Finite Group Databasebeta

← #10725791

Subgroups — #10725791

Order 768.

1550 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#394C211yesmaximal, characteristic
2#72131C212yesmaximal
3#71569C211yesmaximal, characteristic, complemented
4#71564C211yesmaximal, characteristic, complemented
5#71548C212yesmaximal, complemented
6#71545C212yesmaximal, complemented
7#71545C212yesmaximal, complemented
8#71545C212yesmaximal, complemented
9#69401C212yesmaximal, complemented
10#14617—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C22⋊C4C2212yes—
15C22×#217C2212yes—
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C22×C4×Dic3C2212yes—
20C22×C4×Dic3C2211yescharacteristic
21C3×#394C418yescomplemented
22C2×C3×#95C2211yescharacteristic
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×#394D311yescharacteristic
34#1445—31no—
35#1440—32no—
36#1438—32no—
37#1437—32no—
38#1437—32no—
39#1437—32no—
40#1436—31no—
41#1434—32no—
42C25×C3C2312yes—
43C22×C3×D4—42no—
44C22×C3×D4—42no—
45C22×C3×D4—41no—
46C22×C3×D4—41no—
47C22×C3×D4C2×C414yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C2×C3×C4⋊C4—41no—
57C2×C3×C4⋊C4—41no—
58C2×C3×C4⋊C4C2×C414yes—
59C2×C3×C22⋊C4—42no—
60C2×C3×C22⋊C4—42no—
61C2×C3×C22⋊C4—42no—
62C2×C3×C22⋊C4—42no—
63C2×C3×C22⋊C4C2×C418yes—
64C2×C3×C22⋊C4C2×C418yes—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—42no—
72C2×#217—42no—
73C2×#217—42no—
74C2×#217—42no—
75C2×#217—42no—
76C2×#217—42no—
77C2×C4×Dic3—24no—
78C2×C4×Dic3—42no—
79C2×C4×Dic3—42no—
80C2×C4×Dic3—42no—
81C2×C4×Dic3—41no—
82C3×#95C2×C414yes—
83C3⋊#95—24no—
84C3⋊#95—24no—
85C3⋊#95—42no—
86C3⋊#95—42no—
87C3⋊#95—42no—
88C3⋊#95—42no—
89C3⋊#95—41no—
90C3⋊#95—41no—
91C3⋊#95—42no—
92C3⋊#95—41no—
93C23×D4C2×D311yescharacteristic
94C22×C4⋊C4C2×D311yescharacteristic
95C22×C22⋊C4—32no—
96C22×C22⋊C4—32no—
97C22×C22⋊C4—32no—
98C22×C22⋊C4—32no—
99C22×C22⋊C4C2×D312yes—
100C22×C22⋊C4C2×D312yes—
101C22×C42—32no—
102C22×C42—31no—
103#394Dic318yescomplemented
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—31no—
111C2×#95—31no—
112C2×#95—32no—
113C2×#95—31no—
114C2×#95C2×D311yescharacteristic
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3C22×C414yes—
130C24×C3C2411yescharacteristic
131C2×C3×D4—44no—
132C2×C3×D4—48no—
133C2×C3×D4—44no—
134C2×C3×D4—48no—
135C2×C3×D4—44no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4C22×C414yes—
155C22×C3×C4C22×C412yes—
156C22×C3×C4C22×C412yes—
157C22×C3×C4C22×C412yes—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3#4114yes—
184C22×Dic3#4114yes—
185C22×Dic3#4114yes—
186C3×C4⋊C4—44no—
187C3×C22⋊C4—48no—
188C3×C22⋊C4—48no—
189#217—48no—
190#217—48no—
191#217—48no—
192#217—48no—
193C4×Dic3—48no—
194C4×Dic3—44no—
195C25C22×D312yes—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—41no—
199C22×D4—41no—
200C22×D4C2×Dic314yes—
201C23×C4—32no—
202C23×C4—32no—
203C23×C4—32no—
204C23×C4—32no—
205C23×C4C22×D311yescharacteristic
206C23×C4C22×D311yescharacteristic
207C23×C4C22×D312yes—
208C23×C4C22×D311yescharacteristic
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C2×Dic314yes—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—64no—
218C2×C22⋊C4—64no—
219C2×C22⋊C4—64no—
220C2×C22⋊C4—64no—
221C2×C22⋊C4—64no—
222C2×C22⋊C4—122no—
223C2×C22⋊C4—122no—
224C2×C22⋊C4—122no—
225C2×C22⋊C4—122no—
226C2×C22⋊C4—122no—
227C2×C22⋊C4—122no—
228C2×C22⋊C4C2×Dic318yes—
229C2×C22⋊C4C2×Dic318yes—
230C2×C42—64no—
231C2×C42—122no—
232C2×C42—122no—
233C2×C42—122no—
234C2×C42—121no—
235#95—64no—
236#95—64no—
237#95—122no—
238#95—122no—
239#95—122no—
240#95—121no—
241#95—121no—
