Finite Group Databasebeta

← #10827578

Subgroups — #10827578

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic, complemented
2C3×#2408C211yesmaximal, characteristic
3C2×#5218C211yesmaximal, characteristic, complemented
4C2×#5215C211yesmaximal, characteristic, complemented
5#71685C212yesmaximal, complemented
6#71678C212yesmaximal, complemented
7#71611C211yesmaximal, characteristic, complemented
8#69660C212yesmaximal, complemented
9#69638C212yesmaximal, complemented
10#69516C211yesmaximal, characteristic, complemented
11#69503C211yesmaximal, characteristic, complemented
12#23979—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C2×C3×#121C2212yes—
15C2×C3×#120—21no—
16C2×C3×C4×D4C2211yescharacteristic
17C2×#936—21no—
18C22×#214C2212yes—
19C2×#926C2211yescharacteristic
20C2×#926C2211yescharacteristic, complemented
21C2×#924C2211yescharacteristic
22C2×#924C2211yescharacteristic, complemented
23C22×#208C2211yescharacteristic
24#5542—24no—
25#5542—24no—
26C2×#887—21no—
27C2×#879—21no—
28C2×#879—21no—
29C2×#869C2211yescharacteristic, complemented
30C2×C4×Dic6C2211yescharacteristic, complemented
31C3×#394—21no—
32C3×#392—21no—
33C3×#390C2212yes—
34C3×#380C2211yescharacteristic
35C3×#379—21no—
36#5218—24no—
37#5215—24no—
38C2×#832C2212yes—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×#830C2211yescharacteristic
42C2×#830C2212yescomplemented
43C2×#810C2212yescomplemented
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47#4524C2212yescomplemented
48C2×#556—31no—
49C2×#450—31no—
50C2×#418—31no—
51#2408D311yescharacteristic
52#2023—32no—
53#2000—32no—
54#1894—31no—
55#1880—32no—
56#1875—32no—
57#1870—31no—
58#1825—31no—
59C25×C3—21no—
60C22×C3×D4—24no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—42no—
66C22×C3×D4D411yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C22×Dic6—21no—
71C22×Dic6C2311yescharacteristic
72C3×#121—24no—
73C3×#121—42no—
74C3×#120—44no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4D411yescharacteristic
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90#936—28no—
91C2×#214—24no—
92C2×#214—24no—
93C2×#214—42no—
94#926—22no—
95#926—22no—
96#926—22no—
97#926D414yescomplemented
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102#924—22no—
103#924—22no—
104#924—22no—
105#924D414yescomplemented
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×#208—22no—
113C2×#208—42no—
114C22×C3⋊C8C2312yes—
115C22×C3⋊C8C2312yes—
116#887—28no—
117#879—28no—
118#879—28no—
119#869—22no—
120#869—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#95—21no—
124#832—24no—
125#832—24no—
126#832—24no—
127#830—22no—
128#830—42no—
129#810—42no—
130C23×D4C2×D311yescharacteristic
131C22×SD16—32no—
132C22×M16—31no—
133#556—64no—
134#556—64no—
135C2×C4⋊Q8—31no—
136C2×#126—61no—
137C2×#126—61no—
138C2×#124—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—31no—
141C2×#121C2×D312yes—
142C2×#120—21no—
143C2×C4×Q8—31no—
144C2×C4×Q8—61no—
145C2×C4×D4C2×D311yescharacteristic
146C2×#117—31no—
147C2×#117—31no—
148#450—64no—
149#418—64no—
150C2×#103—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×C22⋊C8—31no—
155C2×C22⋊C8—32no—
156#394—21no—
157#392—21no—
158#390C2×D312yes—
159#380C2×D311yescharacteristic
160#379—21no—
161#340—32no—
162#340—32no—
163#340—32no—
164#336—32no—
165C24×C3—22no—
166C24×C3—44no—
167C24×C3—42no—
168C24×C3—42no—
169C24×C3—42no—
170C24×C3—82no—
171C24×C3C2×D411yescharacteristic
172C24×C3C2×D411yescharacteristic
173C2×C3×D4—28no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—48no—
182C2×C3×D4—41no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—42no—
188C2×C3×D4—82no—
189C2×C3×D4#4114yes—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×Dic6—22no—
209C2×Dic6—22no—
210C2×Dic6—24no—
211C2×Dic6—21no—
212C2×Dic6—42no—
213C2×Dic6—42no—
214C2×Dic6C2×D412yes—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—44no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—44no—
223C3×C22⋊C4—82no—
224C3×C42—21no—
225C3×C42—21no—
226#217—24no—
227#217—24no—
228#217—24no—
229#217—24no—
230#214—48no—
231C4⋊Dic3—24no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3C2×D412yes—
236C4⋊Dic3C2×D412yes—
237C4⋊Dic3C2×D412yes—
238Dic3⋊C4—24no—
239Dic3⋊C4—24no—
240Dic3⋊C4—24no—
241C4×Dic3—24no—
242C4×Dic3—24no—
243C4×Dic3—24no—
244#208—44no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C25—21no—
252C22×Q8—31no—
253C22×Q8—61no—
254C22×D4—24no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—42no—
260C22×D4#6711yescharacteristic
261C22×D4C22×D312yes—
262C23×C4C22×D311yescharacteristic
263C23×C4C22×D311yescharacteristic
264C2×SD16—64no—
265C2×SD16—64no—
266C2×SD16—122no—
267C2×M16—62no—
268C2×M16—122no—
269C22×C8—32no—
270C22×C8—32no—
271C4⋊Q8—62no—
272C4⋊Q8—62no—
273#126—68no—
274#126—68no—
275#124—68no—
276C22⋊Q8—34no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—62no—
280#121—24no—
281#121—42no—
282#120—44no—
283C4×Q8—62no—
284C4×Q8—62no—
285C4×Q8—68no—
286C4×D4—22no—
287C4×D4—22no—
288#117—34no—
289#117—62no—
290#117—62no—
291#117—62no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4#6711yescharacteristic
