Finite Group Databasebeta

← #10729284

Subgroups — #10729284

Order 768.

2062 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C2×#4664C212yesmaximal
3C2×D3×#95C211yesmaximal, characteristic
4C3×#1446C211yesmaximal, characteristic, complemented
5#72134C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#95C212yesmaximal
7(C2×D3)⋊#95C211yesmaximal, characteristic
8#69415C211yesmaximal, characteristic, complemented
9#69414C212yesmaximal, complemented
10#69413C212yesmaximal, complemented
11#69412C211yesmaximal, characteristic, complemented
12#14934—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C2×#946—24no—
18C2×#946—28no—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2212yes—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4664—28no—
39D3×#95—24no—
40C22×C22⋊Q8—31no—
41C2×#381—32no—
42C22×#95—31no—
43#1446—31no—
44#1446D311yescharacteristic, complemented
45#1442—31no—
46#1442—32no—
47#1442—32no—
48#1442—31no—
49C22⋊#95—32no—
50C22⋊#95—31no—
51C24×D3C2311yescharacteristic
52C22×C3×Q8—24no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—42no—
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—24no—
70C22×C4×D3—24no—
71C22×C4×D3—22no—
72C22×C4×D3—24no—
73C22×C4×D3—22no—
74C22×C4×D3—22no—
75C22×C4×D3—22no—
76C22×C4×D3—22no—
77C22×C4×D3—24no—
78C22×C4×D3—42no—
79C22×C4×D3—42no—
80C22×C4×D3D414yes—
81C22×C4×D3Q814yescomplemented
82#946—232no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4D418yes—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C3×#95—22no—
95C3×#95—24no—
96C3×#95—22no—
97C3×#95—42no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C23×Q8C2×D311yescharacteristic
110C24×C4—31no—
111C24×C4—31no—
112C24×C4—31no—
113C2×C22⋊Q8—68no—
114C2×C22⋊Q8—64no—
115C22×C4⋊C4—32no—
116C22×C4⋊C4—31no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4—32no—
119#381—68no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—31no—
123C2×#95—31no—
124C2×#95—31no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—64no—
132C2×#95C2×D311yescharacteristic
133C2×#95C2×D311yescharacteristic
134C2×#95C2×D312yes—
135C2×#95C2×D312yes—
136C24×C3C2411yescharacteristic
137C23×D3—21no—
138C23×D3—21no—
139C23×D3—21no—
140C23×D3—21no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3C2×D414yes—
144C23×D3C2×D412yes—
145C23×D3C2×Q812yes—
146C23×D3#4112yes—
147C23×D3#4114yes—
148C2×C3×Q8—28no—
149C2×C3×Q8—48no—
150C2×C3×Q8—48no—
151C2×C3×Q8—44no—
152C2×C3×Q8—44no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
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—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×Q812yes—
179C22×C3×C4#4112yes—
180C22×C3×C4#4112yes—
181C22×C3×C4#4112yes—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
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×Dic3C2×D412yes—
208C22×Dic3C2×D414yes—
209C22×Dic3C2×Q812yes—
210C22×Dic3#4112yes—
211C2×C4×D3—28no—
212C2×C4×D3—28no—
213C2×C4×D3—28no—
214C2×C4×D3—216no—
215C2×C4×D3—28no—
216C2×C4×D3—44no—
217C2×C4×D3—48no—
218C2×C4×D3—48no—
219C2×C4×D3—48no—
220C2×C4×D3—48no—
221C2×C4×D3—44no—
222C2×C4×D3—44no—
223C2×C4×D3—44no—
224C2×C4×D3—48no—
225C2×C4×D3—44no—
226(C2×D3)⋊C4—216no—
227(C2×D3)⋊C4—48no—
228C4⋊Dic3—28no—
229Dic3⋊C4—216no—
