Finite Group Databasebeta

← #10750419

Subgroups — #10750419

Order 768.

1746 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic, complemented
2C2×#4990C211yesmaximal, characteristic
3C2×#4675C212yesmaximal
4C2×#4673C212yesmaximal
5C3×#1441C211yesmaximal, characteristic, complemented
6#71570C211yesmaximal, characteristic
7#71565C211yesmaximal, characteristic, complemented
8#69427C211yesmaximal, characteristic
9#69424C212yesmaximal, complemented
10#69421C212yesmaximal, complemented
11#69401C211yesmaximal, characteristic, complemented
12#16506—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C2×#888—28no—
25C2×#888—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4990—24no—
39#4675—28no—
40#4673—28no—
41C22×C4⋊Q8—31no—
42C2×#397—32no—
43C2×#393—32no—
44C2×#388—31no—
45#1446—32no—
46#1445—32no—
47#1441—31no—
48#1441D311yescharacteristic, complemented
49#1439—31no—
50#1439—31no—
51#1434—31no—
52C24×D3C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C22×Dic6—24no—
64C22×Dic6—24no—
65C22×Dic6—24no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6—42no—
69C22×Dic6—42no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—42no—
87C2×(C2×D3)⋊C4—42no—
88C2×(C2×D3)⋊C4D414yes—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4D418yes—
94C2×C4×Dic3—24no—
95C2×C4×Dic3D414yes—
96#888—232no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—22no—
100C3⋊#95—24no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C23×Q8—32no—
110C2×C4⋊Q8—64no—
111C2×C4⋊Q8—68no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C4⋊C4C2×D311yescharacteristic
116C22×C22⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—31no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4—32no—
121C22×C42—31no—
122#397—68no—
123#393—68no—
124#388—64no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—31no—
130C2×#95—31no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95C2×D312yes—
135C2×#95C2×D312yes—
136C2×#95C2×D311yescharacteristic
137C24×C3C2411yescharacteristic
138C23×D3—21no—
139C23×D3—22no—
140C23×D3—21no—
141C23×D3—21no—
142C23×D3—21no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—21no—
146C23×D3—22no—
147C23×D3—21no—
148C23×D3C2×D412yes—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D414yes—
204C22×Dic3C2×D414yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×D412yes—
208C22×Dic3#4112yes—
209C2×Dic6—216no—
210C2×Dic6—48no—
211C2×Dic6—48no—
212C2×Dic6—48no—
213C2×Dic6—48no—
214C2×Dic6—48no—
215C2×Dic6—48no—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—44no—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—48no—
220(C2×D3)⋊C4—48no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—44no—
223C4⋊Dic3—28no—
224Dic3⋊C4—216no—
225C4×Dic3—28no—
226C25—31no—
227C22×Q8—64no—
228C22×Q8—64no—
229C22×Q8—64no—
230C22×Q8—64no—
231C22×Q8—64no—
232C22×Q8—122no—
233C22×Q8—122no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D312yes—
241C23×C4C22×D312yes—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C4⋊Q8—632no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—38no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—62no—
255C2×C22⋊C4—34no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—122no—
268C2×C22⋊C4—122no—
269C2×C42—34no—
270C2×C42—64no—
271#95—24no—
272#95—24no—
273#95—22no—
274#95—62no—
275#95—62no—
276#95—64no—
277#95—64no—
278#95—64no—
279#95—62no—
280#95—64no—
281#95—122no—
282#95—122no—
283C23×C3C22×D411yescharacteristic
284C23×C3C22×D412yes—
285C23×C3C22×D411yescharacteristic
286C23×C3C2×#4111yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4111yescharacteristic
289C23×C3H(5,𝔽2)12yes—
290C23×C32−1+412yes—
291C23×C32−1+411yescharacteristic
292C23×C32−1+411yescharacteristic
293C23×C32−1+412yes—
294C22×D3—22no—
295C22×D3—24no—
296C22×D3—24no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—41no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—42no—
316C22×D3—42no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4#12414yes—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3#12014yes—
376C2×Dic3#12114yes—
377Dic6—432no—
378C24—32no—
379C24—61no—
380C24—61no—
381C24—62no—
382C24—61no—
383C24—61no—
384C24—62no—
385C24—61no—
386C24—62no—
387C24—61no—
388C24—62no—
389C24C23×D311yescharacteristic
390C2×Q8—616no—
391C2×Q8—128no—
392C2×Q8—128no—
393C2×Q8—128no—
