Finite Group Databasebeta

← #359718

Subgroups — #359718

Order 512.

2306 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3C2×#1818C211yesmaximal, characteristic, complemented
4C2×#1788C212yesmaximal
5#11052C211yesmaximal, characteristic, complemented
6#11051C211yesmaximal, characteristic, complemented
7#11050C211yesmaximal, characteristic, complemented
8#11049C211yesmaximal, characteristic, complemented
9#10932C212yesmaximal, complemented
10#10931C212yesmaximal, complemented
11#10730C212yesmaximal, complemented
12C23×#41C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2212yes—
20#2314—28no—
21#2311—24no—
22#2311C418yescomplemented
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#388—22no—
28C2×#387—22no—
29C2×#386—22no—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380—22no—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35#1818—24no—
36#1788—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44C22×#41—22no—
45C22×#41—22no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—22no—
49C22×#41—24no—
50C22×#41—24no—
51C22×#41—42no—
52C22×#41—42no—
53C22×#41C2×C414yes—
54C23×Q8C2311yescharacteristic
55C23×D4—21no—
56C23×D4C2311yescharacteristic
57C24×C4—21no—
58C24×C4—21no—
59C24×C4C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117C2×C414yes—
65C2×#117D414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2×C414yes—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42—22no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—42no—
95C2×#102—42no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99#391—22no—
100#391—22no—
101#391C2×C414yes—
102#391C2×C414yes—
103#390—22no—
104#390—22no—
105#390C2×C414yes—
106#390C2×C414yes—
107#388—216no—
108#387—216no—
109#386—216no—
110#384—24no—
111#382—24no—
112#380—24no—
113#380—216no—
114#380D418yes—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122#340—42no—
123#340—42no—
124#340—42no—
125#340—42no—
126#336—22no—
127#336—22no—
128#336—24no—
129C25—21no—
130C25C2×D411yescharacteristic
131C2×#41—28no—
132C2×#41—28no—
133C2×#41—28no—
134C2×#41—28no—
135C2×#41—216no—
136C2×#41—48no—
137C2×#41—48no—
138C2×#41—48no—
139C2×#41—48no—
140C2×#41—48no—
141C2×#41—48no—
142C2×#41—48no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—48no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22×C412yes—
155C22×Q8#4112yes—
156C22×Q8#4112yes—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4C22×C412yes—
172C22×D4#4112yes—
173C22×D4#4112yes—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—21no—
183C23×C4—24no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—82no—
196C23×C4—82no—
197C23×C4C22×C412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D412yes—
202C23×C4C2×D414yes—
203C23×C4#4114yes—
204C23×C4C2411yescharacteristic
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219#117—216no—
220#117—216no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4C22⋊C418yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4—82no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4C22⋊C418yes—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42—82no—
270C2×C42—82no—
271C2×C42C22×C412yes—
272C2×C42C22×C412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275#103—48no—
276#103—48no—
277#102—48no—
278#102—48no—
279C22⋊C8—48no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24C2×C22⋊C412yes—
305C24#12012yes—
306C24#12312yes—
307C24C22×D411yescharacteristic
308#41—432no—
309#41—432no—
310#41—432no—
311#41—432no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8#11714yes—
328C2×D4—216no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4#11714yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—216no—
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—24no—
388C22×C4—24no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—88no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—88no—
424C22×C4C2×C22⋊C414yes—
425C22×C4C2×C22⋊C414yes—
426C22×C4C2×C22⋊C412yes—
427C22×C4C4×D418yes—
428C22×C4#12012yes—
429C22×C4#12118yes—
430C22×C4#12312yes—
431C22×C4C23×C411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4112yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C4⋊C4—216no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—88no—
470C22⋊C4—216no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—24no—
478C42—88no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—82no—
515C23—82no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23#38614yes—
525C23C22×C22⋊C411yescharacteristic
526C23C2×#12011yescharacteristic
527C23C2×#12311yescharacteristic
528C23#52511yescharacteristic
529C23#52511yescharacteristic
530C23#53511yescharacteristic
531C23#53511yescharacteristic
532C23#57612yes—
533C23#57712yes—
534C23#57712yes—
535C23#57812yes—
536Q8—28no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540D4—28no—
541D4—432no—
542D4—44no—
543D4—44no—
544D4—44no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—432no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
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—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—816no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—82no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4#38614yes—
603C2×C4C2×#11711yescharacteristic
604C2×C4C2×#11711yescharacteristic
605C2×C4C2×C4×D412yes—
606C2×C4#51812yes—
607C2×C4C2×#12112yes—
608C2×C4#52511yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4#53512yes—
611C2×C4#54612yes—
612C2×C4#54612yes—
613C2×C4#54612yes—
614C2×C4#54612yes—
615C2×C4#55112yes—
616C2×C4#55112yes—
617C2×C4#55112yes—
618C2×C4#55112yes—
619C8—82no—
620C8—82no—
621C22—24no—
622C22—28no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—48no—
633C22—44no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22#228011yescharacteristic
644C22C2×#38611yescharacteristic
645C22#230911yescharacteristic
646C22#240411yescharacteristic
647C22#240411yescharacteristic
648C22#248211yescharacteristic
649C22#248211yescharacteristic
650C22#289712yes—
651C22#289812yes—
652C22#301212yes—
653C22#301312yes—
654C22#309812yes—
655C22#309912yes—
656C22#311912yes—
657C22#312012yes—
658C22#312412yes—
659C22#312512yes—
660C22#322011yescharacteristic
661C22#322111yescharacteristic
662C22#322211yescharacteristic
663C22#322311yescharacteristic
664C22#322412yes—
665C22#322512yes—
666C4—24no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—82no—
671C4—82no—
672C4—84no—
673C4—84no—
674C4#234212yes—
675C4#234212yes—
676C4#241112yes—
677C4#241112yes—
678C2—44no—
679C2—84no—
680C2#1386811yescharacteristic
681C2#2074612yes—
682C2#2091512yes—
683C2#2181311yescharacteristic
684C2#2181411yescharacteristic
685C2#2394111yescharacteristic
686C2#2394411yescharacteristic
687C2#2394712yes—
688C2#2759312yes—
689C2#2759511yescharacteristic
690C2#2759711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2314C2semidirect
3. C2×#1818C2semidirect
5. #11052C2semidirect
6. #11051C2semidirect
7. #11050C2semidirect
8. #11049C2semidirect
9. #10932C2semidirect
10. #10931C2semidirect
11. #10730C2semidirect
22. #2311C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#23112
C2×#18182
C2×#17882
#110522
#110512
#110502
#110492
#109322
#109312
#107302