Finite Group Databasebeta

← #10818109

Subgroups — #10818109

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
1#79209C211yesmaximal, characteristic, complemented
2#78915C211yesmaximal, characteristic, complemented
3#78726C211yesmaximal, characteristic, complemented
4C2×#4749C211yesmaximal, characteristic
5#78560C211yesmaximal, characteristic, complemented
6C3×#1931C211yesmaximal, characteristic, complemented
7#70674C211yesmaximal, characteristic, complemented
8#70328C211yesmaximal, characteristic, complemented
9#70191C211yesmaximal, characteristic, complemented
10#70068C211yesmaximal, characteristic, complemented
11#69890C211yesmaximal, characteristic, complemented
12#69005C211yesmaximal, characteristic, complemented
13#68984C211yesmaximal, characteristic, complemented
14#68931C211yesmaximal, characteristic, complemented
15#68927C211yesmaximal, characteristic, complemented
16#23048—31nomaximal, Sylow
17#5798C2211yescharacteristic
18C2×#932C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×#908C2211yescharacteristic
21C2×#907C2211yescharacteristic
22#5523C2211yescharacteristic
23#5521C2211yescharacteristic
24#5504C2211yescharacteristic
25C3×#428C2211yescharacteristic
26C3×#418C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#100C2211yescharacteristic
29C3×#405C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#825C2211yescharacteristic
32#5131C2211yescharacteristic
33C2×#823C2211yescharacteristic
34C2×#822C2211yescharacteristic
35#5122C2211yescharacteristic
36#5121C2211yescharacteristic
37#5121C2211yescharacteristic
38#5113C2211yescharacteristic
39C2×#821C2211yescharacteristic
40#5101C2211yescharacteristic
41C2×#815C2211yescharacteristic
42#4998C2211yescharacteristic
43#4988C2211yescharacteristic
44#4964C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4881—21no—
47#4880—21no—
48#4861—21no—
49D3×#106—21no—
50#4785—21no—
51#4785—21no—
52#4785—21no—
53#4758—21no—
54#4757—21no—
55#4756—21no—
56#4756—21no—
57#4756—21no—
58#4749C418yescomplemented
59#4711—21no—
60#4710—21no—
61C4×D24—21no—
62C4×#204—21no—
63C3×#343C2211yescharacteristic
64C3×#341C2211yescharacteristic
65#4555C2211yescharacteristic
66#4555C2211yescharacteristic
67#4554C2211yescharacteristic
68#4532C2211yescharacteristic
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#1931D311yescharacteristic, complemented
78#1883—31no—
79#1883—31no—
80#1856—31no—
81#1856—31no—
82#1773—31no—
83#1765—31no—
84C22×C3×D4C2311yescharacteristic
85C2×#235C2311yescharacteristic
86C22×D12C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#117C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×#217—41no—
94C2×#217—41no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97D4×Dic3—21no—
98D4×Dic3—21no—
99C2×#214—21no—
100#932—22no—
101#932C2×C414yescomplemented
102C2×#213—21no—
103C2×#213—21no—
104C2×#213—21no—
105C2×#213—22no—
106C2×#213C2×C414yescomplemented
107C4×#67—41no—
108C2×(C2×D3)⋊C4—41no—
109#926C2311yescharacteristic
110#926C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#908—22no—
116#908C2×C414yescomplemented
117#907—22no—
118#907C2×C414yescomplemented
119C2×D24—21no—
120C2×#204—21no—
121C2×ΓL(1,𝔽25)—21no—
122C2×D3×C8—21no—
123#893—21no—
124#893—21no—
125#893—21no—
126#892—21no—
127#892—21no—
128D3×C4⋊C4—21no—
129D3×C4⋊C4—21no—
130#887—21no—
131#67⋊C4—41no—
132D3×C22⋊C4—41no—
133C4×D12—21no—
134C4×D12—21no—
135C4×D12—21no—
136C4×Dic6—21no—
137C3×#107—21no—
138C3×#106—21no—
139C3×#102—21no—
140C3×#102—21no—
