Finite Group Databasebeta

← #10982787

Subgroups — #10982787

Order 768.

2170 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic, complemented
2C2×#5572C211yesmaximal, characteristic
3C3×#2546C211yesmaximal, characteristic, complemented
4#83261C211yesmaximal, characteristic
5#83232C212yesmaximal
6#83229C212yesmaximal, complemented
7#82481C211yesmaximal, characteristic, complemented
8#77823C212yesmaximal, complemented
9#77816C212yesmaximal
10#77810C212yesmaximal
11#77740C212yesmaximal, complemented
12#76504C212yesmaximal, complemented
13#76491C212yesmaximal
14#76024C214yesmaximal
15#76018C214yesmaximal, complemented
16#73668C211yesmaximal, characteristic
17#73643C211yesmaximal, characteristic, complemented
18#35749—31nomaximal, Sylow
19C22×D3×Q8C2211yescharacteristic
20C2×C3×C4⋊Q8C2212yes—
21C2×C3×C4⋊Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×#946C2212yes—
24C2×Q8×Dic3C2211yescharacteristic
25C2×Q8×Dic3C2212yes—
26C2×#944C2212yes—
27#5721—28no—
28#5572—28no—
29C2×#892C2211yescharacteristic
30C2×#892C2212yes—
31C2×#890C2212yes—
32C2×#889C2214yes—
33C2×#887C2212yes—
34C2×#887C2211yescharacteristic
35C2×#887C2212yes—
36C2×#872C2211yescharacteristic
37C2×#872C2212yes—
38C2×C4×Dic6C2212yes—
39C2×C4×Dic6C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#398C2212yes—
42C3×#398C2212yes—
43C3×#391C2212yes—
44C3×#389C2214yes—
45C3×C4×C4⋊C4C2211yescharacteristic
46#5229C2212yes—
47#5229C2212yes—
48#4994C2212yes—
49#4992C2212yes—
50#4992C2212yes—
51#4989C2214yes—
52#4989C2214yes—
53#4986C2212yes—
54#4984C2214yes—
55#4983C2212yes—
56#4983C2212yes—
57#4981C2212yes—
58#4980C2214yes—
59#4980C2214yes—
60#4979C2214yes—
61#4979C2214yes—
62#4979C2212yes—
63#4978C2214yes—
64Dic3×C4⋊C4C2214yes—
65#4972C2212yes—
66C4×(C2×D3)⋊C4C2212yes—
67#4936C2211yescharacteristic
68C4×C4⋊Dic3C2212yes—
69C4×C4⋊Dic3C2212yes—
70#4933C2214yes—
71#4929C2212yes—
72#4677C2211yescharacteristic
73#4673C2211yescharacteristic
74#4670C2214yes—
75#4669C2214yes—
76#4659C2211yescharacteristic
77#4658C2214yes—
78#4657C2214yes—
79#4656C2211yescharacteristic
80#4655C2214yes—
81#4654C2214yes—
82#4652C2214yes—
83#4651C2212yes—
84#4649C2214yes—
85#4648C2212yes—
86C2×#557—31no—
87C2×#519—31no—
88#2634—32no—
89#2621—31no—
90#2611—32no—
91#2599—32no—
92#2581—31no—
93#2546D311yescharacteristic, complemented
94#2475—34no—
95#2462—34no—
96#2446—32no—
97#2434—32no—
98#2399—32no—
99#2396—31no—
100#2365—32no—
101#2362—32no—
102#2356—31no—
103C22×C3×Q8C2312yes—
104C22×C3×Q8C2311yescharacteristic
105C2×D3×Q8—24no—
106C2×D3×Q8—24no—
107C2×D3×Q8—24no—
108C22×C4×D3C2312yes—
109C22×C4×D3C2311yescharacteristic
110C22×Dic6C2311yescharacteristic
111C22×Dic6C2312yes—
112C22×Dic6C2311yescharacteristic
