Finite Group Databasebeta

← #10818107

Subgroups — #10818107

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5199C211yesmaximal, characteristic
2#83930C211yesmaximal, characteristic, complemented
3#82564C211yesmaximal, characteristic, complemented
4#82561C211yesmaximal, characteristic, complemented
5#78915C211yesmaximal, characteristic, complemented
6C3×#1883C211yesmaximal, characteristic, complemented
7#70370C211yesmaximal, characteristic, complemented
8#70366C211yesmaximal, characteristic, complemented
9#70318C211yesmaximal, characteristic, complemented
10#69648C211yesmaximal, characteristic, complemented
11#69642C211yesmaximal, characteristic, complemented
12#68984C211yesmaximal, characteristic, complemented
13#68983C211yesmaximal, characteristic, complemented
14#68758C211yesmaximal, characteristic, complemented
15#68742C211yesmaximal, characteristic, complemented
16#23048—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18C3×#520C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#949C2211yescharacteristic
21C2×#948C2211yescharacteristic
22#5798C2211yescharacteristic
23C22×#213C2211yescharacteristic
24#5521C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#418C2211yescharacteristic
27C3×#390C2211yescharacteristic
28#5233C2211yescharacteristic
29C2×#833C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5199C418yescomplemented
32#5164—21no—
33Dic3×SD16—21no—
34#5152—21no—
35Dic3×D8—21no—
36#5131C2211yescharacteristic
37C2×#823C2211yescharacteristic
38C2×#822C2211yescharacteristic
39#5122C2211yescharacteristic
40#5122C2211yescharacteristic
41#5121C2211yescharacteristic
42#5026—21no—
43C4×#215—21no—
44#5014—21no—
45C4×#213—21no—
46#4998—21no—
47#4998C2211yescharacteristic
48#4997C2211yescharacteristic
49#4995—21no—
50#4995C2211yescharacteristic
51C2×#809C2211yescharacteristic
52#4964—21no—
53#4964C2211yescharacteristic
54#4964C2211yescharacteristic
55C2×#808—21no—
56C2×#808C2211yescharacteristic
57#4939C2211yescharacteristic
58C2×#803C2211yescharacteristic
59#4785—21no—
60#4785—21no—
61#4756—21no—
62#4756—21no—
63C3×#343C2211yescharacteristic
64C3×#339C2211yescharacteristic
65#4558C2211yescharacteristic
66#4554C2211yescharacteristic
67#4523C2211yescharacteristic
68#4523C2211yescharacteristic
69C2×#472—31no—
70#2948—31no—
71#2946—31no—
72#2946—31no—
73#2912—31no—
74#1972—31no—
75#1951—31no—
76#1931—31no—
77#1931—31no—
78#1883—31no—
79#1883D311yescharacteristic, complemented
80#1856—31no—
81#1856—31no—
82#1773—31no—
83#1765—31no—
84C2×C3×#41C2311yescharacteristic
85C22×C3×D4C2311yescharacteristic
86C22×D12C2311yescharacteristic
87C3×AGL(1,ℤ/8)—22no—
88C3×AGL(1,ℤ/8)C2×C414yescomplemented
89C2×C3×SD16—21no—
90C2×C3×D8—21no—
91C2×C3×M16C2311yescharacteristic
92C2×C3×M16C2311yescharacteristic
93C3×C4×Q8—21no—
94C3×C4×D4—21no—
95C3×C4×D4—41no—
96C3×#117—21no—
97C3×#117C2311yescharacteristic
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—41no—
