Finite Group Databasebeta

← #10900849

Subgroups — #10900849

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic, complemented
2C2×#5238C211yesmaximal, characteristic, complemented
3C2×#4732C211yesmaximal, characteristic, complemented
4#74033C211yesmaximal, characteristic, complemented
5C3×#1788C211yesmaximal, characteristic, complemented
6#70277C212yesmaximal, complemented
7#70245C212yesmaximal, complemented
8#70209C212yesmaximal, complemented
9#70177C212yesmaximal, complemented
10#69911C211yesmaximal, characteristic, complemented
11#69521C211yesmaximal, characteristic, complemented
12#29630—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15#5830—24no—
16#5830—24no—
17C2×#947C2212yes—
18C22×#215C2211yescharacteristic, complemented
19C2×#940C2212yes—
20C2×#937—22no—
21C22×#213C2211yescharacteristic, complemented
22C2×#930—21no—
23C2×#930—21no—
24C2×#930C2211yescharacteristic
25C2×#926C2211yescharacteristic
26C22×D24C2212yes—
27C2×C3×#103C2212yescomplemented
28C2×C3×#102C2212yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#380C2211yescharacteristic
31#5238—24no—
32C2×#835C2211yescharacteristic, complemented
33C2×#832C2211yescharacteristic, complemented
34C2×#830C2211yescharacteristic
35C2×#821C2212yes—
36#4988C2212yescomplemented
37#4984C2212yescomplemented
38C2×#809C2212yes—
39C2×#809C2212yes—
40#4961—22no—
41#4732—24no—
42#4677—21no—
43#4674—21no—
44#4550C2212yescomplemented
45#4550C2212yescomplemented
46C2×#535—31no—
47C2×#449—31no—
48C2×#447—31no—
49#2624—31no—
50#2042—32no—
51#2041—32no—
52#2012—32no—
53#2011—32no—
54#1818—31no—
55#1788D311yescharacteristic, complemented
56#1786—31no—
57C24×D3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C2×C3×#41D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C22×#67—21no—
68C22×#67—22no—
69C22×#67—22no—
70C22×D12—21no—
71C22×D12—22no—
72C22×D12—42no—
73C22×D12—42no—
74C22×D12D414yes—
75C22×D12C2311yescharacteristic
76C22×D12C2312yes—
77C22×C3×C8C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×#217D412yes—
81#947—24no—
82#947—24no—
83C2×#215—22no—
84C2×#215—42no—
85#940—24no—
86#940—24no—
87#937—216no—
88C2×#213—22no—
89C2×#213—42no—
90#930—28no—
91#930—28no—
92#930—22no—
93#930—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4×Dic3C2312yes—
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C2×D24—24no—
109C2×D24—42no—
110C2×D24—42no—
111C3×#103—42no—
112C3×#102—42no—
113C3×C22⋊C8—22no—
114C3×#95—21no—
115#835—22no—
116#832—22no—
117C3⋊#95—22no—
118#830—22no—
119#821—24no—
120#809—42no—
121#809—42no—
122C22×#41C2×D311yescharacteristic
123C23×D4—31no—
124C22×SD16—31no—
125C22×D8—31no—
126C22×D8—32no—
127#535—64no—
128#535—64no—
129C2×#127—32no—
130C2×#124—32no—
131C2×#121—31no—
132C2×#121—61no—
133C2×#121—61no—
134C2×#120—62no—
135C2×#117—31no—
136#449—64no—
137#447—64no—
138C2×#103—31no—
139C2×#103C2×D312yescomplemented
140C2×#102—31no—
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102C2×D312yescomplemented
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147#399—61no—
148#397—61no—
149#394—62no—
150#390—32no—
151#389—32no—
152#380C2×D311yescharacteristic
153#336—32no—
154#336—32no—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C23×D3—22no—
158C23×D3—24no—
159C23×D3—21no—
160C23×D3—44no—
161C23×D3—81no—
162C23×D3—82no—
163C23×D3—81no—
164C3×#41—44no—
165C3×#41—44no—
166C3×#41—48no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—24no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—44no—
176C2×C3×D4C2×D412yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—22no—
182C22×C3×C4—44no—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2411yescharacteristic
