Finite Group Databasebeta

← #411935

Subgroups — #411935

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2#23453C211yesmaximal, characteristic
3#23452C211yesmaximal, characteristic
4C2×#1816C211yesmaximal, characteristic
5C2×#1815C211yesmaximal, characteristic
6#20384C211yesmaximal, characteristic
7#11383C211yesmaximal, characteristic, complemented
8#11382C211yesmaximal, characteristic, complemented
9#11271C211yesmaximal, characteristic, complemented
10#11270C211yesmaximal, characteristic, complemented
11#11107C211yesmaximal, characteristic
12#11106C211yesmaximal, characteristic
13#10870C211yesmaximal, characteristic, complemented
14#10869C211yesmaximal, characteristic, complemented
15#10865C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C2×#501—24no—
20C2×#501—24no—
21C2×#501—22no—
22C2×#501—22no—
23C22×#108C2211yescharacteristic
24C22×#108C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#417C2211yescharacteristic
33C2×#408C2211yescharacteristic
34C2×#405C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2297C2211yescharacteristic
37#2026—22no—
38#2025—22no—
39#2024—22no—
40#2023—22no—
41#1816—24no—
42#1815—24no—
43#1753—22no—
44#1752—22no—
45C2×#343C2211yescharacteristic
46C2×#343C2211yescharacteristic
47C2×#343C2211yescharacteristic
48C2×#343C2211yescharacteristic
49C2×#343C2211yescharacteristic
50C2×#343C2211yescharacteristic
51C2×#343C2211yescharacteristic
52C2×#343C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55#1736C2211yescharacteristic
56#1736C2211yescharacteristic
57#1736C2211yescharacteristic
58#1736C2211yescharacteristic
59#1729C2211yescharacteristic
60#1728C2211yescharacteristic
61#1728C2211yescharacteristic
62#1728C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C23×Q8—21no—
65C2×#131—22no—
66C2×#131—22no—
67C2×#131—22no—
68C2×#131—22no—
69C2×#131—22no—
70C2×#131Q814yescomplemented
71C22×M16—21no—
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C23×C8—21no—
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—41no—
86C2×C4⋊Q8—41no—
87C2×#125—22no—
88C2×#125—22no—
89C2×#125—22no—
90C2×#125—22no—
91C2×#125—41no—
92C2×#125—41no—
93C2×#117C2311yescharacteristic
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#501—216no—
103#501—216no—
104C2×#108—22no—
105C2×#108—22no—
106C2×#108—22no—
107C2×#108—22no—
108C2×#108—22no—
109C2×#108—22no—
110C2×#107—22no—
111C2×#107—21no—
112C2×#107—21no—
113C2×#107—22no—
114C2×#107—22no—
115C2×#107—22no—
116C2×#107D414yes—
117C2×#106—22no—
118C2×#106—21no—
119C2×#106—21no—
120C2×#106—22no—
121C2×#106—22no—
122C2×#106—22no—
123C2×#106D414yes—
124#417—22no—
125#417—22no—
126#417—22no—
127#417—22no—
128#417—41no—
129#417—41no—
130#417—41no—
131#417—41no—
132C2×#103—21no—
133C2×#103—21no—
134C2×#103—21no—
135C2×#103—21no—
136C2×#102—21no—
137C2×#102—21no—
138C2×#102—21no—
139C2×#102—21no—
140#408—22no—
141#408Q814yescomplemented
142C2×C22⋊C8—22no—
143#405—22no—
144#405—22no—
145C2×#97—22no—
146C2×#97—22no—
147C2×#97—21no—
148C2×#97D412yes—
149C2×#97D412yes—
150C2×C4×C8—21no—
151#391D412yes—
152#390D412yes—
153#386—21no—
154#386—21no—
155C4×C22⋊C4—22no—
156#343—22no—
157#343—22no—
158#343—22no—
159#343—22no—
160#343—41no—
161#343—41no—
162#343—41no—
163#343—41no—
164#341—22no—
165#341—22no—
166#340—22no—
167#340—22no—
168#340—22no—
169#340—22no—
170#336—22no—
171#336—22no—
172#336—22no—
173#336—22no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41C2×Q812yes—
185C2×#41C2×Q812yes—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—84no—
190C22×Q8C2×D411yescharacteristic
191C22×D4—21no—
192C22×D4C2×D411yescharacteristic
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4#4111yescharacteristic
199C23×C4#4111yescharacteristic
200C23×C4C2411yescharacteristic
201#131—28no—
202#131—28no—
203#131—44no—
204#131—44no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—24no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—42no—
241C2×M16—42no—
242C2×M16C2×Q812yes—
243C2×M16C2×Q812yes—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—24no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8C2×D412yes—
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D412yes—
261C22×C8C2×D412yes—
262C22×C8C2×Q812yes—
263C22×C8C2×Q812yes—
264C22×C8#4112yes—
265C22×C8#4112yes—
266C22×C8#4112yes—
