Finite Group Databasebeta

← #10858717

Subgroups — #10858717

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2946C211yesmaximal, characteristic
2#82561C211yesmaximal, characteristic, complemented
3#82539C211yesmaximal, characteristic, complemented
4#78824C211yesmaximal, characteristic, complemented
5#72091C211yesmaximal, characteristic
6#72088C211yesmaximal, characteristic
7#71664C211yesmaximal, characteristic, complemented
8#71647C211yesmaximal, characteristic, complemented
9#70671C211yesmaximal, characteristic, complemented
10#70666C211yesmaximal, characteristic, complemented
11#70292C211yesmaximal, characteristic, complemented
12#70197C211yesmaximal, characteristic, complemented
13#70184C211yesmaximal, characteristic, complemented
14#69899C211yesmaximal, characteristic, complemented
15#69893C211yesmaximal, characteristic, complemented
16#26407—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C3×#518C2211yescharacteristic
20#5827C2211yescharacteristic
21C2×#948C2211yescharacteristic
22#5519C2211yescharacteristic
23C3×#442—21no—
24C3×#440—21no—
25C3×C4×SD16—21no—
26C3×C4×D8—21no—
27C3×#428C2211yescharacteristic
28C3×#416C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C3×#405C2211yescharacteristic
31C2×#837C2211yescharacteristic
32#5232C2211yescharacteristic
33#5221C2211yescharacteristic
34C2×#833C2211yescharacteristic
35#5215C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#819C2211yescharacteristic
38C2×#817C2211yescharacteristic
39#5102C2211yescharacteristic
40#5009—21no—
41#5009—21no—
42#5008—21no—
43#5007—21no—
44#4999C2211yescharacteristic
45#4998—21no—
46#4998C2211yescharacteristic
47#4995—21no—
48#4995C2211yescharacteristic
49#4995C2211yescharacteristic
50#4995C2211yescharacteristic
51#4975C2211yescharacteristic
52#4964—21no—
53#4964C2211yescharacteristic
54C2×#808—21no—
55C2×#808C2211yescharacteristic
56#4799—21no—
57#4791—21no—
58#4763—21no—
59#4761—21no—
60#4558C2211yescharacteristic
61#4555C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4550C2211yescharacteristic
65#4532C2211yescharacteristic
66#4532C2211yescharacteristic
67#4530C2211yescharacteristic
68#3084—31no—
69#3081—31no—
70#2946D311yescharacteristic
71#2912—31no—
72#2094—31no—
73#2090—31no—
74#2063—31no—
75#2060—31no—
76#1951—31no—
77#1922—31no—
78#1907—31no—
79#1883—31no—
80#1866—31no—
81#1859—31no—
82#1856—31no—
83C2×C3×#41C2311yescharacteristic
84C22×C3×D4C2311yescharacteristic
85C3×AGL(1,ℤ/8)—22no—
86C3×AGL(1,ℤ/8)Q814yescomplemented
87C2×C3×SD16—21no—
88C2×C3×D8—21no—
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C4×Q8—21no—
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—41no—
98C3×#117—21no—
99C3×#117C2311yescharacteristic
100C3×#117C2311yescharacteristic
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—41no—
104C2×#217—41no—
105#948—22no—
106#948—22no—
107Q8×Dic3—21no—
108D4×Dic3—21no—
109D4×Dic3—41no—
110#926—21no—
111#926C2311yescharacteristic
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—41no—
116C2×C4×Dic3C2311yescharacteristic
117C2×#208—21no—
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C2×#208C2311yescharacteristic
121C22×C3⋊C8—21no—
122C22×C3⋊C8C2311yescharacteristic
123#890—21no—
124#888—21no—
125#878—41no—
126#870—21no—
127#869—21no—
128C3×#107—21no—
129C3×#106—21no—
130C3×#103D411yescharacteristic
131C3×#103D411yescharacteristic
132C3×#102—21no—
133C3×#102—21no—
134C3×#102D411yescharacteristic
135C3×#102D411yescharacteristic
136C3×#97—21no—
137C3×C4×C8—21no—
138#837—22no—
139#835—21no—
140#833—22no—
141#832—21no—
142#832—21no—
143#832—21no—
144#832—21no—
145#832—21no—
146#819—22no—
147#817—21no—
148#817—21no—
149Dic3⋊C8—21no—
150Dic3⋊C8—21no—
151#808—21no—
152#808—21no—
153#808—21no—
154#808—21no—
155#808—21no—
156#808—21no—
157#808D411yescharacteristic
158#808D411yescharacteristic
159#807—21no—
160#807—21no—
161#807—21no—
162#807—21no—
163#807D411yescharacteristic
164#807D411yescharacteristic
165C4⋊(C3⋊C8)—21no—
166C4⋊(C3⋊C8)—21no—
167C2×AGL(1,ℤ/8)C2×D311yescharacteristic
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520C2×D311yescharacteristic
172#518C2×D311yescharacteristic
173#479—61no—
174#478—61no—
