Finite Group Databasebeta

← #10726576

Subgroups — #10726576

Order 768.

3654 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C23×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×#5231C212yesmaximal, complemented
4C2×#4955C212yesmaximal
5#72127C211yesmaximal, characteristic, complemented
6#71570C214yesmaximal, complemented
7#71570C212yesmaximal, complemented
8#71544C212yesmaximal, complemented
9#14717—31nomaximal, Sylow
10C25×D3C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic, complemented
12C24×C3×C4C2211yescharacteristic
13C24×Dic3C2211yescharacteristic
14C2×C3×C22⋊Q8—24no—
15C2×C3×C22⋊Q8—28no—
16C22×C3×C4⋊C4C2212yescomplemented
17C22×C3×C4⋊C4C2211yescharacteristic, complemented
18C22×C3×C22⋊C4C2212yes—
19C22×#217C2212yes—
20C22×(C2×D3)⋊C4—24no—
21C22×(C2×D3)⋊C4—28no—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2214yes—
24C22×(C2×D3)⋊C4C2214yes—
25C22×(C2×D3)⋊C4C2212yes—
26#5231—28no—
27C2×C3⋊#95C2214yescomplemented
28C2×C3⋊#95C2212yescomplemented
29C2×C3⋊#95C2214yescomplemented
30C2×C3⋊#95C2212yescomplemented
31#4955—28no—
32C22×C22⋊Q8D311yescharacteristic, complemented
33C23×C22⋊C4—31no—
34C2×#393—32no—
35C2×#379—32no—
36#1439—32no—
37#1439—34no—
38C22⋊#95—31no—
39C22⋊#95—32no—
40C25×C3C2311yescharacteristic
41C24×D3—22no—
42C24×D3—22no—
43C24×D3—24no—
44C24×D3—24no—
45C24×D3—24no—
46C24×D3—24no—
47C24×D3—42no—
48C24×D3—42no—
49C24×D3Q814yescomplemented
50C24×D3C2312yes—
51C22×C3×Q8—24no—
52C22×C3×Q8—22no—
53C22×C3×Q8—24no—
54C22×C3×Q8—42no—
55C23×C3×C4—24no—
56C23×C3×C4—24no—
57C23×C3×C4—42no—
58C23×C3×C4D414yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—22no—
64C23×Dic3—24no—
65C23×Dic3—24no—
66C23×Dic3—42no—
67C23×Dic3C2314yes—
68C23×Dic3C2312yes—
69C3×C22⋊Q8—232no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4D418yes—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×(C2×D3)⋊C4—232no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—28no—
90C2×(C2×D3)⋊C4—28no—
91C2×(C2×D3)⋊C4—44no—
92C2×(C2×D3)⋊C4—44no—
93C2×(C2×D3)⋊C4D418yes—
94C3⋊#95—24no—
95C3⋊#95—28no—
96C3⋊#95—44no—
97C3⋊#95—42no—
98C26—31no—
99C23×Q8C2×D311yescharacteristic, complemented
100C24×C4—31no—
101C24×C4C2×D311yescharacteristic
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—28no—
104C22×C4⋊C4C2×D311yescharacteristic, complemented
105C22×C4⋊C4C2×D312yescomplemented
106C22×C22⋊C4—32no—
107C22×C22⋊C4—32no—
108C22×C22⋊C4—32no—
109C22×C22⋊C4—34no—
110C22×C22⋊C4—34no—
111C22×C22⋊C4—68no—
112C22×C22⋊C4—64no—
113C22×C22⋊C4C2×D312yes—
114#393—68no—
115#379—68no—
116C2×#95—34no—
117C2×#95—32no—
118C2×#95—34no—
119C2×#95—32no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3C2×D414yes—
125C24×C3C2×D412yes—
126C24×C3C2×Q812yes—
127C24×C3#4112yes—
128C24×C3#4114yes—
129C24×C3C2411yescharacteristic
130C23×D3—28no—
131C23×D3—216no—
132C23×D3—28no—
133C23×D3—28no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—24no—
137C23×D3—22no—
138C23×D3—24no—
139C23×D3—22no—
140C23×D3—24no—
141C23×D3—24no—
142C23×D3—216no—
