Finite Group Databasebeta

← #10900848

Subgroups — #10900848

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×C3×#449C211yesmaximal, characteristic, complemented
3#82678C211yesmaximal, characteristic, complemented
4C2×#5036C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic, complemented
6#72055C212yesmaximal, complemented
7#72044C212yesmaximal, complemented
8#70279C212yesmaximal, complemented
9#70211C212yesmaximal, complemented
10#69959C211yesmaximal, characteristic, complemented
11#69911C211yesmaximal, characteristic, complemented
12#29630—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×SD16C2211yescharacteristic, complemented
16C22×C3×D8C2211yescharacteristic, complemented
17C2×C3×#121C2211yescharacteristic
18#5830—24no—
19#5830—24no—
20C2×#947C2212yes—
21C2×#940C2212yes—
22C2×#937—22no—
23C22×#213C2212yes—
24C2×#930—21no—
25C2×#930—21no—
26C2×#926C2211yescharacteristic
27C3×#449—24no—
28C2×C3×#103C2211yescharacteristic, complemented
29C2×C3×#102C2211yescharacteristic, complemented
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×#835C2212yescomplemented
32#5222—22no—
33#5221C2212yescomplemented
34C2×#832C2212yescomplemented
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37C2×#821C2212yes—
38C2×#821C2212yes—
39#5036—24no—
40#4972C2212yescomplemented
41C2×#809C2212yes—
42#4953—21no—
43#4949—21no—
44#4550C2212yescomplemented
45#4550C2212yescomplemented
46C2×#535—31no—
47C2×#449D311yescharacteristic, complemented
48C2×#447—31no—
49#2624—31no—
50#2042—32no—
51#2041—32no—
52#2012—32no—
53#2011—32no—
54#1818—31no—
55#1788—31no—
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×D4—21no—
65C22×C3×D4C2311yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×#67—22no—
70C22×#67—22no—
71C22×D12—21no—
72C22×D12—22no—
73C22×D12—42no—
74C22×D12—42no—
75C22×D12D414yes—
76C22×D12C2312yes—
77C2×C3×SD16—22no—
78C2×C3×SD16—42no—
79C2×C3×D8—22no—
80C2×C3×D8—42no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#121—22no—
84C3×#121—22no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×#217—21no—
88C2×#217—22no—
89C2×#217—22no—
90C2×#217D412yes—
91#947—24no—
92#947—24no—
93#940—24no—
94#940—24no—
95#937—216no—
96C2×#213—24no—
97C2×#213—42no—
98C2×#213—42no—
99#930—28no—
100#930—28no—
101C2×(C2×D3)⋊C4—22no—
102C2×(C2×D3)⋊C4—22no—
103#926—22no—
104#926—22no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2312yes—
108C2×Dic3⋊C4—22no—
109C2×C4×Dic3C2312yes—
110C22×C3⋊C8C2312yes—
111C3×#103—22no—
112C3×#102—22no—
113C3×C22⋊C8—22no—
114#835—42no—
115#832—42no—
116C3⋊#95—21no—
117C3⋊#95—22no—
118#830—22no—
119#821—42no—
120#821—42no—
121#809—24no—
122C22×#41C2×D311yescharacteristic
123C23×D4—31no—
124C22×SD16C2×D311yescharacteristic, complemented
125C22×D8—32no—
126C22×D8C2×D311yescharacteristic, complemented
127#535—64no—
128#535—64no—
129C2×#127—32no—
130C2×#124—32no—
131C2×#121—61no—
132C2×#121—61no—
133C2×#121C2×D311yescharacteristic
134C2×#120—62no—
135C2×#117—31no—
136#449—24no—
137#447—64no—
138C2×#103—32no—
139C2×#103C2×D311yescharacteristic, complemented
140C2×#102—32no—
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102C2×D311yescharacteristic, complemented
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147#399—61no—
148#397—61no—
149#394—62no—
150#390—32no—
151#389—32no—
152#380—31no—
153#336—32no—
154#336—32no—
155C24×C3—21no—
156C24×C3—21no—
157C24×C3C2×D411yescharacteristic
158C23×D3—22no—
159C23×D3—24no—
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—21no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—44no—
180C2×C3×D4—42no—
