Finite Group Databasebeta

← #11675071

Subgroups — #11675071

Order 768.

2126 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#550C211yesmaximal, characteristic, complemented
2Q8×#237C211yesmaximal, characteristic, complemented
3C3×#3358C211yesmaximal, characteristic, complemented
4#83586C211yesmaximal, characteristic, complemented
5#83562C211yesmaximal, characteristic, complemented
6#81590C213yesmaximal, complemented
7#81564C213yesmaximal, complemented
8#80995C213yesmaximal, complemented
9#80965C213yesmaximal, complemented
10#80333C213yesmaximal, complemented
11#80320C213yesmaximal, complemented
12#79932C211yesmaximal, characteristic, complemented
13Q8×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
14C4×#911C213yesmaximal, complemented
15#79570C213yesmaximal, complemented
16#61321—31nomaximal, Sylow
17C2×D3×#41—21no—
18C3×#550C2211yescharacteristic
19C3×D4×Q8C2211yescharacteristic, complemented
20#5834—21no—
21Q8×#67—21no—
22C2×Q8×Dic3—21no—
23C2×#911C2213yes—
24#5728C2213yes—
25#5721C2213yescomplemented
26#5718C2213yes—
27D3×#127C2213yes—
28#5635—23no—
29#5622—23no—
30D3×#121—23no—
31#5599—23no—
32#5577C2211yescharacteristic
33C4×D3×Q8C2211yescharacteristic
34Q8×Dic6C2211yescharacteristic, complemented
35#5553C2213yes—
36C4×D3×D4C2213yes—
37#5543C2213yescomplemented
38C4×#237C2213yes—
39C3×#486C2213yescomplemented
40C3×#474C2213yescomplemented
41C3×#459C2213yescomplemented
42C3×C8×Q8C2211yescharacteristic, complemented
43C3×C4×D8C2213yescomplemented
44#5152C2213yescomplemented
45#5150C2216yescomplemented
46#5088C2213yescomplemented
47#5087C2213yescomplemented
48#5075C2213yescomplemented
49#5070C2213yescomplemented
50Q8×C3⋊C8C2211yescharacteristic, complemented
51C4×#214C2213yescomplemented
52#5015C2213yescomplemented
53C4×#213C2213yescomplemented
54#5008C2213yescomplemented
55#4891C2216yescomplemented
56#4889C2213yescomplemented
57#4881C2213yes—
58#4848C2213yescomplemented
59#4842C2213yescomplemented
60#4837C2213yes—
61#4831C2213yescomplemented
62#4785C2213yescomplemented
63#4771C2211yescharacteristic
64#4770C2216yes—
65#4770C2213yes—
66#4764C2216yescomplemented
67#4764C2216yescomplemented
68#4763C2216yescomplemented
69#4762C2216yescomplemented
70#4757C2213yescomplemented
71#4694C2213yescomplemented
72C4×#204C2213yescomplemented
73C4×ΓL(1,𝔽25)C2213yes—
74#4679C2213yescomplemented
75Q8×#41—31no—
76C2×#550—31no—
77#3368—33no—
78#3368—33no—
79#3366—33no—
80#3366—33no—
81#3362—31no—
82#3362—31no—