113C3×C4⋊Q8—28no—
114C3×C4⋊Q8—24no—
115C3×C4×Q8—24no—
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2314yes—
118C2×C3×C4⋊C4C2312yes—
119C2×C3×C4⋊C4C2314yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C4⋊C4C2312yes—
123C2×C3×C42C2312yes—
124C2×C3×C42C2312yes—
125C2×C3×C42C2311yescharacteristic
126#946—28no—
127Q8×Dic3—28no—
128Q8×Dic3—24no—
129#944—28no—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2314yes—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2311yescharacteristic
135C2×(C2×D3)⋊C4C2311yescharacteristic
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2314yes—
138C2×C4⋊Dic3C2312yes—
139C2×C4⋊Dic3C2311yescharacteristic
140C2×C4⋊Dic3C2312yes—
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2311yescharacteristic
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2311yescharacteristic
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2312yes—
150C2×Dic3⋊C4C2311yescharacteristic
151C2×Dic3⋊C4C2312yes—
152C2×C4×Dic3C2314yes—
153C2×C4×Dic3C2312yes—
154C2×C4×Dic3C2314yes—
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2311yescharacteristic
157C2×C4×Dic3C2312yes—
158C2×C4×Dic3C2312yes—
159C2×C4×Dic3C2312yes—
160#892—24no—
161#892—28no—
162#890—28no—
163#889—216no—
164#887—24no—
165#887—28no—
166#887—28no—
167#872—24no—
168#872—28no—
169C4×Dic6—24no—
170C4×Dic6—28no—
171C4×Dic6D418yes—
172C3×#95C2311yescharacteristic
173C3×#95C2314yes—
174C3×#95C2311yescharacteristic
175C3×#95C2314yes—
176C3⋊#95C2312yes—
177C3⋊#95C2314yes—
178C3⋊#95C2312yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181C3⋊#95C2312yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2312yes—
185C3⋊#95C2314yes—
186C3⋊#95C2312yes—
187C3⋊#95C2312yes—
188C3⋊#95C2314yes—
189C3⋊#95C2314yes—
190C23×Q8—31no—
191#557—68no—
192C2×C4⋊Q8—32no—
193C2×C4⋊Q8C2×D312yes—
194C2×C4⋊Q8C2×D311yescharacteristic
195C2×#125—32no—
196C2×#125—34no—
197C2×C22⋊Q8—32no—
198#519—68no—
199C2×C4×Q8—32no—
200C2×C4×Q8—31no—
201C2×C4×Q8—32no—
202C2×C4×Q8—32no—
203C2×C4×Q8—31no—
204C2×C4×Q8—31no—
205C2×C4×Q8—31no—
206C2×C4×Q8—32no—
207C2×C4×Q8C2×D311yescharacteristic
208C2×#117—32no—
209C2×#117—31no—
210C2×#117—31no—
211C2×#117—32no—
212#400—32no—
213#400—34no—
214#400—31no—
215#399—31no—
216#398—32no—
217#398—32no—
218#398—32no—
219#398—34no—
220#398C2×D312yes—
221#398C2×D312yes—
222#397—32no—
223#397—32no—
224#396—32no—
225#395—32no—
226#395—34no—
227#395—34no—
228#395—31no—
229#393—31no—
230#391—32no—
231#391—32no—
232#391—32no—
233#391C2×D312yes—
234#389—32no—
235#389—34no—
236#389C2×D314yes—
237#388—34no—
238#388—34no—
239#387—34no—
240#387—34no—
241#387—34no—
242#387—34no—
243#387—34no—
244#387—34no—
245#387—34no—
246#385—34no—
247#385—34no—
248#383—32no—
249#382—31no—
