Finite Group Databasebeta

← #154242

Subgroups — #154242

Order 512.

4666 conjugacy classes of subgroups.

The subgroup lattice has 832 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3C22×C4×C22⋊C4C211yesmaximal, characteristic
4C2×#1441C212yesmaximal, complemented
5C2×#1436C212yesmaximal, complemented
6#7557C212yesmaximal, complemented
7#7555C212yesmaximal
8#7555C212yesmaximal
9#7554C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18C23×C42C2212yes—
19C2×#385—24no—
20C2×#385—24no—
21C2×#385C418yescomplemented
22C2×#380—28no—
23C2×#380—24no—
24C2×C4×C22⋊C4—28no—
25C2×C4×C22⋊C4—24no—
26C22×#95C2214yes—
27C22×#95C2214yes—
28C22×#95C2214yes—
29C22×#95C2211yescharacteristic
30C22×#95C2211yescharacteristic
31C22×#95C2212yes—
32C22×#95C2212yes—
33C22×#95C2214yes—
34#1441—28no—
35#1436—28no—
36C26C2311yescharacteristic
37C24×C4—24no—
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—24no—
41C24×C4—44no—
42C24×C4C2×C414yes—
43C24×C4D414yes—
44C24×C4D414yes—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2314yes—
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C24×C4C2314yes—
51C24×C4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C22×C4⋊C4—24no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—28no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—44no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4C2×C414yes—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—28no—
73C22×C22⋊C4—24no—
74C22×C22⋊C4—24no—
75C22×C22⋊C4—28no—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—28no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4C2×C414yes—
85C22×C22⋊C4C2×C414yes—
86C22×C22⋊C4Q818yes—
87C22×C42—28no—
88C22×C42—24no—
89C22×C42—24no—
90C22×C42—24no—
91C22×C42—44no—
92#385—232no—
93#380—232no—
94C4×C22⋊C4—232no—
95C2×#95—28no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—28no—
105C2×#95—28no—
106C2×#95—28no—
107C2×#95—28no—
108C2×#95—28no—
109C2×#95—28no—
110C2×#95—28no—
111C2×#95—44no—
112C2×#95—44no—
113C2×#95—44no—
114C2×#95—44no—
115C2×#95C2×C414yes—
116C2×#95C2×C414yes—
117C2×#95D418yes—
118C2×#95D418yes—
119C25—22no—
120C25—22no—
121C25—22no—
122C25—22no—
123C25—24no—
124C25—22no—
125C25—22no—
126C25—22no—
127C25—24no—
128C25—22no—
129C25C22×C412yes—
130C25C2×D412yes—
131C25C2×D412yes—
132C25C2×Q814yes—
133C25#4114yes—
134C25C2411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—216no—
167C23×C4—216no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—216no—
171C23×C4—216no—
172C23×C4—28no—
173C23×C4—28no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—48no—
202C23×C4—48no—
203C23×C4—48no—
204C23×C4—416no—
205C23×C4—416no—
206C23×C4—416no—
207C23×C4—416no—
208C23×C4C22⋊C4116yes—
209C23×C4C22×C412yes—
210C23×C4C22×C412yes—
211C23×C4C22×C412yes—
212C23×C4C22×C412yes—
213C23×C4C22×C412yes—
214C23×C4C22×C412yes—
215C23×C4C2×D418yes—
216C23×C4C2×D418yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×Q818yes—
224C23×C4#4114yes—
225C23×C4#4114yes—
226C23×C4#4118yes—
227C23×C4#4114yes—
228C23×C4#4114yes—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C22⋊C4—232no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—232no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—416no—
258C2×C22⋊C4—416no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C42—216no—
264C2×C42—416no—
265C2×C42—416no—
266C2×C42—48no—
267C2×C42—48no—
268#95—232no—
269#95—28no—
270#95—28no—
271#95—216no—
272#95—216no—
273#95—416no—
274#95—416no—
275#95—416no—
