Finite Group Databasebeta

← #154144

Subgroups — #154144

Order 512.

4226 conjugacy classes of subgroups.

The subgroup lattice has 813 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#381C211yesmaximal, characteristic
2C22×C4×C22⋊C4C211yesmaximal, characteristic
3C2×#1434C212yesmaximal, complemented
4C2×C22⋊#95C211yesmaximal, characteristic
5#7556C212yesmaximal, complemented
6#7556C212yesmaximal, complemented
7#7555C212yesmaximal
8#7555C212yesmaximal
9#7554C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C23×C22⋊C4C2211yescharacteristic
12C23×C22⋊C4C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14C23×C22⋊C4C2212yes—
15C23×C42C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#381—28no—
18C2×#381—24no—
19C2×C4×C22⋊C4—24no—
20C2×C4×C22⋊C4—24no—
21C2×C4×C22⋊C4C418yescomplemented
22C22×#95C2212yes—
23C22×#95C2214yes—
24C22×#95C2212yes—
25C22×#95C2214yes—
26C22×#95C2214yes—
27C22×#95C2211yescharacteristic
28C22×#95C2211yescharacteristic
29C22×#95C2212yes—
30C22×#95C2214yes—
31C22×#95C2212yes—
32#1434—28no—
33C22⋊#95—24no—
34C26C2311yescharacteristic
35C24×C4—24no—
36C24×C4—24no—
37C24×C4—24no—
38C24×C4—24no—
39C24×C4—44no—
40C24×C4C2×C414yes—
41C24×C4D414yes—
42C24×C4D414yes—
43C24×C4C2314yes—
44C24×C4C2314yes—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—24no—
57C22×C22⋊C4—24no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—24no—
60C22×C22⋊C4—24no—
61C22×C22⋊C4—28no—
62C22×C22⋊C4—28no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—24no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—44no—
68C22×C22⋊C4—44no—
69C22×C22⋊C4C2×C414yes—
70C22×C22⋊C4C2×C414yes—
71C22×C42—24no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—24no—
77C22×C42—44no—
78C22×C42C2×C414yes—
79C22×C42C2×C414yes—
80#381—232no—
81C4×C22⋊C4—232no—
82C2×#95—28no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—28no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95—24no—
91C2×#95—28no—
92C2×#95—28no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—28no—
96C2×#95—24no—
97C2×#95—28no—
98C2×#95—28no—
99C2×#95—28no—
100C2×#95—28no—
101C2×#95—24no—
102C2×#95—44no—
103C2×#95—44no—
104C2×#95—44no—
105C2×#95—44no—
106C2×#95—44no—
107C2×#95C2×C414yes—
108C2×#95C2×C414yes—
109C2×#95D418yes—
110C25—22no—
111C25—22no—
112C25—22no—
113C25—22no—
114C25—22no—
115C25—24no—
116C25—22no—
117C25—22no—
118C25—24no—
119C25—22no—
120C25C22×C412yes—
121C25C2×D412yes—
122C25C2×D412yes—
123C25#4114yes—
124C25#4114yes—
125C25C2411yescharacteristic
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—28no—
130C23×C4—28no—
131C23×C4—216no—
132C23×C4—216no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—216no—
163C23×C4—28no—
164C23×C4—216no—
165C23×C4—28no—
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—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—48no—
195C23×C4—48no—
196C23×C4—48no—
197C23×C4—416no—
198C23×C4—416no—
199C23×C4—416no—
200C23×C4—416no—
201C23×C4C22⋊C4116yes—
202C23×C4C22×C412yes—
203C23×C4C22×C412yes—
204C23×C4C22×C412yes—
205C23×C4C22×C412yes—
206C23×C4C22×C412yes—
207C23×C4C22×C412yes—
208C23×C4C2×D418yes—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D414yes—
214C23×C4#4118yes—
215C23×C4#4114yes—
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4#4114yes—
219C23×C4#4114yes—
220C2×C22⋊C4—216no—
221C2×C22⋊C4—216no—
222C2×C22⋊C4—216no—
223C2×C22⋊C4—216no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—216no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—416no—
