Finite Group Databasebeta

← #11675282

Subgroups — #11675282

Order 768.

1798 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86280C211yesmaximal, characteristic, complemented
2#86260C211yesmaximal, characteristic, complemented
3C3×#2961C211yesmaximal, characteristic, complemented
4#79974C214yesmaximal, complemented
5#79972C214yesmaximal, complemented
6#79971C214yesmaximal, complemented
7#79970C214yesmaximal, complemented
8#79091C212yesmaximal, complemented
9#79087C212yesmaximal, complemented
10#79068C211yesmaximal, characteristic, complemented
11#79055C211yesmaximal, characteristic
12#78575C212yesmaximal, complemented
13#78560C212yesmaximal, complemented
14#78497C211yesmaximal, characteristic, complemented
15C2×#4710C211yesmaximal, characteristic, complemented
16#61333—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C2×#908C2212yes—
19C22×#204C2211yescharacteristic
20#5732—21no—
21#5730—21no—
22#5730—21no—
23#5727—21no—
24#5694—22no—
25#5690—22no—
26#5645—22no—
27#5634—22no—
28D3×C22⋊Q8—22no—
29#5624—22no—
30#5585C2212yescomplemented
31#5584C2214yescomplemented
32#5579C2214yes—
33#5579C2212yes—
34Q8×D12C2212yes—
35#5570C2212yescomplemented
36#5526C2212yes—
37#5523C2212yes—
38#5521C2212yes—
39#5519C2212yes—
40#5477C2211yescharacteristic
41C2×C4×D12C2211yescharacteristic
42#5470C2211yescharacteristic
43C2×C4×Dic6C2211yescharacteristic
44C3×#446C2214yescomplemented
45C3×#446C2214yescomplemented
46C3×#424C2212yes—
47C3×#423C2211yescharacteristic
48C3×#405C2212yes—
49C2×C3×#97C2211yescharacteristic
50#5133C2212yes—
51#5121C2212yes—
52#5120C2212yes—
53#5113C2212yes—
54#5113C2212yes—
55C2×#821C2211yescharacteristic
56C2×#818C2211yescharacteristic
57C2×#816C2211yescharacteristic
58#4851C2212yescomplemented
59#4850C2212yescomplemented
60#4850C2214yescomplemented
61#4847C2214yescomplemented
62#4846C2214yescomplemented
63#4844C2214yes—
64#4844C2214yes—
65#4844C2214yes—
66#4843C2212yes—
67#4841C2214yes—
68#4841C2212yes—
69#4841C2214yes—
70#4829C2214yescomplemented
71#4829C2214yescomplemented
72#4714C2212yescomplemented
73#4714C2214yescomplemented
74#4714C2212yescomplemented
75#4713C2212yescomplemented
76#4711C2214yes—
77#4710C2214yes—
78#4710C2214yes—
79#4710C2214yes—
80#4703C2212yescomplemented
81#4703C2214yescomplemented
82#4697C2214yescomplemented
83C4×D24C2214yes—
84C4×#204C2214yes—
85#4681C2214yescomplemented
86#3507—31no—
87#3501—31no—
88#3376—34no—
89#3372—34no—
90#3370—34no—
91#3367—34no—
92#3137—32no—
93#3135—31no—
94#3083—32no—
95#3078—31no—
96#3042—32no—
97#3036—31no—
98#2961D311yescharacteristic, complemented
99#2946—32no—
100C2×#440—31no—
