Finite Group Databasebeta

← #10816005

Subgroups — #10816005

Order 768.

1012 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2946C211yesmaximal, characteristic, complemented
2C2×#5201C211yesmaximal, characteristic
3#82570C211yesmaximal, characteristic, complemented
4#82537C211yesmaximal, characteristic, complemented
5#78832C211yesmaximal, characteristic, complemented
6#72091C211yesmaximal, characteristic, complemented
7#71664C211yesmaximal, characteristic, complemented
8#71649C211yesmaximal, characteristic, complemented
9#70680C211yesmaximal, characteristic, complemented
10#70672C211yesmaximal, characteristic, complemented
11#70290C211yesmaximal, characteristic, complemented
12#70218C211yesmaximal, characteristic, complemented
13#70159C211yesmaximal, characteristic, complemented
14#69891C211yesmaximal, characteristic, complemented
15#69888C211yesmaximal, characteristic, complemented
16#22893—31nomaximal, Sylow
17C2×#1010C2211yescharacteristic
18C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
19C3×#520C2211yescharacteristic
20C3×#518C2211yescharacteristic
21C2×#948C2211yescharacteristic
22C2×#936C2211yescharacteristic
23C2×#932C2211yescharacteristic
24C2×#901C2211yescharacteristic
25#5523C2211yescharacteristic
26C3×#442—21no—
27C3×#440—21no—
28C3×C4×SD16—21no—
29C3×C4×D8—21no—
30C3×#428C2211yescharacteristic
31C3×#416C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C3×#405C2211yescharacteristic
34C2×#837C2211yescharacteristic
35#5232C2211yescharacteristic
36C2×#833C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5201C418yescomplemented
39C2×#825C2211yescharacteristic
40C2×#824C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#815C2211yescharacteristic
43#5018—21no—
44C4×#214—21no—
45#5014—21no—
46C4×#213—21no—
47#5004—21no—
48#5004C2211yescharacteristic
49#5004C2211yescharacteristic
50#5002—21no—
51#5002C2211yescharacteristic
52#4997C2211yescharacteristic
53#4995C2211yescharacteristic
54#4975C2211yescharacteristic
55C2×#810—21no—
56C2×#810C2211yescharacteristic
57#4966—21no—
58#4966C2211yescharacteristic
59#4802—21no—
60D3×#103—21no—
61#4772—21no—
62#4771—21no—
63#4558C2211yescharacteristic
64#4554C2211yescharacteristic
65#4552C2211yescharacteristic
66#4532C2211yescharacteristic
67#4532C2211yescharacteristic
68#4530C2211yescharacteristic
69C2×#454—31no—
70#2946—31no—
71#2946D311yescharacteristic, complemented
72#2898—31no—
73#2898—31no—
74#1959—31no—
75#1930—31no—
76#1930—31no—
77#1907—31no—
78#1907—31no—
79#1885—31no—
80#1859—31no—
81#1859—31no—
82#1765—31no—
83#1765—31no—
84C2×C3×#41C2311yescharacteristic
85C22×C3×D4C2311yescharacteristic
86#1010—22no—
