Finite Group Databasebeta

← #360206

Subgroups — #360206

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2930C211yesmaximal, characteristic
2C22×#430C211yesmaximal, characteristic, complemented
3#20971C212yesmaximal
4C2×#1816C211yesmaximal, characteristic, complemented
5C2×#1815C211yesmaximal, characteristic, complemented
6C2×#1809C211yesmaximal, characteristic, complemented
7#11242C212yesmaximal, complemented
8#11105C212yesmaximal, complemented
9#10870C211yesmaximal, characteristic, complemented
10#10869C211yesmaximal, characteristic, complemented
11#10739C212yesmaximal, complemented
12C22×#131C2211yescharacteristic
13C23×M16C2212yes—
14C23×M16C2211yescharacteristic
15C2×#525C2211yescharacteristic
16#2930—24no—
17#2930C418yescomplemented
18C2×#436—22no—
19C2×#435—22no—
20C2×#430—24no—
21C2×#430—24no—
22C2×#430—24no—
23C22×#108C2212yes—
24C22×#108C2211yescharacteristic
25C22×#108C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#417C2212yes—
30C2×#417C2212yes—
31C2×#408C2211yescharacteristic
32C2×#407C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#380C2212yes—
35#1895—22no—
36#1894—22no—
37#1893—22no—
38#1892—22no—
39#1816—24no—
40#1815—24no—
41#1809—22no—
42#1809—22no—
43C2×#343C2212yes—
44C2×#343C2212yes—
45C2×#343C2212yes—
46C2×#343C2212yes—
47#1736C2212yes—
48#1736C2212yes—
49C2×#340C2212yes—
50C22×#41C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C2×#131—22no—
53C2×#131—22no—
54C2×#131—22no—
55C2×#131—22no—
56C2×#131—22no—
57C2×#131C2×C414yes—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—22no—
61C22×M16—21no—
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—24no—
65C22×M16—24no—
66C22×M16—24no—
67C22×M16—24no—
68C22×M16—42no—
69C22×M16—42no—
70C22×M16C2×C414yes—
71C22×M16D412yes—
72C22×M16D412yes—
73C22×M16D414yes—
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2312yes—
77C22×M16C2312yes—
78C23×C8—21no—
79C23×C8—22no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82#525—22no—
83#525—22no—
84#525C2×C414yes—
85#525Q814yes—
86C2×#123—21no—
87C2×C22⋊Q8D412yes—
88C2×#121D412yes—
89C2×#120—21no—
90C2×C4×D4—22no—
91C2×#117C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—22no—
95#436—216no—
96#435—216no—
97#430—232no—
98C2×#108—24no—
99C2×#108—24no—
100C2×#108—24no—
101C2×#108—22no—
102C2×#108—22no—
103C2×#108—22no—
104C2×#108—22no—
105C2×#108D414yescomplemented
106C2×#108D414yescomplemented
107C2×#107—21no—
108C2×#107—21no—
109C2×#107—22no—
110C2×#107—22no—
111C2×#106—21no—
112C2×#106—21no—
113C2×#106—22no—
114C2×#106—22no—
115#424—22no—
116#424C2×C414yes—
117C2×C4⋊C8—22no—
118#417—24no—
119#417—24no—
120#417—42no—
121#417—42no—
122C2×#103—22no—
123C2×#103—22no—
124C2×#102—22no—
125C2×#102—22no—
126#408—22no—
127#408C2×C414yes—
128#407—22no—
129#407C2×C414yes—
130C2×C22⋊C8—22no—
131C2×C22⋊C8—22no—
132#405—22no—
133#405C2×C414yes—
134C2×#97—21no—
135C2×C4×C8—21no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380—22no—
140C2×#95—22no—
141#343—24no—
142#343—24no—
143#343—42no—
144#343—42no—
145#340—24no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150C25#4111yescharacteristic
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41C22×C412yes—
162C2×#41C2×Q812yes—
163C22×Q8C2×D411yescharacteristic
164C22×D4—21no—
165C22×D4—22no—
166C22×D4C2×D411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4C22×C412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×Q812yes—
183C23×C4#4111yescharacteristic
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186#131—28no—
187#131—28no—
188#131—44no—
189#131—44no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—24no—
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—28no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—28no—
203C2×M16—216no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—44no—
221C2×M16—44no—
222C2×M16—48no—
223C2×M16—48no—
224C2×M16—48no—
225C2×M16—44no—
226C2×M16—42no—
227C2×M16—42no—
228C2×M16C22⋊C418yes—
229C2×M16C22⋊C418yes—
230C2×M16C22×C412yes—
231C2×M16C22×C412yes—
232C2×M16C2×D414yes—
233C2×M16C2×D414yes—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—24no—
243C22×C8—44no—
244C22×C8—44no—
245C22×C8—44no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8C22×C412yes—
249C22×C8C22×C412yes—
250C22×C8C2×D412yes—
251C22×C8C2×D412yes—
252C22×C8C2×D412yes—
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C22×C8C2×D412yes—
256C22×C8C2×D412yes—
257C22×C8#4112yes—
258C22×C8#4112yes—
259#123—24no—
260#123—24no—
261C22⋊Q8—24no—
262C22⋊Q8C4⋊C418yes—
263#121—24no—
264#121C4⋊C418yes—
265#120—24no—
266#120—24no—
267C4×D4—28no—
268C4×D4—28no—
269#117—21no—
270#117—21no—
271#117C22×C412yes—