242#95—121no—
243#95—122no—
244#95—122no—
245#95C2×Dic314yes—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3C23×C411yescharacteristic
282C23×C3C2×#4112yes—
283C23×C3C2×#4112yes—
284C23×C3C2×#4112yes—
285C23×C3H(5,𝔽2)12yes—
286C23×C3H(5,𝔽2)11yescharacteristic
287C23×C3H(5,𝔽2)12yes—
288C23×C3H(5,𝔽2)12yes—
289C23×C3H(5,𝔽2)11yescharacteristic
290C3×D4—416no—
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24C22×Dic314yes—
356C24C23×D311yescharacteristic
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—44no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—34no—
381C22×C4—34no—
382C22×C4—34no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4C22×Dic314yes—
409C22×C4C22×Dic312yes—
410C22×C4C22×Dic312yes—
411C22×C4C22×Dic312yes—
412C4⋊C4—44no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—128no—
416C22⋊C4—128no—
417C22⋊C4—128no—
418C22⋊C4—128no—
419C42—128no—
420C42—124no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3C4×#4112yes—
437C22×C3C4×#4111yescharacteristic
438C22×C3#51811yescharacteristic
439C22×C3#51811yescharacteristic
440C22×C3#51812yes—
441C22×C3#53812yes—
442C22×C3#53812yes—
443C22×C3#53812yes—
444C22×C3#54012yes—
445C22×C3#54012yes—
446C22×C3#54012yes—
447C22×C3#55112yes—
448C22×C3#55112yes—
449C22×C3#55312yes—
450C22×C3#55312yes—
451C22×C3#55511yescharacteristic
452C22×C3#55511yescharacteristic
453C22×C3#55912yes—
454C22×C3#56011yescharacteristic
455C22×C3#56012yes—
456C22×C3H(3,𝔽4)11yescharacteristic
457C3×C4—42no—
458C3×C4—44no—
459C3×C4—42no—
460C3×C4—42no—
461Dic3—44no—
462Dic3—44no—
463Dic3—44no—
464Dic3—82no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23#100412yes—
501C23#100412yes—
502C23#100411yescharacteristic
503C23#100411yescharacteristic
504C23#100412yes—
505C23D3×#4112yes—
506C23D3×#4112yes—
507C23D3×#4112yes—
508C23C23×Dic311yescharacteristic
509D4—416no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C3—84no—
561C2×C3#233312yes—
562C2×C3#233311yescharacteristic
563C2×C3#237211yescharacteristic
564C2×C3#237412yes—
565C2×C3#237511yescharacteristic
566C2×C3#275212yes—
567C2×C3#275312yes—
568C2×C3#275612yes—
569C2×C3#280312yes—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22#559411yescharacteristic
586C22#559512yes—
587C22#559511yescharacteristic
588C22#561312yes—
589C22#561312yes—
590C22#561312yes—
591C22#561512yes—
592C22#561512yes—
593C22#561512yes—
594C22#565912yes—
595C22#566112yes—
596C22#566112yes—
597C22#566412yes—
598C22#567011yescharacteristic
599C22#567011yescharacteristic
600C22#567912yes—
601C22#579811yescharacteristic
602C22#579812yes—
603C22#579811yescharacteristic
604C22Dic3×#4112yes—
605C22Dic3×#4111yescharacteristic
606C4—42no—
607C4—42no—
608C4—42no—
609C4—44no—
610C4—124no—
611C4—124no—
612C4—124no—
613C4—242no—
614C3#1461711yescharacteristic, complemented, Sylow
615C2—84no—
616C2#7705812yes—
617C2#7706012yes—
618C2#7708212yes—
619C2#7715512yes—
620C2#8258011yescharacteristic
621C2#8263511yescharacteristic
622C2#8263512yes—
623C2#8283912yes—
624C2#8295611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#394C2
socle, cosocleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#394D3
hypocenterC3#14617
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
3. #71569C2semidirect
4. #71564C2semidirect
5. #71548C2semidirect
6. #71545C2semidirect
7. #71545C2semidirect
8. #71545C2semidirect
9. #69401C2semidirect
21. C3×#394C4semidirect
103. #394Dic3semidirect
614. C3#14617semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3942
#721312
#715692
#715642
#715482
#715452
#694012
#146173