Finite Group Databasebeta

← #11659425

Subgroups — #11659425

Order 768.

2742 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×#5836C211yesmaximal, characteristic
3#85887C211yesmaximal, characteristic, complemented
4C3×#2897C211yesmaximal, characteristic, complemented
5#83806C211yesmaximal, characteristic, complemented
6Dic3×#131C211yesmaximal, characteristic, complemented
7#82561C211yesmaximal, characteristic, complemented
8#78916C213yesmaximal, complemented
9#78914C213yesmaximal, complemented
10#78869C213yesmaximal, complemented
11#78867C213yesmaximal, complemented
12#78821C213yesmaximal, complemented
13#78818C213yesmaximal, complemented
14#78726C213yesmaximal, complemented
15C2×#4758C213yesmaximal, complemented
16C4×#577—31nomaximal, Sylow
17C2×C3×H(5,𝔽2)C2211yescharacteristic
18C2×#1010C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20C3×#520C2211yescharacteristic
21#5836C4116yescomplemented
22#5827—21no—
23#5827C2213yes—
24#5827C2211yescharacteristic
25Dic3×#41—21no—
26Dic3×#41C2213yes—
27Dic3×#41C2211yescharacteristic
28C2×#951C2213yes—
29C2×#950C2213yes—
30C2×#948C2211yescharacteristic
31#5798—23no—
32#5798C2213yes—
33C2×D4×Dic3—23no—
34C2×D4×Dic3C2213yes—
35C22×#214C2213yes—
36C2×#932C2213yes—
37#5572—21no—
38C4×D3×Q8—21no—
39#5546—23no—
40C4×#237—23no—
41#5522C2213yescomplemented
42#5516C2213yescomplemented
43#5504C2213yescomplemented
44C2×#887C2213yescomplemented
45C3×#418C2213yes—
46C3×#417C2213yes—
47C3×#416C2213yes—
48C2×C3×#102C2213yes—
49#5133C2213yescomplemented
50#5122C2213yescomplemented
51#5120C2213yescomplemented
52Dic3×M16C2213yescomplemented
53#5113C2213yescomplemented
54C2×#816C2213yescomplemented
55#5101C2213yescomplemented
56C2×C8×Dic3C2213yescomplemented
57#4998C2213yescomplemented
58#4995C2213yescomplemented
59#4964C2213yescomplemented
60C2×#808C2213yescomplemented
61#4790C2213yescomplemented
62#4790C2213yescomplemented
63#4789C2216yescomplemented
64#4788C2213yescomplemented
65#4788C2211yescharacteristic, complemented
66#4785C2216yescomplemented
67#4758C2216yescomplemented
68#4758C2216yescomplemented
69#4758C2213yescomplemented
70#4758C2213yescomplemented
71#4758C2216yescomplemented
72#4757C2216yescomplemented
73#4757C2216yescomplemented
74#4756C2213yescomplemented
75#4756C2213yescomplemented
76C2×#577—31no—
77C4×2−1+4—31no—
78C4×H(5,𝔽2)—31no—
79#2952—33no—
80#2949—33no—
81#2947—33no—
82#2946—33no—
83C4×#137—33no—
84C4×AGL(1,ℤ/8)—33no—
85C4×#135—33no—
86C2×C4×SD16—33no—
87#2912—31no—
88#2898—31no—
89#2897D311yescharacteristic, complemented
90C4×#131—31no—
91C3×H(5,𝔽2)—24no—
