Finite Group Databasebeta

← C3⋊D3×#137 #10596247

Subgroups — C3⋊D3×#137 #10596247

Order 576.

774 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 181 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#41×C3⋊D3C211yesmaximal, characteristic
2C2×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#137C211yesmaximal, characteristic, complemented
4#64160C211yesmaximal, characteristic, complemented
5#64151C211yesmaximal, characteristic, complemented
6#64127C212yesmaximal, complemented
7Q16×C3⋊D3C212yesmaximal, complemented
8#64124C212yesmaximal, complemented
9SD16×C3⋊D3C212yesmaximal, complemented
10#64118C211yesmaximal, characteristic, complemented
11M16×C3⋊D3C211yesmaximal, characteristic, complemented
12D3×#137—34nomaximal
13C32×#41C2211yescharacteristic
14C2×C32×Q8C2211yescharacteristic, complemented
15#3832C2211yescharacteristic
16Q8×C3⋊D3—21no—
17Q8×C3⋊D3—21no—
18Q8×C3⋊D3C2212yes—
19Q8×C3⋊D3C2211yescharacteristic
20Q8×C3⋊D3C2212yescomplemented
21#3830—21no—
22#3830C2211yescharacteristic
23D4×C3⋊D3—21no—
24D4×C3⋊D3C2211yescharacteristic
25#3828—21no—
26#3828C2211yescharacteristic
27C2×C4×C3⋊D3—21no—
28C2×C4×C3⋊D3—21no—
29C2×C4×C3⋊D3C2211yescharacteristic
30C2×#634—21no—
31C2×#634C2211yescharacteristic, complemented
32C32×Q16C2212yescomplemented
33C32×SD16C2212yescomplemented
34C32×M16C2211yescharacteristic, complemented
35#3756C2212yescomplemented
36#3756C2212yescomplemented
37#3755C2212yescomplemented
38#3754C2212yescomplemented
39#3748C2211yescharacteristic, complemented
40#3746C2212yescomplemented
41#3744C2212yescomplemented
42#3743C2212yescomplemented
43C8×C3⋊D3C2212yes—
44D3×#41—34no—
45C2×D3×Q8—34no—
46C3×#137D314yescomplemented
47#951—34no—
48#942—34no—
49#918—38no—
50D3×Q16—38no—
51#915—38no—
52D3×SD16—38no—
53#909—34no—
54D3×M16—34no—
55C22×C3⋊D3—21no—
56C22×C3⋊D3D411yescharacteristic
57C32×Q8—21no—
58C32×Q8C2312yes—
59C32×Q8C2311yescharacteristic
60C32×D4—21no—
61C32×D4C2311yescharacteristic
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4C2311yescharacteristic
65#638—21no—
66#638—21no—
67#638—21no—
68C2×C3⋊Dic3—21no—
69C2×C3⋊Dic3—21no—
70C2×C3⋊Dic3D411yescharacteristic
71C4⋊(C3⋊D3)—21no—
72C4⋊(C3⋊D3)C2311yescharacteristic
73C4×C3⋊D3—21no—
74C4×C3⋊D3—22no—
75C4×C3⋊D3—22no—
76C4×C3⋊D3—21no—
77C4×C3⋊D3—21no—
78C4×C3⋊D3D411yescharacteristic
79C4×C3⋊D3D411yescharacteristic
80C4×C3⋊D3C2311yescharacteristic
81C4×C3⋊D3C2311yescharacteristic
82#634—21no—
83#634—22no—
84#634—22no—
85#634—21no—
86#634C2312yes—
87#634C2311yescharacteristic
88C32×C8C2312yes—
89C3⋊(C3⋊C8)C2312yes—
90C2×#137—91noSylow
91C3×#41C2×D314yes—
92C2×C3×Q8C2×D314yescomplemented
93#239—34no—
94D3×Q8—38no—
95D3×Q8—38no—
96D3×Q8—34no—
97D3×Q8—64no—
98D3×Q8—64no—
99#237—34no—
100#237—64no—
101D3×D4—34no—
102D3×D4—64no—
103#235—34no—
104#235—64no—
105C2×C4×D3—34no—
106C2×C4×D3—64no—
107C2×C4×D3—64no—
108C2×Dic6—34no—
109C2×Dic6—64no—
110C3×Q16C2×D318yescomplemented
111C3×SD16C2×D318yescomplemented
112C3×M16C2×D314yescomplemented
113#216—38no—
114#216—38no—
115#215—38no—
116#214—38no—
117#208—34no—
118Dic12—38no—
119#204—38no—
120ΓL(1,𝔽25)—38no—
121D3×C8—38no—
122C22×C32—21no—
123C22×C32C2×D411yescharacteristic
124C2×C3⋊D3—22no—
125C2×C3⋊D3—21no—
126C2×C3⋊D3—42no—
127C2×C3⋊D3C2×D411yescharacteristic
128C2×C3⋊D3C2×D411yescharacteristic
129C32×C4—22no—
130C32×C4—21no—
131C32×C4C2×D411yescharacteristic
132C32×C4C2411yescharacteristic
133C3⋊Dic3—22no—
134C3⋊Dic3—21no—
135C3⋊Dic3C2×D411yescharacteristic
136C3⋊Dic3C2×D411yescharacteristic
137C2×#41—91no—
138C22×Q8—91no—
139#137—91no—
140#137—91no—
141#137—92no—
142#137—92no—
143#137—91no—
144#137C3⋊D311yescharacteristic, direct product factor
145C2×Q16—92no—
146C2×SD16—92no—
147C2×M16—91no—
148C22×D3—34no—
149C22×D3—64no—
150C3×Q8—24no—
151C3×Q8C22×D318yes—
152C3×Q8C22×D314yes—
