Finite Group Databasebeta

← #10818113

Subgroups — #10818113

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
1C3×#2946C211yesmaximal, characteristic, complemented
2C2×#5199C211yesmaximal, characteristic
3#82561C211yesmaximal, characteristic, complemented
4#82538C211yesmaximal, characteristic, complemented
5#78820C211yesmaximal, characteristic, complemented
6#72088C211yesmaximal, characteristic, complemented
7#71667C211yesmaximal, characteristic, complemented
8#71647C211yesmaximal, characteristic, complemented
9#70674C211yesmaximal, characteristic, complemented
10#70668C211yesmaximal, characteristic, complemented
11#70290C211yesmaximal, characteristic, complemented
12#70195C211yesmaximal, characteristic, complemented
13#70191C211yesmaximal, characteristic, complemented
14#69892C211yesmaximal, characteristic, complemented
15#69890C211yesmaximal, characteristic, complemented
16#23048—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C3×#518C2211yescharacteristic
20#5827C2211yescharacteristic
21C2×#949C2211yescharacteristic
22C2×#948C2211yescharacteristic
23C22×#213C2211yescharacteristic
24#5521C2211yescharacteristic
25C3×#442—21no—
26C3×#440—21no—
27C3×C4×SD16—21no—
28C3×C4×D8—21no—
29C3×#428C2211yescharacteristic
30C3×#416C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C3×#405C2211yescharacteristic
33C2×#837C2211yescharacteristic
34#5221C2211yescharacteristic
35C2×#833C2211yescharacteristic
36#5215C2211yescharacteristic
37#5199C418yescomplemented
38C2×#823C2211yescharacteristic
39C2×#822C2211yescharacteristic
40C2×#821C2211yescharacteristic
41#5101C2211yescharacteristic
42C2×#815C2211yescharacteristic
43#5014—21no—
44#5014—21no—
45C4×#213—21no—
46C4×#213—21no—
47#5001C2211yescharacteristic
48#4998—21no—
49#4998C2211yescharacteristic
50#4997C2211yescharacteristic
51#4995—21no—
52#4995C2211yescharacteristic
53#4995C2211yescharacteristic
54#4988C2211yescharacteristic
55#4964—21no—
56#4964C2211yescharacteristic
57C2×#808—21no—
58C2×#808C2211yescharacteristic
59#4817—21no—
60#4788—21no—
61#4785—21no—
62#4756—21no—
63#4555C2211yescharacteristic
64#4554C2211yescharacteristic
65#4552C2211yescharacteristic
66#4532C2211yescharacteristic
67#4532C2211yescharacteristic
68#4530C2211yescharacteristic
69C2×#472—31no—
70#2948—31no—
71#2946—31no—
72#2946D311yescharacteristic, complemented
73#2912—31no—
74#1972—31no—
75#1951—31no—
76#1931—31no—
77#1931—31no—
78#1883—31no—
79#1883—31no—
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
92C22×C3×C8C2311yescharacteristic
93C3×C4×Q8—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4—41no—
100C3×#117—21no—
101C3×#117C2311yescharacteristic
102C3×#117C2311yescharacteristic
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—41no—
106C2×#217—41no—
107#949—22no—
108#949C2×C414yescomplemented
109#948—22no—
110#948C2×C414yescomplemented
111Q8×Dic3—21no—
112C2×#215—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
123C2×C4⋊Dic3—21no—
124C2×C4⋊Dic3C2311yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C2×#208—21no—
127C2×#208C2311yescharacteristic
128C2×#208C2311yescharacteristic
129C2×#208C2311yescharacteristic
130C22×C3⋊C8—21no—
131C22×C3⋊C8C2311yescharacteristic
132#893—21no—
133#893—21no—
134D3×C22⋊C4—41no—
135C4×D12—21no—
136C4×D12—21no—
137C3×#107—21no—
138C3×#106—21no—
139C3×#103D411yescharacteristic
140C3×#103D411yescharacteristic
141C3×#102—21no—
142C3×#102—21no—
143C3×#102D411yescharacteristic
144C3×#102D411yescharacteristic
145C3×#97—21no—
146C3×C4×C8—21no—
147#837—22no—
148#833C2×C414yescomplemented
149#832—21no—
150#832—21no—
151#823C2×C414yescomplemented
152#822C2×C414yescomplemented
153#821—21no—
154#821—21no—
155#815—21no—
156#815—22no—
