Finite Group Databasebeta

← #10900850

Subgroups — #10900850

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×#5805C211yesmaximal, characteristic, complemented
2C2×C3×#447C211yesmaximal, characteristic, complemented
3#82678C211yesmaximal, characteristic, complemented
4C2×#5037C211yesmaximal, characteristic, complemented
5#72128C211yesmaximal, characteristic, complemented
6#72045C212yesmaximal, complemented
7#72044C212yesmaximal, complemented
8#70216C212yesmaximal, complemented
9#70211C212yesmaximal, complemented
10#69960C211yesmaximal, characteristic, complemented
11#69911C211yesmaximal, characteristic, complemented
12#29630—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×D8C2212yes—
16C2×C3×#121C2211yescharacteristic
17C2×#953—22no—
18#5805—24no—
19#5805—24no—
20C2×#940C2212yes—
21C2×#938—21no—
22C2×#938—21no—
23C2×#936C2212yes—
24C22×#214C2211yescharacteristic, complemented
25C22×#213C2211yescharacteristic, complemented
26C2×#926C2211yescharacteristic
27C3×#447—24no—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30#5222—22no—
31#5221C2212yescomplemented
32C2×#832C2212yes—
33C2×#832C2212yes—
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#821C2212yescomplemented
37C2×#816C2212yescomplemented
38#5037—24no—
39#4972C2212yescomplemented
40C2×#810C2211yescharacteristic, complemented
41C2×#809C2211yescharacteristic, complemented
42#4953—21no—
43#4949—21no—
44#4550C2212yescomplemented
45#4550C2212yescomplemented
46C2×#535—31no—
47C2×#449—31no—
48C2×#447D311yescharacteristic, complemented
49#2624—31no—
50#2042—32no—
51#2041—32no—
52#2012—32no—
53#2011—32no—
54#1818—31no—
55#1788—31no—
56#1786—31no—
57C25×C3—21no—
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4—22no—
61C22×C3×D4—42no—
62C22×C3×D4—42no—
63C22×C3×D4D414yes—
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C22×#67—21no—
68C22×#67—22no—
69C22×#67—22no—
70C2×#235—22no—
71C2×#235—42no—
72C2×#235—42no—
73C2×#235D414yes—
74C22×D12C2311yescharacteristic
75C22×C4×D3—21no—
76C22×Dic6C2311yescharacteristic
77C2×C3×D8—24no—
78C2×C3×D8—42no—
79C2×C3×D8—42no—
80C22×C3×C8C2312yes—
81C3×#121—22no—
82C3×#121—22no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85#953—216no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—21no—
89C2×#217—22no—
90C2×#217—22no—
91C2×#217D412yes—
92#940—24no—
93#940—24no—
94#938—28no—
95#938—28no—
96#936—24no—
97#936—24no—
98C2×#214—22no—
99C2×#214—42no—
100C2×#213—22no—
101C2×#213—42no—
102#926—22no—
103#926—22no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2312yes—
107C2×Dic3⋊C4—22no—
108C2×C4×Dic3C2312yes—
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C3×#102—24no—
112C3×C22⋊C8—22no—
113#832—42no—
114#832—42no—
115C3⋊#95—21no—
116C3⋊#95—22no—
117#830—22no—
118#821—42no—
119#816—42no—
120#810—22no—
121#809—22no—
122C22×#41—31no—
123C23×D4C2×D311yescharacteristic
124C22×SD16—31no—
125C22×D8—31no—
126C22×D8C2×D312yes—
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—64no—
137#447—24no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#102—31no—
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102C2×D312yes—
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—22no—
156C24×C3—24no—
157C24×C3—21no—
158C24×C3—44no—
159C24×C3—81no—
160C24×C3—82no—
161C24×C3—81no—
162C24×C3C2×D411yescharacteristic
163C23×D3—21no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—48no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C2×C3×D4—44no—
178C2×C3×D4—42no—
179C2×C3×D4—82no—
180C2×C3×D4C2×D414yes—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2411yescharacteristic
188C2×#67—24no—
189C2×#67—28no—
190C2×#67—28no—
191C2×#67—44no—
192C2×#67—82no—
193C2×#67—82no—
194C2×#67—82no—
195C2×#67—82no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3C2×D412yes—
199C22×Dic3C2×D412yes—
200#235—44no—
201#235—44no—
202#235—48no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—21no—
206C2×D12C2×D412yes—
207C2×C4×D3—24no—
208C2×C4×D3—44no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×Dic6—21no—
212C2×Dic6C2×D412yes—
213C3×D8—48no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C22⋊C4—42no—
220#217—28no—
221#217—28no—
222#217—24no—
223#217—82no—
224#217—82no—
225#217—81no—
226#214—44no—
227#213—44no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232Dic3⋊C4—82no—
233C4×Dic3—22no—
234C4×Dic3—22no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C25—21no—
240C2×#41—34no—
241C2×#41—62no—
242C2×#41—122no—
243C2×#41—122no—
244C22×Q8—31no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—31no—
249C22×D4—42no—
250C22×D4—42no—
251C22×D4—62no—
252C22×D4—62no—
253C22×D4—61no—
254C22×D4#6714yes—
255C22×D4C22×D311yescharacteristic
256C22×D4C22×D312yes—
257C23×C4—61no—
258C23×C4C22×D311yescharacteristic
259C2×SD16—62no—
260C2×SD16—122no—
261C2×D8—24no—
262C2×D8—42no—
263C2×D8—42no—
264C2×D8—62no—
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—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—62no—