300C2×C4⋊C4C22×D312yes—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4#6711yescharacteristic
310C2×C22⋊C4#6711yescharacteristic
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#103—122no—
316#102—64no—
317#102—64no—
318#102—64no—
319C22⋊C8—62no—
320C22⋊C8—122no—
321#95—21no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3—21no—
327C23×C3—24no—
328C23×C3—22no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—48no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—84no—
340C23×C3—84no—
341C23×C3—88no—
342C23×C3—88no—
343C23×C3#12012yes—
344C23×C3C22×D411yescharacteristic
345C3×D4—24no—
346C3×D4—24no—
347C3×D4—44no—
348C3×D4—48no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—88no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4#12012yes—
369C2×C3×C4AGL(1,ℤ/8)12yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4112yes—
373C2×C3×C4H(5,𝔽2)12yes—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—42no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—42no—
384Dic6—24no—
385Dic6—48no—
386Dic6—41no—
387Dic6—41no—
388C3⋊C8—42no—
389C3⋊C8—42no—
390C24—22no—
391C24—44no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—82no—
396C24C2×#6711yescharacteristic
397C24C2×#6711yescharacteristic
398C2×Q8—32no—
399C2×Q8—64no—
400C2×Q8—61no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—122no—
404C2×Q8—122no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—48no—
414C2×D4—41no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—82no—
421C2×D4#23714yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4D3×D412yes—
434C22×C4D3×D412yes—
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C23×D311yescharacteristic
440SD16—128no—
441M16—124no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—32no—
452C4⋊C4—32no—
453C4⋊C4—32no—
454C4⋊C4—64no—
455C4⋊C4—64no—
456C4⋊C4—64no—
457C4⋊C4—64no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—64no—
469C22⋊C4—64no—
470C22⋊C4—82no—
471C42—21no—
472C42—21no—
473C42—64no—
474C42—64no—
475C42—64no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—41no—
482C22×C3—44no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—42no—
486C22×C3—44no—
487C22×C3—42no—
488C22×C3—44no—
489C22×C3—816no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3C2×#12011yescharacteristic
495C22×C3#54612yes—
496C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
497C22×C3#57511yescharacteristic
498C22×C3#57712yes—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—82no—
504C3×C4#52511yescharacteristic
505C3×C4#53511yescharacteristic
506C3×C4#54612yes—
507Dic3—42no—
508Dic3—42no—
509Dic3—42no—
510Dic3—42no—
511C23—21no—
512C23—22no—
513C23—21no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—48no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—84no—
529C23—84no—
530C23—88no—
531C23—88no—
532C23#93712yes—
533C23C22×#6711yescharacteristic
534Q8—64no—
535Q8—128no—
536Q8—121no—
537Q8—121no—
538D4—24no—
539D4—24no—
540D4—44no—
541D4—48no—
542D4—42no—
543D4—42no—
544D4—88no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—122no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—122no—
571C2×C4#93212yes—
572C2×C4#93712yes—
573C2×C4C2×D3×D411yescharacteristic
574C2×C4C2×D3×D411yescharacteristic
575C2×C4C2×#23712yes—
576C2×C4#100412yes—
577C8—122no—
578C8—122no—
579C2×C3—42no—
580C2×C3—81no—
581C2×C3—84no—
582C2×C3#240411yescharacteristic
583C2×C3#300611yescharacteristic
584C2×C3#301011yescharacteristic
585C2×C3#331112yes—
586C2×C3#332412yes—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—42no—
595C22—44no—
596C22—41no—
597C22—44no—
598C22—41no—
599C22—41no—
600C22—816no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22C2×#93211yescharacteristic
606C22C2×#93711yescharacteristic
607C22#580612yes—
608C22#582211yescharacteristic
609C22#583612yes—
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—82no—
615C4—122no—
616C4—122no—
617C4—122no—
618C4—122no—
619C4#558611yescharacteristic
620C4#559112yes—
621C4#559211yescharacteristic
622C3#2397911yescharacteristic, complemented, Sylow
623C2—42no—
624C2—81no—
625C2—84no—
626C2#8258111yescharacteristic
627C2#8262411yescharacteristic
628C2#8339011yescharacteristic
629C2#8348612yes—
630C2#8349712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2408C2
socleC23×C3C22×D4
hypercenter#2408D3
hypocenterC3#23979
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C2×#5542C2semidirect
3. C2×#5218C2semidirect
4. C2×#5215C2semidirect
5. #71685C2semidirect
6. #71678C2semidirect
7. #71611C2semidirect
8. #69660C2semidirect
9. #69638C2semidirect
10. #69516C2semidirect
11. #69503C2semidirect
20. C2×#926C22semidirect
22. C2×#924C22semidirect
29. C2×#869C22semidirect
30. C2×C4×Dic6C22semidirect
42. C2×#830C22semidirect
43. C2×#810C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
47. #4524C22semidirect
97. #926D4semidirect
105. #924D4semidirect
622. C3#23979semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C3×#24082
C2×#52182
C2×#52152
#716852
#716782
#716112
#696602
#696382
#695162
#695032
#239793