Finite Group Databasebeta

← #10826319

Subgroups — #10826319

Order 768.

1760 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#216C211yesmaximal, characteristic, complemented
2C2×#5552C211yesmaximal, characteristic
3C3×#2409C211yesmaximal, characteristic, complemented
4C2×#5227C211yesmaximal, characteristic, complemented
5C2×#5047C212yesmaximal, complemented
6#71700C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic, complemented
8#69690C212yesmaximal, complemented
9#69689C212yesmaximal, complemented
10#69525C211yesmaximal, characteristic, complemented
11#69515C211yesmaximal, characteristic, complemented
12#23859—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C22×#216—24no—
19C22×#216—28no—
20C22×#216C2212yes—
21C22×#216C2212yes—
22C22×#215C2212yes—
23C2×#939—21no—
24C2×#930C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C23×C3⋊C8C2211yescharacteristic
27#5552—24no—
28#5552—24no—
29C2×#896—21no—
30C2×#885—21no—
31C2×#67⋊C4—21no—
32C2×#875C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#395—21no—
35C3×#393—21no—
36C3×#391C2212yes—
37C3×#386—21no—
38C3×#385—21no—
39C3×#380C2211yescharacteristic
40#5227—24no—
41C2×#835C2212yes—
42C2×#830C2212yes—
43C2×#830C2211yescharacteristic
44#5047—28no—
45C2×#810C2212yes—
46C2×#810C2212yes—
47C2×#809C2212yes—
48#4529C2212yes—
49#4524C2212yes—
50C23×Q16—31no—
51C2×#548—31no—
52C2×#452—32no—
53C2×#448—31no—
54#2409D311yescharacteristic, complemented
55#2015—32no—
56#2001—32no—
57#1893—31no—
58#1878—32no—
59#1869—31no—
60#1787—31no—
61C22×C3×Q8—21no—
62C22×C3×Q8—24no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8—42no—
66C22×C3×Q8C2312yes—
67C22×C3×D4D411yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C22×#67—21no—
72C22×#67—21no—
73C23×Dic3—21no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235—24no—
77C2×#235—42no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6—24no—
81C22×Dic6—21no—
82C22×Dic6—22no—
83C22×Dic6—42no—
84C22×Dic6D414yes—
85C22×Dic6C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—42no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#216—232no—
101C2×#216—24no—
102C2×#216—24no—
103C2×#216—24no—
104C2×#216—24no—
105C2×#216—24no—
106C2×#216D418yescomplemented
107C2×#215—24no—
108C2×#215—24no—
109C2×#215—42no—
110#939—28no—
111#930—22no—
112#930D414yes—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×C4⋊Dic3C2311yescharacteristic
117#924—22no—
118#924—22no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×C4×Dic3—21no—
122C22×C3⋊C8—22no—
123C22×C3⋊C8—24no—
124C22×C3⋊C8C2312yes—
125C22×C3⋊C8C2312yes—
126#896—28no—
127#885—28no—
128#67⋊C4—28no—
129#875—22no—
130#875—22no—
131C4×Dic6—22no—
132C4×Dic6—22no—
133C3×#95—21no—
134C3×#95—22no—
135C3×#95—22no—
136#835—24no—
137#830—22no—
138#830—24no—
139#810—24no—
140#810—42no—
141#809—24no—
142C22×#41—31no—
143C23×Q8—31no—
144C23×Q8C2×D311yescharacteristic
145C22×Q16—32no—
146C22×Q16—32no—
147C22×Q16—68no—
148C22×Q16—64no—
149C22×SD16—32no—
150C23×C8—31no—
151#548—64no—
152#548—64no—
153C2×#124—31no—
154C2×#124—61no—
155C2×C22⋊Q8—31no—
156C2×C22⋊Q8—61no—
157C2×C22⋊Q8C2×D312yes—
158C2×#121—31no—
159C2×#121—61no—
160C2×C4×Q8—31no—
161C2×C4×D4—61no—
162C2×C4×D4C2×D311yescharacteristic
163#452—68no—
164#448—64no—
165C2×#103—32no—
166C2×#103—32no—
167C2×#103—32no—
168C2×#102—32no—
169C2×C22⋊C8—31no—
170C2×C22⋊C8—32no—
171#395—21no—
172#393—21no—
173#391C2×D312yes—
174#386—21no—
175#385—21no—
176#380C2×D311yescharacteristic
177#340—32no—
178#336—32no—
179C24×C3C2×D411yescharacteristic
180C24×C3C2×D411yescharacteristic
181C23×D3—21no—
182C2×C3×Q8—22no—
183C2×C3×Q8—28no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—44no—
189C2×C3×Q8—48no—
190C2×C3×Q8—44no—
191C2×C3×Q8—42no—
192C2×C3×Q8C2×D414yes—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—44no—
205C22×C3×C4—42no—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×#67—24no—
214C2×#67—24no—
215C2×#67—44no—
216C2×#67—42no—
217C2×#67—42no—
