Finite Group Databasebeta

← #10720097

Subgroups — #10720097

Order 768.

4046 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C22⋊C4C211yesmaximal, characteristic
2C23×#217C211yesmaximal, characteristic
3C2×#5247C211yesmaximal, characteristic
4C2×#5210C212yesmaximal, complemented
5C2×#4946C212yesmaximal, complemented
6#72127C212yesmaximal, complemented
7#71547C214yesmaximal, complemented
8#71544C212yesmaximal, complemented
9#14127—31nomaximal, Sylow
10C26×C3C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C24×Dic3C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C22×C3×C22⋊C4—28no—
16C22×C3×C22⋊C4—24no—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×#217—28no—
20C22×#217—24no—
21C22×#217C2212yes—
22C22×#217C2212yes—
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×Dic3⋊C4C2214yescomplemented
27#5247C418yescomplemented
28#5210—28no—
29C2×C3⋊#95C2214yescomplemented
30C2×C3⋊#95C2214yescomplemented
31C2×C3⋊#95C2214yescomplemented
32#4946—28no—
33C23×C22⋊C4—31no—
34C23×C22⋊C4D311yescharacteristic
35C2×#385—32no—
36C2×#385—32no—
37C2×#379—31no—
38#1439—34no—
39C22⋊#95—32no—
40C22⋊#95—32no—
41C25×C3—24no—
42C25×C3—24no—
43C25×C3—24no—
44C25×C3—24no—
45C25×C3—42no—
46C25×C3—42no—
47C25×C3C2×C414yes—
48C25×C3D414yes—
49C25×C3Q814yes—
50C25×C3C2311yescharacteristic
51C25×C3C2311yescharacteristic
52C25×C3C2311yescharacteristic
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—24no—
56C23×C3×C4—24no—
57C23×C3×C4—24no—
58C23×C3×C4—24no—
59C23×C3×C4—42no—
60C23×C3×C4—42no—
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C23×Dic3—22no—
64C23×Dic3—22no—
65C23×Dic3—24no—
66C23×Dic3—24no—
67C23×Dic3—24no—
68C23×Dic3—24no—
69C23×Dic3—42no—
70C23×Dic3—42no—
71C23×Dic3C2314yes—
72C23×Dic3C2312yes—
73C23×Dic3C2312yes—
74C2×C3×C22⋊C4—232no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4D418yes—
81C2×#217—232no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—24no—
86C2×#217—24no—
87C2×#217—24no—
88C2×#217—24no—
89C2×#217—24no—
90C2×#217—24no—
91C2×#217—42no—
92C2×#217—42no—
93C2×#217C2×C418yescomplemented
94C2×#217C2×C418yescomplemented
95C2×#217C2×C418yescomplemented
96C2×#217D418yes—
97C2×Dic3⋊C4—28no—
98C2×Dic3⋊C4—28no—
99C2×Dic3⋊C4—44no—
100C3⋊#95—28no—
101C3⋊#95—28no—
102C3⋊#95—44no—
103C26C2×D311yescharacteristic
104C24×C4—31no—
105C24×C4—31no—
106C24×C4C2×D311yescharacteristic
107C24×C4C2×D311yescharacteristic
108C22×C4⋊C4—34no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—32no—
112C22×C22⋊C4—32no—
113C22×C22⋊C4—32no—
114C22×C22⋊C4—32no—
115C22×C22⋊C4—32no—
116C22×C22⋊C4—68no—
117C22×C22⋊C4—64no—
118C22×C22⋊C4C2×D312yes—
119C22×C22⋊C4C2×D312yes—
120#385—68no—
121#385—68no—
122#379—38no—
123C2×#95—34no—
124C2×#95—34no—
125C2×#95—34no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—216no—
135C24×C3—216no—
136C24×C3—28no—
137C24×C3—28no—
138C24×C3—216no—
139C24×C3—216no—
140C24×C3—28no—
141C24×C3—28no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—48no—
149C24×C3—44no—
150C24×C3—44no—
151C24×C3—48no—
152C24×C3—416no—
