Finite Group Databasebeta

← #366565

Subgroups — #366565

Order 512.

2602 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#451C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3C2×#1971C212yesmaximal, complemented
4C2×#1872C211yesmaximal, characteristic
5C2×#1853C211yesmaximal, characteristic
6C2×#1787C211yesmaximal, characteristic
7C2×#1749C211yesmaximal, characteristic
8#14702C211yesmaximal, characteristic
9#11122C212yesmaximal, complemented
10#11024C212yesmaximal, complemented
11#10928C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C23×Q16C2212yes—
14C22×C22⋊Q8C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#451—24no—
18C2×#451—24no—
19C2×#451C418yescomplemented
20C22×#106C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×#97C2212yes—
28C2×#391—22no—
29C2×#391—28no—
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C22×#95—22no—
36#1971—28no—
37#1872—24no—
38#1853—24no—
39#1787—24no—
40#1749—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1436—22no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C22⋊#95—22no—
48C23×Q8—24no—
49C23×Q8—24no—
50C23×Q8—21no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8C2×C414yes—
59C23×Q8C2311yescharacteristic
60C23×Q8C2312yes—
61C24×C4—21no—
62C24×C4—21no—
63C24×C4D412yes—
64C24×C4C2311yescharacteristic
65C22×Q16—24no—
66C22×Q16—24no—
67C22×Q16—24no—
68C22×Q16—24no—
69C22×Q16—42no—
70C22×Q16—42no—
71C22×Q16C2×C418yescomplemented
72C23×C8C2312yes—
73C23×C8C2312yes—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8C2×C414yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C42—22no—
93C22×C42—22no—
94C22×C42—24no—
95C22×C42—42no—
96C22×C42D412yes—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#451—232no—
100C2×#106—24no—
101C2×#106—24no—
102C2×#106—42no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—42no—
112C2×#103—42no—
113C2×#103—42no—
114C2×#103C2×C418yescomplemented
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8C2×C414yes—
118C2×C22⋊C8D414yes—
119C2×C22⋊C8D414yes—
120C2×C22⋊C8D414yes—
121C2×#97—24no—
122C2×#97—24no—
123C2×#97—42no—
124#391—216no—
125#391—24no—
126#391—24no—
127#391—24no—
128#391—24no—
129#391—42no—
130#391D418yes—
131#380—22no—
132#380—22no—
133C2×#95—216no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141#340—24no—
142#340—24no—
143#336—24no—
144C25C2×D411yescharacteristic
145C22×Q8—24no—
146C22×Q8—216no—
147C22×Q8—24no—
148C22×Q8—216no—
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—21no—
154C22×Q8—28no—
155C22×Q8—28no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—21no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—48no—
173C22×Q8—48no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—48no—
178C22×Q8—48no—
179C22×Q8—48no—
180C22×Q8—48no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—48no—
184C22×Q8—42no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—42no—
188C22×Q8C22×C412yes—
189C22×Q8C22×C414yes—
190C22×Q8C2×D414yes—
191C22×Q8#4114yes—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4C22×C412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4#4112yes—
229C23×C4#4112yes—
230C23×C4C2411yescharacteristic
231C2×Q16—216no—
232C2×Q16—216no—
233C2×Q16—48no—
234C2×Q16—48no—
235C2×Q16—48no—
236C2×Q16—48no—
237C2×Q16—48no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8C22×C414yes—
249C22×C8C2×D414yes—
250C22×C8C2×D414yes—
251C22×C8C2×D414yes—
252C22⋊Q8—28no—
253C22⋊Q8—28no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4C2×D414yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—48no—
299C2×C42—48no—
300C2×C42—42no—
301C2×C42C2×D414yes—
302#106—48no—
303#103—216no—
304#103—48no—
305#103—48no—
306#103—48no—
307C22⋊C8—28no—
308C22⋊C8C22⋊C4116yes—
309#97—48no—
310#95—216no—
311#95—216no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—44no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24C4×D412yes—
324C24#12112yes—
325C24#12712yes—
326C24#13714yes—
327C24#13714yes—
328C24C22×D411yescharacteristic
329C2×Q8—216no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—432no—
