Finite Group Databasebeta

← #10801039

Subgroups — #10801039

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×#901C211yesmaximal, characteristic, complemented
2#83734C211yesmaximal, characteristic, complemented
3#78610C211yesmaximal, characteristic
4C2×#4717C212yesmaximal, complemented
5C3×#1752C211yesmaximal, characteristic, complemented
6#69954C211yesmaximal, characteristic, complemented
7#69839C212yesmaximal, complemented
8#69454C211yesmaximal, characteristic, complemented
9#69453C211yesmaximal, characteristic, complemented
10#68695C212yesmaximal, complemented
11#68694C212yesmaximal, complemented
12#21467—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15#5791C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C2×#904C2212yes—
18C2×#901—21no—
19C2×#901—21no—
20C2×#901—28no—
21C2×#901C2212yes—
22C2×#901C2212yes—
23C2×#901C2211yescharacteristic, complemented
24C22×ΓL(1,𝔽25)—21no—
25C22×D3×C8—21no—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×#97C2212yescomplemented
28C3×C4×C22⋊C4C2211yescharacteristic
29#5207C2211yescharacteristic
30#5113C2212yes—
31#5111—21no—
32#5111—21no—
33#5110C2211yescharacteristic, complemented
34C8×#67—21no—
35C8×#67—21no—
36C2×#819C2212yescomplemented
37#5102C2211yescharacteristic, complemented
38#5101C2211yescharacteristic, complemented
39C2×#805C2212yes—
40C2×#805C2212yes—
41C2×#805C2212yescomplemented
42#4739C411yescharacteristic, complemented
43#4738C411yescharacteristic, complemented
44#4734C412yescomplemented
45#4733C412yescomplemented
46#4732C411yescharacteristic, complemented
47#4731C411yescharacteristic, complemented
48#4717—28no—
49C3×#336C2212yescomplemented
50#4552C2212yescomplemented
51#4523C2212yescomplemented
52C22×#131—31no—
53#3002—31no—
54C2×#444—32no—
55#2929—31no—
56#1876—32no—
57#1871—31no—
58#1866—32no—
59#1855—31no—
60#1789—31no—
61#1765—32no—
62#1752D311yescharacteristic, complemented
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
75C22×C3×C8—24no—
76C22×C3×C8D411yescharacteristic
77C22×C3×C8D411yescharacteristic
78C22×C3×C8D411yescharacteristic
79C22×C3×C8D411yescharacteristic
80C22×C3×C8C2312yes—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C42C2312yes—
85#953—21no—
86#930C2×C411yescharacteristic
87#930C2×C411yescharacteristic
88#929—21no—
89C4×#67—21no—
90C4×#67—21no—
91#926C2311yescharacteristic
92#924C2×C411yescharacteristic
93#924C2×C411yescharacteristic
94C2×#208—21no—
95C2×#208—24no—
96C2×#208C2311yescharacteristic
97C2×#208C2311yescharacteristic
98C2×#208C2312yes—
99C22×C3⋊C8—21no—
100C2×Dic12C2×C412yes—
101#904—24no—
102#904—24no—
103C2×D24C2×C412yes—
104C2×#204C2×C412yes—
105C2×#204C2×C412yes—
106#901—28no—
107#901—28no—
108#901—24no—
109#901—24no—
110#901—24no—
111#901—24no—
112#901D418yescomplemented
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×C22⋊C8C2×C411yescharacteristic
122C3×C22⋊C8C2×C411yescharacteristic
123C3×C22⋊C8C2×C412yes—
124C3×#97—24no—
125C3×#95—21no—
126#830—21no—
127#830—21no—
128#821C2×C412yes—
129#821C2×C412yes—
130(C2×D3)⋊C8—21no—
131(C2×D3)⋊C8—21no—
132#819—24no—
133#816C2×C412yes—
134#816C2×C412yes—
135#815—21no—
136Dic3⋊C8—21no—
137Dic3⋊C8—21no—
138C8×Dic3—21no—
139#805—24no—
140#805—24no—
141#805—24no—
142C22×#41—31no—
143C2×#135—32no—
144C2×#131—32no—
145C2×#131—32no—
146C2×#131—31no—
147C2×#131—68no—
148C2×#131—61no—
149C2×#131—61no—
150C22×M16—31no—
151C22×M16—61no—
152C23×C8—61no—
153C23×C8C2×D311yescharacteristic
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×#108—32no—
167#424—31no—
168C2×#104—32no—
169C2×#104—32no—
170C2×#104—32no—
171#416—32no—
172#408—31no—
173#407—31no—
174C2×C22⋊C8C2×D311yescharacteristic
175#405—31no—
176C2×#97C2×D312yescomplemented
177C4×C22⋊C4C2×D311yescharacteristic
178#341—32no—
179#339—32no—
180#336C2×D312yescomplemented
181C24×C3#4111yescharacteristic
182C23×D3—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4#4111yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C2×#67—28no—
193C2×#67—22no—
194C2×#67—22no—
195C2×#67—21no—
196C2×#67—21no—
197C2×#67—21no—
198C22×Dic3—21no—
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—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8C22×C412yes—
231C2×C3×C8C22×C412yes—
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D414yes—
237C3×C22⋊C4—42no—
238C3×C42—22no—
239C3×C42—22no—
240#217—21no—
241#217—21no—
242#217—21no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245C4⋊Dic3C22×C411yescharacteristic
