Finite Group Databasebeta

← #10750418

Subgroups — #10750418

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×#944C211yesmaximal, characteristic, complemented
2C2×#5231C211yesmaximal, characteristic
3C2×#4675C212yesmaximal
4C2×#4673C211yesmaximal, characteristic
5C2×#4669C211yesmaximal, characteristic
6C3×#1446C211yesmaximal, characteristic, complemented
7#69428C212yesmaximal
8#69424C211yesmaximal, characteristic, complemented
9#69421C212yesmaximal, complemented
10#69409C212yesmaximal, complemented
11#69403C211yesmaximal, characteristic, complemented
12#16506—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×#944—28no—
16C2×#944—24no—
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30#5231—24no—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39#4675—28no—
40#4673—24no—
41#4669—24no—
42C22×C4⋊Q8—31no—
43C2×#397—32no—
44C2×#393—31no—
45C2×#393—31no—
46C2×#388—31no—
47#1446—31no—
48#1446D311yescharacteristic, complemented
49#1445—32no—
50#1441—32no—
51#1439—32no—
52#1434—31no—
53C24×D3C2311yescharacteristic
54C22×C3×Q8—24no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—42no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C22×Dic6—22no—
70C22×Dic6—24no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—42no—
74#944—232no—
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—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—42no—
86C2×(C2×D3)⋊C4—42no—
87C2×(C2×D3)⋊C4D414yes—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4D418yes—
96C2×C4×Dic3—24no—
97C2×C4×Dic3D414yes—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—24no—
101C3×#95—42no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—42no—
110C23×Q8—31no—
111C23×Q8C2×D311yescharacteristic
112C2×C4⋊Q8—64no—
113C2×C4⋊Q8—68no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—31no—
119C22×C22⋊C4—31no—
120C22×C22⋊C4—32no—
121C22×C22⋊C4—32no—
122C22×C42—31no—
123#397—68no—
124#393—64no—
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—32no—
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D312yes—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C24×C3C2411yescharacteristic
140C23×D3—21no—
141C23×D3—22no—
142C23×D3—21no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—21no—
146C23×D3—21no—
147C23×D3—22no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3C2×D412yes—
151C2×C3×Q8—28no—
152C2×C3×Q8—48no—
153C2×C3×Q8—48no—
154C2×C3×Q8—44no—
155C2×C3×Q8—44no—
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—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4#4112yes—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×D414yes—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D412yes—
212C22×Dic3C2×D412yes—
213C22×Dic3#4114yes—
214C2×Dic6—28no—
215C2×Dic6—48no—
216C2×Dic6—48no—
217C2×Dic6—44no—
218C2×Dic6—44no—
219(C2×D3)⋊C4—28no—
220(C2×D3)⋊C4—48no—
221(C2×D3)⋊C4—48no—
222(C2×D3)⋊C4—44no—
223(C2×D3)⋊C4—44no—
224Dic3⋊C4—216no—
225Dic3⋊C4—216no—
226Dic3⋊C4—44no—
227C4×Dic3—28no—
228C25—31no—
229C22×Q8—22no—
230C22×Q8—24no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—42no—
234C22×Q8—64no—
235C22×Q8—62no—
236C22×Q8—62no—
237C22×Q8—62no—
238C22×Q8—122no—
239C23×C4—31no—
240C23×C4—31no—
241C23×C4—32no—
242C23×C4—31no—
243C23×C4—32no—
244C23×C4C22×D311yescharacteristic
245C23×C4C22×D312yes—
246C23×C4C22×D311yescharacteristic
247C23×C4C22×D312yes—
248C23×C4C22×D311yescharacteristic
249C4⋊Q8—632no—
250C2×C4⋊C4—38no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—64no—
258C2×C22⋊C4—34no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—122no—
270C2×C22⋊C4—122no—
271C2×C42—34no—
272C2×C42—64no—
273#95—22no—
274#95—22no—
275#95—24no—
276#95—42no—
277#95—64no—
278#95—62no—
279#95—64no—
280#95—64no—
281#95—64no—
282#95—64no—
283#95—62no—
284#95—122no—
285C23×C3C22×D411yescharacteristic
286C23×C3C22×D412yes—
287C23×C3C22×D411yescharacteristic
288C23×C3C2×#4111yescharacteristic
289C23×C3C2×#4112yes—
290C23×C3H(5,𝔽2)12yes—
291C23×C32−1+411yescharacteristic
292C23×C32−1+412yes—
293C23×C32−1+411yescharacteristic
294C23×C32−1+411yescharacteristic
295C23×C32−1+411yescharacteristic
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—24no—
300C22×D3—24no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—42no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—42no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C3×Q8—416no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