350C2×Q8—432no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—432no—
357C2×Q8—432no—
358C2×Q8—42no—
359C2×Q8—41no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—48no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8C4×D418yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—84no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4C2×C22⋊C414yes—
466C22×C4C4×D414yes—
467C22×C4C4×D412yes—
468C22×C4#12014yes—
469C22×C4#12114yes—
470C22×C4#12112yes—
471C22×C4#12414yes—
472C22×C4#12712yes—
473C22×C4#13714yes—
474C22×C4C23×C411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4111yescharacteristic
481C22×C4C2×#4111yescharacteristic
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484Q16—432no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8C2×C22⋊C418yes—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—48no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—48no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C42—24no—
516C42—24no—
517C42—48no—
518C42—42no—
519C42—42no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23#39014yes—
539C23#44118yes—
540C23#47018yes—
541C23#49714yes—
542C23#49714yes—
543C23C2×C4×D411yescharacteristic
544C23C2×#12011yescharacteristic
545C23C2×#12111yescharacteristic
546C23#52511yescharacteristic
547C23C2×#12711yescharacteristic
548C23#53411yescharacteristic
549C23#53511yescharacteristic
550C23C2×#13712yes—
551C23C2×#13712yes—
552C23C2×#13712yes—
553C23#57512yes—
554Q8—28no—
555Q8—28no—
556Q8—432no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—44no—
561Q8—44no—
562Q8—44no—
563Q8—44no—
564Q8—44no—
565Q8—82no—
566Q8—82no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—432no—
585C2×C4—432no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—816no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—84no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—82no—
624C2×C4—81no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—81no—
628C2×C4#39014yes—
629C2×C4C22×C22⋊C411yescharacteristic
630C2×C4C2×C4×D412yes—
631C2×C4C2×C4×D411yescharacteristic
632C2×C4#51812yes—
633C2×C4C2×#12111yescharacteristic
634C2×C4#52512yes—
635C2×C4C2×#12411yescharacteristic
636C2×C4#53812yes—
637C2×C4D4212yes—
638C2×C4D4212yes—
639C2×C4D4212yes—
640C2×C4#54612yes—
641C2×C4#54612yes—
642C2×C4#54612yes—
643C2×C4#54612yes—
644C2×C4#55112yes—
645C8—28no—
646C8—44no—
647C22—28no—
648C22—28no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—28no—
655C22—28no—
656C22—24no—
657C22#197118yescomplemented
658C22C2×#39011yescharacteristic
659C22C4×#12011yescharacteristic
660C22#232111yescharacteristic
661C22#241011yescharacteristic
662C22#241111yescharacteristic
663C22#243311yescharacteristic
664C22#253911yescharacteristic
665C22C2×#44112yes—
666C22C4×#13712yes—
667C22#300412yes—
668C22#300712yes—
669C22C2×#47012yes—
670C22#305612yes—
671C22#311312yes—
672C22#311712yes—
673C22C2×#49711yescharacteristic
674C22C2×#49711yescharacteristic
675C22#315211yescharacteristic
676C22#315311yescharacteristic
677C22#319512yes—
678C22#319612yes—
679C22#321512yes—
680C22#321812yes—
681C4—24no—
682C4—44no—
683C4—44no—
684C4—48no—
685C4—84no—
686C4—82no—
687C4—82no—
688C4—161no—
689C4—161no—
690C4D4×C22⋊C412yes—
691C4#235812yes—
692C4#242412yes—
693C4#247212yes—
694C2—28no—
695C2#1399511yescharacteristic
696C2C2×#197111yescharacteristic
697C2#2123711yescharacteristic
698C2#2145512yes—
699C2#2205012yes—
700C2#2438612yes—
701C2#2450812yes—
702C2#2572611yescharacteristic
703C2#2573011yescharacteristic
704C2#2831311yescharacteristic
705C2#2831511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1971C2semidirect
9. #11122C2semidirect
10. #11024C2semidirect
11. #10928C2semidirect
19. C2×#451C4semidirect
71. C22×Q16C2×C4semidirect
114. C2×#103C2×C4semidirect
657. C22#1971semidirect

Maximal subgroups

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

subgroupindex
C22×#4512
C22×#3912
C2×#19712
C2×#18722
C2×#18532
C2×#17872
C2×#17492
#147022
#111222
#110242
#109282