Finite Group Databasebeta

← #154335

Subgroups — #154335

Order 512.

3846 conjugacy classes of subgroups.

The subgroup lattice has 812 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic, complemented
3C2×#1441C212yesmaximal, complemented
4C2×#1439C212yesmaximal
5C2×#1437C212yesmaximal
6C2×C4×#95C211yesmaximal, characteristic, complemented
7#7556C212yesmaximal, complemented
8#7555C212yesmaximal
9#7554C212yesmaximal, complemented
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2212yes—
12C23×C22⋊C4C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2214yes—
16C23×C42C2212yes—
17C23×C42C2211yescharacteristic
18C2×#388—24no—
19C2×#388—24no—
20C2×#388C418yescomplemented
21C2×#382—24no—
22C2×#382—28no—
23C22×#95C2214yes—
24C22×#95C2212yes—
25C22×#95C2212yes—
26C22×#95C2214yes—
27C22×#95C2214yes—
28C22×#95C2211yescharacteristic
29C22×#95C2211yescharacteristic
30C22×#95C2212yes—
31C22×#95C2212yes—
32#1441—28no—
33#1439—28no—
34#1437—28no—
35C4×#95—24no—
36C26C2311yescharacteristic
37C24×C4C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2314yes—
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2314yes—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C22×C4⋊C4—24no—
46C22×C4⋊C4—28no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—44no—
54C22×C4⋊C4C2×C414yes—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—28no—
60C22×C22⋊C4—28no—
61C22×C22⋊C4—24no—
62C22×C22⋊C4—24no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—28no—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—28no—
71C22×C22⋊C4—28no—
72C22×C22⋊C4—44no—
73C22×C22⋊C4—44no—
74C22×C22⋊C4C2×C414yes—
75C22×C22⋊C4C2×C414yes—
76C22×C22⋊C4C2×C414yes—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42—28no—
81C22×C42—24no—
82C22×C42—24no—
83C22×C42—44no—
84C22×C42C2×C414yes—
85C22×C42D414yes—
86C22×C42D414yes—
87#388—232no—
88#382—232no—
89C2×#95—24no—
90C2×#95—24no—
91C2×#95—28no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—28no—
100C2×#95—24no—
101C2×#95—28no—
102C2×#95—28no—
103C2×#95—28no—
104C2×#95—28no—
105C2×#95—28no—
106C2×#95—28no—
107C2×#95—28no—
108C2×#95—44no—
109C2×#95—44no—
110C2×#95C2×C414yes—
111C2×#95C2×C414yes—
112C2×#95D418yes—
113C2×#95D418yes—
114C25—21no—
115C25—21no—
116C25—22no—
117C25—22no—
118C25—22no—
119C25—22no—
120C25—22no—
121C25—24no—
122C25—24no—
123C25—22no—
124C25—22no—
125C25—22no—
126C25—22no—
127C25C22×C412yes—
128C25#4114yes—
129C25C2411yescharacteristic
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
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—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4C22×C412yes—
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C22×C412yes—
193C23×C4C22×C412yes—
194C23×C4C22×C412yes—
195C23×C4C2×D418yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D418yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4#4114yes—
206C23×C4#4118yes—
207C23×C4#4118yes—
208C23×C4#4114yes—
209C23×C4#4114yes—
210C2×C4⋊C4—216no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—216no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—416no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—44no—
220C2×C22⋊C4—216no—
221C2×C22⋊C4—216no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—232no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—216no—
229C2×C22⋊C4—216no—
230C2×C22⋊C4—416no—
231C2×C22⋊C4—416no—
232C2×C22⋊C4—416no—
233C2×C22⋊C4—416no—
234C2×C22⋊C4—416no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—48no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—416no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C42—216no—
