Finite Group Databasebeta

← #11664534

Subgroups — #11664534

Order 768.

2042 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86281C211yesmaximal, characteristic, complemented
2#86243C211yesmaximal, characteristic
3C3×#3131C211yesmaximal, characteristic, complemented
4#83337C211yesmaximal, characteristic, complemented
5#83283C211yesmaximal, characteristic, complemented
6#81639C218yesmaximal, complemented
7#81600C212yesmaximal, complemented
8D3×#487C212yesmaximal, complemented
9#81565C212yesmaximal, complemented
10#81562C212yesmaximal, complemented
11#78832C214yesmaximal, complemented
12#78828C214yesmaximal, complemented
13C2×#4681C211yesmaximal, characteristic, complemented
14C4×#901C211yesmaximal, characteristic, complemented
15#60623—31nomaximal, Sylow
16C2×#1010C2212yes—
17C3×#533C2211yescharacteristic
18C3×#532C2211yescharacteristic
19#5825C2212yes—
20C2×#901C2212yes—
21#5728C2212yes—
22D3×C4⋊Q8C2212yes—
23D3×C4⋊Q8C2212yes—
24#5716C2212yes—
25#5689C2214yes—
26#5675C2214yes—
27#5635—22no—
28#5632—22no—
29#5611—22no—
30#5611—22no—
31#5579—21no—
32Q8×D12—21no—
33#5567—21no—
34Q8×Dic6—21no—
35#5551—21no—
36#5551—21no—
37D4×Dic6—21no—
38D4×Dic6—21no—
39#5523C2212yes—
40#5519C2212yes—
41C4×#235C2211yescharacteristic
42C2×#869C2211yescharacteristic
43C2×#869C2211yescharacteristic
44C3×#488C2214yes—
45C3×#487C2211yescharacteristic
46C3×#487C2211yescharacteristic
47C3×#486C2211yescharacteristic
48C3×#486C2211yescharacteristic
49C3×#416C2214yes—
50C2×C3×C4×C8C2211yescharacteristic
51#5232C2214yes—
52C2×#816C2214yes—
53#5101C2212yes—
54#5093C2212yes—
55#5093C2212yes—
56#5079C2212yes—
57#5070C2212yes—
58#5069C2214yes—
59#5055C2214yes—
60#5004C2214yes—
61C4×#208C2211yescharacteristic
62#4802C2218yes—
63D3×#103C2218yes—
64#4797C2218yes—
65#4792C2218yes—
66#4772C2218yes—
67#4771C2218yes—
68#4768C2218yes—
69#4765C2218yes—
70#4689C2214yes—
71C4×ΓL(1,𝔽25)C2212yes—
72C4×D3×C8C2212yes—
73#4681C2212yes—
74#4681C2212yes—
75#4681C2214yes—
76#3503—31no—
77#3499—31no—
78#3141—32no—
79#3141—31no—
80#3139—32no—
81#3139—31no—
82#3137—38no—
83#3137—34no—
84#3131—32no—
85#3131D311yescharacteristic, complemented
86C2×#487—32no—
87C2×#487—31no—
88#2898—34no—
89C4×#131—31no—
90C2×C3×#41C2312yes—
91#1010—216no—
92C2×D3×Q8—22no—
93C2×D3×Q8C2314yes—
94C2×#237—24no—
95C2×#237C2314yes—
96C2×#235—22no—
97C2×#235C2311yescharacteristic
98C22×Dic6—21no—
99C22×Dic6—22no—
100C22×Dic6C2312yes—
101C22×C3×C8C2312yes—
102C3×C4⋊Q8C2311yescharacteristic
103C3×C4⋊Q8C2311yescharacteristic
104C3×C4⋊Q8C2311yescharacteristic
105C3×#127C2311yescharacteristic
106C3×#125C2312yes—
107C3×#124C2312yes—
108C3×C22⋊Q8—21no—
109C3×C22⋊Q8—21no—
110C3×#121—21no—
111C3×#121—21no—
112C3×C4×Q8—21no—
113C3×C4×Q8—21no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×#117C2312yes—
117C2×C3×C42C2311yescharacteristic
