Finite Group Databasebeta

← #10720193

Subgroups — #10720193

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C22×#892C211yesmaximal, characteristic
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C2×#5229C211yesmaximal, characteristic, complemented
5C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic, complemented
7#71535C211yesmaximal, characteristic, complemented
8#69429C214yesmaximal
9#69406C214yesmaximal, complemented
10#14138—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×#946—24no—
16C2×#946—24no—
17C2×#946C418yescomplemented
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C2×#892—24no—
27C2×#892—24no—
28C2×#892—24no—
29C3×#391—24no—
30C2×C3×#95C2214yes—
31#5229—24no—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2214yes—
34C2×C3⋊#95C2214yes—
35C2×C3⋊#95C2214yes—
36C4×(C2×D3)⋊C4—24no—
37C22×C22⋊Q8—31no—
38C22×#117—31no—
39C2×#391—31no—
40C2×#391D311yescharacteristic, complemented
41C2×C4×C22⋊C4—31no—
42#1440—34no—
43#1436—34no—
44C4⋊#95—31no—
45C4⋊#95—31no—
46C24×D3C2311yescharacteristic
47C22×C3×Q8—22no—
48C22×C3×Q8—22no—
49C22×C3×Q8—22no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8C2×C414yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3C2×C414yes—
67C22×C4×D3D414yes—
68C22×C4×D3D414yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D414yes—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75#946—232no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4C2×C418yes—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3C2×C414yes—
90C2×C4⋊Dic3D414yes—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4C2×C418yes—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100#892—232no—
101C3×#95—28no—
102C3⋊#95—28no—
103C3⋊#95—24no—
104C3⋊#95—28no—
105C3⋊#95—28no—
106C23×Q8C2×D311yescharacteristic
107C24×C4—31no—
108C2×C22⋊Q8—38no—
109C2×C22⋊Q8—64no—
110C2×C22⋊Q8—64no—
111C2×#117—64no—
112C2×#117—64no—
113C2×#117—64no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4C2×D311yescharacteristic
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4—32no—
121C22×C42—32no—
122C22×C42—31no—
123C22×C42C2×D311yescharacteristic
124#391—24no—
125#391—64no—
126C4×C22⋊C4—64no—
127C2×#95—34no—
128C2×#95—34no—
129C2×#95—32no—
130C2×#95—34no—
131C2×#95C2×D314yes—
132C24×C3C2411yescharacteristic
133C23×D3—21no—
134C23×D3—21no—
135C23×D3—22no—
136C23×D3—22no—
137C23×D3—21no—
138C23×D3—21no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3C22×C412yes—
142C23×D3C2×D412yes—
143C23×D3C2×D412yes—
144C2×C3×Q8—28no—
145C2×C3×Q8—28no—
146C2×C3×Q8—28no—
147C2×C3×Q8—44no—
148C2×C3×Q8—44no—
149C2×C3×Q8—44no—
150C2×C3×Q8—44no—
151C22×C3×C4—22no—
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—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4C22×C414yes—
168C22×C3×C4C22×C412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4#4114yes—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—24no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—24no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3C22×C414yes—
196C22×Dic3C22×C412yes—
197C22×Dic3C2×D414yes—
198C22×Dic3C2×D412yes—
199C22×Dic3C2×D412yes—
200C22×Dic3#4114yes—
201C2×C4×D3—28no—
202C2×C4×D3—28no—
203C2×C4×D3—28no—
204C2×C4×D3—28no—
205C2×C4×D3—28no—
206C2×C4×D3—28no—
207C2×C4×D3C22⋊C4116yes—
208C3×C4⋊C4—28no—
209C3×C42—44no—
210(C2×D3)⋊C4—216no—
211(C2×D3)⋊C4—216no—
212(C2×D3)⋊C4—216no—
213C4⋊Dic3—28no—
214C4⋊Dic3—28no—
215Dic3⋊C4—216no—
216C4×Dic3—216no—
217C4×Dic3—44no—