101C2×C3×C42C2311yescharacteristic
102C2×#217—41no—
103#949—22no—
104#949C2×C414yescomplemented
105#948—22no—
106#948C2×C414yescomplemented
107Q8×Dic3—21no—
108C2×#215—21no—
109D4×Dic3—21no—
110D4×Dic3—21no—
111D4×Dic3—21no—
112D4×Dic3—21no—
113D4×Dic3—21no—
114D4×Dic3—41no—
115C2×#213—21no—
116C2×#213—21no—
117C2×#213—21no—
118C2×#213—22no—
119C2×#213C2×C414yescomplemented
120C2×(C2×D3)⋊C4—41no—
121#926—21no—
122#926C2311yescharacteristic
123#926C2311yescharacteristic
124C2×C4⋊Dic3—21no—
125C2×C4⋊Dic3C2311yescharacteristic
126C2×#208—21no—
127C2×#208C2311yescharacteristic
128C2×#208C2311yescharacteristic
129C22×C3⋊C8—21no—
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132#893—21no—
133#893—21no—
134D3×C22⋊C4—41no—
135C4×D12—21no—
136C4×D12—21no—
137C3×#104—22no—
138C3×#102—21no—
139C3×#102—21no—
140#835D411yescharacteristic
141#835D411yescharacteristic
142#833C2×C414yescomplemented
143#832—21no—
144#832—21no—
145#832D411yescharacteristic
146#832D411yescharacteristic
147#823C2×C414yescomplemented
148#822C2×C414yescomplemented
149#821—21no—
150#821—21no—
151#818—21no—
152#817—21no—
153#815—21no—
154#815—21no—
155C8×Dic3—21no—
156C8×Dic3—21no—
157#809—21no—
158#809—21no—
159#808—21no—
160#808—21no—
161#808—21no—
162#808—21no—
163#808—21no—
164#808D411yescharacteristic
165#808D411yescharacteristic
166#807—21no—
167#807—21no—
168#807—21no—
169#807D411yescharacteristic
170#807D411yescharacteristic
171#803—21no—
172#803—22no—
173C4×C3⋊C8—21no—
174C2×AGL(1,ℤ/8)—31no—
175C2×AGL(1,ℤ/8)C2×D311yescharacteristic
176C22×D8—31no—
177C2×#131—31no—
178#520—31no—
179#520C2×D311yescharacteristic
180#518—31no—
181#518—31no—
182#472—38no—
183#442—61no—
184#442—61no—
185#442—61no—
186#442—61no—
187#440—61no—
188#440—61no—
189C4×SD16—61no—
190C4×SD16—61no—
191C4×D8—61no—
192C4×D8—61no—
193C4×D8—61no—
194C4×D8—61no—
195C2×#108—31no—
196#428—31no—
197#428—31no—
198#428—31no—
199#428—31no—
200#428—61no—
201#428—61no—
202#427—31no—
203#427—61no—
204C2×#106—31no—
205C2×#106—61no—
206C2×#104C2×D311yescharacteristic
207#418—31no—
208#418C2×D311yescharacteristic
209#416—31no—
210C2×#102—31no—
211C2×#102—31no—
212C2×#100—31no—
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405—31no—
217C2×#97—31no—
218#390—31no—
219#390C2×D311yescharacteristic
220#343—31no—
221#343C2×D311yescharacteristic
222#341—31no—
223#339—31no—
224#339—31no—
225#339C2×D311yescharacteristic
226C24×C3—21no—
227C23×D3—21no—
228C3×#41—21no—
229C3×#41—21no—
230C3×#41C22×C412yes—
231C3×#41#4112yes—
232C2×C3×Q8C2×D411yescharacteristic
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4—21no—
239C2×C3×D4C22×C412yes—
240C2×C3×D4C2×D411yescharacteristic
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4—21no—
244C22×C3×C4—41no—
245C22×C3×C4C2411yescharacteristic
246C22×Dic3—21no—
247C22×Dic3—41no—
248C22×Dic3—41no—
249C22×Dic3—41no—
250C2×D12—21no—
251C2×D12—21no—
252C2×D12—21no—
253C2×D12—21no—
254C2×D12—21no—
