Finite Group Databasebeta

← #13160240

Subgroups — #13160240

Order 1728.

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

normal subgroups only

This group has 608 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, complemented
2#11723341C211yesmaximal, characteristic, complemented
3#11722198C211yesmaximal, characteristic, complemented
4C32×#829C211yesmaximal, characteristic, complemented
5#11722008C211yesmaximal, characteristic, complemented
6#11722006C211yesmaximal, characteristic, complemented
7C3⋊D3×C3⋊C16C211yesmaximal, characteristic, complemented
8#10592618—31nomaximal
9#10591048—34nomaximal
10C3×#3832—21no—
11#89018—21no—
12C33×Q16C2211yescharacteristic
13#88852C2211yescharacteristic
14C3×#3755—21no—
15C3×#3745C2211yescharacteristic
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×Dic12C2211yescharacteristic
18#88795—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
20C32×C3⋊C16C2211yescharacteristic
21#64128—31no—
22#64128D311yescharacteristic, complemented
23C3×#919—34no—
24#63808—34no—
25C32×Q32—31no—
26#63655—31no—
27#63655—31no—
28#63655—34no—
29#63655—64no—
30#63626—31no—
31C16×C3⋊D3—31no—
32C3×#829D314yescomplemented
33#63547—34no—
34#63546—34no—
35D3×C3⋊C16—34no—
36C33×Q8—21no—
37C4⋊(C3⋊(C3⋊D3))—21no—
38C3×C4⋊(C3⋊D3)—21no—
39C3×C4⋊(C3⋊D3)—41no—
40C3×C4×C3⋊D3—41no—
41C3×C4×C3⋊D3D411yescharacteristic
42C32×Dic6—21no—
43#6292—41no—
44#6289—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47#5194—91no—
48#4919—94no—
49#3832—21no—
50#3832—61no—
51C3×#239—64no—
52#3799—64no—
53C32×Q16—24no—
54C32×Q16—31no—
55C32×Q16C2×D314yes—
56C32×Q16C2×D311yescharacteristic
57#3755—21no—
58#3755—61no—
59#3745—31no—
60#3745—34no—
61#3745—64no—
62#3745C2×D311yescharacteristic
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#215—64no—
65C3×Dic12C2×D314yes—
66C3×D24—34no—
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⋊D3—41no—
76C2×C3×C3⋊D3D811yescharacteristic
77C33×C4—41no—
78C33×C4C2×D411yescharacteristic
79C3×C3⋊Dic3D811yescharacteristic
80C32×Dic3—41no—
81C3×#135—91no—
82#919—34no—
83#919—94no—
84#904—91no—
85C3×Q32—34no—
86#829C3⋊D311yescharacteristic, complemented
87#828—94no—
88#828—94no—
89#828—91no—
90#828—184no—
91#827—91no—
92#826—91no—
93C2×C3⋊C16—91no—
94D48—94no—
95D3×C16—94no—
96C32×Q8—21no—
97C32×Q8—24no—
98C32×Q8—44no—
99C32×Q8—61no—
100C4⋊(C3⋊D3)—21no—
101C4⋊(C3⋊D3)—41no—
102C4⋊(C3⋊D3)—61no—
103C4⋊(C3⋊D3)—64no—
104C4⋊(C3⋊D3)—124no—
105C4⋊(C3⋊D3)—121no—
106C4×C3⋊D3—41no—
107C4×C3⋊D3—121no—
108C4×C3⋊D3#6711yescharacteristic
109C3×D12—64no—
110C3×D12—124no—
111C3×C4×D3—34no—
112C3×C4×D3—124no—
113C3×Dic6—24no—
114#626—124no—
115#624—124no—
116C32×C8—24no—
117C32×C8C22×D314yes—
118C32×C8C22×D311yescharacteristic
119C3⋊(C3⋊C8)#6711yescharacteristic
120C3×C3⋊C8—34no—
121#508—271noSylow
122C2×C33C2×D811yescharacteristic
123C3⋊(C3⋊D3)—81no—
124C3×C3⋊D3—21no—
125C3×C3⋊D3—81no—
126C3×#41—181no—
127#239—64no—
128#239—184no—
129#235—181no—
130C3×Q16—24no—
131C3×Q16—34no—
132C3×Q16D3214yes—
133C3×Q16C2×C3⋊D311yescharacteristic
134C3×SD16—181no—
135C3×D8—91no—
136C2×C3×C8—91no—
137#215—64no—
138#215—184no—
139Dic12C2×C3⋊D311yescharacteristic
140D24—34no—
141D24—94no—
