Finite Group Databasebeta

← #11675075

Subgroups — #11675075

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
1Q8×#239C211yesmaximal, characteristic, complemented
2D3×#550C211yesmaximal, characteristic, complemented
3C3×#3362C211yesmaximal, characteristic, complemented
4#83703C211yesmaximal, characteristic, complemented
5#83580C211yesmaximal, characteristic, complemented
6#81586C213yesmaximal, complemented
7#81564C213yesmaximal, complemented
8#81088C213yesmaximal, complemented
9#81060C213yesmaximal, complemented
10#80384C213yesmaximal, complemented
11#80371C213yesmaximal, complemented
12#79924C211yesmaximal, characteristic, complemented
13Q8×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
14C4×#914C213yesmaximal, complemented
15#79625C213yesmaximal, complemented
16#61321—31nomaximal, Sylow
17C2×D3×#41—21no—
18C3×Q82C2211yescharacteristic, complemented
19C3×#550C2211yescharacteristic
20#5834—21no—
21C2×#914C2213yes—
22#5730—23no—
23#5730C2213yescomplemented
24#5728—23no—
25#5728C2213yes—
26#5718C2213yes—
27D3×#127C2213yes—
28D3×#121—23no—
29#5599—23no—
30#5577C2211yescharacteristic
31Q8×D12—21no—
32Q8×D12C2211yescharacteristic, complemented
33C4×#239C2213yes—
34C4×D3×Q8—21no—
35C4×D3×Q8C2211yescharacteristic
36#5570C2213yescomplemented
37#5553C2213yes—
38C4×D3×D4C2213yes—
39C3×#486C2213yescomplemented
40C3×#476C2213yescomplemented
41C3×#460C2213yescomplemented
42C3×C8×Q8C2211yescharacteristic, complemented
43C3×C4×SD16C2213yescomplemented
44#5164C2213yescomplemented
45#5163C2216yescomplemented
46#5088C2213yescomplemented
47#5084C2213yescomplemented
48#5072C2213yescomplemented
49#5070C2213yescomplemented
50#5027C2213yescomplemented
51C4×#215C2213yescomplemented
52Q8×C3⋊C8C2211yescharacteristic, complemented
53C4×#213C2213yescomplemented
54#5007C2213yescomplemented
55#4871C2216yescomplemented
56#4869C2213yescomplemented
57#4861C2213yes—
58#4846C2213yescomplemented
59#4843C2213yescomplemented
60#4837C2213yes—
61#4831C2213yescomplemented
62#4817C2213yescomplemented
63#4803C2211yescharacteristic
64#4786C2216yescomplemented
65#4785C2213yescomplemented
66#4770C2216yes—
67#4770C2213yes—
68#4766C2216yescomplemented
69#4764C2216yescomplemented
70#4764C2216yescomplemented
71#4694C2213yescomplemented
72C4×D24C2213yescomplemented
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#3362D311yescharacteristic, complemented
83#3358—31no—
84#3358—31no—
85#3138—33no—
86#3077—33no—
87#3038—33no—
88Q8×M16—31no—
89C4×AGL(1,ℤ/8)—33no—
90C2×C3×#41—21no—
91D3×#41—48no—
92C2×#239—21no—
93C2×#239C2311yescharacteristic
94C2×D3×Q8—21no—
95C2×D3×Q8D411yescharacteristic
96C2×#237—23no—
97C2×D3×D4—23no—
98C2×D3×D4—23no—
99C2×D3×D4C2313yes—
100C2×#235—23no—
101C22×C4×D3—23no—
102C2×C3×SD16C2313yes—
103C3×C4⋊Q8—23no—
104C3×C4⋊Q8C2313yes—
105C3×#127C2313yes—
106C3×#121—23no—
107C3×C4×Q8—21no—
108C3×C4×Q8C2313yes—
109C3×C4×Q8C2311yescharacteristic
110C3×C4×D4C2313yes—
