Finite Group Databasebeta

← #153899

Subgroups — #153899

Order 512.

4298 conjugacy classes of subgroups.

The subgroup lattice has 818 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1439C212yesmaximal
4C2×#1437C212yesmaximal
5C2×#1437C212yesmaximal
6C2×#1436C212yesmaximal, complemented
7C2×C4×#95C211yesmaximal, characteristic, complemented
8#7557C212yesmaximal, complemented
9#7554C212yesmaximal, complemented
10C23×C4⋊C4C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C23×C22⋊C4C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2214yes—
15C23×C22⋊C4C2211yescharacteristic
16C23×C42C2212yes—
17C23×C42C2212yes—
18C23×C42C2211yescharacteristic
19C2×#388—24no—
20C2×#388—24no—
21C2×#388C418yescomplemented
22C2×C4×C4⋊C4—24no—
23C2×C4×C4⋊C4—28no—
24C22×#95C2214yes—
25C22×#95C2214yes—
26C22×#95C2214yes—
27C22×#95C2212yes—
28C22×#95C2211yescharacteristic
29C22×#95C2211yescharacteristic
30C22×#95C2212yes—
31C22×#95C2212yes—
32#1439C4116yescomplemented
33#1437—28no—
34#1437—28no—
35#1436—28no—
36C4×#95—24no—
37C26C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2314yes—
42C24×C4C2314yes—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×C4⋊C4—28no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4C2×C418yes—
55C22×C4⋊C4C2×C414yes—
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—28no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—24no—
71C22×C22⋊C4—44no—
72C22×C22⋊C4—44no—
73C22×C22⋊C4C2×C414yes—
74C22×C22⋊C4C2×C414yes—
75C22×C22⋊C4C2×C4116yes—
76C22×C22⋊C4C2×C418yes—
77C22×C22⋊C4C2×C414yes—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42—24no—
81C22×C42—28no—
82C22×C42—24no—
83C22×C42—28no—
84C22×C42—24no—
85C22×C42—24no—
86C22×C42—24no—
87C22×C42—24no—
88C22×C42—44no—
89C22×C42—44no—
90C22×C42C2×C414yes—
91C22×C42D414yes—
92C22×C42D414yes—
93#388—232no—
94C4×C4⋊C4—232no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95—28no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—28no—
105C2×#95—24no—
106C2×#95—24no—
107C2×#95—28no—
108C2×#95—28no—
109C2×#95—28no—
110C2×#95—28no—
111C2×#95—44no—
112C2×#95—44no—
113C2×#95C2×C4116yes—
114C2×#95C2×C418yes—
115C2×#95C2×C414yes—
116C2×#95C2×C414yes—
117C2×#95D418yes—
118C2×#95D418yes—
119C25—21no—
120C25—21no—
121C25—22no—
122C25—22no—
123C25—22no—
124C25—22no—
125C25—22no—
126C25—24no—
127C25—24no—
128C25—22no—
129C25—22no—
130C25—22no—
131C25—22no—
132C25C22×C412yes—
133C25C22×C414yes—
134C25C2411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
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—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C22×C412yes—
193C23×C4C22×C414yes—
194C23×C4C22×C418yes—
195C23×C4C22×C418yes—
196C23×C4C22×C412yes—
197C23×C4C22×C414yes—
198C23×C4C22×C412yes—
199C23×C4C22×C412yes—
200C23×C4C2×D418yes—
201C23×C4C2×D418yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4#4118yes—
211C23×C4#4118yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—232no—
217C2×C4⋊C4—216no—
218C2×C4⋊C4—232no—
219C2×C4⋊C4—216no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—216no—
228C2×C22⋊C4—216no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—232no—
232C2×C22⋊C4—232no—
233C2×C22⋊C4—216no—
234C2×C22⋊C4—216no—
235C2×C22⋊C4—416no—
236C2×C22⋊C4—416no—
237C2×C22⋊C4—416no—
238C2×C22⋊C4—416no—
239C2×C22⋊C4—416no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4C42116yescomplemented
247C2×C42—216no—
248C2×C42—28no—
249C2×C42—216no—
250C2×C42—28no—
251C2×C42—216no—
252C2×C42—216no—
253C2×C42—416no—
254C2×C42—416no—
255C2×C42—416no—
256C2×C42—416no—
257C2×C42—48no—
258C2×C42—48no—
259C2×C42—48no—
