Finite Group Databasebeta

← #154135

Subgroups — #154135

Order 512.

5838 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#381C212yesmaximal
2C22×#380C211yesmaximal, characteristic
3C23×#95C211yesmaximal, characteristic
4C2×C22⋊#95C212yesmaximal
5C2×C22⋊#95C211yesmaximal, characteristic
6#7557C214yesmaximal, complemented
7#7556C214yesmaximal, complemented
8C25×C4C2211yescharacteristic
9C25×C4C2211yescharacteristic
10C25×C4C2211yescharacteristic
11C23×C4⋊C4C2212yes—
12C23×C22⋊C4C2212yes—
13C23×C22⋊C4C2212yes—
14C2×#381—28no—
15C2×#381—28no—
16C2×#381—28no—
17C2×#380—24no—
18C2×#380—28no—
19C22×#95—24no—
20C22×#95—24no—
21C22×#95C418yescomplemented
22C22×#95C2212yes—
23C22×#95C2218yes—
24C22×#95C2212yes—
25C22×#95C2218yes—
26C22×#95C2212yes—
27C22×#95C2212yes—
28C22×#95C2212yes—
29C22⋊#95—24no—
30C22⋊#95—28no—
31C26C2311yescharacteristic
32C24×C4—22no—
33C24×C4—22no—
34C24×C4—22no—
35C24×C4—22no—
36C24×C4—24no—
37C24×C4—24no—
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—22no—
41C24×C4—22no—
42C24×C4—24no—
43C24×C4—24no—
44C24×C4—28no—
45C24×C4—28no—
46C24×C4—24no—
47C24×C4—24no—
48C24×C4—44no—
49C24×C4—44no—
50C24×C4C2×C414yes—
51C24×C4C2×C414yes—
52C24×C4C2×C414yes—
53C24×C4D414yes—
54C24×C4D414yes—
55C24×C4D414yes—
56C24×C4Q814yes—
57C24×C4C2314yes—
58C24×C4C2314yes—
59C24×C4C2311yescharacteristic
60C24×C4C2312yes—
61C24×C4C2312yes—
62C24×C4C2311yescharacteristic
63C22×C4⋊C4—28no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—44no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—24no—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—24no—
73C22×C22⋊C4—24no—
74C22×C22⋊C4—24no—
75C22×C22⋊C4—24no—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4D418yes—
80C22×C22⋊C4D418yes—
81#381—264no—
82#380—232no—
83C2×#95—232no—
84C2×#95—24no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95—28no—
89C2×#95—24no—
90C2×#95—216no—
91C2×#95—216no—
92C2×#95—216no—
93C2×#95—216no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—216no—
97C2×#95—28no—
98C2×#95—48no—
99C2×#95—44no—
100C2×#95C2×C418yes—
101C2×#95C2×C418yes—
102C25—24no—
103C25—24no—
104C25—24no—
105C25—24no—
106C25C22×C412yes—
107C25C2×D412yes—
108C25C2×D412yes—
109C25C2×D412yes—
110C25C2×D414yes—
111C25C2×D414yes—
112C25C2×Q812yes—
113C25#4112yes—
114C25#4112yes—
115C25#4114yes—
116C25#4114yes—
117C25C2411yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—216no—
122C23×C4—216no—
123C23×C4—28no—
124C23×C4—28no—
125C23×C4—28no—
126C23×C4—216no—
127C23×C4—216no—
128C23×C4—216no—
129C23×C4—28no—
130C23×C4—216no—
131C23×C4—216no—
132C23×C4—28no—
133C23×C4—28no—
134C23×C4—28no—
135C23×C4—28no—
136C23×C4—216no—
137C23×C4—216no—
138C23×C4—28no—
139C23×C4—216no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—28no—
158C23×C4—216no—
159C23×C4—216no—
160C23×C4—216no—
161C23×C4—216no—
162C23×C4—216no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—48no—
184C23×C4—48no—
185C23×C4—48no—
186C23×C4—48no—
187C23×C4—432no—
188C23×C4—432no—
189C23×C4—432no—
190C23×C4—432no—
191C23×C4—48no—
192C23×C4—48no—
193C23×C4—44no—
194C23×C4—48no—
195C23×C4—48no—
196C23×C4—48no—
197C23×C4—48no—
198C23×C4—44no—
199C23×C4—48no—
200C23×C4—44no—
201C23×C4—48no—
202C23×C4—44no—
203C23×C4—48no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C22⋊C4116yes—
207C23×C4C4⋊C4116yescomplemented
208C23×C4C22×C412yes—
209C23×C4C22×C414yes—
210C23×C4C22×C414yes—
211C23×C4C22×C412yes—
212C23×C4C2×D418yes—
