Finite Group Databasebeta

← #10801121

Subgroups — #10801121

Order 768.

1422 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85857C211yesmaximal, characteristic
2#82953C211yesmaximal, characteristic, complemented
3C2×#5061C211yesmaximal, characteristic
4#78946C212yesmaximal, complemented
5C3×#1885C211yesmaximal, characteristic, complemented
6#71647C211yesmaximal, characteristic, complemented
7#70348C212yesmaximal, complemented
8#70086C211yesmaximal, characteristic
9Dic3×#100C211yesmaximal, characteristic, complemented
10#68771C212yesmaximal, complemented
11#68729C212yesmaximal, complemented
12#21474—31nomaximal, Sylow
13C2×#1004C2211yescharacteristic
14C2×C3×#124C2211yescharacteristic
15C3×#518C2211yescharacteristic
16C2×Q8×Dic3C2211yescharacteristic, complemented
17C2×#942C2212yes—
18C2×D4×Dic3C2211yescharacteristic
19#5546—22no—
20#5545—22no—
21C4×D3×D4—22no—
22C4×#237—22no—
23#5523C2212yes—
24#5522C2212yes—
25#5486—22no—
26C2×#67⋊C4—22no—
27C3×#419C2212yescomplemented
28C2×C3×#100C2211yescharacteristic
29#5229C2211yescharacteristic, complemented
30#5221C2211yescharacteristic
31C2×#833C2211yescharacteristic
32C2×#825C2212yes—
33#5133C2212yes—
34Dic3×M16C2212yescomplemented
35#5061C418yescomplemented
36#4995C2212yescomplemented
37#4951—22no—
38#4950—22no—
39#4935C2211yescharacteristic, complemented
40#4932C2211yescharacteristic, complemented
41C2×#805C2212yes—
42#4817—22no—
43#4790—22no—
44#4789—22no—
45#4788—22no—
46C3×#339C2212yescomplemented
47#4554C2212yescomplemented
48#4532C2212yescomplemented
49C4×H(5,𝔽2)—31no—
50C2×#455—31no—
51C4×#137—32no—
52#2326—31no—
53#1885D311yescharacteristic, complemented
54#1882—32no—
55#1859—32no—
56#1856—31no—
57#1792—31no—
58#1759—32no—
59C4×#100—31no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C22×#67—22no—
63C23×Dic3D412yes—
64#1004—22no—
65#1004—24no—
66#1004C2×C414yes—
67C2×#237—22no—
68C2×D3×D4—22no—
69C2×#235C2312yes—
70C22×C4×D3—22no—
71C2×C3×M16C2312yes—
72C3×#124—22no—
73C3×#124C2×C414yescomplemented
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×#117C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#217D412yes—
82Q8×Dic3—22no—
83Q8×Dic3—22no—
84C2×#216—22no—
85#942—24no—
86#942C2×C418yescomplemented
87C2×#215—22no—
88D4×Dic3—22no—
89D4×Dic3—22no—
90C4×#67—24no—
91C4×#67—24no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3C2311yescharacteristic
101C2×C4×Dic3C2311yescharacteristic
102C2×#208C2312yes—
103#893D412yes—
104#893D412yes—
105#892—22no—
106D3×C4⋊C4—22no—
107#887D412yes—
108#887D412yes—
109#67⋊C4—24no—
110#67⋊C4—24no—
111#67⋊C4—22no—
112#67⋊C4—24no—
113#67⋊C4—22no—
114D3×C22⋊C4—24no—
115#877—24no—
116C4×D12—22no—
117#872—22no—
118C42×D3—22no—
119C4×Dic6—22no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#100—22no—
123#833C2×C414yescomplemented
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—22no—
127#825—24no—
128#821—22no—
129#816—22no—
130#815—22no—
131C8×Dic3—22no—
132#808—22no—
133#807—22no—
134#805C2×C418yescomplemented
135C2×H(5,𝔽2)—31no—
136C2×#137—32no—
137C2×#124C2×D311yescharacteristic
138#520—32no—
139#520—62no—
140#518—62no—
141#518—62no—
142#518C2×D311yescharacteristic
143C4×#41—32no—
144C4×#41—62no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—62no—
148C2×C4×D4—62no—
149#455—38no—
150#441—62no—
151#440—62no—
152C4×Q16—62no—
153C4×SD16—62no—
154#428—32no—
155C2×#104—32no—
156C2×#104—32no—
157#419C2×D312yescomplemented
158#417—32no—
159C2×#100—31no—
160C2×#100C2×D311yescharacteristic
161C4×M16—32no—
162#391—31no—
163#390—31no—
164#388—62no—
165#385—62no—
166#384—31no—
167#376—31no—
168#343—32no—
169#339—32no—
170#339C2×D312yescomplemented
171C24×C3C2×D412yes—
172C23×D3—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8C22×C412yescomplemented
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4C22×C412yes—
180C2×C3×D4#4112yes—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4C2411yescharacteristic
186C2×#67—24no—
187C2×#67—28no—
188C2×#67—28no—
189C2×#67—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—82no—
196C22×Dic3C2×D412yes—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199#237—28no—
200#237—28no—
201D3×D4—28no—
202D3×D4—28no—
203#235—22no—
204#235—22no—
205#235C22×C414yes—
206#235#4114yes—
207C2×D12C2×D412yes—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—24no—
211C2×C4×D3—22no—
212C2×C4×D3—84no—
213C2×Dic6C2×D412yes—
214C3×M16—22no—
215C3×M16—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4C2×D412yes—
218C3×C4⋊C4C2×D412yes—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—22no—
222C3×C42—22no—
223C3×C42—21no—
224C3×C42C22×C412yescomplemented
225#217—24no—
226#217—82no—
227#216—28no—
228#215—28no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—22no—
232C4⋊Dic3—22no—
233C4⋊Dic3—22no—
234C4⋊Dic3—41no—
235C4⋊Dic3—41no—
236C4⋊Dic3—82no—
237Dic3⋊C4—24no—
238Dic3⋊C4—22no—
239Dic3⋊C4—22no—
240C4×Dic3—24no—
241C4×Dic3—22no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244C4×Dic3—42no—
245C4×Dic3C2×D412yes—
246C4×Dic3C2×D412yes—
247#208—22no—
248#208C22×C414yescomplemented
249C2×C3⋊C8—22no—
250H(5,𝔽2)—34no—
251H(5,𝔽2)—64no—
252H(5,𝔽2)—62no—
253C2×#41—32no—
254C2×#41—62no—
255C22×Q8C22×D311yescharacteristic
256C22×D4—62no—
257C22×D4—62no—
258C22×D4C22×D311yescharacteristic
259C23×C4—32no—
260C23×C4—62no—
261#137—38no—
262#137—64no—
263C2×Q16—62no—
264C2×SD16—62no—
265C2×M16—32no—
266C2×M16C22×D312yes—