101C2×D3×Q8—22no—
102C2×#235C2312yes—
103C22×D12C2311yescharacteristic
104C22×C4×D3—21no—
105C22×Dic6C2311yescharacteristic
106C2×C3×M16C2312yes—
107C22×C3×C8C2311yescharacteristic
108C22×C3×C8C2311yescharacteristic
109C3×C4⋊Q8D411yescharacteristic
110C3×C4⋊Q8D411yescharacteristic
111C3×#125D412yes—
112C3×C22⋊Q8D412yes—
113C3×C22⋊Q8D412yes—
114C3×C4×Q8C2312yes—
115C3×C4×Q8C2312yes—
116C3×#117C2312yes—
117C2×C3×C42C2311yescharacteristic
118#947—22no—
119#946—24no—
120#946—24no—
121#946—22no—
122Q8×Dic3—22no—
123#930—21no—
124#930—21no—
125C2×(C2×D3)⋊C4—21no—
126#926C2312yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2311yescharacteristic
129#924—21no—
130#924—21no—
131C2×Dic3⋊C4—21no—
132C2×C4×Dic3—21no—
133#908—28no—
134C2×D24C2314yes—
135C2×#204—24no—
136C2×#204C2314yes—
137C2×#204C2314yes—
138#898—24no—
139#898—24no—
140#897—24no—
141#897—22no—
142#897—22no—
143#897—22no—
144#896—24no—
145#896—24no—
146#896—24no—
147#894—24no—
148#893—22no—
149#893—22no—
150#893—22no—
151#892—22no—
152#892—24no—
153D3×C4⋊C4—24no—
154D3×C4⋊C4—22no—
155D3×C4⋊C4—22no—
156#890—24no—
157#890—22no—
158#890—22no—
159#888—22no—
160#886—24no—
161#67⋊C4—24no—
162D3×C22⋊C4—24no—
163#878—24no—
164#875C2314yes—
165#875C2312yes—
166#874C2311yescharacteristic
167C4×D12—21no—
168C4×D12—21no—
169C4×D12—21no—
170C4×D12—21no—
171C4×D12C2314yes—
172C4×D12C2314yes—
173C4×D12C2314yes—
174C4×D12C2312yes—
175C4×D12C2312yes—
176#870C2314yes—
177#870C2312yes—
178#869C2314yes—
179#869C2311yescharacteristic
180#869C2314yes—
181#869C2311yescharacteristic
182#869C2311yescharacteristic
183C4×Dic6—21no—
184C4×Dic6—21no—
185C4×Dic6—21no—
186C4×Dic6—21no—
187C4×Dic6C2314yes—
188C4×Dic6C2312yes—
189C4×Dic6C2312yes—
190C3×C4⋊C8C2312yes—
191C3×C4⋊C8C2312yes—
192C3×C4⋊C8C2314yes—
193C3×C4⋊C8C2314yes—
194C3×#97C2312yes—
195C3×#97C2314yes—
196C3×#97C2312yes—
197C3×C4×C8C2314yes—
198#821C2314yes—
199#821C2314yes—
200#821C2312yes—
201#821C2312yes—
202#821C2312yes—
203#821C2312yes—
204#818C2314yes—
205#818C2314yes—
206#817C2314yes—
207#816C2314yes—
208#816C2314yes—
209#816C2314yes—
210#816C2312yes—
211#816C2314yes—
212#816C2312yes—
213#816C2312yes—
214#816C2312yes—
215C2×AGL(1,ℤ/8)—32no—
216C22×SD16—31no—
217#559—32no—
218#557—32no—
219#556—34no—
220#556—61no—
221#554—34no—
222#554—32no—
223#554—61no—
224#554—61no—
225#554—62no—
226#552—62no—
227D4×Q8—32no—
228D4×Q8—61no—
229#544—32no—
230#543—62no—
231#539—62no—
232#536—32no—
233#533—31no—
234#533C2×D311yescharacteristic
235#532—31no—
236#529—62no—
237C2×C22⋊Q8—62no—
238#520—32no—
239#518—32no—
240C2×C4×Q8—31no—
241C2×C4×D4—31no—
242#491—32no—
243#491—34no—