246C4⋊Dic3C22×C411yescharacteristic
247Dic3⋊C4—21no—
248Dic3⋊C4—21no—
249C4×Dic3—21no—
250#208—24no—
251#208—22no—
252#208—22no—
253#208—22no—
254#208C2×D414yes—
255C2×C3⋊C8—24no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
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—68no—
291#131—68no—
292#131—64no—
293#131—64no—
294C2×M16—32no—
295C2×M16—31no—
296C2×M16—31no—
297C2×M16—68no—
298C2×M16—64no—
299C2×M16—61no—
300C2×M16—62no—
301C2×M16—62no—
302C2×M16—61no—
303C22×C8—24no—
304C22×C8—68no—
305C22×C8—61no—
306C22×C8—62no—
307C22×C8—62no—
308C22×C8—61no—
309C22×C8D1211yescharacteristic
310C22×C8D1211yescharacteristic
311C22×C8D1211yescharacteristic
312C22×C8D1211yescharacteristic
313C22×C8C22×D312yes—
314C22×C8C22×D311yescharacteristic
315C22×C8C22×D311yescharacteristic
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—21no—
326C2×C42C22×D312yes—
327#108—64no—
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—61no—
338C22⋊C8—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8C4×D311yescharacteristic
342C22⋊C8C4×D311yescharacteristic
343C22⋊C8C4×D312yes—
344#97—24no—
345#97—61no—
346C4×C8—61no—
347#95—21no—
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×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4C4×D411yescharacteristic
370C2×C3×C4C4×D411yescharacteristic
371C2×C3×C4C23×C411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D412yes—
374C2×C3×C4C2×#4112yes—
375#67—28no—
376#67—24no—
377#67—24no—
378#67—42no—
379#67—42no—
380#67—42no—
381C2×Dic3—24no—
382C2×Dic3—22no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
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—21no—
402C3×C8—21no—
403C3×C8#12012yes—
404C3×C8#12012yes—
405C3⋊C8—24no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C24—61no—
409C24#23511yescharacteristic
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—22no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4C2×C4×D311yescharacteristic
445C22×C4C2×C4×D311yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4#23511yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—64no—
451SD16—64no—
452SD16—64no—
453D8—64no—
454M16—34no—
455M16—68no—
456M16—64no—
457M16—62no—
458M16—68no—
459M16—62no—
460M16—62no—
461M16—122no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—68no—
470C2×C8—64no—
471C2×C8—62no—
472C2×C8—68no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—122no—
476C2×C8C2×C4×D312yes—
477C2×C8C2×C4×D312yes—
478C2×C8C2×D1211yescharacteristic
479C2×C8C2×D1211yescharacteristic
480C2×C8C2×D1211yescharacteristic
481C2×C8C2×D1211yescharacteristic
482C2×C8D3×D414yes—
483C4⋊C4—31no—
484C4⋊C4—31no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C22⋊C4—42no—
488C22⋊C4—61no—
489C22⋊C4—61no—
490C22⋊C4—61no—
491C22⋊C4—61no—
492C22⋊C4—61no—
493C42—22no—
494C42—22no—
495C42—61no—
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×C4—82no—
517C3×C4C2×C4×D412yes—
518C3×C4C2×#12011yescharacteristic
519C3×C4#52511yescharacteristic
520Dic3—24no—
521Dic3—41no—
522Dic3—41no—
523Dic3—41no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—64no—
528C23—62no—
529C23—61no—
530C23—122no—
531C23—122no—
532C23C4×D1211yescharacteristic
533C23C4×D1211yescharacteristic
534C23C2×#23511yescharacteristic
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—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—68no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4C4×D1211yescharacteristic
570C2×C4C4×D1211yescharacteristic
571C2×C4C22×C4×D311yescharacteristic
572C2×C4C22×D1211yescharacteristic
573C2×C4C2×D3×D412yes—
574C2×C4D3×#4112yes—
575C8—21no—
576C8—21no—
577C8—22no—
578C8—22no—
579C8—64no—
580C8—121no—
581C8—121no—
582C8#88212yes—
583C8#88212yes—
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#471712yescomplemented
608C22C2×C4×D1211yescharacteristic
609C22#554712yes—
610C4—21no—
611C4—21no—
612C4—64no—
613C4—82no—
614C4—121no—
615C4—121no—
616C4—121no—
617C4C2×#88211yescharacteristic
618C4#552411yescharacteristic
619C4C4×D3×D412yes—
620C3#2146711yescharacteristic, complemented, Sylow
621C2—22no—
622C2—22no—
623C2—41no—
624C2—124no—
625C2—241no—
626C2C4×#88211yescharacteristic
627C2C2×#471711yescharacteristic
628C2#7857911yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1752C2
socleC22×C3C2×C4×D4
prime radical#78610C2
cosocleC22×C3×C4C24
hypercenter#1752D3
hypocenterC3#21467
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #1752