118#946—22no—
119Q8×Dic3—22no—
120#944—22no—
121#944C2314yes—
122#938—22no—
123#938—22no—
124#936C2314yes—
125#935—24no—
126D4×Dic3—22no—
127D4×Dic3—22no—
128C4×#67D412yes—
129C4×#67D412yes—
130#926C2311yescharacteristic
131C2×C4⋊Dic3—21no—
132C2×C4⋊Dic3—21no—
133C2×C4⋊Dic3—21no—
134C2×C4⋊Dic3C2312yes—
135#924—22no—
136#924—22no—
137C2×#208C2312yes—
138#901—28no—
139C2×ΓL(1,𝔽25)C2314yes—
140C2×D3×C8C2314yes—
141#897—22no—
142#897—24no—
143#896—24no—
144#892C2314yes—
145D3×C4⋊C4—22no—
146D3×C4⋊C4—22no—
147D3×C4⋊C4—22no—
148D3×C4⋊C4C2314yes—
149#890C2314yes—
150#889—24no—
151#888—24no—
152#888—22no—
153#888—22no—
154#888—22no—
155#888C2314yes—
156#887—22no—
157#883—24no—
158#883—24no—
159#67⋊C4—24no—
160#878—24no—
161#878—24no—
162#878—24no—
163C4×D12—22no—
164C4×D12D411yescharacteristic
165C4×D12D411yescharacteristic
166#872C2312yes—
167C42×D3C2312yes—
168#870—22no—
169#869—21no—
170#869—21no—
171#869—21no—
172#869—21no—
173#869—22no—
174#869C2312yes—
175#869C2312yes—
176#869C2312yes—
177#869C2312yes—
178C4×Dic6—22no—
179C4×Dic6D411yescharacteristic
180C4×Dic6D411yescharacteristic
181C3×#103C2314yes—
182C3×#103C2314yes—
183C3×#102C2314yes—
184C3×#102C2314yes—
185C3×C4×C8C2311yescharacteristic
186C3×C4×C8C2311yescharacteristic
187C3×C4×C8C2312yes—
188#835C2318yes—
189#832C2318yes—
190#816C2318yes—
191#816C2318yes—
192#815C2314yes—
193C8×Dic3C2314yes—
194#810C2318yes—
195#810C2318yes—
196#803C2312yes—
197C4×C3⋊C8C2312yes—
198C2×2−1+4—32no—
199C2×#131—32no—
200Q82—61no—
201#557—61no—
202#554—61no—
203#554—62no—
204D4×Q8—61no—
205D4×Q8—61no—
206D4×Q8—61no—
207D4×Q8—62no—
208#547—61no—
209#547—61no—
210#547—62no—
211#547—62no—
212#544—34no—
213#544—32no—
214#536—32no—
215#536—34no—
216#533—32no—
217#533C2×D311yescharacteristic
218#532—32no—
219#532C2×D311yescharacteristic
220C2×C4⋊Q8—32no—
221C2×C4⋊Q8—32no—
222C2×C4⋊Q8—31no—
223C2×C4⋊Q8—31no—
224#520—32no—
225C4×#41—31no—
226#491—34no—
227#491—38no—
228#489—34no—
229#489—38no—
230#488—38no—
231#488C2×D314yes—
232#487—32no—
233#487—32no—
234#487—32no—
235#487—38no—
236#487—32no—
237#487—34no—
238#487—32no—
239#487C2×D311yescharacteristic
240#487C2×D311yescharacteristic
241#486—32no—
242#486C2×D311yescharacteristic
243#486C2×D311yescharacteristic
244#419—34no—
245#419—38no—
246#417—34no—
247#417—38no—
248#416—38no—
249#416C2×D314yes—
250C2×#103—34no—
251C2×#103—38no—
252#405—32no—
253#405—34no—
254C4×M16—31no—
255C4×M16—32no—
256C2×C4×C8—32no—
257C2×C4×C8C2×D311yescharacteristic
258C3×#41—28no—
259C2×C3×Q8—21no—
260C2×C3×Q8—21no—
261C2×C3×Q8C2412yes—
262C2×C3×D4—22no—
263C2×C3×D4—21no—
264C2×C3×D4—21no—
265C2×C3×D4C2412yes—
266C22×C3×C4—22no—
267C22×C3×C4C2×D411yescharacteristic
268C22×C3×C4C2×D411yescharacteristic
269C22×C3×C4C2411yescharacteristic
270C2×#67—24no—
271C2×#67C2×D412yes—
272C22×Dic3—22no—
273C22×Dic3—22no—