87#1010—22no—
88#1010—41no—
89#1010C2×C414yescomplemented
90C2×D3×Q8—21no—
91C2×#237—21no—
92C2×#237—41no—
93C2×#235—21no—
94C2×#235C2311yescharacteristic
95C22×Dic6—21no—
96C22×Dic6C2311yescharacteristic
97C3×AGL(1,ℤ/8)—22no—
98C3×AGL(1,ℤ/8)C2×C414yescomplemented
99C2×C3×SD16—21no—
100C2×C3×D8—21no—
101C2×C3×M16C2311yescharacteristic
102C22×C3×C8C2311yescharacteristic
103C3×C4×Q8—21no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4—21no—
107C3×C4×D4—21no—
108C3×C4×D4—21no—
109C3×C4×D4—41no—
110C3×#117—21no—
111C3×#117C2311yescharacteristic
112C3×#117C2311yescharacteristic
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C22⋊C4—41no—
116C2×#217—41no—
117#948—22no—
118#948—41no—
119#936—41no—
120#936C2×C414yescomplemented
121C2×#214—21no—
122#932—22no—
123#932C2×C414yescomplemented
124C2×#213—21no—
125C2×C4⋊Dic3C2311yescharacteristic
126C2×Dic3⋊C4—41no—
127C2×C4×Dic3C2311yescharacteristic
128C2×#208—21no—
129C2×#208C2311yescharacteristic
130C2×#208C2311yescharacteristic
131C2×#208C2311yescharacteristic
132C22×C3⋊C8—21no—
133C22×C3⋊C8C2311yescharacteristic
134#901—22no—
135#901—41no—
136C2×ΓL(1,𝔽25)—21no—
137C2×D3×C8—21no—
138#892—21no—
139D3×C4⋊C4—21no—
140#67⋊C4—41no—
141C4×D12—21no—
142C4×Dic6—21no—
143C3×#107—21no—
144C3×#106—21no—
145C3×#103D411yescharacteristic
146C3×#103D411yescharacteristic
147C3×#102—21no—
148C3×#102—21no—
149C3×#102D411yescharacteristic
150C3×#102D411yescharacteristic
151C3×#97—21no—
152C3×C4×C8—21no—
153#837C2×C414yescomplemented
154#835—21no—
155#833—41no—
156#832—21no—
157#832—21no—
158#832—21no—
159#825C2×C414yescomplemented
160#824C2×C414yescomplemented
161#816—21no—
162#816—21no—
163#815—41no—
164#810—21no—
165#810—21no—
166#810—21no—
167#810—21no—
168#810—21no—
169#810—21no—
170#810—21no—
171#810D411yescharacteristic
172#810D411yescharacteristic
173#809—21no—
174#809D411yescharacteristic
175#809D411yescharacteristic
176#808—21no—
177#807—21no—
178#803—21no—
179C4×C3⋊C8—21no—
180C2×2−1+4—31no—
181C2×AGL(1,ℤ/8)—31no—
182C2×AGL(1,ℤ/8)C2×D311yescharacteristic
183C2×#131—31no—
184C2×#131—31no—
185C2×#124—31no—
186#520—31no—
187#520C2×D311yescharacteristic
188#518C2×D311yescharacteristic
189#454—38no—
190#442—21no—
191#442—61no—
192#440—21no—
193#440—61no—
194C4×SD16—21no—
195C4×SD16—61no—
196C4×D8—21no—
197C4×D8—61no—
198#428—31no—
199#428C2×D311yescharacteristic
200C2×#104—31no—
201C2×#104—31no—
202#419—31no—
203#419—31no—
204#419—61no—
205#419—61no—
206#417—31no—
207#417—31no—
208#417—61no—
209#417—61no—
210#416—31no—
211#416—61no—
212#416—61no—
213#416C2×D311yescharacteristic
214C2×#103—31no—
215C2×#103—31no—
216C2×#103—61no—
217C2×#103—61no—