218C25—31no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8C4×D314yes—
226C23×C4—31no—
227C23×C4—34no—
228C23×C4—34no—
229C23×C4—34no—
230C23×C4—32no—
231C23×C4—32no—
232C23×C4—32no—
233C23×C4—62no—
234C23×C4—62no—
235C23×C4—62no—
236C23×C4—62no—
237C23×C4C22×D312yes—
238C23×C4C22×D312yes—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C22⋊Q8—632no—
242#117—632no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—38no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4#6714yes—
255C2×C22⋊C4—38no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—64no—
268C2×C42—64no—
269C2×C42—64no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42—62no—
273#95—28no—
274#95—68no—
275#95—64no—
276#95—68no—
277#95—68no—
278C23×C3C23×C411yescharacteristic
279C23×C3C22×D411yescharacteristic
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D411yescharacteristic
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4112yes—
286C23×C3C2×#4112yes—
287C23×C32−1+412yes—
288C23×C32−1+412yes—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—24no—
292C22×D3—24no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—24no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—24no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3C2×C22⋊C414yes—
312C3×Q8—416no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—28no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4C2×C22⋊C414yes—
333C2×C3×C4C4×D414yes—
334C2×C3×C4C4×D414yes—
335C2×C3×C4#12014yes—
336C2×C3×C4#12114yes—
337C2×C3×C4#12114yes—
338C2×C3×C4#12314yes—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—28no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—28no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—42no—
353C2×Dic3—44no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—44no—
361C2×Dic3—42no—
362C2×Dic3C2×C22⋊C414yes—
363C4×D3—232no—
364C24—32no—
365C24—32no—
366C24—32no—
367C24—61no—
368C24—61no—
369C24—61no—
370C24—62no—
371C24—62no—
372C24—62no—
373C24—62no—
374C24—61no—
375C24C23×D311yescharacteristic
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—32no—
400C22×C4—32no—
401C22×C4—32no—
402C22×C4—316no—
403C22×C4—34no—
404C22×C4—34no—
405C22×C4—34no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—68no—
411C22×C4—68no—
412C22×C4—68no—
413C22×C4—68no—
414C22×C4—68no—
415C22×C4—68no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—64no—
423C22×C4—64no—
424C22×C4—62no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—64no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4C2×C4×D314yes—
431C22×C4C2×C4×D312yes—
432C22×C4#23512yes—
433C22×C4D3×D412yes—
434C22×C4D3×D412yes—
435C22×C4D3×D412yes—
436C22×C4#23712yes—
437C22×C4#23914yes—
438C22×C4C2×#6714yes—
439C22×C4C2×#6712yes—
440C4⋊C4—28no—
441C4⋊C4—616no—
442C4⋊C4—68no—
443C4⋊C4—68no—
444C22⋊C4—616no—
445C22⋊C4—616no—
446C22⋊C4—616no—
447C42—44no—
448C42—616no—
449C42—124no—
450C22×C3C22×C22⋊C411yescharacteristic
451C22×C3C2×C4×D411yescharacteristic
452C22×C3C2×C4×D411yescharacteristic
453C22×C3C4×#4112yes—
454C22×C3#51912yes—
455C22×C3C2×#12011yescharacteristic
456C22×C3C2×#12111yescharacteristic
457C22×C3C2×#12111yescharacteristic
458C22×C3C2×#12311yescharacteristic
459C22×C3#52512yes—
460C22×C3#53211yescharacteristic
461C22×C3#53211yescharacteristic
462C22×C3#53611yescharacteristic
463C22×C3#53611yescharacteristic
464C22×C3#53612yes—
465C22×C3#54712yes—
466C22×C3#54712yes—
467C22×C3#54812yes—
468C22×C3#54812yes—
469C22×C3#55212yes—
