Finite Group Databasebeta

← #10729070

Subgroups — #10729070

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×#870C211yesmaximal, characteristic, complemented
2C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
3C2×#5210C212yesmaximal
4#71547C212yesmaximal
5#71546C211yesmaximal, characteristic
6#71546C211yesmaximal, characteristic
7#71536C212yesmaximal, complemented
8#71534C211yesmaximal, characteristic, complemented
9#69420C211yesmaximal, characteristic, complemented
10#69415C211yesmaximal, characteristic, complemented
11#69409C212yesmaximal, complemented
12#14916—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C2×#870—24no—
26C2×#870—28no—
27C3×C4×C22⋊C4—24no—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30#5210—28no—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C22×#125—31no—
41C2×#385—32no—
42C2×C4×C22⋊C4D311yescharacteristic
43#1444—31no—
44#1442—31no—
45#1441—32no—
46#1439—31no—
47#1439—31no—
48#1439—32no—
49#1436—32no—
50#1434—31no—
51C25×C3C2311yescharacteristic
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—24no—
56C23×C3×C4—42no—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C23×Dic3C2312yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4Q814yes—
73C2×C3×C42—24no—
74C2×C3×C42—22no—
75C2×C3×C42—42no—
76C2×C3×C42D414yes—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—24no—
82C2×#217—42no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—42no—
95C2×Dic3⋊C4D418yes—
96#870—232no—
97C3×#95—22no—
98C3×#95—22no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—42no—
107C3⋊#95—42no—
108C24×C4C2×D311yescharacteristic
109C2×#125—68no—
110C2×#125—64no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—32no—
116C22×C22⋊C4—32no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4C2×D311yescharacteristic
119C22×C22⋊C4C2×D311yescharacteristic
120C22×C42C2×D311yescharacteristic
121C22×C42C2×D311yescharacteristic
122#385—68no—
123C4×C22⋊C4—24no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95C2×D311yescharacteristic
134C2×#95C2×D311yescharacteristic
135C24×C3—22no—
136C24×C3—21no—
137C24×C3—22no—
138C24×C3—21no—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3C2×Q812yes—
144C24×C3#4112yes—
145C24×C3#4112yes—
146C24×C3C2411yescharacteristic
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—28no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—48no—
170C22×C3×C4—48no—
171C22×C3×C4—44no—
172C22×C3×C4—44no—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×Q812yes—
178C22×C3×C4C2×Q812yes—
179C22×C3×C4#4112yes—
180C22×C3×C4#4112yes—
181C22×C3×C4#4112yes—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
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×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3C2×D414yes—
208C22×Dic3C2×D414yes—
209C22×Dic3#4114yes—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—28no—
212C3×C42—28no—
213C3×C42—44no—
214#217—216no—
215#217—48no—
216C4⋊Dic3—28no—
217C4⋊Dic3—28no—
218Dic3⋊C4—216no—
219Dic3⋊C4—216no—
220Dic3⋊C4—48no—
221C25C22×D311yescharacteristic
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—32no—
229C23×C4—32no—
230C23×C4—42no—
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D311yescharacteristic
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D311yescharacteristic
237#125—632no—
238C2×C4⋊C4—38no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—122no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—122no—
262C2×C22⋊C4Dic614yes—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42#6714yes—
267#95—22no—
268#95—22no—
269#95—64no—
270#95—64no—
271#95—64no—
272#95—62no—
273#95—64no—
274#95—62no—
275#95—64no—
276#95—122no—
277#95—122no—
278C23×C3—22no—
279C23×C3—24no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—41no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3#12514yescomplemented
301C23×C3C22×D412yes—
302C23×C3C22×D411yescharacteristic
303C23×C3C22×Q811yescharacteristic
304C23×C3C2×#4111yescharacteristic
305C23×C3C2×#4111yescharacteristic
306C23×C3C2×#4111yescharacteristic
307C23×C3C2×#4111yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4112yes—
310C23×C3H(5,𝔽2)12yes—
311C23×C32−1+412yes—
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—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—416no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4#12014yes—
342C2×C3×C4C22⋊Q814yes—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
