Finite Group Databasebeta

← C2×D3×#137 #87193

Subgroups — C2×D3×#137 #87193

Order 384.

1464 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#41C211yesmaximal, characteristic
2C22×D3×Q8C211yesmaximal, characteristic, complemented
3C2×C3×#137C211yesmaximal, characteristic, complemented
4C2×#951C211yesmaximal, characteristic, complemented
5C2×#942C211yesmaximal, characteristic, complemented
6D3×#137C2116yesmaximal, direct product factor
7C2×#918C212yesmaximal, complemented
8C2×D3×Q16C212yesmaximal, complemented
9C2×#915C212yesmaximal, complemented
10C2×D3×SD16C212yesmaximal, complemented
11C2×#909C211yesmaximal, characteristic, complemented
12C2×D3×M16C211yesmaximal, characteristic, complemented
13C22×#137—31nomaximal, Sylow
14C2×C3×#41C2211yescharacteristic
15C22×C3×Q8C2211yescharacteristic, complemented
16D3×#41—24no—
17D3×#41C2218yes—
18C2×#239C2211yescharacteristic
19C2×D3×Q8—21no—
20C2×D3×Q8—21no—
21C2×D3×Q8—24no—
22C2×D3×Q8C2212yes—
23C2×D3×Q8C2211yescharacteristic
24C2×D3×Q8C2212yescomplemented
25C2×D3×Q8C2218yescomplemented
26C2×#237—21no—
27C2×#237C2211yescharacteristic
28C2×D3×D4—21no—
29C2×D3×D4C2211yescharacteristic
30C2×#235—21no—
31C2×#235C2211yescharacteristic
32C22×C4×D3—21no—
33C22×C4×D3—21no—
34C22×C4×D3C2211yescharacteristic
35C22×Dic6—21no—
36C22×Dic6C2211yescharacteristic, complemented
37C3×#137C2218yescomplemented
38C2×C3×Q16C2212yescomplemented
39C2×C3×SD16C2212yescomplemented
40C2×C3×M16C2211yescharacteristic, complemented
41#951C2218yescomplemented
42C2×#216C2212yescomplemented
43C2×#216C2212yescomplemented
44#942C2218yescomplemented
45C2×#215C2212yescomplemented
46C2×#214C2212yescomplemented
47C2×#208C2211yescharacteristic, complemented
48#918C22116yescomplemented
49D3×Q16C22116yescomplemented
50#915C22116yescomplemented
51D3×SD16C22116yescomplemented
52#909C2218yescomplemented
53D3×M16C2218yescomplemented
54C2×Dic12C2212yescomplemented
55C2×#204C2212yescomplemented
56C2×ΓL(1,𝔽25)C2212yescomplemented
57C2×D3×C8C2212yes—
58C22×#41—31no—
59C23×Q8—31no—
60C2×#137—31no—
61C2×#137—31no—
62C2×#137—316no—
63C2×#137—32no—
64C2×#137—32no—
65C2×#137—31no—
66C2×#137D311yescharacteristic, direct product factor
67C22×Q16—32no—
68C22×SD16—32no—
69C22×M16—31no—
70C23×D3—21no—
71C23×D3D411yescharacteristic
72C3×#41—22no—
73C3×#41C2314yes—
74C2×C3×Q8—21no—
75C2×C3×Q8—22no—
76C2×C3×Q8C2312yes—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2314yescomplemented
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89C22×Dic3D411yescharacteristic
90#239—22no—
91#239C2314yes—
92D3×Q8—24no—
93D3×Q8—24no—
94D3×Q8—28no—
95D3×Q8—28no—
96D3×Q8—22no—
97D3×Q8C2318yes—
98D3×Q8C2314yes—
99D3×Q8C2318yescomplemented
100#237—28no—
101#237—22no—
102#237C2314yes—
103D3×D4—28no—
104D3×D4—22no—
