Finite Group Databasebeta

← #11660864

Subgroups — #11660864

Order 768.

2004 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86239C211yesmaximal, characteristic
2C3×#2931C211yesmaximal, characteristic, complemented
3#83794C214yesmaximal, complemented
4#83278C211yesmaximal, characteristic, complemented
5#79542C214yesmaximal, complemented
6#79527C214yesmaximal, complemented
7D3×C8×D4C214yesmaximal, complemented
8#79401C214yesmaximal, complemented
9#79051C212yesmaximal, complemented
10#78665C214yesmaximal, complemented
11C8×#235C212yesmaximal, complemented
12#60305—31nomaximal, Sylow
13C2×C3×#131C2212yescomplemented
14C3×#532C2211yescharacteristic
15#5828C2214yes—
16C22×D3×C8C2214yescomplemented
17#5730C412yes—
18#5717C411yescharacteristic
19#5713C411yescharacteristic
20#5679C414yes—
21#5614C414yes—
22#5612C414yes—
23#5557C2214yes—
24C4×D3×D4C2214yes—
25C4×#235C2212yes—
26C3×#436C2214yescomplemented
27C3×C8×D4C2214yescomplemented
28C3×#423C2211yescharacteristic, complemented
29C3×#408C2212yescomplemented
30C2×C3×C4×C8C2211yescharacteristic, complemented
31#5234C2214yescomplemented
32#5127C2218yescomplemented
33#5127C2218yescomplemented
34#5126C2214yescomplemented
35#5119C2214yescomplemented
36C8×#67C2218yescomplemented
37C8×#67C2214yescomplemented
38C8×#67C2214yescomplemented
39#5101C2214yescomplemented
40#5012C2214yescomplemented
41D4×C3⋊C8C2214yescomplemented
42#4925C2212yescomplemented
43#4835C2214yescomplemented
44#4834C2214yes—
45D3×C4⋊C8C2214yescomplemented
46#4831C2214yescomplemented
47#4728C2218yescomplemented
48#67⋊C8C2218yescomplemented
49D3×C22⋊C8C2218yescomplemented
50#4720C2218yescomplemented
51C8×D12C2212yes—
52C8×D12C2212yes—
53#4685C2214yescomplemented
54C4×D3×C8C2214yescomplemented
55C8×Dic6C2212yescomplemented
56C8×Dic6C2212yescomplemented
57#3481—31no—
58#2994—34no—
59#2990—32no—
60#2979—31no—
61#2977—34no—
62#2974—34no—
63#2966—34no—
64C8×#41—32no—
65#2931D311yescharacteristic, complemented
66#2929—34no—
67C2×C8×D4—34no—
68C2×C3×#41C2312yes—
69C2×D3×D4C2×C414yes—
70C2×#235C2×C412yes—
71C22×C4×D3C2314yes—
72C3×#131—44no—
73C2×C3×M16C2314yes—
74C2×C3×M16C2312yes—
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2314yes—
77C22×C3×C8C2311yescharacteristic
78C3×C4⋊Q8C2×C411yescharacteristic
79C3×#127C2×C411yescharacteristic
80C3×#124C2×C412yes—
81C3×#121C2×C414yes—
82C3×C4×D4C2314yes—
83C2×C3×C42C2311yescharacteristic
84#946C2×C418yes—
85#938C2×C414yes—
86#938C2×C418yes—
87#937C2×C414yes—
88#937C2×C414yes—
89#936C2×C414yes—
90#935C2×C414yes—
91#935C2×C414yes—
92D4×Dic3C2×C414yes—
93C4×#67C2×C414yes—
94C4×#67C2314yes—
95C4×#67C2318yes—
96#926C2×C412yes—
97#926C2314yes—
98C2×#208C2314yes—
99C22×C3⋊C8C2314yes—
100C2×D3×C8—48no—
101C2×D3×C8C2314yes—
102C2×D3×C8C2318yes—
103C2×D3×C8C2314yes—
104#897C2×C414yes—
105#893C2×C414yes—
106D3×C4⋊C4C2×C414yes—
107#890C2×C414yes—
108#883C2×C418yes—
109#67⋊C4C2×C418yes—
110D3×C22⋊C4C2×C418yes—
111#879C2×C418yes—
112C4×D12C2×C412yes—
113C4×D12C2312yes—
114#872C2×C414yes—
115C42×D3C2314yes—
116C4×Dic6C2×C412yes—
117C4×Dic6C2312yes—
118C3×C4⋊C8C2314yes—
119C3×C22⋊C8C2318yes—
120C3×C4×C8C2312yes—
121C3×C4×C8C2312yes—
122#830C2314yes—
123#830C2318yes—
124(C2×D3)⋊C8C2314yes—
125(C2×D3)⋊C8C2318yes—
126(C2×D3)⋊C8C2314yes—
127#815C2318yes—
128Dic3⋊C8C2314yes—
129Dic3⋊C8C2318yes—
130Dic3⋊C8C2314yes—
131C8×Dic3C2314yes—
132C8×Dic3C2314yes—
133C4⋊(C3⋊C8)C2312yes—
134C4⋊(C3⋊C8)C2312yes—
135C4×C3⋊C8C2314yes—
