Finite Group Databasebeta

← #10728197

Subgroups — #10728197

Order 768.

1534 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
2C2×#5210C211yesmaximal, characteristic
3C2×#5212C211yesmaximal, characteristic
4C2×#4659C211yesmaximal, characteristic, complemented
5#71549C212yesmaximal
6#71549C212yesmaximal
7#71536C212yesmaximal, complemented
8#71534C212yesmaximal, complemented
9#69422C211yesmaximal, characteristic, complemented
10#69412C211yesmaximal, characteristic, complemented
11#69410C211yesmaximal, characteristic, complemented
12#14841—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C42C2212yes—
16C22×#217C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C3×C4×C22⋊C4—24no—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26#5212—24no—
27#5210—24no—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4659—24no—
40C2×#400—31no—
41C2×#385—31no—
42C2×#380—31no—
43C2×C4×C22⋊C4D311yescharacteristic
44#1445—31no—
45#1442—31no—
46#1441—31no—
47#1440—32no—
48#1440—32no—
49#1436—32no—
50#1434—32no—
51C25×C3C2311yescharacteristic
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—24no—
55C23×C3×C4—42no—
56C23×C3×C4D414yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C42—24no—
71C2×C3×C42—24no—
72C2×C3×C42—42no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—42no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—42no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—24no—
91C3×#95—22no—
92C3×#95D414yes—
93C3⋊#95—24no—
94C3⋊#95—24no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C24×C4C2×D311yescharacteristic
105C22×C4⋊C4—31no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—31no—
109C22×C22⋊C4—32no—
110C22×C22⋊C4—31no—
111C22×C22⋊C4—31no—
112C22×C22⋊C4C2×D312yes—
113C22×C42C2×D312yes—
114#400—64no—
115#385—64no—
116#380—64no—
117C4×C22⋊C4—24no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95C2×D311yescharacteristic
130C2×#95C2×D311yescharacteristic
131C24×C3—21no—
132C24×C3—21no—
133C24×C3—21no—
134C24×C3—22no—
135C24×C3—21no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3—21no—
139C24×C3—21no—
140C24×C3C2×D412yes—
141C24×C3#4114yes—
142C24×C3C2411yescharacteristic
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
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—28no—
161C22×C3×C4—28no—
162C22×C3×C4—44no—
163C22×C3×C4—48no—
164C22×C3×C4—48no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4#4114yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4114yes—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
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—21no—
205C22×Dic3—21no—
206C22×Dic3#4112yes—
207C22×Dic3#4112yes—
208C3×C22⋊C4—216no—
209C3×C42—48no—
210#217—28no—
211#217—44no—
212#217—48no—
213C4⋊Dic3—44no—
214C4⋊Dic3—44no—
215Dic3⋊C4—44no—
216Dic3⋊C4—44no—
217C25C22×D311yescharacteristic
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4—32no—
225C23×C4—32no—
226C23×C4—31no—
227C23×C4—42no—
228C23×C4D1214yes—
229C23×C4C22×D312yes—
230C23×C4C22×D312yes—
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D311yescharacteristic
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—122no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—64no—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—122no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—42no—
257#95—22no—
258#95—64no—
259#95—64no—
260#95—64no—
261#95—62no—
262#95—62no—
263#95—64no—
264#95—64no—
265#95—64no—
266#95—64no—
267#95—122no—
268#95—122no—
269#95#6714yes—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3#12414yes—
293C23×C3C22×D411yescharacteristic
294C23×C3C22×D411yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4112yes—
297C23×C3C2×#4112yes—
298C23×C3C2×#4112yes—
299C23×C3C2×#4111yescharacteristic
300C23×C3C2×#4111yescharacteristic
301C23×C3H(5,𝔽2)12yes—
302C23×C32−1+411yescharacteristic
303C23×C32−1+411yescharacteristic
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—416no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
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#12114yes—
328C2×C3×C4#12314yes—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—42no—
343C2×Dic3—41no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—41no—
353C2×Dic3—42no—