230C25—31no—
231C22×Q8—22no—
232C22×Q8—24no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—42no—
236C23×C4—32no—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4—31no—
240C23×C4—34no—
241C23×C4—34no—
242C23×C4—32no—
243C23×C4—62no—
244C23×C4—64no—
245C23×C4—62no—
246C23×C4—62no—
247C23×C4—64no—
248C23×C4—62no—
249C23×C4—64no—
250C23×C4—64no—
251C23×C4—62no—
252C23×C4—62no—
253C23×C4—62no—
254C23×C4—122no—
255C23×C4—122no—
256C23×C4C22×D312yes—
257C23×C4C22×D312yes—
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C22⋊Q8—632no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—62no—
267C2×C22⋊C4—38no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273#95—22no—
274#95—24no—
275#95—22no—
276#95—42no—
277#95—64no—
278#95—62no—
279#95—62no—
280#95—64no—
281#95—64no—
282#95—62no—
283#95—62no—
284#95—64no—
285#95—64no—
286#95—122no—
287#95—122no—
288C23×C3C22×D411yescharacteristic
289C23×C3C22×D412yes—
290C23×C3C22×Q811yescharacteristic
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4112yes—
296C23×C3C2×#4111yescharacteristic
297C23×C3H(5,𝔽2)12yes—
298C23×C32−1+412yes—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3#12014yes—
318C22×D3C22⋊Q814yes—
319C22×D3C22⋊Q818yes—
320C22×D3#12418yes—
321C22×D3#12414yes—
322C3×Q8—416no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
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—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4#12314yes—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—41no—
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—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C4×D3—416no—
382C4×D3—416no—
383C4×D3—416no—
384C24—34no—
385C24—32no—
386C24—32no—
387C24—32no—
388C24—34no—
389C24—61no—
390C24—61no—
391C24—62no—
392C24—61no—
393C24—62no—
394C24—61no—
395C24C23×D311yescharacteristic
396C2×Q8—28no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—44no—
400C2×Q8—44no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—68no—
429C22×C4—68no—
430C22×C4—616no—
431C22×C4—68no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—61no—
450C22×C4—68no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—124no—
459C22×C4—128no—
460C22×C4—128no—
461C22×C4—128no—
462C22×C4—128no—
463C22×C4—128no—
464C22×C4—124no—
465C22×C4—124no—
466C22×C4—124no—
467C22×C4—124no—
468C22×C4#23512yes—
469C22×C4D3×D414yes—
470C22×C4D3×D412yes—
471C22×C4#23712yes—
472C22×C4D3×Q812yes—
473C22×C4#23912yes—
474C4⋊C4—616no—
475C4⋊C4—68no—
476C22⋊C4—616no—
477C22⋊C4—128no—
478C22×C3C2×#12011yescharacteristic
479C22×C3C2×C22⋊Q811yescharacteristic
480C22×C3C2×C22⋊Q812yes—
481C22×C3C2×#12311yescharacteristic
482C22×C3#52511yescharacteristic
483C22×C3C2×#12411yescharacteristic
484C22×C3C2×#12412yes—
485C22×C3#52911yescharacteristic
486C22×C3#52911yescharacteristic
487C22×C3#53311yescharacteristic
488C22×C3#53311yescharacteristic
489C22×C3#53411yescharacteristic
490C22×C3#53611yescharacteristic
491C22×C3#53912yes—
492C22×C3#54412yes—
493C22×C3#54612yes—
494C22×C3#54612yes—
495C22×C3#54712yes—
496C22×C3#54712yes—
497C22×C3#55112yes—
498C22×C3#55112yes—
499C22×C3#55212yes—
500C22×C3#55212yes—
501C2×D3—24no—
502C2×D3—24no—
503C2×D3—24no—
504C2×D3—24no—
505C2×D3—24no—