143C23×D3—216no—
144C23×D3—48no—
145C23×D3—416no—
146C23×D3—416no—
147C23×D3—416no—
148C23×D3—416no—
149C23×D3—44no—
150C23×D3—44no—
151C23×D3—48no—
152C23×D3—44no—
153C23×D3—48no—
154C23×D3—44no—
155C23×D3C2×D414yes—
156C23×D3C2×Q814yes—
157C23×D3#4114yes—
158C2×C3×Q8—28no—
159C2×C3×Q8—416no—
160C2×C3×Q8—44no—
161C2×C3×Q8—44no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—28no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—28no—
177C22×C3×C4—44no—
178C22×C3×C4—416no—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×Dic3—28no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—416no—
197C22×Dic3—44no—
198C22×Dic3—44no—
199C22×Dic3C2×D414yes—
200C3×C4⋊C4—28no—
201C3×C4⋊C4—216no—
202C3×C22⋊C4—216no—
203#217—48no—
204(C2×D3)⋊C4—216no—
205(C2×D3)⋊C4—216no—
206(C2×D3)⋊C4—416no—
207(C2×D3)⋊C4—416no—
208C25—32no—
209C25—34no—
210C25—64no—
211C25—64no—
212C25—62no—
213C25—62no—
214C25—64no—
215C25—64no—
216C25—122no—
217C25—122no—
218C25C22×D311yescharacteristic
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—22no—
222C22×Q8—42no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—32no—
226C23×C4—34no—
227C23×C4—42no—
228C23×C4—64no—
229C23×C4—62no—
230C23×C4—64no—
231C23×C4—122no—
232C23×C4#6714yes—
233C23×C4C22×D312yes—
234C23×C4C22×D312yes—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D311yescharacteristic
237C22⋊Q8—232no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—38no—
246C2×C22⋊C4—64no—
247C2×C22⋊C4—64no—
248C2×C22⋊C4—632no—
249C2×C22⋊C4—64no—
250C2×C22⋊C4—64no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—68no—
253C2×C22⋊C4—68no—
254C2×C22⋊C4—64no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—68no—
258C2×C22⋊C4—68no—
259C2×C22⋊C4—124no—
260C2×C22⋊C4—124no—
261C2×C22⋊C4#6718yes—
262#95—68no—
263#95—64no—
264#95—122no—
265#95—124no—
266C23×C3—22no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—24no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—28no—
275C23×C3—28no—
276C23×C3—28no—
277C23×C3—28no—
278C23×C3#12014yes—
279C23×C3C22⋊Q814yes—
280C23×C3C22⋊Q818yes—
281C23×C3#12418yes—
282C23×C3#12414yes—
283C23×C3C22×D412yes—
284C23×C3C22×D411yescharacteristic
285C23×C3C22×D412yes—
286C23×C3C22×Q811yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4112yes—
289C23×C3C2×#4112yes—
290C23×C3H(5,𝔽2)12yes—
291C23×C3H(5,𝔽2)12yes—
292C22×D3—232no—
293C22×D3—24no—
294C22×D3—24no—
295C22×D3—28no—
296C22×D3—24no—
297C22×D3—24no—
298C22×D3—24no—
299C22×D3—24no—
300C22×D3—28no—
301C22×D3—28no—
302C22×D3—28no—
303C22×D3—28no—
304C22×D3—28no—
305C22×D3—464no—
306C22×D3—416no—
307C22×D3—416no—
308C22×D3—416no—
309C22×D3—416no—
310C22×D3—416no—
311C22×D3—464no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—44no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—44no—
318C22×D3—44no—
319C22×D3—44no—
320C22×D3—44no—
321C22×D3—44no—
322C22×D3—416no—
323C22×D3C22⋊Q818yescomplemented
324C3×Q8—416no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—28no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—28no—
