Finite Group Databasebeta

← #10972844

Subgroups — #10972844

Order 768.

2294 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5680C211yesmaximal, characteristic
2C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
3C3×#2632C211yesmaximal, characteristic, complemented
4#82977C211yesmaximal, characteristic, complemented
5#82486C211yesmaximal, characteristic, complemented
6#78045C212yesmaximal, complemented
7#78029C212yesmaximal, complemented
8#77255C212yesmaximal, complemented
9#77191C212yesmaximal, complemented
10#77121C212yesmaximal
11#76943C212yesmaximal, complemented
12#76806C212yesmaximal, complemented
13#76780C212yesmaximal, complemented
14#75959C214yesmaximal, complemented
15#75460C214yesmaximal, complemented
16#74278C211yesmaximal, characteristic, complemented
17#74258C211yesmaximal, characteristic, complemented
18#35245—31nomaximal, Sylow
19C22×D3×Q8C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×#946C2211yescharacteristic
23C2×Q8×Dic3C2211yescharacteristic
24C2×Q8×Dic3C2211yescharacteristic
25C2×#938C2211yescharacteristic
26C2×D4×Dic3C2211yescharacteristic
27#5680—28no—
28C4×D3×Q8—28no—
29C2×D3×C4⋊C4C2212yes—
30C2×D3×C4⋊C4C2211yescharacteristic
31C2×#887C2212yes—
32C2×#887C2211yescharacteristic
33C2×#883C2212yes—
34C2×#67⋊C4C2212yes—
35C2×#879C2212yes—
36C2×#878C2214yes—
37C2×#877C2212yes—
38C2×C42×D3C2212yes—
39C2×C42×D3C2211yescharacteristic
40C2×#870C2211yescharacteristic
41C2×C4×Dic6C2211yescharacteristic
42C2×C4×Dic6C2212yes—
43C3×#399C2212yes—
44C3×#395C2212yes—
45C3×#394C2212yes—
46C3×#393C2212yes—
47C3×#388C2214yes—
48C3×#383C2211yescharacteristic
49#5230C2212yes—
50#5230C2212yes—
51#5222C2212yes—
52#4989C2214yes—
53#4986C2212yes—
54#4985C2212yes—
55#4981C2212yes—
56#4980C2214yes—
57#4976C2214yes—
58#4972C2214yes—
59#4956C2214yes—
60#4956C2214yes—
61#4951C2214yes—
62#4951C2214yes—
63#4949C2212yes—
64#4947C2212yes—
65#4945C2214yes—
66#4944C2214yes—
67#4944C2214yes—
68#4942C2214yes—
69#4937C2211yescharacteristic
70#4933C2214yes—
71#4668C2214yes—
72#4667C2212yes—
73#4666C2212yes—
74#4664C2212yes—
75#4661C2212yes—
76#4659C2212yes—
77#4659C2212yes—
78#4656C2212yes—
79#4655C2214yes—
80#4653C2214yes—
81#4650C2212yes—
82#4649C2212yes—
83#4648C2212yes—
84#4647C2214yes—
85#4646C2214yes—
86C2×#529—31no—
87C22×C4×Q8—31no—
88#2853—34no—
89#2634—31no—
90#2634—31no—
91#2632—31no—
92#2632D311yescharacteristic, complemented
93#2604—32no—
94#2604—32no—
95#2591—32no—
96#2583—32no—
97#2581—32no—
98#2536—32no—
99#2487—32no—
100#2409—32no—
101#2323—34no—
102#2283—31no—
103C22×C3×Q8C2311yescharacteristic
104C22×C3×Q8C2311yescharacteristic
105C22×C3×D4C2311yescharacteristic
106C22×#67C2312yes—
107C23×Dic3C2312yes—
108C2×D3×Q8—24no—
109C2×D3×Q8—28no—
110C22×C4×D3C2312yes—
111C22×C4×D3C2311yescharacteristic