83#3358—31no—
84#3358D311yescharacteristic, complemented
85#3138—33no—
86#3077—33no—
87#3038—33no—
88Q8×M16—31no—
89C4×AGL(1,ℤ/8)—33no—
90C2×C3×#41—21no—
91C22×C3×Q8—21no—
92D3×#41—48no—
93C2×#239—21no—
94C2×D3×Q8D411yescharacteristic
95C2×#237—23no—
96C2×#237C2311yescharacteristic
97C2×D3×D4—23no—
98C2×D3×D4—23no—
99C2×D3×D4C2313yes—
100C2×#235—23no—
101C22×C4×D3—23no—
102C2×C3×D8C2313yes—
103C3×C4⋊Q8C2313yes—
104C3×#127C2313yes—
105C3×C22⋊Q8—23no—
106C3×#121—23no—
107C3×C4×Q8C2311yescharacteristic
108C3×C4×D4C2313yes—
109C3×C4×D4C2313yes—
110#946—23no—
111Q8×Dic3—21no—
112Q8×Dic3—23no—
113Q8×Dic3—23no—
114Q8×Dic3D411yescharacteristic
115#944—23no—
116#940—23no—
117#940—23no—
118#940C2316yes—
119#939—26no—
120#938—23no—
121D4×Dic3C2313yes—
122D4×Dic3C2313yes—
123C2×#214C2313yes—
124C2×#213C2313yes—
125#930—23no—
126C4×#67—23no—
127C4×#67—23no—
128C2×C4⋊Dic3—23no—
129#924—23no—
130C2×C4×Dic3—23no—
131#911—212no—
132C2×#204C2313yes—
133C2×ΓL(1,𝔽25)C2313yes—
134C2×ΓL(1,𝔽25)C2311yescharacteristic
135#897—23no—
136#895—23no—
137#893C2313yes—
138#892C2313yes—
139D3×C4⋊C4D413yes—
140D3×C4⋊C4C2311yescharacteristic
141#890C2316yes—
142#889—26no—
143#888—23no—
144#887—23no—
145#887D413yes—
146#887C2313yes—
147#884—26no—
148#67⋊C4—23no—
149#67⋊C4—23no—
150D3×C22⋊C4—23no—
151#878—26no—
152#877—23no—
153#874C2313yes—
154C4×D12C2313yes—
155C42×D3C2313yes—
156#869C2313yes—
157C4×Dic6C2313yes—
158C4×Dic6C2313yes—
159C3×#107C2313yes—
160C3×C4⋊C8C2313yes—
161C3×#102C2313yes—
162C3×#102C2316yes—
163C3×#102C2311yescharacteristic
164C3×C4×C8C2313yes—
165#832C2316yes—
166#832C2311yescharacteristic
167#832C2313yes—
168#821C2316yes—
169#821C2313yes—
170#817C2313yes—
171#816C2311yescharacteristic
172#815C2313yes—
173#815C2313yes—
174Dic3⋊C8C2316yes—
175#810C2311yescharacteristic
176#809C2313yes—
177#809C2316yes—
178#808C2313yes—
179#807C2313yes—
180C4⋊(C3⋊C8)C2313yes—
181C4×C3⋊C8C2313yes—
182C22×#41—61no—
183C2×AGL(1,ℤ/8)—33no—
184Q82—31no—
185#557—33no—
186#554—33no—
187#554—63no—
188#550—33no—
189#550—31no—
190#550—33no—
191#550—63no—
192#550—61no—
193#550C2×D311yescharacteristic
194D4×Q8—61no—
195D4×Q8C2×D311yescharacteristic, complemented
196#533—33no—
197#533—63no—
198C2×#127—33no—
199C2×#121—63no—
200C4×#41—33no—
201C2×C4×Q8—31no—
202C2×C4×Q8—61no—
203C2×C4×D4—33no—
204#490—36no—
205#490—36no—
206#486—33no—
207#486—33no—
208#486—33no—
209#486C2×D313yescomplemented
210#478—36no—
211#478—36no—
212#476—33no—
213#476—33no—
