Finite Group Databasebeta

← #13155247

Subgroups — #13155247

Order 1728.

492 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#11723647C211yesmaximal, characteristic, complemented
2#11723339C211yesmaximal, characteristic, complemented
3#11722011C212yesmaximal, complemented
4C3×#63546C212yesmaximal, complemented
5C3×#63544C211yesmaximal, characteristic, complemented
6#10591026—31nomaximal
7#10590232—32nomaximal
8C4⋊#1113—21no—
9#89014—21no—
10#88853C2211yescharacteristic, complemented
11C3×#3745C2212yes—
12C3×C8×C3⋊D3C2211yescharacteristic
13C32×D24C2211yescharacteristic, complemented
14#88810—21no—
15#88794—21no—
16C32×C3⋊C16C2212yescomplemented
17#64121—31no—
18#63807—32no—
19#63807—61no—
20#63797—31no—
21D3×D24—32no—
22#63627—32no—
23C3×#826—32no—
24C3×D48—32no—
25C3×#798—32no—
26#63548—32no—
27#63548—32no—
28#63548—62no—
29#63546D312yescomplemented
30#63544D311yescharacteristic, complemented
31C2×#1113—41no—
32#6308—21no—
33C3×C4⋊(C3⋊D3)—22no—
34C3×C4×C3⋊D3D411yescharacteristic
35C32×D12—21no—
36#6295—41no—
37#6290—41no—
38D3×C3⋊Dic3—41no—
39C33×C8C2311yescharacteristic
40C3×C3⋊(C3⋊C8)D411yescharacteristic
41#4911—92no—
42#4908—91no—
43#3830—61no—
44#3802—61no—
45D3×D12—62no—
46#3795—62no—
47#3795—62no—
48C32×D8—31no—
49#3754—61no—
50#3746—31no—
51#3746—32no—
52#3746—61no—
53#3746—62no—
54#3746—121no—
55#3745C2×D312yes—
56C8×C3⋊D3C2×D311yescharacteristic
57C3×D24—21no—
58C3×D24—32no—
59C3×D24—32no—
60C3×D24—62no—
61C3×D24C2×D312yescomplemented
62C3×D3×C8—32no—
63C3×D3×C8—61no—
64#3716—62no—
65#3716—62no—
66#3714—62no—
67#3713—61no—
68C32×C16—32no—
69C3×C3⋊C16—22no—
70C3×C3⋊C16C2×D312yescomplemented
71C3×C3⋊C16C2×D312yescomplemented
72C2×C3×C3⋊D3—42no—
73C2×C3×C3⋊D3D811yescharacteristic
74C2×C32×D3—41no—
75#1113—82no—
76C33×C4C2×D411yescharacteristic
77C3⋊(C3⋊Dic3)—41no—
78C3×C3⋊Dic3D811yescharacteristic
79#912—92no—
80#912—181no—
81D3×D8—92no—
82#904—91no—
83C2×D24—91no—
84C3×D16—92no—
85C3×M32—91no—
86#827—92no—
87#826—32no—
88#800—92no—
89#800—92no—
90#800—182no—
91D48—32no—
92#798—32no—
93C2×D32—122no—
94C2×D32—121no—
95C32×D4—61no—
96#638—121no—
97C2×C3⋊Dic3—121no—
98C4⋊(C3⋊D3)—22no—
99C4×C3⋊D3D1211yescharacteristic
100#634—61no—
101#634—62no—
102#634—62no—
103#634—122no—
104#634—122no—
105C3×D12—22no—
106C3×D12—22no—
107C3×D12—62no—
108C3×D12—62no—
109C3×D12—122no—
110C3×C4×D3—32no—
111C3×C4×D3—61no—
112#626—122no—
113#625—122no—
114#625—122no—
115#625—121no—
116D3×Dic3—122no—
117D3×Dic3—122no—
118C32×C8—22no—
119C32×C8—21no—
120C32×C8—41no—
121C32×C8C22×D311yescharacteristic
122C32×C8C22×D312yes—
123C3⋊(C3⋊C8)D1211yescharacteristic
124C3×C3⋊C8—32no—
125C3×C3⋊C8—61no—
126#509—271noSylow
127C2×C33C2×D811yescharacteristic
128C3×C3⋊D3—21no—
129C3×C3⋊D3—82no—
130C32×D3—81no—
131#237—182no—
132#237—182no—
133D3×D4—182no—
134#235—181no—
135C2×D12—181no—
136C3×D8—32no—
137C3×D8—61no—
138C3×D8—92no—
139C2×C3×C8—91no—
140#214—182no—
141#214—182no—
142#213—182no—
143Dic12—91no—
144Dic12—92no—
145Dic12—181no—
146Dic12—182no—
147Dic12—361no—
148D24—32no—
149D24—32no—
150D24—62no—
151D24—181no—
152D24D3211yescharacteristic, complemented
153#204—181no—
154D3×C8—32no—
155D3×C8—61no—
156C3×C16—32no—
157C3×C16—32no—
158C3×C16—62no—
159C3⋊C16D3212yescomplemented
160C22×C32—121no—
161C2×C3⋊D3—42no—
162C2×C3⋊D3D2411yescharacteristic
163C2×C3×D3—32no—
164C2×C3×D3—42no—
165C2×C3×D3—42no—
166C2×C3×D3—61no—
167C2×C3×D3—122no—
168C2×C3×D3—122no—