112C22×C4×D3C2311yescharacteristic
113C22×C4×D3C2312yes—
114C22×C4×D3C2311yescharacteristic
115C22×Dic6C2311yescharacteristic
116C22×Dic6C2312yes—
117C3×#124—24no—
118C3×C4×Q8—24no—
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2314yes—
122C2×C3×C22⋊C4C2312yes—
123C2×C3×C22⋊C4C2314yes—
124C2×C3×C22⋊C4C2312yes—
125C2×C3×C22⋊C4C2314yes—
126C2×C3×C42C2312yes—
127C2×C3×C42C2311yescharacteristic
128C2×#217C2314yes—
129C2×#217C2312yes—
130C2×#217C2312yes—
131C2×#217C2314yes—
132#946—24no—
133Q8×Dic3—24no—
134Q8×Dic3—24no—
135#938—24no—
136D4×Dic3—24no—
137C2×(C2×D3)⋊C4C2314yes—
138C2×(C2×D3)⋊C4C2312yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2314yes—
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2311yescharacteristic
143C2×C4⋊Dic3C2311yescharacteristic
144C2×C4⋊Dic3C2311yescharacteristic
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2312yes—
150C2×Dic3⋊C4C2312yes—
151C2×Dic3⋊C4C2312yes—
152C2×Dic3⋊C4C2312yes—
153C2×Dic3⋊C4C2314yes—
154C2×C4×Dic3C2312yes—
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2312yes—
157C2×C4×Dic3C2312yes—
158C2×C4×Dic3C2314yes—
159C2×C4×Dic3C2311yescharacteristic
160C2×C4×Dic3C2311yescharacteristic
161C2×C4×Dic3C2311yescharacteristic
162D3×C4⋊C4—28no—
163D3×C4⋊C4—24no—
164#887—24no—
165#887—28no—
166#883—28no—
167#67⋊C4—28no—
168#879—28no—
169#878—216no—
170#877—28no—
171C42×D3—28no—
172C42×D3D418yes—
173#870—24no—
174C4×Dic6—24no—
175C4×Dic6—28no—
176C3×#95C2314yes—
177C3×#95C2312yes—
178C3×#95C2312yes—
179C3×#95C2312yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2312yes—
186C3⋊#95C2312yes—
187C3⋊#95C2312yes—
188C3⋊#95C2314yes—
189C3⋊#95C2314yes—
190C3⋊#95C2314yes—
191C23×Q8—31no—
192#529—68no—
193C2×#126—32no—
194C2×#125—31no—
195C2×#124C2×D311yescharacteristic
196C2×#123—32no—
197C2×C22⋊Q8—31no—
198C2×C22⋊Q8—34no—
199C2×#121—31no—
200C2×C4×Q8—31no—
201C2×C4×Q8—31no—
202C2×C4×Q8—32no—
203C2×C4×Q8—31no—
204C2×C4×Q8—32no—
205C2×C4×Q8—31no—
206C2×C4×Q8—68no—
207C2×C4×Q8C2×D311yescharacteristic
208C2×C4×D4—31no—
209C2×C4×D4—32no—
210C2×#117—32no—
211C22×C4⋊C4—32no—
212C22×C4⋊C4—31no—
213C22×C42—31no—
214C22×C42—32no—
215#400—32no—
216#400—32no—
217#400—32no—
218#400—32no—
219#399—32no—
220#399C2×D312yes—
221#398—32no—
222#398—32no—
223#395—32no—
224#395—32no—
225#395—32no—
226#395C2×D312yes—
227#394—32no—
228#394C2×D312yes—
229#393—32no—
230#393C2×D312yes—
231#391—34no—
232#389—34no—
233#389—34no—
234#389—34no—
235#388—34no—
236#388—34no—
237#388—34no—
238#388—34no—
239#388—34no—
240#388C2×D314yes—
241#387—34no—
242#387—34no—
243#387—34no—
244#387—34no—
245#386—34no—
246#386—34no—
247#385—34no—
248#385—34no—
249#383—31no—
250#383C2×D311yescharacteristic
251#382—32no—
252#381—32no—
253#380—34no—
254#380—32no—
255C4×C4⋊C4—32no—
256C4×C22⋊C4—32no—
257#376—32no—