181C2×C3×D4—82no—
182C2×C3×D4C2×D412yes—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—44no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2411yescharacteristic
193C2×#67—28no—
194C2×#67—28no—
195C2×#67—82no—
196C2×#67—82no—
197C2×#67—82no—
198C2×#67—82no—
199C22×Dic3—22no—
200C22×Dic3C2×D412yes—
201C22×Dic3C2×D412yes—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—24no—
207C2×D12—22no—
208C2×D12—48no—
209C2×D12—44no—
210C2×D12—44no—
211C2×D12—44no—
212C2×D12C2×D414yes—
213C3×SD16—44no—
214C3×D8—44no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C22⋊C4—42no—
222#217—24no—
223#217—82no—
224#217—82no—
225#217—81no—
226#213—48no—
227(C2×D3)⋊C4—28no—
228(C2×D3)⋊C4—28no—
229C4⋊Dic3—22no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233Dic3⋊C4—82no—
234C4×Dic3—22no—
235C4×Dic3—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C25—61no—
240C2×#41—22no—
241C2×#41—42no—
242C2×#41—42no—
243C2×#41#6714yes—
244C22×Q8C22×D311yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—34no—
248C22×D4—32no—
249C22×D4—62no—
250C22×D4—62no—
251C22×D4—61no—
252C22×D4—62no—
253C22×D4—122no—
254C22×D4—122no—
255C22×D4C22×D311yescharacteristic
256C22×D4C22×D311yescharacteristic
257C23×C4—21no—
258C23×C4C22×D311yescharacteristic
259C2×SD16—22no—
260C2×SD16—42no—
261C2×D8—22no—
262C2×D8—42no—
263C2×D8—64no—
264C2×D8—122no—
265C2×D8—122no—
266C22×C8—32no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269#127—64no—
270#127—64no—
271#124—64no—
272#124—64no—
273#121—22no—
274#121—22no—
275#121—68no—
276#121—68no—
277#120—616no—
278#117—62no—
279#117—62no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—32no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—61no—
292C2×C42—32no—
293#103—22no—
294#103—122no—
295#102—22no—
296#102—64no—
297#102—122no—
298#102—122no—
299#102—122no—
300C22⋊C8—22no—
301C22⋊C8—62no—
302#95—62no—
303#95—61no—
304C23×C3—22no—
305C23×C3—21no—
306C23×C3—22no—
307C23×C3—21no—
308C23×C3—22no—
309C23×C3—42no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—42no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—41no—
318C23×C3#12012yes—
319C23×C3C22×D411yescharacteristic
320C22×D3—24no—
321C22×D3—48no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
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—
339C3×D4—42no—
340C3×D4—41no—
341C3×D4—41no—
342C3×D4—84no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—48no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4#12012yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4H(5,𝔽2)12yes—
356C2×C3×C4H(5,𝔽2)12yes—
357#67—88no—
358#67—88no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368D12—24no—
369D12—24no—
370D12—48no—
371D12—42no—
372D12—42no—
373C3×C8—41no—
374C3×C8—41no—
375C3⋊C8—42no—
376C24—21no—
377C24—21no—
378C24—62no—
379C24—64no—
380C24—124no—
381C24—241no—
382C24—241no—
383C24—242no—
384C24D3×D411yescharacteristic
385#41—44no—
386#41—44no—
387#41—48no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8C2×#6712yes—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—34no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—68no—
404C2×D4—68no—
405C2×D4—64no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—82no—
412C2×D4—128no—
413C2×D4—124no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4—242no—
417C2×D4—242no—
418C2×D4—242no—
419C2×D4—242no—
420C2×D4C2×#6712yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—44no—
429C22×C4—62no—
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D412yes—
432C22×C4C2×#6712yes—
433C22×C4C23×D311yescharacteristic
