Finite Group Databasebeta

← #10801034

Subgroups — #10801034

Order 768.

1342 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#907C211yesmaximal, characteristic, complemented
2C2×#4894C212yesmaximal, complemented
3#78626C211yesmaximal, characteristic, complemented
4#78610C211yesmaximal, characteristic
5C3×#1855C211yesmaximal, characteristic, complemented
6#71984C211yesmaximal, characteristic, complemented
7#70722C212yesmaximal, complemented
8#69991C211yesmaximal, characteristic, complemented
9#69948C211yesmaximal, characteristic, complemented
10#68946C212yesmaximal, complemented
11#68927C212yesmaximal, complemented
12#21467—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15#5791C2211yescharacteristic
16C23×C3⋊C8C2211yescharacteristic
17C2×#907—21no—
18C2×#907—21no—
19C2×#907—28no—
20C2×#907C2212yes—
21C2×#907C2212yes—
22C2×#907C2211yescharacteristic, complemented
23C2×#904C2212yes—
24C22×ΓL(1,𝔽25)—21no—
25C22×D3×C8—21no—
26C2×C3×#108C2212yescomplemented
27C3×#407C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C4×#217C2211yescharacteristic
30C2×#825C2212yes—
31C2×#825C2212yes—
32C2×#825C2212yescomplemented
33#5129C2211yescharacteristic, complemented
34#5122C2211yescharacteristic, complemented
35#5119C2211yescharacteristic, complemented
36#5113C2212yes—
37C2×#815C2212yescomplemented
38#4894—28no—
39#4739C411yescharacteristic, complemented
40#4738C411yescharacteristic, complemented
41#4734C412yescomplemented
42#4733C412yescomplemented
43#4732C411yescharacteristic, complemented
44#4731C411yescharacteristic, complemented
45#4730—21no—
46#4730—21no—
47#67⋊C8—21no—
48#67⋊C8—21no—
49C3×#341C2212yescomplemented
50#4555C2212yescomplemented
51#4550C2212yescomplemented
52C22×#131—31no—
53#3002—31no—
54C2×#444—32no—
55#2929—31no—
56#1876—32no—
57#1871—31no—
58#1866—32no—
59#1855D311yescharacteristic, complemented
60#1789—31no—
61#1765—32no—
62#1752—31no—
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C2×#235—21no—
66C2×#235—21no—
67C2×#235—24no—
68C2×#235—24no—
69C2×#235C2×C412yes—
70C2×#235C2311yescharacteristic
71C2×#235C2312yes—
72C22×D12C2×C411yescharacteristic
73C22×C4×D3—21no—
74C22×Dic6C2×C411yescharacteristic
75C2×C3×M16—21no—
76C2×C3×M16—24no—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2312yes—
80C22×C3×C8—21no—
81C22×C3×C8C2312yes—
82#953—21no—
83C2×#217—21no—
84#930C2×C411yescharacteristic
85#930C2×C411yescharacteristic
86#929—21no—
87C4×#67—21no—
88C4×#67—21no—
89#926C2311yescharacteristic
90#924C2×C411yescharacteristic
91#924C2×C411yescharacteristic
92C2×C4×Dic3C2312yes—
93C22×C3⋊C8—24no—
94C22×C3⋊C8D411yescharacteristic
95C22×C3⋊C8D411yescharacteristic
96C22×C3⋊C8D411yescharacteristic
97C22×C3⋊C8D411yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100#907—28no—
101#907—28no—
102#907—24no—
103#907—24no—
104#907—24no—
105#907—24no—
106#907D418yescomplemented
107C2×Dic12C2×C412yes—
108#904—24no—
109#904—24no—
110C2×D24C2×C412yes—
111C2×#204C2×C412yes—
112C2×#204C2×C412yes—
113C2×ΓL(1,𝔽25)—28no—
114C2×ΓL(1,𝔽25)—22no—
115C2×ΓL(1,𝔽25)—22no—
116C2×ΓL(1,𝔽25)—21no—
117C2×D3×C8—28no—
118C2×D3×C8—22no—
119C2×D3×C8—22no—
120C2×D3×C8—21no—
121C3×#108—24no—