153C24×C3—416no—
154C24×C3—416no—
155C24×C3—416no—
156C24×C3C4⋊C4116yescomplemented
157C24×C3C22×C412yes—
158C24×C3C22×C412yes—
159C24×C3C22×C412yes—
160C24×C3C2×D414yes—
161C24×C3C2×D412yes—
162C24×C3C2×D412yes—
163C24×C3C2×D412yes—
164C24×C3C2×D414yes—
165C24×C3C2×Q812yes—
166C24×C3C2×Q812yes—
167C24×C3C2×Q812yes—
168C24×C3#4114yes—
169C24×C3#4114yes—
170C24×C3C2411yescharacteristic
171C22×C3×C4—28no—
172C22×C3×C4—28no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—416no—
182C22×C3×C4—416no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4—44no—
186C22×C3×C4—44no—
187C22×C3×C4C2×D414yes—
188C22×C3×C4C2×D414yes—
189C22×Dic3—28no—
190C22×Dic3—28no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—416no—
206C22×Dic3—416no—
207C22×Dic3—44no—
208C22×Dic3—44no—
209C22×Dic3—44no—
210C22×Dic3—44no—
211C22×Dic3C22×C418yes—
212C22×Dic3C2×D414yes—
213C22×Dic3C2×D414yes—
214C3×C22⋊C4—216no—
215C3×C22⋊C4—216no—
216#217—216no—
217#217—216no—
218#217—216no—
219#217—216no—
220#217—48no—
221Dic3⋊C4—416no—
222C25—24no—
223C25—24no—
224C25—24no—
225C25—24no—
226C25—42no—
227C25—42no—
228C25Dic614yes—
229C25C4×D314yes—
230C25#6714yes—
231C25C22×D311yescharacteristic
232C25C22×D311yescharacteristic
233C25C22×D311yescharacteristic
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—34no—
241C23×C4—32no—
242C23×C4—32no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—64no—
246C23×C4—64no—
247C23×C4—62no—
248C23×C4—62no—
249C23×C4—64no—
250C23×C4—64no—
251C23×C4—122no—
252C23×C4—122no—
253C23×C4C22×D312yes—
254C23×C4C22×D312yes—
255C2×C4⋊C4—68no—
256C2×C4⋊C4—68no—
257C2×C4⋊C4—124no—
258C2×C22⋊C4—232no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—38no—
265C2×C22⋊C4—38no—
266C2×C22⋊C4—38no—
267C2×C22⋊C4—38no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—632no—
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—64no—
278C2×C22⋊C4—122no—
279C2×C22⋊C4—122no—
280C2×C22⋊C4#6718yes—
281#95—68no—
282#95—68no—
283#95—124no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—28no—
301C23×C3—28no—
302C23×C3—28no—
303C23×C3—28no—
304C23×C3—28no—
305C23×C3—28no—
306C23×C3—28no—
307C23×C3—28no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—24no—
320C23×C3—232no—
321C23×C3—232no—
322C23×C3—416no—
323C23×C3—464no—
324C23×C3—416no—
325C23×C3—416no—
326C23×C3—416no—
327C23×C3—464no—
328C23×C3—416no—
329C23×C3—44no—
330C23×C3—44no—
331C23×C3—44no—
332C23×C3—44no—
333C23×C3C2×C4⋊C414yes—
334C23×C3C2×C4⋊C414yes—
335C23×C3C2×C4⋊C414yes—
336C23×C3C4×D418yes—
337C23×C3C4×D418yes—
338C23×C3#12014yes—
339C23×C3#12014yes—
340C23×C3C22⋊Q818yes—
341C23×C3C22⋊Q818yes—
342C23×C3#12314yes—
343C23×C3#12314yes—
344C23×C3C23×C411yescharacteristic
345C23×C3C22×D412yes—
346C23×C3C22×D411yescharacteristic
347C23×C3C22×D412yes—
348C23×C3C22×Q811yescharacteristic
349C23×C3C2×#4112yes—
350C23×C3C2×#4112yes—
351C23×C3H(5,𝔽2)12yes—
352C23×C3H(5,𝔽2)12yes—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—28no—
356C2×C3×C4—28no—
357C2×C3×C4—416no—
358C2×C3×C4—416no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—44no—