260C2×C42—48no—
261C2×C42C22⋊C4116yescomplemented
262#95—28no—
263#95—28no—
264#95—232no—
265#95—216no—
266#95—216no—
267#95—216no—
268#95—416no—
269#95—416no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—28no—
275C24—28no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—28no—
280C24—28no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—41no—
289C24—42no—
290C24—42no—
291C24—41no—
292C24—44no—
293C24—42no—
294C24—42no—
295C24—41no—
296C24—44no—
297C24—44no—
298C24—41no—
299C24—44no—
300C24—42no—
301C24—42no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—44no—
318C24—44no—
319C24—44no—
320C24—44no—
321C24—44no—
322C24—44no—
323C24—44no—
324C24—44no—
325C24—44no—
326C24—44no—
327C24—44no—
328C24C2×C4214yes—
329C24C23×C411yescharacteristic
330C24C23×C412yes—
331C24C22×D411yescharacteristic
332C24C22×D411yescharacteristic
333C24C22×D412yes—
334C24C22×D412yes—
335C24C2×#4112yes—
336C24C2×#4112yes—
337C24C2×#4112yes—
338C24C2×#4114yes—
339C24C2×#4114yes—
340C24H(5,𝔽2)12yes—
341C24H(5,𝔽2)12yes—
342C242−1+412yes—
343C242−1+412yes—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—24no—
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—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—41no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
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—41no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4—42no—
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×C4214yes—
495C22×C4C2×C4214yes—
496C22×C4C2×C22⋊C414yes—
497C22×C4C2×C22⋊C414yes—
498C22×C4C2×C22⋊C414yes—
499C22×C4#11714yes—
500C22×C4C4×D4116yes—
501C22×C4C4×D4116yes—
502C22×C4C4×D418yes—
503C22×C4C4×D418yes—
504C22×C4C4×D418yes—
505C22×C4C4×D418yes—
506C22×C4#12014yes—
507C22×C4#12118yes—
508C22×C4#12118yes—
509C22×C4#12314yes—
510C22×C4#12414yes—
511C22×C4#12714yes—
512C4⋊C4—264no—
513C4⋊C4—416no—
514C22⋊C4—232no—
515C22⋊C4—464no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C42—232no—
519C42—432no—
520C42—432no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—48no—
553C23—48no—
554C23—44no—
555C23—44no—
556C23—82no—
557C23—84no—
558C23—82no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—84no—
563C23—82no—
564C23—81no—
565C23—81no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—82no—
571C23—81no—
572C23—81no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—82no—
578C23—82no—
579C23—82no—
580C23C22×C4211yescharacteristic
581C23C22×C22⋊C411yescharacteristic
582C23C2×#11711yescharacteristic
583C23C2×C4×D412yes—
584C23C2×C4×D414yes—
585C23C2×C4×D412yes—
586C23C2×C4×D414yes—
587C23C2×C4×D412yes—
588C23C2×C4×D412yes—
589C23C4×#4112yes—
590C23C4×#4114yes—
591C23C4×#4114yes—
592C23C4×#4112yes—
593C23#51812yes—
594C23#51912yes—
595C23#52012yes—
596C23#52012yes—
597C23#52012yes—
598C23#52012yes—
599C23C2×#12011yescharacteristic
600C23C2×#12112yes—
601C23C2×#12112yes—
602C23C2×#12311yescharacteristic
603C23#52512yes—
604C23#52512yes—
605C23#52514yes—
606C23C2×#12411yescharacteristic
607C23#52914yes—
608C23#52914yes—
609C23C2×#12711yescharacteristic
610C23#53212yes—
611C23#53214yes—
612C23#53212yes—
613C23#53511yescharacteristic
614C23#53511yescharacteristic
615C23#53611yescharacteristic
616C23#53611yescharacteristic
617C23#53712yes—
618C23#53712yes—
619C23#54012yes—
620C23#54112yes—
621C23#54212yes—
622C23#54212yes—
623C23D4214yes—
624C23#54614yes—
625C23#54614yes—
626C23#54714yes—
627C23#54714yes—
628C23#54814yes—
629C23#54814yes—
630C23#55114yes—
631C23#55114yes—
632C23#55214yes—
633C23#55214yes—
634C23#55314yes—
635C23#55314yes—
636C23#55614yes—
