Finite Group Databasebeta

← #10876200

Subgroups — #10876200

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2297C211yesmaximal, characteristic
2#82060C211yesmaximal, characteristic, complemented
3#72135C211yesmaximal, characteristic
4#72135C211yesmaximal, characteristic
5#71576C211yesmaximal, characteristic, complemented
6#71576C211yesmaximal, characteristic, complemented
7#71531C211yesmaximal, characteristic, complemented
8#69686C211yesmaximal, characteristic, complemented
9#69671C211yesmaximal, characteristic, complemented
10#69667C211yesmaximal, characteristic, complemented
11#69666C211yesmaximal, characteristic, complemented
12#69654C211yesmaximal, characteristic, complemented
13#69641C211yesmaximal, characteristic, complemented
14#69637C211yesmaximal, characteristic, complemented
15#69636C211yesmaximal, characteristic, complemented
16#27590—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×#391C2211yescharacteristic
21C3×#391C2211yescharacteristic
22C3×#390C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#386—21no—
25C3×#386—21no—
26C3×C4×C22⋊C4—21no—
27C3×C4×C22⋊C4—21no—
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36#5212C2211yescharacteristic
37#5212C2211yescharacteristic
38#5211—21no—
39C4×#217—21no—
40C2×#830C2211yescharacteristic
41C2×#830C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4937—21no—
47#4936C2211yescharacteristic
48#4936C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4932—21no—
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2297D311yescharacteristic
61#2283—31no—
62#2097—31no—
63#2092—31no—
64#2074—31no—
65#2073—31no—
66#1928—31no—
67#1927—31no—
68#1926—31no—
69#1925—31no—
70#1854—31no—
71#1854—31no—
72#1817—31no—
73#1788—31no—
74#1788—31no—
75C2×C3×#41—22no—
76C2×C3×#41—22no—
77C2×C3×#41—22no—
78C2×C3×#41—22no—
79C2×C3×#41—41no—
80C2×C3×#41—41no—
81C22×C3×Q8C2311yescharacteristic
82C22×C3×D4—21no—
83C22×C3×D4C2311yescharacteristic
84C23×C3×C4—21no—
85C23×C3×C4D411yescharacteristic
86C23×C3×C4D411yescharacteristic
87C23×C3×C4C2311yescharacteristic
88C3×#117—41no—
89C3×#117—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—22no—
98C2×C3×C42—22no—
99C2×C3×C42—21no—
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105C2×#217—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120#835—22no—
121#835—22no—
122#835—41no—
123#835—41no—
124#832—22no—
125#832—22no—
126#832—41no—
127#832—41no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—22no—
137#830—22no—
138#808—41no—
139#808—41no—
140#807—41no—
141#807—41no—
142C22×#41C2×D311yescharacteristic
143C2×#117C2×D311yescharacteristic
144C22×C42C2×D311yescharacteristic
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#391C2×D311yescharacteristic
160#391C2×D311yescharacteristic
161#390C2×D311yescharacteristic
162#390C2×D311yescharacteristic
163#386—21no—
164#386—21no—
165#383—61no—
166#382—31no—
167#382—31no—
168#381—61no—
169#380—31no—
170#380—31no—
171C4×C4⋊C4—31no—
172C4×C4⋊C4—31no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—61no—
176#376—61no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3—21no—
186C24×C3C2×D411yescharacteristic
187C3×#41—44no—
188C3×#41—44no—
189C3×#41—44no—
190C3×#41—44no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8#4112yes—
197C2×C3×D4—24no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—42no—
204C2×C3×D4—42no—
205C2×C3×D4#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—24no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—21no—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—41no—
218C22×C3×C4—41no—
219C22×C3×C4—41no—
220C22×C3×C4—42no—
221C22×C3×C4—42no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4#4112yes—
228C22×C3×C4#4112yes—
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250#217—42no—
251#217—42no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—22no—
278C2×#41—41no—
279C2×#41—41no—
280C22×Q8C22×D311yescharacteristic
281C22×D4—21no—
282C22×D4C22×D311yescharacteristic
283C23×C4—21no—
284C23×C4#6711yescharacteristic
285C23×C4#6711yescharacteristic
286C23×C4C22×D311yescharacteristic
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8—31no—
291#117—41no—
292#117—41no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42#6711yescharacteristic
312C2×C42#6711yescharacteristic
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#107—121no—