218C22×Dic3—21no—
219C22×Dic3—24no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—44no—
223#235—48no—
224#235—44no—
225#235—44no—
226C2×D12—22no—
227C2×D12—22no—
228C2×D12—21no—
229C2×D12C2×D412yes—
230C2×C4×D3—24no—
231C2×C4×D3—44no—
232C2×Dic6—21no—
233C2×Dic6—22no—
234C2×Dic6—22no—
235C2×Dic6—22no—
236C2×Dic6—28no—
237C2×Dic6—24no—
238C2×Dic6—21no—
239C2×Dic6—44no—
240C2×Dic6—48no—
241C2×Dic6—44no—
242C2×Dic6C2×D412yes—
243C2×Dic6C2×D414yes—
244C2×Dic6C2×D412yes—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—44no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—44no—
252C3×C22⋊C4—42no—
253C3×C42—21no—
254C3×C42—21no—
255#217—24no—
256#216—216no—
257#216—216no—
258#215—48no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—24no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2×D412yes—
264Dic3⋊C4—24no—
265Dic3⋊C4—24no—
266C4×Dic3—24no—
267C2×C3⋊C8—28no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273C2×C3⋊C8C2×D414yes—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—64no—
277C2×#41—122no—
278C22×Q8—21no—
279C22×Q8—24no—
280C22×Q8—22no—
281C22×Q8—22no—
282C22×Q8—34no—
283C22×Q8—31no—
284C22×Q8—31no—
285C22×Q8—42no—
286C22×Q8—64no—
287C22×Q8—62no—
288C22×Q8—61no—
289C22×Q8—122no—
290C22×Q8C22×D312yes—
291C22×D4—31no—
292C22×D4—61no—
293C22×D4—61no—
294C22×D4#6711yescharacteristic
295C23×C4—21no—
296C23×C4—61no—
297C23×C4—61no—
298C23×C4C22×D311yescharacteristic
299C23×C4C22×D311yescharacteristic
300C2×Q16—38no—
301C2×Q16—64no—
302C2×Q16—64no—
303C2×Q16—64no—
304C2×Q16—64no—
305C2×Q16—632no—
306C2×Q16—64no—
307C2×SD16—64no—
308C2×SD16—64no—
309C2×SD16—122no—
310C22×C8—32no—
311C22×C8—32no—
312C22×C8—64no—
313C22×C8—62no—
314#124—62no—
315#124—62no—
316#124—68no—
317C22⋊Q8—24no—
318C22⋊Q8—42no—
319C22⋊Q8—68no—
320C22⋊Q8—62no—
321C22⋊Q8—62no—
322#121—34no—
323#121—68no—
324#121—62no—
325C4×Q8—62no—
326C4×Q8—62no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—68no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4#6711yescharacteristic
335C2×C4⋊C4C22×D312yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4#6711yescharacteristic
343C2×C22⋊C4#6711yescharacteristic
344C2×C42—61no—
345C2×C42C22×D311yescharacteristic
346#103—64no—
347#103—64no—
348#103—122no—
349#102—64no—
350C22⋊C8—64no—
351C22⋊C8—62no—
352#95—21no—
353#95—22no—
354#95—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—21no—
358C23×C3—22no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—22no—
362C23×C3#12012yes—
363C23×C3#13512yes—
364C23×C3#13712yes—
365C23×C3C22×D411yescharacteristic
366C22×D3—22no—
367C22×D3—42no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—42no—
371C3×Q8—24no—
372C3×Q8—24no—
373C3×Q8—48no—
374C3×Q8—42no—
375C3×Q8—42no—
376C3×D4—44no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—24no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—48no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4#12012yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D412yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4H(5,𝔽2)12yes—
403#67—48no—
404#67—48no—
405C2×Dic3—22no—
406C2×Dic3—24no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—48no—
410C2×Dic3—42no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413D12—24no—
414D12—41no—
415D12—41no—
416C4×D3—48no—
417Dic6—22no—
418Dic6—24no—
419Dic6—22no—
420Dic6—22no—
421Dic6—48no—
422Dic6—42no—
423Dic6#12014yes—
424C3⋊C8—24no—
425C3⋊C8—42no—
426C24—61no—
427C24C2×#6711yescharacteristic
428C24C2×#6711yescharacteristic
429#41—128no—
430#41—124no—
431#41—124no—
432C2×Q8—22no—
433C2×Q8—28no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—34no—
439C2×Q8—32no—
440C2×Q8—32no—
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—68no—
446C2×Q8—64no—
447C2×Q8—61no—
448C2×Q8—61no—
449C2×Q8—62no—
450C2×Q8—62no—
451C2×Q8—62no—
452C2×Q8—124no—
453C2×Q8—124no—
454C2×Q8—128no—
455C2×Q8D3×D414yes—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—32no—
460C2×D4—64no—