258C24×C3C2412yes—
259C23×D3C2411yescharacteristic
260C2×C3×Q8—22no—
261C2×C3×Q8—22no—
262C2×C3×Q8—24no—
263C2×C3×Q8—42no—
264C2×C3×D4—22no—
265C2×C3×D4—42no—
266C22×C3×C4C2414yes—
267C22×C3×C4C2412yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2411yescharacteristic
270C22×C3×C4C2411yescharacteristic
271C22×C3×C4C2411yescharacteristic
272C22×C3×C4C2412yes—
273C2×#67—24no—
274C2×#67—28no—
275C22×Dic3—24no—
276C22×Dic3—28no—
277C22×Dic3C2414yes—
278C22×Dic3C2412yes—
279C22×Dic3C2412yes—
280C22×Dic3C2412yes—
281C22×Dic3C2411yescharacteristic
282C22×Dic3C2411yescharacteristic
283C22×Dic3C2412yes—
284C22×Dic3C2411yescharacteristic
285D3×Q8—232no—
286C2×C4×D3—24no—
287C2×C4×D3—24no—
288C2×C4×D3—24no—
289C2×C4×D3—24no—
290C2×C4×D3—24no—
291C2×C4×D3—24no—
292C2×C4×D3—42no—
293C2×C4×D3—44no—
294C2×C4×D3C2×D414yes—
295C2×C4×D3C2×D414yes—
296C2×C4×D3C2×D414yes—
297C2×Dic6—22no—
298C2×Dic6—24no—
299C2×Dic6—24no—
300C2×Dic6—24no—
301C2×Dic6#4118yes—
302C3×C4⋊C4—24no—
303C3×C4⋊C4—22no—
304C3×C4⋊C4—44no—
305C3×C22⋊C4—24no—
306C3×C22⋊C4—24no—
307C3×C22⋊C4—28no—
308C3×C22⋊C4—44no—
309C3×C42—24no—
310C3×C42C2×D414yes—
311#217—24no—
312#217—24no—
313#217—28no—
314#217—44no—
315(C2×D3)⋊C4—24no—
316(C2×D3)⋊C4—28no—
317C4⋊Dic3—22no—
318C4⋊Dic3—22no—
319C4⋊Dic3—22no—
320C4⋊Dic3—24no—
321C4⋊Dic3—24no—
322C4⋊Dic3—28no—
323Dic3⋊C4—24no—
324Dic3⋊C4—24no—
325Dic3⋊C4—24no—
326Dic3⋊C4—28no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4—24no—
330Dic3⋊C4—28no—
331Dic3⋊C4—44no—
332C4×Dic3—24no—
333C4×Dic3—24no—
334C4×Dic3—24no—
335C4×Dic3—24no—
336C4×Dic3—44no—
337C4×Dic3C2×D414yes—
338C4×Dic3C2×D414yes—
339C4×Dic3C2×D414yes—
340C22×Q8—32no—
341C22×Q8—31no—
342C22×Q8—64no—
343C22×Q8—68no—
344C22×Q8C22×D311yescharacteristic
345C22×Q8C22×D311yescharacteristic
346C22×D4—32no—
347C22×D4C22×D311yescharacteristic
348C23×C4—31no—
349C23×C4—31no—
350C23×C4—31no—
351C23×C4—32no—
352C23×C4—32no—
353C23×C4—32no—
354#126—68no—
355#125—64no—
356#124—24no—
357#123—68no—
358C22⋊Q8—616no—
359C22⋊Q8—64no—
360#121—64no—
361C4×Q8—24no—
362C4×Q8—68no—
363C4×Q8—68no—
364C4×Q8—64no—
365C4×Q8—64no—
366C4×Q8—64no—
367C4×Q8—64no—
368C4×D4—64no—
369C4×D4—68no—
370#117—68no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—32no—
373C2×C4⋊C4—32no—
374C2×C4⋊C4—32no—
375C2×C4⋊C4—31no—
376C2×C4⋊C4—32no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—31no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—31no—
381C2×C4⋊C4—32no—
382C2×C4⋊C4—34no—
383C2×C4⋊C4—34no—
384C2×C4⋊C4—34no—
385C2×C4⋊C4—34no—
386C2×C4⋊C4—64no—
387C2×C4⋊C4—68no—
388C2×C4⋊C4C22×D312yes—
389C2×C4⋊C4C22×D311yescharacteristic
390C2×C4⋊C4C22×D314yes—
391C2×C22⋊C4—32no—
392C2×C22⋊C4—32no—
393C2×C22⋊C4—34no—
394C2×C22⋊C4—32no—
395C2×C22⋊C4—34no—
396C2×C22⋊C4—34no—
397C2×C22⋊C4C22×D312yes—
