Finite Group Databasebeta

← #13160244

Subgroups — #13160244

Order 1728.

596 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C32×D3×Q16C211yesmaximal, characteristic, complemented
2#11723337C211yesmaximal, characteristic
3#11722198C211yesmaximal, characteristic, complemented
4C3×#63656C211yesmaximal, characteristic, complemented
5#11722013C211yesmaximal, characteristic, complemented
6#11722010C211yesmaximal, characteristic, complemented
7#11722001C211yesmaximal, characteristic, complemented
8#10592878—31nomaximal
9#10591047—34nomaximal
10C32×D3×Q8—21no—
11#89022—21no—
12C33×Q16C2211yescharacteristic
13#88852C2211yescharacteristic
14C3×#3746C2211yescharacteristic
15C32×#216—21no—
16C32×Dic12C2211yescharacteristic
17C32×D3×C8C2211yescharacteristic
18#88797—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
20C3×C3⋊(C3⋊C16)C2211yescharacteristic
21C2×C32×Q16—31no—
22#64113—31no—
23C3×D3×Q16D314yescomplemented
24#63808—34no—
25#63656—31no—
26#63656D311yescharacteristic, complemented
27#63655—31no—
28#63655—31no—
29#63655—34no—
30#63655—64no—
31#63639—31no—
32C3×#829—34no—
33#63550—34no—
34#63549—34no—
35#63543—34no—
36C33×Q8—21no—
37C4⋊(C3⋊(C3⋊D3))—21no—
38C3×#634—21no—
39C32×C4×D3—41no—
40C32×C4×D3D411yescharacteristic
41C32×Dic6—21no—
42C32×Dic6—41no—
43#6291—41no—
44#6289—41no—
45C33×C8C2311yescharacteristic
46C32×C3⋊C8D411yescharacteristic
47#5179—94no—
48#4918—91no—
49C2×C32×Q8—61no—
50#3828—61no—
51C3×D3×Q8—24no—
52#3799—64no—
53C32×Q16—24no—
54C32×Q16—31no—
55C32×Q16—61no—
56C32×Q16C2×D314yes—
57C32×Q16C2×D311yescharacteristic
58C2×C32×C8—31no—
59#3746C2×D311yescharacteristic
60#3745—31no—
61#3745—34no—
62#3745—64no—
63#3744—61no—
64C3×#216—24no—
65C3×Dic12—34no—
66C3×Dic12C2×D314yes—
67C3×D3×C8C2×D314yes—
68#3717—64no—
69C3⋊(C3⋊C16)—31no—
70C3⋊(C3⋊C16)—34no—
71C3⋊(C3⋊C16)—64no—
72C3⋊(C3⋊C16)C2×D311yescharacteristic
73C3×C3⋊C16—34no—
74C2×C3⋊(C3⋊D3)—41no—
75C2×C32×D3D811yescharacteristic
76C33×C4—41no—
77C33×C4C2×D411yescharacteristic
78C3×C3⋊Dic3—41no—
79C32×Dic3—41no—
80C32×Dic3D811yescharacteristic
81C2×C3×Q16—34no—
82#919—91no—
83D3×Q16C3⋊D311yescharacteristic, complemented
84#904—94no—
85C3×Q32—91no—
86#829—34no—
87#829—94no—
88#829—91no—
89#828—94no—
90#828—94no—
91#828—91no—
92#828—184no—
93#812—94no—
94#800—91no—
95#798—91no—
96C32×Q8—21no—
97C32×Q8—24no—
98C32×Q8—44no—
99C32×Q8—61no—
100C32×Q8—121no—
101C2×C32×C4—31no—
102C2×C32×C4—121no—
103#638—121no—
104C4⋊(C3⋊D3)—61no—
105C4⋊(C3⋊D3)—64no—
106C4⋊(C3⋊D3)—124no—
107C4×C3⋊D3—121no—
108#634—21no—
109C3×C4×D3—44no—
110C3×C4×D3#6714yes—
111C3×Dic6—24no—
112C3×Dic6—44no—
113C3×Dic6—64no—
114#626—124no—
115#624—124no—
116C32×C8—24no—
117C32×C8—31no—
118C32×C8C22×D314yes—
119C32×C8C22×D311yescharacteristic
120C3×C3⋊C8#6714yes—
121#510—271noSylow
122C2×C33C2×D811yescharacteristic
123C3⋊(C3⋊D3)—81no—
124C32×D3—21no—
125C2×C3×Q8—64no—
126#239—181no—
127D3×Q8—21no—
128#235—184no—
129C3×Q16—24no—
130C3×Q16—34no—
131C3×Q16—64no—
132C3×Q16—91no—
133C3×Q16D3214yes—
134C3×Q16C2×C3⋊D311yescharacteristic
135C2×C3×C8—34no—
136#216—21no—
