Finite Group Databasebeta

← #10742729

Subgroups — #10742729

Order 768.

2462 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#878C212yesmaximal, complemented
2C2×#5220C212yesmaximal
3C2×#4950C212yesmaximal, complemented
4C3×#1438C211yesmaximal, characteristic
5#71547C212yesmaximal, complemented
6#71547C212yesmaximal, complemented
7#69424C211yesmaximal, characteristic, complemented
8#69420C211yesmaximal, characteristic, complemented
9#69411C212yesmaximal, complemented
10#15951—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×Dic3C2212yes—
13C23×Dic6C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2212yes—
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C2×#878—28no—
24C2×#878—28no—
25C2×#878—28no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28#5220—28no—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36#4950—28no—
37C22×C22⋊Q8—32no—
38C2×#386—32no—
39C2×#385—32no—
40#1446—31no—
41#1444—31no—
42#1441—32no—
43#1439—32no—
44#1439—32no—
45#1438D311yescharacteristic
46C25×C3C2312yes—
47C22×C3×D4—24no—
48C22×C3×D4—24no—
49C22×C3×D4—41no—
50C22×C3×D4—41no—
51C22×C3×D4Q814yescomplemented
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—24no—
57C23×Dic3—24no—
58C23×Dic3—24no—
59C23×Dic3—24no—
60C23×Dic3—24no—
61C23×Dic3—42no—
62C23×Dic3—42no—
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C23×Dic3C2312yes—
67C22×Dic6—24no—
68C22×Dic6—24no—
69C22×Dic6—41no—
70C22×Dic6—41no—
71C22×Dic6D414yescomplemented
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—42no—
85C2×#217—42no—
86C2×#217D418yes—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—41no—
91C2×C4⋊Dic3D418yescomplemented
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98#878—264no—
99C3×#95—41no—
100C3×#95—41no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—42no—
107C3⋊#95—41no—
108C23×Q8—31no—
109C23×D4C2×D311yescharacteristic
110C24×C4—32no—
111C2×C22⋊Q8—68no—
112C2×C22⋊Q8—68no—
113C2×C22⋊Q8—68no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4—32no—
121C22×C22⋊C4C2×D312yes—
122C22×C22⋊C4C2×D312yes—
123#386—68no—
124#385—68no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95C2×D311yescharacteristic
133C2×#95C2×D311yescharacteristic
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3C2×D414yes—
147C24×C3C2×Q814yes—
148C24×C3#4114yes—
149C24×C3C2411yescharacteristic
150C2×C3×D4—216no—
151C2×C3×D4—44no—
152C2×C3×D4—44no—
153C2×C3×D4—48no—
154C2×C3×D4—44no—
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—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×Q812yes—
176C22×C3×C4#4112yes—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—216no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—48no—
202C22×Dic3—48no—
203C22×Dic3—48no—
204C22×Dic3—48no—
205C22×Dic3—48no—
206C22×Dic3—48no—
207C22×Dic3C2×D414yes—
208C22×Dic3C2×D414yes—
209C22×Dic3C2×D414yes—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D414yes—
212C2×Dic6—216no—
213C2×Dic6—44no—
214C2×Dic6—44no—
215C2×Dic6—44no—
216C2×Dic6—48no—
217C3×C22⋊C4—216no—
218C3×C22⋊C4—216no—
219#217—216no—
220#217—48no—
221#217—48no—
222C4⋊Dic3—216no—
223C4⋊Dic3—44no—
224Dic3⋊C4—216no—
225Dic3⋊C4—216no—
226C25C22×D312yes—
227C22×Q8—34no—
228C22×Q8—64no—
229C22×Q8—64no—
230C22×Q8—121no—
231C22×Q8—121no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—41no—
235C22×D4—41no—
236C22×D4Dic614yescomplemented
237C23×C4—32no—
238C23×C4—32no—
239C23×C4—32no—
240C23×C4—32no—
241C23×C4—64no—
242C23×C4—64no—
243C23×C4—64no—
244C23×C4—64no—
245C23×C4—64no—
246C23×C4—122no—
247C23×C4—122no—
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D312yes—
251C23×C4C22×D311yescharacteristic
252C22⋊Q8—664no—
253C2×C4⋊C4—38no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—121no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—38no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—122no—
278C2×C22⋊C4—122no—
279#95—41no—
280#95—41no—
281#95—64no—
282#95—64no—
283#95—64no—
284#95—64no—
285#95—62no—
286#95—122no—
287#95—121no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3C22⋊Q818yescomplemented
323C23×C3C22×D411yescharacteristic
324C23×C3C22×D412yes—
325C23×C3C22×D412yes—
326C23×C3C22×Q811yescharacteristic
327C23×C3C2×#4112yes—
328C23×C3C2×#4111yescharacteristic
329C23×C3H(5,𝔽2)11yescharacteristic
330C23×C3H(5,𝔽2)11yescharacteristic
331C23×C3H(5,𝔽2)11yescharacteristic
332C23×C32−1+411yescharacteristic
333C23×C32−1+412yes—
334C3×D4—416no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—432no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3#12018yes—