243C2×C42—28no—
244C2×C42—216no—
245C2×C42—28no—
246C2×C42—216no—
247C2×C42—416no—
248C2×C42—416no—
249C2×C42—48no—
250C2×C42—48no—
251C2×C42C22⋊C4116yescomplemented
252#95—216no—
253#95—216no—
254#95—28no—
255#95—28no—
256#95—216no—
257#95—216no—
258#95—416no—
259#95—416no—
260#95—416no—
261C24—22no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—28no—
272C24—28no—
273C24—24no—
274C24—24no—
275C24—28no—
276C24—28no—
277C24—24no—
278C24—24no—
279C24—41no—
280C24—42no—
281C24—42no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24—41no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—41no—
291C24—42no—
292C24—44no—
293C24—42no—
294C24—44no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—44no—
302C24—44no—
303C24—44no—
304C24—44no—
305C24—44no—
306C24—44no—
307C24—42no—
308C24—44no—
309C24—44no—
310C24—42no—
311C24—44no—
312C24—44no—
313C24—42no—
314C24—44no—
315C24—44no—
316C24—42no—
317C24—44no—
318C24—44no—
319C24#11714yes—
320C24C23×C411yescharacteristic
321C24C22×D411yescharacteristic
322C24C22×D411yescharacteristic
323C24C22×D412yes—
324C24C22×D412yes—
325C24C2×#4112yes—
326C24C2×#4112yes—
327C24C2×#4112yes—
328C24C2×#4112yes—
329C24C2×#4114yes—
330C24C2×#4114yes—
331C24H(5,𝔽2)12yes—
332C24H(5,𝔽2)12yes—
333C242−1+412yes—
334C242−1+412yes—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—41no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—41no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—41no—
459C22×C4—44no—
460C22×C4—41no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—42no—
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×C4C2×C22⋊C414yes—
495C22×C4C2×C22⋊C414yes—
496C22×C4C2×C22⋊C414yes—
497C22×C4#11714yes—
498C22×C4C4×D418yes—
499C22×C4C4×D418yes—
500C22×C4#12014yes—
501C22×C4#12118yes—
502C22×C4#12118yes—
503C22×C4#12314yes—
504C22×C4#124116yes—
505C22×C4#12414yes—
506C22×C4#12714yes—
507C4⋊C4—432no—
508C4⋊C4—416no—
509C22⋊C4—232no—
510C22⋊C4—464no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C42—232no—
514C42—432no—
515C23—28no—
516C23—28no—
517C23—24no—
518C23—28no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—48no—
523C23—48no—
524C23—44no—
525C23—44no—
526C23—48no—
527C23—44no—
528C23—44no—
529C23—48no—
530C23—48no—
531C23—44no—
532C23—48no—
533C23—44no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—48no—
549C23—48no—
550C23—84no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23—82no—
558C23—82no—
559C23—84no—
560C23—82no—
561C23—81no—
562C23—81no—
563C23—84no—
564C23—84no—
565C23—82no—
566C23—81no—
567C23—81no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23C22×C22⋊C411yescharacteristic
575C23C2×#11711yescharacteristic
576C23C2×#11711yescharacteristic
577C23C2×C4×D412yes—
578C23C2×C4×D412yes—
579C23C4×#4112yes—
580C23C4×#4112yes—
581C23#52012yes—
582C23#52012yes—
583C23C2×#12011yescharacteristic
584C23C2×#12112yes—
585C23C2×#12112yes—
586C23C2×#12311yescharacteristic
587C23#52512yes—
588C23#52512yes—
589C23#52512yes—
590C23#52512yes—
591C23#52514yes—
592C23C2×#12411yescharacteristic
593C23C2×#12414yes—
594C23#52914yes—
595C23#52914yes—
596C23#52914yes—
597C23#52914yes—
598C23C2×#12711yescharacteristic
599C23#53214yes—
600C23#53212yes—
601C23#53212yes—
602C23#53214yes—
603C23#53511yescharacteristic
604C23#53511yescharacteristic
605C23#53611yescharacteristic
606C23#53611yescharacteristic
607C23#53712yes—
608C23#53712yes—
609C23#53814yes—
610C23#53914yes—
611C23#54114yes—
612C23#54214yes—
613C23D4214yes—
614C23#54612yes—
615C23#54612yes—
616C23#54614yes—