276C24—22no—
277C24—22no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—28no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—28no—
294C24—28no—
295C24—24no—
296C24—28no—
297C24—24no—
298C24—24no—
299C24—22no—
300C24—22no—
301C24—24no—
302C24—28no—
303C24—28no—
304C24—28no—
305C24—28no—
306C24—28no—
307C24—28no—
308C24—28no—
309C24—28no—
310C24—44no—
311C24—44no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—44no—
318C24—44no—
319C24—44no—
320C24—44no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—44no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24C2×C22⋊C414yes—
331C24C4×Q818yes—
332C24C22⋊Q818yes—
333C24#12414yes—
334C24C4⋊Q814yes—
335C24C23×C411yescharacteristic
336C24C22×D411yescharacteristic
337C24C22×D411yescharacteristic
338C24C22×D412yes—
339C24C22×D412yes—
340C24C22×Q812yes—
341C24C2×#4112yes—
342C24C2×#4112yes—
343C24C2×#4112yes—
344C24C2×#4112yes—
345C24C2×#4112yes—
346C24C2×#4114yes—
347C24H(5,𝔽2)14yes—
348C24H(5,𝔽2)12yes—
349C242−1+412yes—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—232no—
409C22×C4—232no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—416no—
426C22×C4—464no—
427C22×C4—432no—
428C22×C4—432no—
429C22×C4—432no—
430C22×C4—432no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—42no—
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×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4C2×C22⋊C414yes—
516C22×C4C2×C22⋊C414yes—
517C22×C4#11714yes—
518C22×C4#11714yes—
519C22×C4C4×D418yes—
520C22×C4C4×D418yes—
521C22×C4#12014yes—
522C22×C4#12014yes—
523C22×C4#12118yes—
524C22×C4#12118yes—
525C22×C4C22⋊Q8116yes—
526C22×C4#12314yes—
527C22×C4#12314yes—
528C4⋊C4—432no—
529C4⋊C4—416no—
530C4⋊C4—416no—
531C22⋊C4—264no—
532C22⋊C4—232no—
533C22⋊C4—416no—
534C22⋊C4—432no—
535C42—432no—
536C23—24no—
537C23—24no—
538C23—216no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—216no—
548C23—216no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—216no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—44no—
562C23—44no—
563C23—48no—
564C23—48no—
565C23—44no—
566C23—48no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—48no—
581C23—48no—
582C23—48no—
583C23—48no—
584C23—48no—
585C23—48no—
586C23—48no—
587C23—44no—
588C23—48no—
589C23—48no—
590C23—44no—
591C23—44no—
592C23—48no—
593C23—48no—
594C23#39118yes—
595C23C22×C22⋊C411yescharacteristic
596C23C2×#11711yescharacteristic
597C23C2×#11711yescharacteristic
598C23C2×C4×D412yes—
599C23C2×C4×D412yes—
600C23C2×C4×Q812yes—
601C23C4×#4112yes—
602C23#51812yes—
603C23#52012yes—
604C23C2×#12011yescharacteristic
605C23C2×#12011yescharacteristic
606C23C2×#12112yes—
607C23C2×#12112yes—
608C23C2×C22⋊Q812yes—
609C23C2×C22⋊Q814yes—
610C23C2×#12311yescharacteristic
611C23C2×#12311yescharacteristic
612C23#52512yes—
613C23#52512yes—
614C23#52514yes—
615C23#52512yes—
616C23#52514yes—
617C23C2×#12411yescharacteristic
618C23#52914yes—
619C23#52914yes—
620C23#52914yes—
621C23C2×C4⋊Q811yescharacteristic
622C23#53212yes—
623C23#53214yes—
624C23#53314yes—
625C23#53412yes—
626C23#53412yes—
627C23#53511yescharacteristic
628C23#53611yescharacteristic
629C23#53712yes—
630C23#53814yes—
631C23#53914yes—
632C23#54014yes—
633C23#54214yes—
634C23D4214yes—
635C23#54612yes—
636C23#54614yes—
637C23#54614yes—
638C23#54614yes—
639C23#54712yes—
640C23#54812yes—
641C23D4×Q814yes—
642C23#55012yes—
643C23#55114yes—
644C23#55214yes—
645C23#55212yes—
646C23#55314yes—
647C23#55314yes—
648C23#55312yes—
649C23#55414yes—