218C2×#102—31no—
219C2×#102C2×D311yescharacteristic
220C2×#101—31no—
221C2×#100—31no—
222#405—31no—
223#405C2×D311yescharacteristic
224C2×#97—31no—
225#384—31no—
226#343—31no—
227#343—31no—
228#341—31no—
229#341—31no—
230#339—31no—
231#339—31no—
232C24×C3—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41C22×C412yes—
236C3×#41#4112yes—
237C2×C3×Q8C2×D411yescharacteristic
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4C22×C412yes—
245C2×C3×D4C2×D411yescharacteristic
246C22×C3×C4—21no—
247C22×C3×C4—21no—
248C22×C3×C4—41no—
249C22×C3×C4—41no—
250C22×C3×C4—41no—
251C22×C3×C4C2411yescharacteristic
252C2×#67—21no—
253C2×#67—41no—
254C22×Dic3—21no—
255C22×Dic3—21no—
256C22×Dic3—21no—
257C22×Dic3—41no—
258D3×Q8—24no—
259D3×Q8—42no—
260#237—24no—
261#237—42no—
262#237—44no—
263#237—44no—
264#235—21no—
265#235—21no—
266#235—24no—
267#235—42no—
268#235C22×C412yes—
269#235#4112yes—
270C2×D12C2×D411yescharacteristic
271C2×C4×D3—21no—
272C2×C4×D3—41no—
273C2×C4×D3—41no—
274C2×Dic6—21no—
275C2×Dic6—21no—
276C2×Dic6—21no—
277C2×Dic6—21no—
278C2×Dic6—24no—
279C2×Dic6—21no—
280C2×Dic6—42no—
281C2×Dic6—41no—
282C2×Dic6C22×C412yes—
283C2×Dic6C2×D411yescharacteristic
284C3×SD16—24no—
285C3×D8—24no—
286C3×M16—21no—
287C3×M16C22×C412yes—
288C2×C3×C8—21no—
289C2×C3×C8—21no—
290C2×C3×C8—21no—
291C2×C3×C8C2×D411yescharacteristic
292C2×C3×C8C2×D411yescharacteristic
293C3×C4⋊C4—21no—
294C3×C4⋊C4—21no—
295C3×C4⋊C4—41no—
296C3×C4⋊C4C2×D411yescharacteristic
297C3×C4⋊C4C2×D411yescharacteristic
298C3×C4⋊C4C2×D411yescharacteristic
299C3×C4⋊C4C2×D411yescharacteristic
300C3×C22⋊C4—21no—
301C3×C22⋊C4—21no—
302C3×C22⋊C4—41no—
303C3×C22⋊C4—41no—
304C3×C22⋊C4—41no—
305C3×C42—21no—
306C3×C42—21no—
307C3×C42—41no—
308#217—44no—
309#214—24no—
310#213—24no—
311(C2×D3)⋊C4—41no—
312C4⋊Dic3—21no—
313C4⋊Dic3—21no—
314Dic3⋊C4—41no—
315Dic3⋊C4—82no—
316C4×Dic3—21no—
317C4×Dic3—41no—
318C4×Dic3C22×C412yes—
319#208—21no—
320#208—21no—
321#208—21no—
322#208—21no—
323#208—21no—
324#208—42no—
325#208C22×C412yes—
326C2×C3⋊C8—21no—
327C2×C3⋊C8—21no—
328C2×C3⋊C8—21no—
329C2×C3⋊C8—21no—
330C2×C3⋊C8—21no—
331C2×C3⋊C8—42no—
332C2×C3⋊C8C2×D411yescharacteristic
333C2×C3⋊C8C2×D411yescharacteristic
334ΓL(1,𝔽25)—42no—
335D3×C8—42no—
3362−1+4—34no—
3372−1+4—62no—
3382−1+4—62no—
3392−1+4—121no—
340C2×#41—31no—
341C2×#41—61no—
342C2×#41—61no—
343C2×#41—121no—
344C2×#41C22×D311yescharacteristic
345C22×Q8—31no—
346C22×Q8—61no—
347C22×Q8—61no—
348C22×D4C22×D311yescharacteristic
349AGL(1,ℤ/8)—22no—
350AGL(1,ℤ/8)—34no—
351AGL(1,ℤ/8)—62no—
352AGL(1,ℤ/8)C4×D314yescomplemented
353C2×SD16—21no—
354C2×SD16—61no—
355C2×D8—21no—
356C2×D8—61no—
357#131—62no—
