Finite Group Databasebeta

← #10728713

Subgroups — #10728713

Order 768.

1742 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic, complemented
2C2×C3×#397C211yesmaximal, characteristic
3C2×#4947C211yesmaximal, characteristic
4C2×#4944C211yesmaximal, characteristic
5#71569C211yesmaximal, characteristic, complemented
6#71547C212yesmaximal
7#71545C212yesmaximal
8#69424C211yesmaximal, characteristic, complemented
9#69414C212yesmaximal, complemented
10#69406C212yesmaximal, complemented
11#69402C211yesmaximal, characteristic, complemented
12#14882—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C2×#887—24no—
25C2×#887—24no—
26C2×#887—24no—
27C3×#397—24no—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4947—24no—
41#4944—24no—
42C22×C4×Q8—31no—
43C2×#397D311yescharacteristic
44C2×#393—31no—
45C2×#388—31no—
46#1446—31no—
47#1445—31no—
48#1442—32no—
49#1439—32no—
50#1437—32no—
51#1436—32no—
52#1434—31no—
53C25×C3C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2311yescharacteristic
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6D414yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D414yes—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—42no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81C2×#217—22no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—24no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—24no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—24no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3D414yes—
97#887—232no—
98C3×#95—24no—
99C3×#95—22no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—42no—
109C3⋊#95—42no—
110C23×Q8—31no—
111C2×C4×Q8—64no—
112C2×C4×Q8—64no—
113C2×C4×Q8—64no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C22⋊C4—32no—
118C22×C22⋊C4—31no—
119C22×C22⋊C4—31no—
120C22×C22⋊C4C2×D311yescharacteristic
121C22×C22⋊C4C2×D312yes—
122C22×C42—31no—
123C22×C42—32no—
124#397—24no—
125#393—64no—
126#388—64no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—21no—
147C24×C3—21no—
148C24×C3—22no—
149C24×C3#4112yes—
150C24×C3C2411yescharacteristic
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D412yes—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×D414yes—
206C22×Dic3C2×D412yes—
207C22×Dic3#4112yes—
208C22×Dic3#4114yes—
209C22×Dic3#4114yes—
210C22×Dic3#4114yes—
211C2×Dic6—28no—
212C2×Dic6—28no—
213C2×Dic6—216no—
214C2×Dic6—28no—
215C2×Dic6—216no—
216C3×C4⋊C4—28no—
217C3×C22⋊C4—48no—
218C3×C22⋊C4—44no—
219#217—28no—
220#217—48no—
221#217—44no—
222Dic3⋊C4—216no—
223Dic3⋊C4—44no—
224C4×Dic3—216no—
225C4×Dic3—28no—
226C25C22×D311yescharacteristic
227C22×Q8—34no—
228C22×Q8—64no—
229C22×Q8—64no—
230C22×Q8—62no—
231C22×Q8—62no—
232C23×C4—32no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4—31no—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D312yes—
240C23×C4C22×D312yes—
241C23×C4C22×D311yescharacteristic
242C4×Q8—632no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4#6714yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C42—34no—
266C2×C42—64no—
267C2×C42—64no—
268C2×C42—62no—
269C2×C42—64no—
270C2×C42—62no—
271#95—24no—
272#95—22no—
273#95—64no—
274#95—64no—
275#95—64no—
276#95—64no—
277#95—62no—
278#95—64no—
279#95—62no—
280#95—62no—
281#95—122no—
282#95—122no—
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—24no—
288C23×C3—41no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3C22×D411yescharacteristic
307C23×C3C22×D411yescharacteristic
308C23×C3C22×D411yescharacteristic
309C23×C3C2×#4111yescharacteristic
310C23×C3C2×#4112yes—
311C23×C3C2×#4112yes—
312C23×C3C2×#4111yescharacteristic
313C23×C3C2×#4112yes—
314C23×C3H(5,𝔽2)11yescharacteristic
315C23×C32−1+412yes—
316C23×C32−1+411yescharacteristic
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4#12014yes—
341C2×Dic3—22no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—22no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3#12318yes—
376C2×Dic3#12414yes—
377Dic6—232no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—21no—