213C23×C4C2×D418yes—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D414yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×Q814yes—
220C23×C4#4118yes—
221C23×C4#4118yes—
222C2×C4⋊C4—216no—
223C2×C4⋊C4—432no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C22⋊C4—216no—
227C2×C22⋊C4—216no—
228C2×C22⋊C4—216no—
229C2×C22⋊C4—216no—
230C2×C22⋊C4—216no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4—48no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—48no—
240#95—216no—
241#95—264no—
242#95—216no—
243#95—216no—
244#95—432no—
245#95—48no—
246#95—48no—
247C24—24no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—24no—
255C24—28no—
256C24—28no—
257C24—28no—
258C24—28no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24—28no—
264C24—28no—
265C24—28no—
266C24—28no—
267C24—216no—
268C24—216no—
269C24—216no—
270C24—216no—
271C24—216no—
272C24—216no—
273C24—216no—
274C24—216no—
275C24C2×C22⋊C414yes—
276C24C2×C4⋊C414yes—
277C24#11714yes—
278C24C4×D418yes—
279C24C4×D418yes—
280C24#12018yes—
281C24#12014yes—
282C24#12114yes—
283C24#12114yes—
284C24#12118yes—
285C24#12118yes—
286C24#12118yes—
287C24C22⋊Q814yes—
288C24C22⋊Q814yes—
289C24C22⋊Q818yes—
290C24C22⋊Q818yes—
291C24#12318yes—
292C24#12314yes—
293C24#12318yes—
294C24#12418yes—
295C24#12414yes—
296C24#12618yes—
297C24#12714yes—
298C24C23×C411yescharacteristic
299C24C22×D411yescharacteristic
300C24C22×D411yescharacteristic
301C24C22×D411yescharacteristic
302C24C22×D412yes—
303C24C22×D412yes—
304C24C22×Q811yescharacteristic
305C24C2×#4111yescharacteristic
306C24C2×#4111yescharacteristic
307C24C2×#4114yes—
308C24C2×#4114yes—
309C24C2×#4112yes—
310C24C2×#4112yes—
311C24H(5,𝔽2)14yes—
312C242−1+414yes—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—216no—
321C22×C4—216no—
322C22×C4—232no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—22no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—232no—
354C22×C4—264no—
355C22×C4—232no—
356C22×C4—232no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—416no—
368C22×C4—464no—
369C22×C4—432no—
370C22×C4—464no—
371C22×C4—432no—
372C22×C4—464no—
373C22×C4—432no—
374C22×C4—416no—
375C22×C4—464no—
376C22×C4—432no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—416no—
397C22×C4—432no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—432no—
403C22×C4—432no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4C2×C22⋊C414yes—
443C22×C4C2×C22⋊C414yes—
444C22×C4C2×C4⋊C418yes—
445C22×C4#11718yes—
446C22×C4#11718yes—
447C22×C4#12318yes—
448C22×C4#12318yes—
449C4⋊C4—432no—
450C22⋊C4—432no—
451C22⋊C4—432no—
452C23—28no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—216no—
458C23—216no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—216no—
464C23—216no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—216no—
472C23—216no—
473C23—232no—
474C23—232no—
475C23—232no—
476C23—232no—
477C23—232no—
478C23—28no—
479C23—28no—
480C23—216no—
481C23—232no—
482C23—232no—
483C23—232no—
484C23—28no—
485C23—28no—
486C23—216no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23#38018yes—
493C23#385116yes—
494C23#38618yes—
495C23#388116yes—
496C23#39018yes—
497C23#39218yes—
498C23#39318yes—
499C23#394116yes—
500C23#396116yes—
501C23#39718yes—
502C23#39918yes—
503C23C22×C22⋊C411yescharacteristic
504C23C22×C4⋊C411yescharacteristic
505C23C2×#11711yescharacteristic
506C23C2×#11712yes—
507C23C2×#11712yes—
508C23C2×C4×D412yes—
509C23C2×C4×D412yes—
510C23#51812yes—
511C23#51912yes—
512C23C2×#12011yescharacteristic