214#474—33no—
215#474C2×D313yescomplemented
216#460—33no—
217#460—33no—
218#460—36no—
219#459—36no—
220#459—33no—
221#459C2×D313yescomplemented
222#446—33no—
223#446—33no—
224C8×Q8—31no—
225C8×Q8C2×D311yescharacteristic, complemented
226#442—33no—
227#442—33no—
228#440—33no—
229#440—33no—
230C4×SD16—33no—
231C4×SD16—33no—
232C4×D8—33no—
233C4×D8C2×D313yescomplemented
234#428—33no—
235#423—33no—
236#418—36no—
237#418—33no—
238#417—31no—
239C4×M16—33no—
240C23×D3—23no—
241C3×#41—44no—
242C2×C3×Q8—23no—
243C2×C3×Q8—44no—
244C2×C3×Q8C2×D411yescharacteristic
245C2×C3×D4—23no—
246C2×C3×D4—23no—
247C2×C3×D4C2413yes—
248C2×C3×D4C2411yescharacteristic
249C22×C3×C4—23no—
250C22×C3×C4—23no—
251C2×#67—21no—
252C2×#67—23no—
253C2×#67—26no—
254C2×#67—26no—
255C22×Dic3—21no—
256C22×Dic3—23no—
257C22×Dic3—23no—
258#239—44no—
259D3×Q8—24no—
260#237—22no—
261#237—412no—
262#237#4114yescomplemented
263D3×D4—26no—
264D3×D4—26no—
265D3×D4—412no—
266D3×D4—86no—
267#235—412no—
268C2×D12—23no—
269C2×D12—23no—
270C2×D12C2413yes—
271C2×C4×D3—21no—
272C2×C4×D3—23no—
273C2×C4×D3—23no—
274C2×C4×D3—412no—
275C2×C4×D3C2×D413yes—
276C2×C4×D3C2×D413yes—
277C2×C4×D3C2411yescharacteristic
278C2×Dic6—23no—
279C2×Dic6C2×D413yes—
280C2×Dic6C2411yescharacteristic
281C3×D8—26no—
282C2×C3×C8C2413yes—
283C2×C3×C8C2411yescharacteristic
284C3×C4⋊C4—23no—
285C3×C4⋊C4C2×D413yes—
286C3×C4⋊C4C2413yes—
287C3×C4⋊C4C2411yescharacteristic
288C3×C22⋊C4—23no—
289C3×C22⋊C4—23no—
290C3×C42C2413yes—
291#217—23no—
292#217—23no—
293#214—26no—
294#213—26no—
295(C2×D3)⋊C4—23no—
296(C2×D3)⋊C4—23no—
297(C2×D3)⋊C4—23no—
298C4⋊Dic3—23no—
299C4⋊Dic3—26no—
300C4⋊Dic3C2×D413yes—
301C4⋊Dic3C2411yescharacteristic
302C4⋊Dic3C2413yes—
303Dic3⋊C4—23no—
304Dic3⋊C4—23no—
305Dic3⋊C4—26no—
306Dic3⋊C4—21no—
307Dic3⋊C4C2×D416yes—
308C4×Dic3—23no—
309C4×Dic3—23no—
310C4×Dic3—23no—
311C4×Dic3C2×D413yes—
312C4×Dic3C2×D413yes—
313C4×Dic3C2413yes—
314C2×C3⋊C8C2411yescharacteristic
315C2×C3⋊C8C2413yes—
316#204—26no—
317ΓL(1,𝔽25)—26no—
318ΓL(1,𝔽25)—22no—
319C2×#41—21no—
320C2×#41—31no—
321C2×#41—63no—
322C2×#41—63no—
323C2×#41—61no—
324C2×#41—128no—
325C22×Q8—21no—
326C22×Q8—31no—
327C22×D4—33no—
328C22×D4—63no—
329C22×D4—63no—
330C23×C4—63no—
331AGL(1,ℤ/8)—612no—
332C2×SD16—33no—
333C2×SD16—33no—
334C2×D8—33no—
335C2×D8C22×D313yes—
336C2×M16—33no—
337C2×M16—31no—
338C4⋊Q8—33no—
339C4⋊Q8—63no—
340C4⋊Q8—63no—
341C4⋊Q8C22×D313yes—
342#127—36no—
343#127—33no—
344#127—63no—
345#127—63no—
346#127C22×D313yes—