386Dic6—416no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24C2×Dic614yes—
400C24D3×D414yes—
401C24#23714yes—
402C24C23×D311yescharacteristic
403C2×Q8—616no—
404C2×Q8—124no—
405C2×Q8—124no—
406C2×Q8—124no—
407C2×Q8—128no—
408C2×D4—216no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—48no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—34no—
432C22×C4—34no—
433C22×C4—34no—
434C22×C4—34no—
435C22×C4—34no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—616no—
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—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—128no—
461C22×C4—128no—
462C22×C4—128no—
463C22×C4—128no—
464C22×C4—128no—
465C22×C4—128no—
466C22×C4C2×Dic612yes—
467C22×C4D3×D414yes—
468C22×C4D3×D412yes—
469C22×C4#23712yes—
470C4⋊C4—616no—
471C4⋊C4—616no—
472C4⋊C4—616no—
473C4⋊C4—124no—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—616no—
477C22⋊C4—128no—
478C22⋊C4—128no—
479C22×C3—28no—
480C22×C3—28no—
481C22×C3—28no—
482C22×C3—44no—
483C22×C3—44no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3C2×#12012yes—
495C22×C3C2×C22⋊Q812yes—
496C22×C3#52512yes—
497C22×C3#53411yescharacteristic
498C22×C3#53612yes—
499C22×C3#53712yes—
500C22×C3#53912yes—
501C22×C3#53911yescharacteristic
502C22×C3#54311yescharacteristic
503C22×C3D4211yescharacteristic
504C22×C3D4211yescharacteristic
505C22×C3D4212yes—
506C22×C3#54612yes—
507C22×C3#54712yes—
508C22×C3#54812yes—
509C22×C3#54812yes—
510C22×C3D4×Q811yescharacteristic
511C22×C3D4×Q812yes—
512C22×C3#55011yescharacteristic
513C22×C3#55111yescharacteristic
514C22×C3#55411yescharacteristic
515C22×C3#56211yescharacteristic
516C22×C3#56311yescharacteristic
517C3×C4—42no—
518C3×C4—44no—
519C3×C4—42no—
520C3×C4—42no—
521Dic3—28no—
522Dic3—44no—
523Dic3—44no—
524Dic3—82no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23#87818yescomplemented
560C23C22×Dic611yescharacteristic
561C23C2×D3×D412yes—
562C23C2×D3×D412yes—
563C23C2×D3×D411yescharacteristic
564C23C2×#23712yes—
565C23C2×#23711yescharacteristic
566C23#100411yescharacteristic
567C23#100411yescharacteristic
568C23#100911yescharacteristic
569C23#101011yescharacteristic
570C23#101012yes—
571Q8—1216no—
572D4—416no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—38no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—1232no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C3—48no—
625C2×C3#240311yescharacteristic
626C2×C3#240712yes—
627C2×C3#240912yes—
628C2×C3#243711yescharacteristic
629C2×C3#244912yes—
630C2×C3#246311yescharacteristic
631C2×C3#268011yescharacteristic
632C2×C3#268112yes—
633C2×C3#268311yescharacteristic
634C2×C3#272811yescharacteristic
635C2×C3#273211yescharacteristic
636C22—28no—
637C22—28no—
638C22—28no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22C2×#87812yes—
652C22#548311yescharacteristic
653C22D4×Dic612yes—
654C22D4×Dic611yescharacteristic
655C22#554311yescharacteristic
656C22#558612yes—
657C22#558711yescharacteristic
658C22D3×#12012yes—
659C22#559011yescharacteristic
660C22#559112yes—
661C22#559712yes—
662C22#559812yes—
663C22#559812yes—
664C22#560211yescharacteristic
665C22#560511yescharacteristic
666C22#560912yes—
667C22#561112yes—
668C22#564812yes—
669C22#566512yes—
670C22#567211yescharacteristic
671C22#568411yescharacteristic
672C22#571711yescharacteristic
673C22#571811yescharacteristic
674C4—42no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—68no—
679C4—124no—
680C4—124no—
681C4—242no—
682C3#1595111yescharacteristic, complemented, Sylow
683C2—48no—
684C2#7382811yescharacteristic
685C2#7468912yes—
686C2#7471012yes—
687C2#7638011yescharacteristic
688C2#7640111yescharacteristic
689C2#7666812yes—
690C2#7668111yescharacteristic
691C2#7675311yescharacteristic
692C2#7700411yescharacteristic
693C2#7700612yes—
694C2#7704311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1438C2
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#15951
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
1. C22×#878C2semidirect
3. C2×#4950C2semidirect
5. #71547C2semidirect
6. #71547C2semidirect
7. #69424C2semidirect
8. #69420C2semidirect
9. #69411C2semidirect
51. C22×C3×D4Q8semidirect
71. C22×Dic6D4semidirect
91. C2×C4⋊Dic3D4semidirect
236. C22×D4Dic6semidirect
322. C23×C3C22⋊Q8semidirect
559. C23#878semidirect
682. C3#15951semidirect

Maximal subgroups

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

subgroupindex
C22×#8782
C2×#52202
C2×#49502
C3×#14382
#715472
#694242
#694202
#694112
#159513