267#124—22no—
268#124C4×D314yescomplemented
269C4×Q8—32no—
270C4×Q8—32no—
271C4×Q8—62no—
272C4×Q8—62no—
273C4×Q8—62no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—32no—
277C4×D4—32no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—64no—
283C4×D4—62no—
284C4×D4—64no—
285C4×D4—64no—
286C4×D4—64no—
287C4×D4—64no—
288#117—62no—
289#117—64no—
290#117—62no—
291#117C22×D312yes—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4—62no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—62no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—32no—
301C2×C22⋊C4—64no—
302C2×C22⋊C4—62no—
303C2×C22⋊C4—62no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—62no—
307C2×C42—62no—
308C2×C42—61no—
309C2×C42C22×D311yescharacteristic
310#107—62no—
311#106—62no—
312#104—38no—
313#104—64no—
314#103—22no—
315#103—22no—
316#103—62no—
317#102—62no—
318#100—22no—
319#100—34no—
320#97—62no—
321C4×C8—62no—
322#95—61no—
323#95—61no—
324#95—62no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3C4×D414yes—
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—28no—
333C22×D3—82no—
334C22×D3—82no—
335C3×Q8—24no—
336C3×Q8—42no—
337C3×D4—24no—
338C3×D4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—82no—
348C2×C3×C4—82no—
349C2×C3×C4C23×C411yescharacteristic
350C2×C3×C4C22×D412yes—
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4112yes—
353#67—216no—
354#67—24no—
355#67—24no—
356C2×Dic3—28no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—44no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—82no—
369C2×Dic3#12011yescharacteristic
370C2×Dic3#12012yes—
371C2×Dic3#12011yescharacteristic
372D12—22no—
373D12C4×D414yes—
374C4×D3—24no—
375C4×D3—24no—
376C4×D3—44no—
377C4×D3—82no—
378C4×D3—84no—
379Dic6—22no—
380Dic6C4×D414yes—
381C3×C8—42no—
382C3⋊C8—24no—
383C24—62no—
384C24D3×D412yes—
385#41—34no—
386#41—34no—
387#41—68no—
388#41—68no—
389#41—62no—
390#41—62no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—32no—
395C2×Q8C2×C4×D312yescomplemented
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—32no—
399C2×D4—68no—
400C2×D4—68no—
401C2×D4—64no—
402C2×D4—68no—
403C2×D4—68no—
404C2×D4—62no—
405C2×D4C2×C4×D312yes—
406C2×D4#23712yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—32no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—64no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—244no—
424C22×C4—242no—
425C22×C4C23×D311yescharacteristic
426Q16—68no—
427SD16—68no—
428M16—22no—
429M16—22no—
430M16—34no—
431M16—62no—
432C2×C8—22no—
433C2×C8—62no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—64no—
437C4⋊C4—62no—
438C4⋊C4—62no—
439C4⋊C4—121no—
440C4⋊C4—121no—
441C4⋊C4—242no—
442C4⋊C4D3×D412yes—
443C4⋊C4D3×D412yes—
444C22⋊C4—24no—
445C22⋊C4—28no—
446C22⋊C4—22no—
447C22⋊C4—62no—
448C22⋊C4—62no—
449C22⋊C4—64no—
450C22⋊C4—64no—
451C22⋊C4—242no—
452C42—22no—
453C42—21no—
454C42—32no—
455C42—32no—
456C42—62no—
457C42—62no—
458C42—64no—
459C42—62no—
460C42—122no—
461C42C2×C4×D312yescomplemented
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—24no—
465C22×C3—44no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—82no—
469C22×C3C2×C4×D411yescharacteristic
470C22×C3C2×#12011yescharacteristic
471C22×C3#52511yescharacteristic
472C2×D3—24no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—88no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4C2×C4×D412yes—
481C3×C4#52512yes—
482Dic3—21no—
483Dic3—21no—
484Dic3—22no—
485Dic3—24no—
486Dic3—42no—
487Dic3—81no—
488Dic3—81no—
489C23—22no—
490C23—22no—
491C23—42no—
492C23—42no—
493C23—68no—
494C23—62no—
495C23—242no—
496C23—242no—
497C23#67⋊C414yes—
498C23C2×D3×D411yescharacteristic
499Q8—24no—
500Q8—34no—
501Q8—42no—
502Q8—62no—
503D4—24no—
504D4—24no—
505D4—34no—
506D4—616no—
507D4—64no—
508D4—64no—
509D4—62no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—31no—
515C2×C4—32no—
516C2×C4—31no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—68no—
522C2×C4—62no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—124no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—124no—
532C2×C4—122no—
533C2×C4—121no—
534C2×C4—122no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—242no—
539C2×C4—244no—
540C2×C4—242no—
541C2×C4C22×C4×D311yescharacteristic
542C2×C4C2×D3×D412yes—
543C2×C4C2×#23711yescharacteristic
544C2×C4D3×#4112yes—
545C8—42no—
546C8—64no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—44no—
550C2×C3C4×#12011yescharacteristic
551C2×C3#301911yescharacteristic
552C2×C3#302211yescharacteristic
553D3—44no—
554C22—21no—
555C22—21no—
556C22—24no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—64no—
561C22—82no—
562C22—122no—
563C22—122no—
564C22—248no—
565C22C2×#67⋊C411yescharacteristic
566C22D3×#12011yescharacteristic
567C22#565611yescharacteristic
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—64no—
573C4—61no—
574C4—61no—
575C4—62no—
576C4—122no—
577C4—241no—
578C4—241no—
579C4C4×D3×D412yes—
580C4#560612yes—
581C3#2147411yescharacteristic, complemented, Sylow
582C2—21no—
583C2—21no—
584C2—44no—
585C2—124no—
586C2#7465911yescharacteristic
587C2#8029211yescharacteristic
588C2#8029311yescharacteristic
subgroupG/H
centerC22D3×#120
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1885C2
socleC22×C3C2×#120
prime radicalC2×#1004C22
cosocleC22×C3×C4C24
hypercenter#1885D3
hypocenterC3#21474
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1885