347#125—36no—
348#125—66no—
349C22⋊Q8—23no—
350C22⋊Q8—66no—
351C22⋊Q8—63no—
352C22⋊Q8—63no—
353C22⋊Q8—63no—
354#121—23no—
355#121—66no—
356#121—63no—
357#121—63no—
358#121—66no—
359#121—63no—
360C4×Q8—31no—
361C4×Q8—33no—
362C4×Q8—33no—
363C4×Q8—33no—
364C4×Q8—33no—
365C4×Q8—63no—
366C4×Q8—63no—
367C4×Q8—63no—
368C4×Q8—61no—
369C4×Q8C22×D311yescharacteristic
370C4×D4—33no—
371C4×D4—33no—
372C4×D4—33no—
373C4×D4—33no—
374C4×D4—63no—
375C4×D4—63no—
376C4×D4—63no—
377C4×D4—63no—
378C4×D4C22×D313yes—
379C4×D4C22×D313yes—
380#117—33no—
381#117—63no—
382C2×C4⋊C4—33no—
383C2×C4⋊C4—31no—
384C2×C4⋊C4—63no—
385C2×C22⋊C4—63no—
386C2×C42—33no—
387C2×C42—63no—
388#107—33no—
389#107C22×D313yes—
390#106—33no—
391#106—33no—
392C4⋊C8—36no—
393C4⋊C8—33no—
394C4⋊C8C22×D313yes—
395#103—31no—
396#103—31no—
397#102—31no—
398#102—33no—
399#102—36no—
400#102—33no—
401#102—36no—
402#102—33no—
403#102—36no—
404#102C22×D311yescharacteristic
405#102C22×D313yes—
406#102C22×D316yes—
407#97—33no—
408#97—33no—
409C4×C8—33no—
410C4×C8C22×D313yes—
411C23×C3—21no—
412C23×C3—23no—
413C22×D3—23no—
414C22×D3—46no—
415C22×D3—46no—
416C22×D3—46no—
417C22×D3C22×D411yescharacteristic
418C3×Q8—46no—
419C3×Q8AGL(1,ℤ/8)14yescomplemented
420C3×D4—23no—
421C3×D4—21no—
422C3×D4—23no—
423C3×D4—46no—
424C3×D4—83no—
425C3×D4C2×#4112yescomplemented
426C2×C3×C4—23no—
427C2×C3×C4—21no—
428C2×C3×C4—46no—
429C2×C3×C4—46no—
430C2×C3×C4C22×D413yes—
431C2×C3×C4C22×D413yes—
432C2×C3×C4C2511yescharacteristic
433#67—24no—
434#67—412no—
435#67—46no—
436#67—86no—
437C2×Dic3—21no—
438C2×Dic3—24no—
439C2×Dic3—23no—
440C2×Dic3—21no—
441C2×Dic3—46no—
442C2×Dic3—83no—
443C2×Dic3C22×D413yes—
444C2×Dic3C22×D413yes—
445C2×Dic3C22×D411yescharacteristic
446D12—23no—
447D12—23no—
448D12—46no—
449D12—83no—
450C4×D3—26no—
451C4×D3—43no—
452C4×D3—46no—
453C4×D3—42no—
454C4×D3—46no—
455C4×D3C2×#4112yes—
456C4×D32−1+412yes—
457Dic6—26no—
458Dic6—21no—
459Dic6—83no—
460Dic6C2×#4112yescomplemented
461C3×C8—23no—
462C3×C8—21no—
463C3⋊C8—23no—
464C3⋊C8—21no—
465C24—63no—
466#41—34no—
467#41—44no—
468#41—62no—
469#41—1212no—
470#41—1212no—
471#41—124no—
472C2×Q8—23no—
473C2×Q8—33no—
474C2×Q8—31no—
475C2×Q8—44no—
476C2×Q8—64no—
477C2×Q8—63no—
478C2×Q8D3×D411yescharacteristic
479C2×D4—23no—
480C2×D4—23no—
481C2×D4—33no—
482C2×D4—66no—
483C2×D4—66no—
484C2×D4—63no—
485C2×D4—66no—
486C2×D4—66no—
487C2×D4—61no—
488C2×D4—63no—
489C2×D4—63no—
490C2×D4—1212no—
491C2×D4—246no—
492C2×D4C23×D313yes—