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—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24C2×D1212yes—
375C24#23514yes—
376C24C23×D311yescharacteristic
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
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—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—32no—
397C22×C4—32no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—61no—
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—61no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4C2×D1212yes—
436C22×C4C2×D1212yes—
437C22×C4#23514yes—
438C22×C4#23512yes—
439C22×C4#23714yes—
440C22×C4C2×#6714yes—
441C22×C4C2×#6712yes—
442C4⋊C4—124no—
443C4⋊C4—124no—
444C4⋊C4—124no—
445C4⋊C4—124no—
446C22⋊C4—216no—
447C22⋊C4—68no—
448C22⋊C4—124no—
449C22⋊C4—128no—
450C42—48no—
451C22×C3—24no—
452C22×C3—24no—
453C22×C3—24no—
454C22×C3—44no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3C2×#12111yescharacteristic
463C22×C3C2×#12311yescharacteristic
464C22×C3#52511yescharacteristic
465C22×C3#52512yes—
466C22×C3C2×#12411yescharacteristic
467C22×C3#52912yes—
468C22×C3#52912yes—
469C22×C3#52912yes—
470C22×C3#53211yescharacteristic
471C22×C3#53411yescharacteristic
472C22×C3#53611yescharacteristic
473C22×C3#53912yes—
474C22×C3#54212yes—
475C22×C3#54612yes—
476C22×C3#54711yescharacteristic
477C22×C3#54811yescharacteristic
478C22×C3#55112yes—
479C22×C3#55211yescharacteristic
480C22×C3#55212yes—
481C22×C3#55212yes—
482C22×C3#55312yes—
483C22×C3#55312yes—
484C22×C3#55611yescharacteristic
485C3×C4—24no—
486C3×C4—24no—
487C3×C4—44no—
488C3×C4—44no—
489Dic3—42no—
490Dic3—42no—
491Dic3—82no—
492Dic3—82no—
493Dic3—81no—
494Dic3—81no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23#87514yes—
518C23C22×D1211yescharacteristic
519C23C2×#23512yes—
520C23C2×#23512yes—
521C23C2×#23511yescharacteristic
522C23C2×#23712yes—
523C23#100412yes—
524C23D3×#4111yescharacteristic
525C23D3×#4111yescharacteristic
526C23#101011yescharacteristic
527C23#101011yescharacteristic
528C23C22×#6711yescharacteristic
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—122no—
563C2×C4—121no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—121no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—121no—
577C2×C4—122no—
578C2×C4—121no—
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#88614yes—
589C2×C4#93014yes—
590C2×C3—44no—
591C2×C3#239511yescharacteristic
592C2×C3#241211yescharacteristic
593C2×C3#241511yescharacteristic
594C2×C3#246012yes—
595C2×C3#246812yes—
596C2×C3#250311yescharacteristic
597C2×C3#253012yes—
598C2×C3#257712yes—
599C2×C3#259111yescharacteristic
600C2×C3#266311yescharacteristic
601C2×C3#285011yescharacteristic
602C22—24no—
603C22—24no—
604C22—24no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22C2×#87511yescharacteristic
614C22#547711yescharacteristic
615C22#547912yes—
616C22C2×#88611yescharacteristic
617C22#549311yescharacteristic
618C22#549512yes—
619C22#552411yescharacteristic
620C22#552611yescharacteristic
621C22#552911yescharacteristic
622C22#553311yescharacteristic
623C22#553812yes—
624C22#554012yes—
625C22#554012yes—
626C22#554112yes—
627C22#554112yes—
628C22#555712yes—
629C22#556112yes—
630C22#556412yes—
631C22C2×#93011yescharacteristic
632C22#579112yes—
633C22#580612yes—
634C22#582411yescharacteristic
635C22#583111yescharacteristic
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—122no—
641C4—122no—
642C4—242no—
643C4—241no—
644C4—242no—
645C4—241no—
646C3#1484111yescharacteristic, complemented, Sylow
647C2—44no—
648C2#7351811yescharacteristic
649C2#7353212yes—
650C2#7354512yes—
651C2#7355011yescharacteristic
652C2#7358311yescharacteristic
653C2#7361211yescharacteristic
654C2#8207711yescharacteristic
655C2#8210811yescharacteristic
656C2#8227711yescharacteristic
657C2#8236612yes—
658C2#8237112yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×C4×C22⋊C4C2
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#14841
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#4659C2semidirect
7. #71536C2semidirect
8. #71534C2semidirect
9. #69422C2semidirect
10. #69412C2semidirect
11. #69410C2semidirect
646. C3#14841semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C42
C2×#52102
C2×#52122
C2×#46592
#715492
#715362
#715342
#694222
#694122
#694102
#148413