506C2×D3—28no—
507C2×D3—28no—
508C2×D3—28no—
509C2×D3—28no—
510C2×D3—24no—
511C2×D3#39318yes—
512C3×C4—24no—
513C3×C4—42no—
514C3×C4—42no—
515C3×C4—44no—
516C3×C4—82no—
517Dic3—42no—
518Dic3—42no—
519Dic3—42no—
520Dic3—44no—
521Dic3—82no—
522C23—38no—
523C23—34no—
524C23—34no—
525C23—38no—
526C23—34no—
527C23—64no—
528C23—64no—
529C23—64no—
530C23—64no—
531C23—62no—
532C23—62no—
533C23—64no—
534C23—62no—
535C23—64no—
536C23—62no—
537C23—64no—
538C23—62no—
539C23—62no—
540C23—64no—
541C23—62no—
542C23—64no—
543C23—62no—
544C23—64no—
545C23C2×#23511yescharacteristic
546C23C2×D3×D411yescharacteristic
547C23C2×D3×D412yes—
548C23C2×#23711yescharacteristic
549C23#100412yes—
550C23C2×D3×Q811yescharacteristic
551C23C2×#23911yescharacteristic
552C23#100712yes—
553C23D3×#4112yes—
554C23D3×#4111yescharacteristic
555C23D3×#4111yescharacteristic
556Q8—416no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—64no—
586C2×C4—64no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—64no—
595C2×C4—1216no—
596C2×C4—1216no—
597C2×C4—1216no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—122no—
608C2×C4—122no—
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#88314yes—
618C2×C3C2×#39311yescharacteristic
619C2×C3#240511yescharacteristic
620C2×C3#240611yescharacteristic
621C2×C3#243812yes—
622C2×C3#243812yes—
623C2×C3#248812yes—
624C2×C3#248812yes—
625C2×C3#250311yescharacteristic
626C2×C3#256111yescharacteristic
627C2×C3#262711yescharacteristic
628C2×C3#285511yescharacteristic
629D3—28no—
630C22—38no—
631C22—68no—
632C22—64no—
633C22—68no—
634C22—68no—
635C22—64no—
636C22—64no—
637C22—64no—
638C22—68no—
639C22—64no—
640C22—64no—
641C22C2×#88311yescharacteristic
642C22#549012yes—
643C22#557811yescharacteristic
644C22#558011yescharacteristic
645C22#558112yes—
646C22D3×#12011yescharacteristic
647C22#558911yescharacteristic
648C22#559112yes—
649C22D3×C22⋊Q812yes—
650C22D3×C22⋊Q811yescharacteristic
651C22#562711yescharacteristic
652C22#563112yes—
653C22#563712yes—
654C22#563812yes—
655C22#563912yes—
656C22D3×#12412yes—
657C22D3×#12411yescharacteristic
658C22#567912yes—
659C22#568011yescharacteristic
660C22#572411yescharacteristic
661C22#572612yes—
662C22#572811yescharacteristic
663C22#572912yes—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—82no—
669C4—122no—
670C4—122no—
671C4—122no—
672C4—124no—
673C4—242no—
674C3#1493411yescharacteristic, complemented, Sylow
675C2—68no—
676C2#7385311yescharacteristic
677C2#7652011yescharacteristic
678C2#7662712yes—
679C2#7662912yes—
680C2#7666311yescharacteristic
681C2D3×#39311yescharacteristic
682C2#7686712yes—
683C2#7686811yescharacteristic
684C2#7724911yescharacteristic
685C2#7725112yes—
686C2#7725711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1446C2
prime radicalC24×D3C23
hypercenter#1446D3
hypocenterC3#14934
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
4. C3×#1446C2semidirect
5. #72134C2semidirect
8. #69415C2semidirect
9. #69414C2semidirect
10. #69413C2semidirect
11. #69412C2semidirect
44. #1446D3semidirect
81. C22×C4×D3Q8semidirect
674. C3#14934semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
C2×#46642
C2×D3×#952
C3×#14462
#721342
(C2×D3)⋊#952
#694152
#694142
#694132
#694122
#149343