358#131—62no—
359#131—121no—
360#131—121no—
361C2×M16—31no—
362C2×M16—31no—
363C2×M16—31no—
364C2×M16—61no—
365C2×M16—61no—
366C2×M16C22×D311yescharacteristic
367C22×C8—31no—
368C22×C8—61no—
369C22×C8—61no—
370C22×C8C22×D311yescharacteristic
371#124—34no—
372#124—121no—
373C4×Q8—21no—
374C4×Q8—61no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—41no—
381C4×D4—61no—
382C4×D4—121no—
383#117—21no—
384#117—61no—
385#117C22×D311yescharacteristic
386#117C22×D311yescharacteristic
387C2×C4⋊C4—21no—
388C2×C4⋊C4—31no—
389C2×C4⋊C4—61no—
390C2×C4⋊C4—121no—
391C2×C4⋊C4C22×D311yescharacteristic
392C2×C22⋊C4—41no—
393C2×C22⋊C4—121no—
394C2×C42—31no—
395#107—21no—
396#107—61no—
397#106—21no—
398#106—61no—
399#104—34no—
400#104—34no—
401#103—31no—
402#103—31no—
403#103—61no—
404#103—61no—
405#103—61no—
406#103—61no—
407#103—61no—
408#103—61no—
409#103—61no—
410#103—61no—
411#103—61no—
412#103—61no—
413#103#6711yescharacteristic
414#103#6711yescharacteristic
415#102—21no—
416#102—21no—
417#102—31no—
418#102—31no—
419#102—61no—
420#102—61no—
421#102—61no—
422#102—61no—
423#102#6711yescharacteristic
424#102#6711yescharacteristic
425#101—34no—
426#100—121no—
427#97—21no—
428#97—61no—
429#97—121no—
430C4×C8—21no—
431C4×C8—61no—
432C23×C3—21no—
433C23×C3—22no—
434C23×C3—41no—
435C23×C3—41no—
436C23×C3—81no—
437C23×C3—81no—
438C23×C3H(5,𝔽2)11yescharacteristic
439C22×D3—21no—
440C3×Q8—21no—
441C3×Q8C4×D412yes—
442C3×D4—22no—
443C3×D4—22no—
444C3×D4—22no—
445C3×D4—22no—
446C3×D4—21no—
447C3×D4—41no—
448C3×D4—41no—
449C3×D4C4×D412yes—
450C2×C3×C4—21no—
451C2×C3×C4—22no—
452C2×C3×C4—22no—
453C2×C3×C4—21no—
454C2×C3×C4—21no—
455C2×C3×C4—21no—
456C2×C3×C4—21no—
457C2×C3×C4—41no—
458C2×C3×C4—81no—
459C2×C3×C4—161no—
460C2×C3×C4—161no—
461C2×C3×C4—161no—
462C2×C3×C4C23×C411yescharacteristic
463C2×C3×C4C22×D411yescharacteristic
464C2×C3×C4C22×D411yescharacteristic
465C2×C3×C4C2×#4111yescharacteristic
466C2×C3×C4C2×#4111yescharacteristic
467C2×C3×C4H(5,𝔽2)11yescharacteristic
468#67—22no—
469#67—22no—
470#67—44no—
471C2×Dic3—21no—
472C2×Dic3—22no—
473C2×Dic3—22no—
474C2×Dic3—44no—
475C2×Dic3—41no—
476C2×Dic3—41no—
477C2×Dic3—41no—
478C2×Dic3—41no—
479C2×Dic3—41no—
480C2×Dic3—81no—
481D12—21no—
482D12C4×D412yes—
483C4×D3—22no—
484C4×D3—22no—
485C4×D3—44no—
486C4×D3—161no—
487Dic6—21no—
488Dic6—22no—
489Dic6—22no—
490Dic6—44no—
491Dic6—41no—
492Dic6—41no—
493Dic6C4×D412yes—
494C3×C8—22no—
495C3×C8—21no—
496C3×C8—21no—
497C3⋊C8—22no—
498C3⋊C8—21no—
499C3⋊C8—21no—
500C3⋊C8—41no—
501C3⋊C8—41no—
502C24—21no—
503#41—21no—
504#41—21no—
505#41—32no—
506#41—32no—
507#41—64no—
508#41—61no—
509#41—61no—
510#41—64no—
511#41—122no—