244#490—32no—
245#489—32no—
246#489—34no—
247#489—32no—
248#488—34no—
249#487—34no—
250#479—34no—
251#479—34no—
252#478—34no—
253#477—32no—
254#477—34no—
255#475—32no—
256#474—34no—
257#463—34no—
258#463—34no—
259#463—34no—
260#461—34no—
261#461—32no—
262#461—34no—
263#459—32no—
264#446C2×D314yescomplemented
265#446C2×D314yescomplemented
266#442—34no—
267#440—34no—
268#440—34no—
269#440—34no—
270C4×SD16—34no—
271C4×D8—34no—
272#428—32no—
273C2×#107—31no—
274#424C2×D312yes—
275#423C2×D311yescharacteristic
276#419—32no—
277#417—32no—
278#416—32no—
279#416—32no—
280C2×#103—31no—
281C2×#102—31no—
282#405C2×D312yes—
283C2×#97C2×D311yescharacteristic
284C23×D3—21no—
285C2×C3×Q8C2×D412yes—
286C2×C3×Q8C2×D412yes—
287C22×C3×C4C2×D411yescharacteristic
288C22×C3×C4C2×D411yescharacteristic
289C22×C3×C4C2411yescharacteristic
290C2×#67—22no—
291C22×Dic3—21no—
292C22×Dic3—21no—
293C22×Dic3—21no—
294D3×Q8—84no—
295#235—24no—
296#235—24no—
297C2×D12—21no—
298C2×D12—22no—
299C2×D12—21no—
300C2×D12—21no—
301C2×D12#4114yes—
302C2×D12C2412yes—
303C2×D12C2412yes—
304C2×C4×D3—22no—
305C2×C4×D3—22no—
306C2×C4×D3—22no—
307C2×C4×D3—24no—
308C2×C4×D3—24no—
309C2×C4×D3—24no—
310C2×C4×D3—82no—
311C2×Dic6—22no—
312C2×Dic6—22no—
313C2×Dic6—21no—
314C2×Dic6—21no—
315C2×Dic6—21no—
316C2×Dic6—24no—
317C2×Dic6C2412yes—
318C2×Dic6C2412yes—
319C3×M16—24no—
320C2×C3×C8—22no—
321C2×C3×C8—22no—
322C2×C3×C8C2412yes—
323C2×C3×C8C2414yes—
324C2×C3×C8C2412yes—
325C3×C4⋊C4C2×D412yes—
326C3×C4⋊C4C2×D414yes—
327C3×C4⋊C4C2×D412yes—
328C3×C4⋊C4C2×D412yes—
329C3×C4⋊C4C2×D412yes—
330C3×C4⋊C4C2×D414yes—
331C3×C22⋊C4C2×D414yes—
332C3×C42C2×D411yescharacteristic
333C3×C42C2×D411yescharacteristic
334C3×C42C2414yes—
335C3×C42C2411yescharacteristic
336C3×C42C2411yescharacteristic
337#217—22no—
338(C2×D3)⋊C4—22no—
339(C2×D3)⋊C4—22no—
340(C2×D3)⋊C4—22no—
341(C2×D3)⋊C4—22no—
342(C2×D3)⋊C4—24no—
343(C2×D3)⋊C4—24no—
344(C2×D3)⋊C4—24no—
345(C2×D3)⋊C4—24no—
346(C2×D3)⋊C4—24no—
347C4⋊Dic3—21no—
348C4⋊Dic3—21no—
349C4⋊Dic3—21no—
350C4⋊Dic3—21no—
351C4⋊Dic3—21no—
352C4⋊Dic3—24no—
353C4⋊Dic3—24no—
354C4⋊Dic3—24no—
355C4⋊Dic3—21no—
356C4⋊Dic3C2412yes—
357C4⋊Dic3C2412yes—
358C4⋊Dic3C2412yes—
359C4⋊Dic3C2412yes—
360Dic3⋊C4—22no—
361Dic3⋊C4—22no—
362Dic3⋊C4—22no—
363Dic3⋊C4—22no—
364Dic3⋊C4—24no—
365Dic3⋊C4—24no—
366Dic3⋊C4—24no—
367C4×Dic3—22no—
368C4×Dic3—22no—
369C4×Dic3—22no—
370C4×Dic3—24no—
371D24—28no—
372#204—28no—
373#204—28no—
374C2×#41—32no—
375C22×Q8—31no—
376C22×Q8—62no—
377C22×D4—31no—
378C23×C4—61no—
379AGL(1,ℤ/8)—68no—
380C2×SD16—34no—
381C2×SD16—34no—
382C2×SD16—64no—
383C2×D8—34no—
384C2×M16C22×D312yes—