294#103—122no—
295#102—24no—
296#102—62no—
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—24no—
309C23×C3—48no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—41no—
317C23×C3—42no—
318C23×C3—41no—
319C23×C3—42no—
320C23×C3—41no—
321C23×C3—84no—
322C23×C3—84no—
323C23×C3—88no—
324C23×C3—88no—
325C23×C3#12012yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—42no—
329C22×D3—42no—
330C22×D3—41no—
331C22×D3—41no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—48no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—42no—
340C3×D4—84no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4#12012yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4H(5,𝔽2)12yes—
351C2×C3×C4H(5,𝔽2)12yes—
352#67—48no—
353#67—88no—
354#67—88no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366D12—22no—
367D12—22no—
368D12—42no—
369C4×D3—48no—
370Dic6—22no—
371Dic6—22no—
372Dic6—42no—
373C3×C8—42no—
374C3⋊C8—41no—
375C3⋊C8—41no—
376C24—22no—
377C24—24no—
378C24—21no—
379C24—44no—
380C24—61no—
381C24—82no—
382C24—81no—
383C24—81no—
384C24D3×D411yescharacteristic
385#41—124no—
386#41—124no—
387#41—128no—
388C2×Q8—32no—
389C2×Q8—61no—
390C2×Q8—62no—
391C2×Q8—62no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—32no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—64no—
409C2×D4—68no—
410C2×D4—68no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—61no—
414C2×D4—82no—
415C2×D4—124no—
416C2×D4—242no—
417C2×D4—242no—
418C2×D4—242no—
419C2×D4—242no—
420C2×D4C2×#6714yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—61no—
427C22×C4—64no—
428C22×C4—62no—
429C22×C4—124no—
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D412yes—
432C22×C4C2×#6712yes—
433C22×C4C23×D311yescharacteristic
434SD16—124no—
435D8—48no—
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—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—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—816no—
469C22×C3—82no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3—81no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3C2×#12011yescharacteristic
479C22×C3#53512yes—
480C22×C3#57612yes—
481C22×C3#57611yescharacteristic
482C22×C3#57711yescharacteristic
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—82no—
487C2×D3—82no—
488C3×C4—22no—
489C3×C4—81no—
490C3×C4#53412yes—
491C3×C4#53411yescharacteristic
492C3×C4#53411yescharacteristic
493Dic3—42no—
494Dic3—42no—
495Dic3—42no—
496Dic3—82no—
497C23—22no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—24no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—62no—
515C23—84no—
516C23—84no—
517C23—88no—
518C23—88no—
519C23—122no—
520C23—121no—
521C23—121no—
522C23—122no—
523C23#93712yes—
524C23C2×D3×D411yescharacteristic
525Q8—62no—
526Q8—62no—
527Q8—122no—
528D4—24no—
529D4—24no—
530D4—48no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—41no—
535D4—41no—
536D4—62no—
537D4—62no—
538D4—84no—
539D4—128no—
540D4—122no—
541D4—248no—
542D4—248no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—128no—
557C2×C4—122no—
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—42no—
567C8—121no—
568C8—121no—
569C2×C3—42no—
570C2×C3—84no—
571C2×C3—161no—
572C2×C3#261311yescharacteristic
573C2×C3#300811yescharacteristic
574C2×C3#301211yescharacteristic
575C2×C3#322012yes—
576C2×C3#322412yes—
577D3—82no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—816no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—81no—
592C22—82no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—122no—
597C22—122no—
598C22—122no—
599C22—242no—
600C22—242no—
601C22#559212yes—
602C22#575712yes—
603C22#577411yescharacteristic
604C22#577511yescharacteristic
605C22C2×#93711yescharacteristic
606C4—22no—
607C4—81no—
608C4—122no—
609C4—122no—
610C4—122no—
611C4—242no—
612C4#558911yescharacteristic
613C4#558911yescharacteristic
614C4#580412yes—
615C3#2963011yescharacteristic, complemented, Sylow
616C2—42no—
617C2—84no—
618C2—161no—
619C2—242no—
620C2#7999812yes—
621C2#7999912yes—
622C2#8260711yescharacteristic
623C2#8383311yescharacteristic
624C2#8394911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#447C2
socleC23×C3C22×D4
hypercenterC2×#447D3
hypocenterC3#29630
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C2×#5805C2semidirect
2. C2×C3×#447C2semidirect
3. #82678C2semidirect
4. C2×#5037C2semidirect
5. #72128C2semidirect
6. #72045C2semidirect
7. #72044C2semidirect
8. #70216C2semidirect
9. #70211C2semidirect
10. #69960C2semidirect
11. #69911C2semidirect
24. C22×#214C22semidirect
25. C22×#213C22semidirect
31. #5221C22semidirect
36. C2×#821C22semidirect
37. C2×#816C22semidirect
39. #4972C22semidirect
40. C2×#810C22semidirect
41. C2×#809C22semidirect
44. #4550C22semidirect
45. #4550C22semidirect
48. C2×#447D3semidirect
615. C3#29630semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
C2×C3×#4472
#826782
C2×#50372
#721282
#720452
#720442
#702162
#702112
#699602
#699112
#296303