92C3×H(5,𝔽2)C2×C418yescomplemented
93C2×C3×#41—21no—
94C2×C3×#41C2313yes—
95C2×C3×#41C2311yescharacteristic
96C22×C3×D4—23no—
97C22×C3×D4C2313yes—
98C23×Dic3—23no—
99#1010—24no—
100#1010C2×C418yescomplemented
101C2×D3×Q8—21no—
102C2×#237—23no—
103C2×#235C2313yes—
104C22×Dic6C2313yes—
105C3×#131—24no—
106C2×C3×M16C2313yes—
107C22×C3×C8C2313yes—
108C3×C4×Q8—21no—
109C3×C4×D4—23no—
110C3×#117C2313yes—
111C2×C3×C4⋊C4C2313yes—
112C2×#217—23no—
113C2×#217—26no—
114#951C2×C4124yescomplemented
115#950C2×C4124yescomplemented
116#948C2×C418yescomplemented
117Q8×Dic3—21no—
118Q8×Dic3D411yescharacteristic
119Q8×Dic3D411yescharacteristic
120Q8×Dic3C2313yes—
121Q8×Dic3C2311yescharacteristic
122C2×#216C2313yes—
123C2×#215C2311yescharacteristic
124D4×Dic3—23no—
125D4×Dic3—23no—
126D4×Dic3—23no—
127D4×Dic3—26no—
128D4×Dic3—23no—
129D4×Dic3—23no—
130D4×Dic3—23no—
131D4×Dic3—23no—
132D4×Dic3D413yes—
133D4×Dic3D413yes—
134D4×Dic3C2313yes—
135D4×Dic3C2313yes—
136D4×Dic3C2316yes—
137C2×#214C2×C4124yescomplemented
138C2×#214C2316yes—
139C2×#214C2313yes—
140#932C2×C4124yescomplemented
141C2×#213C2313yes—
142C4×#67—26no—
143#926—23no—
144#926—23no—
145#926C2313yes—
146#926C2313yes—
147C2×C4⋊Dic3—23no—
148C2×C4⋊Dic3C2313yes—
149C2×Dic3⋊C4—23no—
150C2×C4×Dic3—23no—
151C2×C4×Dic3—23no—
152C2×C4×Dic3C2313yes—
153C2×#208C2313yes—
154C22×C3⋊C8C2313yes—
155#893C2313yes—
156#893C2311yescharacteristic
157#892—23no—
158D3×C4⋊C4—23no—
159#887—23no—
160#887—23no—
161#887C2313yes—
162#887C2316yes—
163#887C2313yes—
164#67⋊C4—23no—
165#67⋊C4—23no—
166#877—26no—
167#872—23no—
168C42×D3—23no—
169C4×Dic6—23no—
170C4×Dic6—23no—
171C3×#103C2311yescharacteristic
172C3×#103C2313yes—
173C3×#102C2313yes—
174C3×#102C2313yes—
175C3×#102C2316yes—
176#821C2313yes—
177#821C2311yescharacteristic
178#816C2313yes—
179#816C2316yes—
180#816C2313yes—
181#815C2316yes—
182C8×Dic3C2313yes—
183C8×Dic3C2313yes—
184C8×Dic3C2313yes—
185C8×Dic3C2311yescharacteristic
186#808C2313yes—
187#808C2316yes—
188#808C2311yescharacteristic
189#807C2313yes—
190#807C2313yes—
191C2×2−1+4—31no—
192C2×H(5,𝔽2)C2×D311yescharacteristic
193#577—316no—
194C2×#137—33no—
195C2×AGL(1,ℤ/8)—33no—
196C2×#135—33no—
197C22×SD16—33no—
198C2×#131—31no—
199C2×#131C2×D311yescharacteristic
200#520—31no—
201#520—33no—
202#520—33no—
203#520—63no—
204#520—61no—
205#520C2×D311yescharacteristic
206#519—33no—
207#519—61no—
208#518—33no—
209#518—63no—
210C4×#41—31no—
211C4×#41—33no—
212C4×#41—33no—
213C4×#41—63no—
214C4×#41—61no—
215C2×C4×Q8—33no—
216C2×C4×Q8—61no—
217C2×C4×D4—33no—
218C2×C4×D4—63no—
219#442—36no—
220#441—36no—
221#440—36no—
