Finite Group Databasebeta

← #368334

Subgroups — #368334

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#21258C211yesmaximal, characteristic
3C2×#1980C211yesmaximal, characteristic
4C2×#1860C212yesmaximal, complemented
5C4×#428C211yesmaximal, characteristic, complemented
6#13911C211yesmaximal, characteristic, complemented
7#11113C212yesmaximal, complemented
8#11079C212yesmaximal, complemented
9#10857C211yesmaximal, characteristic, complemented
10#10852C211yesmaximal, characteristic, complemented
11#10781C212yesmaximal, complemented
12C2×#532C2211yescharacteristic
13C2×#520C2212yes—
14C2×#428C2211yescharacteristic
15C2×#428C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C22×#104C2212yes—
18C22×#104C2212yes—
19C2×#417C2212yes—
20C2×#417C2212yes—
21C2×C4×M16C2211yescharacteristic
22#2361—22no—
23#2361—22no—
24#2354—21no—
25#2354—21no—
26D4×C4⋊C4—21no—
27D4×C4⋊C4—21no—
28#2326—22no—
29#2311C2212yes—
30C4×C4⋊Q8—21no—
31C4×#127—21no—
32C2×#384C2211yescharacteristic
33C2×#382C2212yes—
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1983—22no—
38#1982—21no—
39#1981—21no—
40#1980C418yescomplemented
41#1860—28no—
42#1776—21no—
43#1776—21no—
44C4×#107—21no—
45C4×#106—21no—
46C2×#343C2212yes—
47C2×#343C2212yes—
48#1736C2211yescharacteristic
49#1736C2211yescharacteristic
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1442—22no—
53C4⋊#95—22no—
54C4⋊#95C2212yes—
55C22×#41C2312yes—
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59#532—22no—
60#532—22no—
61#532C2×C414yes—
62#532Q814yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×#127D411yescharacteristic
65C2×#124D412yes—
66C2×#121—21no—
67C2×#121—21no—
68#520—24no—
69#520—24no—
70#520—42no—
71#520—42no—
72C2×C4×Q8—22no—
73C2×C4×D4—22no—
74C2×C4×D4—24no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×#117—22no—
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C42—21no—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#428—22no—
93#428—22no—
94#428—22no—
95#428D414yescomplemented
96C2×#107—21no—
97C2×#107—21no—
98C2×#106—21no—
99C2×#106—21no—
100#423—22no—
101#423—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×#104—24no—
105C2×#104—24no—
106C2×#104—42no—
107C2×#104—42no—
108C2×#104C2×C418yescomplemented
109C2×#104C2×C418yescomplemented
110#417—24no—
111#417—24no—
112#417—42no—
113#417—42no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#102—22no—
117C2×#102—22no—
118C4×M16—22no—
119C4×M16C2×C414yescomplemented
120C2×#97—21no—
121C2×C4×C8—21no—
122#391—22no—
123#390—22no—
124#390—22no—
125#390—22no—
126#389—22no—
127#389—22no—
128#388—24no—
129#388—24no—
130#386—24no—
131#385—24no—
132#385—24no—
133#384—21no—
134#384—21no—
135#384—21no—
136#384—21no—
137#382—24no—
138#382—42no—
139#380—22no—
140#380—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2×C414yescomplemented
149C4×C4⋊C4D411yescharacteristic
150C4×C4⋊C4D411yescharacteristic
151C4×C4⋊C4D411yescharacteristic
152C4×C4⋊C4D411yescharacteristic
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155#376—21no—
156C2×#95—21no—
157C2×#95—24no—
158C2×#95—21no—
159C2×#95—24no—
160C43—21no—
161#343—24no—
162#343—42no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#339—24no—
168#339—42no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41C22×C414yes—
180C2×#41C2×Q814yes—
181C22×Q8C2×D412yes—
182C22×D4—22no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C2×D412yes—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—24no—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D414yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2411yescharacteristic
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16C22×C412yes—
210C2×M16C22×C412yes—
211C2×M16C2×D412yes—
212C2×M16#4112yes—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—22no—
216C4⋊Q8—22no—
217C4⋊Q8C4⋊C414yes—
218#127—22no—
219#127C4⋊C414yes—
220#124—24no—
221#124C4⋊C418yes—
222#121—24no—
223#121—24no—
224#121—24no—
225#121—24no—
226C4×Q8—44no—
227C4×Q8—44no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—44no—
233C4×D4—44no—
234C4×D4—48no—
235C4×D4—48no—
236C4×D4—84no—
237C4×D4—84no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—44no—
245#117—44no—
246#117C2×D412yes—
247#117#4112yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4#4112yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—44no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42C22×C414yes—
307C2×C42C22×C412yes—
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×Q812yes—
313C2×C42#4112yes—
314C2×C42#4112yes—
315#107—24no—
316#107—24no—
317#106—24no—
318#106—24no—
319C4⋊C8—42no—
320C4⋊C8—42no—
321#104—216no—
322#104—48no—
323#103—44no—
324#103—44no—
325#102—44no—
326#102—44no—
327#97—24no—
328C4×C8—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—48no—
337#95—41no—
338#95—41no—
339#95—84no—
