Finite Group Databasebeta

← #154100

Subgroups — #154100

Order 512.

5650 conjugacy classes of subgroups.

The subgroup lattice has 852 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#380C211yesmaximal, characteristic
2C23×#95C211yesmaximal, characteristic
3C2×#1442C212yesmaximal, complemented
4C2×C22⋊#95C212yesmaximal
5C2×C22⋊#95C212yesmaximal
6C2×C22⋊#95C211yesmaximal, characteristic
7#7557C212yesmaximal, complemented
8#7556C214yesmaximal, complemented
9C25×C4C2212yes—
10C25×C4C2211yescharacteristic
11C23×C4⋊C4C2212yes—
12C23×C22⋊C4C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14C23×C22⋊C4C2211yescharacteristic
15C2×#380—24no—
16C2×#380—28no—
17C22×#95—28no—
18C22×#95C418yescomplemented
19C22×#95C2212yes—
20C22×#95C2212yes—
21C22×#95C2214yes—
22C22×#95C2214yes—
23C22×#95C2214yes—
24C22×#95C2214yes—
25C22×#95C2212yes—
26C22×#95C2211yescharacteristic
27C22×#95C2212yes—
28C22×#95C2211yescharacteristic
29#1442—28no—
30C22⋊#95—24no—
31C22⋊#95—28no—
32C22⋊#95—28no—
33C26C2311yescharacteristic
34C24×C4—24no—
35C24×C4—24no—
36C24×C4—24no—
37C24×C4—24no—
38C24×C4—28no—
39C24×C4—24no—
40C24×C4—28no—
41C24×C4—24no—
42C24×C4—24no—
43C24×C4—24no—
44C24×C4—24no—
45C24×C4—24no—
46C24×C4—44no—
47C24×C4—44no—
48C24×C4—44no—
49C24×C4C2×C418yes—
50C24×C4C2×C414yes—
51C24×C4D414yes—
52C24×C4Q814yes—
53C24×C4C2314yes—
54C24×C4C2312yes—
55C24×C4C2312yes—
56C24×C4C2312yes—
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2311yescharacteristic
60C22×C4⋊C4—28no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—44no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—24no—
71C22×C22⋊C4—24no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—24no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—24no—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4D414yes—
80C22×C22⋊C4D418yes—
81C22×C22⋊C4Q814yes—
82#380—232no—
83C2×#95—28no—
84C2×#95—24no—
85C2×#95—28no—
86C2×#95—24no—
87C2×#95—232no—
88C2×#95—28no—
89C2×#95—28no—
90C2×#95—28no—
91C2×#95—28no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—28no—
95C2×#95—28no—
96C2×#95—24no—
97C2×#95—28no—
98C2×#95—28no—
99C2×#95—24no—
100C2×#95—44no—
101C2×#95—44no—
102C2×#95—44no—
103C2×#95—44no—
104C2×#95—44no—
105C2×#95C2×C418yes—
106C2×#95C2×C414yes—
107C2×#95C2×C414yes—
108C2×#95D418yes—
109C25—22no—
110C25—22no—
111C25—24no—
112C25—24no—
113C25—22no—
114C25—22no—
115C25C22×C412yes—
116C25C2×D412yes—
117C25C2×D414yes—
118C25C2×D412yes—
119C25C2×Q812yes—
120C25C2×Q812yes—
121C25#4114yes—
122C25#4114yes—
123C25#4114yes—
124C25#4112yes—
125C25#4112yes—
126C25C2411yescharacteristic
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—232no—
131C23×C4—232no—
132C23×C4—216no—
133C23×C4—216no—
134C23×C4—28no—
135C23×C4—216no—
136C23×C4—28no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—216no—
161C23×C4—28no—
162C23×C4—216no—
163C23×C4—28no—
164C23×C4—216no—
165C23×C4—216no—
166C23×C4—216no—
167C23×C4—216no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—48no—
203C23×C4—48no—
204C23×C4—48no—
205C23×C4—48no—
206C23×C4—48no—
207C23×C4—416no—
208C23×C4—416no—
209C23×C4—416no—