142D24—91no—
143D24—184no—
144#204—181no—
145D3×C8—34no—
146C3×C16—34no—
147C3⋊C16—94no—
148C3⋊C16—91no—
149C3⋊C16—184no—
150C3⋊C16C2×C3⋊D311yescharacteristic
151C2×C3⋊D3—41no—
152C2×C3⋊D3—121no—
153C2×C3⋊D3—124no—
154C2×C3⋊D3—244no—
155C2×C3⋊D3#21311yescharacteristic
156C2×C3×D3—34no—
157C2×C3×D3—124no—
158C32×C4—24no—
159C32×C4—41no—
160C32×C4—44no—
161C32×C4—84no—
162C32×C4—121no—
163C32×C4D3×D414yes—
164C32×C4C2×#6711yescharacteristic
165C3⋊Dic3#21311yescharacteristic
166C3×Dic3—34no—
167C3×Dic3—44no—
168#135—91no—
169#135—271no—
170Q32—31no—
171SD32—271no—
172SD32—271no—
173D16—271no—
174C2×C16—271no—
175C33#50811yescharacteristic, complemented, Sylow
176C3×Q8—24no—
177C3×Q8—21no—
178C3×Q8—44no—
179C3×Q8—64no—
180C3×D4—181no—
181C3×D4—361no—
182C2×C3×C4—91no—
183C2×C3×C4—361no—
184#67—361no—
185D12—64no—
186D12—124no—
187D12—184no—
188D12—181no—
189D12—364no—
190D12—364no—
191C4×D3—34no—
192C4×D3—124no—
193C4×D3—364no—
194C4×D3—361no—
195Dic6—21no—
196C3×C8—24no—
197C3×C8—91no—
198C3×C8C2×D3214yes—
199C3×C8C22×C3⋊D311yescharacteristic
200C3⋊C8—34no—
201C2×C32—24no—
202C2×C32D3×D814yes—
203C2×C32C2×#21311yescharacteristic
204C3⋊D3—21no—
205C3⋊D3—81no—
206C3⋊D3—241no—
207C3⋊D3—244no—
208C3⋊D3—484no—
209C3×D3—64no—
210C3×D3—244no—
211#41—181no—
212#41—541no—
213Q16—31no—
214Q16D3×C3⋊D311yescharacteristic
215SD16—181no—
216SD16—541no—
217D8—91no—
218D8—271no—
219C2×C8—91no—
220C16—31no—
221C16—271no—
222C22×C3—91no—
223C22×C3—361no—
224C2×D3—34no—
225C2×D3—124no—
226C2×D3—361no—
227C2×D3—364no—
228C2×D3—724no—
229C3×C4—24no—
230C3×C4—41no—
231C3×C4—44no—
232C3×C4—84no—
233C3×C4—91no—
234C3×C4—124no—
235C3×C4D3×#6714yes—
236C3×C4D4×C3⋊D311yescharacteristic
237Dic3—34no—
238Dic3—41no—
239C32—24no—
240C32#491914yescomplemented
241C32#519411yescharacteristic, complemented
242Q8—21no—
243Q8—61no—
244D4—181no—
245D4—361no—
246D4—541no—
247D4—1081no—
248C2×C4—91no—
249C2×C4—361no—
250C2×C4—1081no—
251C8—91no—
252C8C2×D3×C3⋊D311yescharacteristic
253C2×C3—24no—
254C2×C3—181no—
255C2×C3—721no—
256C2×C3D3×#21314yes—
257C2×C3D8×C3⋊D311yescharacteristic
258D3—64no—
259D3—244no—
260D3—724no—
261D3—721no—
262D3—1444no—
263C22—91no—
264C22—361no—
265C22—1081no—
266C4—41no—
267C4—91no—
268C4—121no—
269C4C3⋊D3×#6711yescharacteristic
270C3—24no—
271C3#1059104814yescomplemented
272C3#1059261811yescharacteristic, complemented
273C2—181no—
274C2—721no—
275C2—2161no—
276C2C3⋊D3×#21311yescharacteristic
subgroupG/H
centerC2C3⋊D3×#213
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
hypercenterQ16D3×C3⋊D3
hypocenterC33#508
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×#64128C2semidirect
2. #11723341C2semidirect
3. #11722198C2semidirect
4. C32×#829C2semidirect
5. #11722008C2semidirect
6. #11722006C2semidirect
7. C3⋊D3×C3⋊C16C2semidirect
22. #64128D3semidirect
32. C3×#829D3semidirect
86. #829C3⋊D3semidirect
175. C33#508semidirect
240. C32#4919semidirect
241. C32#5194semidirect
271. C3#10591048semidirect
272. C3#10592618semidirect

Maximal subgroups

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

subgroupindex
C3×#641282
#117233412
#117221982
C32×#8292
#117220082
#117220062
C3⋊D3×C3⋊C162
#105926183
#105910483