387C24—21no—
388C24#23712yes—
389C24C23×D311yescharacteristic
390C2×Q8—616no—
391C2×Q8—68no—
392C2×Q8—68no—
393C2×Q8—616no—
394C2×Q8—68no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—34no—
416C22×C4—32no—
417C22×C4—34no—
418C22×C4—34no—
419C22×C4—34no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—61no—
451C22×C4D3×D412yes—
452C22×C4D3×D412yes—
453C22×C4C2×#6714yes—
454C22×C4C2×#6712yes—
455C4⋊C4—28no—
456C4⋊C4—616no—
457C4⋊C4—124no—
458C22⋊C4—48no—
459C22⋊C4—44no—
460C22⋊C4—68no—
461C22⋊C4—128no—
462C22⋊C4—124no—
463C42—616no—
464C42—68no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3C2×#12011yescharacteristic
477C22×C3C2×#12312yes—
478C22×C3#52511yescharacteristic
479C22×C3#52511yescharacteristic
480C22×C3#52512yes—
481C22×C3#52511yescharacteristic
482C22×C3C2×#12411yescharacteristic
483C22×C3#52912yes—
484C22×C3#52912yes—
485C22×C3#53611yescharacteristic
486C22×C3#53711yescharacteristic
487C22×C3#54112yes—
488C22×C3#54212yes—
489C22×C3#54611yescharacteristic
490C22×C3#54711yescharacteristic
491C22×C3#54812yes—
492C22×C3#54812yes—
493C22×C3#54811yescharacteristic
494C22×C3#55011yescharacteristic
495C22×C3#55212yes—
496C22×C3#55612yes—
497C22×C3#55612yes—
498C22×C3#55912yes—
499C3×C4—24no—
500C3×C4—44no—
501C3×C4—81no—
502C3×C4—82no—
503Dic3—28no—
504Dic3—24no—
505Dic3—42no—
506Dic3—42no—
507Dic3—81no—
508Dic3—82no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23C2×D3×D411yescharacteristic
533C23C2×D3×D411yescharacteristic
534C23C2×#23711yescharacteristic
535C23#100411yescharacteristic
536C23D3×#4112yes—
537C23D3×#4112yes—
538C23D3×#4112yes—
539C23D3×#4111yescharacteristic
540C23#101012yes—
541C23#101011yescharacteristic
542C23C22×#6711yescharacteristic
543Q8—632no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—38no—
552C2×C4—34no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—122no—
592C2×C4—121no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—121no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—121no—
603C2×C4#93714yes—
604C2×C3—82no—
605C2×C3#239811yescharacteristic
606C2×C3#240711yescharacteristic
607C2×C3#242112yes—
608C2×C3#243611yescharacteristic
609C2×C3#244812yes—
610C2×C3#245812yes—
611C2×C3#246912yes—
612C2×C3#259211yescharacteristic
613C2×C3#259911yescharacteristic
614C2×C3#263711yescharacteristic
615C2×C3#284911yescharacteristic
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—81no—
622C22—82no—
623C22—82no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22#559811yescharacteristic
628C22#559911yescharacteristic
629C22#560611yescharacteristic
630C22#560811yescharacteristic
631C22#564912yes—
632C22#565012yes—
633C22D3×#12312yes—
634C22#565611yescharacteristic
635C22#565611yescharacteristic
636C22#566212yes—
637C22#566412yes—
638C22D3×#12411yescharacteristic
639C22#567511yescharacteristic
640C22#567712yes—
641C22#570112yes—
642C22#570212yes—
643C22#570212yes—
644C22#570612yes—
645C22C2×#93711yescharacteristic
646C22#580611yescharacteristic
647C22#582411yescharacteristic
648C22#582812yes—
649C22#583112yes—
650C4—24no—
651C4—44no—
652C4—64no—
653C4—68no—
654C4—81no—
655C4—82no—
656C4—122no—
657C4—122no—
658C4—242no—
659C4—241no—
660C3#1488211yescharacteristic, complemented, Sylow
661C2—82no—
662C2#7746011yescharacteristic
663C2#7746412yes—
664C2#7746912yes—
665C2#7752611yescharacteristic
666C2#7753311yescharacteristic
667C2#7754411yescharacteristic
668C2#8270111yescharacteristic
669C2#8285112yes—
670C2#8291211yescharacteristic
671C2#8296111yescharacteristic
672C2#8307812yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#397C2
prime radicalC25×C3C23
hypercenterC2×#397D3
hypocenterC3#14882
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C22×#887C2semidirect
5. #71569C2semidirect
8. #69424C2semidirect
9. #69414C2semidirect
10. #69406C2semidirect
11. #69402C2semidirect
660. C3#14882semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
C2×C3×#3972
C2×#49472
C2×#49442
#715692
#715472
#715452
#694242
#694142
#694062
#694022
#148823