210C23×C4—416no—
211C23×C4—416no—
212C23×C4—416no—
213C23×C4—416no—
214C23×C4—416no—
215C23×C4—416no—
216C23×C4—416no—
217C23×C4—416no—
218C23×C4—416no—
219C23×C4—416no—
220C23×C4—416no—
221C23×C4—48no—
222C23×C4—48no—
223C23×C4—48no—
224C23×C4—48no—
225C23×C4C4⋊C4116yes—
226C23×C4C22×C414yes—
227C23×C4C22×C412yes—
228C23×C4C22×C414yes—
229C23×C4C22×C412yes—
230C23×C4C2×D418yes—
231C23×C4C2×D414yes—
232C23×C4C2×D418yes—
233C23×C4C2×D412yes—
234C23×C4C2×D414yes—
235C23×C4C2×D412yes—
236C23×C4C2×Q814yes—
237C23×C4C2×Q812yes—
238C23×C4C2×Q812yes—
239C23×C4#4118yes—
240C23×C4#4114yes—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—48no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—48no—
263#95—216no—
264#95—216no—
265#95—28no—
266#95—416no—
267#95—416no—
268#95—416no—
269#95—416no—
270#95—48no—
271#95—48no—
272#95C22⋊C4116yescomplemented
273C24—22no—
274C24—22no—
275C24—24no—
276C24—22no—
277C24—24no—
278C24—22no—
279C24—24no—
280C24—24no—
281C24—28no—
282C24—28no—
283C24—28no—
284C24—28no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—28no—
290C24—28no—
291C24—28no—
292C24—28no—
293C24—24no—
294C24—24no—
295C24—28no—
296C24—24no—
297C24—28no—
298C24—24no—
299C24—28no—
300C24—28no—
301C24—28no—
302C24—24no—
303C24—28no—
304C24—24no—
305C24—24no—
306C24—28no—
307C24—28no—
308C24—28no—
309C24—24no—
310C24—28no—
311C24—28no—
312C24—24no—
313C24—28no—
314C24—24no—
315C24C2×C4⋊C414yes—
316C24#11718yes—
317C24C4×D418yes—
318C24C4×D414yes—
319C24C4×Q814yes—
320C24#12014yes—
321C24#12118yes—
322C24#12114yes—
323C24C22⋊Q818yes—
324C24C22⋊Q818yes—
325C24C22⋊Q814yes—
326C24#12314yes—
327C24#12318yes—
328C24#12318yes—
329C24#12414yes—
330C24#12414yes—
331C24#12418yes—
332C24#12518yes—
333C24#12518yes—
334C24#12514yes—
335C24#12618yes—
336C24#12618yes—
337C24C4⋊Q814yes—
338C24C23×C411yescharacteristic
339C24C22×D412yes—
340C24C22×D411yescharacteristic
341C24C22×D412yes—
342C24C22×D411yescharacteristic
343C24C22×Q811yescharacteristic
344C24C22×Q811yescharacteristic
345C24C2×#4112yes—
346C24C2×#4112yes—
347C24C2×#4112yes—
348C24C2×#4114yes—
349C24C2×#4112yes—
350C24C2×#4111yescharacteristic
351C24C2×#4111yescharacteristic
352C24H(5,𝔽2)14yes—
353C24H(5,𝔽2)12yes—
354C242−1+412yes—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—264no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
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—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
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—232no—
403C22×C4—232no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—416no—
422C22×C4—432no—
423C22×C4—432no—
424C22×C4—432no—
425C22×C4—432no—
426C22×C4—464no—
427C22×C4—432no—
428C22×C4—432no—
429C22×C4—432no—
430C22×C4—432no—
431C22×C4—432no—
432C22×C4—464no—
433C22×C4—464no—
434C22×C4—464no—
435C22×C4—464no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
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—432no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
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×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—42no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—42no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—44no—