398C2×C22⋊C4C22×D312yes—
399C2×C22⋊C4C22×D314yes—
400C2×C22⋊C4C22×D314yes—
401C2×C42—31no—
402C2×C42—31no—
403C2×C42—31no—
404C2×C42—38no—
405C2×C42—32no—
406C2×C42—32no—
407C2×C42—32no—
408C2×C42—32no—
409C2×C42—34no—
410C2×C42—68no—
411C2×C42C22×D312yes—
412C2×C42C22×D311yescharacteristic
413#95—32no—
414#95—34no—
415#95—32no—
416#95—32no—
417#95—34no—
418#95—34no—
419#95—34no—
420#95—34no—
421#95—34no—
422#95—34no—
423#95—34no—
424#95C22×D312yes—
425#95C22×D312yes—
426#95C22×D312yes—
427#95C22×D314yes—
428C23×C3—22no—
429C23×C3—24no—
430C23×C3—24no—
431C23×C3—24no—
432C23×C3C2511yescharacteristic
433C22×D3—22no—
434C22×D3—22no—
435C22×D3C22×D412yes—
436C22×D32−1+414yes—
437C3×Q8—28no—
438C3×Q8—44no—
439C3×D4—44no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—22no—
446C2×C3×C4—22no—
447C2×C3×C4—24no—
448C2×C3×C4—24no—
449C2×C3×C4—24no—
450C2×C3×C4—24no—
451C2×C3×C4—24no—
452C2×C3×C4C22×D412yes—
453C2×C3×C4C22×D412yes—
454C2×C3×C4C22×D412yes—
455C2×C3×C4C2×#4114yes—
456#67—48no—
457C2×Dic3—22no—
458C2×Dic3—22no—
459C2×Dic3—22no—
460C2×Dic3—22no—
461C2×Dic3—22no—
462C2×Dic3—22no—
463C2×Dic3—22no—
464C2×Dic3—24no—
465C2×Dic3—24no—
466C2×Dic3—24no—
467C2×Dic3—24no—
468C2×Dic3—24no—
469C2×Dic3—24no—
470C2×Dic3—48no—
471C2×Dic3C22×D412yes—
472C2×Dic3C22×D412yes—
473C2×Dic3C22×D412yes—
474C2×Dic3C2×#4114yes—
475C2×Dic3C2×#4114yes—
476C4×D3—216no—
477C4×D3—28no—
478C4×D3—28no—
479C4×D3—44no—
480C4×D3—48no—
481Dic6—28no—
482Dic6—216no—
483C24—31no—
484C24C23×D312yes—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—24no—
488C2×Q8—38no—
489C2×Q8—42no—
490C2×Q8—64no—
491C2×Q8—64no—
492C2×Q8—64no—
493C2×Q8—632no—
494C2×Q8—62no—
495C2×D4—22no—
496C2×D4—42no—
497C2×D4—64no—
498C2×D4—68no—
499C22×C4—31no—
500C22×C4—31no—
501C22×C4—32no—
502C22×C4—31no—
503C22×C4—34no—
504C22×C4—34no—
505C22×C4—34no—
506C22×C4—34no—
507C22×C4—32no—
508C22×C4—32no—
509C22×C4—32no—
510C22×C4—64no—
511C22×C4—64no—
512C22×C4—64no—
513C22×C4—64no—
514C22×C4—64no—
515C22×C4—64no—
516C22×C4—68no—
517C22×C4—64no—
518C22×C4—122no—
519C22×C4—124no—
520C22×C4C23×D314yes—
521C22×C4C23×D312yes—
522C22×C4C23×D312yes—
523C22×C4C23×D311yescharacteristic
524C22×C4C23×D311yescharacteristic
525C22×C4C23×D311yescharacteristic
526C22×C4C23×D312yes—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—64no—
531C4⋊C4—64no—
532C4⋊C4—64no—
533C4⋊C4—64no—
534C4⋊C4—64no—
535C4⋊C4—68no—
536C4⋊C4—68no—
537C4⋊C4—64no—
538C4⋊C4—62no—
539C4⋊C4—64no—
540C4⋊C4—62no—
541C4⋊C4—64no—
542C4⋊C4—62no—
543C4⋊C4—68no—
544C4⋊C4—124no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—28no—
548C22⋊C4—44no—
549C22⋊C4—64no—
550C22⋊C4—68no—
551C22⋊C4—64no—
552C22⋊C4—64no—
553C22⋊C4—68no—
554C22⋊C4—124no—
555C42—24no—
556C42—34no—
557C42—34no—
558C42—34no—
559C42—64no—
560C42—64no—
561C42—64no—