250C4×C4⋊C4—34no—
251C4×C4⋊C4—32no—
252C4×C4⋊C4—32no—
253C4×C4⋊C4—34no—
254C4×C4⋊C4C2×D311yescharacteristic
255C4×C22⋊C4—32no—
256#376—32no—
257C23×D3C2411yescharacteristic
258C2×C3×Q8—22no—
259C2×C3×Q8—24no—
260C2×C3×Q8—22no—
261C2×C3×Q8—24no—
262C2×C3×Q8—24no—
263C2×C3×Q8#4114yes—
264C22×C3×C4C2412yes—
265C22×C3×C4C2412yes—
266C22×C3×C4C2414yes—
267C22×C3×C4C2411yescharacteristic
268C22×C3×C4C2411yescharacteristic
269C22×C3×C4C2412yes—
270C22×C3×C4C2412yes—
271C22×C3×C4C2411yescharacteristic
272C22×Dic3C2414yes—
273C22×Dic3C2411yescharacteristic
274C22×Dic3C2411yescharacteristic
275C22×Dic3C2412yes—
276C22×Dic3C2412yes—
277C22×Dic3C2412yes—
278C22×Dic3C2411yescharacteristic
279C22×Dic3C2412yes—
280D3×Q8—232no—
281C2×C4×D3—24no—
282C2×C4×D3—22no—
283C2×C4×D3—24no—
284C2×C4×D3—22no—
285C2×C4×D3—24no—
286C2×C4×D3#4114yes—
287C2×Dic6—22no—
288C2×Dic6—24no—
289C2×Dic6—22no—
290C2×Dic6—22no—
291C2×Dic6—24no—
292C2×Dic6—24no—
293C2×Dic6—24no—
294C2×Dic6C2×D414yes—
295C3×C4⋊C4—24no—
296C3×C4⋊C4—24no—
297C3×C4⋊C4—22no—
298C3×C4⋊C4—24no—
299C3×C4⋊C4—28no—
300C3×C4⋊C4—28no—
301C3×C4⋊C4—24no—
302C3×C42—24no—
303C3×C42—24no—
304C3×C42C2×D414yes—
305(C2×D3)⋊C4—28no—
306(C2×D3)⋊C4—22no—
307(C2×D3)⋊C4—22no—
308(C2×D3)⋊C4—24no—
309(C2×D3)⋊C4—24no—
310(C2×D3)⋊C4—24no—
311C4⋊Dic3—22no—
312C4⋊Dic3—24no—
313C4⋊Dic3—24no—
314C4⋊Dic3—24no—
315C4⋊Dic3—28no—
316C4⋊Dic3—24no—
317C4⋊Dic3C2×D414yes—
318Dic3⋊C4—22no—
319Dic3⋊C4—24no—
320Dic3⋊C4—22no—
321Dic3⋊C4—24no—
322Dic3⋊C4—28no—
323Dic3⋊C4—28no—
324Dic3⋊C4—28no—
325Dic3⋊C4—28no—
326Dic3⋊C4C2×D418yes—
327C4×Dic3—24no—
328C4×Dic3—22no—
329C4×Dic3—28no—
330C4×Dic3—24no—
331C4×Dic3—24no—
332C4×Dic3—24no—
333C4×Dic3—28no—
334C4×Dic3C2×D418yes—
335C22×Q8—31no—
336C22×Q8—31no—
337C22×Q8—32no—
338C22×Q8—64no—
339C22×Q8—64no—
340C22×Q8—64no—
341C22×Q8C22×D312yes—
342C22×Q8C22×D311yescharacteristic
343C23×C4—31no—
344C23×C4—32no—
345C4⋊Q8—28no—
346C4⋊Q8—24no—
347C4⋊Q8—68no—
348#125—68no—
349#125—616no—
350C22⋊Q8—68no—
351C4×Q8—24no—
352C4×Q8—38no—
353C4×Q8—68no—
354C4×Q8—68no—
355C4×Q8—68no—
356C4×Q8—64no—
357C4×Q8—64no—
358C4×Q8—68no—
359C4×Q8—64no—
360#117—68no—
361#117—68no—
362#117—64no—
363#117—64no—
364C2×C4⋊C4—31no—
365C2×C4⋊C4—32no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—31no—
369C2×C4⋊C4—32no—
370C2×C4⋊C4—34no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—31no—
373C2×C4⋊C4—34no—
374C2×C4⋊C4—32no—
375C2×C4⋊C4—34no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—34no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4C22×D311yescharacteristic
381C2×C4⋊C4C22×D312yes—
382C2×C4⋊C4C22×D312yes—
383C2×C4⋊C4C22×D312yes—
384C2×C4⋊C4C22×D314yes—
385C2×C4⋊C4C22×D314yes—
386C2×C4⋊C4C22×D312yes—
387C2×C22⋊C4—32no—