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×C22⋊C8C2×C411yescharacteristic
125C3×C22⋊C8C2×C411yescharacteristic
126C3×C22⋊C8C2×C412yes—
127C3⋊#95—21no—
128#825—24no—
129#825—24no—
130#825—24no—
131#821C2×C412yes—
132#821C2×C412yes—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135#816C2×C412yes—
136#816C2×C412yes—
137#815—21no—
138#815—24no—
139Dic3⋊C8—21no—
140Dic3⋊C8—21no—
141C8×Dic3—21no—
142C22×#41—31no—
143C2×#135—32no—
144C2×#131—32no—
145C2×#131—32no—
146C2×#131—31no—
147C2×#131—61no—
148C2×#131—61no—
149C2×#131—68no—
150C22×M16—61no—
151C22×M16C2×D311yescharacteristic
152C23×C8—31no—
153C23×C8—61no—
154#525—31no—
155#452—32no—
156#451—31no—
157#450—31no—
158#449—32no—
159#448—31no—
160#447—31no—
161#444—68no—
162#435—61no—
163#435—61no—
164C8×D4—61no—
165C8×D4—61no—
166C2×#108C2×D312yescomplemented
167#424—31no—
168C2×#104—32no—
169C2×#104—32no—
170C2×#104—32no—
171#416—32no—
172#408—31no—
173#407C2×D311yescharacteristic
174C2×C22⋊C8C2×D311yescharacteristic
175#405—31no—
176C2×#97—32no—
177C4×C22⋊C4—31no—
178#341C2×D312yescomplemented
179#339—32no—
180#336—32no—
181C24×C3#4111yescharacteristic
182C23×D3—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4C22×C411yescharacteristic
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4#4111yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×#67—28no—
192C2×#67—22no—
193C2×#67—22no—
194C2×#67—21no—
195C2×#67—21no—
196C2×#67—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199#235—24no—
200#235—28no—
201#235—24no—
202#235—22no—
203#235—22no—
204#235—24no—
205#235—22no—
206#235—22no—
207#235—24no—
208#235C2×D414yes—
209#235#4114yes—
210C2×D12—21no—
211C2×D12—24no—
212C2×D12C22×C411yescharacteristic
213C2×D12C22×C412yes—
214C2×C4×D3—24no—
215C2×C4×D3—24no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—21no—
219C2×Dic6—21no—
220C2×Dic6—24no—
221C2×Dic6C22×C411yescharacteristic
222C2×Dic6C22×C412yes—
223C3×M16—24no—
224C3×M16—22no—
225C3×M16—22no—
226C3×M16—22no—
227C3×M16C2×D414yes—
228C2×C3×C8—24no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8C22×C412yes—
235C2×C3×C8C22×C412yes—
236#217—21no—
237#217—21no—
238#217—21no—
239#217—42no—
240(C2×D3)⋊C4—21no—
241(C2×D3)⋊C4—21no—
242C4⋊Dic3C22×C411yescharacteristic
243C4⋊Dic3C22×C411yescharacteristic
244Dic3⋊C4—21no—
245Dic3⋊C4—21no—
246C4×Dic3—21no—
247C4×Dic3—22no—
248C4×Dic3—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8C2×D411yescharacteristic
255C2×C3⋊C8C2×D411yescharacteristic
256C2×C3⋊C8C2×D411yescharacteristic
257C2×C3⋊C8C2×D411yescharacteristic
258C2×C3⋊C8C2×D414yes—
259Dic12—24no—
260D24—24no—
261#204—24no—
262#204—24no—
263ΓL(1,𝔽25)—28no—
264ΓL(1,𝔽25)—28no—
265ΓL(1,𝔽25)—42no—
266D3×C8—28no—
267D3×C8—28no—
268D3×C8—42no—
269C2×#41—32no—
270C2×#41—32no—
271C2×#41—31no—
272C2×#41—64no—
273C2×#41—64no—
274C2×#41—61no—
275C2×#41—61no—
276C22×Q8—31no—
277C22×D4—31no—
278C22×D4—61no—
279C23×C4—61no—
280C23×C4C22×D311yescharacteristic
281#135—64no—
282#135—64no—
283C2×Q16—32no—
284C2×SD16—32no—
285C2×SD16—32no—
286C2×D8—32no—
287#131—38no—
288#131—64no—
289#131—64no—
290#131—64no—
291#131—68no—
292#131—68no—
293#131—64no—
