Finite Group Databasebeta

← #86276

Subgroups — #86276

Order 384.

1132 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5734C211yesmaximal, characteristic, complemented
3#5732C211yesmaximal, characteristic, complemented
4#5730C211yesmaximal, characteristic, complemented
5#5726C211yesmaximal, characteristic, complemented
6#5695C212yesmaximal, complemented
7#5694C212yesmaximal, complemented
8#5642C212yesmaximal, complemented
9#5637C214yesmaximal, complemented
10#5630C212yesmaximal, complemented
11#5580C214yesmaximal, complemented
12#5580C212yesmaximal, complemented
13C4×#239C212yesmaximal, complemented
14#5524C212yesmaximal, complemented
15D3×#117C212yesmaximal, complemented
16C4×#235C211yesmaximal, characteristic, complemented
17C2×C4×D12C211yesmaximal, characteristic, complemented
18#3475—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×#235C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×C4×D3C2212yes—
23C3×C4⋊Q8C2211yescharacteristic, complemented
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×#125C2212yescomplemented
26C3×C22⋊Q8C2212yes—
27C3×C22⋊Q8C2212yes—
28C3×C4×Q8C2212yescomplemented
29C3×C4×Q8C2212yescomplemented
30C3×#117C2212yes—
31C2×C3×C42C2211yescharacteristic
32#947C2212yes—
33#947C2214yescomplemented
34#946C2214yescomplemented
35#946C2212yescomplemented
36Q8×Dic3C2212yescomplemented
37C4×#67C2212yes—
38C4×#67C2212yes—
39C2×(C2×D3)⋊C4C2212yes—
40#926C2211yescharacteristic
41C2×C4⋊Dic3C2211yescharacteristic
42#898C2214yescomplemented
43#898C2214yescomplemented
44#898C2214yescomplemented
45#897C2212yescomplemented
46#897C2212yescomplemented
47#895C2214yes—
48#895C2212yes—
49#895C2212yescomplemented
50#895C2212yescomplemented
51#894C2214yes—
52#894C2214yescomplemented
53#894C2214yescomplemented
54#893C2212yes—
55#893C2214yescomplemented
56#892C2214yescomplemented
57#892C2212yescomplemented
58#892C2212yescomplemented
59#892C2212yescomplemented
60D3×C4⋊C4C2212yescomplemented
61D3×C4⋊C4C2212yescomplemented
62#890C2214yescomplemented
63#890C2212yescomplemented
64#886C2214yes—
65#882C2214yes—
66D3×C22⋊C4C2214yes—
67#877C2214yes—
68C4×D12C2212yes—
69C4×D12C2211yescharacteristic
70C4×D12C2211yescharacteristic
71C4×D12C2211yescharacteristic
72C4×D12C2212yes—
73C4×D12C2211yescharacteristic
74C4×D12C2211yescharacteristic, complemented
75C4×D12C2211yescharacteristic, complemented
76C4×D12C2214yescomplemented
77C4×D12C2214yescomplemented
78#872C2212yescomplemented
79#872C2214yescomplemented
80C42×D3C2212yescomplemented
81C42×D3C2214yescomplemented
82C4×Dic6C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#556—31no—
85#555—31no—
86#554—31no—
87#553—32no—
88#552—32no—
89#552—32no—
90#551—34no—
91#547—31no—
92#546—32no—
93#533D311yescharacteristic, complemented
94#529—34no—
95#529—32no—
96#525—32no—
97C4×#41—32no—
98C4×#41—31no—
99C2×C4×D4—31no—
100C2×#117—32no—
101C23×D3C2312yes—
102C2×C3×Q8C2312yes—
103C2×C3×Q8C2312yes—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C2×#67C2312yes—
108C22×Dic3C2312yes—
109#239—28no—
110#235—24no—
111C2×D12—24no—
112C2×D12C2312yes—
113C2×D12C2311yescharacteristic
114C2×D12C2311yescharacteristic
115C2×D12C2314yes—
116C2×D12C2314yes—
117C2×D12C2311yescharacteristic
118C2×C4×D3—28no—
119C2×C4×D3C2312yes—
120C2×C4×D3C2312yes—
121C2×C4×D3C2312yes—
122C2×C4×D3C2314yes—
123C2×C4×D3C2312yes—
124C2×C4×D3C2312yes—
125C2×C4×D3C2312yes—
126C2×C4×D3C2312yes—
127C2×C4×D3C2314yes—
128C2×Dic6C2311yescharacteristic
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2314yes—
131C3×C4⋊C4C2312yes—
132C3×C4⋊C4C2312yes—
133C3×C4⋊C4C2314yes—
134C3×C4⋊C4C2312yes—
135C3×C22⋊C4C2314yes—