517C22×C4—44no—
518C22×C4—42no—
519C22×C4—44no—
520C22×C4—42no—
521C22×C4—44no—
522C22×C4—44no—
523C22×C4—44no—
524C22×C4—44no—
525C22×C4C2×C22⋊C418yes—
526C22×C4C2×C22⋊C414yes—
527C22×C4C2×C4⋊C414yes—
528C22×C4C2×C4⋊C414yes—
529C22×C4#11718yes—
530C22×C4#11714yes—
531C22×C4#12118yes—
532C22×C4C22⋊Q818yes—
533C4⋊C4—432no—
534C22⋊C4—416no—
535C22⋊C4—432no—
536C22⋊C4—416no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—216no—
548C23—24no—
549C23—28no—
550C23—28no—
551C23—28no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—24no—
556C23—28no—
557C23—28no—
558C23—216no—
559C23—28no—
560C23—28no—
561C23—28no—
562C23—28no—
563C23—216no—
564C23—216no—
565C23—216no—
566C23—216no—
567C23—28no—
568C23—216no—
569C23—216no—
570C23—28no—
571C23—28no—
572C23—216no—
573C23—216no—
574C23—216no—
575C23—216no—
576C23—216no—
577C23—216no—
578C23—216no—
579C23—216no—
580C23—216no—
581C23—216no—
582C23—28no—
583C23—216no—
584C23—216no—
585C23—24no—
586C23—28no—
587C23—28no—
588C23—24no—
589C23—28no—
590C23—28no—
591C23#38218yes—
592C23#38518yes—
593C23#387116yes—
594C23#388116yes—
595C23#38918yes—
596C23#39318yes—
597C23#39418yes—
598C23#39718yes—
599C23#398116yes—
600C23#400116yes—
601C23#40018yes—
602C23C22×C22⋊C411yescharacteristic
603C23C22×C4⋊C411yescharacteristic
604C23C2×#11712yes—
605C23C2×#11711yescharacteristic
606C23C2×#11712yes—
607C23C2×C4×D412yes—
608C23C2×C4×D411yescharacteristic
609C23C2×C4×Q811yescharacteristic
610C23#51811yescharacteristic
611C23#51812yes—
612C23#51911yescharacteristic
613C23C2×#12011yescharacteristic
614C23C2×#12112yes—
615C23C2×#12111yescharacteristic
616C23C2×#12112yes—
617C23C2×C22⋊Q811yescharacteristic
618C23C2×C22⋊Q812yes—
619C23C2×C22⋊Q812yes—
620C23C2×C22⋊Q812yes—
621C23C2×#12312yes—
622C23C2×#12312yes—
623C23C2×#12311yescharacteristic
624C23#52514yes—
625C23#52514yes—
626C23C2×#12411yescharacteristic
627C23C2×#12411yescharacteristic
628C23C2×#12412yes—
629C23C2×#12511yescharacteristic
630C23C2×#12512yes—
631C23C2×#12512yes—
632C23C2×#12612yes—
633C23C2×#12612yes—
634C23#52914yes—
635C23#52914yes—
636C23C2×C4⋊Q811yescharacteristic
637C23#53212yes—
638C23#53312yes—
639C23#53412yes—
640C23#53712yes—
641C23#53814yes—
642C23#53814yes—
643C23#53914yes—
644C23#53914yes—
645C23#54312yes—
646C23#54412yes—
647C23D4214yes—
648C23D4212yes—
649C23#54614yes—
650C23#54612yes—
651C23#54614yes—
652C23#54614yes—
653C23#54614yes—
654C23#54712yes—
655C23#54714yes—
656C23#54812yes—
657C23D4×Q812yes—
658C23#55114yes—
659C23#55114yes—
660C23#55114yes—
661C23#55212yes—
662C23#55214yes—
663C23#55314yes—
664C23#55312yes—
665C23#55414yes—
666C23#55412yes—
667C2×C4—24no—
668C2×C4—28no—
669C2×C4—28no—
670C2×C4—28no—
671C2×C4—28no—
672C2×C4—216no—
673C2×C4—24no—
674C2×C4—24no—
675C2×C4—28no—
676C2×C4—28no—
677C2×C4—28no—
678C2×C4—28no—
679C2×C4—24no—
680C2×C4—28no—
681C2×C4—28no—
682C2×C4—24no—
683C2×C4—464no—
684C2×C4—4128no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—48no—
696C2×C4—48no—
697C2×C4—48no—