493C2×D4C23×D311yescharacteristic
494C22×C4—23no—
495C22×C4—23no—
496C22×C4—31no—
497C22×C4—33no—
498C22×C4—33no—
499C22×C4—63no—
500C22×C4—61no—
501C22×C4—61no—
502C22×C4—63no—
503C22×C4—63no—
504C22×C4—63no—
505C22×C4—1212no—
506SD16—66no—
507SD16—66no—
508D8—26no—
509D8—66no—
510M16—66no—
511M16—62no—
512C2×C8—31no—
513C2×C8—33no—
514C2×C8C23×D311yescharacteristic
515C2×C8C23×D313yes—
516C4⋊C4—23no—
517C4⋊C4—33no—
518C4⋊C4—36no—
519C4⋊C4—33no—
520C4⋊C4—31no—
521C4⋊C4—61no—
522C4⋊C4—66no—
523C4⋊C4—63no—
524C4⋊C4—66no—
525C4⋊C4—63no—
526C4⋊C4—63no—
527C4⋊C4D3×D413yes—
528C4⋊C4C23×D313yes—
529C4⋊C4C23×D311yescharacteristic
530C22⋊C4—23no—
531C22⋊C4—23no—
532C22⋊C4—63no—
533C22⋊C4—63no—
534C22⋊C4—63no—
535C22⋊C4—63no—
536C22⋊C4—63no—
537C42—33no—
538C42—33no—
539C42—33no—
540C42—63no—
541C42—63no—
542C42—63no—
543C42C23×D313yes—
544C22×C3—22no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3—43no—
548C22×C3—43no—
549C22×C3—43no—
550C22×C3C23×D411yescharacteristic
551C2×D3—21no—
552C2×D3—21no—
553C2×D3—43no—
554C2×D3—43no—
555C2×D3—43no—
556C2×D3—812no—
557C2×D3#54812yes—
558C3×C4—23no—
559C3×C4—41no—
560C3×C4—43no—
561C3×C4C2×AGL(1,ℤ/8)16yes—
562C3×C4C22×#4111yescharacteristic
563C3×C4C2×2−1+411yescharacteristic
564Dic3—23no—
565Dic3—22no—
566Dic3—23no—
567Dic3—43no—
568Dic3—41no—
569Dic3#54812yes—
570C23—21no—
571C23—23no—
572C23—31no—
573C23—63no—
574C23—126no—
575C23—126no—
576C23—126no—
577Q8—32no—
578Q8—46no—
579Q8—61no—
580Q8—66no—
581Q8—243no—
582Q8#91114yescomplemented
583D4—23no—
584D4—21no—
585D4—23no—
586D4—46no—
587D4—64no—
588D4—63no—
589D4—63no—
590D4—83no—
591D4—126no—
592D4—1212no—
593D4—126no—
594D4—243no—
595D4—246no—
596D4D3×#4112yescomplemented
597C2×C4—23no—
598C2×C4—21no—
599C2×C4—31no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—33no—
603C2×C4—33no—
604C2×C4—46no—
605C2×C4—46no—
606C2×C4—64no—
607C2×C4—61no—
608C2×C4—66no—
609C2×C4—63no—
610C2×C4—61no—
611C2×C4—123no—
612C2×C4—126no—
613C2×C4—122no—
614C2×C4—126no—
615C2×C4—126no—
616C2×C4—243no—
617C2×C4C2×D3×D413yes—
618C2×C4C2×D3×D413yes—
619C2×C4C24×D311yescharacteristic
620C8—23no—
621C8—21no—
622C8—63no—
623C8—61no—
624C2×C3—42no—
625C2×C3—83no—
626C2×C3C2×#54811yescharacteristic
627C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
628C2×C3#358511yescharacteristic
629D3—22no—
630D3—83no—
631C22—22no—
632C22—32no—
633C22—41no—
634C22—41no—
635C22—43no—
636C22—43no—
637C22—43no—
638C22—61no—
639C22—61no—
640C22—123no—