187C2×#67—28no—
188C2×#67—28no—
189C2×#67—42no—
190C2×#67—82no—
191C2×#67—82no—
192C2×#67—82no—
193C2×#67—82no—
194C2×#67—82no—
195C22×Dic3—21no—
196C22×Dic3C2×D412yes—
197C22×Dic3C2×D412yes—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—24no—
203C2×D12—22no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—48no—
209C2×D12—44no—
210C2×D12—44no—
211C2×D12—44no—
212C2×D12C2×D414yes—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C22⋊C4—81no—
219#217—24no—
220#215—44no—
221#213—44no—
222(C2×D3)⋊C4—28no—
223(C2×D3)⋊C4—28no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—82no—
226(C2×D3)⋊C4—82no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—82no—
232C4×Dic3—22no—
233C4×Dic3—22no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238D24—48no—
239C25—61no—
240C2×#41—22no—
241C2×#41—42no—
242C2×#41—42no—
243C2×#41D1214yes—
244C22×Q8C22×D311yescharacteristic
245C22×D4—21no—
246C22×D4—34no—
247C22×D4—32no—
248C22×D4—31no—
249C22×D4—62no—
250C22×D4—62no—
251C22×D4—61no—
252C22×D4—62no—
253C22×D4—61no—
254C22×D4—122no—
255C22×D4—122no—
256C22×D4C22×D311yescharacteristic
257C23×C4—21no—
258C23×C4C22×D311yescharacteristic
259C2×SD16—62no—
260C2×SD16—122no—
261C2×D8—64no—
262C2×D8—62no—
263C2×D8—122no—
264C2×D8—122no—
265C2×D8—122no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8C22×D312yes—
269#127—64no—
270#127—64no—
271#124—64no—
272#124—64no—
273#121—68no—
274#121—68no—
275#121—62no—
276#121—62no—
277#120—616no—
278#117—62no—
279#117—62no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4C22×D312yes—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—32no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—62no—
292C2×C42—32no—
293#103—42no—
294#103—62no—
295#102—42no—
296#102—64no—
297#102—62no—
298#102—122no—
299#102—122no—
300C22⋊C8—22no—
301C22⋊C8—62no—
302#95—21no—
303#95—62no—
304C23×C3—22no—
305C23×C3—21no—
306C23×C3—22no—
307C23×C3—21no—
308C23×C3—22no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3#12012yes—
314C23×C3C22×D411yescharacteristic
315C22×D3—24no—
316C22×D3—48no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—41no—
324C22×D3—42no—
325C22×D3—41no—
326C22×D3—42no—
327C22×D3—41no—
328C22×D3—84no—
329C22×D3—84no—
330C22×D3—88no—
331C22×D3—88no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—42no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—48no—
338C3×D4—42no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—48no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4#12012yes—
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4H(5,𝔽2)12yes—
353C2×C3×C4H(5,𝔽2)12yes—
354#67—88no—
355#67—88no—
356#67—84no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365D12—24no—
366D12—24no—
367D12—48no—
368D12—42no—
369D12—42no—
370D12—42no—
371D12—41no—
372D12—41no—
373C3×C8—42no—
374C3⋊C8—41no—
375C3⋊C8—41no—
376C24—21no—
377C24—62no—
378C24—64no—
379C24—61no—
380C24—124no—
381C24—242no—
382C24—241no—
383C24—241no—
384C24D3×D411yescharacteristic
385#41—44no—
386#41—44no—
387#41—48no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8C2×D1212yes—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—34no—
397C2×D4—44no—
398C2×D4—68no—
399C2×D4—68no—
400C2×D4—64no—
401C2×D4—62no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—128no—
411C2×D4—124no—
412C2×D4—124no—
413C2×D4—124no—
414C2×D4—122no—
415C2×D4—242no—
416C2×D4—242no—
417C2×D4—242no—
418C2×D4—242no—
419C2×D4—242no—
420C2×D4C2×D1212yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—44no—
429C22×C4—61no—
430C22×C4C2×D1212yes—
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D412yes—
433C22×C4C23×D311yescharacteristic