175#478—61no—
176#478—61no—
177#476—61no—
178#476—61no—
179#475—61no—
180#474—61no—
181#442—21no—
182#440—21no—
183C4×SD16—21no—
184C4×D8—21no—
185C2×#108—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428—61no—
191#428—61no—
192#428C2×D311yescharacteristic
193#427—31no—
194#427—61no—
195C2×#106—31no—
196C2×#106—31no—
197C2×#106—61no—
198#424—31no—
199#424—31no—
200C2×#104—31no—
201#418—31no—
202#417—31no—
203#416C2×D311yescharacteristic
204C2×#102—31no—
205C2×#102C2×D311yescharacteristic
206C2×#100—31no—
207#405C2×D311yescharacteristic
208#390—31no—
209#384—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#341—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#336—31no—
218C24×C3—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41C2×Q812yes—
222C3×#41#4112yes—
223C2×C3×Q8C2×D411yescharacteristic
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4C2×D411yescharacteristic
231C2×C3×D4C2×Q812yes—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—41no—
235C22×C3×C4—41no—
236C22×C3×C4—41no—
237C22×C3×C4C2411yescharacteristic
238C22×Dic3—21no—
239C22×Dic3—21no—
240C22×Dic3—21no—
241C22×Dic3—41no—
242C2×Dic6—41no—
243C3×SD16—24no—
244C3×D8—24no—
245C3×M16—21no—
246C3×M16C2×Q812yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8C2×D411yescharacteristic
251C2×C3×C8C2×D411yescharacteristic
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C22⋊C4—21no—
260C3×C22⋊C4—21no—
261C3×C22⋊C4—41no—
262C3×C22⋊C4—41no—
263C3×C22⋊C4—41no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—41no—
267#217—21no—
268#217—41no—
269#217—82no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—41no—
275C4⋊Dic3—41no—
276C4⋊Dic3C2×D411yescharacteristic
277C4⋊Dic3C2×D411yescharacteristic
278Dic3⋊C4—41no—
279Dic3⋊C4—41no—
280Dic3⋊C4—82no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—41no—
285#208—24no—
286#208—21no—
287#208—21no—
288#208—21no—
289#208—21no—
290#208—21no—
291#208#4112yes—
292C2×C3⋊C8—24no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8C2×D411yescharacteristic
299C2×C3⋊C8C2×D411yescharacteristic
300C2×#41C22×D311yescharacteristic
301C22×D4C22×D311yescharacteristic
302AGL(1,ℤ/8)—22no—
303AGL(1,ℤ/8)Dic614yescomplemented
304C2×SD16—21no—
305C2×D8—21no—
306#131—62no—
307#131—62no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16—61no—
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8—61no—
315C22×C8C22×D311yescharacteristic
316C4⋊Q8—61no—
317C4⋊Q8—61no—
318#125—61no—
319#125—61no—
320C22⋊Q8—121no—
321C4×Q8—21no—
322C4×Q8—61no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—41no—
329C4×D4—61no—
330C4×D4—121no—
331#117—21no—
332#117—31no—
333#117—61no—
334#117C22×D311yescharacteristic
335#117C22×D311yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—121no—
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—41no—
343C2×C22⋊C4—121no—
344C2×C42—31no—
345#108—62no—
346#107—21no—
347#107—31no—
348#107—31no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#106—21no—
354#106—31no—
355#106—31no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106—61no—
363#106—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366C4⋊C8—61no—
367C4⋊C8—61no—
368#104—62no—
369#103—61no—
370#103#6711yescharacteristic
371#103#6711yescharacteristic
372#102—21no—
373#102—21no—
374#102—61no—
375#102—61no—
376#102—61no—
377#102—61no—
378#102—61no—
379#102#6711yescharacteristic
380#102#6711yescharacteristic
381#100—62no—
382#97—21no—
383C4×C8—21no—
384C23×C3—21no—
385C23×C3—22no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3—81no—
389C23×C3—81no—
390C23×C3H(5,𝔽2)11yescharacteristic
391C3×Q8—21no—
392C3×Q8C22⋊Q812yes—
393C3×D4—22no—
394C3×D4—22no—
395C3×D4—22no—
396C3×D4—21no—
397C3×D4—22no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4C22⋊Q812yes—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—41no—
409C2×C3×C4—81no—
410C2×C3×C4—161no—
411C2×C3×C4—161no—
412C2×C3×C4—161no—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×Q811yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—81no—
427C2×Dic3—81no—
428C2×Dic3—161no—
429Dic6—161no—
430C3×C8—22no—
431C3×C8—21no—
432C3×C8—21no—
433C3⋊C8—21no—
434C3⋊C8—22no—