105D3×D4C2314yes—
106#235—22no—
107#235—28no—
108#235C2314yes—
109C2×D12—21no—
110C2×D12C2311yescharacteristic
111C2×C4×D3—21no—
112C2×C4×D3—22no—
113C2×C4×D3—28no—
114C2×C4×D3—22no—
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×C4×D3—28no—
118C2×C4×D3D411yescharacteristic
119C2×C4×D3D411yescharacteristic
120C2×C4×D3D414yes—
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2311yescharacteristic
123C2×C4×D3C2314yes—
124C2×Dic6—22no—
125C2×Dic6—21no—
126C2×Dic6—22no—
127C2×Dic6—28no—
128C2×Dic6—22no—
129C2×Dic6—21no—
130C2×Dic6C2312yes—
131C2×Dic6C2311yescharacteristic
132C2×Dic6C2314yescomplemented
133C3×Q16C2318yescomplemented
134C3×SD16C2318yescomplemented
135C3×M16C2314yescomplemented
136C2×C3×C8C2312yes—
137#216C2318yescomplemented
138#216C2318yescomplemented
139#215C2318yescomplemented
140#214C2318yescomplemented
141#208C2314yescomplemented
142C2×C3⋊C8C2312yes—
143Dic12C2318yescomplemented
144#204C2318yescomplemented
145ΓL(1,𝔽25)C2318yescomplemented
146D3×C8C2318yes—
147C2×#41—38no—
148C2×#41—31no—
149C2×#41—31no—
150C2×#41—31no—
151C2×#41—61no—
152C2×#41—61no—
153C2×#41—64no—
154C2×#41C2×D311yescharacteristic
155C22×Q8—32no—
156C22×Q8—31no—
157C22×Q8—32no—
158C22×Q8—38no—
159C22×Q8—31no—
160C22×Q8—61no—
161C22×Q8—61no—
162C22×Q8—64no—
163C22×Q8—61no—
164C22×Q8C2×D311yescharacteristic, complemented
165C22×D4—31no—
166C22×D4—61no—
167C23×C4—31no—
168C23×C4—61no—
169C23×C4—61no—
170#137—38no—
171#137—38no—
172#137—316no—
173#137—316no—
174#137—38no—
175#137C2×D318yesdirect product factor
176C2×Q16—32no—
177C2×Q16—32no—
178C2×Q16—316no—
179C2×Q16—32no—
180C2×Q16C2×D312yescomplemented
181C2×SD16—32no—
182C2×SD16—32no—
183C2×SD16—316no—
184C2×SD16—32no—
185C2×SD16C2×D312yescomplemented
186C2×M16—31no—
187C2×M16—38no—
188C2×M16—32no—
189C2×M16C2×D311yescharacteristic, complemented
190C22×C8—32no—
191C23×C3—21no—
192C23×C3C2×D411yescharacteristic
193C22×D3—24no—
194C22×D3—21no—
195C22×D3—24no—
196C22×D3—44no—
197C22×D3C2×D411yescharacteristic
198C22×D3C2×D414yes—
199C22×D3C2×D411yescharacteristic
200C3×Q8—22no—
201C3×Q8—24no—
202C3×Q8—21no—
203C3×Q8C2414yes—
204C3×Q8C2412yes—
205C3×D4—24no—
206C3×D4—21no—
207C3×D4C2412yes—
208C2×C3×C4—22no—
209C2×C3×C4—24no—
210C2×C3×C4—21no—
211C2×C3×C4—24no—
212C2×C3×C4C2×D411yescharacteristic
213C2×C3×C4C2×D412yes—
214C2×C3×C4C2411yescharacteristic
215C2×C3×C4C2412yes—
216#67—24no—
217#67—24no—
218#67—24no—
219C2×Dic3—22no—
220C2×Dic3—24no—
221C2×Dic3—21no—
222C2×Dic3—24no—
223C2×Dic3C2×D411yescharacteristic
224C2×Dic3C2×D411yescharacteristic
225C2×Dic3C2×D414yes—
226D12—24no—
227D12—21no—
228D12C2412yes—
229C4×D3—24no—
230C4×D3—28no—
231C4×D3—28no—
232C4×D3—24no—
233C4×D3—24no—
234C4×D3C2×D414yes—
235C4×D3C2×D414yes—
236C4×D3C2×D412yes—
237C4×D3C2×D412yes—
