Finite Group Databasebeta

← #10830937

Subgroups — #10830937

Order 768.

1776 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×D8C211yesmaximal, characteristic, complemented
2#82640C212yesmaximal, complemented
3C2×#5039C212yesmaximal, complemented
4C2×#4728C211yesmaximal, characteristic, complemented
5C3×#1999C211yesmaximal, characteristic, complemented
6#72040C211yesmaximal, characteristic, complemented
7#70198C212yesmaximal, complemented
8#70191C212yesmaximal, complemented
9#69961C211yesmaximal, characteristic, complemented
10#68868C211yesmaximal, characteristic, complemented
11#68862C211yesmaximal, characteristic, complemented
12#24272—31nomaximal, Sylow
13C22×D3×D4C2212yes—
14C22×C3×D8C2211yescharacteristic
15C2×C3×#121C2212yescomplemented
16C2×#937—22no—
17C22×#213C2212yes—
18C2×#930C2212yescomplemented
19C2×C4×#67C2211yescharacteristic
20C22×#208C2211yescharacteristic, complemented
21C2×D3×D8—28no—
22C2×D3×D8—24no—
23C22×D24C2211yescharacteristic
24C22×D3×C8C2211yescharacteristic
25C2×C3×#102C2212yescomplemented
26C2×C3×C22⋊C8C2211yescharacteristic, complemented
27#5223—22no—
28#5221C2212yescomplemented
29C2×#832C2212yescomplemented
30C2×#821C2212yescomplemented
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#815C2211yescharacteristic, complemented
33C2×Dic3⋊C8C2211yescharacteristic, complemented
34#5039—28no—
35#4988C2212yescomplemented
36#4987C2212yes—
37C2×#809C2212yescomplemented
38C2×#808C2212yescomplemented
39#4959—22no—
40#4955—22no—
41#4728—24no—
42C3×#340C2211yescharacteristic, complemented
43#4551C2211yescharacteristic, complemented
44C23×D8—31no—
45C2×#469—32no—
46C2×#435—31no—
47#2408—32no—
48#2054—31no—
49#2054—31no—
50#1999—31no—
51#1999D311yescharacteristic, complemented
52#1972—32no—
53#1807—31no—
54#1786—32no—
55C24×D3—22no—
56C22×C3×D4—22no—
57C22×C3×D4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67—22no—
60C22×#67—22no—
61C22×#67D411yescharacteristic
62C2×D3×D4—24no—
63C2×D3×D4—24no—
64C2×D3×D4—24no—
65C2×D3×D4—24no—
66C2×D3×D4—42no—
67C2×D3×D4—42no—
68C22×D12C2312yes—
69C22×C4×D3C2311yescharacteristic
70C2×C3×D8—24no—
71C2×C3×D8—22no—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#121—24no—
75C3×#121—42no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217D411yescharacteristic
81#937—48no—
82C2×#213—24no—
83C2×#213—24no—
84C2×#213D418yescomplemented
85#930—24no—
86#930—42no—
87C4×#67—22no—
88C4×#67—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×C4⋊Dic3C2312yes—
95C2×Dic3⋊C4D411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C2×#208—22no—
98C2×#208—24no—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101D3×D8—232no—
102C2×D24—24no—
103C2×D24—22no—
104C2×D3×C8—24no—
105C2×D3×C8D414yescomplemented
106C3×#102—42no—
107C3×C22⋊C8—22no—
108#832—24no—
109C3⋊#95—22no—
110#821—42no—
111(C2×D3)⋊C8—22no—
112#815—22no—
113Dic3⋊C8—22no—
114#809—24no—
115#808—24no—
116C23×D4—32no—
117C22×D8—32no—
118C22×D8—31no—
119C22×D8—64no—
120C22×D8—68no—
121C22×D8C2×D311yescharacteristic
122C22×M16—31no—
123C23×C8—31no—
124C2×#121—32no—
125C2×#121C2×D312yescomplemented
126C2×#120—62no—
127C2×C4×D4—31no—
128#469—68no—
129#435—64no—
130C2×#106—32no—
131C2×C4⋊C8—31no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102C2×D312yescomplemented
136C2×C22⋊C8—31no—
137C2×C22⋊C8C2×D311yescharacteristic, complemented
138C2×#97—31no—
139#394—62no—
140#392—62no—
141#390—32no—
142#390—32no—
143#380—32no—
144#379—62no—
145#340—31no—
146#340C2×D311yescharacteristic, complemented
147C24×C3—22no—
148C24×C3C2×D411yescharacteristic
149C23×D3—22no—
150C23×D3—44no—
151C23×D3—44no—
152C23×D3—44no—
153C23×D3—44no—
154C23×D3—44no—
155C23×D3—82no—
156C23×D3—82no—
157C23×D3C2×D411yescharacteristic
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—82no—
166C2×C3×D4C2×D414yes—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C2×#67—22no—
175C2×#67—24no—