274C22×Dic3—24no—
275D3×Q8—216no—
276D3×Q8—84no—
277#237—216no—
278#237—416no—
279#235—48no—
280#235#4118yes—
281C2×D12C2×D411yescharacteristic
282C2×C4×D3—24no—
283C2×C4×D3—24no—
284C2×C4×D3C2×D412yes—
285C2×C4×D3C2×D412yes—
286C2×C4×D3C2412yes—
287C2×Dic6—28no—
288C2×Dic6—22no—
289C2×Dic6—22no—
290C2×Dic6—24no—
291C2×Dic6—22no—
292C2×Dic6—24no—
293C2×Dic6—48no—
294C2×Dic6—82no—
295C2×Dic6C2×D411yescharacteristic
296C2×Dic6C2414yes—
297C2×C3×C8—24no—
298C2×C3×C8C2412yes—
299C2×C3×C8C2412yes—
300C3×C4⋊C4—21no—
301C3×C4⋊C4—21no—
302C3×C4⋊C4—21no—
303C3×C4⋊C4—21no—
304C3×C4⋊C4—22no—
305C3×C4⋊C4—22no—
306C3×C4⋊C4C2412yes—
307C3×C4⋊C4C2412yes—
308C3×C22⋊C4—22no—
309C3×C22⋊C4—22no—
310C3×C22⋊C4—22no—
311C3×C42—22no—
312C3×C42C2×D411yescharacteristic
313C3×C42C2×D411yescharacteristic
314C3×C42C2411yescharacteristic
315C3×C42C2411yescharacteristic
316#217—24no—
317#217—24no—
318#217C2×D412yes—
319(C2×D3)⋊C4—24no—
320(C2×D3)⋊C4C2×D412yes—
321(C2×D3)⋊C4C2×D412yes—
322C4⋊Dic3—22no—
323C4⋊Dic3—22no—
324C4⋊Dic3—22no—
325C4⋊Dic3—22no—
326C4⋊Dic3—22no—
327C4⋊Dic3—22no—
328C4⋊Dic3—22no—
329C4⋊Dic3—24no—
330C4⋊Dic3C2×D411yescharacteristic
331C4⋊Dic3C2×D411yescharacteristic
332C4⋊Dic3C2414yes—
333Dic3⋊C4—24no—
334Dic3⋊C4—24no—
335Dic3⋊C4—24no—
336Dic3⋊C4—24no—
337Dic3⋊C4—24no—
338Dic3⋊C4C2×D412yes—
339Dic3⋊C4C2×D412yes—
340C4×Dic3—24no—
341C4×Dic3C2×D412yes—
342C4×Dic3C2×D412yes—
343C4×Dic3C2412yes—
344#208—24no—
345C2×C3⋊C8C2414yes—
346ΓL(1,𝔽25)—28no—
347D3×C8—28no—
3482−1+4—616no—
349C2×#41—31no—
350C2×#41—34no—
351C2×#41—62no—
352C2×#41—64no—
353C2×#41C22×D312yes—
354C22×Q8—34no—
355C22×Q8—32no—
356C22×Q8—62no—
357C22×Q8—62no—
358C22×Q8—61no—
359#131—68no—
360C2×M16—32no—
361C2×M16—34no—
362C22×C8—34no—
363C22×C8C22×D312yes—
364C4⋊Q8—34no—
365C4⋊Q8—34no—
366C4⋊Q8—32no—
367C4⋊Q8—32no—
368C4⋊Q8—32no—
369C4⋊Q8—32no—
370C4⋊Q8—62no—
371C4⋊Q8—62no—
372C4⋊Q8—62no—
373C4⋊Q8—62no—
374C4⋊Q8—62no—
375C4⋊Q8—64no—
376C4⋊Q8—61no—
377C4⋊Q8—61no—
378C4⋊Q8—61no—
379C4⋊Q8—61no—
380C4⋊Q8C22×D311yescharacteristic
381C4⋊Q8C22×D311yescharacteristic
382C4⋊Q8C22×D311yescharacteristic
383#127C22×D311yescharacteristic
384#125—34no—
385#125—62no—
386#125—64no—
387#125C22×D312yes—
388#124—34no—
389#124C22×D312yes—
390#123—64no—
391#123—64no—
392#123—64no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8—64no—
396C22⋊Q8—64no—
397C22⋊Q8—64no—
398C22⋊Q8—64no—
399C22⋊Q8—64no—
400C22⋊Q8—62no—
401C22⋊Q8—62no—
402C22⋊Q8—62no—
403C22⋊Q8—62no—
404#121—21no—
405#121—21no—
406#121—62no—
407#121—62no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—31no—
411C4×Q8—31no—
412C4×Q8—62no—
413C4×Q8—62no—
414C4×Q8—62no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—32no—
418C4×D4—32no—
419C4×D4—31no—
420C4×D4—31no—
421C4×D4—62no—
422C4×D4—62no—
423C4×D4—62no—
424C4×D4—64no—
425#117—31no—
426#117—32no—
427#117—34no—