136C3×C42C2311yescharacteristic
137C3×C42C2311yescharacteristic
138C3×C42C2311yescharacteristic
139C3×C42C2314yes—
140C3×C42C2311yescharacteristic
141#217C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2314yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2311yescharacteristic
156C4⋊Dic3C2312yes—
157C4⋊Dic3C2311yescharacteristic
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2314yes—
160C4⋊Dic3C2314yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2314yes—
169C4×Dic3C2314yes—
170C2×#41—32no—
171C2×#41—31no—
172C22×D4—31no—
173C23×C4—32no—
174C4⋊Q8C2×D311yescharacteristic, complemented
175C4⋊Q8C2×D311yescharacteristic, complemented
176#126—34no—
177#126—34no—
178#126—34no—
179#125—34no—
180#125—32no—
181#125C2×D312yescomplemented
182#124—32no—
183#124—34no—
184#123—34no—
185#123—34no—
186#123—34no—
187#123—34no—
188C22⋊Q8—32no—
189C22⋊Q8—34no—
190C22⋊Q8—32no—
191C22⋊Q8—32no—
192C22⋊Q8C2×D312yes—
193C22⋊Q8C2×D312yes—
194#121—32no—
195#121—32no—
196#121—34no—
197#121—32no—
198#120—34no—
199C4×Q8—32no—
200C4×Q8—31no—
201C4×Q8—31no—
202C4×Q8C2×D312yescomplemented
203C4×Q8C2×D312yescomplemented
204C4×D4—32no—
205C4×D4—32no—
206C4×D4—32no—
207C4×D4—34no—
208C4×D4—31no—
209C4×D4—31no—
210C4×D4—32no—
211C4×D4—31no—
212C4×D4—32no—
213C4×D4—31no—
214C4×D4—34no—
215C4×D4—31no—
216C4×D4—31no—
217C4×D4—34no—
218#117—31no—
219#117—32no—
220#117—32no—
221#117—34no—
222#117—32no—
223#117—34no—
224#117—34no—
225#117—32no—
226#117C2×D312yes—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—32no—
230C2×C22⋊C4—32no—
231C2×C22⋊C4—34no—
232C2×C42—34no—
233C2×C42—32no—
234C2×C42C2×D311yescharacteristic
235C23×C3C2411yescharacteristic
236C22×D3—24no—
237C22×D3—24no—
238C22×D3—42no—
239C22×D3C2412yes—
240C22×D3C2412yes—
241C22×D3C2412yes—
242C3×Q8—24no—
243C3×Q8—42no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4#4114yescomplemented
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2414yes—
249C2×C3×C4C2414yes—
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2411yescharacteristic
255#67—24no—
256C2×Dic3—24no—
257C2×Dic3C2412yes—
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262D12—24no—
263D12—28no—
264D12—22no—
265D12—22no—
266D12—44no—
267D12#4114yescomplemented
268C4×D3—28no—
269C4×D3—24no—
270C4×D3—24no—
271C4×D3—24no—
272C4×D3—24no—
273C4×D3—24no—
274C4×D3—24no—
275C4×D3—42no—
276C4×D3—44no—
277Dic6—22no—
278C24—32no—
279#41—68no—
280#41—64no—
281C2×Q8—31no—
282C2×Q8C22×D312yes—
283C2×Q8C22×D312yes—
284C2×D4—32no—
285C2×D4—31no—
286C2×D4—34no—
287C2×D4—31no—
288C2×D4—32no—
289C2×D4—31no—
290C2×D4—34no—
291C2×D4—64no—
292C22×C4—32no—
293C22×C4—34no—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—34no—
301C22×C4—32no—
302C22×C4—68no—
303C22×C4C22×D311yescharacteristic
304C22×C4C22×D311yescharacteristic
305C22×C4C22×D311yescharacteristic
306C4⋊C4—31no—
307C4⋊C4—31no—
308C4⋊C4—31no—
309C4⋊C4—34no—
310C4⋊C4—34no—
311C4⋊C4—31no—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—34no—
315C4⋊C4—32no—
316C4⋊C4—34no—
317C4⋊C4C22×D312yes—
318C4⋊C4C22×D312yes—
319C4⋊C4C22×D312yes—
320C4⋊C4C22×D314yes—
321C4⋊C4C22×D314yes—
322C4⋊C4C22×D312yes—
323C22⋊C4—32no—
324C22⋊C4—32no—
325C22⋊C4—34no—
326C22⋊C4—34no—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—32no—
330C22⋊C4—34no—
331C22⋊C4—32no—
332C22⋊C4—34no—
333C22⋊C4—32no—
334C22⋊C4—34no—
335C22⋊C4—34no—
336C22⋊C4C22×D314yes—
337C42—34no—
338C42—34no—
339C42—32no—
340C42—32no—
341C42—32no—
342C42C22×D311yescharacteristic
343C42C22×D311yescharacteristic
344C42C22×D311yescharacteristic
345C42C22×D314yes—
346C42C22×D311yescharacteristic
347C22×C3—21no—
348C22×C3—21no—