637C23#55614yes—
638C23#55914yes—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—216no—
642C2×C4—216no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—216no—
646C2×C4—28no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—28no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—28no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—28no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—28no—
659C2×C4—216no—
660C2×C4—28no—
661C2×C4—216no—
662C2×C4—216no—
663C2×C4—216no—
664C2×C4—216no—
665C2×C4—24no—
666C2×C4—28no—
667C2×C4—216no—
668C2×C4—216no—
669C2×C4—48no—
670C2×C4—44no—
671C2×C4—48no—
672C2×C4—48no—
673C2×C4—48no—
674C2×C4—48no—
675C2×C4—48no—
676C2×C4—48no—
677C2×C4—48no—
678C2×C4—48no—
679C2×C4—48no—
680C2×C4—48no—
681C2×C4—48no—
682C2×C4—48no—
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—42no—
694C2×C4—44no—
695C2×C4—42no—
696C2×C4—48no—
697C2×C4—48no—
698C2×C4—44no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—48no—
702C2×C4—48no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—42no—
707C2×C4—82no—
708C2×C4—84no—
709C2×C4—81no—
710C2×C4—81no—
711C2×C4—81no—
712C2×C4—81no—
713C2×C4—82no—
714C2×C4—82no—
715C2×C4—82no—
716C2×C4C4×C22⋊C418yes—
717C2×C4#38618yes—
718C2×C4#39018yes—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—416no—
723C22—82no—
724C22—82no—
725C22—88no—
726C22—82no—
727C22—82no—
728C22—84no—
729C22—84no—
730C22—88no—
731C22—88no—
732C22—84no—
733C22—84no—
734C22C2×C4×C22⋊C411yescharacteristic
735C22C42×D412yes—
736C22C42×D412yes—
737C22#227412yes—
738C22C2×#38611yescharacteristic
739C22#229012yes—
740C22C2×#39011yescharacteristic
741C22#229312yes—
742C22#229711yescharacteristic
743C22#229711yescharacteristic
744C22C4×#12012yes—
745C22C4×#12114yes—
746C22C4×#12114yes—
747C22C4×#12312yes—
748C22C4×#12412yes—
749C22C4×#12712yes—
750C22#231111yescharacteristic
751C22#231111yescharacteristic
752C22#231412yes—
753C22#232312yes—
754C22#232312yes—
755C22#232412yes—
756C22#232412yes—
757C22#235814yes—
758C22#235914yes—
759C22#235914yes—
760C22#236114yes—
761C22#236114yes—
762C22#236214yes—
763C22#236214yes—
764C22#236714yes—
765C22#236714yes—
766C22#237314yes—
767C22#237314yes—
768C22#237414yes—
769C22#241112yes—
770C22#241312yes—
771C22#241614yes—
772C22#243012yes—
773C22#243311yescharacteristic
774C22#243611yescharacteristic
775C22#245814yes—
776C22#246114yes—
777C22#246414yes—
778C22#246914yes—
779C22#249812yes—
780C22#250111yescharacteristic
781C22#250411yescharacteristic
782C22#251614yes—
783C22#251914yes—
784C22#252114yes—
785C22#252514yes—
786C22#254012yes—
787C22#254412yes—
788C22#254512yes—
789C22#259212yes—
790C22#260114yes—
791C22#260312yes—
792C22#260412yes—
793C4—28no—
794C4—28no—
795C4—28no—
796C4—416no—
797C4—416no—
798C4—44no—
799C4—82no—
800C4—82no—
801C2—84no—
802C2C4×#38611yescharacteristic
803C2C4×#39011yescharacteristic
804C2#1377711yescharacteristic
805C2#1382314yes—
806C2#1413511yescharacteristic
807C2#1413711yescharacteristic
808C2#1414614yes—
809C2#1415311yescharacteristic
810C2#1416611yescharacteristic
811C2#1418812yes—
812C2#1418912yes—
813C2#1420014yes—
814C2#1420114yes—
815C2#1423112yes—
816C2#1424912yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC23C22×C42
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C4⋊C4C2semidirect
6. C2×#1436C2semidirect
7. C2×C4×#95C2semidirect
8. #7557C2semidirect
9. #7554C2semidirect
21. C2×#388C4semidirect
32. #1439C4semidirect
246. C2×C22⋊C4C42semidirect
261. 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×C4×C4⋊C42
C2×#14392
C2×#14372
C2×#14362
C2×C4×#952
#75572
#75542