385C22×C8C22×D311yescharacteristic
386C22×C8C22×D311yescharacteristic
387C4⋊Q8—31no—
388C4⋊Q8—31no—
389C4⋊Q8—34no—
390C4⋊Q8—31no—
391C4⋊Q8—34no—
392C4⋊Q8—62no—
393C4⋊Q8D1211yescharacteristic
394C4⋊Q8D1211yescharacteristic
395#127—31no—
396#126—64no—
397#126—64no—
398#125—32no—
399#125—34no—
400#125—62no—
401#125—64no—
402#125—62no—
403#125D1212yes—
404#124—32no—
405#124—34no—
406#124—62no—
407#123—64no—
408#123—64no—
409C22⋊Q8—62no—
410C22⋊Q8—64no—
411C22⋊Q8—64no—
412C22⋊Q8—61no—
413C22⋊Q8—61no—
414C22⋊Q8—62no—
415C22⋊Q8—64no—
416C22⋊Q8—62no—
417C22⋊Q8—62no—
418C22⋊Q8—64no—
419C22⋊Q8—64no—
420C22⋊Q8—64no—
421C22⋊Q8—64no—
422C22⋊Q8D1212yes—
423C22⋊Q8D1212yes—
424#121—61no—
425#121—61no—
426C4×Q8—32no—
427C4×Q8—32no—
428C4×Q8—34no—
429C4×Q8—62no—
430C4×Q8—61no—
431C4×Q8—61no—
432C4×Q8—61no—
433C4×Q8—61no—
434C4×Q8C22×D312yes—
435C4×Q8C22×D312yes—
436C4×D4—32no—
437C4×D4—32no—
438C4×D4—34no—
439C4×D4—34no—
440C4×D4—34no—
441C4×D4—62no—
442C4×D4—62no—
443C4×D4—62no—
444C4×D4—64no—
445C4×D4—61no—
446C4×D4—61no—
447C4×D4—61no—
448C4×D4—61no—
449#117—32no—
450#117—62no—
451#117—64no—
452#117C22×D312yes—
453C2×C4⋊C4—31no—
454C2×C4⋊C4—31no—
455C2×C4⋊C4—61no—
456C2×C4⋊C4—62no—
457C2×C4⋊C4—64no—
458C2×C4⋊C4—62no—
459C2×C22⋊C4—61no—
460C2×C22⋊C4—64no—
461C2×C42—61no—
462C2×C42C22×D311yescharacteristic
463#107—34no—
464#107—34no—
465#106—34no—
466C4⋊C8C22×D312yes—
467C4⋊C8C22×D312yes—
468C4⋊C8C22×D314yes—
469C4⋊C8C22×D314yes—
470#103—34no—
471#103—32no—
472#103—32no—
473#103—32no—
474#103—32no—
475#103—34no—
476#103—34no—
477#103—34no—
478#102—34no—
479#102—32no—
480#102—32no—
481#102—32no—
482#102—32no—
483#102—34no—
484#97C22×D312yes—
485#97C22×D312yes—
486#97C22×D314yes—
487C4×C8C22×D314yes—
488C23×C3C22×D411yescharacteristic
489C22×D3—22no—
490C22×D3—24no—
491C22×D3—22no—
492C22×D3—81no—
493C22×D3—81no—
494C3×Q8—42no—
495C3×Q8—42no—
496C2×C3×C4—41no—
497C2×C3×C4—41no—
498C2×C3×C4C22×D411yescharacteristic
499C2×C3×C4C22×D411yescharacteristic
500C2×C3×C4C22×D414yes—
501C2×C3×C4C22×D411yescharacteristic
502C2×C3×C4C22×D411yescharacteristic
503C2×C3×C4C22×D414yes—
504C2×C3×C4C22×D411yescharacteristic
505C2×C3×C4C2×#4112yes—
506C2×C3×C42−1+412yes—
507C2×C3×C4C2511yescharacteristic
508#67—44no—
509C2×Dic3—22no—
510C2×Dic3—22no—
511C2×Dic3—22no—
512C2×Dic3—22no—
513C2×Dic3—22no—
514C2×Dic3—22no—
515C2×Dic3—42no—
516C2×Dic3—81no—
517C2×Dic3—81no—
518D12—24no—
519D12—22no—
520D12—22no—
521D12—22no—
522D12—42no—
523D12—81no—
524D12C2×#4114yes—
525C4×D3—44no—
526C4×D3—44no—
527C4×D3—84no—
528C4×D3—84no—
529C4×D3—82no—
530C4×D3—84no—
531Dic6—22no—
532Dic6—22no—
533Dic6—22no—