434SD16—124no—
435D8—128no—
436D8—124no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—62no—
443C2×C8—62no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—62no—
447C4⋊C4—62no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—242no—
451C22⋊C4—68no—
452C22⋊C4—68no—
453C22⋊C4—64no—
454C22⋊C4—81no—
455C22⋊C4—122no—
456C22⋊C4—242no—
457C22⋊C4—242no—
458C42—62no—
459C42—62no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3C2×#12011yescharacteristic
471C22×C3#53512yes—
472C22×C3#57612yes—
473C22×C3#57611yescharacteristic
474C22×C3#57711yescharacteristic
475C2×D3—44no—
476C2×D3—44no—
477C2×D3—44no—
478C2×D3—816no—
479C2×D3—82no—
480C2×D3—82no—
481C2×D3—82no—
482C2×D3—82no—
483C2×D3—81no—
484C2×D3—82no—
485C2×D3—82no—
486C2×D3—81no—
487C2×D3—81no—
488C3×C4—22no—
489C3×C4—42no—
490C3×C4—82no—
491C3×C4#53411yescharacteristic
492C3×C4#53411yescharacteristic
493C3×C4#53412yes—
494Dic3—42no—
495Dic3—42no—
496Dic3—81no—
497C23—22no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—64no—
507C23—128no—
508C23—122no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23—122no—
513C23—122no—
514C23—121no—
515C23—122no—
516C23—121no—
517C23—121no—
518C23—122no—
519C23—244no—
520C23—244no—
521C23—248no—
522C23—248no—
523C23#88212yes—
524C23C2×D3×D411yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—42no—
528D4—22no—
529D4—22no—
530D4—48no—
531D4—42no—
532D4—64no—
533D4—64no—
534D4—128no—
535D4—122no—
536D4—122no—
537D4—121no—
538D4—121no—
539D4—122no—
540D4—248no—
541D4—248no—
542D4—244no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—121no—
560C2×C4—122no—
561C2×C4#88212yes—
562C2×C4C22×D1211yescharacteristic
563C2×C4C2×D3×D411yescharacteristic
564C2×C4#100412yes—
565C2×C4#100412yes—
566C8—42no—
567C8—121no—
568C8—121no—
569C2×C3—42no—
570C2×C3—82no—
571C2×C3#261311yescharacteristic
572C2×C3#300811yescharacteristic
573C2×C3#301211yescharacteristic
574C2×C3#322012yes—
575C2×C3#322412yes—
576D3—84no—
577D3—161no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22—124no—
589C22—124no—
590C22—124no—
591C22—2416no—
592C22—242no—
593C22—242no—
594C22—242no—
595C22—242no—
596C22—241no—
597C22—241no—
598C22—242no—
599C22—241no—
600C22—242no—
601C22C2×#88211yescharacteristic
602C22#559212yes—
603C22#576912yes—
604C22#577411yescharacteristic
605C22#577811yescharacteristic
606C4—22no—
607C4—42no—
608C4—82no—
609C4—122no—
610C4—122no—
611C4—241no—
612C4#549311yescharacteristic
613C4#549311yescharacteristic
614C4#558912yes—
615C3#2963011yescharacteristic, complemented, Sylow
616C2—42no—
617C2—82no—
618C2—244no—
619C2—481no—
620C2#7385511yescharacteristic
621C2#7883411yescharacteristic
622C2#7888611yescharacteristic
623C2#8011812yes—
624C2#8026312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1788C2
socleC23×C3C22×D4
hypercenter#1788D3
hypocenterC3#29630
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
1. C2×#5830C2semidirect
2. C2×#5238C2semidirect
3. C2×#4732C2semidirect
4. #74033C2semidirect
5. C3×#1788C2semidirect
6. #70277C2semidirect
7. #70245C2semidirect
8. #70209C2semidirect
9. #70177C2semidirect
10. #69911C2semidirect
11. #69521C2semidirect
18. C22×#215C22semidirect
21. C22×#213C22semidirect
27. C2×C3×#103C22semidirect
28. C2×C3×#102C22semidirect
32. C2×#835C22semidirect
33. C2×#832C22semidirect
36. #4988C22semidirect
37. #4984C22semidirect
44. #4550C22semidirect
45. #4550C22semidirect
55. #1788D3semidirect
139. C2×#103C2×D3semidirect
144. C2×#102C2×D3semidirect
615. C3#29630semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
C2×#52382
C2×#47322
#740332
C3×#17882
#702772
#702452
#702092
#701772
#699112
#695212
#296303