136C2×#131C2×D312yescomplemented
137C23×C8—34no—
138#559—31no—
139#554—32no—
140#553—34no—
141#551—34no—
142#546—34no—
143D42—31no—
144#532C2×D311yescharacteristic
145#529—34no—
146#525—34no—
147C4×#41—32no—
148C2×C4×D4—34no—
149#446—34no—
150C8×Q8—32no—
151C8×Q8—32no—
152#436—34no—
153#436C2×D314yescomplemented
154#435—38no—
155#435—38no—
156#435—38no—
157C8×D4—38no—
158C8×D4—34no—
159C8×D4—34no—
160C8×D4—34no—
161C8×D4—34no—
162C8×D4—38no—
163C8×D4—32no—
164C8×D4—32no—
165C8×D4C2×D314yescomplemented
166#433—38no—
167#432—34no—
168#431—32no—
169#431—34no—
170#424—34no—
171#423C2×D311yescharacteristic, complemented
172C2×C4⋊C8—34no—
173#408—34no—
174#408C2×D312yescomplemented
175#407—34no—
176C2×C22⋊C8—38no—
177#405—34no—
178C2×C4×C8—34no—
179C2×C4×C8C2×D311yescharacteristic, complemented
180C23×D3C22×C414yes—
181C3×#41—42no—
182C3×#41—42no—
183C2×C3×Q8C22×C412yes—
184C2×C3×D4C22×C414yes—
185C2×C3×D4C22×C412yes—
186C2×C3×D4C22×C414yes—
187C22×C3×C4C22×C412yes—
188C22×C3×C4C2411yescharacteristic
189C22×C3×C4C2414yes—
190C2×#67C22×C418yes—
191C2×#67C22×C414yes—
192C22×Dic3C22×C414yes—
193D3×D4—216no—
194#235—28no—
195C2×D12C22×C412yes—
196C2×C4×D3—44no—
197C2×C4×D3—44no—
198C2×C4×D3C22×C414yes—
199C2×C4×D3C22×C414yes—
200C2×C4×D3C22×C418yes—
201C2×C4×D3C2414yes—
202C2×Dic6C22×C412yes—
203C3×M16—42no—
204C3×M16—44no—
205C2×C3×C8—28no—
206C2×C3×C8—22no—
207C2×C3×C8—41no—
208C2×C3×C8C2412yes—
209C2×C3×C8C2411yescharacteristic
210C2×C3×C8C2414yes—
211C2×C3×C8C2411yescharacteristic
212C3×C4⋊C4C22×C414yes—
213C3×C22⋊C4C22×C418yes—
214C3×C42C22×C411yescharacteristic
215C3×C42C22×C411yescharacteristic
216C3×C42C2412yes—
217#217C22×C414yes—
218#217C22×C418yes—
219#217C22×C414yes—
220#217C22×C414yes—
221(C2×D3)⋊C4C22×C414yes—
222(C2×D3)⋊C4C22×C414yes—
223(C2×D3)⋊C4C22×C418yes—
224C4⋊Dic3C22×C412yes—
225C4⋊Dic3C22×C412yes—
226C4⋊Dic3C22×C418yes—
227Dic3⋊C4C22×C414yes—
228Dic3⋊C4C22×C414yes—
229Dic3⋊C4C22×C418yes—
230C4×Dic3C22×C414yes—
231C4×Dic3C22×C414yes—
232C4×Dic3C2414yes—
233#208—44no—
234C2×C3⋊C8—44no—
235C2×C3⋊C8C2414yes—
236C2×C3⋊C8C2414yes—
237D3×C8—28no—
238D3×C8—48no—
239D3×C8—44no—
240C2×#41—32no—
241C2×#41C22×D312yes—
242C22×D4—34no—
243C23×C4—34no—
244#131—44no—
245C2×M16—34no—
246C2×M16C22×D314yes—
247C2×M16C22×D312yes—
248C22×C8—34no—
249C22×C8—34no—
250C22×C8—38no—
251C22×C8—34no—
252C22×C8—128no—
253C22×C8C22×D314yes—
254C22×C8C22×D311yescharacteristic
255C22×C8C22×D311yescharacteristic
256C4⋊Q8C4×D311yescharacteristic
257#127C4×D311yescharacteristic
258#126—38no—
259#125—34no—
260#124—34no—
261#124C4×D312yes—
262#123—34no—
263#123—34no—
264#123—38no—
265C22⋊Q8—38no—
266C22⋊Q8—34no—
267#121—38no—
268#121—34no—
269#121C4×D314yes—
270#120—34no—
271#120—34no—
272C4×Q8—32no—
273C4×Q8—32no—
274C4×D4—34no—
275C4×D4—34no—
276C4×D4—38no—
277C4×D4—34no—
278C4×D4—34no—
279C4×D4—38no—
280C4×D4—32no—
281C4×D4—32no—
282C4×D4C22×D314yes—
283#117—32no—
284#117—34no—
285#117—34no—
286C2×C4⋊C4—34no—
287C2×C22⋊C4—38no—
288C2×C42—34no—
289C2×C42C22×D311yescharacteristic
290C4⋊C8—38no—
291C4⋊C8—34no—
292C4⋊C8—34no—
293C4⋊C8—32no—
294C4⋊C8—32no—
295C4⋊C8C22×D314yes—
296C22⋊C8—38no—
297C22⋊C8—34no—
298C22⋊C8—34no—
299C22⋊C8—38no—
300C22⋊C8—34no—
301C22⋊C8C22×D318yes—
302#97—38no—
303C4×C8—34no—
304C4×C8—34no—
305C4×C8—34no—
306C4×C8C22×D312yes—
307C4×C8C22×D312yes—
308C23×C3C23×C414yes—
309C23×C3C23×C411yescharacteristic
310C22×D3—216no—
311C22×D3—44no—
312C22×D3C23×C414yes—
313C3×Q8—42no—
314C3×D4—28no—