316#107—121no—
317#106—121no—
318#106—121no—
319#103—62no—
320#103—62no—
321#103—121no—
322#103—121no—
323#102—62no—
324#102—62no—
325#102—121no—
326#102—121no—
327C22⋊C8—62no—
328C22⋊C8—62no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—22no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3#12312yes—
355C23×C3C22×D411yescharacteristic
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—48no—
363C3×D4—41no—
364C3×D4—41no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—44no—
384C2×C3×C4—48no—
385C2×C3×C4—44no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4#12312yes—
395C2×C3×C4#13512yes—
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401C2×C3×C4H(5,𝔽2)11yescharacteristic
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—21no—
419C24C2×#6711yescharacteristic
420#41—44no—
421#41—44no—
422#41—44no—
423#41—44no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#23512yes—
430C2×D4—24no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4#23512yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4#23712yes—
460C22×C4#23712yes—
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C2×#6711yescharacteristic
464C22×C4C2×#6711yescharacteristic
465C22×C4C2×#6711yescharacteristic
466C22×C4C23×D311yescharacteristic
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C42—22no—
499C42—22no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—122no—
506C42—122no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3C2×#12311yescharacteristic
522C22×C3#52511yescharacteristic
523C22×C3#53511yescharacteristic
524C22×C3C2×#13511yescharacteristic
525C22×C3#57511yescharacteristic
526C22×C3#57511yescharacteristic
527C22×C3#57511yescharacteristic
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—41no—
533C3×C4—42no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4#55111yescharacteristic
537C3×C4#55111yescharacteristic
538C3×C4#55111yescharacteristic
539C3×C4#55111yescharacteristic
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543Dic3—81no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23#93512yes—
558C23C22×#6711yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563D4—22no—
564D4—22no—
565D4—48no—
566D4—41no—
567D4—41no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4#93512yes—
610C2×C4#95012yes—
611C2×C4C2×#23511yescharacteristic
612C2×C4C2×#23511yescharacteristic
613C2×C4C2×#23711yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4#100411yescharacteristic
616C2×C4#100411yescharacteristic
617C8—121no—
618C8—121no—
619C8—121no—
620C8—121no—
621C2×C3—42no—
622C2×C3—82no—
623C2×C3#248211yescharacteristic
624C2×C3#309111yescharacteristic
625C2×C3#309311yescharacteristic
626C2×C3#310211yescharacteristic
627C2×C3#311111yescharacteristic
628C2×C3#321211yescharacteristic
629C2×C3#321911yescharacteristic
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22#579111yescharacteristic
645C22C2×#93511yescharacteristic
646C22#580511yescharacteristic
647C22C2×#95011yescharacteristic
648C22#582211yescharacteristic
649C22#582211yescharacteristic
650C22#582211yescharacteristic
651C4—22no—
652C4—22no—
653C4—42no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—81no—
658C4—81no—
659C4—241no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4#549511yescharacteristic
664C4#549511yescharacteristic
665C4#549511yescharacteristic
666C4#549511yescharacteristic
667C3#2759011yescharacteristic, complemented, Sylow
668C2—42no—
669C2—82no—
670C2#8211311yescharacteristic
671C2#8252811yescharacteristic
672C2#8253011yescharacteristic
673C2#8291311yescharacteristic
674C2#8291511yescharacteristic
675C2#8330511yescharacteristic
676C2#8330911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2297C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#2297D3
hypocenterC3#27590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #82060C2semidirect
5. #71576C2semidirect
6. #71576C2semidirect
7. #71531C2semidirect
8. #69686C2semidirect
9. #69671C2semidirect
10. #69667C2semidirect
11. #69666C2semidirect
12. #69654C2semidirect
13. #69641C2semidirect
14. #69637C2semidirect
15. #69636C2semidirect
667. C3#27590semidirect

Maximal subgroups

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

subgroupindex
C3×#22972
#820602
#721352
#715762
#715312
#696862
#696712
#696672
#696662
#696542
#696412
#696372
#696362
#275903