111#946—23no—
112#946—26no—
113Q8×Dic3—23no—
114Q8×Dic3D411yescharacteristic
115Q8×Dic3C2313yes—
116C2×#215C2313yes—
117#940—23no—
118#940—23no—
119#940C2316yes—
120#939—26no—
121#938—23no—
122D4×Dic3C2313yes—
123C2×#213C2313yes—
124#930—23no—
125C4×#67—23no—
126#914—212no—
127C2×D24C2313yes—
128C2×ΓL(1,𝔽25)C2311yescharacteristic
129C2×ΓL(1,𝔽25)C2313yes—
130#897—23no—
131#897—23no—
132#897—23no—
133#895—23no—
134#893—23no—
135#893C2313yes—
136#893C2313yes—
137#892—23no—
138#892C2313yes—
139D3×C4⋊C4—23no—
140D3×C4⋊C4D413yes—
141D3×C4⋊C4C2311yescharacteristic
142#890—26no—
143#890C2316yes—
144#887—23no—
145#887D413yes—
146#884—26no—
147#67⋊C4—23no—
148D3×C22⋊C4—23no—
149#874C2313yes—
150C4×D12—23no—
151C4×D12C2313yes—
152C4×D12C2313yes—
153C42×D3—23no—
154C42×D3C2313yes—
155#869—23no—
156#869C2313yes—
157C4×Dic6—23no—
158C4×Dic6C2313yes—
159C3×#106C2313yes—
160C3×C4⋊C8C2313yes—
161C3×#103C2311yescharacteristic
162C3×#102C2313yes—
163C3×#102C2316yes—
164C3×C4×C8C2313yes—
165#835C2311yescharacteristic
166#832C2316yes—
167#832C2313yes—
168#821C2316yes—
169#821C2311yescharacteristic
170#821C2313yes—
171#818C2313yes—
172#815C2313yes—
173#815C2313yes—
174Dic3⋊C8C2316yes—
175#809C2311yescharacteristic
176#809C2313yes—
177#809C2316yes—
178#808C2313yes—
179#807C2313yes—
180C4⋊(C3⋊C8)C2313yes—
181C4×C3⋊C8C2313yes—
182C22×#41—61no—
183C2×AGL(1,ℤ/8)—33no—
184Q82C2×D311yescharacteristic, complemented
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—31no—
195D4×Q8—61no—
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#476C2×D313yescomplemented
214#474—33no—
215#474—33no—
216#460—36no—
217#460—33no—
218#460C2×D313yescomplemented
219#459—33no—
220#459—33no—
221#459—36no—
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×SD16C2×D313yescomplemented
232C4×D8—33no—
233C4×D8—33no—
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×Q8C2×D411yescharacteristic
244C2×C3×Q8C2411yescharacteristic
245C2×C3×D4—23no—
246C2×C3×D4—23no—
247C2×C3×D4C2413yes—
248C22×C3×C4—23no—
249C2×#67—23no—
250C2×#67—26no—
251C2×#67—26no—
252C22×Dic3—23no—
253#239—22no—
254#239—44no—
255#239#4114yescomplemented
256D3×Q8—24no—
257D3×Q8—44no—
258#237—412no—
259D3×D4—26no—
260D3×D4—26no—
261D3×D4—412no—
262D3×D4—86no—
263#235—412no—
264C2×D12—21no—
265C2×D12—23no—
266C2×D12—23no—
267C2×D12C2413yes—
268C2×D12C2411yescharacteristic
269C2×C4×D3—21no—
270C2×C4×D3—21no—
271C2×C4×D3—23no—
272C2×C4×D3—23no—
273C2×C4×D3—23no—
274C2×C4×D3—23no—
275C2×C4×D3—412no—
276C2×C4×D3C2×D413yes—
277C2×C4×D3C2×D413yes—
278C2×C4×D3C2411yescharacteristic
279C2×Dic6—23no—
280C2×Dic6C2×D413yes—
281C3×SD16—26no—
282C2×C3×C8C2413yes—
283C2×C3×C8C2411yescharacteristic
284C3×C4⋊C4—23no—
285C3×C4⋊C4—23no—
286C3×C4⋊C4C2×D413yes—
287C3×C4⋊C4C2411yescharacteristic
288C3×C4⋊C4C2413yes—
289C3×C22⋊C4—23no—
290C3×C42—23no—
291C3×C42C2413yes—
292#217—23no—
293#215—26no—
294#213—26no—
295(C2×D3)⋊C4—23no—