272#117C2×Q812yes—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4C2×D412yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—84no—
293C2×C42—21no—
294#108—216no—
295#108—28no—
296#108—28no—
297#107—42no—
298#107—42no—
299#106—42no—
300#106—42no—
301C4⋊C8—28no—
302#103—44no—
303#103—44no—
304#102—44no—
305#102—44no—
306C22⋊C8—28no—
307C22⋊C8—28no—
308#97—24no—
309C4×C8—24no—
310#95—42no—
311#95—42no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—42no—
318C24—42no—
319C24#11712yes—
320C24#12112yes—
321C24C22⋊Q812yes—
322C24C2×#4111yescharacteristic
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8C2×C4⋊C412yes—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—28no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4C2×C4⋊C412yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4C2×C22⋊C414yes—
407C22×C4C2×C4⋊C414yes—
408C22×C4#11712yes—
409C22×C4#12114yes—
410C22×C4#12112yes—
411C22×C4C22⋊Q814yes—
412C22×C4C22⋊Q812yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427M16—216no—
428M16—216no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—24no—
434M16—24no—
435M16—24no—
436M16—24no—
437M16—416no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—42no—
442M16—42no—
443M16—41no—
444M16—42no—
445M16—42no—
446M16—48no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—416no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—48no—
467C2×C8C2×C22⋊C414yes—
468C2×C8C2×C22⋊C412yes—
469C2×C8C2×C22⋊C412yes—
470C2×C8C4×D414yes—
471C2×C8C4×D414yes—
472C2×C8#12014yes—
473C2×C8#12112yes—
474C2×C8#12112yes—
475C2×C8#12112yes—
476C2×C8#12112yes—
477C2×C8#12314yes—
478C4⋊C4—22no—
479C4⋊C4—28no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4C2×C4⋊C414yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—22no—
496C42—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—44no—
511C23—44no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—82no—
517C23—82no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23#38014yes—
523C23C2×#11711yescharacteristic
524C23C2×#12111yescharacteristic
525C23C2×C22⋊Q811yescharacteristic
526C23#54811yescharacteristic
527C23#54811yescharacteristic
528C23#55311yescharacteristic
529C23#55311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534D4—22no—
535D4—22no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—41no—
541D4—41no—
542D4—88no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—162no—
574C2×C4—162no—
575C2×C4—162no—
576C2×C4—162no—
577C2×C4#38014yes—
578C2×C4C22×C22⋊C411yescharacteristic
579C2×C4C22×C4⋊C411yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4C2×C4×D411yescharacteristic
582C2×C4#52011yescharacteristic
583C2×C4#52011yescharacteristic
584C2×C4C2×#12011yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12111yescharacteristic
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×C22⋊Q811yescharacteristic
589C2×C4C2×#12311yescharacteristic
590C2×C4#53511yescharacteristic
591C2×C4#53611yescharacteristic
592C2×C4#53711yescharacteristic
593C2×C4#53711yescharacteristic
594C2×C4D4211yescharacteristic
595C2×C4D4211yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4#54711yescharacteristic
600C2×C4D4×Q811yescharacteristic
601C2×C4D4×Q811yescharacteristic
602C2×C4#55211yescharacteristic
603C2×C4#55211yescharacteristic
604C2×C4#55411yescharacteristic
605C2×C4#55411yescharacteristic
606C8—24no—
607C8—42no—
608C8—44no—
609C8—44no—
610C8—42no—
611C8#38614yes—
612C8#38614yes—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—81no—
636C22—81no—
637C22C2×#38011yescharacteristic
638C22#235211yescharacteristic
639C22#235211yescharacteristic
640C22#240911yescharacteristic
641C22#241611yescharacteristic
642C22#241611yescharacteristic
643C22#248711yescharacteristic
644C22#306812yes—
645C22#306912yes—
646C4—24no—
647C4—42no—
648C4—44no—
649C4—162no—
650C4—162no—
651C4C2×#38611yescharacteristic
652C4#231111yescharacteristic
653C4D4×C4⋊C411yescharacteristic
654C4D4×C4⋊C411yescharacteristic
655C4#240811yescharacteristic
656C4#241711yescharacteristic
657C4#241711yescharacteristic
658C4#248611yescharacteristic
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—44no—
664C2—82no—
665C2#1387211yescharacteristic
666C2#2067912yes—
667C2#2480512yes—
668C2#2480611yescharacteristic
669C2#2480711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#430C2semidirect
4. C2×#1816C2semidirect
5. C2×#1815C2semidirect
6. C2×#1809C2semidirect
7. #11242C2semidirect
8. #11105C2semidirect
9. #10870C2semidirect
10. #10869C2semidirect
11. #10739C2semidirect
17. #2930C4semidirect
105. C2×#108D4semidirect
106. C2×#108D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29302
C22×#4302
#209712
C2×#18162
C2×#18152
C2×#18092
#112422
#111052
#108702
#108692
#107392