562C42—64no—
563C42—124no—
564C42D3×D414yes—
565C22×C3—42no—
566C22×C3—42no—
567C22×C3—42no—
568C22×C3—48no—
569C22×C3C23×D411yescharacteristic
570C22×C3C22×#4112yes—
571C22×C3C2×2−1+412yes—
572C22×C3#58512yes—
573C2×D3—22no—
574C2×D3—22no—
575C2×D3—28no—
576C2×D3#53614yes—
577C3×C4—22no—
578C3×C4—24no—
579C3×C4—22no—
580C3×C4—22no—
581C3×C4—42no—
582C3×C4—42no—
583C3×C4—44no—
584Dic3—24no—
585Dic3—22no—
586Dic3—22no—
587Dic3—42no—
588Dic3—42no—
589Dic3—42no—
590Dic3—44no—
591Dic3#52514yes—
592C23—22no—
593C23—24no—
594C23—24no—
595C23—24no—
596C23—32no—
597C23—34no—
598C23—62no—
599C23—62no—
600C23C24×D311yescharacteristic
601Q8—28no—
602Q8—44no—
603Q8—616no—
604Q8—68no—
605D4—44no—
606D4—128no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—32no—
620C2×C4—34no—
621C2×C4—32no—
622C2×C4—32no—
623C2×C4—34no—
624C2×C4—68no—
625C2×C4—68no—
626C2×C4—616no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—64no—
635C2×C4—64no—
636C2×C4—64no—
637C2×C4—64no—
638C2×C4—64no—
639C2×C4—64no—
640C2×C4—124no—
641C2×C4—128no—
642C2×C4—128no—
643C2×C4C2×D3×D412yes—
644C2×C4C2×D3×D412yes—
645C2×C4C2×D3×D412yes—
646C2×C4D3×#4114yes—
647C2×C3—82no—
648C2×C3C2×#52511yescharacteristic
649C2×C3C2×#53611yescharacteristic
650C2×C3#344911yescharacteristic
651C2×C3#351614yes—
652D3—24no—
653C22—34no—
654C22—42no—
655C22—48no—
656C22—42no—
657C22—42no—
658C22—62no—
659C22—68no—
660C22—62no—
661C22C22×D3×D411yescharacteristic
662C22C2×D3×#4112yes—
663C22#596612yes—
664C22D3×2−1+412yes—
665C4—22no—
666C4—24no—
667C4—22no—
668C4—22no—
669C4—34no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—64no—
674C4—62no—
675C4—62no—
676C4—122no—
677C4—124no—
678C4—122no—
679C4—122no—
680C3#3524511yescharacteristic, complemented, Sylow
681C2—64no—
682C2—82no—
683C2D3×#52511yescharacteristic
684C2#8631311yescharacteristic
685C2D3×#53611yescharacteristic
686C2#8655514yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2632C2
prime radicalC2×#938C22
hypercenter#2632D3
hypocenterC3#35245
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2632

Complements

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

NH ≅ G/Ntype
2. C2×C4×D3×Q8C2semidirect
3. C3×#2632C2semidirect
4. #82977C2semidirect
5. #82486C2semidirect
6. #78045C2semidirect
7. #78029C2semidirect
8. #77255C2semidirect
9. #77191C2semidirect
11. #76943C2semidirect
12. #76806C2semidirect
13. #76780C2semidirect
14. #75959C2semidirect
15. #75460C2semidirect
16. #74278C2semidirect
17. #74258C2semidirect
92. #2632D3semidirect
680. C3#35245semidirect

Maximal subgroups

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

subgroupindex
C2×#56802
C2×C4×D3×Q82
C3×#26322
#829772
#824862
#780452
#780292
#772552
#771912
#771212
#769432
#768062
#767802
#759592
#754602
#742782
#742582
#352453