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—42no—
348C2×C3×C4#12114yes—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—42no—
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—42no—
376C2×Dic3#12014yes—
377C2×Dic3#12414yes—
378Dic6—416no—
379C24—32no—
380C24—61no—
381C24—61no—
382C24—62no—
383C24—62no—
384C24—61no—
385C24—61no—
386C24—62no—
387C24—62no—
388C24—61no—
389C24—61no—
390C24C23×D311yescharacteristic
391C2×Q8—28no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—68no—
397C2×Q8—128no—
398C2×Q8—128no—
399C2×Q8—124no—
400C2×Q8—124no—
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—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—32no—
431C22×C4—34no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4#23512yes—
456C22×C4D3×D414yes—
457C22×C4D3×D412yes—
458C22×C4D3×D412yes—
459C4⋊C4—616no—
460C4⋊C4—616no—
461C4⋊C4—124no—
462C22⋊C4—68no—
463C22⋊C4—128no—
464C22⋊C4—124no—
465C22⋊C4—124no—
466C22⋊C4—128no—
467C42—68no—
468C22×C3C2×#12011yescharacteristic
469C22×C3C2×#12111yescharacteristic
470C22×C3C2×#12411yescharacteristic
471C22×C3#53511yescharacteristic
472C22×C3#53611yescharacteristic
473C22×C3#53611yescharacteristic
474C22×C3#53612yes—
475C22×C3#53611yescharacteristic
476C22×C3#53611yescharacteristic
477C22×C3#53611yescharacteristic
478C22×C3#53612yes—
479C22×C3#54111yescharacteristic
480C22×C3#54111yescharacteristic
481C22×C3#54212yes—
482C22×C3#54212yes—
483C22×C3D4212yes—
484C22×C3#54612yes—
485C22×C3#54712yes—
486C22×C3#54812yes—
487C22×C3#54812yes—
488C22×C3#55612yes—
489C22×C3#56312yes—
490C22×C3#56312yes—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—44no—
494C2×D3—44no—
495C2×D3—42no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—82no—
501C2×D3—82no—
502C3×C4—24no—
503C3×C4—44no—
504C3×C4—81no—
505C3×C4—82no—
506C3×C4—81no—
507Dic3—24no—
508Dic3—24no—
509Dic3—44no—
510Dic3—42no—
511Dic3—82no—
512C23—62no—
513C23—62no—
514C23—62no—
515C23—64no—
516C23—64no—
517C23—122no—
518C23—122no—
519C23—122no—
520C23—122no—
521C23—121no—
522C23—122no—
523C23—122no—
524C23—121no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—122no—
533C23—121no—
534C23—121no—
535C23C2×#23511yescharacteristic
536C23C2×D3×D412yes—
537C23C2×D3×D411yescharacteristic
538C23C2×D3×D411yescharacteristic
539C23#100412yes—
540C23#100711yescharacteristic
541C23#100711yescharacteristic
542C23#100712yes—
543C23D3×#4112yes—
544C23#101011yescharacteristic
545C23#101011yescharacteristic
546Q8—416no—
547Q8—1216no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—34no—
556C2×C4—34no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4#88414yes—
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—124no—
619C22—122no—
620C22—122no—
621C22—124no—
622C22—122no—
623C22—241no—
624C22—242no—
625C22—242no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22C2×#88411yescharacteristic
630C22#549012yes—
631C22#556812yes—
632C22#558311yescharacteristic
633C22#558311yescharacteristic
634C22D3×#12011yescharacteristic
635C22#559012yes—
636C22#559211yescharacteristic
637C22#562811yescharacteristic
638C22#562812yes—
639C22#562811yescharacteristic
640C22#563112yes—
641C22#563312yes—
642C22#564012yes—
643C22#564612yes—
644C22#566912yes—
645C22D3×#12411yescharacteristic
646C22#567511yescharacteristic
647C22#567712yes—
648C22#572412yes—
649C22#572411yescharacteristic
650C22#572411yescharacteristic
651C22#573512yes—
652C4—24no—
653C4—44no—
654C4—64no—
655C4—64no—
656C4—81no—
657C4—82no—
658C4—81no—
659C4—124no—
660C4—122no—
661C4—242no—
662C3#1650611yescharacteristic, complemented, Sylow
663C2—242no—
664C2#7382411yescharacteristic
665C2#7653611yescharacteristic
666C2#7656912yes—
667C2#7657012yes—
668C2#7664111yescharacteristic
669C2#7678111yescharacteristic
670C2#7681511yescharacteristic
671C2#7687412yes—
672C2#7725611yescharacteristic
673C2#7726012yes—
674C2#7727611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1446C2
prime radicalC24×D3C23
hypercenter#1446D3
hypocenterC3#16506
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
1. C22×#944C2semidirect
6. C3×#1446C2semidirect
8. #69424C2semidirect
9. #69421C2semidirect
10. #69409C2semidirect
11. #69403C2semidirect
48. #1446D3semidirect
662. C3#16506semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C2×#52312
C2×#46752
C2×#46732
C2×#46692
C3×#14462
#694282
#694242
#694212
#694092
#694032
#165063