222#440—36no—
223C4×Q16—33no—
224C4×Q16—33no—
225C4×SD16—31no—
226C4×SD16—33no—
227C4×SD16—33no—
228C4×SD16—36no—
229C4×SD16—36no—
230C4×SD16—36no—
231C4×SD16—33no—
232C4×D8—33no—
233C4×D8—33no—
234#428—33no—
235#428—33no—
236#427—33no—
237C2×#106—33no—
238#419—33no—
239#418C2×D313yes—
240#417—33no—
241#417C2×D313yes—
242#416—33no—
243#416C2×D313yes—
244C2×#103—33no—
245C2×#102C2×D313yes—
246#405—33no—
247#405—33no—
248C4×M16—33no—
249C2×C4×C8—33no—
250C24×C3—23no—
251C3×#41—28no—
252C3×#41—26no—
253C3×#41C22×C4112yes—
254C3×#41C22×C414yes—
255C3×#41#4114yes—
256C2×C3×Q8C2×D411yescharacteristic
257C2×C3×Q8C2411yescharacteristic
258C2×C3×D4—23no—
259C2×C3×D4—224no—
260C2×C3×D4—23no—
261C2×C3×D4—26no—
262C2×C3×D4C22×C4112yes—
263C2×C3×D4C2×D413yes—
264C2×C3×D4C2413yes—
265C22×C3×C4—23no—
266C22×C3×C4—23no—
267C22×C3×C4C2413yes—
268C2×#67—23no—
269C22×Dic3—23no—
270C22×Dic3—23no—
271C22×Dic3—23no—
272C22×Dic3—23no—
273C22×Dic3—26no—
274C22×Dic3—86no—
275C22×Dic3C2×D413yes—
276C22×Dic3C2×D413yes—
277D3×Q8—28no—
278#237—224no—
279#235—26no—
280#235C22×C4112yescomplemented
281C2×D12C2411yescharacteristic
282C2×C4×D3—23no—
283C2×C4×D3—21no—
284C2×C4×D3—23no—
285C2×Dic6—23no—
286C2×Dic6—26no—
287C2×Dic6C22×C4112yescomplemented
288C2×Dic6C2413yes—
289C3×M16—26no—
290C2×C3×C8—26no—
291C2×C3×C8C2411yescharacteristic
292C2×C3×C8C2413yes—
293C3×C4⋊C4—23no—
294C3×C4⋊C4C2411yescharacteristic
295C3×C4⋊C4C2413yes—
296C3×C22⋊C4—23no—
297C3×C42—23no—
298#217—23no—
299#217—23no—
300#217—26no—
301#217—23no—
302#217C2×D416yes—
303#216C22×C4112yescomplemented
304#215C22×C414yescomplemented
305#214C22×C4124yescomplemented
306#214C22×C4112yescomplemented
307#213C22×C4112yescomplemented
308(C2×D3)⋊C4—21no—
309(C2×D3)⋊C4—23no—
310C4⋊Dic3—23no—
311C4⋊Dic3—23no—
312C4⋊Dic3—21no—
313C4⋊Dic3C2×D413yes—
314C4⋊Dic3C2×D413yes—
315C4⋊Dic3C2411yescharacteristic
316C4⋊Dic3C2413yes—
317Dic3⋊C4—23no—
318Dic3⋊C4—26no—
319Dic3⋊C4—23no—
320C4×Dic3—23no—
321C4×Dic3—23no—
322C4×Dic3—26no—
323C4×Dic3—23no—
324C4×Dic3—21no—
325C4×Dic3—23no—
326C4×Dic3C2×D413yes—
327C4×Dic3C2×D413yes—
328C4×Dic3C2411yescharacteristic
329C4×Dic3C2413yes—
330#208C22×C4112yescomplemented
331C2×C3⋊C8C22×C4112yescomplemented
332C2×C3⋊C8C2413yes—
333C2×C3⋊C8C2411yescharacteristic
3342−1+4—38no—
3352−1+4—64no—
336H(5,𝔽2)—24no—
337H(5,𝔽2)C4×D318yescomplemented
338C2×#41—21no—
339C2×#41—33no—
340C2×#41—63no—
341C2×#41C22×D313yes—
342C2×#41C22×D311yescharacteristic
343C22×Q8—33no—
344C22×Q8—61no—
345C22×D4—23no—
346C22×D4C22×D313yes—
347C23×C4—63no—
348#137—324no—
349AGL(1,ℤ/8)—324no—
350#135—324no—