315C3×D4—44no—
316C3×D4—42no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—21no—
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4C23×C412yes—
323C2×C3×C4C23×C411yescharacteristic
324C2×C3×C4C23×C414yes—
325C2×C3×C4C23×C412yes—
326C2×C3×C4C2511yescharacteristic
327#67—28no—
328#67—216no—
329C2×Dic3—44no—
330C2×Dic3C23×C414yes—
331C2×Dic3C23×C414yes—
332C2×Dic3C23×C414yes—
333D12#13118yescomplemented
334C4×D3—28no—
335C4×D3—28no—
336C4×D3—24no—
337C4×D3—24no—
338C4×D3—48no—
339Dic6—24no—
340C3×C8—24no—
341C3×C8—21no—
342C3×C8—22no—
343C3×C8H(5,𝔽2)12yes—
344C3⋊C8—24no—
345C3⋊C8—24no—
346C24—34no—
347#41—42no—
348#41—42no—
349#41—68no—
350C2×Q8—32no—
351C2×Q8C2×C4×D312yes—
352C2×D4—34no—
353C2×D4—38no—
354C2×D4—32no—
355C2×D4—616no—
356C2×D4C2×C4×D314yes—
357C2×D4C2×C4×D312yes—
358C2×D4C2×C4×D314yes—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—34no—
362C22×C4—34no—
363C22×C4—38no—
364C22×C4—124no—
365C22×C4—124no—
366C22×C4C2×C4×D312yes—
367C22×C4C23×D311yescharacteristic
368C22×C4C23×D314yes—
369M16—42no—
370M16—44no—
371M16—124no—
372C2×C8—28no—
373C2×C8—22no—
374C2×C8—34no—
375C2×C8—34no—
376C2×C8—41no—
377C2×C8—68no—
378C2×C8—128no—
379C2×C8—124no—
380C2×C8—124no—
381C2×C8C23×D312yes—
382C2×C8C23×D311yescharacteristic
383C2×C8C23×D314yes—
384C2×C8C23×D311yescharacteristic
385C4⋊C4—32no—
386C4⋊C4—38no—
387C4⋊C4—32no—
388C4⋊C4—34no—
389C4⋊C4—34no—
390C4⋊C4—38no—
391C4⋊C4C2×C4×D314yes—
392C22⋊C4—38no—
393C22⋊C4—34no—
394C22⋊C4—34no—
395C22⋊C4—34no—
396C22⋊C4—34no—
397C22⋊C4—38no—
398C22⋊C4—34no—
399C22⋊C4C2×C4×D318yes—
400C42—34no—
401C42—34no—
402C42—34no—
403C42C2×C4×D311yescharacteristic
404C42C2×C4×D311yescharacteristic
405C42C23×D312yes—
406C22×C3—24no—
407C22×C3—21no—
408C22×C3—22no—
409C22×C3—28no—
410C22×C3C24×C411yescharacteristic
411C2×D3—24no—
412C2×D3—24no—
413C2×D3—416no—
414C2×D3C2×#13118yes—
415C3×C4—24no—
416C3×C4—21no—
417C3×C4—22no—
418C3×C4C2×#13114yes—
419C3×C4C2×H(5,𝔽2)11yescharacteristic
420C3×C4#58511yescharacteristic
421Dic3—24no—
422Dic3—24no—
423Dic3—24no—
424C23—34no—
425C23—616no—
426C23—124no—
427C23C22×C4×D314yes—
428C23C22×C4×D311yescharacteristic
429Q8—42no—
430Q8—64no—
431D4—28no—
432D4—38no—
433D4—44no—
434D4—42no—
435D4—616no—
436D4—68no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—34no—
443C2×C4—34no—
444C2×C4—34no—
445C2×C4—68no—
446C2×C4—64no—
447C2×C4—64no—
448C2×C4—68no—
449C2×C4—128no—
450C2×C4—124no—
451C2×C4C22×C4×D312yes—
452C2×C4C22×C4×D311yescharacteristic
453C2×C4C22×C4×D314yes—
454C2×C4C22×C4×D312yes—
455C2×C4C24×D311yescharacteristic
456C8—24no—
457C8—21no—
458C8—22no—
459C8—64no—
460C8—64no—
461C8#100412yes—
462C2×C3—44no—
463C2×C3—41no—
464C2×C3C4×H(5,𝔽2)11yescharacteristic
465C2×C3C22×#13112yes—
466D3—28no—
467C22—22no—
468C22—28no—
469C22—24no—
470C22—21no—
471C22—38no—
472C22—64no—
473C22—64no—
474C22—1216no—
475C22C23×C4×D311yescharacteristic
476C4—22no—
477C4—24no—
478C4—21no—
479C4—64no—
480C4—64no—
481C4—64no—
482C4D3×#13114yes—
483C4C2×#100411yescharacteristic
484C4#596411yescharacteristic
485C3#6030511yescharacteristic, complemented, Sylow
486C2—44no—
487C2—41no—
488C2—68no—
489C2C4×#100411yescharacteristic
490C2C2×D3×#13112yes—
subgroupG/H
centerC2×C8C23×D3
commutator, socleC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC3×#2931C2
prime radical#86239C2
cosocleC2×C3×C4C25
hypercenter#2931D3
hypocenterC3#60305
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ #2931