394C2×Q8—128no—
395C2×Q8—128no—
396C2×Q8—128no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—32no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—34no—
427C22×C4—32no—
428C22×C4—34no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4#23512yes—
453C22×C4D3×D412yes—
454C22×C4D3×D414yes—
455C22×C4D3×D412yes—
456C22×C4#23712yes—
457C4⋊C4—28no—
458C4⋊C4—44no—
459C4⋊C4—616no—
460C4⋊C4—68no—
461C22⋊C4—68no—
462C22⋊C4—128no—
463C22⋊C4—124no—
464C22⋊C4—128no—
465C22⋊C4—124no—
466C42—68no—
467C22×C3C2×#12011yescharacteristic
468C22×C3C2×#12111yescharacteristic
469C22×C3C2×#12411yescharacteristic
470C22×C3#53511yescharacteristic
471C22×C3#53611yescharacteristic
472C22×C3#53611yescharacteristic
473C22×C3#53612yes—
474C22×C3#53611yescharacteristic
475C22×C3#53612yes—
476C22×C3#53612yes—
477C22×C3#54112yes—
478C22×C3#54212yes—
479C22×C3#54212yes—
480C22×C3D4212yes—
481C22×C3#54612yes—
482C22×C3#54711yescharacteristic
483C22×C3#54711yescharacteristic
484C22×C3#54812yes—
485C22×C3#54812yes—
486C22×C3#55611yescharacteristic
487C22×C3#55611yescharacteristic
488C22×C3#56312yes—
489C22×C3#56312yes—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—42no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—82no—
500C2×D3—82no—
501C3×C4—24no—
502C3×C4—44no—
503C3×C4—42no—
504C3×C4—82no—
505C3×C4—81no—
506Dic3—24no—
507Dic3—24no—
508Dic3—44no—
509Dic3—81no—
510Dic3—82no—
511C23—62no—
512C23—62no—
513C23—64no—
514C23—62no—
515C23—64no—
516C23—121no—
517C23—122no—
518C23—121no—
519C23—122no—
520C23—121no—
521C23—122no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—122no—
529C23—122no—
530C23—121no—
531C23—122no—
532C23—122no—
533C23—121no—
534C23C2×#23511yescharacteristic
535C23C2×D3×D411yescharacteristic
536C23C2×D3×D412yes—
537C23C2×D3×D411yescharacteristic
538C23C2×#23711yescharacteristic
539C23#100711yescharacteristic
540C23#100712yes—
541C23D3×#4111yescharacteristic
542C23#100912yes—
543C23#101012yes—
544C23#101011yescharacteristic
545Q8—1232no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—34no—
557C2×C4—34no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—64no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—122no—
596C2×C4—121no—
597C2×C4—122no—
598C2×C4—121no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4#88514yes—
606C2×C3#240811yescharacteristic
607C2×C3#243611yescharacteristic
608C2×C3#254711yescharacteristic
609C2×C3#261411yescharacteristic
610C2×C3#262111yescharacteristic
611C2×C3#264011yescharacteristic
612C2×C3#267412yes—
613C2×C3#267412yes—
614C2×C3#271612yes—
615C2×C3#271612yes—
616C2×C3#282111yescharacteristic
617D3—82no—
618C22—122no—
619C22—124no—
620C22—122no—
621C22—124no—
622C22—122no—
623C22—242no—
624C22—241no—
625C22—242no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22C2×#88511yescharacteristic
630C22#550711yescharacteristic
631C22#553311yescharacteristic
632C22#555212yes—
633C22#558212yes—
634C22D3×#12011yescharacteristic
635C22#559812yes—
636C22D3×#12111yescharacteristic
637C22#560912yes—
638C22#561111yescharacteristic
639C22#562812yes—
640C22#562812yes—
641C22#562811yescharacteristic
642C22#564111yescharacteristic
643C22#564312yes—
644C22#564811yescharacteristic
645C22#565412yes—
646C22#567411yescharacteristic
647C22#569612yes—
648C22#569912yes—
649C22#571212yes—
650C22#572412yes—
651C22#572411yescharacteristic
652C4—24no—
653C4—44no—
654C4—42no—
655C4—64no—
656C4—64no—
657C4—82no—
658C4—81no—
659C4—124no—
660C4—241no—
661C4—242no—
662C3#1650611yescharacteristic, complemented, Sylow
663C2—242no—
664C2#7393711yescharacteristic
665C2#7470111yescharacteristic
666C2#7516212yes—
667C2#7625812yes—
668C2#7664311yescharacteristic
669C2#7689511yescharacteristic
670C2#7725611yescharacteristic
671C2#7742712yes—
672C2#7781811yescharacteristic
673C2#7785712yes—
674C2#7831711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1441C2
prime radicalC24×D3C23
hypercenter#1441D3
hypocenterC3#16506
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
1. C22×#888C2semidirect
5. C3×#1441C2semidirect
7. #71565C2semidirect
9. #69424C2semidirect
10. #69421C2semidirect
11. #69401C2semidirect
48. #1441D3semidirect
662. C3#16506semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#49902
C2×#46752
C2×#46732
C3×#14412
#715702
#715652
#694272
#694242
#694212
#694012
#165063