428#117C22×D312yes—
429C2×C4⋊C4—32no—
430C2×C4⋊C4—34no—
431C2×C4⋊C4—62no—
432C2×C4⋊C4—62no—
433C2×C4⋊C4—62no—
434C2×C4⋊C4—61no—
435C2×C4⋊C4—61no—
436C2×C4⋊C4—61no—
437C2×C42—32no—
438C2×C42C22×D311yescharacteristic
439#103—38no—
440#103—38no—
441#103—38no—
442#103—38no—
443#103—38no—
444#103C22×D314yes—
445#103C22×D314yes—
446#102—38no—
447#102C22×D314yes—
448#102C22×D314yes—
449#97—34no—
450#97—32no—
451C4×C8—34no—
452C4×C8—32no—
453C4×C8C22×D311yescharacteristic
454C4×C8C22×D311yescharacteristic
455C4×C8C22×D312yes—
456C23×C3—22no—
457C23×C3C22×D411yescharacteristic
458C22×D3C22×D412yes—
459C3×Q8—24no—
460C3×Q8—42no—
461C3×Q8—42no—
462C3×D4—24no—
463C3×D4—44no—
464C3×D4—42no—
465C3×D4—42no—
466C2×C3×C4—22no—
467C2×C3×C4—22no—
468C2×C3×C4—22no—
469C2×C3×C4—22no—
470C2×C3×C4—22no—
471C2×C3×C4—44no—
472C2×C3×C4C22×D411yescharacteristic
473C2×C3×C4C22×D411yescharacteristic
474C2×C3×C4C22×D411yescharacteristic
475C2×C3×C4C22×D411yescharacteristic
476C2×C3×C4C22×D411yescharacteristic
477C2×C3×C4C2×#4114yes—
478C2×C3×C4C2511yescharacteristic
479#67—48no—
480#67#12418yes—
481C2×Dic3—24no—
482C2×Dic3—24no—
483C2×Dic3—44no—
484C2×Dic3—48no—
485C2×Dic3—82no—
486C2×Dic3C22×D412yes—
487C2×Dic3C22×D412yes—
488C2×Dic3C22×D412yes—
489D12#12414yes—
490C4×D3—24no—
491C4×D3—24no—
492C4×D3—48no—
493C4×D3—84no—
494C4×D3C2×#4118yes—
495Dic6—28no—
496Dic6—44no—
497Dic6—44no—
498Dic6—48no—
499Dic6—84no—
500Dic6—82no—
501Dic6#12414yes—
502C3×C8—22no—
503C3×C8—22no—
504C3⋊C8—24no—
505#41—28no—
506#41—38no—
507#41—616no—
508#41—128no—
509#41—1216no—
510C2×Q8—21no—
511C2×Q8—21no—
512C2×Q8—31no—
513C2×Q8—34no—
514C2×Q8—64no—
515C2×Q8—64no—
516C2×Q8—616no—
517C2×Q8—62no—
518C2×Q8—62no—
519C2×Q8—62no—
520C2×Q8—68no—
521C2×Q8—128no—
522C2×Q8—242no—
523C2×Q8—244no—
524C2×Q8C23×D312yes—
525C2×D4—22no—
526C2×D4—21no—
527C2×D4—21no—
528C2×D4—32no—
529C2×D4—31no—
530C2×D4—64no—
531C2×D4C23×D312yes—
532C22×C4—22no—
533C22×C4—32no—
534C22×C4—32no—
535C22×C4—32no—
536C22×C4—64no—
537C22×C4—64no—
538C22×C4—64no—
539C22×C4—62no—
540C22×C4—62no—
541C22×C4D3×D411yescharacteristic
542C22×C4D3×D411yescharacteristic
543C22×C4C23×D311yescharacteristic
544M16—68no—
545M16—64no—
546C2×C8—24no—
547C2×C8—34no—
548C2×C8—68no—
549C2×C8C23×D312yes—
550C2×C8C23×D312yes—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—32no—
558C4⋊C4—31no—
559C4⋊C4—32no—
560C4⋊C4—31no—
561C4⋊C4—34no—
562C4⋊C4—64no—
563C4⋊C4—64no—
564C4⋊C4—64no—
565C4⋊C4—64no—
566C4⋊C4—64no—
567C4⋊C4—64no—
568C4⋊C4—62no—
569C4⋊C4—62no—
570C4⋊C4—62no—
571C4⋊C4—62no—
572C4⋊C4—62no—
573C4⋊C4—62no—
574C4⋊C4—62no—
575C4⋊C4C23×D312yes—
576C4⋊C4C23×D312yes—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—22no—
580C22⋊C4—32no—
581C22⋊C4—32no—
582C22⋊C4—32no—
583C22⋊C4—64no—
584C22⋊C4—64no—
585C22⋊C4—64no—
586C42—22no—
587C42—32no—
588C42—32no—