255C2×D12C22×C412yes—
256C2×C4×D3—41no—
257C2×C4×D3—41no—
258C3×SD16—24no—
259C3×D8—24no—
260C3×M16—21no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16C22×C412yes—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—41no—
269C3×C4⋊C4C2×D411yescharacteristic
270C3×C4⋊C4C2×D411yescharacteristic
271C3×C22⋊C4—21no—
272C3×C22⋊C4—41no—
273C3×C22⋊C4—82no—
274C3×C42—21no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42—41no—
278#217—21no—
279#217—21no—
280#217—41no—
281#217—41no—
282#217—41no—
283#215—24no—
284#213—24no—
285#213—24no—
286#213—24no—
287(C2×D3)⋊C4—41no—
288(C2×D3)⋊C4—41no—
289(C2×D3)⋊C4—82no—
290C4⋊Dic3—21no—
291C4⋊Dic3—21no—
292C4⋊Dic3—41no—
293C4⋊Dic3C2×D411yescharacteristic
294C4⋊Dic3C2×D411yescharacteristic
295C4⋊Dic3C2×D411yescharacteristic
296C4⋊Dic3C2×D411yescharacteristic
297C4×Dic3—21no—
298C4×Dic3—21no—
299C4×Dic3—41no—
300#208—21no—
301#208—21no—
302#208—24no—
303#208C22×C412yes—
304#208C22×C412yes—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—21no—
308C2×C3⋊C8—21no—
309C2×C3⋊C8—24no—
310C2×C3⋊C8C22×C412yes—
311C2×C3⋊C8C2×D411yescharacteristic
312C2×C3⋊C8C2×D411yescharacteristic
313C2×C3⋊C8C2×D411yescharacteristic
314C2×C3⋊C8C2×D411yescharacteristic
315C2×C3⋊C8#4112yes—
316C2×#41C22×D311yescharacteristic
317C22×D4—31no—
318C22×D4C22×D311yescharacteristic
319AGL(1,ℤ/8)—22no—
320AGL(1,ℤ/8)—34no—
321AGL(1,ℤ/8)—62no—
322AGL(1,ℤ/8)C4×D314yescomplemented
323C2×SD16—21no—
324C2×SD16—61no—
325C2×D8—21no—
326C2×D8—34no—
327C2×D8—62no—
328C2×D8—61no—
329C2×D8—61no—
330C2×D8—61no—
331#131—34no—
332#131—62no—
333C2×M16—31no—
334C2×M16—31no—
335C2×M16—61no—
336C2×M16C22×D311yescharacteristic
337C2×M16C22×D311yescharacteristic
338C22×C8—31no—
339C22×C8—31no—
340C22×C8—61no—
341C4×Q8—21no—
342C4×Q8—61no—
343C4×D4—21no—
344C4×D4—41no—
345C4×D4—61no—
346C4×D4—61no—
347C4×D4—61no—
348C4×D4—61no—
349C4×D4—61no—
350C4×D4—61no—
351C4×D4—61no—
352C4×D4—61no—
353C4×D4—61no—
354C4×D4—121no—
355#117—21no—
356#117—31no—
357#117—31no—
358#117—61no—
359#117C22×D311yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—41no—
365C2×C22⋊C4—121no—
366C2×C22⋊C4—121no—
367C2×C22⋊C4—121no—
368C2×C42C22×D311yescharacteristic
369#108—34no—
370#107—31no—
371#107—31no—
372#107—61no—
373#107—61no—
374#107—61no—
375#107—61no—
376#106—31no—
377#106—31no—
378#106—61no—
379#106—61no—
380#106—61no—
381#106—61no—
382#106—61no—
383#106—61no—
384#104—22no—
385#103—31no—
386#103—31no—
387#102—21no—
388#102—21no—
389#102—31no—
390#102—31no—
391#102—61no—
392#102—61no—
393#102—61no—
394#102—61no—
395#102—61no—
396#102—61no—
397#100—34no—
398#100—34no—
399#97—62no—
400#97—61no—
401#97—61no—
402#97—61no—
403C4×C8—61no—
404C4×C8—61no—
405C4×C8—61no—
406C23×C3—21no—
407C23×C3—22no—
408C23×C3—41no—
409C23×C3—41no—
410C23×C3—81no—
411C23×C3—81no—
412C23×C3H(5,𝔽2)11yescharacteristic
413C22×D3—22no—
414C22×D3—41no—
415C22×D3—41no—
416C22×D3—81no—
417C22×D3—81no—
418C3×Q8—21no—