388C2×C22⋊C4—31no—
389C2×C22⋊C4—31no—
390C2×C22⋊C4—34no—
391C2×C22⋊C4—32no—
392C2×C22⋊C4—32no—
393C2×C42—32no—
394C2×C42—32no—
395C2×C42—32no—
396C2×C42—31no—
397C2×C42—32no—
398C2×C42—34no—
399C2×C42—32no—
400C2×C42—34no—
401C2×C42C22×D312yes—
402C2×C42C22×D312yes—
403C2×C42C22×D311yescharacteristic
404#95—32no—
405#95—34no—
406#95—32no—
407#95—32no—
408#95—34no—
409#95—34no—
410#95—32no—
411#95—32no—
412#95—34no—
413#95—32no—
414#95—34no—
415#95—34no—
416#95—34no—
417#95—34no—
418#95C22×D311yescharacteristic
419#95C22×D314yes—
420#95C22×D311yescharacteristic
421#95C22×D314yes—
422C23×C3C2511yescharacteristic
423C22×D3—21no—
424C22×D3—21no—
425C22×D3—22no—
426C22×D3—22no—
427C22×D3C2×#4112yes—
428C3×Q8—28no—
429C3×Q8—48no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—22no—
437C2×C3×C4—24no—
438C2×C3×C4—24no—
439C2×C3×C4—22no—
440C2×C3×C4—24no—
441C2×C3×C4—24no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4C22×D412yes—
445C2×C3×C4C22×D412yes—
446C2×C3×C4C22×D412yes—
447C2×C3×C4C2×#4114yes—
448C2×C3×C4C2×#4112yes—
449C2×C3×C4C2×#4112yes—
450C2×C3×C42−1+414yes—
451C2×Dic3—22no—
452C2×Dic3—21no—
453C2×Dic3—22no—
454C2×Dic3—21no—
455C2×Dic3—22no—
456C2×Dic3—24no—
457C2×Dic3—22no—
458C2×Dic3—21no—
459C2×Dic3—21no—
460C2×Dic3—22no—
461C2×Dic3—24no—
462C2×Dic3—24no—
463C2×Dic3—24no—
464C2×Dic3—22no—
465C2×Dic3—22no—
466C2×Dic3—22no—
467C2×Dic3—22no—
468C2×Dic3C22×D414yes—
469C2×Dic3C22×D414yes—
470C2×Dic3C2×#4112yes—
471C2×Dic32−1+414yes—
472C4×D3—216no—
473C4×D3—28no—
474Dic6—28no—
475Dic6—216no—
476Dic6—28no—
477C24—31no—
478C2×Q8—22no—
479C2×Q8—24no—
480C2×Q8—24no—
481C2×Q8—22no—
482C2×Q8—24no—
483C2×Q8—34no—
484C2×Q8—64no—
485C2×Q8—64no—
486C2×Q8—64no—
487C2×Q8—632no—
488C2×Q8—62no—
489C2×Q8—64no—
490C2×Q8—62no—
491C2×Q8—62no—
492C2×Q8#23714yes—
493C22×C4—32no—
494C22×C4—32no—
495C22×C4—31no—
496C22×C4—32no—
497C22×C4—34no—
498C22×C4—34no—
499C22×C4—31no—
500C22×C4—31no—
501C22×C4—32no—
502C22×C4—64no—
503C22×C4—64no—
504C22×C4—62no—
505C22×C4—64no—
506C22×C4—62no—
507C22×C4C23×D312yes—
508C22×C4C23×D312yes—
509C22×C4C23×D314yes—
510C22×C4C23×D311yescharacteristic
511C22×C4C23×D311yescharacteristic
512C22×C4C23×D312yes—
513C22×C4C23×D312yes—
514C22×C4C23×D311yescharacteristic
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—34no—
523C4⋊C4—38no—
524C4⋊C4—64no—
525C4⋊C4—64no—
526C4⋊C4—68no—
527C4⋊C4—68no—
528C4⋊C4—68no—
529C4⋊C4—68no—
530C4⋊C4—68no—
531C4⋊C4—64no—
532C4⋊C4—62no—
533C4⋊C4—62no—
534C4⋊C4—64no—
535C4⋊C4—62no—
536C4⋊C4—64no—
537C4⋊C4—64no—
538C22⋊C4—64no—
539C22⋊C4—64no—
540C22⋊C4—68no—
541C22⋊C4—62no—
542C22⋊C4—62no—
543C22⋊C4—64no—
544C42—24no—
545C42—24no—
546C42—38no—
547C42—64no—
548C42—64no—
549C42—64no—
550C42—68no—
