Finite Group Databasebeta

← #10826838

Subgroups — #10826838

Order 768.

1814 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3485C211yesmaximal, characteristic, complemented
2C2×#5832C211yesmaximal, characteristic, complemented
3C2×#5242C211yesmaximal, characteristic, complemented
4C2×#5241C211yesmaximal, characteristic, complemented
5#82928C212yesmaximal, complemented
6#72137C211yesmaximal, characteristic, complemented
7#71868C212yesmaximal, complemented
8#71867C212yesmaximal, complemented
9#71711C212yesmaximal, complemented
10#71650C211yesmaximal, characteristic, complemented
11#71649C211yesmaximal, characteristic, complemented
12#23908—31nomaximal, Sylow
13C2×C3×H(5,𝔽2)C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C22×#235C2211yescharacteristic, complemented
16C3×#547—24no—
17C3×#546—24no—
18C3×#536C2212yes—
19C3×#534—22no—
20C3×#525C2212yes—
21C2×C3×#123—22no—
22C3×#518C2211yescharacteristic
23#5832—24no—
24#5832—24no—
25C2×#951C2212yes—
26C2×#949C2212yes—
27C2×#947C2211yescharacteristic, complemented
28C2×#944C2211yescharacteristic, complemented
29C2×#940C2211yescharacteristic
30C2×#939—22no—
31C2×#936C2211yescharacteristic
32C2×C4×#67—22no—
33C22×C4×Dic3C2211yescharacteristic, complemented
34#5242—24no—
35#5241—24no—
36C2×#837C2212yes—
37C2×#837C2212yes—
38C2×#836C2212yescomplemented
39C2×#833C2211yescharacteristic
40C2×#833C2211yescharacteristic
41#5207C2212yescomplemented
42#5047—24no—
43#5037—24no—
44#5004C2212yescomplemented
45#5001C2212yescomplemented
46#4532C2212yescomplemented
47#4532C2212yescomplemented
48#3485D311yescharacteristic, complemented
49C2×#532—31no—
50C2×#455—31no—
51C2×#453—31no—
52#3005—32no—
53#2005—32no—
54#2003—32no—
55#1896—32no—
56#1887—31no—
57#1885—31no—
58#1785—31no—
59C3×H(5,𝔽2)—22no—
60C3×H(5,𝔽2)—22no—
61C3×H(5,𝔽2)—42no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—24no—
66C2×C3×#41—42no—
67C2×C3×#41C2312yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—22no—
70C22×C3×D4—22no—
71C22×C3×D4D412yes—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4D412yes—
74C23×C3×C4C2311yescharacteristic
75C22×#67—21no—
76C22×#67—22no—
77C23×Dic3—21no—
78C2×#235—22no—
79C2×#235—24no—
80C2×#235—22no—
81C2×#235—42no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C22×Dic6C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×#124—22no—
87C3×#123—28no—
88C3×#123—24no—
89C3×#123—22no—
90C3×#123—24no—
91C3×C22⋊Q8—24no—
92C3×C22⋊Q8D414yescomplemented
93C3×#121—28no—
94C3×#121D414yescomplemented
95C3×#120—24no—
96C3×#120—22no—
97C3×C4×D4—24no—
98C3×C4×D4—24no—
99C3×#117C2312yes—
100C2×C3×C4⋊C4—24no—
101C2×C3×C22⋊C4—24no—
102C2×C3×C22⋊C4—22no—
103C2×#217—22no—
104#951—24no—
105#951—42no—
106#949—24no—
107#949—42no—
108#947—22no—
109#947—22no—
110#944—22no—
111#944—22no—
112C2×#216—22no—
113C2×#215—22no—
114#940—22no—
115#940D414yescomplemented
116#939—216no—
117#936—22no—
118#936D414yescomplemented
119C2×#214—22no—
120C2×#213—22no—
121C4×#67—216no—
122C2×(C2×D3)⋊C4—22no—
123C2×Dic3⋊C4—22no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3—22no—
126C2×C4×Dic3—22no—
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C2×#208C2312yes—
130C2×#208C2312yes—
131#837—24no—
132#837—24no—
133#836—42no—
134#833—22no—
135#833—22no—
136#830—24no—
137#810—24no—
138#809—24no—
139C2×H(5,𝔽2)C2×D311yescharacteristic
140C22×#41—31no—
141C22×#41C2×D311yescharacteristic
142C2×#137—32no—
143C2×AGL(1,ℤ/8)—32no—
144#547—24no—
145#546—24no—
146#536C2×D312yes—
147#534—22no—
148#532—64no—
149#532—64no—
150C2×C4⋊Q8—31no—
151C2×#127—31no—
152C2×#124—31no—
153C2×#124—31no—
154#525C2×D312yes—
155C2×#123—22no—
156C2×#121—62no—
157#518C2×D311yescharacteristic
158C2×C4×D4—62no—
159C22×C42—31no—
160#455—64no—
161#453—64no—
162#452—64no—
163#449—64no—
164C2×#104—32no—
165C2×#104—32no—
166#419—32no—
167#418—32no—
168C2×#101—32no—
169C2×#100—31no—
170C2×#100—31no—
171#407—32no—
172#339—32no—
173#339—32no—
174C24×C3—22no—
175C24×C3C2×D412yes—
176C24×C3C2×D411yescharacteristic
177C23×D3—21no—
178C3×#41—22no—
179C3×#41—22no—
180C3×#41—22no—
181C3×#41—22no—
182C3×#41—48no—
183C3×#41—48no—
184C3×#41—44no—