419C3×Q8C4×D412yes—
420C3×D4—22no—
421C3×D4—22no—
422C3×D4—22no—
423C3×D4—22no—
424C3×D4—21no—
425C3×D4—41no—
426C3×D4—41no—
427C3×D4C4×D412yes—
428C2×C3×C4—21no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—41no—
434C2×C3×C4—41no—
435C2×C3×C4—41no—
436C2×C3×C4—41no—
437C2×C3×C4—81no—
438C2×C3×C4—161no—
439C2×C3×C4C23×C411yescharacteristic
440C2×C3×C4C22×D411yescharacteristic
441C2×C3×C4C22×D411yescharacteristic
442C2×C3×C4C2×#4111yescharacteristic
443C2×C3×C4C2×#4111yescharacteristic
444C2×C3×C42−1+411yescharacteristic
445C2×Dic3—21no—
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—21no—
449C2×Dic3—41no—
450C2×Dic3—81no—
451C2×Dic3—161no—
452C2×Dic3—161no—
453C2×Dic3—161no—
454D12—22no—
455D12—22no—
456D12—41no—
457D12—41no—
458C4×D3—161no—
459C4×D3—161no—
460C3×C8—22no—
461C3×C8—41no—
462C3⋊C8—22no—
463C3⋊C8—22no—
464C3⋊C8—21no—
465C3⋊C8—21no—
466C3⋊C8C4×D412yes—
467C3⋊C8C4×D412yes—
468C24—21no—
469C24—61no—
470#41—21no—
471#41—21no—
472#41C2×C4×D312yes—
473#41#23712yes—
474C2×Q8D3×D411yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—32no—
482C2×D4—61no—
483C2×D4—61no—
484C2×D4—61no—
485C2×D4—61no—
486C2×D4—61no—
487C2×D4C2×C4×D312yes—
488C2×D4D3×D411yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—41no—
493C22×C4—61no—
494C22×C4—121no—
495C22×C4—121no—
496C22×C4—121no—
497C22×C4—121no—
498C22×C4—121no—
499C22×C4C23×D311yescharacteristic
500SD16—24no—
501SD16—64no—
502D8—24no—
503D8—64no—
504D8—64no—
505D8—64no—
506M16—21no—
507M16—21no—
508M16—21no—
509M16—32no—
510M16—32no—
511M16—64no—
512M16—61no—
513M16—61no—
514M16C2×C4×D312yes—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—32no—
518C2×C8—31no—
519C2×C8—31no—
520C2×C8—32no—
521C2×C8—31no—
522C2×C8—31no—
523C2×C8—64no—
524C2×C8—61no—
525C2×C8—61no—
526C2×C8—61no—
527C2×C8—61no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—31no—
531C4⋊C4—31no—
532C4⋊C4—31no—
533C4⋊C4—31no—
534C4⋊C4—41no—
535C4⋊C4—61no—
536C4⋊C4—61no—
537C4⋊C4—121no—
538C4⋊C4D3×D411yescharacteristic
539C4⋊C4D3×D411yescharacteristic
540C22⋊C4—21no—
541C22⋊C4—41no—
542C22⋊C4—61no—
543C22⋊C4—61no—
544C22⋊C4—82no—
545C22⋊C4—121no—
546C22⋊C4—121no—
547C22⋊C4—121no—
548C22⋊C4—121no—
549C22⋊C4—121no—
550C22⋊C4—242no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—41no—
555C42—61no—
556C42—61no—
557C42—121no—
558C22×C3—21no—
559C22×C3—21no—
560C22×C3—22no—
561C22×C3—41no—
562C22×C3—41no—
563C22×C3—42no—
564C22×C3—82no—
565C22×C3—81no—
566C22×C3—81no—
567C22×C3—161no—
568C22×C3#52011yescharacteristic
569C22×C3D4211yescharacteristic
570C22×C3#55311yescharacteristic
571C2×D3—42no—
572C2×D3—81no—
573C2×D3—81no—
574C2×D3—82no—
575C2×D3—161no—
576C3×C4—22no—
577C3×C4—41no—
578C3×C4—41no—
579C3×C4—81no—
580C3×C4—81no—
581C3×C4C2×C4×D411yescharacteristic
582C3×C4C2×C4×D411yescharacteristic
583C3×C4#54711yescharacteristic
584C3×C4#54811yescharacteristic
585Dic3—41no—
586Dic3—41no—
587Dic3—41no—
588Dic3—41no—
