Finite Group Databasebeta

← #10729064

Subgroups — #10729064

Order 768.

1782 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic, complemented
2C2×#4989C212yesmaximal
3C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
4#71570C212yesmaximal
5#71570C211yesmaximal, characteristic
6#71570C211yesmaximal, characteristic
7#71568C211yesmaximal, characteristic, complemented
8#71567C211yesmaximal, characteristic, complemented
9#71566C212yesmaximal, complemented
10#71564C212yesmaximal, complemented
11#71534C211yesmaximal, characteristic, complemented
12#14916—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C2×C3×#125—28no—
15C2×C3×#125—24no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C42C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2212yes—
26C22×C4×Dic3C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4989—28no—
39C4×(C2×D3)⋊C4—24no—
40C22×#125D311yescharacteristic, complemented
41C2×#385—32no—
42C2×C4×C22⋊C4—31no—
43#1444—31no—
44#1442—31no—
45#1441—32no—
46#1439—31no—
47#1439—31no—
48#1439—32no—
49#1436—32no—
50#1434—31no—
51C24×D3C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C22×C4×D3—22no—
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—24no—
66C22×C4×D3—42no—
67C3×#125—232no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4D418yes—
78C2×C3×C42—24no—
79C2×C3×C42D414yes—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—42no—
91C2×(C2×D3)⋊C4Q814yes—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—42no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—42no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—42no—
107C3⋊#95—42no—
108C24×C4—31no—
109C2×#125—28no—
110C2×#125—24no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4C2×D312yes—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C4⋊C4C2×D312yes—
116C22×C22⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C42—31no—
121C22×C42C2×D311yescharacteristic
122#385—68no—
123C4×C22⋊C4—64no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95—32no—
135C24×C3C2411yescharacteristic
136C23×D3—21no—
137C23×D3—22no—
138C23×D3—21no—
139C23×D3—21no—
140C23×D3—22no—
141C23×D3—21no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3C2×Q812yes—
145C23×D3#4112yes—
146C23×D3#4112yes—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
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—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D414yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×Q812yes—
175C22×C3×C4#4112yes—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—21no—
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×Dic3C2×Q812yes—
202C22×Dic3#4112yes—
203C22×Dic3#4114yes—
204C22×Dic3#4112yes—
205C2×C4×D3—28no—
206C2×C4×D3—48no—
207C2×C4×D3—48no—
208C2×C4×D3—44no—
209C2×C4×D3—44no—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—216no—
213C3×C4⋊C4—216no—
214C3×C42—28no—
215(C2×D3)⋊C4—28no—
216(C2×D3)⋊C4—216no—
217(C2×D3)⋊C4—28no—
218(C2×D3)⋊C4—48no—
219Dic3⋊C4—48no—
220C4×Dic3—44no—
221C25—31no—
222C23×C4—32no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4—62no—
230C23×C4—64no—
231C23×C4—122no—
232C23×C4C22×D312yes—
233C23×C4C22×D312yes—
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D311yescharacteristic
237#125—232no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—122no—
250C2×C4⋊C4#6718yes—
251C2×C22⋊C4—34no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—64no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—122no—
263C2×C42—24no—
264C2×C42—62no—
265C2×C42—122no—
266C2×C42#6714yes—
267#95—64no—
268#95—64no—
269#95—64no—
270#95—62no—
271#95—62no—
272#95—62no—
273#95—62no—
274#95—64no—
275#95—64no—
276#95—122no—
277#95—122no—
278C23×C3C22×D411yescharacteristic
279C23×C3C22×D412yes—
280C23×C3C22×Q811yescharacteristic
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4111yescharacteristic
286C23×C3C2×#4112yes—
287C23×C3H(5,𝔽2)12yes—
288C23×C32−1+412yes—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—24no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—24no—
297C22×D3—24no—
298C22×D3—24no—
299C22×D3—24no—
300C22×D3—24no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3#12514yescomplemented
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4#12014yes—
340C2×C3×C4C22⋊Q814yes—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—41no—
351C2×Dic3—42no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C4×D3—416no—
371C24—32no—
372C24—32no—
373C24—32no—
374C24—61no—
375C24—62no—