470C22×C3#55212yes—
471C22×C3#55612yes—
472C22×C3#55612yes—
473C2×D3—24no—
474C2×D3—24no—
475C2×D3—24no—
476C2×D3—44no—
477C2×D3—44no—
478C2×D3—44no—
479C2×D3—44no—
480C2×D3—42no—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C3×C4—44no—
485C3×C4—44no—
486C3×C4—44no—
487C3×C4#38618yes—
488Dic3—24no—
489Dic3—44no—
490Dic3—44no—
491Dic3—44no—
492C23—34no—
493C23—62no—
494C23—62no—
495C23—62no—
496C23—62no—
497C23—62no—
498C23—62no—
499C23—62no—
500C23—62no—
501C23—62no—
502C23—62no—
503C23—64no—
504C23—64no—
505C23—64no—
506C23—64no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23—122no—
513C23—122no—
514C23—122no—
515C23C22×C4×D311yescharacteristic
516C23C2×#23511yescharacteristic
517C23C2×D3×D411yescharacteristic
518C23C2×D3×D411yescharacteristic
519C23C2×D3×D411yescharacteristic
520C23C2×#23711yescharacteristic
521C23C2×#23912yes—
522C23#100712yes—
523C23D3×#4112yes—
524C23#101012yes—
525C23C22×#6711yescharacteristic
526Q8—416no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—34no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—632no—
554C2×C4—64no—
555C2×C4—68no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—68no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—124no—
568C2×C4—122no—
569C2×C4—124no—
570C2×C4—122no—
571C2×C4D3×C22⋊C414yes—
572C2×C4#67⋊C414yes—
573C2×C4#88314yes—
574C2×C4#88414yes—
575C2×C4C4×#6714yes—
576C2×C4#93714yes—
577C2×C4#93814yes—
578C2×C3C2×#38611yescharacteristic
579C2×C3#229711yescharacteristic
580C2×C3#231011yescharacteristic
581C2×C3#236212yes—
582C2×C3#236212yes—
583C2×C3#241512yes—
584C2×C3#241512yes—
585C2×C3#243211yescharacteristic
586C2×C3#243211yescharacteristic
587C2×C3#249911yescharacteristic
588C2×C3#249911yescharacteristic
589D3—44no—
590C22—64no—
591C22—64no—
592C22—64no—
593C22—122no—
594C22—124no—
595C22—122no—
596C22—124no—
597C22—122no—
598C22—124no—
599C22—124no—
600C22—122no—
601C22C2×D3×C22⋊C411yescharacteristic
602C22C2×#67⋊C411yescharacteristic
603C22C2×#88311yescharacteristic
604C22C2×#88411yescharacteristic
605C22#557212yes—
606C22C4×#23912yes—
607C22#558112yes—
608C22#558412yes—
609C22#562712yes—
610C22#562812yes—
611C22#564112yes—
612C22#564212yes—
613C22#566811yescharacteristic
614C22#567511yescharacteristic
615C22#567712yes—
616C22#572411yescharacteristic
617C22#572511yescharacteristic
618C22#572612yes—
619C22C2×C4×#6711yescharacteristic
620C22C2×#93711yescharacteristic
621C22C2×#93811yescharacteristic
622C22#581712yes—
623C22#582412yes—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—64no—
628C4—124no—
629C4—124no—
630C4—124no—
631C4#495618yes—
632C3#1413811yescharacteristic, complemented, Sylow
633C2—124no—
634C2#7647811yescharacteristic
635C2#7649112yes—
636C2#7651811yescharacteristic
637C2#7661011yescharacteristic
638C2#7662012yes—
639C2#7663411yescharacteristic
640C2C2×#495611yescharacteristic
641C2#8246412yes—
642C2#8281312yes—
643C2#8295911yescharacteristic
644C2#8325511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#391C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenterC2×#391D3
hypocenterC3#14138
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×C3×#391C2semidirect
4. C2×#5229C2semidirect
6. #71563C2semidirect
7. #71535C2semidirect
9. #69406C2semidirect
17. C2×#946C4semidirect
40. C2×#391D3semidirect
632. C3#14138semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
C22×#8922
C2×C3×#3912
C2×#52292
C2×C4×(C2×D3)⋊C42
#715632
#715352
#694292
#694062
#141383