551C42—62no—
552C42—64no—
553C42—68no—
554C42D3×D414yes—
555C22×C3C23×D411yescharacteristic
556C22×C3C22×#4111yescharacteristic
557C22×C3C22×#4111yescharacteristic
558C22×C3C2×2−1+412yes—
559C22×C3C2×2−1+412yes—
560C2×D3—22no—
561C2×D3—24no—
562C2×D3—41no—
563C2×D3—41no—
564C2×D3—42no—
565C3×C4—24no—
566C3×C4—22no—
567C3×C4—22no—
568C3×C4—24no—
569C3×C4—44no—
570C3×C4—42no—
571C3×C4—42no—
572C3×C4#52514yes—
573Dic3—24no—
574Dic3—22no—
575Dic3—24no—
576Dic3—24no—
577Dic3—41no—
578Dic3—41no—
579Dic3—42no—
580Dic3—44no—
581C23—32no—
582C23—61no—
583C23—61no—
584C23—62no—
585C23—62no—
586C23C24×D311yescharacteristic
587Q8—28no—
588Q8—48no—
589Q8—616no—
590Q8—68no—
591Q8—68no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—34no—
607C2×C4—34no—
608C2×C4—32no—
609C2×C4—34no—
610C2×C4—616no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—61no—
615C2×C4—64no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—68no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—62no—
622C2×C4—64no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—64no—
627C2×C4—62no—
628C2×C4—64no—
629C2×C4C2×D3×D412yes—
630C2×C4C2×D3×D412yes—
631C2×C4C2×D3×D412yes—
632C2×C4C2×#23714yes—
633C2×C4C2×#23712yes—
634C2×C4#100714yes—
635C2×C4D3×#4112yes—
636C2×C3C2×#52511yescharacteristic
637C2×C3#348611yescharacteristic
638C2×C3#349911yescharacteristic
639C2×C3#350912yes—
640C2×C3#352312yes—
641D3—42no—
642C22—62no—
643C22—64no—
644C22—121no—
645C22—121no—
646C22—122no—
647C22C22×D3×D411yescharacteristic
648C22C22×#23711yescharacteristic
649C22C2×#100712yes—
650C22C2×D3×#4111yescharacteristic
651C22D3×2−1+412yes—
652C4—22no—
653C4—24no—
654C4—22no—
655C4—24no—
656C4—42no—
657C4—44no—
658C4—42no—
659C4—64no—
660C4—64no—
661C4—62no—
662C4—64no—
663C4—121no—
664C4—121no—
665C4—122no—
666C4—124no—
667C4#560614yes—
668C3#3574911yescharacteristic, complemented, Sylow
669C2—122no—
670C2C2×#560611yescharacteristic
671C2#8621411yescharacteristic
672C2#8635511yescharacteristic
673C2#8675212yes—
674C2#8702012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2546C2
prime radicalC23×D3C24
hypercenter#2546D3
hypocenterC3#35749
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2546

Complements

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

NH ≅ G/Ntype
1. C2×#5721C2semidirect
3. C3×#2546C2semidirect
6. #83229C2semidirect
7. #82481C2semidirect
8. #77823C2semidirect
11. #77740C2semidirect
12. #76504C2semidirect
15. #76018C2semidirect
17. #73643C2semidirect
93. #2546D3semidirect
668. C3#35749semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×#55722
C3×#25462
#832612
#832322
#832292
#824812
#778232
#778162
#778102
#777402
#765042
#764912
#760242
#760182
#736682
#736432
#357493