617C23#54714yes—
618C23#54714yes—
619C23#54812yes—
620C23#54812yes—
621C23#54814yes—
622C23#55114yes—
623C23#55114yes—
624C23#55212yes—
625C23#55212yes—
626C23#55214yes—
627C23#55214yes—
628C23#55214yes—
629C23#55312yes—
630C23#55312yes—
631C23#55314yes—
632C23#55514yes—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—216no—
636C2×C4—216no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—216no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—216no—
649C2×C4—28no—
650C2×C4—216no—
651C2×C4—216no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—48no—
659C2×C4—44no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—44no—
663C2×C4—48no—
664C2×C4—48no—
665C2×C4—44no—
666C2×C4—48no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—48no—
670C2×C4—48no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—48no—
678C2×C4—48no—
679C2×C4—48no—
680C2×C4—42no—
681C2×C4—48no—
682C2×C4—48no—
683C2×C4—48no—
684C2×C4—48no—
685C2×C4—48no—
686C2×C4—48no—
687C2×C4—48no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—48no—
694C2×C4—48no—
695C2×C4—84no—
696C2×C4—84no—
697C2×C4—82no—
698C2×C4—82no—
699C2×C4—84no—
700C2×C4—84no—
701C2×C4—84no—
702C2×C4—81no—
703C2×C4—81no—
704C2×C4—84no—
705C2×C4—84no—
706C2×C4—84no—
707C2×C4—81no—
708C2×C4—81no—
709C2×C4—84no—
710C2×C4—84no—
711C2×C4#38618yes—
712C2×C4#39018yes—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—416no—
717C22—82no—
718C22—82no—
719C22—88no—
720C22—84no—
721C22—84no—
722C22—82no—
723C22—84no—
724C22—84no—
725C22—88no—
726C22—82no—
727C22—88no—
728C22#228011yescharacteristic
729C22C2×#38611yescharacteristic
730C22#229012yes—
731C22C2×#39011yescharacteristic
732C22#229312yes—
733C22#229711yescharacteristic
734C22#229711yescharacteristic
735C22C4×#12414yes—
736C22#231111yescharacteristic
737C22#231111yescharacteristic
738C22#233414yes—
739C22#234312yes—
740C22#234612yes—
741C22#235212yes—
742C22#235212yes—
743C22#235914yes—
744C22#236114yes—
745C22#238812yes—
746C22#239214yes—
747C22#239512yes—
748C22#241112yes—
749C22#241312yes—
750C22#241614yes—
751C22#243012yes—
752C22#243311yescharacteristic
753C22#243611yescharacteristic
754C22#246014yes—
755C22#246114yes—
756C22#247214yes—
757C22#247514yes—
758C22#248814yes—
759C22#248914yes—
760C22#249114yes—
761C22#249812yes—
762C22#250111yescharacteristic
763C22#250411yescharacteristic
764C22#251514yes—
765C22#251814yes—
766C22#251914yes—
767C22#251914yes—
768C22#252914yes—
769C22#253512yes—
770C22#253612yes—
771C22#254012yes—
772C22#255314yes—
773C22#255514yes—
774C22#255814yes—
775C22#256212yes—
776C22#257614yes—
777C22#260914yes—
778C22#261012yes—
779C22#261112yes—
780C22#263612yes—
781C22#263712yes—
782C22#266212yes—
783C22#266412yes—
784C22#285014yes—
785C22#285812yes—
786C22#286112yes—
787C4—28no—
788C4—28no—
789C4—44no—
790C4—416no—
791C4—44no—
792C4—82no—
793C4—88no—
794C4—82no—
795C2—84no—
796C2#1383911yescharacteristic
797C2#1390511yescharacteristic
798C2#1395711yescharacteristic
799C2#1398614yes—
800C2#1401014yes—
801C2#1413511yescharacteristic
802C2#1413711yescharacteristic
803C2#1429111yescharacteristic
804C2#1470811yescharacteristic
805C2#1492214yes—
806C2#1492812yes—
807C2#1493312yes—
808C2#1502012yes—
809C2#1502512yes—
810C2#1507414yes—
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
2. C22×#382C2semidirect
3. C2×#1441C2semidirect
6. C2×C4×#95C2semidirect
7. #7556C2semidirect
9. #7554C2semidirect
20. C2×#388C4semidirect
251. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
C22×#3822
C2×#14412
C2×#14392
C2×#14372
C2×C4×#952
#75562
#75552
#75542