176C2×#67—44no—
177C2×#67—44no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—44no—
181C2×#67—82no—
182C22×Dic3—22no—
183C22×Dic3C2×D411yescharacteristic
184C22×Dic3C2×D411yescharacteristic
185C22×Dic3C2×D411yescharacteristic
186D3×D4—216no—
187D3×D4—48no—
188D3×D4—48no—
189D3×D4—48no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12C2×D414yes—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3C2×D412yes—
199C2×C4×D3#4112yes—
200C3×D8—28no—
201C2×C3×C8—22no—
202C2×C3×C8—21no—
203C2×C3×C8—22no—
204C2×C3×C8C2×D412yes—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C22⋊C4—44no—
208#217—22no—
209#217—44no—
210#217—82no—
211#213—216no—
212(C2×D3)⋊C4—22no—
213(C2×D3)⋊C4—44no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—44no—
216(C2×D3)⋊C4—82no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219Dic3⋊C4—22no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222#208—44no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8C2×D414yes—
226C2×C3⋊C8C2×D412yes—
227D24—28no—
228D3×C8—28no—
229C25—62no—
230C22×D4—22no—
231C22×D4—31no—
232C22×D4—32no—
233C22×D4—64no—
234C22×D4—64no—
235C22×D4—62no—
236C22×D4—64no—
237C22×D4—64no—
238C22×D4—62no—
239C22×D4—122no—
240C22×D4—122no—
241C22×D4C22×D312yes—
242C23×C4—31no—
243C23×C4C22×D311yescharacteristic
244C2×D8—24no—
245C2×D8—22no—
246C2×D8—38no—
247C2×D8—64no—
248C2×D8—64no—
249C2×D8—64no—
250C2×D8—632no—
251C2×D8—62no—
252C2×M16—62no—
253C2×M16—64no—
254C22×C8—31no—
255C22×C8—34no—
256C22×C8—31no—
257C22×C8—64no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#121—24no—
261#121—42no—
262#121—64no—
263#121—122no—
264#120—128no—
265C4×D4—62no—
266C4×D4—62no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4C22×D312yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—31no—
272C2×C22⋊C4—31no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C42—31no—
280#106—64no—
281C4⋊C8—62no—
282#102—42no—
283#102—64no—
284#102—64no—
285#102—122no—
286C22⋊C8—22no—
287C22⋊C8—62no—
288#97—62no—
289#95—62no—
290C23×C3—21no—
291C23×C3—24no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3C22×D411yescharacteristic
303C22×D3—22no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—24no—
307C22×D3—416no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—88no—
315C22×D3—88no—
316C22×D3—88no—
317C22×D3—88no—
318C22×D3—88no—
319C22×D3—88no—
320C22×D3#12112yes—
321C22×D3AGL(1,ℤ/8)14yes—
322C3×D4—24no—
323C3×D4—24no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—88no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D412yes—
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4H(5,𝔽2)12yes—
339#67—416no—
340#67—44no—
341#67—88no—
342C2×Dic3—22no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—22no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3#12112yes—
352D12—24no—
353D12—24no—
354D12—42no—
355D12—42no—
356C4×D3—22no—
357C4×D3—22no—
358C4×D3—24no—
359C3×C8—22no—
360C3×C8—41no—
361C3⋊C8—41no—
362C3⋊C8#12014yescomplemented
363C24—22no—
364C24—31no—
365C24—62no—
366C24—124no—
367C24—124no—
368C24—124no—
369C24—124no—
370C24—124no—
371C24—242no—
372C24—242no—
373C24D3×D411yescharacteristic
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—34no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—616no—
383C2×D4—62no—
384C2×D4—64no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—82no—
391C2×D4—124no—
392C2×D4—128no—
393C2×D4—128no—
394C2×D4—128no—
395C2×D4—124no—
396C2×D4—124no—
397C2×D4—124no—
398C2×D4—124no—
399C2×D4—242no—
400C2×D4D3×D414yes—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—31no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4D3×D411yescharacteristic
415C22×C4C23×D311yescharacteristic
416D8—28no—
417D8—68no—
418D8—616no—
419M16—124no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—32no—
424C2×C8—34no—
425C2×C8—68no—
426C2×C8—61no—
427C2×C8—62no—
