Finite Group Databasebeta

← #10732189

Subgroups — #10732189

Order 768.

1842 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic, complemented
2C2×#4947C211yesmaximal, characteristic
3C2×#4945C211yesmaximal, characteristic
4C2×#4951C212yesmaximal
5C2×#4656C211yesmaximal, characteristic, complemented
6C3×#1439C211yesmaximal, characteristic
7#71566C211yesmaximal, characteristic, complemented
8#71547C212yesmaximal
9#69411C212yesmaximal, complemented
10#69403C211yesmaximal, characteristic, complemented
11#69401C212yesmaximal, complemented
12#15176—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C2×#887—24no—
26C2×#887—28no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38#4951—28no—
39#4947—24no—
40#4945—24no—
41#4656—24no—
42C22×C4×Q8—31no—
43C2×#395—31no—
44C2×#393—31no—
45C2×#388—32no—
46C2×#388—31no—
47#1441—32no—
48#1441—31no—
49#1439—32no—
50#1439D311yescharacteristic
51#1434—32no—
52#1434—31no—
53C25×C3C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6D414yes—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×#217—22no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—24no—
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3D414yes—
98C2×C4×Dic3D418yes—
99#887—232no—
100C3×#95—22no—
101C3×#95—24no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C23×Q8—31no—
112C2×C4×Q8—64no—
113C2×C4×Q8—68no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C22⋊C4—32no—
119C22×C22⋊C4—31no—
120C22×C22⋊C4—31no—
121C22×C22⋊C4C2×D311yescharacteristic
122C22×C22⋊C4C2×D312yes—
123C22×C42—32no—
124C22×C42—31no—
125#395—64no—
126#393—64no—
127#388—64no—
128#388—68no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—31no—
137C2×#95—32no—
138C2×#95C2×D311yescharacteristic
139C2×#95C2×D312yes—
140C24×C3—21no—
141C24×C3—22no—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3—22no—
145C24×C3—21no—
146C24×C3—22no—
147C24×C3—21no—
148C24×C3—21no—
149C24×C3—22no—
150C24×C3#4112yes—
151C24×C3C2411yescharacteristic
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—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4114yes—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—22no—
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—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×D414yes—
208C22×Dic3C2×D414yes—
209C22×Dic3C2×D414yes—
210C22×Dic3#4112yes—
211C22×Dic3#4114yes—
212C2×Dic6—28no—
213C2×Dic6—216no—
214C2×Dic6—28no—
215C2×Dic6—28no—
216C2×Dic6—216no—
217C3×C4⋊C4—28no—
218C3×C22⋊C4—216no—
219C3×C22⋊C4—44no—
220#217—28no—
221#217—44no—
222#217—48no—
223C4⋊Dic3—48no—
224Dic3⋊C4—216no—
225Dic3⋊C4—44no—
226C4×Dic3—216no—
227C4×Dic3—28no—
228C25C22×D311yescharacteristic
229C22×Q8—34no—
230C22×Q8—64no—
231C22×Q8—64no—
232C22×Q8—62no—
233C22×Q8—62no—
234C23×C4—32no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4—32no—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D312yes—
242C23×C4C22×D312yes—
243C23×C4C22×D311yescharacteristic
244C4×Q8—632no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C42—38no—
271C2×C42—34no—
272C2×C42—64no—
273C2×C42—64no—
274C2×C42—64no—
275#95—22no—
276#95—24no—
277#95—64no—
278#95—64no—
279#95—62no—
280#95—62no—
281#95—64no—
282#95—64no—
283#95—64no—
284#95—62no—
285#95—64no—
286C23×C3—22no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—24no—
290C23×C3—22no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—42no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3C22×D411yescharacteristic
310C23×C3C22×D412yes—
311C23×C3C22×D411yescharacteristic
312C23×C3C2×#4111yescharacteristic
313C23×C3C2×#4111yescharacteristic
314C23×C3C2×#4112yes—
315C23×C3C2×#4112yes—
316C23×C3C2×#4111yescharacteristic
317C23×C32−1+411yescharacteristic
318C23×C32−1+411yescharacteristic
319C23×C32−1+412yes—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4#12114yes—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—41no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—41no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3#12014yes—
379C2×Dic3#12118yes—
380Dic6—232no—
381C24—21no—
382C24—22no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24—22no—