141C3×#102—21no—
142C3×#102—21no—
143C3×#100C2×C414yescomplemented
144C3×#97—21no—
145C3×C4×C8—21no—
146#825—22no—
147#823C2×C414yescomplemented
148#822C2×C414yescomplemented
149#821—21no—
150#821—21no—
151#821—21no—
152#821—21no—
153#821D411yescharacteristic
154#821D411yescharacteristic
155#818—21no—
156#818—21no—
157#817—21no—
158#817—21no—
159#816D411yescharacteristic
160#816D411yescharacteristic
161#815—21no—
162#815—21no—
163#815—22no—
164C8×Dic3—21no—
165C8×Dic3—21no—
166#808—21no—
167#808—21no—
168#808—21no—
169#808D411yescharacteristic
170#808D411yescharacteristic
171#807—21no—
172#807D411yescharacteristic
173#807D411yescharacteristic
174C2×AGL(1,ℤ/8)—31no—
175C2×AGL(1,ℤ/8)—31no—
176C22×D8—31no—
177C2×#131—31no—
178#520—31no—
179#520—31no—
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—61no—
200#428—61no—
201#428C2×D311yescharacteristic
202#427—31no—
203#427—61no—
204C2×#106—31no—
205C2×#106—61no—
206C2×#104—31no—
207#418—31no—
208#418C2×D311yescharacteristic
209#416—31no—
210C2×#102—31no—
211C2×#102C2×D311yescharacteristic
212C2×#100—31no—
213C2×#100C2×D311yescharacteristic
214#405—31no—
215#405—31no—
216#405C2×D311yescharacteristic
217C2×#97—31no—
218#390—31no—
219#390—31no—
220#343—31no—
221#343C2×D311yescharacteristic
222#341C2×D311yescharacteristic
223#339—31no—
224#339—31no—
225#339—31no—
226C24×C3—21no—
227C23×D3—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4C22×C412yes—
234C22×C3×C4—21no—
235C22×C3×C4C2411yescharacteristic
236C2×#67—21no—
237C22×Dic3—21no—
238C22×Dic3—21no—
239C22×Dic3—41no—
240C22×Dic3—41no—
241#235—21no—
242#235—21no—
243#235C22×C412yes—
244#235#4112yes—
245C2×D12—21no—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12—21no—
249C2×D12—21no—
250C2×D12C22×C412yes—
251C2×D12C2×D411yescharacteristic
252C2×C4×D3—21no—
253C2×C4×D3—41no—
254C2×C4×D3—41no—
255C2×C4×D3—41no—
256C2×C4×D3—41no—
257C2×Dic6C2×D411yescharacteristic
258C3×M16—21no—
259C3×M16—21no—
260C3×M16—21no—
261C3×M16—21no—
262C3×M16C22×C412yes—
263C3×M16C22×C412yes—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C2×C3×C8—21no—
267C2×C3×C8—21no—
268C2×C3×C8—21no—
269C2×C3×C8C2×D411yescharacteristic
270C2×C3×C8C2×D411yescharacteristic
271C3×C4⋊C4—21no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4C2×D411yescharacteristic
274C3×C4⋊C4C2×D411yescharacteristic
275C3×C22⋊C4—21no—
276C3×C42—21no—
277#217—21no—
278#217—21no—
279#217—41no—
280#217—41no—
281#217—82no—
282#214—24no—
283#213—24no—
284#213—24no—
285#213—24no—
286(C2×D3)⋊C4—41no—
287(C2×D3)⋊C4—41no—
288(C2×D3)⋊C4—41no—
289(C2×D3)⋊C4—41no—
290(C2×D3)⋊C4—82no—
291C4⋊Dic3—21no—
292C4⋊Dic3—21no—
293C4⋊Dic3C2×D411yescharacteristic
294C4⋊Dic3C2×D411yescharacteristic
295C4⋊Dic3C2×D411yescharacteristic
296C4⋊Dic3C2×D411yescharacteristic
297Dic3⋊C4—41no—
298Dic3⋊C4—41no—
299C4×Dic3—21no—
300C4×Dic3—21no—
301C4×Dic3—21no—
302C4×Dic3—21no—
303C4×Dic3—41no—
304C4×Dic3—41no—
305#208—21no—
306#208C22×C412yes—
307C2×C3⋊C8—21no—
308C2×C3⋊C8C22×C412yes—
309C2×C3⋊C8C2×D411yescharacteristic
310C2×C3⋊C8C2×D411yescharacteristic
311C2×C3⋊C8#4112yes—
312D24—24no—
313#204—24no—
314ΓL(1,𝔽25)—24no—
315D3×C8—24no—
316C2×#41—31no—
317C22×D4—31no—
318C22×D4C22×D311yescharacteristic
319AGL(1,ℤ/8)—34no—
320AGL(1,ℤ/8)—34no—
321AGL(1,ℤ/8)—62no—
322AGL(1,ℤ/8)—62no—
323C2×SD16—61no—
324C2×SD16—61no—
325C2×D8—34no—
326C2×D8—61no—
327C2×D8—62no—
328C2×D8—61no—
329C2×D8—61no—
330C2×D8—61no—
331#131—34no—
332#131—62no—
333C2×M16—31no—
334C2×M16—61no—