434SD16—44no—
435D8—44no—
436D8—128no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—62no—
442C2×C8—62no—
443C2×C8—62no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—62no—
447C4⋊C4—62no—
448C4⋊C4—62no—
449C4⋊C4—62no—
450C4⋊C4—242no—
451C22⋊C4—42no—
452C22⋊C4—68no—
453C22⋊C4—68no—
454C22⋊C4—64no—
455C22⋊C4—242no—
456C22⋊C4—242no—
457C22⋊C4—241no—
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×C3—81no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3C2×#12011yescharacteristic
476C22×C3#53512yes—
477C22×C3#57611yescharacteristic
478C22×C3#57612yes—
479C22×C3#57711yescharacteristic
480C2×D3—44no—
481C2×D3—44no—
482C2×D3—44no—
483C2×D3—816no—
484C2×D3—82no—
485C2×D3—82no—
486C2×D3—82no—
487C2×D3—82no—
488C3×C4—22no—
489C3×C4—42no—
490C3×C4—81no—
491C3×C4#53411yescharacteristic
492C3×C4#53411yescharacteristic
493C3×C4#53412yes—
494Dic3—42no—
495Dic3—42no—
496Dic3—82no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—64no—
512C23—128no—
513C23—122no—
514C23—122no—
515C23—122no—
516C23—122no—
517C23—122no—
518C23—122no—
519C23—244no—
520C23—244no—
521C23—248no—
522C23—248no—
523C23#93712yes—
524C23C2×D3×D411yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—42no—
528D4—22no—
529D4—22no—
530D4—48no—
531D4—41no—
532D4—41no—
533D4—42no—
534D4—42no—
535D4—64no—
536D4—64no—
537D4—84no—
538D4—128no—
539D4—122no—
540D4—122no—
541D4—248no—
542D4—248no—
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—41no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4#93712yes—
562C2×C4C2×D3×D411yescharacteristic
563C2×C4#100412yes—
564C2×C4#100412yes—
565C2×C4C22×#6711yescharacteristic
566C8—41no—
567C8—41no—
568C8—122no—
569C2×C3—42no—
570C2×C3—82no—
571C2×C3—161no—
572C2×C3#261311yescharacteristic
573C2×C3#300811yescharacteristic
574C2×C3#301211yescharacteristic
575C2×C3#322012yes—
576C2×C3#322412yes—
577D3—84no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—82no—
592C22—82no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—2416no—
597C22—242no—
598C22—242no—
599C22—242no—
600C22—242no—
601C22#559212yes—
602C22#576911yescharacteristic
603C22#577011yescharacteristic
604C22#577412yes—
605C22C2×#93711yescharacteristic
606C4—22no—
607C4—42no—
608C4—81no—
609C4—122no—
610C4—122no—
611C4—242no—
612C4#558912yes—
613C4#580411yescharacteristic
614C4#580411yescharacteristic
615C3#2963011yescharacteristic, complemented, Sylow
616C2—42no—
617C2—82no—
618C2—161no—
619C2—244no—
620C2#8011812yes—
621C2#8011912yes—
622C2#8260711yescharacteristic
623C2#8391211yescharacteristic
624C2#8394711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#449C2
socleC23×C3C22×D4
hypercenterC2×#449D3
hypocenterC3#29630
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#449

Complements

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

NH ≅ G/Ntype
1. C2×#5830C2semidirect
2. C2×C3×#449C2semidirect
3. #82678C2semidirect
4. C2×#5036C2semidirect
5. #72135C2semidirect
6. #72055C2semidirect
7. #72044C2semidirect
8. #70279C2semidirect
9. #70211C2semidirect
10. #69959C2semidirect
11. #69911C2semidirect
15. C22×C3×SD16C22semidirect
16. C22×C3×D8C22semidirect
28. C2×C3×#103C22semidirect
29. C2×C3×#102C22semidirect
31. C2×#835C22semidirect
33. #5221C22semidirect
34. C2×#832C22semidirect
40. #4972C22semidirect
44. #4550C22semidirect
45. #4550C22semidirect
47. C2×#449D3semidirect
124. C22×SD16C2×D3semidirect
126. C22×D8C2×D3semidirect
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×C3×#4492
#826782
C2×#50362
#721352
#720552
#720442
#702792
#702112
#699592
#699112
#296303