364C2×C3×C4—44no—
365C2×Dic3—28no—
366C2×Dic3—28no—
367C2×Dic3—28no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—28no—
371C2×Dic3—28no—
372C2×Dic3—416no—
373C2×Dic3—416no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—44no—
379C2×Dic3—44no—
380C2×Dic3—44no—
381C2×Dic3—44no—
382C2×Dic3—44no—
383C2×Dic3—44no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—216no—
393C24—216no—
394C24—216no—
395C24—216no—
396C24—28no—
397C24—28no—
398C24—28no—
399C24—28no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—48no—
407C24—48no—
408C24—44no—
409C24—44no—
410C24—416no—
411C24—416no—
412C24—416no—
413C24—416no—
414C24Dic3⋊C4116yescomplemented
415C24C2×Dic612yes—
416C24C2×Dic612yes—
417C24C2×Dic612yes—
418C24C2×C4×D312yes—
419C24C2×C4×D312yes—
420C24C2×C4×D312yes—
421C24#23514yes—
422C24D3×D414yes—
423C24#23714yes—
424C24C2×#6712yes—
425C24C2×#6712yes—
426C24C2×#6712yes—
427C24C2×#6714yes—
428C24C23×D311yescharacteristic
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—38no—
440C22×C4—34no—
441C22×C4—34no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—68no—
449C22×C4—68no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—64no—
456C22×C4—64no—
457C22×C4—64no—
458C22×C4—64no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—1216no—
465C22×C4—1216no—
466C22×C4—124no—
467C22×C4—124no—
468C22×C4—124no—
469C22×C4—124no—
470C22×C4C2×#6714yes—
471C22×C4C2×#6714yes—
472C4⋊C4—1216no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—616no—
476C22⋊C4—616no—
477C22⋊C4—616no—
478C22⋊C4—616no—
479C22⋊C4—128no—
480C22×C3—24no—
481C22×C3—24no—
482C22×C3—216no—
483C22×C3—216no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—24no—
487C22×C3—24no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—216no—
491C22×C3—216no—
492C22×C3—24no—
493C22×C3—24no—
494C22×C3—24no—
495C22×C3—24no—
496C22×C3—24no—
497C22×C3—24no—
498C22×C3—24no—
499C22×C3—464no—
500C22×C3—48no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—48no—
505C22×C3—42no—
506C22×C3#38518yescomplemented
507C22×C3#38518yescomplemented
508C22×C3C22×C4⋊C411yescharacteristic
509C22×C3C2×C4×D412yes—
510C22×C3C2×C4×D412yes—
511C22×C3#51812yes—
512C22×C3C2×#12011yescharacteristic
513C22×C3C2×#12011yescharacteristic
514C22×C3C2×C22⋊Q812yes—
515C22×C3C2×C22⋊Q812yes—
516C22×C3C2×#12311yescharacteristic
517C22×C3C2×#12311yescharacteristic
518C22×C3#53411yescharacteristic
519C22×C3#53411yescharacteristic
520C22×C3#54312yes—
521C22×C3D4214yes—
522C22×C3#54614yes—
523C22×C3#54614yes—
524C22×C3#55114yes—
525C3×C4—44no—
526C3×C4—44no—
527Dic3—48no—
528Dic3—44no—
529Dic3—44no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—28no—
546C23—28no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—28no—
552C23—28no—
553C23—22no—
554C23—232no—
555C23—232no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—416no—
569C23—416no—
570C23—416no—
571C23—464no—
572C23—464no—
573C23—416no—
574C23—416no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23#87818yes—
580C23#67⋊C418yes—