335C2×M16C22×D311yescharacteristic
336C2×M16C22×D311yescharacteristic
337C2×M16C22×D311yescharacteristic
338C22×C8—31no—
339C22×C8—61no—
340C22×C8C22×D311yescharacteristic
341C4×Q8—61no—
342C4×Q8—61no—
343C4×D4—61no—
344C4×D4—61no—
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—121no—
354C4×D4—121no—
355#117—31no—
356#117—31no—
357#117—61no—
358#117—61no—
359#117C22×D311yescharacteristic
360C2×C4⋊C4—31no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—121no—
365C2×C22⋊C4—121no—
366C2×C22⋊C4—121no—
367C2×C22⋊C4—121no—
368C2×C42—31no—
369#108—34no—
370#107—21no—
371#107—31no—
372#107—31no—
373#107—61no—
374#107—61no—
375#107—61no—
376#106—21no—
377#106—31no—
378#106—31no—
379#106—61no—
380#106—61no—
381#106—61no—
382#106—61no—
383#106—61no—
384#104—62no—
385#103—31no—
386#103—31no—
387#102—21no—
388#102—21no—
389#102—21no—
390#102—21no—
391#102—31no—
392#102—31no—
393#102—61no—
394#102—61no—
395#102—61no—
396#102—61no—
397#100—34no—
398#100C4×D314yescomplemented
399#97—21no—
400#97—62no—
401#97—61no—
402#97—61no—
403C4×C8—21no—
404C4×C8—61no—
405C4×C8—61no—
406C23×C3—22no—
407C23×C3—41no—
408C23×C3—41no—
409C23×C3—81no—
410C23×C3—81no—
411C23×C3H(5,𝔽2)11yescharacteristic
412C22×D3—21no—
413C22×D3—22no—
414C22×D3—41no—
415C22×D3—41no—
416C22×D3—81no—
417C22×D3—81no—
418C3×D4—22no—
419C3×D4—22no—
420C3×D4—41no—
421C3×D4—41no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—41no—
425C2×C3×C4—81no—
426C2×C3×C4C23×C411yescharacteristic
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C2×#4111yescharacteristic
430C2×C3×C4C2×#4111yescharacteristic
431C2×C3×C42−1+411yescharacteristic
432#67—22no—
433#67—22no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—81no—
444C2×Dic3—161no—
445C2×Dic3—161no—
446D12—22no—
447D12—22no—
448D12—21no—
449D12—41no—
450D12—41no—
451D12C4×D412yes—
452C4×D3—22no—
453C4×D3—22no—
454C4×D3—161no—
455C4×D3—161no—
456C4×D3—161no—
457C4×D3—161no—
458Dic6—21no—
459Dic6C4×D412yes—
460C3×C8—22no—
461C3×C8—22no—
462C3×C8—21no—
463C3×C8—21no—
464C3×C8—41no—
465C3⋊C8—22no—
466C3⋊C8C4×D412yes—
467C3⋊C8C4×D412yes—
468C24—21no—
469C24—61no—
470#41—32no—
471#41—32no—
472#41—61no—
473#41—61no—
474C2×Q8—31no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—32no—
481C2×D4—31no—
482C2×D4—61no—
483C2×D4—61no—
484C2×D4—61no—
485C2×D4—61no—
486C2×D4—61no—
487C2×D4—61no—
488C2×D4C2×C4×D312yes—
489C22×C4—21no—
490C22×C4—61no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—121no—
494C22×C4—121no—
495C22×C4—121no—
496C22×C4—121no—
497C22×C4—121no—
498C22×C4—121no—
499C22×C4C23×D311yescharacteristic
500SD16—64no—
501SD16—64no—
502D8—64no—
503D8—64no—
504D8—64no—
505D8—64no—
506M16—21no—
507M16—21no—
508M16—21no—
509M16—21no—
510M16—32no—
511M16—64no—
512M16—61no—
513M16C2×C4×D312yes—
514M16C2×C4×D312yes—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—32no—
521C2×C8—31no—
522C2×C8—31no—
523C2×C8—32no—
524C2×C8—64no—
525C2×C8—61no—
526C2×C8D3×D411yescharacteristic
527C2×C8D3×D411yescharacteristic
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—31no—
531C4⋊C4—31no—
532C4⋊C4—31no—
533C4⋊C4—31no—
534C4⋊C4—61no—
535C4⋊C4—61no—
536C4⋊C4—121no—
537C4⋊C4—121no—
538C4⋊C4D3×D411yescharacteristic
539C4⋊C4D3×D411yescharacteristic
540C22⋊C4—21no—
541C22⋊C4—61no—
542C22⋊C4—61no—
543C22⋊C4—121no—
544C22⋊C4—121no—
545C22⋊C4—121no—
546C22⋊C4—121no—
547C22⋊C4—121no—
548C22⋊C4—121no—
549C22⋊C4—242no—
550C22⋊C4—242no—
551C42—21no—
552C42—61no—
553C42—61no—
554C42—61no—
555C42—61no—