461C2×D4—64no—
462C2×D4—62no—
463C2×D4—61no—
464C2×D4—62no—
465C2×D4—124no—
466C2×D4—122no—
467C2×D4—122no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—64no—
479C22×C4—61no—
480C22×C4—64no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—124no—
484C22×C4—124no—
485C22×C4D3×D412yes—
486C22×C4D3×D412yes—
487C22×C4C2×#6711yescharacteristic
488C22×C4C2×#6711yescharacteristic
489C22×C4C2×#6711yescharacteristic
490C22×C4C2×#6711yescharacteristic
491C22×C4C23×D311yescharacteristic
492Q16—616no—
493Q16—616no—
494SD16—128no—
495C2×C8—34no—
496C2×C8—68no—
497C2×C8—62no—
498C2×C8—62no—
499C2×C8—62no—
500C2×C8—62no—
501C2×C8—62no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—32no—
506C4⋊C4—44no—
507C4⋊C4—64no—
508C4⋊C4—64no—
509C4⋊C4—61no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—64no—
515C22⋊C4—64no—
516C22⋊C4—64no—
517C22⋊C4—64no—
518C42—21no—
519C42—21no—
520C42—64no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—24no—
524C22×C3—24no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3#45214yescomplemented
531C22×C3C2×#12011yescharacteristic
532C22×C3D4212yes—
533C22×C3C2×#13511yescharacteristic
534C22×C3C2×#13711yescharacteristic
535C22×C3#57812yes—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—82no—
540C2×D3—82no—
541C3×C4—22no—
542C3×C4—41no—
543C3×C4—44no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—82no—
547C3×C4C2×#12011yescharacteristic
548C3×C4#53411yescharacteristic
549C3×C4D4212yes—
550Dic3—24no—
551Dic3—42no—
552Dic3—42no—
553C23—22no—
554C23—22no—
555C23—21no—
556C23—22no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—62no—
561C23—122no—
562C23—121no—
563C23—121no—
564C23—122no—
565C23#93712yes—
566C23#94212yes—
567C23#95012yes—
568C23C22×#6711yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—34no—
572Q8—48no—
573Q8—42no—
574Q8—42no—
575Q8—62no—
576Q8—64no—
577Q8—62no—
578Q8—62no—
579Q8—128no—
580Q8—122no—
581D4—44no—
582D4—64no—
583D4—128no—
584D4—128no—
585D4—121no—
586D4—121no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—128no—
613C2×C4—128no—
614C2×C4—122no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4#93712yes—
618C2×C4C2×D3×D411yescharacteristic
619C2×C4C2×D3×D412yes—
620C2×C4C2×D3×D411yescharacteristic
621C2×C4#100412yes—
622C8—64no—
623C8—122no—
624C2×C3—24no—
625C2×C3—81no—
626C2×C3#240311yescharacteristic
627C2×C3C2×#45211yescharacteristic
628C2×C3#301111yescharacteristic
629C2×C3#327012yes—
630C2×C3#328512yes—
631D3—82no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—42no—
638C22—42no—
639C22—41no—
640C22—41no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—242no—
645C22—242no—
646C22#504714yescomplemented
647C22C2×#93711yescharacteristic
648C22D4×#6712yes—
649C22C2×#94211yescharacteristic
650C22C2×#95011yescharacteristic
651C22#583812yes—
652C4—22no—
653C4—41no—
654C4—41no—
655C4—44no—
656C4—41no—
657C4—64no—
658C4—82no—
659C4—122no—
660C4—122no—
661C4D3×#12011yescharacteristic
662C4#558911yescharacteristic
663C4#559012yes—
664C3#2385911yescharacteristic, complemented, Sylow
665C2—24no—
666C2—81no—
667C2—242no—
668C2#8260411yescharacteristic
669C2C2×#504711yescharacteristic
670C2#8341111yescharacteristic
671C2#8354912yes—
672C2#8356512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2409C2
socleC23×C3C22×D4
prime radicalC2×#5552C2
hypercenter#2409D3
hypocenterC3#23859
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
1. C23×#216C2semidirect
3. C3×#2409C2semidirect
4. C2×#5227C2semidirect
5. C2×#5047C2semidirect
6. #71700C2semidirect
7. #71610C2semidirect
8. #69690C2semidirect
9. #69689C2semidirect
10. #69525C2semidirect
11. #69515C2semidirect
54. #2409D3semidirect
106. C2×#216D4semidirect
530. C22×C3#452semidirect
646. C22#5047semidirect
664. C3#23859semidirect

Maximal subgroups

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

subgroupindex
C23×#2162
C2×#55522
C3×#24092
C2×#52272
C2×#50472
#717002
#716102
#696902
#696892
#695252
#695152
#238593