698C2×C4—48no—
699C2×C4—48no—
700C2×C4—48no—
701C2×C4—48no—
702C2×C4—48no—
703C2×C4—48no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—48no—
709C2×C4—44no—
710C2×C4—48no—
711C2×C4—44no—
712C2×C4—44no—
713C2×C4—48no—
714C2×C4—48no—
715C2×C4—48no—
716C2×C4—48no—
717C2×C4—48no—
718C2×C4—48no—
719C2×C4—48no—
720C2×C4—48no—
721C2×C4—48no—
722C2×C4—44no—
723C2×C4—82no—
724C2×C4—84no—
725C2×C4—82no—
726C2×C4—82no—
727C2×C4—84no—
728C2×C4—82no—
729C2×C4—82no—
730C2×C4—82no—
731C2×C4—84no—
732C2×C4—84no—
733C2×C4—84no—
734C2×C4—82no—
735C2×C4—84no—
736C2×C4—84no—
737C2×C4—84no—
738C2×C4—84no—
739C2×C4—84no—
740C2×C4—82no—
741C2×C4—84no—
742C2×C4—84no—
743C2×C4#38018yes—
744C22—28no—
745C22—216no—
746C22—216no—
747C22—28no—
748C22—28no—
749C22—216no—
750C22—216no—
751C22—28no—
752C22—28no—
753C22—28no—
754C22—216no—
755C22—232no—
756C22—216no—
757C22—216no—
758C22—232no—
759C22—216no—
760C22#1442116yescomplemented
761C22C2×#38011yescharacteristic
762C22#228012yes—
763C22C2×#38211yescharacteristic
764C22#228312yes—
765C22#228511yescharacteristic
766C22C2×#38511yescharacteristic
767C22C2×#38712yes—
768C22C2×#38812yes—
769C22C2×#38911yescharacteristic
770C22#231212yes—
771C22#231312yes—
772C22#231511yescharacteristic
773C22#231611yescharacteristic
774C22D4×C22⋊C411yescharacteristic
775C22D4×C22⋊C412yes—
776C22Q8×C22⋊C411yescharacteristic
777C22#234212yes—
778C22#234212yes—
779C22#234212yes—
780C22#234411yescharacteristic
781C22#234412yes—
782C22#234512yes—
783C22#234511yescharacteristic
784C22C2×#39311yescharacteristic
785C22C2×#39411yescharacteristic
786C22C2×#39711yescharacteristic
787C22C2×#39812yes—
788C22C2×#40012yes—
789C22C2×#40011yescharacteristic
790C22#241012yes—
791C22#241212yes—
792C22#241612yes—
793C22#241712yes—
794C22#241814yes—
795C22#241914yes—
796C22#242414yes—
797C22#242514yes—
798C22#243012yes—
799C22#243714yes—
800C22#243814yes—
801C22#244012yes—
802C22#244212yes—
803C22#244314yes—
804C22#244414yes—
805C22#244914yes—
806C22#245014yes—
807C22#247214yes—
808C22#247314yes—
809C22#248014yes—
810C22#248114yes—
811C22#248814yes—
812C22#249314yes—
813C22#249812yes—
814C22#250514yes—
815C22#251014yes—
816C22#254312yes—
817C22#254912yes—
818C22#256412yes—
819C22#256912yes—
820C22#263912yes—
821C22#263912yes—
822C22#264614yes—
823C22#285012yes—
824C22#285012yes—
825C22#285114yes—
826C4—28no—
827C4—48no—
828C4—48no—
829C4—44no—
830C4—84no—
831C4—88no—
832C4—84no—
833C2—216no—
834C2C2×#144211yescharacteristic
835C2#1387312yes—
836C2#1387412yes—
837C2#1387711yescharacteristic
838C2#1387811yescharacteristic
839C2#1388212yes—
840C2#1388312yes—
841C2#1390711yescharacteristic
842C2#1390811yescharacteristic
843C2#1434711yescharacteristic
844C2#1468211yescharacteristic
845C2#1473512yes—
846C2#1473614yes—
847C2#1474712yes—
848C2#1475912yes—
849C2#1476714yes—
850C2#1482612yes—
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×#1442C2semidirect
7. #7557C2semidirect
8. #7556C2semidirect
18. C22×#95C4semidirect
272. #95C22⋊C4semidirect
760. C22#1442semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
C23×#952
C2×#14422
C2×C22⋊#952
#75572
#75562