Finite Group Databasebeta

← #366549

Subgroups — #366549

Order 512.

2644 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#448C211yesmaximal, characteristic
2C23×#103C211yesmaximal, characteristic, complemented
3C2×#1973C212yesmaximal, complemented
4C2×#1893C211yesmaximal, characteristic
5C2×#1852C211yesmaximal, characteristic, complemented
6C2×#1752C211yesmaximal, characteristic, complemented
7#14137C211yesmaximal, characteristic
8#13922C211yesmaximal, characteristic, complemented
9#11124C212yesmaximal, complemented
10#11023C212yesmaximal, complemented
11#10928C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C23×SD16C2212yes—
14C24×C8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#448—24no—
18C2×#448—24no—
19C2×#448C418yescomplemented
20C22×#107C2212yes—
21C22×#103—28no—
22C22×#103—24no—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2212yes—
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×#97C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#386—22no—
34C2×#380C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1973—28no—
37#1893—24no—
38#1852—24no—
39#1752—24no—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1446—22no—
45#1439—22no—
46C4⋊#95C2212yes—
47C22⋊#95—22no—
48C23×Q8—21no—
49C23×Q8—22no—
50C23×Q8—24no—
51C23×Q8—24no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8C2×C414yes—
59C23×Q8C2312yes—
60C23×D4—21no—
61C23×D4C2311yescharacteristic
62C24×C4—21no—
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C22×SD16—24no—
66C22×SD16—24no—
67C22×SD16—24no—
68C22×SD16—24no—
69C22×SD16—42no—
70C22×SD16—42no—
71C22×SD16C2×C418yescomplemented
72C23×C8—22no—
73C23×C8—22no—
74C23×C8—24no—
75C23×C8—42no—
76C23×C8D414yes—
77C23×C8C2311yescharacteristic
78C23×C8C2312yes—
79C23×C8C2311yescharacteristic
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—22no—
84C2×#121—22no—
85C2×#121C2×C414yes—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4D412yes—
99C22×C42C2312yes—
100#448—232no—
101C2×#107—24no—
102C2×#107—24no—
103C2×#107—42no—
104C2×#103—232no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103C2×C418yescomplemented
113C2×#103D418yescomplemented
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—24no—
117C2×#102—42no—
118C2×C22⋊C8C2×C414yes—
119C2×C22⋊C8D414yes—
120C2×C22⋊C8D414yes—
121C2×#97—24no—
122C2×#97—24no—
123C2×#97—42no—
124#391—24no—
125#391—24no—
126#391—42no—
127#391—42no—
128#390—24no—
129#390—42no—
130#386—48no—
131#380—22no—
132#380—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—21no—
139C2×#95D412yes—
140#340—24no—
141#340—22no—
142#340—22no—
143#336—24no—
144C25—21no—
145C25C2×D411yescharacteristic
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—28no—
151C22×Q8—216no—
152C22×Q8—28no—
153C22×Q8—216no—
154C22×Q8—28no—
155C22×Q8—28no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—48no—
167C22×Q8—48no—
168C22×Q8—48no—
169C22×Q8—48no—
170C22×Q8—48no—
171C22×Q8—48no—
172C22×Q8—48no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8—48no—
176C22×Q8—42no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8—42no—
180C22×Q8C22×C414yes—
181C22×Q8C2×D414yes—
182C22×Q8#4114yes—
183C22×D4—24no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—22no—
188C22×D4—21no—
189C22×D4—22no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4C22×C412yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—82no—
216C23×C4C22×C412yes—
217C23×C4C2×D414yes—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D412yes—
222C23×C4#4112yes—
223C23×C4#4112yes—
224C23×C4C2411yescharacteristic
225C2×SD16—216no—
226C2×SD16—216no—
227C2×SD16—48no—
228C2×SD16—48no—
229C2×SD16—48no—
230C2×SD16—48no—
231C2×SD16—48no—
232C22×C8—21no—
233C22×C8—22no—
234C22×C8—21no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—28no—
240C22×C8—21no—
241C22×C8—22no—
242C22×C8—21no—
243C22×C8—22no—
244C22×C8—28no—
245C22×C8—44no—
246C22×C8—48no—
247C22×C8—48no—
248C22×C8C22×C414yes—
249C22×C8C2×D412yes—
250C22×C8C2×D414yes—
251C22×C8C2×D414yes—
252C22×C8C2×D414yes—
253C22×C8C2×D412yes—
254#121—28no—
255#121—28no—
256#121—44no—
257#121—44no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C2×D414yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—84no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300#107—48no—
301#103—216no—
302#103—216no—
303#103—216no—
304#102—44no—
305#102—44no—
306C22⋊C8C22⋊C4116yescomplemented
307#97—48no—
308#95—24no—
309#95—48no—
310#95—48no—
311#95—48no—
312#95—42no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24C4×D412yes—
327C24#12112yes—
328C24#12712yes—
329C24AGL(1,ℤ/8)14yes—