238C4×D3C2412yes—
239C4×D3C2412yes—
240Dic6—22no—
241Dic6—24no—
242Dic6—21no—
243Dic6—28no—
244Dic6—28no—
245Dic6—24no—
246Dic6C2414yes—
247Dic6C2412yes—
248C3×C8C2414yes—
249C3⋊C8C2414yes—
250C24—31no—
251C24—61no—
252#41—22no—
253#41—34no—
254#41—34no—
255#41—34no—
256#41—68no—
257#41—68no—
258#41—62no—
259#41—62no—
260#41—62no—
261#41C22×D314yes—
262C2×Q8—22no—
263C2×Q8—21no—
264C2×Q8—38no—
265C2×Q8—34no—
266C2×Q8—38no—
267C2×Q8—32no—
268C2×Q8—31no—
269C2×Q8—34no—
270C2×Q8—68no—
271C2×Q8—62no—
272C2×Q8—62no—
273C2×Q8—61no—
274C2×Q8—68no—
275C2×Q8—64no—
276C2×Q8—64no—
277C2×Q8—68no—
278C2×Q8—62no—
279C2×Q8—62no—
280C2×Q8—61no—
281C2×Q8C22×D312yes—
282C2×Q8C22×D311yescharacteristic
283C2×Q8C22×D314yescomplemented
284C2×D4—21no—
285C2×D4—34no—
286C2×D4—31no—
287C2×D4—61no—
288C2×D4—62no—
289C2×D4—61no—
290C2×D4—68no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4C22×D311yescharacteristic
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—34no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—34no—
302C22×C4—31no—
303C22×C4—61no—
304C22×C4—68no—
305C22×C4—61no—
306C22×C4—68no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4C22×D311yescharacteristic
313Q16—38no—
314Q16—38no—
315Q16—38no—
316Q16C22×D318yescomplemented
317SD16—38no—
318SD16—38no—
319SD16—38no—
320SD16C22×D318yescomplemented
321M16—34no—
322M16—38no—
323M16C22×D314yescomplemented
324C2×C8—32no—
325C2×C8—38no—
326C2×C8C22×D312yes—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—42no—
330C22×C3C22×D412yes—
331C22×C3C22×D411yescharacteristic
332C2×D3—22no—
333C2×D3—28no—
334C2×D3—22no—
335C2×D3—42no—
336C2×D3—48no—
337C2×D3#13714yesdirect product factor
338C2×D3C22×D412yes—
339C2×D3C22×D412yes—
340C3×C4—24no—
341C3×C4—22no—
342C3×C4C22×D411yescharacteristic
343C3×C4C22×D412yes—
344C3×C4C2511yescharacteristic
345Dic3—24no—
346Dic3—22no—
347Dic3C22×D412yes—
348Dic3C22×D412yes—
349C23—21no—
350C23—31no—
351C23—34no—
352C23—31no—
353C23—64no—
354C23—64no—
355C23—61no—
356C23—124no—
357C23D3×D411yescharacteristic
358Q8—22no—
359Q8—24no—
360Q8—21no—
361Q8—34no—
362Q8—32no—
363Q8—68no—
364Q8—68no—
365Q8—64no—
366Q8—64no—
367Q8—62no—
368Q8—61no—
369Q8C23×D314yes—
370Q8C23×D312yes—
371D4—24no—
372D4—21no—
373D4—32no—
374D4—64no—
375D4—64no—
376D4—64no—
377D4—64no—
378D4—61no—
379D4C23×D312yes—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—34no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—34no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—34no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—64no—
394C2×C4—64no—
395C2×C4—68no—
396C2×C4—68no—
397C2×C4—64no—
398C2×C4—64no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—64no—
402C2×C4D3×D412yes—
403C2×C4D3×D411yescharacteristic
404C2×C4C23×D312yes—