233C2×C22⋊C4—416no—
234C2×C22⋊C4—416no—
235C2×C22⋊C4—416no—
236C2×C22⋊C4—48no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—48no—
240C2×C42—28no—
241C2×C42—28no—
242C2×C42—216no—
243C2×C42—48no—
244C2×C42—48no—
245C2×C42—48no—
246C2×C42—44no—
247C2×C42—48no—
248C2×C42—44no—
249#95—232no—
250#95—216no—
251#95—28no—
252#95—28no—
253#95—216no—
254#95—416no—
255#95—416no—
256#95—48no—
257#95—416no—
258#95—48no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—22no—
268C24—24no—
269C24—22no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—28no—
276C24—28no—
277C24—28no—
278C24—28no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—28no—
286C24—28no—
287C24—28no—
288C24—28no—
289C24—28no—
290C24—28no—
291C24—28no—
292C24—28no—
293C24—44no—
294C24—44no—
295C24—44no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24—44no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—44no—
306C24—42no—
307C24—44no—
308C24—42no—
309C24—44no—
310C24—42no—
311C24—42no—
312C24—44no—
313C24C2×C22⋊C414yes—
314C24#11714yes—
315C24#11714yes—
316C24#12318yes—
317C24#12318yes—
318C24C23×C411yescharacteristic
319C24C22×D411yescharacteristic
320C24C22×D411yescharacteristic
321C24C22×D412yes—
322C24C2×#4112yes—
323C24C2×#4112yes—
324C24C2×#4112yes—
325C24C2×#4112yes—
326C24C2×#4112yes—
327C24C2×#4112yes—
328C24C2×#4112yes—
329C24C2×#4114yes—
330C24H(5,𝔽2)14yes—
331C24H(5,𝔽2)12yes—
332C242−1+412yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—232no—
390C22×C4—232no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—416no—
406C22×C4—432no—
407C22×C4—432no—
408C22×C4—432no—
409C22×C4—464no—
410C22×C4—432no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4C2×C22⋊C414yes—
502C22×C4C2×C22⋊C414yes—
503C22×C4#11714yes—
504C22×C4#11714yes—
505C22×C4#11714yes—
506C22×C4#11714yes—
507C22×C4C4×D418yes—
508C22×C4#12014yes—
509C22×C4#12118yes—
510C22×C4#12314yes—
511C22⋊C4—232no—
512C22⋊C4—232no—
513C22⋊C4—432no—
514C22⋊C4—432no—
515C42—416no—
516C42—416no—
517C23—24no—
518C23—24no—
519C23—216no—
520C23—216no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—216no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—216no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—48no—
540C23—48no—
541C23—44no—
542C23—44no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—48no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—48no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—44no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—44no—
570C23—44no—
571C23—48no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23#38118yes—
576C23C22×C22⋊C411yescharacteristic
577C23C2×#11711yescharacteristic
578C23C2×#11711yescharacteristic
579C23C2×#11711yescharacteristic
580C23C2×#11711yescharacteristic
581C23C2×#11711yescharacteristic
582C23C2×#11711yescharacteristic
583C23C2×C4×D412yes—
584C23C4×#4112yes—
585C23#51812yes—
586C23#52012yes—
587C23C2×#12011yescharacteristic
588C23C2×#12112yes—
589C23C2×#12312yes—
590C23C2×#12312yes—
591C23C2×#12311yescharacteristic
592C23#52512yes—
593C23#52512yes—
594C23#52512yes—
595C23#52512yes—
596C23#52512yes—
597C23#52514yes—
598C23#52914yes—
599C23#52914yes—
600C23#52914yes—
601C23#52914yes—
602C23#52914yes—
603C23#52914yes—
604C23#53212yes—
605C23#53214yes—
606C23#53412yes—
607C23#53412yes—
608C23#53511yescharacteristic
609C23#53611yescharacteristic
610C23#53712yes—
611C23#53814yes—