330C24#13714yes—
331C24C22×D411yescharacteristic
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—232no—
336C2×Q8—24no—
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—416no—
349C2×Q8—432no—
350C2×Q8—416no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—416no—
354C2×Q8—416no—
355C2×Q8—416no—
356C2×Q8—432no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—48no—
370C2×Q8C4×D418yes—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—88no—
392C2×D4—84no—
393C2×D4—84no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—816no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4C2×C22⋊C414yes—
452C22×C4C4×D414yes—
453C22×C4C4×D412yes—
454C22×C4#12114yes—
455C22×C4#12112yes—
456C22×C4#12414yes—
457C22×C4#12712yes—
458C22×C4C23×C411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4C2×#4111yescharacteristic
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)12yes—
468SD16—432no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—416no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8C2×C22⋊C418yes—
486C2×C8#12014yes—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—48no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—816no—
505C22⋊C4—84no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—42no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—22no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—81no—
527C23—81no—
528C23—81no—
529C23—81no—
530C23—82no—
531C23—81no—
532C23—84no—
533C23—84no—
534C23—81no—
535C23—82no—
536C23—84no—
537C23—82no—
538C23—84no—
539C23#39014yes—
540C23#44018yes—
541C23#46818yes—
542C23#49614yes—
543C23#49714yes—
544C23C2×C4×D411yescharacteristic
545C23C2×#12111yescharacteristic
546C23C2×#12711yescharacteristic
547C23D4212yes—
548C23#54612yes—
549C23C2×AGL(1,ℤ/8)12yes—
550C23C2×#13712yes—
551C23#57712yes—
552C23#57812yes—
553Q8—28no—
554Q8—28no—
555Q8—432no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—44no—
561Q8—44no—
562D4—42no—
563D4—44no—
564D4—42no—
565D4—816no—
566D4—81no—
567D4—81no—
568D4—82no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—432no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—816no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—84no—
604C2×C4—82no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—84no—
611C2×C4—81no—
612C2×C4—84no—
613C2×C4#39014yes—
614C2×C4C22×C22⋊C411yescharacteristic
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C2×C4×D412yes—
617C2×C4#51812yes—
618C2×C4C2×#12011yescharacteristic
619C2×C4C2×#12111yescharacteristic
620C2×C4#52511yescharacteristic
621C2×C4#52512yes—
622C2×C4C2×#12411yescharacteristic
623C2×C4#53411yescharacteristic
624C2×C4#53511yescharacteristic
625C2×C4#53812yes—
626C2×C4D4212yes—
627C2×C4D4212yes—
628C2×C4#54612yes—
629C2×C4#54612yes—
630C2×C4#54612yes—
631C2×C4#55112yes—
632C8—28no—
633C8—24no—
634C8—42no—
635C22—216no—
636C22—216no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—88no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—164no—
648C22—161no—
649C22—161no—
650C22—161no—
651C22—161no—
652C22#197318yescomplemented
653C22C2×#39011yescharacteristic
654C22#235812yes—
655C22#242412yes—
656C22#243311yescharacteristic
657C22#253911yescharacteristic
658C22C2×#44012yes—
659C22#294712yes—
660C22C2×#46812yes—
661C22#306312yes—
662C22C2×#49611yescharacteristic
663C22C2×#49711yescharacteristic
664C22#315411yescharacteristic
665C22#315611yescharacteristic
666C22#326514yes—
667C22#327014yes—
668C22#330814yes—
669C22#332014yes—
670C4—24no—
671C4—48no—
672C4—84no—
673C4—84no—
674C4—82no—
675C4—161no—
676C4—161no—
677C4C4×#12011yescharacteristic
678C4#232111yescharacteristic
679C4D4×C22⋊C412yes—
680C4#241011yescharacteristic
681C4#241111yescharacteristic
682C4#247212yes—
683C2—28no—
684C2—162no—
685C2#1399511yescharacteristic
686C2C2×#197311yescharacteristic
687C2#2123411yescharacteristic
688C2#2285514yes—
689C2#2513014yes—
690C2#2578311yescharacteristic
691C2#2578911yescharacteristic
692C2#2834611yescharacteristic
693C2#2834811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C23×#103C2semidirect
3. C2×#1973C2semidirect
5. C2×#1852C2semidirect
6. C2×#1752C2semidirect
8. #13922C2semidirect
9. #11124C2semidirect
10. #11023C2semidirect
11. #10928C2semidirect
19. C2×#448C4semidirect
71. C22×SD16C2×C4semidirect
112. C2×#103C2×C4semidirect
113. C2×#103D4semidirect
306. C22⋊C8C22⋊C4semidirect
652. C22#1973semidirect

Maximal subgroups

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

subgroupindex
C22×#4482
C23×#1032
C2×#19732
C2×#18932
C2×#18522
C2×#17522
#141372
#139222
#111242
#110232
#109282