334C2×C3×C4—28no—
335C2×C3×C4—416no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—44no—
341C2×C3×C4—44no—
342C2×C3×C4—44no—
343C2×C3×C4#12014yes—
344C2×Dic3—24no—
345C2×Dic3—28no—
346C2×Dic3—416no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—44no—
350C2×Dic3—44no—
351C2×Dic3—44no—
352C2×Dic3—44no—
353C2×Dic3—44no—
354C2×Dic3—44no—
355C2×Dic3—44no—
356C2×Dic3—44no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—34no—
362C24—34no—
363C24—34no—
364C24—616no—
365C24—68no—
366C24—68no—
367C24—616no—
368C24—68no—
369C24—62no—
370C24—64no—
371C24—64no—
372C24—62no—
373C24—62no—
374C24—64no—
375C24—62no—
376C24—64no—
377C24—616no—
378C24—128no—
379C24—1216no—
380C24—1216no—
381C24—1216no—
382C24—1216no—
383C24—128no—
384C24—124no—
385C24—124no—
386C24—128no—
387C24—124no—
388C24—124no—
389C24D3×Q812yes—
390C24#23912yes—
391C24#23914yes—
392C24C2×#6712yes—
393C24C2×#6714yes—
394C24C23×D311yescharacteristic
395C2×Q8—28no—
396C2×Q8—416no—
397C2×Q8—44no—
398C2×Q8—44no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—34no—
415C22×C4—44no—
416C22×C4—416no—
417C22×C4—68no—
418C22×C4—62no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—62no—
422C22×C4—64no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—1216no—
430C22×C4—124no—
431C22×C4—124no—
432C22×C4D3×D414yes—
433C22×C4C2×#6714yes—
434C22×C4C2×#6714yes—
435C22×C4C2×#6712yes—
436C22×C4C2×#6712yes—
437C4⋊C4—28no—
438C4⋊C4—216no—
439C22⋊C4—216no—
440C22⋊C4—616no—
441C22⋊C4—616no—
442C22⋊C4—1216no—
443C22⋊C4—1216no—
444C22⋊C4—128no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—24no—
448C22×C3—24no—
449C22×C3—24no—
450C22×C3—24no—
451C22×C3—216no—
452C22×C3—216no—
453C22×C3#39318yescomplemented
454C22×C3C2×#12011yescharacteristic
455C22×C3C2×#12011yescharacteristic
456C22×C3C2×C22⋊Q811yescharacteristic
457C22×C3C2×C22⋊Q812yes—
458C22×C3C2×C22⋊Q812yes—
459C22×C3#52511yescharacteristic
460C22×C3#52512yes—
461C22×C3C2×#12411yescharacteristic
462C22×C3C2×#12412yes—
463C22×C3#53411yescharacteristic
464C22×C3#53511yescharacteristic
465C22×C3#53812yes—
466C22×C3#54312yes—
467C22×C3D4214yes—
468C22×C3#54614yes—
469C22×C3#54614yes—
470C22×C3#55114yes—
471C2×D3—28no—
472C2×D3—216no—
473C2×D3—464no—
474C2×D3—44no—
475C2×D3—44no—
476C2×D3—44no—
477C2×D3—44no—
478C2×D3—44no—
479C2×D3—44no—
480C2×D3—48no—
481C2×D3—48no—
482C2×D3—48no—
483C3×C4—24no—
484C3×C4—42no—
485C3×C4—44no—
486C3×C4—44no—
487Dic3—44no—
488Dic3—84no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—22no—
493C23—28no—
494C23—22no—
495C23—22no—
496C23—28no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—38no—
502C23—632no—
503C23—68no—
504C23—68no—
505C23—64no—
506C23—68no—
507C23—68no—
508C23—64no—
509C23—64no—
510C23—68no—
511C23—64no—
512C23—64no—
513C23—64no—
514C23—68no—
515C23—1264no—
516C23—1216no—
517C23—1264no—
518C23—1216no—
519C23—1216no—
520C23—1216no—
521C23—1216no—
522C23—124no—
523C23—124no—
524C23—1216no—
525C23—124no—
526C23—122no—
527C23—122no—