556C42—121no—
557C42—121no—
558C22×C3—21no—
559C22×C3—21no—
560C22×C3—42no—
561C22×C3—82no—
562C22×C3—81no—
563C22×C3—81no—
564C22×C3—161no—
565C22×C3#52011yescharacteristic
566C22×C3D4211yescharacteristic
567C22×C3#55311yescharacteristic
568C2×D3—22no—
569C2×D3—41no—
570C2×D3—41no—
571C2×D3—42no—
572C2×D3—81no—
573C2×D3—81no—
574C2×D3—82no—
575C2×D3—161no—
576C3×C4—41no—
577C3×C4—41no—
578C3×C4—81no—
579C3×C4C2×C4×D411yescharacteristic
580C3×C4C2×C4×D411yescharacteristic
581C3×C4#54711yescharacteristic
582C3×C4#54811yescharacteristic
583Dic3—22no—
584Dic3—41no—
585Dic3—41no—
586Dic3—41no—
587Dic3—41no—
588Dic3—81no—
589Dic3—81no—
590C23—22no—
591C23—41no—
592C23—41no—
593C23—61no—
594C23—62no—
595C23—81no—
596C23—81no—
597C23—121no—
598C23—121no—
599C23—241no—
600C23—241no—
601C23#100911yescharacteristic
602Q8—32no—
603Q8—61no—
604D4—22no—
605D4—22no—
606D4—32no—
607D4—41no—
608D4—41no—
609D4—62no—
610D4—62no—
611D4—61no—
612D4—62no—
613D4—62no—
614D4—121no—
615D4—121no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—41no—
619C2×C4—61no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—81no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—241no—
632C2×C4—481no—
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×C4D3×#4111yescharacteristic
642C2×C4D3×#4111yescharacteristic
643C2×C4#101011yescharacteristic
644C8—21no—
645C8—21no—
646C8—22no—
647C8—22no—
648C8—32no—
649C8—32no—
650C8—41no—
651C8—62no—
652C2×C3—21no—
653C2×C3—21no—
654C2×C3—82no—
655C2×C3#236111yescharacteristic
656C2×C3#328811yescharacteristic
657C2×C3#328811yescharacteristic
658D3—42no—
659D3—82no—
660C22—21no—
661C22—21no—
662C22—42no—
663C22—62no—
664C22—82no—
665C22—81no—
666C22—81no—
667C22—121no—
668C22—121no—
669C22—122no—
670C22—161no—
671C22—242no—
672C22—241no—
673C22—241no—
674C22—481no—
675C22#552311yescharacteristic
676C22#560911yescharacteristic
677C22#570911yescharacteristic
678C4—41no—
679C4—41no—
680C4—62no—
681C4—81no—
682C4—121no—
683C4—121no—
684C4—121no—
685C4—121no—
686C4—241no—
687C4—241no—
688C4C4×D3×D411yescharacteristic
689C4C4×D3×D411yescharacteristic
690C4#565511yescharacteristic
691C4#567711yescharacteristic
692C3#2304811yescharacteristic, complemented, Sylow
693C2—21no—
694C2—21no—
695C2—82no—
696C2—122no—
697C2—242no—
698C2#7547711yescharacteristic
699C2#8093211yescharacteristic
700C2#8093911yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1931C2
socleC22×C3D42
prime radicalC2×#4749C2
cosocleC22×C3×C4C24
hypercenter#1931D3
hypocenterC3#23048
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1931

Complements

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

NH ≅ G/Ntype
1. #79209C2semidirect
2. #78915C2semidirect
3. #78726C2semidirect
5. #78560C2semidirect
6. C3×#1931C2semidirect
7. #70674C2semidirect
8. #70328C2semidirect
9. #70191C2semidirect
10. #70068C2semidirect
11. #69890C2semidirect
12. #69005C2semidirect
13. #68984C2semidirect
14. #68931C2semidirect
15. #68927C2semidirect
58. #4749C4semidirect
77. #1931D3semidirect
101. #932C2×C4semidirect
106. C2×#213C2×C4semidirect
116. #908C2×C4semidirect
118. #907C2×C4semidirect
143. C3×#100C2×C4semidirect
147. #823C2×C4semidirect
148. #822C2×C4semidirect
398. #100C4×D3semidirect
692. C3#23048semidirect

Maximal subgroups

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

subgroupindex
#792092
#789152
#787262
C2×#47492
#785602
C3×#19312
#706742
#703282
#701912
#700682
#698902
#690052
#689842
#689312
#689272
#230483