—64no—
102C4⋊(C3⋊D3)—124no—
103C4⋊(C3⋊D3)—121no—
104C4×C3⋊D3—41no—
105C4×C3⋊D3—121no—
106C4×C3⋊D3#6711yescharacteristic
107#634—21no—
108#634—41no—
109C3×C4×D3—34no—
110C3×C4×D3—124no—
111C3×Dic6—24no—
112C3×Dic6—64no—
113C3×Dic6—124no—
114#626—124no—
115#624—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)—81no—
124C3×C3⋊D3—21no—
125C2×C3×Q8—181no—
126#239—184no—
127D3×Q8—64no—
128#235—181no—
129C3×Q16—24no—
130C3×Q16—34no—
131C3×Q16—91no—
132C3×Q16—181no—
133C3×Q16D3214yes—
134C3×Q16C2×C3⋊D311yescharacteristic
135C2×C3×C8—91no—
136#216—64no—
137#215—184no—
138Dic12—34no—
139Dic12C2×C3⋊D311yescharacteristic
140D24—94no—
141D24—91no—
142D24—184no—
143#204—181no—
144D3×C8—34no—
145C3×C16—34no—
146C3⋊C16—94no—
147C3⋊C16—91no—
148C3⋊C16—184no—
149C3⋊C16C2×C3⋊D311yescharacteristic
150C2×C3⋊D3—121no—
151C2×C3⋊D3—124no—
152C2×C3⋊D3—244no—
153C2×C3⋊D3#21311yescharacteristic
154C2×C3×D3—34no—
155C32×C4—24no—
156C32×C4—41no—
157C32×C4—44no—
158C32×C4—84no—
159C32×C4—121no—
160C32×C4D3×D414yes—
161C32×C4C2×#6711yescharacteristic
162C3⋊Dic3—41no—
163C3⋊Dic3#21311yescharacteristic
164C3×Dic3—34no—
165C3×Dic3—44no—
166C3×Dic3—124no—
167#135—271no—
168C2×Q16—91no—
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×Q8—181no—
180C3×Q8—361no—
181C2×C3×C4—91no—
182C2×C3×C4—361no—
183#67—361no—
184D12—184no—
185D12—181no—
186D12—364no—
187D12—364no—
188C4×D3—34no—
189C4×D3—124no—
190C4×D3—364no—
191C4×D3—361no—
192Dic6—21no—
193Dic6—64no—
194Dic6—124no—
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—241no—
205C3⋊D3—244no—
206C3⋊D3—484no—
207C3×D3—64no—
208#41—541no—
209C2×Q8—181no—
210Q16—31no—
211Q16—91no—
212Q16—181no—
213Q16D3×C3⋊D311yescharacteristic
214SD16—541no—
215D8—271no—
216C2×C8—91no—
217C16—31no—
218C16—271no—
219C22×C3—91no—
220C2×D3—34no—
221C2×D3—361no—
222C2×D3—364no—
223C2×D3—724no—
224C3×C4—24no—
225C3×C4—41no—
226C3×C4—44no—
227C3×C4—84no—
228C3×C4—91no—
229C3×C4—124no—
230C3×C4—361no—
231C3×C4D3×#6714yes—
232C3×C4D4×C3⋊D311yescharacteristic
233Dic3—34no—
234Dic3—41no—
235Dic3—124no—
236C32—24no—
237C32#491814yescomplemented
238C32#517911yescharacteristic, complemented
239Q8—21no—
240Q8—61no—
241Q8—181no—
242Q8—361no—
243D4—541no—
244D4—1081no—
245C2×C4—91no—
246C2×C4—361no—
247C2×C4—1081no—
248C8—91no—
249C8C2×D3×C3⋊D311yescharacteristic
250C2×C3—24no—
251C2×C3—181no—
252C2×C3D3×#21314yes—
253C2×C3D8×C3⋊D311yescharacteristic
254D3—64no—
255D3—724no—
256D3—721no—
257D3—1444no—
258C22—91no—
259C22—1081no—
260C4—41no—
261C4—91no—
262C4—121no—
263C4—361no—
264C4C3⋊D3×#6711yescharacteristic
265C3—24no—
266C3#1059104714yescomplemented
267C3#1059261711yescharacteristic, complemented
268C2—181no—
269C2—2161no—
270C2C3⋊D3×#21311yescharacteristic
subgroupG/H
centerC2C3⋊D3×#213
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
prime radical#11723341C2
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
1. C3×Q16×C3⋊D3C2semidirect
3. #11722198C2semidirect
4. C32×#829C2semidirect
5. #11722013C2semidirect
6. #11722012C2semidirect
7. #11722002C2semidirect
22. Q16×C3⋊D3D3semidirect
32. C3×#829D3semidirect
88. #829C3⋊D3semidirect
174. C33#510semidirect
237. C32#4918semidirect
238. C32#5179semidirect
266. C3#10591047semidirect
267. C3#10592617semidirect

Maximal subgroups

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

subgroupindex
C3×Q16×C3⋊D32
#117233412
#117221982
C32×#8292
#117220132
#117220122
#117220022
#105926173
#105910473