294C2×M16—21no—
295C2×M16—24no—
296C2×M16—62no—
297C2×M16—62no—
298C2×M16—61no—
299C2×M16—68no—
300C2×M16C22×D311yescharacteristic
301C2×M16C22×D311yescharacteristic
302C2×M16C22×D312yes—
303C22×C8—21no—
304C22×C8—31no—
305C22×C8—31no—
306C22×C8—31no—
307C22×C8—31no—
308C22×C8—31no—
309C22×C8—31no—
310C22×C8—62no—
311C22×C8—62no—
312C22×C8—61no—
313C22×C8—68no—
314C22×C8—64no—
315C22×C8C22×D312yes—
316#123—61no—
317C22⋊Q8—31no—
318C22⋊Q8—31no—
319#121—31no—
320#121—31no—
321#120—61no—
322C4×D4—61no—
323C4×D4—61no—
324#117—31no—
325C2×C22⋊C4—61no—
326C2×C42—32no—
327#108—24no—
328C4⋊C8—61no—
329C4⋊C8—61no—
330#104—64no—
331#104—64no—
332#104—64no—
333#103—32no—
334#103—32no—
335#102—32no—
336#102—32no—
337C22⋊C8—21no—
338C22⋊C8—21no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8C4×D311yescharacteristic
342C22⋊C8C4×D311yescharacteristic
343C22⋊C8C4×D312yes—
344#97—64no—
345#97—61no—
346C4×C8—61no—
347#95—61no—
348C23×C3—22no—
349C23×C3—22no—
350C23×C3—41no—
351C23×C3C4×D411yescharacteristic
352C23×C3C4×D411yescharacteristic
353C23×C3C2×#4111yescharacteristic
354C22×D3—24no—
355C22×D3—22no—
356C22×D3—21no—
357C22×D3—42no—
358C22×D3—42no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4C4×D411yescharacteristic
367C2×C3×C4C4×D411yescharacteristic
368C2×C3×C4C23×C411yescharacteristic
369C2×C3×C4C22×D412yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4112yes—
372#67—28no—
373#67—24no—
374#67—24no—
375#67—42no—
376#67—42no—
377#67—42no—
378C2×Dic3—24no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386D12—24no—
387D12—22no—
388D12—22no—
389D12C4×D414yes—
390C4×D3—28no—
391C4×D3—24no—
392C4×D3—24no—
393C4×D3—41no—
394C4×D3—41no—
395Dic6—24no—
396Dic6—22no—
397Dic6—22no—
398Dic6C4×D414yes—
399C3×C8—22no—
400C3×C8—22no—
401C3×C8—24no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8#12012yes—
407C3⋊C8#12012yes—
408C24—61no—
409C24#23911yescharacteristic
410#41—34no—
411#41—34no—
412#41—64no—
413#41—64no—
414#41—64no—
415#41—68no—
416#41—64no—
417#41—62no—
418#41—62no—
419#41—62no—
420#41—62no—
421C2×Q8—32no—
422C2×Q8—31no—
423C2×Q8—64no—
424C2×Q8—61no—
425C2×D4—32no—
426C2×D4—31no—
427C2×D4—64no—
428C2×D4—68no—
429C2×D4—62no—
430C2×D4—62no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4C2×C4×D311yescharacteristic
445C22×C4C2×C4×D311yescharacteristic
446C22×C4D3×D411yescharacteristic
447C22×C4D3×D411yescharacteristic
448C22×C4#23911yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—64no—
451SD16—64no—
452SD16—64no—
453D8—64no—
454M16—24no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16—68no—
459M16—68no—
460M16—122no—
461M16D3×D414yes—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—34no—
469C2×C8—31no—
470C2×C8—31no—
471C2×C8—31no—
472C2×C8—31no—
473C2×C8—68no—
474C2×C8—68no—
475C2×C8—62no—
476C2×C8—62no—
477C2×C8—62no—
478C2×C8—62no—
479C2×C8—62no—
480C2×C8—122no—
481C2×C8C2×C4×D312yes—
482C2×C8C2×C4×D312yes—
483C4⋊C4—31no—
484C4⋊C4—31no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C22⋊C4—61no—
488C22⋊C4—61no—
489C22⋊C4—61no—
490C22⋊C4—61no—
491C22⋊C4—61no—
492C22⋊C4—122no—
493C42—61no—
494C42—62no—
495C42—62no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—44no—