296(C2×D3)⋊C4—23no—
297(C2×D3)⋊C4—23no—
298(C2×D3)⋊C4—23no—
299(C2×D3)⋊C4—23no—
300C4⋊Dic3—23no—
301C4⋊Dic3—23no—
302C4⋊Dic3—26no—
303C4⋊Dic3C2×D413yes—
304C4⋊Dic3C2413yes—
305C4⋊Dic3C2411yescharacteristic
306Dic3⋊C4—26no—
307Dic3⋊C4—21no—
308Dic3⋊C4C2×D416yes—
309C4×Dic3—23no—
310C4×Dic3—23no—
311C4×Dic3C2×D413yes—
312C4×Dic3C2×D413yes—
313C4×Dic3C2413yes—
314C2×C3⋊C8C2411yescharacteristic
315C2×C3⋊C8C2413yes—
316D24—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—31no—
326C22×Q8—61no—
327C22×D4—33no—
328C22×D4—63no—
329C22×D4—63no—
330C23×C4—63no—
331AGL(1,ℤ/8)—612no—
332C2×SD16—33no—
333C2×SD16C22×D313yes—
334C2×D8—33no—
335C2×D8—33no—
336C2×M16—33no—
337C2×M16—31no—
338C4⋊Q8—23no—
339C4⋊Q8—33no—
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—63no—
350C22⋊Q8—63no—
351C22⋊Q8—66no—
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—21no—
361C4×Q8—31no—
362C4×Q8—33no—
363C4×Q8—33no—
364C4×Q8—33no—
365C4×Q8—63no—
366C4×Q8—63no—
367C4×Q8—63no—
368C4×Q8C22×D313yes—
369C4×Q8C22×D311yescharacteristic
370C4×D4—33no—
371C4×D4—33no—
372C4×D4—33no—
373C4×D4—33no—
374C4×D4—33no—
375C4×D4—63no—
376C4×D4—63no—
377C4×D4—63no—
378C4×D4—63no—
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#107—33no—
390#106—33no—
391#106C22×D313yes—
392C4⋊C8—36no—
393C4⋊C8—33no—
394C4⋊C8C22×D313yes—
395#103—31no—
396#103C22×D311yescharacteristic
397#102—33no—
398#102—36no—
399#102—31no—
400#102—33no—
401#102—36no—
402#102—31no—
403#102—33no—
404#102—36no—
405#102C22×D313yes—
406#102C22×D316yes—
407#97—33no—
408#97—33no—
409C4×C8—33no—
410C4×C8C22×D313yes—
411C23×C3—23no—
412C22×D3—21no—
413C22×D3—23no—
414C22×D3—46no—
415C22×D3—46no—
416C22×D3—46no—
417C22×D3C22×D411yescharacteristic
418C3×Q8—21no—
419C3×Q8—83no—
420C3×Q8AGL(1,ℤ/8)14yescomplemented
421C3×Q8C2×#4112yescomplemented
422C3×D4—23no—
423C3×D4—23no—
424C3×D4—46no—
425C3×D4—83no—
426C2×C3×C4—21no—
427C2×C3×C4—23no—
428C2×C3×C4—21no—
429C2×C3×C4—46no—
430C2×C3×C4C22×D413yes—
431C2×C3×C4C22×D413yes—
432C2×C3×C4C2511yescharacteristic
433#67—412no—
434#67—46no—
435#67—86no—
436C2×Dic3—23no—
437C2×Dic3—21no—
438C2×Dic3—46no—
439C2×Dic3C22×D413yes—
440C2×Dic3C22×D413yes—
441C2×Dic3C22×D411yescharacteristic
442D12—24no—
443D12—23no—
444D12—21no—
445D12—23no—
446D12—46no—
447D12—83no—
448D12C2×#4112yescomplemented
449C4×D3—26no—
450C4×D3—24no—
451C4×D3—43no—
452C4×D3—46no—
453C4×D3—42no—
454C4×D3—46no—
455C4×D3—46no—
456C4×D3—83no—
457C4×D3C2×#4112yes—
458C4×D32−1+412yes—
459Dic6—26no—
460Dic6—46no—
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—64no—
475C2×Q8—63no—
476C2×Q8—124no—
477C2×Q8D3×D411yescharacteristic
478C2×Q8C23×D311yescharacteristic
479C2×D4—23no—
480C2×D4—23no—
481C2×D4—31no—
482C2×D4—33no—
483C2×D4—61no—
484C2×D4—66no—
485C2×D4—66no—
486C2×D4—63no—
487C2×D4—66no—
488C2×D4—66no—
489C2×D4—63no—
490C2×D4—63no—
491C2×D4—1212no—
492C2×D4—246no—
493C2×D4C23×D313yes—
494C22×C4—23no—
495C22×C4—31no—