534Dic6—22no—
535Dic6—42no—
536Dic6—42no—
537Dic6—81no—
538Dic6—84no—
539C3×C8—22no—
540C3×C8—24no—
541C3×C8—22no—
542C24—61no—
543#41—64no—
544#41—64no—
545C2×Q8—32no—
546C2×Q8—32no—
547C2×Q8—61no—
548C2×Q8—61no—
549C2×Q8—61no—
550C2×Q8—64no—
551C2×Q8—62no—
552C2×Q8—62no—
553C2×Q8—244no—
554C2×Q8C2×D1212yes—
555C2×Q8C2×D1212yes—
556C2×D4—34no—
557C2×D4—32no—
558C2×D4—32no—
559C2×D4—62no—
560C2×D4—61no—
561C2×D4—61no—
562C2×D4—61no—
563C2×D4—62no—
564C22×C4—61no—
565C22×C4—61no—
566C22×C4—61no—
567C22×C4—62no—
568C22×C4—62no—
569C22×C4—64no—
570C22×C4—64no—
571C22×C4—62no—
572C22×C4—64no—
573C22×C4—242no—
574C22×C4C2×D1211yescharacteristic
575C22×C4C2×D1211yescharacteristic
576C22×C4C23×D311yescharacteristic
577SD16—68no—
578SD16—68no—
579D8—68no—
580M16—24no—
581C2×C8—22no—
582C2×C8—22no—
583C2×C8C23×D312yes—
584C2×C8C23×D314yes—
585C2×C8C23×D312yes—
586C4⋊C4—32no—
587C4⋊C4—32no—
588C4⋊C4—32no—
589C4⋊C4—32no—
590C4⋊C4—61no—
591C4⋊C4—61no—
592C4⋊C4—62no—
593C4⋊C4—61no—
594C4⋊C4—64no—
595C4⋊C4—62no—
596C4⋊C4—62no—
597C4⋊C4—64no—
598C4⋊C4—64no—
599C4⋊C4—64no—
600C4⋊C4—64no—
601C4⋊C4—61no—
602C4⋊C4—61no—
603C4⋊C4—62no—
604C4⋊C4—61no—
605C4⋊C4—64no—
606C4⋊C4C2×D1212yes—
607C4⋊C4C2×D1214yes—
608C4⋊C4C2×D1212yes—
609C4⋊C4C2×D1212yes—
610C4⋊C4C2×D1214yes—
611C4⋊C4C2×D1212yes—
612C22⋊C4—62no—
613C22⋊C4—64no—
614C22⋊C4—64no—
615C22⋊C4—62no—
616C22⋊C4—62no—
617C22⋊C4—62no—
618C22⋊C4—64no—
619C22⋊C4—64no—
620C22⋊C4—62no—
621C22⋊C4—64no—
622C22⋊C4C2×D1214yes—
623C42—62no—
624C42—62no—
625C42—62no—
626C42—64no—
627C42C2×D1211yescharacteristic
628C42C2×D1211yescharacteristic
629C42C23×D314yes—
630C42C23×D311yescharacteristic
631C42C23×D311yescharacteristic
632C22×C3—21no—
633C22×C3—21no—
634C22×C3#54812yes—
635C22×C3C23×D411yescharacteristic
636C2×D3—24no—
637C2×D3—42no—
638C2×D3—42no—
639C2×D3—42no—
640C2×D3—42no—
641C2×D3—42no—
642C2×D3—84no—
643C3×C4—21no—
644C3×C4—24no—
645C3×C4—21no—
646C3×C4—21no—
647C3×C4—24no—
648C3×C4—21no—
649C3×C4#54812yes—
650C3×C4C22×#4111yescharacteristic
651C3×C4C2×2−1+411yescharacteristic
652Dic3—42no—
653Dic3—42no—
654Dic3—42no—
655Dic3—42no—
656Dic3—42no—
657Dic3—42no—
658C23—62no—
659C23—64no—
660C23—62no—
661C23—241no—
662C23—241no—
663C23C22×D1211yescharacteristic
664Q8—42no—
665Q8—42no—
666Q8—62no—
667Q8—62no—
668Q8—62no—
669Q8—62no—
670Q8—122no—
671Q8—122no—
672Q8—244no—
673Q8—241no—
674D4—34no—
675D4—62no—
676D4—62no—
677D4—62no—
678D4—64no—
679D4—122no—
680D4—124no—
681D4—241no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—62no—
685C2×C4—62no—
686C2×C4—62no—
687C2×C4—62no—
688C2×C4—62no—
689C2×C4—62no—
690C2×C4—124no—