351C2×Q16—33no—
352C2×SD16—31no—
353C2×SD16—324no—
354C2×SD16—33no—
355C2×SD16—36no—
356C2×D8—33no—
357#131—24no—
358#131—38no—
359C2×M16—33no—
360C2×M16C22×D313yes—
361C22×C8—33no—
362C22×C8C22×D313yes—
363C4×Q8—21no—
364C4×Q8—31no—
365C4×Q8—31no—
366C4×Q8—31no—
367C4×Q8—33no—
368C4×Q8—33no—
369C4×Q8—36no—
370C4×Q8—33no—
371C4×Q8—63no—
372C4×Q8—63no—
373C4×Q8—63no—
374C4×Q8—63no—
375C4×Q8—61no—
376C4×D4—23no—
377C4×D4—33no—
378C4×D4—33no—
379C4×D4—33no—
380C4×D4—36no—
381C4×D4—33no—
382C4×D4—31no—
383C4×D4—33no—
384C4×D4—63no—
385C4×D4—63no—
386C4×D4—63no—
387C4×D4—63no—
388C4×D4—66no—
389C4×D4—66no—
390C4×D4—63no—
391C4×D4—63no—
392C4×D4—63no—
393C4×D4—63no—
394C4×D4—63no—
395#117—33no—
396#117—33no—
397#117—63no—
398#117—63no—
399#117—66no—
400#117—63no—
401#117—63no—
402#117C22×D313yes—
403C2×C4⋊C4—33no—
404C2×C4⋊C4—63no—
405C2×C4⋊C4—63no—
406C2×C4⋊C4—63no—
407C2×C4⋊C4C22×D313yes—
408C2×C22⋊C4—66no—
409C2×C22⋊C4—63no—
410C2×C42—33no—
411C2×C42—63no—
412C2×C42—63no—
413C2×C42—63no—
414#107—33no—
415#107—33no—
416#106—33no—
417#106—36no—
418#106—31no—
419#103—33no—
420#103—36no—
421#103—33no—
422#103C22×D311yescharacteristic
423#103C22×D313yes—
424#102—31no—
425#102—33no—
426#102C22×D313yes—
427#102C22×D316yes—
428#102C22×D313yes—
429#97—36no—
430C4×C8—31no—
431C4×C8—33no—
432C4×C8—33no—
433C4×C8—33no—
434C23×C3—23no—
435C23×C3—212no—
436C23×C3—83no—
437C23×C3—83no—
438C23×C3C22×D413yes—
439C22×D3—21no—
440C3×Q8—21no—
441C3×Q8C4×D414yes—
442C3×Q8C23×C412yes—
443C3×D4—212no—
444C3×D4—212no—
445C3×D4—23no—
446C3×D4C4×D4112yes—
447C3×D4C23×C416yes—
448C2×C3×C4—21no—
449C2×C3×C4—212no—
450C2×C3×C4—21no—
451C2×C3×C4—23no—
452C2×C3×C4—83no—
453C2×C3×C4C23×C416yes—
454C2×C3×C4C22×D413yes—
455C2×C3×C4C2×#4116yes—
456C2×C3×C4C2511yescharacteristic
457#67—212no—
458C2×Dic3—212no—
459C2×Dic3—23no—
460C2×Dic3—26no—
461C2×Dic3—21no—
462C2×Dic3—23no—
463C2×Dic3—43no—
464C2×Dic3—46no—
465C2×Dic3—83no—
466C2×Dic3—86no—
467C2×Dic3—83no—
468C2×Dic3C22×D413yes—
469C2×Dic3C22×D413yes—
470C2×Dic3C22×D411yescharacteristic
471C2×Dic3C22×D411yescharacteristic
472D12—21no—
473D12C23×C412yes—
474C4×D3—212no—
475C4×D3—42no—
476C4×D3—83no—
477Dic6—212no—
478Dic6—23no—
479Dic6C23×C416yes—
480C3×C8—21no—
481C3×C8—23no—
482C3⋊C8C23×C412yes—
483C3⋊C8C23×C416yes—
484C24—23no—
485#41—28no—
486#41—26no—
487#41—312no—
488#41—624no—
489#41—66no—
490#41C2×C4×D3112yes—
491#41C2×C4×D314yes—
492#41#23714yes—
493C2×Q8—312no—
494C2×Q8—33no—
495C2×Q8—68no—
496C2×Q8—63no—
497C2×Q8—66no—
498C2×Q8D3×D411yescharacteristic
499C2×Q8C23×D311yescharacteristic
500C2×D4—23no—