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—
370C2×Dic3—42no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—21no—
379C24C2×Dic612yes—
380C24#23512yes—
381C24#23512yes—
382C24C23×D311yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
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—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—34no—
406C22×C4—34no—
407C22×C4—34no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
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—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4C2×Dic612yes—
438C22×C4C2×Dic612yes—
439C22×C4#23512yes—
440C22×C4#23512yes—
441C22×C4#23512yes—
442C22×C4D3×D414yes—
443C22×C4C2×#6712yes—
444C22×C4C2×#6712yes—
445C22×C4C2×#6712yes—
446C4⋊C4—616no—
447C4⋊C4—68no—
448C4⋊C4—68no—
449C4⋊C4—616no—
450C4⋊C4—128no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—616no—
454C22⋊C4—128no—
455C42—28no—
456C42—44no—
457C22×C3—24no—
458C22×C3—28no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3C2×#12011yescharacteristic
469C22×C3C2×C22⋊Q811yescharacteristic
470C22×C3#52511yescharacteristic
471C22×C3#52511yescharacteristic
472C22×C3#52512yes—
473C22×C3C2×#12511yescharacteristic
474C22×C3#52911yescharacteristic
475C22×C3#52911yescharacteristic
476C22×C3#53212yes—
477C22×C3#53211yescharacteristic
478C22×C3#53311yescharacteristic
479C22×C3#53511yescharacteristic
480C22×C3#53611yescharacteristic
481C22×C3#53912yes—
482C22×C3#53912yes—
483C22×C3#54612yes—
484C22×C3#54612yes—
485C22×C3#54812yes—
486C22×C3D4×Q812yes—
487C22×C3#55212yes—
488C22×C3#55212yes—
489C22×C3#55312yes—
490C22×C3#55412yes—
491C3×C4—24no—
492C3×C4—24no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—42no—
497Dic3—44no—
498Dic3—44no—
499Dic3—44no—
500Dic3—82no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23#87014yescomplemented
524C23C22×Dic611yescharacteristic
525C23C2×#23511yescharacteristic
526C23C2×#23511yescharacteristic
527C23C2×#23511yescharacteristic
528C23C2×#23511yescharacteristic
529C23C2×#23511yescharacteristic
530C23C2×D3×D412yes—
531C23#100412yes—
532C23D3×#4112yes—
533C23#101012yes—
534C23C22×#6711yescharacteristic
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—416no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—64no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4#92414yes—
593C2×C4#93714yes—
594C2×C3—44no—
595C2×C3#238811yescharacteristic
596C2×C3#240411yescharacteristic
597C2×C3#240911yescharacteristic
598C2×C3#246012yes—
599C2×C3#247312yes—
600C2×C3#255612yes—
601C2×C3#255812yes—
602C2×C3#256611yescharacteristic
603C2×C3#261111yescharacteristic
604C2×C3#263311yescharacteristic
605C2×C3#285111yescharacteristic
606C22—28no—
607C22—24no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C2×#87011yescharacteristic
618C22#547011yescharacteristic
619C22#547911yescharacteristic
620C22#547911yescharacteristic
621C22#549012yes—
622C22#549012yes—
623C22#552912yes—
624C22#553112yes—
625C22D4×Dic612yes—
626C22#553912yes—
627C22#554712yes—
628C22#554812yes—
629C22#555212yes—
630C22#555812yes—
631C22#556412yes—
632C22#556412yes—
633C22C2×#92411yescharacteristic
634C22#579111yescharacteristic
635C22#579111yescharacteristic
636C22C2×#93711yescharacteristic
637C22#580511yescharacteristic
638C22#582511yescharacteristic
639C22#583211yescharacteristic
640C4—24no—
641C4—24no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—124no—
647C4—124no—
648C4—124no—
649C4—242no—
650C3#1491611yescharacteristic, complemented, Sylow
651C2—44no—
652C2#7349712yes—
653C2#7350112yes—
654C2#7353612yes—
655C2#7353712yes—
656C2#8207411yescharacteristic
657C2#8212511yescharacteristic
658C2#8226111yescharacteristic
659C2#8234911yescharacteristic
660C2#8238611yescharacteristic
661C2#8239111yescharacteristic
662C2#8240011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×C4×C22⋊C4C2
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#14916
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#870C2semidirect
7. #71536C2semidirect
8. #71534C2semidirect
9. #69420C2semidirect
10. #69415C2semidirect
11. #69409C2semidirect
300. C23×C3#125semidirect
523. C23#870semidirect
650. C3#14916semidirect

Maximal subgroups

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

subgroupindex
C22×#8702
C2×C3×C4×C22⋊C42
C2×#52102
#715472
#715462
#715362
#715342
#694202
#694152
#694092
#149163