512#41—124no—
513#41—122no—
514#41—124no—
515#41C2×C4×D312yes—
516#41#23512yes—
517C2×Q8—32no—
518C2×Q8—31no—
519C2×Q8—64no—
520C2×Q8—61no—
521C2×Q8—64no—
522C2×Q8—61no—
523C2×Q8—61no—
524C2×Q8—61no—
525C2×Q8—61no—
526C2×Q8—121no—
527C2×Q8—122no—
528C2×Q8—122no—
529C2×Q8C2×#6711yescharacteristic
530C2×D4—21no—
531C2×D4—21no—
532C2×D4—21no—
533C2×D4—21no—
534C2×D4—21no—
535C2×D4—21no—
536C2×D4—31no—
537C2×D4—61no—
538C2×D4—121no—
539C2×D4C2×C4×D312yes—
540C2×D4C2×#6711yescharacteristic
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—41no—
544C22×C4—41no—
545C22×C4—41no—
546C22×C4—61no—
547C22×C4—61no—
548C22×C4—61no—
549C22×C4—61no—
550C22×C4—121no—
551C22×C4—121no—
552C22×C4—121no—
553C22×C4C23×D311yescharacteristic
554SD16—24no—
555SD16—64no—
556D8—24no—
557D8—64no—
558M16—21no—
559M16—32no—
560M16—61no—
561M16—61no—
562M16—61no—
563M16—61no—
564M16—61no—
565M16—122no—
566M16—122no—
567M16C2×C4×D312yes—
568C2×C8—21no—
569C2×C8—21no—
570C2×C8—21no—
571C2×C8—31no—
572C2×C8—31no—
573C2×C8—61no—
574C2×C8—61no—
575C2×C8—61no—
576C2×C8—61no—
577C2×C8—61no—
578C2×C8—122no—
579C2×C8—122no—
580C2×C8C2×#6711yescharacteristic
581C2×C8C2×#6711yescharacteristic
582C4⋊C4—21no—
583C4⋊C4—21no—
584C4⋊C4—41no—
585C4⋊C4—61no—
586C4⋊C4—61no—
587C4⋊C4—121no—
588C4⋊C4—242no—
589C4⋊C4D3×D411yescharacteristic
590C4⋊C4D3×D411yescharacteristic
591C4⋊C4C2×#6711yescharacteristic
592C4⋊C4C2×#6711yescharacteristic
593C22⋊C4—21no—
594C22⋊C4—21no—
595C22⋊C4—41no—
596C22⋊C4—41no—
597C22⋊C4—41no—
598C22⋊C4—121no—
599C22⋊C4—124no—
600C42—21no—
601C42—21no—
602C42—32no—
603C42—41no—
604C42—61no—
605C42—121no—
606C22×C3—21no—
607C22×C3—21no—
608C22×C3—22no—
609C22×C3—41no—
610C22×C3—41no—
611C22×C3—42no—
612C22×C3—82no—
613C22×C3—81no—
614C22×C3—81no—
615C22×C3—161no—
616C22×C3#51811yescharacteristic
617C22×C3D4211yescharacteristic
618C22×C3#55111yescharacteristic
619C2×D3—22no—
620C2×D3—41no—
621C2×D3—41no—
622C3×C4—22no—
623C3×C4—41no—
624C3×C4—41no—
625C3×C4—41no—
626C3×C4—41no—
627C3×C4—81no—
628C3×C4C2×C4×D411yescharacteristic
629C3×C4C2×C4×D411yescharacteristic
630C3×C4#54611yescharacteristic
631C3×C4#54611yescharacteristic
632Dic3—22no—
633Dic3—42no—
634Dic3—42no—
635Dic3—81no—
636C23—21no—
637C23—22no—
638C23—41no—
639C23—41no—
640C23—61no—
641C23—81no—
642C23—81no—
643C23#100411yescharacteristic
644Q8—21no—
645Q8—32no—
646Q8—61no—
647Q8—62no—
648Q8—62no—
649Q8—124no—
650Q8—121no—
651Q8—121no—
652Q8C4×#6712yes—
653D4—22no—
654D4—22no—
655D4—22no—
656D4—22no—
657D4—21no—
658D4—32no—
659D4—41no—
660D4—41no—
661D4—62no—
662D4—62no—
663D4—61no—
664D4—124no—
665D4C4×#6712yes—