501C2×D4—224no—
502C2×D4—23no—
503C2×D4—26no—
504C2×D4—31no—
505C2×D4—63no—
506C2×D4C2×C4×D3112yes—
507C2×D4D3×D413yes—
508C2×D4C23×D313yes—
509C22×C4—23no—
510C22×C4—23no—
511C22×C4—33no—
512C22×C4—33no—
513C22×C4—61no—
514C22×C4—63no—
515C22×C4—63no—
516C22×C4—63no—
517C22×C4—66no—
518C22×C4—63no—
519C22×C4—63no—
520C22×C4—63no—
521C22×C4—246no—
522C22×C4C23×D313yes—
523Q16—312no—
524SD16—34no—
525SD16—312no—
526SD16—324no—
527D8—312no—
528M16—26no—
529M16—312no—
530C2×C8—26no—
531C2×C8—312no—
532C2×C8—31no—
533C2×C8—33no—
534C2×C8C23×D311yescharacteristic
535C2×C8C23×D313yes—
536C4⋊C4—23no—
537C4⋊C4—33no—
538C4⋊C4—33no—
539C4⋊C4—33no—
540C4⋊C4—31no—
541C4⋊C4—63no—
542C4⋊C4—63no—
543C4⋊C4—66no—
544C4⋊C4—61no—
545C4⋊C4—63no—
546C4⋊C4—63no—
547C4⋊C4C23×D313yes—
548C4⋊C4C23×D311yescharacteristic
549C22⋊C4—23no—
550C22⋊C4—36no—
551C22⋊C4—61no—
552C22⋊C4—63no—
553C22⋊C4—63no—
554C22⋊C4—63no—
555C22⋊C4—66no—
556C22⋊C4—63no—
557C42—23no—
558C42—33no—
559C42—33no—
560C42—31no—
561C42—33no—
562C42—61no—
563C42—63no—
564C42—63no—
565C42—63no—
566C42—66no—
567C42—63no—
568C22×C3—23no—
569C22×C3—23no—
570C22×C3—26no—
571C22×C3—43no—
572C22×C3—43no—
573C22×C3—812no—
574C22×C3C2×C4×D416yes—
575C22×C3C23×D411yescharacteristic
576C2×D3—22no—
577C2×D3—41no—
578C2×D3—41no—
579C3×C4—22no—
580C3×C4—43no—
581C3×C4—41no—
582C3×C4C2×C4×D416yes—
583C3×C4C24×C411yescharacteristic
584C3×C4C22×#4111yescharacteristic
585Dic3—23no—
586Dic3—21no—
587Dic3—26no—
588Dic3—23no—
589Dic3—41no—
590Dic3—43no—
591Dic3#57712yescomplemented
592C23—23no—
593C23—212no—
594C23—61no—
595C23—83no—
596C23—83no—
597C23C2×D3×D413yes—
598Q8—21no—
599Q8—36no—
600Q8—63no—
601Q8—612no—
602Q8#67⋊C414yes—
603Q8C22×C4×D312yes—
604D4—212no—
605D4—23no—
606D4—212no—
607D4—32no—
608D4—612no—
609D4—61no—
610D4#67⋊C4112yes—
611D4C22×C4×D316yes—
612C2×C4—21no—
613C2×C4—212no—
614C2×C4—21no—
615C2×C4—23no—
616C2×C4—31no—
617C2×C4—31no—
618C2×C4—33no—
619C2×C4—33no—
620C2×C4—66no—
621C2×C4—63no—
622C2×C4—612no—
623C2×C4—612no—
624C2×C4—61no—
625C2×C4—63no—
626C2×C4—83no—
627C2×C4—122no—
628C2×C4—126no—
629C2×C4—123no—
630C2×C4—243no—
631C2×C4—246no—
632C2×C4—243no—
633C2×C4—243no—
634C2×C4C22×C4×D316yes—
635C2×C4C2×D3×D413yes—
636C2×C4C2×#23716yes—
637C2×C4C24×D311yescharacteristic
638C8—21no—
639C8—23no—
640C8—32no—
641C8—36no—
642C2×C3—26no—
643C2×C3—46no—
644C2×C3C22×C4×D411yescharacteristic
645C2×C3C2×#57711yescharacteristic
646C2×C3#358411yescharacteristic
647D3—42no—
648C22—23no—
649C22—23no—
650C22—26no—
651C22—43no—
652C22—43no—
653C22—62no—
654C22—812no—
655C22—121no—
656C22—121no—