641C22—123no—
642C22—123no—
643C22—2412no—
644C22C22×D3×D411yescharacteristic
645C4—23no—
646C4—32no—
647C4—41no—
648C4—43no—
649C4—63no—
650C4—63no—
651C4—62no—
652C4—121no—
653C4—123no—
654C4C2×#91116yes—
655C4C2×D3×#4111yescharacteristic
656C4D3×2−1+411yescharacteristic
657C3#6132111yescharacteristic, complemented, Sylow
658C2—42no—
659C2—62no—
660C2—83no—
661C2—243no—
662C2D3×#54811yescharacteristic
663C2C22×#91111yescharacteristic
664C2#8720711yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3358C2
socleC22×C3C23×D4
hypercenter#3358D3
hypocenterC3#61321
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #3358

Complements

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

NH ≅ G/Ntype
1. D3×#550C2semidirect
2. Q8×#237C2semidirect
3. C3×#3358C2semidirect
4. #83586C2semidirect
5. #83562C2semidirect
6. #81590C2semidirect
7. #81564C2semidirect
8. #80995C2semidirect
9. #80965C2semidirect
10. #80333C2semidirect
11. #80320C2semidirect
12. #79932C2semidirect
13. Q8×ΓL(1,𝔽25)C2semidirect
14. C4×#911C2semidirect
15. #79570C2semidirect
19. C3×D4×Q8C22semidirect
25. #5721C22semidirect
34. Q8×Dic6C22semidirect
37. #5543C22semidirect
39. C3×#486C22semidirect
40. C3×#474C22semidirect
41. C3×#459C22semidirect
42. C3×C8×Q8C22semidirect
43. C3×C4×D8C22semidirect
44. #5152C22semidirect
45. #5150C22semidirect
46. #5088C22semidirect
47. #5087C22semidirect
48. #5075C22semidirect
49. #5070C22semidirect
50. Q8×C3⋊C8C22semidirect
51. C4×#214C22semidirect
52. #5015C22semidirect
53. C4×#213C22semidirect
54. #5008C22semidirect
55. #4891C22semidirect
56. #4889C22semidirect
58. #4848C22semidirect
59. #4842C22semidirect
61. #4831C22semidirect
62. #4785C22semidirect
66. #4764C22semidirect
67. #4764C22semidirect
68. #4763C22semidirect
69. #4762C22semidirect
70. #4757C22semidirect
71. #4694C22semidirect
72. C4×#204C22semidirect
74. #4679C22semidirect
84. #3358D3semidirect
195. D4×Q8C2×D3semidirect
209. #486C2×D3semidirect
215. #474C2×D3semidirect
221. #459C2×D3semidirect
225. C8×Q8C2×D3semidirect
233. C4×D8C2×D3semidirect
262. #237#41semidirect
419. C3×Q8AGL(1,ℤ/8)semidirect
425. C3×D4C2×#41semidirect
460. Dic6C2×#41semidirect
582. Q8#911semidirect
596. D4D3×#41semidirect
657. C3#61321semidirect

Maximal subgroups

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

subgroupindex
D3×#5502
Q8×#2372
C3×#33582
#835862
#835622
#815902
#815642
#809952
#809652
#803332
#803202
#799322
Q8×ΓL(1,𝔽25)2
C4×#9112
#795702
#613213