496C22×C4—33no—
497C22×C4—33no—
498C22×C4—61no—
499C22×C4—63no—
500C22×C4—63no—
501C22×C4—61no—
502C22×C4—63no—
503C22×C4—63no—
504C22×C4—63no—
505C22×C4—1212no—
506SD16—26no—
507SD16—66no—
508D8—66no—
509D8—66no—
510M16—66no—
511M16—62no—
512C2×C8—31no—
513C2×C8—33no—
514C2×C8C23×D311yescharacteristic
515C2×C8C23×D313yes—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—33no—
519C4⋊C4—36no—
520C4⋊C4—33no—
521C4⋊C4—31no—
522C4⋊C4—61no—
523C4⋊C4—66no—
524C4⋊C4—63no—
525C4⋊C4—66no—
526C4⋊C4—63no—
527C4⋊C4D3×D413yes—
528C4⋊C4C23×D313yes—
529C4⋊C4C23×D311yescharacteristic
530C22⋊C4—23no—
531C22⋊C4—63no—
532C22⋊C4—63no—
533C22⋊C4—63no—
534C22⋊C4—63no—
535C22⋊C4—63no—
536C22⋊C4—63no—
537C42—23no—
538C42—33no—
539C42—33no—
540C42—33no—
541C42—63no—
542C42—63no—
543C42C23×D313yes—
544C22×C3—43no—
545C22×C3—43no—
546C22×C3—43no—
547C22×C3C23×D411yescharacteristic
548C2×D3—21no—
549C2×D3—21no—
550C2×D3—22no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—43no—
554C2×D3—43no—
555C2×D3—43no—
556C2×D3—812no—
557C2×D3#54812yes—
558C3×C4—23no—
559C3×C4—22no—
560C3×C4—43no—
561C3×C4—41no—
562C3×C4C2×AGL(1,ℤ/8)16yes—
563C3×C4C22×#4111yescharacteristic
564C3×C4C2×2−1+411yescharacteristic
565Dic3—23no—
566Dic3—23no—
567Dic3—43no—
568Dic3—41no—
569Dic3#54812yes—
570C23—23no—
571C23—31no—
572C23—61no—
573C23—63no—
574C23—126no—
575C23—126no—
576C23—126no—
577Q8—21no—
578Q8—66no—
579Q8—83no—
580Q8—126no—
581Q8#91414yescomplemented
582Q8D3×#4112yescomplemented
583D4—23no—
584D4—23no—
585D4—32no—
586D4—46no—
587D4—64no—
588D4—63no—
589D4—61no—
590D4—63no—
591D4—83no—
592D4—126no—
593D4—1212no—
594D4—126no—
595D4—243no—
596D4—246no—
597C2×C4—21no—
598C2×C4—23no—
599C2×C4—21no—
600C2×C4—31no—
601C2×C4—32no—
602C2×C4—32no—
603C2×C4—33no—
604C2×C4—33no—
605C2×C4—46no—
606C2×C4—64no—
607C2×C4—66no—
608C2×C4—63no—
609C2×C4—61no—
610C2×C4—123no—
611C2×C4—126no—
612C2×C4—122no—
613C2×C4—126no—
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—83no—
625C2×C3C2×#54811yescharacteristic
626C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
627C2×C3#358511yescharacteristic
628D3—22no—
629D3—42no—
630D3—83no—
631C22—32no—
632C22—43no—
633C22—43no—
634C22—43no—
635C22—61no—
636C22—61no—
637C22—62no—
638C22—121no—
639C22—121no—
640C22—123no—
641C22—123no—
642C22—123no—
643C22—2412no—
644C22C22×D3×D411yescharacteristic
645C4—23no—
646C4—22no—
647C4—32no—
648C4—41no—
649C4—43no—
650C4—63no—
651C4—63no—
652C4—121no—
653C4—123no—
654C4C2×#91416yes—
655C4C2×D3×#4111yescharacteristic
656C4D3×2−1+411yescharacteristic
657C3#6132111yescharacteristic, complemented, Sylow
658C2—62no—
659C2—83no—
660C2—122no—
661C2—243no—
662C2D3×#54811yescharacteristic
663C2C22×#91411yescharacteristic
664C2#8720411yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3362C2
socleC22×C3C23×D4
hypercenter#3362D3
hypocenterC3#61321
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #3362