589C42—32no—
590C42—64no—
591C42D3×D411yescharacteristic
592C42D3×D411yescharacteristic
593C42C23×D311yescharacteristic
594C42C23×D311yescharacteristic
595C22×C3—21no—
596C22×C3—21no—
597C22×C3—42no—
598C22×C3—42no—
599C22×C3—42no—
600C22×C3C2×#12412yes—
601C22×C3C23×D411yescharacteristic
602C2×D3—22no—
603C2×D3—22no—
604C2×D3C2×#12414yes—
605C3×C4—21no—
606C3×C4—21no—
607C3×C4—42no—
608C3×C4—42no—
609C3×C4—42no—
610C3×C4C2×#12412yes—
611C3×C4#57712yes—
612C3×C4#57812yes—
613C3×C4C22×#4112yes—
614Dic3—22no—
615Dic3—22no—
616Dic3—44no—
617Dic3—44no—
618Dic3C2×#12414yes—
619C23—22no—
620C23—32no—
621C23C2×D3×D411yescharacteristic
622Q8—24no—
623Q8—34no—
624Q8—42no—
625Q8—42no—
626Q8—68no—
627Q8—128no—
628Q8—124no—
629Q8—124no—
630Q8—242no—
631Q8—244no—
632D4—24no—
633D4—38no—
634D4—34no—
635D4—44no—
636D4—42no—
637D4—42no—
638D4—128no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—32no—
645C2×C4—38no—
646C2×C4—32no—
647C2×C4—32no—
648C2×C4—44no—
649C2×C4—64no—
650C2×C4—64no—
651C2×C4—64no—
652C2×C4—64no—
653C2×C4—128no—
654C2×C4—128no—
655C2×C4—124no—
656C2×C4—244no—
657C2×C4—242no—
658C2×C4C2×D3×D411yescharacteristic
659C2×C4C2×D3×D411yescharacteristic
660C2×C4C2×D3×D411yescharacteristic
661C2×C4C2×D3×D411yescharacteristic
662C2×C4C2×D3×D411yescharacteristic
663C2×C4D3×#4114yes—
664C2×C4C24×D311yescharacteristic
665C8—22no—
666C8—22no—
667C8—64no—
668C2×C3—22no—
669C2×C3—82no—
670C2×C3C22×#12411yescharacteristic
671C2×C3C2×#57711yescharacteristic
672C2×C3C2×#57811yescharacteristic
673D3—24no—
674C22—21no—
675C22—21no—
676C22—34no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—62no—
681C22—62no—
682C22D3×#12412yes—
683C22C22×D3×D411yescharacteristic
684C4—21no—
685C4—21no—
686C4—34no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—62no—
691C4—62no—
692C4—124no—
693C4—124no—
694C4D3×#12412yes—
695C4#577012yes—
696C4#577112yes—
697C4C2×D3×#4112yes—
698C3#6062311yescharacteristic, complemented, Sylow
699C2—22no—
700C2—64no—
701C2—82no—
702C2C2×D3×#12411yescharacteristic
703C2C2×#577011yescharacteristic
704C2C2×#577111yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3131C2
socleC22×C3C23×D4
prime radical#86243C2
hypercenter#3131D3
hypocenterC3#60623
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #86281C2semidirect
3. C3×#3131C2semidirect
4. #83337C2semidirect
5. #83283C2semidirect
6. #81639C2semidirect
7. #81600C2semidirect
8. D3×#487C2semidirect
9. #81565C2semidirect
10. #81562C2semidirect
11. #78832C2semidirect
12. #78828C2semidirect
13. C2×#4681C2semidirect
14. C4×#901C2semidirect
85. #3131D3semidirect
698. C3#60623semidirect

Maximal subgroups

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

subgroupindex
#862812
#862432
C3×#31312
#833372
#832832
#816392
#816002
D3×#4872
#815652
#815622
#788322
#788282
C2×#46812
C4×#9012
#606233