Finite Group Databasebeta

← #10724481

Subgroups — #10724481

Order 768.

1666 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#386C211yesmaximal, characteristic
2#72131C211yesmaximal, characteristic
3#72131C211yesmaximal, characteristic
4#72127C212yesmaximal, complemented
5#71548C212yesmaximal, complemented
6#71545C212yesmaximal, complemented
7#71545C212yesmaximal, complemented
8#69416C212yesmaximal, complemented
9#69401C212yesmaximal, complemented
10#14516—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×#217C2211yescharacteristic
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×C4×Dic3C2212yes—
21C3×#386C418yescomplemented
22C2×C3×#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2211yescharacteristic
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×#386D311yescharacteristic
34#1443—32no—
35#1440—32no—
36#1438—31no—
37#1438—31no—
38#1437—32no—
39#1437—32no—
40#1434—32no—
41C22⋊#95—32no—
42C25×C3C2311yescharacteristic
43C25×C3C2311yescharacteristic
44C22×C3×D4—24no—
45C22×C3×D4—42no—
46C22×C3×D4—41no—
47C22×C3×D4—41no—
48C22×C3×D4C2×C414yes—
49C23×C3×C4—41no—
50C23×C3×C4—41no—
51C23×C3×C4—42no—
52C23×C3×C4—42no—
53C23×C3×C4C2×C414yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C2×C3×C22⋊C4—42no—
62C2×C3×C22⋊C4—42no—
63C2×C3×C22⋊C4—42no—
64C2×C3×C22⋊C4C2×C414yes—
65C2×C3×C22⋊C4C2×C418yes—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—24no—
72C2×#217—41no—
73C2×#217—42no—
74C2×#217—42no—
75C2×#217—42no—
76C2×#217—42no—
77C2×#217—41no—
78C2×#217—42no—
79C2×C4×Dic3—24no—
80C2×C4×Dic3—42no—
81C2×C4×Dic3—42no—
82C3×#95C2×C418yes—
83C3⋊#95—24no—
84C3⋊#95—41no—
85C3⋊#95—42no—
86C3⋊#95—42no—
87C3⋊#95—42no—
88C3⋊#95—42no—
89C3⋊#95—42no—
90C3⋊#95—41no—
91C3⋊#95—42no—
92C3⋊#95—42no—
93C23×D4C2×D311yescharacteristic
94C24×C4C2×D311yescharacteristic
95C22×C22⋊C4—32no—
96C22×C22⋊C4—32no—
97C22×C22⋊C4—31no—
98C22×C22⋊C4—32no—
99C22×C22⋊C4—31no—
100C22×C22⋊C4C2×D311yescharacteristic
101C22×C22⋊C4C2×D312yes—
102C22×C42—32no—
103#386Dic318yescomplemented
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—31no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—31no—
114C2×#95C2×D312yes—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—21no—
129C24×C3—21no—
130C24×C3C22×C412yes—
131C24×C3C22×C412yes—
132C24×C3#4114yes—
133C24×C3C2411yescharacteristic
134C2×C3×D4—28no—
135C2×C3×D4—44no—
136C2×C3×D4—44no—
137C2×C3×D4—48no—
138C2×C3×D4—48no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—48no—
156C22×C3×C4—44no—
157C22×C3×C4—48no—
158C22×C3×C4—44no—
159C22×C3×C4—44no—
160C22×C3×C4C22×C412yes—
161C22×C3×C4C22×C414yes—
162C22×C3×C4C22×C414yes—
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—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3#4114yes—
190C22×Dic3#4114yes—
191C3×C22⋊C4—48no—
192C3×C22⋊C4—44no—
193#217—44no—
194#217—48no—
195#217—48no—
196#217—48no—
197#217—44no—
198C4×Dic3—48no—
199C25C22×D311yescharacteristic
200C25C22×D311yescharacteristic
201C22×D4—24no—
202C22×D4—42no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4C2×Dic314yes—
206C23×C4—32no—
207C23×C4—32no—
208C23×C4—32no—
209C23×C4—32no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4C2×Dic314yes—
215C23×C4C22×D312yes—
216C23×C4C22×D312yes—
217C23×C4C22×D311yescharacteristic
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—64no—
222C2×C22⋊C4—64no—
223C2×C22⋊C4—64no—
224C2×C22⋊C4—64no—
225C2×C22⋊C4—64no—
226C2×C22⋊C4—64no—
227C2×C22⋊C4—121no—
228C2×C22⋊C4—122no—
229C2×C22⋊C4—122no—
230C2×C22⋊C4—122no—
231C2×C22⋊C4—122no—
232C2×C22⋊C4—121no—
233C2×C22⋊C4—122no—
234C2×C22⋊C4C2×Dic314yes—
235C2×C22⋊C4C2×Dic318yes—
236C2×C42—64no—
237C2×C42—122no—
238C2×C42—122no—
239#95—64no—
240#95—122no—
241#95—122no—
242#95—122no—
243#95—122no—
244#95—122no—
245#95—122no—
246#95—121no—
247#95—122no—
248#95—121no—
249#95C2×Dic318yes—
250C23×C3—24no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—24no—
260C23×C3—22no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—41no—