657C22C2×#67⋊C416yes—
658C22C22×D3×D411yescharacteristic
659C4—22no—
660C4—32no—
661C4—43no—
662C4—41no—
663C4—61no—
664C4—63no—
665C4—63no—
666C4—66no—
667C4—121no—
668C4—123no—
669C4C2×#67⋊C416yes—
670C4C23×C4×D311yescharacteristic
671C4C22×#23711yescharacteristic
672C3C4×#57711yescharacteristic, complemented, Sylow
673C2—26no—
674C2—46no—
675C2—122no—
676C2C22×#67⋊C411yescharacteristic
677C2#8720311yescharacteristic
678C2D3×#57711yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC3×#2897C2
socleC22×C3C23×D4
prime radicalC2×#5836C2
cosocleC2×C3×C4C25
hypercenter#2897D3
hypocenterC3C4×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2897

Complements

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

NH ≅ G/Ntype
1. Dic3×H(5,𝔽2)C2semidirect
3. #85887C2semidirect
4. C3×#2897C2semidirect
5. #83806C2semidirect
6. Dic3×#131C2semidirect
7. #82561C2semidirect
8. #78916C2semidirect
9. #78914C2semidirect
10. #78869C2semidirect
11. #78867C2semidirect
12. #78821C2semidirect
13. #78818C2semidirect
14. #78726C2semidirect
15. C2×#4758C2semidirect
21. #5836C4semidirect
41. #5522C22semidirect
42. #5516C22semidirect
43. #5504C22semidirect
44. C2×#887C22semidirect
49. #5133C22semidirect
50. #5122C22semidirect
51. #5120C22semidirect
52. Dic3×M16C22semidirect
53. #5113C22semidirect
54. C2×#816C22semidirect
55. #5101C22semidirect
56. C2×C8×Dic3C22semidirect
57. #4998C22semidirect
58. #4995C22semidirect
59. #4964C22semidirect
60. C2×#808C22semidirect
61. #4790C22semidirect
62. #4790C22semidirect
63. #4789C22semidirect
64. #4788C22semidirect
65. #4788C22semidirect
66. #4785C22semidirect
67. #4758C22semidirect
68. #4758C22semidirect
69. #4758C22semidirect
70. #4758C22semidirect
71. #4758C22semidirect
72. #4757C22semidirect
73. #4757C22semidirect
74. #4756C22semidirect
75. #4756C22semidirect
89. #2897D3semidirect
92. C3×H(5,𝔽2)C2×C4semidirect
100. #1010C2×C4semidirect
114. #951C2×C4semidirect
115. #950C2×C4semidirect
116. #948C2×C4semidirect
137. C2×#214C2×C4semidirect
140. #932C2×C4semidirect
280. #235C22×C4semidirect
287. C2×Dic6C22×C4semidirect
303. #216C22×C4semidirect
304. #215C22×C4semidirect
305. #214C22×C4semidirect
306. #214C22×C4semidirect
307. #213C22×C4semidirect
330. #208C22×C4semidirect
331. C2×C3⋊C8C22×C4semidirect
337. H(5,𝔽2)C4×D3semidirect
591. Dic3#577semidirect
672. C3C4×#577semidirect

Maximal subgroups

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

subgroupindex
Dic3×H(5,𝔽2)2
C2×#58362
#858872
C3×#28972
#838062
Dic3×#1312
#825612
#789162
#789142
#788692
#788672
#788212
#788182
#787262
C2×#47582
C4×#5773