691C2×C4—122no—
692C2×C4—124no—
693C2×C4—242no—
694C2×C4—244no—
695C2×C4—244no—
696C2×C4—244no—
697C2×C4—241no—
698C2×C4—241no—
699C2×C4C22×D1211yescharacteristic
700C2×C4C22×D1211yescharacteristic
701C2×C4C22×D1214yes—
702C2×C4C22×D1211yescharacteristic
703C2×C4C22×D1211yescharacteristic
704C2×C4C22×D1214yes—
705C2×C4C22×D1211yescharacteristic
706C2×C4#100712yes—
707C2×C4D3×#4112yes—
708C2×C4C24×D311yescharacteristic
709C8—22no—
710C8—22no—
711C8—24no—
712C2×C3—22no—
713C2×C3C2×#54811yescharacteristic
714C2×C3#358511yescharacteristic
715C2×C3#358611yescharacteristic
716D3—44no—
717D3—82no—
718C22—21no—
719C22—21no—
720C22—64no—
721C22—122no—
722C22—122no—
723C22—122no—
724C22—122no—
725C22—122no—
726C22—244no—
727C22#557612yes—
728C22C23×D1211yescharacteristic
729C4—21no—
730C4—21no—
731C4—24no—
732C4—21no—
733C4—21no—
734C4—24no—
735C4—122no—
736C4—122no—
737C4—122no—
738C4—122no—
739C4—122no—
740C4—122no—
741C4#557612yes—
742C4C2×#100711yescharacteristic
743C4C2×D3×#4111yescharacteristic
744C3#6133311yescharacteristic, complemented, Sylow
745C2—22no—
746C2—124no—
747C2—242no—
748C2C2×#557611yescharacteristic
749C2#8716711yescharacteristic
750C2#8716811yescharacteristic
subgroupG/H
centerC22C23×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2961C2
socleC22×C3C23×D4
prime radical#79055C2
hypercenter#2961D3
hypocenterC3#61333
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533 ◁ #2961

Complements

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

NH ≅ G/Ntype
1. #86280C2semidirect
2. #86260C2semidirect
3. C3×#2961C2semidirect
4. #79974C2semidirect
5. #79972C2semidirect
6. #79971C2semidirect
7. #79970C2semidirect
8. #79091C2semidirect
9. #79087C2semidirect
10. #79068C2semidirect
12. #78575C2semidirect
13. #78560C2semidirect
14. #78497C2semidirect
15. C2×#4710C2semidirect
30. #5585C22semidirect
31. #5584C22semidirect
35. #5570C22semidirect
44. C3×#446C22semidirect
45. C3×#446C22semidirect
58. #4851C22semidirect
59. #4850C22semidirect
60. #4850C22semidirect
61. #4847C22semidirect
62. #4846C22semidirect
70. #4829C22semidirect
71. #4829C22semidirect
72. #4714C22semidirect
73. #4714C22semidirect
74. #4714C22semidirect
75. #4713C22semidirect
80. #4703C22semidirect
81. #4703C22semidirect
82. #4697C22semidirect
85. #4681C22semidirect
98. #2961D3semidirect
264. #446C2×D3semidirect
265. #446C2×D3semidirect
744. C3#61333semidirect

Maximal subgroups

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

subgroupindex
#862802
#862602
C3×#29612
#799742
#799722
#799712
#799702
#790912
#790872
#790682
#790552
#785752
#785602
#784972
C2×#47102
#613333