185C3×#41—44no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8C2×D412yes—
191C2×C3×D4—24no—
192C2×C3×D4—24no—
193C2×C3×D4—24no—
194C2×C3×D4—22no—
195C2×C3×D4—24no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—42no—
199C2×C3×D4—44no—
200C2×C3×D4—44no—
201C2×C3×D4—44no—
202C2×C3×D4—44no—
203C2×C3×D4C2×D412yes—
204C2×C3×D4C2×D412yes—
205C2×C3×D4C2×D412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4—22no—
214C22×C3×C4—42no—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D414yes—
218C22×C3×C4C2411yescharacteristic
219C2×#67—24no—
220C2×#67—28no—
221C2×#67—48no—
222C2×#67—42no—
223C2×#67—42no—
224C22×Dic3—21no—
225C22×Dic3—24no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—44no—
229#235—48no—
230#235—48no—
231C2×D12—22no—
232C2×D12—22no—
233C2×D12—21no—
234C2×D12C2×D412yes—
235C2×C4×D3—24no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×Dic6—22no—
239C2×Dic6—22no—
240C2×Dic6—21no—
241C2×Dic6C2×D412yes—
242C3×C4⋊C4—28no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4C2×D414yes—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—24no—
250C3×C42—22no—
251#217—28no—
252#216—44no—
253#215—44no—
254#214—44no—
255#213—44no—
256(C2×D3)⋊C4—28no—
257Dic3⋊C4—28no—
258C4×Dic3—24no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3C2×D412yes—
262C4×Dic3C2×D412yes—
263#208—22no—
264#208—22no—
265#208—22no—
266#208—22no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269H(5,𝔽2)—22no—
270H(5,𝔽2)—22no—
271H(5,𝔽2)—42no—
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—22no—
275C2×#41—24no—
276C2×#41—42no—
277C2×#41—62no—
278C2×#41—62no—
279C2×#41—64no—
280C2×#41—122no—
281C2×#41C22×D312yes—
282C22×Q8—31no—
283C22×Q8C22×D311yescharacteristic
284C22×D4—22no—
285C22×D4—22no—
286C22×D4—31no—
287C22×D4—62no—
288C22×D4—61no—
289C22×D4#6712yes—
290C22×D4C22×D311yescharacteristic
291C23×C4—61no—
292C23×C4—61no—
293C23×C4#6712yes—
294C23×C4C22×D311yescharacteristic
295#137—64no—
296#137—122no—
297AGL(1,ℤ/8)—64no—
298AGL(1,ℤ/8)—122no—
299C2×Q16—62no—
300C2×SD16—62no—
301C2×SD16—62no—
302C2×D8—62no—
303C2×M16—32no—
304C2×M16—32no—
305C4⋊Q8—22no—
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308#127—34no—
309#127—62no—
310#124—22no—
311#124—34no—
312#124—62no—
313#124—62no—
314#124—62no—
315#123—28no—
316#123—24no—
317#123—22no—
318#123—24no—
319C22⋊Q8—24no—
320C22⋊Q8#6714yescomplemented
321#121—28no—
322#121—616no—
323#121#6714yescomplemented
324#120—24no—
325#120—22no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—616no—
329#117C22×D312yes—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—62no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—62no—
335C2×C22⋊C4—62no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—62no—
339C2×C42—62no—
340C2×C42—61no—
341#104—64no—
342#104—64no—
343#103—64no—
344#102—64no—
345#101—122no—
346#100—62no—
347#100—62no—
348C22⋊C8—64no—
349C23×C3—22no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—42no—
355C23×C3—44no—
356C23×C3—44no—
357C23×C3—84no—
358C23×C3#12014yes—
359C23×C3#12012yes—
360C23×C3C22×D411yescharacteristic
361C22×D3—22no—
362C22×D3—44no—
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—42no—
368C3×Q8—42no—
369C3×D4—24no—
370C3×D4—24no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—48no—
374C3×D4—48no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—84no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—48no—
388C2×C3×C4—44no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—84no—
392C2×C3×C4#12012yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C22×D412yes—
396C2×C3×C4H(5,𝔽2)12yes—
397#67—416no—
398#67—48no—
399C2×Dic3—24no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—48no—
404C2×Dic3—42no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407D12—24no—
408D12—42no—
409C4×D3—48no—
410Dic6—24no—
411Dic6—42no—
412C3⋊C8—42no—
413C3⋊C8—42no—
414C24—22no—
415C24—61no—
416C24C2×#6712yes—
417C24C2×#6711yescharacteristic
418#41—22no—
419#41—22no—
420#41—22no—
421#41—22no—
422#41—48no—
423#41—48no—
424#41—44no—
425#41—44no—
426#41—128no—
427#41—128no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—32no—
433C2×Q8—61no—
434C2×Q8—62no—