Complements

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

NH ≅ G/Ntype
2. C3×#2931C2semidirect
3. #83794C2semidirect
4. #83278C2semidirect
5. #79542C2semidirect
6. #79527C2semidirect
7. D3×C8×D4C2semidirect
8. #79401C2semidirect
9. #79051C2semidirect
10. #78665C2semidirect
11. C8×#235C2semidirect
13. C2×C3×#131C22semidirect
16. C22×D3×C8C22semidirect
26. C3×#436C22semidirect
27. C3×C8×D4C22semidirect
28. C3×#423C22semidirect
29. C3×#408C22semidirect
30. C2×C3×C4×C8C22semidirect
31. #5234C22semidirect
32. #5127C22semidirect
33. #5127C22semidirect
34. #5126C22semidirect
35. #5119C22semidirect
36. C8×#67C22semidirect
37. C8×#67C22semidirect
38. C8×#67C22semidirect
39. #5101C22semidirect
40. #5012C22semidirect
41. D4×C3⋊C8C22semidirect
42. #4925C22semidirect
43. #4835C22semidirect
45. D3×C4⋊C8C22semidirect
46. #4831C22semidirect
47. #4728C22semidirect
48. #67⋊C8C22semidirect
49. D3×C22⋊C8C22semidirect
50. #4720C22semidirect
53. #4685C22semidirect
54. C4×D3×C8C22semidirect
55. C8×Dic6C22semidirect
56. C8×Dic6C22semidirect
65. #2931D3semidirect
136. C2×#131C2×D3semidirect
153. #436C2×D3semidirect
165. C8×D4C2×D3semidirect
171. #423C2×D3semidirect
174. #408C2×D3semidirect
179. C2×C4×C8C2×D3semidirect
333. D12#131semidirect
485. C3#60305semidirect

Maximal subgroups

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

subgroupindex
#862392
C3×#29312
#837942
#832782
#795422
#795272
D3×C8×D42
#794012
#790512
#786652
C8×#2352
#603053