405C2×C4C23×D311yescharacteristic
406C8—34no—
407C8C23×D314yes—
408C2×C3—22no—
409C2×C3—42no—
410C2×C3C2×#13712yescomplemented
411C2×C3C23×D411yescharacteristic
412D3—22no—
413D3—42no—
414D3C2×#13714yesdirect product factor
415C22—22no—
416C22—22no—
417C22—32no—
418C22—34no—
419C22—32no—
420C22—42no—
421C22—68no—
422C22—62no—
423C22—62no—
424C22—128no—
425C22—122no—
426C22C2×D3×D412yes—
427C22C2×D3×D411yescharacteristic
428C4—22no—
429C4—24no—
430C4—32no—
431C4—32no—
432C4—62no—
433C4—64no—
434C4C2×D3×D412yes—
435C4C2×D3×D411yescharacteristic
436C4C24×D311yescharacteristic
437C3C22×#13711yescharacteristic, complemented, Sylow
438C2—22no—
439C2—34no—
440C2—42no—
441C2—62no—
442C2—122no—
443C2D3×#13712yesdirect product factor
444C2C22×D3×D411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC2×C3×#137C2
socleC22×C3C22×D4
prime radicalC2×D3×#41C2
hypercenterC2×#137D3
hypocenterC3C22×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×D3×Q8C2semidirect
3. C2×C3×#137C2semidirect
4. C2×#951C2semidirect
5. C2×#942C2semidirect
6. D3×#137C2direct (N × H)
7. C2×#918C2semidirect
8. C2×D3×Q16C2semidirect
9. C2×#915C2semidirect
10. C2×D3×SD16C2semidirect
11. C2×#909C2semidirect
12. C2×D3×M16C2semidirect
15. C22×C3×Q8C22semidirect
24. C2×D3×Q8C22semidirect
25. C2×D3×Q8C22semidirect
36. C22×Dic6C22semidirect
37. C3×#137C22semidirect
38. C2×C3×Q16C22semidirect
39. C2×C3×SD16C22semidirect
40. C2×C3×M16C22semidirect
41. #951C22semidirect
42. C2×#216C22semidirect
43. C2×#216C22semidirect
44. #942C22semidirect
45. C2×#215C22semidirect
46. C2×#214C22semidirect
47. C2×#208C22semidirect
48. #918C22semidirect
49. D3×Q16C22semidirect
50. #915C22semidirect
51. D3×SD16C22semidirect
52. #909C22semidirect
53. D3×M16C22semidirect
54. C2×Dic12C22semidirect
55. C2×#204C22semidirect
56. C2×ΓL(1,𝔽25)C22semidirect
66. C2×#137D3direct (N × H)
78. C2×C3×Q8C23semidirect
99. D3×Q8C23semidirect
132. C2×Dic6C23semidirect
133. C3×Q16C23semidirect
134. C3×SD16C23semidirect
135. C3×M16C23semidirect
137. #216C23semidirect
138. #216C23semidirect
139. #215C23semidirect
140. #214C23semidirect
141. #208C23semidirect
143. Dic12C23semidirect
144. #204C23semidirect
145. ΓL(1,𝔽25)C23semidirect
164. C22×Q8C2×D3semidirect
175. #137C2×D3direct (N × H)
180. C2×Q16C2×D3semidirect
185. C2×SD16C2×D3semidirect
189. C2×M16C2×D3semidirect
283. C2×Q8C22×D3semidirect
316. Q16C22×D3semidirect
320. SD16C22×D3semidirect
323. M16C22×D3semidirect
337. C2×D3#137direct (N × H)
410. C2×C3C2×#137semidirect
414. D3C2×#137direct (N × H)
437. C3C22×#137semidirect
443. C2D3×#137direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×#412
C22×D3×Q82
C2×C3×#1372
C2×#9512
C2×#9422
D3×#1372
C2×#9182
C2×D3×Q162
C2×#9152
C2×D3×SD162
C2×#9092
C2×D3×M162
C22×#1373