435C2×Q8—62no—
436C2×Q8C2×#6712yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—32no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—64no—
451C2×D4—68no—
452C2×D4—61no—
453C2×D4—62no—
454C2×D4—62no—
455C2×D4—128no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4D3×D412yes—
459C2×D4D3×D412yes—
460C2×D4C2×#6712yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—64no—
471C22×C4—61no—
472C22×C4—64no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—122no—
476C22×C4—124no—
477C22×C4—122no—
478C22×C4C2×#6712yes—
479C22×C4C2×#6711yescharacteristic
480C22×C4C2×#6714yes—
481C22×C4C23×D311yescharacteristic
482Q16—124no—
483SD16—124no—
484SD16—124no—
485D8—124no—
486M16—62no—
487M16—62no—
488M16—62no—
489M16—62no—
490C2×C8—62no—
491C2×C8—62no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—68no—
495C4⋊C4C2×#6714yes—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—24no—
501C22⋊C4—68no—
502C22⋊C4—68no—
503C42—22no—
504C42—32no—
505C42—32no—
506C42—64no—
507C42—61no—
508C42—61no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—24no—
512C22×C3—24no—
513C22×C3—44no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—44no—
517C22×C3—42no—
518C22×C3—44no—
519C22×C3—82no—
520C22×C3—164no—
521C22×C3C2×#12011yescharacteristic
522C22×C3C2×#12011yescharacteristic
523C22×C3#53411yescharacteristic
524C2×D3—44no—
525C2×D3—42no—
526C2×D3—82no—
527C2×D3—82no—
528C3×C4—22no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4—44no—
532C3×C4D4212yes—
533C3×C4D4212yes—
534Dic3—24no—
535Dic3—42no—
536Dic3—42no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—62no—
546C23—84no—
547C23—124no—
548C23—121no—
549C23—121no—
550C23#93714yes—
551C23#95312yes—
552C23C22×#6711yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—42no—
556Q8—42no—
557Q8—64no—
558Q8—122no—
559D4—24no—
560D4—24no—
561D4—22no—
562D4—22no—
563D4—48no—
564D4—48no—
565D4—42no—
566D4—42no—
567D4—64no—
568D4—84no—
569D4—1216no—
570D4—128no—
571D4—122no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—64no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—84no—
590C2×C4—128no—
591C2×C4—128no—
592C2×C4—122no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4#95312yes—
596C2×C4C2×D3×D411yescharacteristic
597C2×C4C2×D3×D411yescharacteristic
598C2×C4#100412yes—
599C2×C4C22×#6712yes—
600C8—122no—
601C8—122no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—42no—
605C2×C3—44no—
606C2×C3—82no—
607C2×C3#240311yescharacteristic
608C2×C3#301812yes—
609D3—82no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—24no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—82no—
621C22—124no—
622C22—122no—
623C22—164no—
624C22—242no—
625C22—242no—
626C22C2×#93711yescharacteristic
627C22#580411yescharacteristic
628C22C2×#95311yescharacteristic
629C4—22no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—64no—
634C4—122no—
635C4—122no—
636C4D4×#6712yes—
637C4D4×#6712yes—
638C3#2390811yescharacteristic, complemented, Sylow
639C2—21no—
640C2—21no—
641C2—42no—
642C2—44no—
643C2—82no—
644C2—242no—
645C2#8405211yescharacteristic
646C2#8412112yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3485C2
socleC22×C3C2×#120
hypercenter#3485D3
hypocenterC3#23908
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. C3×#3485C2semidirect
2. C2×#5832C2semidirect
3. C2×#5242C2semidirect
4. C2×#5241C2semidirect
5. #82928C2semidirect
6. #72137C2semidirect
7. #71868C2semidirect
8. #71867C2semidirect
9. #71711C2semidirect
10. #71650C2semidirect
11. #71649C2semidirect
15. C22×#235C22semidirect
27. C2×#947C22semidirect
28. C2×#944C22semidirect
33. C22×C4×Dic3C22semidirect
38. C2×#836C22semidirect
41. #5207C22semidirect
44. #5004C22semidirect
45. #5001C22semidirect
46. #4532C22semidirect
47. #4532C22semidirect
48. #3485D3semidirect
92. C3×C22⋊Q8D4semidirect
94. C3×#121D4semidirect
115. #940D4semidirect
118. #936D4semidirect
320. C22⋊Q8#67semidirect
323. #121#67semidirect
638. C3#23908semidirect

Maximal subgroups

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

subgroupindex
C3×#34852
C2×#58322
C2×#52422
C2×#52412
#829282
#721372
#718682
#718672
#717112
#716502
#716492
#239083