349C22×C3C2×#4112yes—
350C22×C3C2511yescharacteristic
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—22no—
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—48no—
360C2×D3C2×#4114yes—
361C3×C4—24no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—24no—
365C3×C4—21no—
366C3×C4C2×#4112yes—
367C3×C4C2×#4112yes—
368C3×C4C2×#4112yes—
369Dic3—22no—
370Dic3—22no—
371Dic3—22no—
372Dic3—22no—
373Dic3—22no—
374C23—32no—
375C23—32no—
376C23—32no—
377C23—64no—
378C23—64no—
379C23—122no—
380C23C23×D311yescharacteristic
381Q8—24no—
382Q8—42no—
383Q8—62no—
384D4—34no—
385D4—64no—
386D4—68no—
387D4—64no—
388D4—62no—
389D4—62no—
390D4—124no—
391C2×C4—22no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—41no—
398C2×C4—68no—
399C2×C4—64no—
400C2×C4—64no—
401C2×C4—64no—
402C2×C4—64no—
403C2×C4—64no—
404C2×C4—64no—
405C2×C4—64no—
406C2×C4—122no—
407C2×C4—124no—
408C2×C4#23914yescomplemented
409C2×C4C23×D311yescharacteristic
410C2×C4C23×D314yes—
411C2×C4C23×D314yes—
412C2×C4C23×D311yescharacteristic
413C2×C4C23×D311yescharacteristic
414C2×C4C23×D311yescharacteristic
415C2×C4C23×D311yescharacteristic
416C2×C4C23×D311yescharacteristic
417C2×C3—22no—
418C2×C3C22×#4111yescharacteristic
419C2×C3C22×#4111yescharacteristic
420C2×C3#58511yescharacteristic
421D3—24no—
422D3—42no—
423D3—42no—
424C22—21no—
425C22—21no—
426C22—34no—
427C22—62no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—128no—
436C22C2×#23912yes—
437C22C24×D311yescharacteristic
438C4—21no—
439C4—24no—
440C4—21no—
441C4—21no—
442C4—24no—
443C4—62no—
444C4—62no—
445C4—62no—
446C4—62no—
447C4—62no—
448C4C2×#23912yes—
449C4C2×#23912yes—
450C4D3×#4112yes—
451C3#347511yescharacteristic, complemented, Sylow
452C2—22no—
453C2—64no—
454C2—122no—
455C2—122no—
456C2C22×#23911yescharacteristic
457C2C2×D3×#4111yescharacteristic
458C2#596411yescharacteristic
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC3×#533C2
hypercenter#533D3
hypocenterC3#3475
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5734C2semidirect
3. #5732C2semidirect
4. #5730C2semidirect
5. #5726C2semidirect
6. #5695C2semidirect
7. #5694C2semidirect
8. #5642C2semidirect
9. #5637C2semidirect
10. #5630C2semidirect
11. #5580C2semidirect
12. #5580C2semidirect
13. C4×#239C2semidirect
14. #5524C2semidirect
15. D3×#117C2semidirect
16. C4×#235C2semidirect
17. C2×C4×D12C2semidirect
23. C3×C4⋊Q8C22semidirect
24. C3×C4⋊Q8C22semidirect
25. C3×#125C22semidirect
28. C3×C4×Q8C22semidirect
29. C3×C4×Q8C22semidirect
33. #947C22semidirect
34. #946C22semidirect
35. #946C22semidirect
36. Q8×Dic3C22semidirect
42. #898C22semidirect
43. #898C22semidirect
44. #898C22semidirect
45. #897C22semidirect
46. #897C22semidirect
49. #895C22semidirect
50. #895C22semidirect
52. #894C22semidirect
53. #894C22semidirect
55. #893C22semidirect
56. #892C22semidirect
57. #892C22semidirect
58. #892C22semidirect
59. #892C22semidirect
60. D3×C4⋊C4C22semidirect
61. D3×C4⋊C4C22semidirect
62. #890C22semidirect
63. #890C22semidirect
74. C4×D12C22semidirect
75. C4×D12C22semidirect
76. C4×D12C22semidirect
77. C4×D12C22semidirect
78. #872C22semidirect
79. #872C22semidirect
80. C42×D3C22semidirect
81. C42×D3C22semidirect
82. C4×Dic6C22semidirect
83. C4×Dic6C22semidirect
93. #533D3semidirect
174. C4⋊Q8C2×D3semidirect
175. C4⋊Q8C2×D3semidirect
181. #125C2×D3semidirect
202. C4×Q8C2×D3semidirect
203. C4×Q8C2×D3semidirect
246. C2×C3×C4#41semidirect
267. D12#41semidirect
408. C2×C4#239semidirect
451. C3#3475semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57342
#57322
#57302
#57262
#56952
#56942
#56422
#56372
#56302
#55802
C4×#2392
#55242
D3×#1172
C4×#2352
C2×C4×D122
#34753