267C22×C8#4112yes—
268C4⋊Q8—28no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#125—28no—
273#125—44no—
274#125—44no—
275#125—44no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324#108—44no—
325#108—44no—
326#107—28no—
327#107—24no—
328#107—24no—
329#106—28no—
330#106—24no—
331#106—24no—
332#103—42no—
333#103—42no—
334#103—42no—
335#103—42no—
336#102—42no—
337#102—42no—
338#102—42no—
339#102—42no—
340C22⋊C8—28no—
341#97—24no—
342#97—24no—
343#97—42no—
344C4×C8—42no—
345#95—22no—
346#95—22no—
347C24—21no—
348C24C2×#4111yescharacteristic
349#41—22no—
350#41—22no—
351#41—22no—
352#41—22no—
353#41—22no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—48no—
371C2×Q8—84no—
372C2×Q8—84no—
373C2×Q8—88no—
374C2×Q8—88no—
375C2×Q8C4⋊Q812yes—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4C4⋊Q812yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q811yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C42−1+411yescharacteristic
433C22×C42−1+411yescharacteristic
434M16—22no—
435M16—22no—
436M16—22no—
437M16—22no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—48no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—48no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8#12012yes—
485C2×C8#12012yes—
486C2×C8C22⋊Q814yes—
487C2×C8C22⋊Q812yes—
488C2×C8C22⋊Q812yes—
489C2×C8C22⋊Q814yes—
490C2×C8C22⋊Q814yes—
491C2×C8#12412yes—
492C2×C8#12412yes—
493C2×C8#12414yes—
494C2×C8#12414yes—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—41no—
525C42—41no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23C2×#12111yescharacteristic
534C23#53311yescharacteristic
535C23#53311yescharacteristic
536C23#54611yescharacteristic
537C23#54611yescharacteristic
538C23#55611yescharacteristic
539C23#55611yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—41no—
543Q8—41no—
544Q8—816no—
545D4—22no—
546D4—22no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—41no—
552D4—41no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4C2×#12011yescharacteristic
590C2×C4C2×C22⋊Q811yescharacteristic
591C2×C4C2×C22⋊Q811yescharacteristic
592C2×C4C2×C22⋊Q811yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4C2×#12411yescharacteristic
595C2×C4C2×#12411yescharacteristic
596C2×C4C2×#12411yescharacteristic
597C2×C4C2×C4⋊Q811yescharacteristic
598C2×C4#53211yescharacteristic
599C2×C4#53511yescharacteristic
600C2×C4#53611yescharacteristic
601C2×C4#54211yescharacteristic
602C2×C4#54211yescharacteristic
603C2×C4#54411yescharacteristic
604C2×C4#54411yescharacteristic
605C2×C4D4211yescharacteristic
606C2×C4D4211yescharacteristic
607C2×C4#54711yescharacteristic
608C2×C4#54711yescharacteristic
609C2×C4#54811yescharacteristic
610C2×C4#54811yescharacteristic
611C2×C4D4×Q811yescharacteristic
612C2×C4D4×Q811yescharacteristic
613C2×C4#55411yescharacteristic
614C2×C4#55411yescharacteristic
615C2×C4#55511yescharacteristic
616C2×C4#55511yescharacteristic
617C8—24no—
618C8—42no—
619C8—42no—
620C8—81no—
621C8—81no—
622C8—81no—
623C8—81no—
624C8#39314yes—
625C8#39314yes—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—42no—
635C22—44no—
636C22—82no—
637C22—81no—
638C22—81no—
639C22#240811yescharacteristic
640C22#243911yescharacteristic
641C22#243911yescharacteristic
642C22#249211yescharacteristic
643C22#249211yescharacteristic
644C22#254711yescharacteristic
645C22#285711yescharacteristic
646C22#306812yes—
647C22#306912yes—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4C2×#39311yescharacteristic
657C4#240911yescharacteristic
658C4#244011yescharacteristic
659C4#244011yescharacteristic
660C4#249111yescharacteristic
661C4#249111yescharacteristic
662C4#256311yescharacteristic
663C4#262811yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—82no—
669C2#1512511yescharacteristic
670C2#2428311yescharacteristic
671C2#2428411yescharacteristic
672C2#2873512yes—
673C2#3322312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
7. #11383C2semidirect
8. #11382C2semidirect
9. #11271C2semidirect
10. #11270C2semidirect
13. #10870C2semidirect
14. #10869C2semidirect
15. #10865C2semidirect
70. C2×#131Q8semidirect
141. #408Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
#234532
#234522
C2×#18162
C2×#18152
#203842
#113832
#113822
#112712
#112702
#111072
#111062
#108702
#108692
#108652