528C23—124no—
529C23—124no—
530C23—122no—
531C23—122no—
532C23—124no—
533C23#94618yes—
534C23#94614yes—
535C23#94718yes—
536C23#94714yes—
537C23#95314yes—
538C23C2×D3×D412yes—
539C23#100412yes—
540C23C2×D3×Q811yescharacteristic
541C23C2×#23912yes—
542C23C2×#23911yescharacteristic
543C23D3×#4112yes—
544C23#100912yes—
545C23C22×#6711yescharacteristic
546C23C22×#6712yes—
547Q8—416no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—416no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—64no—
567C2×C4—68no—
568C2×C4—1216no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—124no—
572C2×C4—124no—
573C2×C4—124no—
574C2×C4—124no—
575C2×C4—124no—
576C2×C4—124no—
577C2×C4—124no—
578C2×C4—124no—
579C2×C4#93714yes—
580C2×C3—28no—
581C2×C3C2×#39311yescharacteristic
582C2×C3#240311yescharacteristic
583C2×C3#240411yescharacteristic
584C2×C3#243511yescharacteristic
585C2×C3#243714yes—
586C2×C3#247214yes—
587C2×C3#256111yescharacteristic
588C2×C3#262611yescharacteristic
589C2×C3#284811yescharacteristic
590D3—48no—
591C22—24no—
592C22—216no—
593C22—24no—
594C22—216no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—616no—
600C22—68no—
601C22—1264no—
602C22—124no—
603C22—128no—
604C22—128no—
605C22—124no—
606C22—124no—
607C22—128no—
608C22—124no—
609C22—124no—
610C22—124no—
611C22#523118yescomplemented
612C22D3×C22⋊Q812yes—
613C22#562712yes—
614C22#563014yes—
615C22#563412yes—
616C22#563714yes—
617C22#564412yes—
618C22C2×#93711yescharacteristic
619C22D4×#6714yes—
620C22#580411yescharacteristic
621C22C2×#94612yes—
622C22C2×#94611yescharacteristic
623C22C2×#94711yescharacteristic
624C22C2×#94712yes—
625C22#582811yescharacteristic
626C22#582914yes—
627C22#583011yescharacteristic
628C22C2×#95311yescharacteristic
629C4—24no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—124no—
634C4—244no—
635C3#1471711yescharacteristic, complemented, Sylow
636C2—28no—
637C2—128no—
638C2#8276111yescharacteristic
639C2#8276914yes—
640C2#8281614yes—
641C2#8281911yescharacteristic
642C2#8282211yescharacteristic
643C2#8283411yescharacteristic
644C2#8405211yescharacteristic
645C2C2×#523111yescharacteristic
646C2#8409611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×C22⋊Q8C2
prime radicalC25×D3C22
hypercenterC22×C22⋊Q8D3
hypocenterC3#14717
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊Q8C2semidirect
3. C2×#5231C2semidirect
5. #72127C2semidirect
6. #71570C2semidirect
7. #71570C2semidirect
8. #71544C2semidirect
11. C23×C3×Q8C22semidirect
16. C22×C3×C4⋊C4C22semidirect
17. C22×C3×C4⋊C4C22semidirect
27. C2×C3⋊#95C22semidirect
28. C2×C3⋊#95C22semidirect
29. C2×C3⋊#95C22semidirect
30. C2×C3⋊#95C22semidirect
32. C22×C22⋊Q8D3semidirect
49. C24×D3Q8semidirect
99. C23×Q8C2×D3semidirect
104. C22×C4⋊C4C2×D3semidirect
105. C22×C4⋊C4C2×D3semidirect
323. C22×D3C22⋊Q8semidirect
453. C22×C3#393semidirect
611. C22#5231semidirect
635. C3#14717semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C23×(C2×D3)⋊C42
C2×#52312
C2×#49552
#721272
#715702
#715442
#147173