153C3×D4—24no—
154C3×D4C22×D314yes—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4C22×D314yes—
158#67—64no—
159#67—64no—
160#67—64no—
161C2×Dic3—34no—
162C2×Dic3—64no—
163C2×Dic3—64no—
164D12—34no—
165D12—64no—
166C4×D3—34no—
167C4×D3—34no—
168C4×D3—34no—
169C4×D3—34no—
170C4×D3—64no—
171C4×D3—64no—
172C4×D3—68no—
173C4×D3—68no—
174C4×D3—64no—
175Dic6—38no—
176Dic6—34no—
177Dic6—68no—
178Dic6—68no—
179Dic6—64no—
180Dic6—64no—
181C3×C8C22×D318yes—
182C3⋊C8—38no—
183C2×C32—21no—
184C2×C32—41no—
185C2×C32C22×D411yescharacteristic
186C3⋊D3—21no—
187C3⋊D3—41no—
188C3⋊D3#13712yesdirect product factor
189#41—91no—
190#41—91no—
191#41—91no—
192#41—181no—
193#41—181no—
194#41C2×C3⋊D311yescharacteristic
195C2×Q8—92no—
196C2×Q8—91no—
197C2×Q8—92no—
198C2×Q8—91no—
199C2×Q8—181no—
200C2×Q8—181no—
201C2×Q8—181no—
202C2×Q8C2×C3⋊D311yescharacteristic, complemented
203C2×D4—91no—
204C2×D4—181no—
205C22×C4—91no—
206C22×C4—181no—
207C22×C4—181no—
208Q16—92no—
209Q16—92no—
210Q16—92no—
211Q16C2×C3⋊D312yescomplemented
212SD16—92no—
213SD16—92no—
214SD16—92no—
215SD16C2×C3⋊D312yescomplemented
216M16—91no—
217M16—92no—
218M16C2×C3⋊D311yescharacteristic, complemented
219C2×C8—92no—
220C22×C3—24no—
221C22×C3D3×D414yes—
222C2×D3—34no—
223C2×D3—34no—
224C2×D3—68no—
225C2×D3—64no—
226C2×D3—128no—
227C3×C4—28no—
228C3×C4—24no—
229C3×C4D3×D414yes—
230C3×C4C23×D314yes—
231Dic3—34no—
232Dic3—34no—
233Dic3—68no—
234Dic3—64no—
235C32C2×#13711yescharacteristic, complemented, Sylow
236C23—91no—
237C23—181no—
238Q8—21no—
239Q8—92no—
240Q8—91no—
241Q8—182no—
242Q8—182no—
243Q8—181no—
244Q8—181no—
245Q8C22×C3⋊D312yes—
246Q8C22×C3⋊D311yescharacteristic
247D4—21no—
248D4—91no—
249D4—181no—
250D4—181no—
251D4—181no—
252D4—181no—
253D4C22×C3⋊D311yescharacteristic
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—181no—
262C2×C4—181no—
263C2×C4—182no—
264C2×C4—182no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—181no—
268C2×C4C22×C3⋊D311yescharacteristic
269C8—92no—
270C8C22×C3⋊D312yes—
271C2×C3—24no—
272C2×C3—44no—
273C2×C3C2×D3×D414yes—
274D3—38no—
275D3—64no—
276D3—124no—
277C22—21no—
278C22—91no—
279C22—91no—
280C22—182no—
281C22—181no—
282C22—362no—
283C22D4×C3⋊D311yescharacteristic
284C4—21no—
285C4—22no—
286C4—91no—
287C4—91no—
288C4—181no—
289C4—182no—
290C4D4×C3⋊D311yescharacteristic
291C4C23×C3⋊D311yescharacteristic
292C3D3×#13714yescomplemented
293C2—21no—
294C2—41no—
295C2—92no—
296C2—181no—
297C2—361no—
298C2C2×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2C2×D4×C3⋊D3
commutator, cosocleC32×C4C24
FrattiniC4C23×C3⋊D3
FittingC32×#137C2
socleC2×C32C22×D4
prime radical#41×C3⋊D3C2
hypercenter#137C3⋊D3
hypocenterC32C2×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C3⋊D3C2semidirect
3. C32×#137C2semidirect
4. #64160C2semidirect
5. #64151C2semidirect
6. #64127C2semidirect
7. Q16×C3⋊D3C2semidirect
8. #64124C2semidirect
9. SD16×C3⋊D3C2semidirect
10. #64118C2semidirect
11. M16×C3⋊D3C2semidirect
14. C2×C32×Q8C22semidirect
20. Q8×C3⋊D3C22semidirect
31. C2×#634C22semidirect
32. C32×Q16C22semidirect
33. C32×SD16C22semidirect
34. C32×M16C22semidirect
35. #3756C22semidirect
36. #3756C22semidirect
37. #3755C22semidirect
38. #3754C22semidirect
39. #3748C22semidirect
40. #3746C22semidirect
41. #3744C22semidirect
42. #3743C22semidirect
46. C3×#137D3semidirect
92. C2×C3×Q8C2×D3semidirect
110. C3×Q16C2×D3semidirect
111. C3×SD16C2×D3semidirect
112. C3×M16C2×D3semidirect
144. #137C3⋊D3direct (N × H)
188. C3⋊D3#137direct (N × H)
202. C2×Q8C2×C3⋊D3semidirect
211. Q16C2×C3⋊D3semidirect
215. SD16C2×C3⋊D3semidirect
218. M16C2×C3⋊D3semidirect
235. C32C2×#137semidirect
292. C3D3×#137semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊D32
C2×Q8×C3⋊D32
C32×#1372
#641602
#641512
#641272
Q16×C3⋊D32
#641242
SD16×C3⋊D32
#641182
M16×C3⋊D32
D3×#1373