589Dic3—81no—
590C23—21no—
591C23—22no—
592C23—41no—
593C23—41no—
594C23—62no—
595C23—81no—
596C23—81no—
597C23—121no—
598C23—121no—
599C23—241no—
600C23—241no—
601C23#100411yescharacteristic
602Q8—21no—
603Q8#67⋊C412yes—
604D4—22no—
605D4—22no—
606D4—21no—
607D4—22no—
608D4—22no—
609D4—41no—
610D4—41no—
611D4—62no—
612D4—62no—
613D4—121no—
614D4—121no—
615D4#67⋊C412yes—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—81no—
630C2×C4—121no—
631C2×C4—161no—
632C2×C4—241no—
633C2×C4—481no—
634C2×C4—481no—
635C2×C4—481no—
636C2×C4—481no—
637C2×C4—481no—
638C2×C4C22×C4×D311yescharacteristic
639C2×C4C2×D3×D411yescharacteristic
640C2×C4C2×D3×D411yescharacteristic
641C2×C4C2×#23711yescharacteristic
642C2×C4D3×#4111yescharacteristic
643C2×C4#101011yescharacteristic
644C8—22no—
645C8—32no—
646C8—32no—
647C8—41no—
648C8—61no—
649C8—61no—
650C8—62no—
651C8—62no—
652C2×C3—21no—
653C2×C3—21no—
654C2×C3—42no—
655C2×C3—82no—
656C2×C3#236111yescharacteristic
657C2×C3#328811yescharacteristic
658C2×C3#328811yescharacteristic
659D3—82no—
660C22—21no—
661C22—21no—
662C22—22no—
663C22—41no—
664C22—41no—
665C22—42no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22—122no—
670C22—161no—
671C22—242no—
672C22—241no—
673C22—241no—
674C22—481no—
675C22#554611yescharacteristic
676C22#559011yescharacteristic
677C22#560111yescharacteristic
678C4—22no—
679C4—41no—
680C4—41no—
681C4—81no—
682C4—81no—
683C4—121no—
684C4—121no—
685C4—121no—
686C4—121no—
687C4—241no—
688C4C2×#67⋊C411yescharacteristic
689C4C4×D3×D411yescharacteristic
690C4#559811yescharacteristic
691C4#565511yescharacteristic
692C3#2304811yescharacteristic, complemented, Sylow
693C2—21no—
694C2—21no—
695C2—42no—
696C2—82no—
697C2—242no—
698C2#7467611yescharacteristic
699C2#8028011yescharacteristic
700C2#8028711yescharacteristic
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1883C2
socleC22×C3D42
prime radicalC2×#5199C2
cosocleC22×C3×C4C24
hypercenter#1883D3
hypocenterC3#23048
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1883

Complements

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

NH ≅ G/Ntype
2. #83930C2semidirect
3. #82564C2semidirect
4. #82561C2semidirect
5. #78915C2semidirect
6. C3×#1883C2semidirect
7. #70370C2semidirect
8. #70366C2semidirect
9. #70318C2semidirect
10. #69648C2semidirect
11. #69642C2semidirect
12. #68984C2semidirect
13. #68983C2semidirect
14. #68758C2semidirect
15. #68742C2semidirect
31. #5199C4semidirect
79. #1883D3semidirect
88. C3×AGL(1,ℤ/8)C2×C4semidirect
104. #949C2×C4semidirect
106. #948C2×C4semidirect
119. C2×#213C2×C4semidirect
142. #833C2×C4semidirect
147. #823C2×C4semidirect
148. #822C2×C4semidirect
322. AGL(1,ℤ/8)C4×D3semidirect
692. C3#23048semidirect

Maximal subgroups

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

subgroupindex
C2×#51992
#839302
#825642
#825612
#789152
C3×#18832
#703702
#703662
#703182
#696482
#696422
#689842
#689832
#687582
#687422
#230483