340C24—22no—
341C24C22×D411yescharacteristic
342#41—24no—
343#41—24no—
344#41—24no—
345#41—24no—
346#41—24no—
347#41—24no—
348#41—24no—
349#41—24no—
350#41—42no—
351#41—42no—
352#41—42no—
353#41—42no—
354#41—42no—
355#41—42no—
356#41—42no—
357#41—42no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8C2×C4⋊C414yes—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4C2×C4⋊C414yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—48no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4C4×D412yes—
423C22×C4C4×D412yes—
424C22×C4#12011yescharacteristic
425C22×C4#12011yescharacteristic
426C22×C4#12011yescharacteristic
427C22×C4#12011yescharacteristic
428C22×C4C22⋊Q812yes—
429C22×C4C22⋊Q812yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×Q811yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C42−1+411yescharacteristic
444M16—22no—
445M16—22no—
446M16—22no—
447M16—22no—
448M16—22no—
449M16—41no—
450M16—41no—
451M16—42no—
452M16—42no—
453M16—42no—
454M16—42no—
455M16C4×D414yescomplemented
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—41no—
488C4⋊C4—44no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4#12112yes—
492C4⋊C4#12112yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—48no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—42no—
507C42—48no—
508C42—42no—
509C42—41no—
510C42—41no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—82no—
516C42—82no—
517C42C2×C4⋊C412yes—
518C42C2×C4⋊C412yes—
519C42C4×D412yes—
520C42C4×D412yes—
521C42C22⋊Q812yes—
522C42C22⋊Q812yes—
523C42#12312yes—
524C42#12312yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C2×C4×D411yescharacteristic
537C23C2×#12011yescharacteristic
538C23C2×C22⋊Q811yescharacteristic
539C23D4211yescharacteristic
540C23#54611yescharacteristic
541C23#54811yescharacteristic
542C23#55011yescharacteristic
543Q8—24no—
544Q8—24no—
545Q8—42no—
546Q8—42no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—44no—
554D4—48no—
555D4—42no—
556D4—42no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—168no—
609C2×C4#38512yes—
610C2×C4#38512yes—
611C2×C4#38512yes—
612C2×C4#38512yes—
613C2×C4#45412yes—
614C2×C4#45512yes—
615C2×C4C22×C4⋊C411yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C4×#4111yescharacteristic
619C2×C4#51811yescharacteristic
620C2×C4#52011yescharacteristic
621C2×C4C2×#12111yescharacteristic
622C2×C4C2×C22⋊Q811yescharacteristic
623C2×C4C2×#12311yescharacteristic
624C2×C4#52511yescharacteristic
625C2×C4#52911yescharacteristic
626C2×C4#53211yescharacteristic
627C2×C4#53511yescharacteristic
628C2×C4#53711yescharacteristic
629C2×C4#54011yescharacteristic
630C2×C4#54211yescharacteristic
631C2×C4D4211yescharacteristic
632C2×C4#54611yescharacteristic
633C2×C4#54611yescharacteristic
634C2×C4#54711yescharacteristic
635C2×C4D4×Q811yescharacteristic
636C2×C4#55011yescharacteristic
637C2×C4#55411yescharacteristic
638C2×C4#55411yescharacteristic
639C2×C4#55411yescharacteristic
640C2×C4#55611yescharacteristic
641C2×C4#55911yescharacteristic
642C2×C4#55911yescharacteristic
643C8—24no—
644C8—42no—
645C8—82no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C2×#38511yescharacteristic
666C22#236611yescharacteristic
667C22#236711yescharacteristic
668C22#240711yescharacteristic
669C22#240811yescharacteristic
670C22#245211yescharacteristic
671C22#245411yescharacteristic
672C22C2×#45511yescharacteristic
673C22C2×#45411yescharacteristic
674C22#301911yescharacteristic
675C22#302011yescharacteristic
676C4—22no—
677C4—22no—
678C4—22no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—84no—
683C4—81no—
684C4—82no—
685C4—81no—
686C4—82no—
687C4—84no—
688C4#186014yescomplemented
689C4C4×#12111yescharacteristic
690C4#232411yescharacteristic
691C4D4×C4⋊C411yescharacteristic
692C4#234911yescharacteristic
693C4#244511yescharacteristic
694C4#244711yescharacteristic
695C4#253811yescharacteristic
696C4#258011yescharacteristic
697C2—21no—
698C2—21no—
699C2—21no—
700C2—21no—
701C2—84no—
702C2#1401811yescharacteristic
703C2C2×#186011yescharacteristic
704C2#2086411yescharacteristic
705C2#2417111yescharacteristic
706C2#2417211yescharacteristic
707C2#2423111yescharacteristic
708C2#2423211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×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
4. C2×#1860C2semidirect
5. C4×#428C2semidirect
6. #13911C2semidirect
7. #11113C2semidirect
8. #11079C2semidirect
9. #10857C2semidirect
10. #10852C2semidirect
11. #10781C2semidirect
40. #1980C4semidirect
95. #428D4semidirect
108. C2×#104C2×C4semidirect
109. C2×#104C2×C4semidirect
119. C4×M16C2×C4semidirect
148. C4×C4⋊C4C2×C4semidirect
455. M16C4×D4semidirect
688. C4#1860semidirect

Maximal subgroups

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

subgroupindex
#342072
#212582
C2×#19802
C2×#18602
C4×#4282
#139112
#111132
#110792
#108572
#108522
#107812