581C23C2×Dic3⋊C414yes—
582C23C2×Dic3⋊C414yes—
583C23C2×Dic3⋊C414yes—
584C23#92418yes—
585C23C4×#6718yes—
586C23#92914yes—
587C23#93514yes—
588C23#93714yes—
589C23#95314yes—
590C23C22×Dic611yescharacteristic
591C23C22×C4×D311yescharacteristic
592C23C2×#23512yes—
593C23C2×D3×D412yes—
594C23C2×#23712yes—
595C23#100412yes—
596C23#100412yes—
597C23C22×#6711yescharacteristic
598C23C22×#6712yes—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—68no—
612C2×C4—68no—
613C2×C4—68no—
614C2×C4—68no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—68no—
618C2×C4—1216no—
619C2×C4—1216no—
620C2×C4—124no—
621C2×C4—124no—
622C2×C4—124no—
623C2×C4—122no—
624C2×C4—124no—
625C2×C4—124no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—124no—
629C2×C4—122no—
630C2×C3—28no—
631C2×C3—28no—
632C2×C3—44no—
633C2×C3C2×#38511yescharacteristic
634C2×C3C2×#38511yescharacteristic
635C2×C3#231511yescharacteristic
636C2×C3#235814yes—
637C2×C3#240311yescharacteristic
638C2×C3#240511yescharacteristic
639C2×C3#240511yescharacteristic
640C2×C3#243714yes—
641C2×C3#248311yescharacteristic
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—216no—
651C22—216no—
652C22—216no—
653C22—216no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—464no—
662C22—42no—
663C22—42no—
664C22—48no—
665C22—48no—
666C22—42no—
667C22—42no—
668C22#494618yescomplemented
669C22#521018yescomplemented
670C22C2×#87812yes—
671C22#548312yes—
672C22C2×#67⋊C412yes—
673C22#548612yes—
674C22#549014yes—
675C22#549514yes—
676C22C22×Dic3⋊C411yescharacteristic
677C22C2×#92412yes—
678C22C2×C4×#6712yes—
679C22C2×#92911yescharacteristic
680C22C2×#93511yescharacteristic
681C22C2×#93711yescharacteristic
682C22D4×#6714yes—
683C22#580411yescharacteristic
684C22#580411yescharacteristic
685C22#580614yes—
686C22C2×#95311yescharacteristic
687C4—44no—
688C4—44no—
689C4—124no—
690C4—128no—
691C4—124no—
692C3#1412711yescharacteristic, complemented, Sylow
693C2—28no—
694C2—28no—
695C2—44no—
696C2C2×#494611yescharacteristic
697C2#8209511yescharacteristic
698C2#8209914yes—
699C2#8211014yes—
700C2#8211411yescharacteristic
701C2#8212611yescharacteristic
702C2C2×#521011yescharacteristic
703C2#8404711yescharacteristic
704C2#8405211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC23×C3×C22⋊C4C2
socle, cosocleC24×C3C24
prime radicalC26×C3C22
hypercenterC23×C22⋊C4D3
hypocenterC3#14127
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#5210C2semidirect
5. C2×#4946C2semidirect
6. #72127C2semidirect
7. #71547C2semidirect
8. #71544C2semidirect
26. C22×Dic3⋊C4C22semidirect
27. #5247C4semidirect
29. C2×C3⋊#95C22semidirect
30. C2×C3⋊#95C22semidirect
31. C2×C3⋊#95C22semidirect
93. C2×#217C2×C4semidirect
94. C2×#217C2×C4semidirect
95. C2×#217C2×C4semidirect
156. C24×C3C4⋊C4semidirect
414. C24Dic3⋊C4semidirect
506. C22×C3#385semidirect
507. C22×C3#385semidirect
668. C22#4946semidirect
669. C22#5210semidirect
692. C3#14127semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C42
C23×#2172
C2×#52472
C2×#52102
C2×#49462
#721272
#715472
#715442
#141273