Finite Group Databasebeta

← #360501

Subgroups — #360501

Order 512.

3394 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#407C211yesmaximal, characteristic
2C22×C4×C22⋊C4C211yesmaximal, characteristic
3C2×#1851C212yesmaximal, complemented
4C2×#1797C212yesmaximal, complemented
5C2×#1779C211yesmaximal, characteristic
6#11042C214yesmaximal, complemented
7#10715C212yesmaximal, complemented
8#10712C212yesmaximal, complemented
9C25×C4C2211yescharacteristic
10C23×M16C2212yes—
11C23×C22⋊C4C412yes—
12C23×C42C2212yes—
13C22×C4⋊C8C2214yescomplemented
14C2×#407—28no—
15C2×#407—24no—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C2×C4×C22⋊C4—28no—
23C2×C4×C22⋊C4C411yescharacteristic
24C2×C4×C22⋊C4C412yes—
25C2×C4×C22⋊C4C411yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C22×#95C411yescharacteristic
30C22×#95C411yescharacteristic
31#1851—28no—
32#1797—28no—
33#1779—24no—
34C2×#336C2214yescomplemented
35C2×#336C2214yescomplemented
36C2×#336C2214yescomplemented
37C26C2×C411yescharacteristic
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—24no—
41C24×C4—24no—
42C24×C4—44no—
43C24×C4C2×C412yes—
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C412yes—
48C24×C4D414yes—
49C24×C4Q814yes—
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C22×M16—24no—
54C22×M16—24no—
55C22×M16—24no—
56C22×M16—28no—
57C22×M16—44no—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C23×C8C2314yes—
61C22×C22⋊C4—28no—
62C22×C22⋊C4—28no—
63C22×C22⋊C4—28no—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C42—24no—
69C22×C42—24no—
70C22×C42—28no—
71C22×C42—44no—
72C22×C42C2×C412yes—
73C22×C42C2×C412yes—
74C22×C42C2312yes—
75C22×C42C2312yes—
76C2×C4⋊C8—28no—
77C2×C4⋊C8—28no—
78C2×C4⋊C8—44no—
79#407—232no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—28no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—44no—
93C2×C22⋊C8D418yes—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—28no—
96C4×C22⋊C4—28no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4D418yes—
100C2×#95—24no—
101C2×#95—24no—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108#336—28no—
109#336—28no—
110#336—44no—
111C25—22no—
112C25—22no—
113C25—24no—
114C25—24no—
115C25—24no—
116C25—22no—
117C25—22no—
118C25—42no—
119C25—41no—
120C25—41no—
121C25C4⋊C414yes—
122C25C22×C411yescharacteristic
123C25C22×C411yescharacteristic
124C25C22×C411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—28no—
146C23×C4—216no—
147C23×C4—216no—
148C23×C4—216no—
149C23×C4—28no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—48no—
160C23×C4—48no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—416no—
169C23×C4—416no—
170C23×C4—416no—
171C23×C4—416no—
172C23×C4C4⋊C414yes—
173C23×C4C4⋊C414yes—
174C23×C4C4⋊C414yes—
175C23×C4M1618yescomplemented
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D414yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D414yes—
188C23×C4C2×Q812yes—
189C23×C4C2×Q812yes—
190C23×C4C2×Q812yes—
191C23×C4#4114yes—
192C23×C4#4114yes—
193C23×C4C2411yescharacteristic
194C2×M16—216no—
195C2×M16—416no—
196C2×M16—416no—
197C2×M16—48no—
198C2×M16—48no—
199C22×C8—24no—
200C22×C8—24no—
201C22×C8—24no—
202C22×C8—24no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8—22no—
206C22×C8—24no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—24no—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8C2×D418yes—
216C2×C22⋊C4—264no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4C22⋊C418yes—
228C2×C22⋊C4M1618yescomplemented
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—416no—
241C2×C42—416no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42C2×D418yes—
245C4⋊C8—416no—
246C22⋊C8—216no—
247C22⋊C8—216no—
248C22⋊C8—48no—
249C22⋊C8—44no—
250C22⋊C8—48no—
251C22⋊C8—44no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95C22⋊C414yes—
257#95C22⋊C414yes—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24—21no—
263C24—21no—
264C24—216no—
265C24—216no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—28no—
284C24—28no—
285C24—28no—
286C24—28no—
287C24—28no—
288C24—44no—
289C24—44no—
290C24—48no—
291C24—48no—
292C24—44no—
293C24—44no—
294C24—44no—
295C24—44no—
296C24—48no—
297C24—48no—
298C24—48no—
299C24—48no—
300C24—48no—
301C24—48no—
302C24—48no—
303C24—48no—
304C24C2×C22⋊C412yes—
305C24C2×C4⋊C412yes—
306C24C2×C4⋊C412yes—
307C24C2×C4⋊C412yes—
308C24#11712yes—
309C24C4×D414yes—
310C24C2×M1614yes—
311C24C2×M1614yes—