Complements

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

NH ≅ G/Ntype
1. Q8×#239C2semidirect
2. D3×#550C2semidirect
3. C3×#3362C2semidirect
4. #83703C2semidirect
5. #83580C2semidirect
6. #81586C2semidirect
7. #81564C2semidirect
8. #81088C2semidirect
9. #81060C2semidirect
10. #80384C2semidirect
11. #80371C2semidirect
12. #79924C2semidirect
13. Q8×ΓL(1,𝔽25)C2semidirect
14. C4×#914C2semidirect
15. #79625C2semidirect
18. C3×Q82C22semidirect
23. #5730C22semidirect
32. Q8×D12C22semidirect
36. #5570C22semidirect
39. C3×#486C22semidirect
40. C3×#476C22semidirect
41. C3×#460C22semidirect
42. C3×C8×Q8C22semidirect
43. C3×C4×SD16C22semidirect
44. #5164C22semidirect
45. #5163C22semidirect
46. #5088C22semidirect
47. #5084C22semidirect
48. #5072C22semidirect
49. #5070C22semidirect
50. #5027C22semidirect
51. C4×#215C22semidirect
52. Q8×C3⋊C8C22semidirect
53. C4×#213C22semidirect
54. #5007C22semidirect
55. #4871C22semidirect
56. #4869C22semidirect
58. #4846C22semidirect
59. #4843C22semidirect
61. #4831C22semidirect
62. #4817C22semidirect
64. #4786C22semidirect
65. #4785C22semidirect
68. #4766C22semidirect
69. #4764C22semidirect
70. #4764C22semidirect
71. #4694C22semidirect
72. C4×D24C22semidirect
74. #4679C22semidirect
82. #3362D3semidirect
184. Q82C2×D3semidirect
209. #486C2×D3semidirect
213. #476C2×D3semidirect
218. #460C2×D3semidirect
225. C8×Q8C2×D3semidirect
231. C4×SD16C2×D3semidirect
255. #239#41semidirect
420. C3×Q8AGL(1,ℤ/8)semidirect
421. C3×Q8C2×#41semidirect
448. D12C2×#41semidirect
581. Q8#914semidirect
582. Q8D3×#41semidirect
657. C3#61321semidirect

Maximal subgroups

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

subgroupindex
Q8×#2392
D3×#5502
C3×#33622
#837032
#835802
#815862
#815642
#810882
#810602
#803842
#803712
#799242
Q8×ΓL(1,𝔽25)2
C4×#9142
#796252
#613213