157C8×Dic3—21no—
158#809—21no—
159#809—21no—
160#808—21no—
161#808—21no—
162#808—21no—
163#808—21no—
164#808—21no—
165#808D411yescharacteristic
166#808D411yescharacteristic
167#807—21no—
168#807—21no—
169#807—21no—
170#807D411yescharacteristic
171#807D411yescharacteristic
172#803—21no—
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#518C2×D311yescharacteristic
182#472—38no—
183#442—21no—
184#442—61no—
185#442—61no—
186#442—61no—
187#440—21no—
188#440—61no—
189C4×SD16—21no—
190C4×SD16—61no—
191C4×D8—21no—
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#418—31no—
209#416C2×D311yescharacteristic
210C2×#102—31no—
211C2×#102C2×D311yescharacteristic
212C2×#100—31no—
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405C2×D311yescharacteristic
217C2×#97—31no—
218#390—31no—
219#390—31no—
220#343—31no—
221#343—31no—
222#341—31no—
223#339—31no—
224#339—31no—
225#339—31no—
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—41no—
244C22×C3×C4—41no—
245C22×C3×C4—41no—
246C22×C3×C4C2411yescharacteristic
247C22×Dic3—21no—
248C22×Dic3—21no—
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×M16C22×C412yes—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C2×C3×C8C2×D411yescharacteristic
266C2×C3×C8C2×D411yescharacteristic
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—41no—
270C3×C4⋊C4C2×D411yescharacteristic
271C3×C4⋊C4C2×D411yescharacteristic
272C3×C4⋊C4C2×D411yescharacteristic
273C3×C4⋊C4C2×D411yescharacteristic
274C3×C22⋊C4—21no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—41no—
277C3×C22⋊C4—41no—
278C3×C22⋊C4—41no—
279C3×C42—21no—
280C3×C42—21no—
281C3×C42—41no—
282#217—21no—
283#217—41no—
284#217—82no—
285#215—24no—
286#213—24no—
287#213—24no—
288#213—24no—
289(C2×D3)⋊C4—41no—
290(C2×D3)⋊C4—41no—
291(C2×D3)⋊C4—82no—
292C4⋊Dic3—21no—
293C4⋊Dic3—21no—
294C4⋊Dic3—41no—
295C4⋊Dic3C2×D411yescharacteristic
296C4⋊Dic3C2×D411yescharacteristic
297C4×Dic3—21no—
298C4×Dic3—21no—
299C4×Dic3—21no—
300C4×Dic3—41no—
301#208—21no—
302#208—21no—
303#208—24no—
304#208—21no—
305#208—21no—
306#208C22×C412yes—
307#208C22×C412yes—
308C2×C3⋊C8—21no—
309C2×C3⋊C8—21no—
310C2×C3⋊C8—24no—
311C2×C3⋊C8—21no—
312C2×C3⋊C8C22×C412yes—
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—31no—
336C2×M16—61no—
337C2×M16C22×D311yescharacteristic
338C22×C8—31no—
339C22×C8—61no—
340C22×C8C22×D311yescharacteristic
341C4×Q8—21no—
342C4×Q8—61no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—41no—
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—61no—
358#117C22×D311yescharacteristic
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×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#6711yescharacteristic
386#103#6711yescharacteristic
387#102—21no—
388#102—21no—
389#102—61no—
390#102—61no—
391#102—61no—
392#102—61no—
393#102—61no—
394#102—61no—
395#102#6711yescharacteristic
396#102#6711yescharacteristic
397#100—34no—
398#100—34no—
399#97—21no—
400#97—61no—
401#97—62no—
402#97—61no—
403C4×C8—21no—
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—21no—
434C2×C3×C4—21no—
435C2×C3×C4—41no—
436C2×C3×C4—81no—
437C2×C3×C4—161no—
438C2×C3×C4—161no—
439C2×C3×C4—161no—
440C2×C3×C4C23×C411yescharacteristic
441C2×C3×C4C22×D411yescharacteristic
442C2×C3×C4C22×D411yescharacteristic
443C2×C3×C4C2×#4111yescharacteristic
444C2×C3×C4C2×#4111yescharacteristic
445C2×C3×C42−1+411yescharacteristic
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451C2×Dic3—41no—
452C2×Dic3—81no—
453C2×Dic3—161no—
454D12—22no—
455D12—22no—
456D12—41no—
457D12—41no—
458C4×D3—161no—
459C4×D3—161no—
460C3×C8—22no—
461C3×C8—21no—
462C3×C8—21no—
463C3⋊C8—22no—
464C3⋊C8—22no—
465C3⋊C8—41no—
466C3⋊C8C4×D412yes—
467C3⋊C8C4×D412yes—
468C24—21no—