513C23C2×#12012yes—
514C23C2×#12111yescharacteristic
515C23C2×#12111yescharacteristic
516C23C2×#12112yes—
517C23C2×#12112yes—
518C23C2×#12112yes—
519C23C2×C22⋊Q811yescharacteristic
520C23C2×C22⋊Q811yescharacteristic
521C23C2×C22⋊Q812yes—
522C23C2×C22⋊Q812yes—
523C23C2×#12312yes—
524C23C2×#12312yes—
525C23C2×#12312yes—
526C23C2×#12312yes—
527C23C2×#12311yescharacteristic
528C23#52512yes—
529C23#52512yes—
530C23C2×#12411yescharacteristic
531C23C2×#12412yes—
532C23C2×#12612yes—
533C23#52914yes—
534C23#52914yes—
535C23C2×#12711yescharacteristic
536C23#53214yes—
537C23#53314yes—
538C23#53412yes—
539C23#53412yes—
540C23#53612yes—
541C23#53612yes—
542C23#53714yes—
543C23#53914yes—
544C23#53914yes—
545C23#54414yes—
546C23#54618yes—
547C23#54618yes—
548C23#54718yes—
549C23#54718yes—
550C23#55118yes—
551C23#55118yes—
552C23#55218yes—
553C23#55218yes—
554C2×C4—24no—
555C2×C4—216no—
556C2×C4—28no—
557C2×C4—216no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—216no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—216no—
566C2×C4—24no—
567C2×C4—464no—
568C2×C4—464no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—464no—
582C2×C4—44no—
583C2×C4—416no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—88no—
606C2×C4—88no—
607C2×C4—88no—
608C2×C4—88no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4#38118yescomplemented
612C22—28no—
613C22—28no—
614C22—28no—
615C22—232no—
616C22—232no—
617C22—216no—
618C22—28no—
619C22—232no—
620C22—28no—
621C22—216no—
622C22—232no—
623C22—216no—
624C22—28no—
625C22—216no—
626C22#1439116yes—
627C22C2×#38011yescharacteristic
628C22C2×#38111yescharacteristic
629C22#228011yescharacteristic
630C22#228312yes—
631C22#228512yes—
632C22C2×#38512yes—
633C22C2×#38611yescharacteristic
634C22C2×#38812yes—
635C22C2×#39011yescharacteristic
636C22#230811yescharacteristic
637C22#231011yescharacteristic
638C22#231312yes—
639C22#231612yes—
640C22#234214yes—
641C22#234214yes—
642C22#234414yes—
643C22#234414yes—
644C22C2×#39211yescharacteristic
645C22C2×#39311yescharacteristic
646C22C2×#39412yes—
647C22C2×#39612yes—
648C22C2×#39711yescharacteristic
649C22C2×#39911yescharacteristic
650C22#240512yes—
651C22#240612yes—
652C22#241214yes—
653C22#241314yes—
654C22#241918yes—
655C22#242518yes—
656C22#243018yes—
657C22#243818yes—
658C22#243818yes—
659C22#248312yes—
660C22#248412yes—
661C22#248614yes—
662C22#248818yes—
663C22#249318yes—
664C22#249818yes—
665C22#250112yes—
666C22#250312yes—
667C22#253414yes—
668C22#253912yes—
669C22#256112yes—
670C22#261812yes—
671C22#261812yes—
672C22#262712yes—
673C22#262712yes—
674C22#285412yes—
675C22#285412yes—
676C22#285512yes—
677C22#285512yes—
678C4—28no—
679C4—48no—
680C4—44no—
681C4—48no—
682C4—88no—
683C4—88no—
684C2—216no—
685C2C2×#143911yescharacteristic
686C2#1386911yescharacteristic
687C2#1387011yescharacteristic
688C2#1387414yes—
689C2#1387814yes—
690C2#1390311yescharacteristic
691C2#1390411yescharacteristic
692C2#1433911yescharacteristic
693C2#1468511yescharacteristic
694C2#1492612yes—
695C2#1492714yes—
696C2#1493412yes—
697C2#1493514yes—
698C2#1493612yes—
699C2#1501912yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC24C23×C4
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #7557C2semidirect
7. #7556C2semidirect
21. C22×#95C4semidirect
207. C23×C4C4⋊C4semidirect
611. C2×C4#381semidirect

Maximal subgroups

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

subgroupindex
C22×#3812
C22×#3802
C23×#952
C2×C22⋊#952
#75572
#75562