666C2×C4—21no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—41no—
674C2×C4—61no—
675C2×C4—62no—
676C2×C4—62no—
677C2×C4—62no—
678C2×C4—62no—
679C2×C4—81no—
680C2×C4—124no—
681C2×C4—124no—
682C2×C4—121no—
683C2×C4—121no—
684C2×C4—121no—
685C2×C4—121no—
686C2×C4—121no—
687C2×C4—161no—
688C2×C4—161no—
689C2×C4—161no—
690C2×C4—241no—
691C2×C4—481no—
692C2×C4C22×C4×D311yescharacteristic
693C2×C4C2×#23511yescharacteristic
694C2×C4C2×D3×D411yescharacteristic
695C2×C4D3×#4111yescharacteristic
696C2×C4#100911yescharacteristic
697C2×C4C22×#6711yescharacteristic
698C8—22no—
699C8—21no—
700C8—21no—
701C8—62no—
702C8—61no—
703C8—61no—
704C8—121no—
705C8—121no—
706C2×C3—21no—
707C2×C3—21no—
708C2×C3—42no—
709C2×C3—82no—
710C2×C3#235811yescharacteristic
711C2×C3#328811yescharacteristic
712C2×C3#328911yescharacteristic
713D3—42no—
714C22—21no—
715C22—21no—
716C22—22no—
717C22—41no—
718C22—41no—
719C22—42no—
720C22—62no—
721C22—82no—
722C22—81no—
723C22—81no—
724C22—121no—
725C22—121no—
726C22—161no—
727C22#554511yescharacteristic
728C22#555611yescharacteristic
729C22D4×#6711yescharacteristic
730C4—22no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—62no—
736C4—81no—
737C4—122no—
738C4—122no—
739C4—241no—
740C4C4×D3×D411yescharacteristic
741C4#555011yescharacteristic
742C4C2×C4×#6711yescharacteristic
743C4#582911yescharacteristic
744C3#2289311yescharacteristic, complemented, Sylow
745C2—21no—
746C2—21no—
747C2—42no—
748C2—82no—
749C2—122no—
750C2#8239811yescharacteristic
751C2#8400911yescharacteristic
752C2#8401311yescharacteristic
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2946C2
socleC22×C3D42
prime radicalC2×#5201C2
cosocleC22×C3×C4C24
hypercenter#2946D3
hypocenterC3#22893
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2946

Complements

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

NH ≅ G/Ntype
1. C3×#2946C2semidirect
3. #82570C2semidirect
4. #82537C2semidirect
5. #78832C2semidirect
6. #72091C2semidirect
7. #71664C2semidirect
8. #71649C2semidirect
9. #70680C2semidirect
10. #70672C2semidirect
11. #70290C2semidirect
12. #70218C2semidirect
13. #70159C2semidirect
14. #69891C2semidirect
15. #69888C2semidirect
38. #5201C4semidirect
71. #2946D3semidirect
89. #1010C2×C4semidirect
98. C3×AGL(1,ℤ/8)C2×C4semidirect
120. #936C2×C4semidirect
123. #932C2×C4semidirect
153. #837C2×C4semidirect
159. #825C2×C4semidirect
160. #824C2×C4semidirect
352. AGL(1,ℤ/8)C4×D3semidirect
744. C3#22893semidirect

Maximal subgroups

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

subgroupindex
C3×#29462
C2×#52012
#825702
#825372
#788322
#720912
#716642
#716492
#706802
#706722
#702902
#702182
#701592
#698912
#698882
#228933