312C24#13112yes—
313C24#13112yes—
314C24C23×C411yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—216no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
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—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—216no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—216no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—432no—
386C22×C4—432no—
387C22×C4—464no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4C2×C22⋊C414yes—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C414yes—
409C22×C4C2×C4⋊C412yes—
410C22×C4C2×C4⋊C412yes—
411C22×C4C2×C4⋊C412yes—
412C22×C4#11712yes—
413C22×C4C4×D414yes—
414C22×C4#12014yes—
415C22×C4#12118yes—
416C22×C4C22⋊Q818yes—
417C22×C4C22⋊Q818yes—
418C22×C4#12314yes—
419C22×C4C2×M1614yes—
420C22×C4C2×M1614yes—
421C22×C4C2×M1614yes—
422C22×C4C2×M1614yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×Q811yescharacteristic
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)14yes—
430M16—432no—
431C2×C8—28no—
432C2×C8—28no—
433C2×C8—28no—
434C2×C8—44no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—44no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C42—28no—
453C42—28no—
454C42—48no—
455C42—48no—
456C42—44no—
457C42—44no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—232no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—416no—
504C23—416no—
505C23—416no—
506C23—416no—
507C23—432no—
508C23—432no—
509C23—432no—
510C23—432no—
511C23—416no—
512C23—42no—
513C23—42no—
514C23—416no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23#38014yes—
532C23#38514yes—
533C23#42314yes—
534C23#42314yes—
535C23#42414yes—
536C23#43518yes—
537C23#43618yes—
538C23C22×C22⋊C411yescharacteristic
539C23C22×C4⋊C411yescharacteristic
540C23C2×#11711yescharacteristic
541C23C2×C4×D412yes—
542C23#51812yes—
543C23C22×M1612yes—
544C23C22×M1612yes—
545C23C2×#13111yescharacteristic
546C23C2×#13111yescharacteristic
547C23#56811yescharacteristic
548C23#56811yescharacteristic
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—216no—
568C2×C4—216no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—416no—
582C2×C4—432no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—432no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4#38014yes—
604C2×C4#38514yes—
605C2×C4#40718yescomplemented
606C2×C4C2×#12011yescharacteristic
607C2×C4C2×#12112yes—
608C2×C4C2×C22⋊Q812yes—
609C2×C4C2×C22⋊Q812yes—
610C2×C4C2×#12311yescharacteristic
611C2×C4#53412yes—
612C2×C4#54312yes—
613C2×C4D4214yes—
614C2×C4#54614yes—
615C2×C4#54614yes—
616C2×C4#55114yes—
617C8—48no—
618C8—82no—
619C8—82no—
620C22—28no—
621C22—28no—
622C22—24no—
623C22—24no—
624C22—28no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—464no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22#185118yescomplemented
654C22C2×#38011yescharacteristic
655C22C2×#38511yescharacteristic
656C22#231511yescharacteristic
657C22D4×C22⋊C412yes—
658C22#234212yes—
659C22C2×#40712yes—
660C22#288011yescharacteristic
661C22#288011yescharacteristic
662C22C2×#42311yescharacteristic
663C22C2×#42311yescharacteristic
664C22C2×#42411yescharacteristic
665C22#290511yescharacteristic
666C22#292012yes—
667C22#292411yescharacteristic
668C22#292411yescharacteristic
669C22C2×#43512yes—
670C22C2×#43612yes—
671C22D4×M1612yes—
672C22D4×M1612yes—
673C22#297312yes—
674C22#297412yes—
675C22#297412yes—
676C22#297512yes—
677C4—24no—
678C4—24no—
679C4—28no—
680C4—42no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4#241012yes—
685C4#241212yes—
686C4#243714yes—
687C2—28no—
688C2—44no—
689C2—44no—
690C2#1387711yescharacteristic
691C2#2062611yescharacteristic
692C2#2062811yescharacteristic
693C2C2×#185111yescharacteristic
694C2#2083711yescharacteristic
695C2#2193211yescharacteristic
696C2#2193411yescharacteristic
697C2#2239912yes—
698C2#2240112yes—
699C2#2255312yes—
700C2#2255512yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1851C2semidirect
4. C2×#1797C2semidirect
6. #11042C2semidirect
7. #10715C2semidirect
8. #10712C2semidirect
13. C22×C4⋊C8C22semidirect
34. C2×#336C22semidirect
35. C2×#336C22semidirect
36. C2×#336C22semidirect
175. C23×C4M16semidirect
228. C2×C22⋊C4M16semidirect
605. C2×C4#407semidirect
653. C22#1851semidirect

Maximal subgroups

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

subgroupindex
C22×#4072
C22×C4×C22⋊C42
C2×#18512
C2×#17972
C2×#17792
#110422
#107152
#107122