650C23#55414yes—
651C23#55414yes—
652C23#55512yes—
653C23#55612yes—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—216no—
657C2×C4—216no—
658C2×C4—28no—
659C2×C4—28no—
660C2×C4—28no—
661C2×C4—28no—
662C2×C4—28no—
663C2×C4—28no—
664C2×C4—28no—
665C2×C4—28no—
666C2×C4—24no—
667C2×C4—24no—
668C2×C4—216no—
669C2×C4—28no—
670C2×C4—216no—
671C2×C4—216no—
672C2×C4—24no—
673C2×C4—24no—
674C2×C4—28no—
675C2×C4—464no—
676C2×C4—42no—
677C2×C4—48no—
678C2×C4—44no—
679C2×C4—48no—
680C2×C4—44no—
681C2×C4—48no—
682C2×C4—48no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—48no—
686C2×C4—48no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—48no—
690C2×C4—48no—
691C2×C4—48no—
692C2×C4—48no—
693C2×C4—48no—
694C2×C4—48no—
695C2×C4—48no—
696C2×C4—44no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—44no—
700C2×C4—42no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—48no—
705C2×C4—48no—
706C2×C4—48no—
707C2×C4—48no—
708C2×C4—48no—
709C2×C4—48no—
710C2×C4—48no—
711C2×C4—48no—
712C2×C4—42no—
713C2×C4—42no—
714C2×C4—42no—
715C2×C4—48no—
716C2×C4—84no—
717C2×C4—84no—
718C2×C4—84no—
719C2×C4—84no—
720C2×C4—82no—
721C2×C4—82no—
722C2×C4—82no—
723C2×C4—82no—
724C2×C4—82no—
725C2×C4—82no—
726C2×C4—84no—
727C2×C4—84no—
728C2×C4—84no—
729C2×C4—84no—
730C2×C4—84no—
731C2×C4#38618yes—
732C2×C4#38618yes—
733C22—28no—
734C22—28no—
735C22—28no—
736C22—44no—
737C22—44no—
738C22—44no—
739C22—416no—
740C22—416no—
741C22—416no—
742C22—48no—
743C22—48no—
744C22—416no—
745C22—44no—
746C22—48no—
747C22—48no—
748C22#228011yescharacteristic
749C22C2×#38611yescharacteristic
750C22C2×#38611yescharacteristic
751C22#229012yes—
752C22#229312yes—
753C22C2×#39111yescharacteristic
754C22#229711yescharacteristic
755C22C4×C22⋊Q814yes—
756C22#230811yescharacteristic
757C22#231111yescharacteristic
758C22#233514yes—
759C22#234512yes—
760C22#234512yes—
761C22#235212yes—
762C22#235412yes—
763C22#235814yes—
764C22#236014yes—
765C22#238812yes—
766C22#239214yes—
767C22#239512yes—
768C22#240712yes—
769C22#241012yes—
770C22#241012yes—
771C22#241212yes—
772C22#241212yes—
773C22#241412yes—
774C22#241512yes—
775C22#241714yes—
776C22#243411yescharacteristic
777C22#243511yescharacteristic
778C22#244914yes—
779C22#245014yes—
780C22#245214yes—
781C22#245414yes—
782C22#245614yes—
783C22#246014yes—
784C22#246414yes—
785C22#246514yes—
786C22#246814yes—
787C22#247614yes—
788C22#248512yes—
789C22#250211yescharacteristic
790C22#250311yescharacteristic
791C22#252214yes—
792C22#253014yes—
793C22#255614yes—
794C22#255914yes—
795C22#257414yes—
796C22#257714yes—
797C22#257914yes—
798C22#258312yes—
799C22#259112yes—
800C22#263014yes—
801C22#266112yes—
802C22#266312yes—
803C22#284812yes—
804C22#285012yes—
805C22#285512yes—
806C22#285712yes—
807C4—28no—
808C4—28no—
809C4—44no—
810C4—44no—
811C4—416no—
812C4—82no—
813C4—88no—
814C4—82no—
815C2—48no—
816C2#1383911yescharacteristic
817C2#1387111yescharacteristic
818C2#1392311yescharacteristic
819C2#1398714yes—
820C2#1401114yes—
821C2#1413411yescharacteristic
822C2#1413911yescharacteristic
823C2#1428711yescharacteristic
824C2#1470211yescharacteristic
825C2#1473314yes—
826C2#1476312yes—
827C2#1476412yes—
828C2#1484012yes—
829C2#1484112yes—
830C2#1499714yes—
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
4. C2×#1441C2semidirect
5. C2×#1436C2semidirect
6. #7557C2semidirect
9. #7554C2semidirect
21. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
C22×#3802
C22×C4×C22⋊C42
C2×#14412
C2×#14362
#75572
#75552
#75542