Complements

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

NH ≅ G/Ntype
2. #82953C2semidirect
4. #78946C2semidirect
5. C3×#1885C2semidirect
6. #71647C2semidirect
7. #70348C2semidirect
9. Dic3×#100C2semidirect
10. #68771C2semidirect
11. #68729C2semidirect
16. C2×Q8×Dic3C22semidirect
27. C3×#419C22semidirect
29. #5229C22semidirect
34. Dic3×M16C22semidirect
35. #5061C4semidirect
36. #4995C22semidirect
39. #4935C22semidirect
40. #4932C22semidirect
46. C3×#339C22semidirect
47. #4554C22semidirect
48. #4532C22semidirect
53. #1885D3semidirect
73. C3×#124C2×C4semidirect
86. #942C2×C4semidirect
123. #833C2×C4semidirect
134. #805C2×C4semidirect
157. #419C2×D3semidirect
170. #339C2×D3semidirect
176. C2×C3×Q8C22×C4semidirect
224. C3×C42C22×C4semidirect
248. #208C22×C4semidirect
268. #124C4×D3semidirect
395. C2×Q8C2×C4×D3semidirect
461. C42C2×C4×D3semidirect
581. C3#21474semidirect

Maximal subgroups

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

subgroupindex
#858572
#829532
C2×#50612
#789462
C3×#18852
#716472
#703482
#700862
Dic3×#1002
#687712
#687292
#214743