Complements

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

NH ≅ G/Ntype
1. C22×#901C2semidirect
2. #83734C2semidirect
4. C2×#4717C2semidirect
5. C3×#1752C2semidirect
6. #69954C2semidirect
7. #69839C2semidirect
8. #69454C2semidirect
9. #69453C2semidirect
10. #68695C2semidirect
11. #68694C2semidirect
23. C2×#901C22semidirect
27. C2×C3×#97C22semidirect
33. #5110C22semidirect
36. C2×#819C22semidirect
37. #5102C22semidirect
38. #5101C22semidirect
41. C2×#805C22semidirect
42. #4739C4semidirect
43. #4738C4semidirect
44. #4734C4semidirect
45. #4733C4semidirect
46. #4732C4semidirect
47. #4731C4semidirect
49. C3×#336C22semidirect
50. #4552C22semidirect
51. #4523C22semidirect
62. #1752D3semidirect
112. #901D4semidirect
176. C2×#97C2×D3semidirect
180. #336C2×D3semidirect
504. C22×C3#444semidirect
607. C22#4717semidirect
620. C3#21467semidirect

Maximal subgroups

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

subgroupindex
C22×#9012
#837342
#786102
C2×#47172
C3×#17522
#699542
#698392
#694542
#694532
#686952
#686942
#214673