504C22×C3#44412yescomplemented
505C22×C3C2×C4×D411yescharacteristic
506C22×C3#52512yes—
507C2×D3—24no—
508C2×D3—48no—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—41no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4C2×C4×D412yes—
517C3×C4C2×#12011yescharacteristic
518C3×C4#52511yescharacteristic
519Dic3—24no—
520Dic3—41no—
521Dic3—41no—
522Dic3—41no—
523Dic3—82no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—64no—
528C23—62no—
529C23—61no—
530C23—122no—
531C23—122no—
532C23#89311yescharacteristic
533C23#89311yescharacteristic
534C23C2×#23911yescharacteristic
535Q8—34no—
536Q8—64no—
537Q8—62no—
538Q8—62no—
539D4—34no—
540D4—64no—
541D4—68no—
542D4—64no—
543D4—64no—
544D4—62no—
545D4—62no—
546D4—122no—
547D4—122no—
548D4—122no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—68no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4#89311yescharacteristic
570C2×C4#89311yescharacteristic
571C2×C4C22×C4×D311yescharacteristic
572C2×C4C2×D3×D412yes—
573C2×C4C2×D3×D411yescharacteristic
574C2×C4D3×#4112yes—
575C8—22no—
576C8—22no—
577C8—24no—
578C8—32no—
579C8—32no—
580C8—41no—
581C8—41no—
582C8—61no—
583C8—61no—
584C2×C3—22no—
585C2×C3—22no—
586C2×C3—41no—
587C2×C3C4×#12011yescharacteristic
588C2×C3C2×#44411yescharacteristic
589C2×C3#295511yescharacteristic
590D3—44no—
591D3—81no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—44no—
600C22—64no—
601C22—128no—
602C22—122no—
603C22—122no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22#489412yescomplemented
608C22C2×#89311yescharacteristic
609C22#562712yes—
610C4—21no—
611C4—21no—
612C4—64no—
613C4—121no—
614C4—121no—
615C4—121no—
616C4—242no—
617C4C4×D3×D412yes—
618C4D3×#12011yescharacteristic
619C4#565611yescharacteristic
620C3#2146711yescharacteristic, complemented, Sylow
621C2—22no—
622C2—22no—
623C2—41no—
624C2—124no—
625C2—241no—
626C2#7437911yescharacteristic
627C2C2×#489411yescharacteristic
628C2#7925711yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1855C2
socleC22×C3C2×C4×D4
prime radical#78610C2
cosocleC22×C3×C4C24
hypercenter#1855D3
hypocenterC3#21467
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
1. C22×#907C2semidirect
2. C2×#4894C2semidirect
3. #78626C2semidirect
5. C3×#1855C2semidirect
6. #71984C2semidirect
7. #70722C2semidirect
8. #69991C2semidirect
9. #69948C2semidirect
10. #68946C2semidirect
11. #68927C2semidirect
22. C2×#907C22semidirect
26. C2×C3×#108C22semidirect
32. C2×#825C22semidirect
33. #5129C22semidirect
34. #5122C22semidirect
35. #5119C22semidirect
37. C2×#815C22semidirect
39. #4739C4semidirect
40. #4738C4semidirect
41. #4734C4semidirect
42. #4733C4semidirect
43. #4732C4semidirect
44. #4731C4semidirect
49. C3×#341C22semidirect
50. #4555C22semidirect
51. #4550C22semidirect
59. #1855D3semidirect
106. #907D4semidirect
166. C2×#108C2×D3semidirect
178. #341C2×D3semidirect
504. C22×C3#444semidirect
607. C22#4894semidirect
620. C3#21467semidirect

Maximal subgroups

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

subgroupindex
C22×#9072
C2×#48942
#786262
#786102
C3×#18552
#719842
#707222
#699912
#699482
#689462
#689272
#214673