270C23×C3—42no—
271C23×C3—41no—
272C23×C3—42no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3#11714yes—
292C23×C3C23×C411yescharacteristic
293C23×C3C2×#4112yes—
294C23×C3C2×#4112yes—
295C23×C3C2×#4112yes—
296C23×C3H(5,𝔽2)12yes—
297C23×C3H(5,𝔽2)12yes—
298C23×C3H(5,𝔽2)11yescharacteristic
299C23×C3H(5,𝔽2)11yescharacteristic
300C23×C3H(5,𝔽2)12yes—
301C3×D4—416no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—416no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
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—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24#23514yes—
366C24C22×Dic312yes—
367C24C22×Dic312yes—
368C24C23×D311yescharacteristic
369C2×D4—28no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—34no—
391C22×C4—34no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—48no—
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×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4C22×Dic312yes—
424C22×C4C22×Dic314yes—
425C22×C4C22×Dic314yes—
426C22⋊C4—48no—
427C22⋊C4—44no—
428C22⋊C4—124no—
429C22⋊C4—128no—
430C22⋊C4—128no—
431C22⋊C4—128no—
432C22⋊C4—124no—
433C42—128no—
434C22×C3—24no—
435C22×C3—28no—
436C22×C3—44no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—44no—
443C22×C3—42no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3C2×#11711yescharacteristic
453C22×C3C4×#4112yes—
454C22×C3#51811yescharacteristic
455C22×C3#51811yescharacteristic
456C22×C3#51812yes—
457C22×C3#53812yes—
458C22×C3#53812yes—
459C22×C3#53812yes—
460C22×C3#53812yes—
461C22×C3#54012yes—
462C22×C3#54012yes—
463C22×C3#55111yescharacteristic
464C22×C3#55112yes—
465C22×C3#55112yes—
466C22×C3#55312yes—
467C22×C3#55312yes—
468C22×C3#55312yes—
469C22×C3#55911yescharacteristic
470C22×C3#56012yes—
471C22×C3#56011yescharacteristic
472C22×C3H(3,𝔽4)11yescharacteristic
473C3×C4—42no—
474C3×C4—44no—
475C3×C4—44no—
476Dic3—44no—
477Dic3—44no—
478Dic3—82no—
479Dic3—82no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23#92614yes—
522C23C2×#23512yes—
523C23#100411yescharacteristic
524C23#100411yescharacteristic
525C23#100412yes—
526C23#100412yes—
527C23#100412yes—
528C23D3×#4112yes—
529C23D3×#4112yes—
530C23C23×Dic311yescharacteristic
531D4—416no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C3—44no—
581C2×C3—82no—
582C2×C3#231211yescharacteristic
583C2×C3#233312yes—
584C2×C3#235312yes—
585C2×C3#237411yescharacteristic
586C2×C3#237511yescharacteristic
587C2×C3#274912yes—
588C2×C3#275312yes—
589C2×C3#277812yes—
590C2×C3#279712yes—
591C22—28no—
592C22—24no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22#549412yes—
610C22#549412yes—
611C22#554012yes—
612C22#554012yes—
613C22#555612yes—
614C22#555612yes—
615C22#559411yescharacteristic
616C22#559511yescharacteristic
617C22#559512yes—
618C22#561312yes—
619C22#561312yes—
620C22#561512yes—
621C22#561512yes—
622C22#565911yescharacteristic
623C22#566112yes—
624C22#566411yescharacteristic
625C22C2×#92611yescharacteristic
626C22#579811yescharacteristic
627C22#579812yes—
628C22#579811yescharacteristic
629C22Dic3×#4112yes—
630C4—44no—
631C4—42no—
632C4—44no—
633C4—124no—
634C4—124no—
635C4—242no—
636C4—242no—
637C3#1451611yescharacteristic, complemented, Sylow
638C2—44no—
639C2—82no—
640C2#7484712yes—
641C2#7485512yes—
642C2#7486512yes—
643C2#7493112yes—
644C2#8209311yescharacteristic
645C2#8233612yes—
646C2#8258011yescharacteristic
647C2#8263512yes—
648C2#8283911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#386C2
socle, cosocleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#386D3
hypocenterC3#14516
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
4. #72127C2semidirect
5. #71548C2semidirect
6. #71545C2semidirect
7. #71545C2semidirect
8. #69416C2semidirect
9. #69401C2semidirect
21. C3×#386C4semidirect
103. #386Dic3semidirect
637. C3#14516semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3862
#721312
#721272
#715482
#715452
#694162
#694012
#145163