390C24—21no—
391C24#23712yes—
392C24C23×D311yescharacteristic
393C2×Q8—616no—
394C2×Q8—68no—
395C2×Q8—68no—
396C2×Q8—616no—
397C2×Q8—68no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—34no—
418C22×C4—32no—
419C22×C4—34no—
420C22×C4—34no—
421C22×C4—34no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4#23512yes—
454C22×C4D3×D414yes—
455C22×C4D3×D412yes—
456C22×C4D3×D412yes—
457C22×C4#23714yes—
458C4⋊C4—28no—
459C4⋊C4—616no—
460C4⋊C4—128no—
461C4⋊C4—124no—
462C22⋊C4—216no—
463C22⋊C4—44no—
464C22⋊C4—68no—
465C22⋊C4—124no—
466C22⋊C4—128no—
467C42—68no—
468C42—616no—
469C22×C3—44no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—44no—
473C22×C3—42no—
474C22×C3—81no—
475C22×C3—82no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—82no—
479C22×C3—81no—
480C22×C3C2×#12011yescharacteristic
481C22×C3C2×#12111yescharacteristic
482C22×C3C2×#12112yes—
483C22×C3#52511yescharacteristic
484C22×C3#52512yes—
485C22×C3#52512yes—
486C22×C3#52511yescharacteristic
487C22×C3#53211yescharacteristic
488C22×C3#53212yes—
489C22×C3#53611yescharacteristic
490C22×C3#53612yes—
491C22×C3#53612yes—
492C22×C3#53611yescharacteristic
493C22×C3#54811yescharacteristic
494C22×C3#54811yescharacteristic
495C22×C3#54812yes—
496C22×C3#55212yes—
497C22×C3#55212yes—
498C22×C3#55211yescharacteristic
499C22×C3#55211yescharacteristic
500C22×C3#55612yes—
501C22×C3#55612yes—
502C22×C3#55612yes—
503C3×C4—24no—
504C3×C4—44no—
505C3×C4—44no—
506C3×C4—81no—
507Dic3—28no—
508Dic3—24no—
509Dic3—42no—
510Dic3—42no—
511Dic3—81no—
512Dic3—82no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23C2×#23511yescharacteristic
537C23C2×D3×D411yescharacteristic
538C23C2×D3×D411yescharacteristic
539C23C2×D3×D412yes—
540C23C2×#23712yes—
541C23C2×#23711yescharacteristic
542C23#100711yescharacteristic
543C23D3×#4111yescharacteristic
544C23D3×#4112yes—
545C23#101011yescharacteristic
546C23#101012yes—
547Q8—632no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—34no—
559C2×C4—38no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—122no—
599C2×C4—121no—
600C2×C4—122no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4#88414yes—
608C2×C3—82no—
609C2×C3#238811yescharacteristic
610C2×C3#253612yes—
611C2×C3#253611yescharacteristic
612C2×C3#254512yes—
613C2×C3#254511yescharacteristic
614C2×C3#259912yes—
615C2×C3#259911yescharacteristic
616C2×C3#260412yes—
617C2×C3#260411yescharacteristic
618C2×C3#261411yescharacteristic
619C2×C3#284711yescharacteristic
620C22—44no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—82no—
626C22—81no—
627C22—81no—
628C22—82no—
629C22—81no—
630C22—81no—
631C22C2×#88411yescharacteristic
632C22#551311yescharacteristic
633C22#552911yescharacteristic
634C22#554212yes—
635C22#554712yes—
636C22#558612yes—
637C22D3×#12011yescharacteristic
638C22#559612yes—
639C22#559812yes—
640C22#559811yescharacteristic
641C22D3×#12112yes—
642C22#560611yescharacteristic
643C22#562812yes—
644C22#562811yescharacteristic
645C22#563311yescharacteristic
646C22#564112yes—
647C22#565012yes—
648C22#565012yes—
649C22#565611yescharacteristic
650C22#567511yescharacteristic
651C22#567512yes—
652C22#570212yes—
653C22#571611yescharacteristic
654C4—24no—
655C4—44no—
656C4—44no—
657C4—64no—
658C4—68no—
659C4—81no—
660C4—122no—
661C4—122no—
662C4—242no—
663C4—241no—
664C3#1517611yescharacteristic, complemented, Sylow
665C2—82no—
666C2#7393411yescharacteristic
667C2#7449712yes—
668C2#7493511yescharacteristic
669C2#7564812yes—
670C2#7635111yescharacteristic
671C2#7666511yescharacteristic
672C2#7687011yescharacteristic
673C2#7694312yes—
674C2#7731512yes—
675C2#7753311yescharacteristic
676C2#7804511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1439C2
prime radicalC25×C3C23
hypercenter#1439D3
hypocenterC3#15176
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#887C2semidirect
5. C2×#4656C2semidirect
7. #71566C2semidirect
9. #69411C2semidirect
10. #69403C2semidirect
11. #69401C2semidirect
664. C3#15176semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
C2×#49472
C2×#49452
C2×#49512
C2×#46562
C3×#14392
#715662
#715472
#694112
#694032
#694012
#151763