428C2×C8D3×D412yes—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C22⋊C4—44no—
435C22⋊C4—62no—
436C22⋊C4—62no—
437C22⋊C4—124no—
438C22⋊C4—124no—
439C22⋊C4—124no—
440C22⋊C4—124no—
441C22⋊C4—242no—
442C22⋊C4—242no—
443C42—61no—
444C42—61no—
445C22×C3—44no—
446C22×C3—41no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3C2×#12111yescharacteristic
458C22×C3D4212yes—
459C22×C3C2×AGL(1,ℤ/8)12yes—
460C22×C3#57611yescharacteristic
461C22×C3#57611yescharacteristic
462C2×D3—22no—
463C2×D3—24no—
464C2×D3—24no—
465C2×D3—22no—
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—832no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—82no—
474C2×D3#46914yescomplemented
475C3×C4—41no—
476C3×C4—82no—
477C3×C4C2×#12011yescharacteristic
478C3×C4#53511yescharacteristic
479C3×C4#54612yes—
480Dic3—22no—
481Dic3—41no—
482Dic3—41no—
483Dic3—82no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—21no—
490C23—34no—
491C23—32no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—61no—
499C23—61no—
500C23—62no—
501C23—64no—
502C23—1216no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—248no—
510C23—248no—
511C23—248no—
512C23—248no—
513C23—248no—
514C23—248no—
515C23C2×D3×D411yescharacteristic
516D4—24no—
517D4—24no—
518D4—42no—
519D4—42no—
520D4—64no—
521D4—64no—
522D4—88no—
523D4—1216no—
524D4—124no—
525D4—122no—
526D4—122no—
527D4—248no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—32no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4C2×D3×D411yescharacteristic
550C2×C4C2×D3×D412yes—
551C2×C4#100412yes—
552C2×C4D3×#4111yescharacteristic
553C8—22no—
554C8—34no—
555C8—41no—
556C8—121no—
557C2×C3—84no—
558C2×C3—81no—
559C2×C3#240811yescharacteristic
560C2×C3C2×#46911yescharacteristic
561C2×C3#305811yescharacteristic
562C2×C3#326412yes—
563C2×C3#326412yes—
564D3—24no—
565D3—84no—
566C22—34no—
567C22—41no—
568C22—41no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—41no—
573C22—42no—
574C22—44no—
575C22—62no—
576C22—62no—
577C22—64no—
578C22—64no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—124no—
584C22—124no—
585C22—124no—
586C22—2432no—
587C22—242no—
588C22—242no—
589C22—242no—
590C22—242no—
591C22#559012yes—
592C22D3×#12111yescharacteristic
593C22#575711yescharacteristic
594C22#576911yescharacteristic
595C22D3×AGL(1,ℤ/8)12yes—
596C4—41no—
597C4—62no—
598C4—82no—
599C4—121no—
600C4—121no—
601C4—242no—
602C4D3×#12011yescharacteristic
603C4#561012yes—
604C4#571511yescharacteristic
605C3#2427211yescharacteristic, complemented, Sylow
606C2—64no—
607C2—81no—
608C2—84no—
609C2—244no—
610C2#7666911yescharacteristic
611C2#8001212yes—
612C2#8012012yes—
613C2#8071611yescharacteristic
614C2D3×#46911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1999C2
socleC23×C3C22×D4
hypercenter#1999D3
hypocenterC3#24272
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. C22×D3×D8C2semidirect
2. #82640C2semidirect
3. C2×#5039C2semidirect
4. C2×#4728C2semidirect
5. C3×#1999C2semidirect
6. #72040C2semidirect
7. #70198C2semidirect
8. #70191C2semidirect
9. #69961C2semidirect
10. #68868C2semidirect
11. #68862C2semidirect
15. C2×C3×#121C22semidirect
18. C2×#930C22semidirect
20. C22×#208C22semidirect
25. C2×C3×#102C22semidirect
26. C2×C3×C22⋊C8C22semidirect
28. #5221C22semidirect
29. C2×#832C22semidirect
30. C2×#821C22semidirect
32. C2×#815C22semidirect
33. C2×Dic3⋊C8C22semidirect
35. #4988C22semidirect
37. C2×#809C22semidirect
38. C2×#808C22semidirect
42. C3×#340C22semidirect
43. #4551C22semidirect
51. #1999D3semidirect
84. C2×#213D4semidirect
105. C2×D3×C8D4semidirect
125. C2×#121C2×D3semidirect
135. C2×#102C2×D3semidirect
137. C2×C22⋊C8C2×D3semidirect
146. #340C2×D3semidirect
362. C3⋊C8#120semidirect
474. C2×D3#469semidirect
605. C3#24272semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D82
#826402
C2×#50392
C2×#47282
C3×#19992
#720402
#701982
#701912
#699612
#688682
#688622
#242723