376C24—61no—
377C24—62no—
378C24—61no—
379C24—62no—
380C24—61no—
381C24—62no—
382C24C23×D311yescharacteristic
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—34no—
407C22×C4—32no—
408C22×C4—68no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—128no—
435C22×C4—128no—
436C22×C4—124no—
437C22×C4—124no—
438C22×C4D3×Q812yes—
439C22×C4#23912yes—
440C22×C4C2×#6712yes—
441C22×C4C2×#6714yes—
442C22×C4C2×#6714yes—
443C22×C4C2×#6712yes—
444C22×C4C2×#6714yes—
445C22×C4C2×#6712yes—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—128no—
451C22⋊C4—68no—
452C22⋊C4—616no—
453C22⋊C4—68no—
454C22⋊C4—128no—
455C42—28no—
456C42—124no—
457C22×C3C2×#12011yescharacteristic
458C22×C3C2×C22⋊Q811yescharacteristic
459C22×C3#52511yescharacteristic
460C22×C3#52511yescharacteristic
461C22×C3#52512yes—
462C22×C3C2×#12511yescharacteristic
463C22×C3#52911yescharacteristic
464C22×C3#52911yescharacteristic
465C22×C3#53211yescharacteristic
466C22×C3#53212yes—
467C22×C3#53311yescharacteristic
468C22×C3#53511yescharacteristic
469C22×C3#53611yescharacteristic
470C22×C3#53912yes—
471C22×C3#53912yes—
472C22×C3#54612yes—
473C22×C3#54612yes—
474C22×C3#54812yes—
475C22×C3D4×Q812yes—
476C22×C3#55212yes—
477C22×C3#55212yes—
478C22×C3#55312yes—
479C22×C3#55412yes—
480C2×D3—24no—
481C2×D3—28no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—44no—
489C2×D3—44no—
490C2×D3—44no—
491C3×C4—24no—
492C3×C4—24no—
493C3×C4—42no—
494C3×C4—44no—
495C3×C4—44no—
496Dic3—42no—
497Dic3—42no—
498Dic3—42no—
499Dic3—44no—
500Dic3—82no—
501C23—34no—
502C23—62no—
503C23—64no—
504C23—62no—
505C23—62no—
506C23—64no—
507C23—64no—
508C23—62no—
509C23—64no—
510C23—64no—
511C23—62no—
512C23—62no—
513C23—64no—
514C23—122no—
515C23—122no—
516C23—122no—
517C23—121no—
518C23—122no—
519C23—122no—
520C23—121no—
521C23—122no—
522C23—121no—
523C23—121no—
524C23C2×D3×Q811yescharacteristic
525C23C2×#23911yescharacteristic
526C23#100712yes—
527C23D3×#4111yescharacteristic
528C23D3×#4111yescharacteristic
529C23D3×#4111yescharacteristic
530C23D3×#4111yescharacteristic
531C23D3×#4112yes—
532C23#100912yes—
533C23C22×#6711yescharacteristic
534C23C22×#6712yes—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—62no—
567C2×C4—64no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—64no—
571C2×C4—1216no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—121no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4#94614yes—
593C2×C4#95314yes—
594C2×C3#238811yescharacteristic
595C2×C3#240411yescharacteristic
596C2×C3#240911yescharacteristic
597C2×C3#246012yes—
598C2×C3#247312yes—
599C2×C3#255612yes—
600C2×C3#255812yes—
601C2×C3#256611yescharacteristic
602C2×C3#261111yescharacteristic
603C2×C3#263311yescharacteristic
604C2×C3#285111yescharacteristic
605D3—44no—
606C22—68no—
607C22—64no—
608C22—124no—
609C22—122no—
610C22—122no—
611C22—124no—
612C22—124no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—122no—
617C22D3×#12511yescharacteristic
618C22#568811yescharacteristic
619C22#569012yes—
620C22#569111yescharacteristic
621C22#569111yescharacteristic
622C22#569212yes—
623C22#569212yes—
624C22#569312yes—
625C22#569312yes—
626C22#569512yes—
627C22C2×#94611yescharacteristic
628C22Q8×#6712yes—
629C22#581611yescharacteristic
630C22#581712yes—
631C22#582812yes—
632C22#582811yescharacteristic
633C22#582811yescharacteristic
634C22#582912yes—
635C22#582912yes—
636C22#583011yescharacteristic
637C22#583212yes—
638C22#583211yescharacteristic
639C22C2×#95311yescharacteristic
640C4—24no—
641C4—24no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—122no—
646C4—124no—
647C4—122no—
648C4—122no—
649C4—242no—
650C3#1491611yescharacteristic, complemented, Sylow
651C2—124no—
652C2#8303212yes—
653C2#8303812yes—
654C2#8304011yescharacteristic
655C2#8305912yes—
656C2#8306212yes—
657C2#8306411yescharacteristic
658C2#8306611yescharacteristic
659C2#8307211yescharacteristic
660C2#8406811yescharacteristic
661C2#8409511yescharacteristic
662C2#8409611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#125C2
prime radicalC24×D3C23
hypercenterC22×#125D3
hypocenterC3#14916
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C22×C3×#125C2semidirect
7. #71568C2semidirect
8. #71567C2semidirect
9. #71566C2semidirect
10. #71564C2semidirect
11. #71534C2semidirect
40. C22×#125D3semidirect
311. C22×D3#125semidirect
650. C3#14916semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×#49892
C2×C4×(C2×D3)⋊C42
#715702
#715682
#715672
#715662
#715642
#715342
#149163