435C3⋊C8—21no—
436C3⋊C8—41no—
437C3⋊C8—41no—
438C24—21no—
439#41—21no—
440#41—21no—
441#41C2×Dic612yes—
442#41#23512yes—
443C2×Q8—121no—
444C2×Q8C2×#6711yescharacteristic
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4C2×Dic612yes—
452C2×D4C2×#6711yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—121no—
462C22×C4C23×D311yescharacteristic
463SD16—24no—
464D8—24no—
465M16—21no—
466M16—32no—
467M16—64no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16C2×Dic612yes—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—31no—
478C2×C8—31no—
479C2×C8—64no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8C2×#6711yescharacteristic
486C2×C8C2×#6711yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—31no—
490C4⋊C4—31no—
491C4⋊C4—41no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—121no—
497C4⋊C4—121no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—242no—
501C4⋊C4D3×D411yescharacteristic
502C4⋊C4D3×D411yescharacteristic
503C4⋊C4C2×#6711yescharacteristic
504C4⋊C4C2×#6711yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—61no—
511C22⋊C4—121no—
512C22⋊C4—242no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—61no—
517C42—61no—
518C42—61no—
519C42—121no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—22no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—42no—
526C22×C3—82no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—161no—
530C22×C3#53911yescharacteristic
531C22×C3D4211yescharacteristic
532C22×C3#55411yescharacteristic
533C3×C4—22no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—81no—
539C3×C4C2×C22⋊Q811yescharacteristic
540C3×C4#52511yescharacteristic
541C3×C4#54711yescharacteristic
542C3×C4D4×Q811yescharacteristic
543Dic3—42no—
544Dic3—41no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—81no—
553C23—81no—
554C23#100411yescharacteristic
555Q8—21no—
556Q8—481no—
557Q8#92412yes—
558D4—22no—
559D4—22no—
560D4—22no—
561D4—22no—
562D4—21no—
563D4—41no—
564D4—41no—
565D4#92412yes—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—62no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—81no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—161no—
583C2×C4—161no—
584C2×C4—161no—
585C2×C4—241no—
586C2×C4—241no—
587C2×C4—481no—
588C2×C4C22×Dic611yescharacteristic
589C2×C4C2×#23511yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4#101011yescharacteristic
593C2×C4C22×#6711yescharacteristic
594C8—22no—
595C8—21no—
596C8—21no—
597C8—61no—
598C8—61no—
599C8—62no—
600C8—121no—
601C8—121no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—42no—
605C2×C3—82no—
606C2×C3#244911yescharacteristic
607C2×C3#328811yescharacteristic
608C2×C3#329111yescharacteristic
609C22—21no—
610C22—21no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—82no—
616C22—81no—
617C22—81no—
618C22—161no—
619C22#553911yescharacteristic
620C22#556411yescharacteristic
621C22D4×#6711yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—122no—
629C4—121no—
630C4—121no—
631C4—241no—
632C4—241no—
633C4D4×Dic611yescharacteristic
634C4#554711yescharacteristic
635C4C2×#92411yescharacteristic
636C4#582411yescharacteristic
637C3#2640711yescharacteristic, complemented, Sylow
638C2—21no—
639C2—21no—
640C2—42no—
641C2—82no—
642C2#8236011yescharacteristic
643C2#8400911yescharacteristic
644C2#8401411yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2946C2
socleC22×C3D42
prime radicalC2×C3×AGL(1,ℤ/8)C22
hypercenter#2946D3
hypocenterC3#26407
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2946

Complements

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

NH ≅ G/Ntype
2. #82561C2semidirect
3. #82539C2semidirect
4. #78824C2semidirect
7. #71664C2semidirect
8. #71647C2semidirect
9. #70671C2semidirect
10. #70666C2semidirect
11. #70292C2semidirect
12. #70197C2semidirect
13. #70184C2semidirect
14. #69899C2semidirect
15. #69893C2semidirect
86. C3×AGL(1,ℤ/8)Q8semidirect
303. AGL(1,ℤ/8)Dic6semidirect
637. C3#26407semidirect

Maximal subgroups

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

subgroupindex
C3×#29462
#825612
#825392
#788242
#720912
#720882
#716642
#716472
#706712
#706662
#702922
#701972
#701842
#698992
#698932
#264073