169C2×C3×D3—242no—
170D32—244no—
171D32—242no—
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—41no—
175C32×C4C2×D1211yescharacteristic
176C32×C4D3×D412yes—
177C3⋊Dic3—121no—
178C3⋊Dic3—122no—
179C3⋊Dic3—122no—
180C3⋊Dic3—242no—
181C3⋊Dic3—242no—
182C3⋊Dic3D2411yescharacteristic
183C3×Dic3—32no—
184C3×Dic3—61no—
185#135—271no—
186C2×D8—271no—
187SD32—272no—
188D16—92no—
189M32—91no—
190C33#50911yescharacteristic, complemented, Sylow
191C22×D3—362no—
192C22×D3—361no—
193C3×D4—62no—
194C3×D4—62no—
195C3×D4—182no—
196C2×C3×C4—91no—
197#67—362no—
198#67—362no—
199#67—361no—
200#67—362no—
201C2×Dic3—362no—
202C2×Dic3—362no—
203D12—21no—
204D12—62no—
205D12—62no—
206D12—122no—
207D12—361no—
208C4×D3—32no—
209C4×D3—61no—
210C4×D3—361no—
211Dic6—182no—
212Dic6—182no—
213Dic6—181no—
214Dic6—362no—
215Dic6—362no—
216C3×C8—22no—
217C3×C8—21no—
218C3×C8—41no—
219C3×C8—91no—
220C3×C8C2×D3212yes—
221C3×C8C2×D3211yescharacteristic
222C3⋊C8—32no—
223C3⋊C8—61no—
224C2×C32—22no—
225C2×C32—21no—
226C2×C32—41no—
227C2×C32—241no—
228C2×C32C2×D2411yescharacteristic
229C2×C32D3×D812yes—
230C3⋊D3—21no—
231C3⋊D3—82no—
232C3×D3—62no—
233C3×D3—82no—
234C3×D3—82no—
235C3×D3—121no—
236C3×D3—242no—
237C3×D3—242no—
238C3×D3—482no—
239#41—541no—
240C2×D4—541no—
241Q16—271no—
242SD16—541no—
243D8—31no—
244D8—92no—
245D8—541no—
246C2×C8—91no—
247C16—32no—
248C22×C3—91no—
249C22×C3—122no—
250C22×C3—122no—
251C22×C3—362no—
252C2×D3—32no—
253C2×D3—41no—
254C2×D3—61no—
255C2×D3—122no—
256C2×D3—122no—
257C2×D3—242no—
258C2×D3—724no—
259C2×D3—722no—
260C3×C4—21no—
261C3×C4—22no—
262C3×C4—41no—
263C3×C4—91no—
264C3×C4D3×D1212yes—
265C3×C4#381111yescharacteristic
266Dic3—32no—
267Dic3—61no—
268Dic3—362no—
269Dic3—362no—
270Dic3—361no—
271Dic3—722no—
272Dic3—722no—
273C32—22no—
274C32—21no—
275C32—41no—
276C32#490811yescharacteristic, complemented
277C32#491112yescomplemented
278C23—1081no—
279Q8—541no—
280D4—61no—
281D4—182no—
282D4—1081no—
283D4—1081no—
284C2×C4—91no—
285C2×C4—1081no—
286C8—91no—
287C8D3311yescharacteristic
288C2×C3—21no—
289C2×C3—22no—
290C2×C3—41no—
291C2×C3—181no—
292C2×C3—242no—
293C2×C3—242no—
294C2×C3—722no—
295C2×C3D3×D2412yes—
296C2×C3#6392611yescharacteristic
297D3—62no—
298D3—81no—
299D3—121no—
300D3—242no—
301D3—242no—
302D3—482no—
303C22—91no—
304C22—121no—
305C22—362no—
306C22—2162no—
307C4—91no—
308C4—1081no—
309C4#8895511yescharacteristic
310C3—21no—
311C3—22no—
312C3—41no—
313C3#1059023212yescomplemented
314C3#1059102611yescharacteristic, complemented
315C2—181no—
316C2—241no—
317C2—722no—
318C2#1172283311yescharacteristic
subgroupG/H
centerC2#11722833
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
hypocenterC33#509
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11723647C2semidirect
2. #11723339C2semidirect
3. #11722011C2semidirect
4. C3×#63546C2semidirect
5. C3×#63544C2semidirect
10. #88853C22semidirect
13. C32×D24C22semidirect
16. C32×C3⋊C16C22semidirect
29. #63546D3semidirect
30. #63544D3semidirect
61. C3×D24C2×D3semidirect
70. C3×C3⋊C16C2×D3semidirect
71. C3×C3⋊C16C2×D3semidirect
152. D24D32semidirect
159. C3⋊C16D32semidirect
190. C33#509semidirect
276. C32#4908semidirect
277. C32#4911semidirect
313. C3#10590232semidirect
314. C3#10591026semidirect

Maximal subgroups

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

subgroupindex
#117236472
#117233392
#117220112
C3×#635462
C3×#635442
#105910263
#105902323