612C23#53914yes—
613C23#54014yes—
614C23#54214yes—
615C23#54614yes—
616C23#54614yes—
617C23#54612yes—
618C23#54712yes—
619C23#54812yes—
620C23#55012yes—
621C23#55114yes—
622C23#55114yes—
623C23#55114yes—
624C23#55114yes—
625C23#55114yes—
626C23#55214yes—
627C23#55212yes—
628C23#55214yes—
629C23#55314yes—
630C23#55312yes—
631C23#55314yes—
632C23#55512yes—
633C23#55612yes—
634C23#55914yes—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—216no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—28no—
653C2×C4—464no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—48no—
662C2×C4—48no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—48no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—48no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—48no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—48no—
684C2×C4—48no—
685C2×C4—48no—
686C2×C4—48no—
687C2×C4—48no—
688C2×C4—48no—
689C2×C4—48no—
690C2×C4—48no—
691C2×C4—48no—
692C2×C4—48no—
693C2×C4—48no—
694C2×C4—48no—
695C2×C4—48no—
696C2×C4—48no—
697C2×C4—84no—
698C2×C4—84no—
699C2×C4—84no—
700C2×C4—84no—
701C2×C4—84no—
702C2×C4—84no—
703C2×C4—84no—
704C2×C4—84no—
705C2×C4—84no—
706C2×C4—84no—
707C2×C4—84no—
708C2×C4—84no—
709C2×C4—84no—
710C2×C4—84no—
711C2×C4—84no—
712C2×C4—84no—
713C2×C4#38618yes—
714C22—28no—
715C22—28no—
716C22—28no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—48no—
721C22—48no—
722C22—48no—
723C22—48no—
724C22—44no—
725C22—416no—
726C22—416no—
727C22—416no—
728C22—416no—
729C22C2×#38111yescharacteristic
730C22#228011yescharacteristic
731C22#228011yescharacteristic
732C22#228311yescharacteristic
733C22#228311yescharacteristic
734C22#228311yescharacteristic
735C22#228311yescharacteristic
736C22C2×#38611yescharacteristic
737C22#229012yes—
738C22#229312yes—
739C22#229711yescharacteristic
740C22#230811yescharacteristic
741C22#231111yescharacteristic
742C22#231812yes—
743C22#231912yes—
744C22#234212yes—
745C22#234212yes—
746C22#234312yes—
747C22#234312yes—
748C22#235212yes—
749C22#235312yes—
750C22#235412yes—
751C22#235712yes—
752C22#238812yes—
753C22#239214yes—
754C22#239512yes—
755C22#240512yes—
756C22#240712yes—
757C22#241214yes—
758C22#241412yes—
759C22#241512yes—
760C22#245714yes—
761C22#245814yes—
762C22#247014yes—
763C22#247114yes—
764C22#248212yes—
765C22#248312yes—
766C22#248412yes—
767C22#248512yes—
768C22#249814yes—
769C22#251514yes—
770C22#251614yes—
771C22#252614yes—
772C22#252714yes—
773C22#255314yes—
774C22#255414yes—
775C22#255714yes—
776C22#255814yes—
777C22#257214yes—
778C22#257314yes—
779C22#257514yes—
780C22#257614yes—
781C22#260014yes—
782C22#260214yes—
783C22#263014yes—
784C22#263512yes—
785C22#263812yes—
786C22#285314yes—
787C22#285314yes—
788C4—28no—
789C4—44no—
790C4—44no—
791C4—44no—
792C4—44no—
793C4—44no—
794C4—88no—
795C4—88no—
796C2—48no—
797C2#1383911yescharacteristic
798C2#1386911yescharacteristic
799C2#1387111yescharacteristic
800C2#1388712yes—
801C2#1389512yes—
802C2#1391312yes—
803C2#1391712yes—
804C2#1394011yescharacteristic
805C2#1395811yescharacteristic
806C2#1434411yescharacteristic
807C2#1468411yescharacteristic
808C2#1496414yes—
809C2#1496514yes—
810C2#1497414yes—
811C2#1504314yes—
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
3. C2×#1434C2semidirect
5. #7556C2semidirect
6. #7556C2semidirect
9. #7554C2semidirect
21. C2×C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3812
C22×C4×C22⋊C42
C2×#14342
C2×C22⋊#952
#75562
#75552
#75542