469C24—61no—
470#41—21no—
471#41—21no—
472#41C2×C4×D312yes—
473#41#23512yes—
474C2×Q8C2×#6711yescharacteristic
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×D4C2×#6711yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—61no—
495C22×C4—61no—
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—32no—
508M16—32no—
509M16—64no—
510M16—61no—
511M16—61no—
512M16—61no—
513M16—61no—
514M16C2×C4×D312yes—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—32no—
519C2×C8—31no—
520C2×C8—31no—
521C2×C8—32no—
522C2×C8—64no—
523C2×C8—61no—
524C2×C8—61no—
525C2×C8—61no—
526C2×C8C2×#6711yescharacteristic
527C2×C8C2×#6711yescharacteristic
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—31no—
531C4⋊C4—31no—
532C4⋊C4—41no—
533C4⋊C4—61no—
534C4⋊C4—61no—
535C4⋊C4—121no—
536C4⋊C4D3×D411yescharacteristic
537C4⋊C4D3×D411yescharacteristic
538C4⋊C4C2×#6711yescharacteristic
539C4⋊C4C2×#6711yescharacteristic
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—61no—
546C22⋊C4—121no—
547C22⋊C4—121no—
548C22⋊C4—121no—
549C22⋊C4—242no—
550C22⋊C4—242no—
551C42—21no—
552C42—21no—
553C42—41no—
554C42—61no—
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—41no—
580C3×C4—41no—
581C3×C4—81no—
582C3×C4C2×C4×D411yescharacteristic
583C3×C4C2×C4×D411yescharacteristic
584C3×C4#54711yescharacteristic
585C3×C4#54811yescharacteristic
586Dic3—41no—
587Dic3—41no—
588Dic3—81no—
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—
603Q8C4×#6712yes—
604D4—22no—
605D4—22no—
606D4—22no—
607D4—22no—
608D4—21no—
609D4—41no—
610D4—41no—
611D4—62no—
612D4—62no—
613D4—121no—
614D4—121no—
615D4C4×#6712yes—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—41no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—81no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—161no—
632C2×C4—161no—
633C2×C4—161no—
634C2×C4—241no—
635C2×C4—481no—
636C2×C4—481no—
637C2×C4—481no—
638C2×C4C22×C4×D311yescharacteristic
639C2×C4C2×#23511yescharacteristic
640C2×C4C2×D3×D411yescharacteristic
641C2×C4D3×#4111yescharacteristic
642C2×C4#101011yescharacteristic
643C2×C4C22×#6711yescharacteristic
644C8—22no—
645C8—21no—
646C8—21no—
647C8—32no—
648C8—32no—
649C8—62no—
650C8—62no—
651C8—121no—
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#554011yescharacteristic
676C22#554611yescharacteristic
677C22D4×#6711yescharacteristic
678C4—22no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—81no—
684C4—121no—
685C4—121no—
686C4—241no—
687C4—241no—
688C4C4×D3×D411yescharacteristic
689C4#555211yescharacteristic
690C4C2×C4×#6711yescharacteristic
691C4#582411yescharacteristic
692C3#2304811yescharacteristic, complemented, Sylow
693C2—21no—
694C2—21no—
695C2—42no—
696C2—82no—
697C2—242no—
698C2#8234011yescharacteristic
699C2#8400911yescharacteristic
700C2#8401511yescharacteristic
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2946C2
socleC22×C3D42
prime radicalC2×#5199C2
cosocleC22×C3×C4C24
hypercenter#2946D3
hypocenterC3#23048
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. #82561C2semidirect
4. #82538C2semidirect
5. #78820C2semidirect
6. #72088C2semidirect
7. #71667C2semidirect
8. #71647C2semidirect
9. #70674C2semidirect
10. #70668C2semidirect
11. #70290C2semidirect
12. #70195C2semidirect
13. #70191C2semidirect
14. #69892C2semidirect
15. #69890C2semidirect
37. #5199C4semidirect
72. #2946D3semidirect
88. C3×AGL(1,ℤ/8)C2×C4semidirect
108. #949C2×C4semidirect
110. #948C2×C4semidirect
119. C2×#213C2×C4semidirect
148. #833C2×C4semidirect
151. #823C2×C4semidirect
152. #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
C3×#29462
C2×#51992
#825612
#825382
#788202
#720882
#716672
#716472
#706742
#706682
#702902
#701952
#701912
#698922
#698902
#230483