137#215—181no—
138Dic12—34no—
139Dic12C2×C3⋊D311yescharacteristic
140D24—94no—
141D24—91no—
142D24—184no—
143#204—184no—
144D3×C8C2×C3⋊D311yescharacteristic
145C3×C16—91no—
146C3⋊C16—34no—
147C3⋊C16—94no—
148C3⋊C16—91no—
149C3⋊C16—184no—
150C22×C32—31no—
151C2×C3⋊D3—121no—
152C2×C3⋊D3—124no—
153C2×C3⋊D3—244no—
154C2×C3×D3#21314yes—
155C32×C4—24no—
156C32×C4—31no—
157C32×C4—41no—
158C32×C4—44no—
159C32×C4—84no—
160C32×C4—121no—
161C32×C4D3×D411yescharacteristic
162C32×C4C2×#6714yes—
163C3⋊Dic3—41no—
164C3×Dic3—44no—
165C3×Dic3—124no—
166C3×Dic3#21314yes—
167#135—271no—
168C2×Q16—31no—
169Q32—91no—
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—124no—
180C3×Q8—181no—
181C2×C3×C4—34no—
182C2×C3×C4—124no—
183#67—364no—
184D12—184no—
185D12—181no—
186D12—364no—
187D12—361no—
188C4×D3—41no—
189C4×D3—364no—
190C4×D3—361no—
191C4×D3#63811yescharacteristic
192Dic6—21no—
193Dic6—41no—
194Dic6—64no—
195C3×C8—24no—
196C3×C8—34no—
197C3×C8C2×D3214yes—
198C3×C8C22×C3⋊D311yescharacteristic
199C3⋊C8#63811yescharacteristic
200C2×C32—24no—
201C2×C32—61no—
202C2×C32D3×D811yescharacteristic
203C2×C32C2×#21314yes—
204C3⋊D3—241no—
205C3⋊D3—244no—
206C3⋊D3—484no—
207C3×D3—24no—
208#41—541no—
209C2×Q8—61no—
210Q16—31no—
211Q16—61no—
212Q16—91no—
213Q16D3×C3⋊D311yescharacteristic
214SD16—541no—
215D8—271no—
216C2×C8—31no—
217C16—91no—
218C16—271no—
219C22×C3—34no—
220C2×D3—361no—
221C2×D3—364no—
222C2×D3—724no—
223C2×D3#375311yescharacteristic
224C3×C4—24no—
225C3×C4—34no—
226C3×C4—41no—
227C3×C4—44no—
228C3×C4—84no—
229C3×C4—124no—
230C3×C4—361no—
231C3×C4D3×#6714yes—
232C3×C4C2×#63811yescharacteristic
233Dic3—41no—
234Dic3—124no—
235Dic3#375311yescharacteristic
236C32—24no—
237C32#491811yescharacteristic, complemented
238C32#517914yescomplemented
239Q8—21no—
240Q8—61no—
241Q8—121no—
242Q8—181no—
243D4—541no—
244D4—1081no—
245C2×C4—31no—
246C2×C4—121no—
247C2×C4—1081no—
248C8—31no—
249C8C2×D3×C3⋊D311yescharacteristic
250C2×C3—24no—
251C2×C3—64no—
252C2×C3D3×#21314yes—
253C2×C3C2×#375311yescharacteristic
254D3—21no—
255D3—724no—
256D3—721no—
257D3—1444no—
258C22—31no—
259C22—1081no—
260C4—31no—
261C4—41no—
262C4—121no—
263C4—361no—
264C4D3×#63811yescharacteristic
265C3—24no—
266C3#1059104714yescomplemented
267C3#1059287811yescharacteristic, complemented
268C2—61no—
269C2—2161no—
270C2D3×#375311yescharacteristic
subgroupG/H
centerC2D3×#3753
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
prime radical#11723337C2
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. C32×D3×Q16C2semidirect
3. #11722198C2semidirect
4. C3×#63656C2semidirect
5. #11722013C2semidirect
6. #11722010C2semidirect
7. #11722001C2semidirect
23. C3×D3×Q16D3semidirect
26. #63656D3semidirect
83. D3×Q16C3⋊D3semidirect
174. C33#510semidirect
237. C32#4918semidirect
238. C32#5179semidirect
266. C3#10591047semidirect
267. C3#10592878semidirect

Maximal subgroups

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

subgroupindex
C32×D3×Q162
#117233372
#117221982
C3×#636562
#117220132
#117220102
#117220012
#105928783
#105910473