Finite Group Databasebeta

← #366607

Subgroups — #366607

Order 512.

2664 conjugacy classes of subgroups.

The subgroup lattice has 780 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic, complemented
2C2×#2436C211yesmaximal, characteristic
3C2×#1983C212yesmaximal, complemented
4C2×#1893C211yesmaximal, characteristic
5C2×#1874C211yesmaximal, characteristic
6C2×#1850C211yesmaximal, characteristic, complemented
7C2×#1787C211yesmaximal, characteristic, complemented
8C22⋊C4×C4⋊C4C211yesmaximal, characteristic, complemented
9#11077C212yesmaximal, complemented
10#11009C212yesmaximal, complemented
11#10932C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#124C2212yes—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×#103—28no—
19C22×#103—24no—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#102C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28#2436—24no—
29#2436C418yescomplemented
30C2×#393—22no—
31C2×#391C2212yes—
32C2×#386—21no—
33C2×#386—21no—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1983—28no—
39#1893—24no—
40#1874—24no—
41#1850—24no—
42#1787—24no—
43C2×#340C2211yescharacteristic
44C2×#340C2212yes—
45C2×#340C2211yescharacteristic
46C2×#336C2212yes—
47#1436—22no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C23×Q8—21no—
51C23×Q8—22no—
52C23×Q8—24no—
53C23×Q8—24no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—22no—
57C23×Q8—22no—
58C23×Q8—42no—
59C23×Q8—42no—
60C23×Q8C2×C414yes—
61C23×Q8C2312yes—
62C23×D4—21no—
63C23×D4C2311yescharacteristic
64C24×C4—21no—
65C24×C4—21no—
66C24×C4C2311yescharacteristic
67C23×C8—22no—
68C23×C8—24no—
69C23×C8—22no—
70C23×C8—22no—
71C23×C8—42no—
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2312yes—
75C2×#124—24no—
76C2×#124—24no—
77C2×#124—24no—
78C2×#124—42no—
79C2×#124—42no—
80C2×#124C2×C418yescomplemented
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—22no—
84C2×#121—22no—
85C2×#121—22no—
86C2×#121C2×C414yes—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4D414yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4D411yescharacteristic
101C22×C22⋊C4D411yescharacteristic
102C22×C42—22no—
103C22×C42C2312yes—
104C2×C4⋊C8—24no—
105C2×C4⋊C8—24no—
106C2×C4⋊C8—42no—
107C2×#103—232no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—24no—
116C2×#103—24no—
117C2×#103—42no—
118C2×#103—42no—
119C2×#103D418yescomplemented
120C2×#102—24no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—24no—
124C2×#102—24no—
125C2×#102—42no—
126C2×#102—42no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—22no—
130#393—216no—
131#391—24no—
132#391C2×C418yescomplemented
133#386—28no—
134#386—28no—
135#380—22no—
136#380D414yes—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—42no—
139C4×C22⋊C4—44no—
140C4×C22⋊C4C2×C414yes—
141C4×C22⋊C4D414yes—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95D412yes—
149#340—22no—
150#340—22no—
151#340—24no—
152#336—24no—
153C25—21no—
154C25C2×D411yescharacteristic
155C22×Q8—24no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—28no—
160C22×Q8—216no—
161C22×Q8—28no—
162C22×Q8—216no—
163C22×Q8—28no—
164C22×Q8—28no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—48no—
178C22×Q8—48no—
179C22×Q8—48no—
180C22×Q8—48no—
181C22×Q8—48no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—48no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C22×C414yes—
190C22×Q8C2×D414yes—
191C22×Q8#4114yes—
192C22×D4—24no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4C22×C412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—21no—
215C23×C4—24no—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4—21no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—82no—
228C23×C4C22×C412yes—
229C23×C4C2×D412yes—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D412yes—
233C23×C4C2×D412yes—
234C23×C4C2×D412yes—
235C23×C4#4112yes—
236C23×C4#4112yes—
237C23×C4C2411yescharacteristic
238C22×C8—28no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—48no—
256C22×C8—48no—
257C22×C8—44no—
258C22×C8—44no—
259C22×C8C2×D414yes—
260#124—216no—
261#124—48no—
262#124—48no—
263#124—48no—
264#121—28no—
265#121—28no—
266#121—44no—
267#121—44no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C22×C412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D412yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4C22⋊C414yes—
310C2×C22⋊C4C22⋊C414yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—82no—
318C2×C42C22×C414yes—
319C2×C42C2×D414yes—
320C4⋊C8—48no—
321#103—216no—
322#103—216no—
323#103—48no—
324#103—48no—
325#102—48no—
326#102—44no—
327#102—44no—
328C22⋊C8—44no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—44no—
333#95—44no—
334#95—82no—
335#95C22⋊C418yes—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—41no—
342C24—41no—
343C24—42no—
344C24—42no—
345C24—41no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—42no—
350C24—42no—
351C24C2×C22⋊C412yes—
352C24#12112yes—
353C24#12112yes—
354C24#13512yes—
355C24#13512yes—
356C24AGL(1,ℤ/8)12yes—
357C24#13712yes—
358C24C22×D411yescharacteristic
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—232no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—416no—
376C2×Q8—432no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—416no—
380C2×Q8—416no—
381C2×Q8—416no—
382C2×Q8—432no—
383C2×Q8—416no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8C4×D418yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—84no—
422C2×D4—84no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—48no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—48no—
489C22×C4—84no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4C2×C22⋊C414yes—
493C22×C4C2×C22⋊C414yes—
494C22×C4C2×C22⋊C412yes—
495C22×C4C4×D414yes—
496C22×C4C4×D414yes—
497C22×C4#12112yes—
498C22×C4#12112yes—
499C22×C4#12414yes—
500C22×C4#12714yes—
501C22×C4C23×C411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4111yescharacteristic
507C22×C4C2×#4112yes—
508C22×C4C2×#4111yescharacteristic
509C22×C4H(5,𝔽2)12yes—
510C22×C4H(5,𝔽2)12yes—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—416no—
515C2×C8—42no—
516C2×C8—42no—
517C2×C8—42no—
518C2×C8—42no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C2×C8—41no—
524C2×C8—41no—
525C2×C8—42no—
526C2×C8—42no—
527C2×C8—42no—
528C2×C8—42no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—48no—
538C4⋊C4—42no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4#12014yes—
544C22⋊C4—28no—
545C22⋊C4—28no—
546C22⋊C4—416no—
547C22⋊C4—416no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C42—24no—
551C42—24no—
552C42—24no—
553C42—24no—
554C42—88no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—42no—
576C23—81no—
577C23—81no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23—82no—
582C23—82no—
583C23—81no—
584C23—81no—
585C23—81no—
586C23—81no—
587C23—81no—
588C23—81no—
589C23—82no—
590C23—82no—
591C23—82no—
592C23—82no—
593C23—82no—
594C23#39014yes—
595C23#41614yes—
596C23#41714yes—
597C23#46314yes—
598C23#46314yes—
599C23#46414yes—
600C23#46414yes—
601C23C22×C22⋊C411yescharacteristic
602C23C2×#12111yescharacteristic
603C23C2×#12111yescharacteristic
604C23D4212yes—
605C23#54612yes—
606C23C2×#13511yescharacteristic
607C23C2×#13511yescharacteristic
608C23C2×AGL(1,ℤ/8)11yescharacteristic
609C23C2×#13711yescharacteristic
610C23#57712yes—
611C23#57812yes—
612Q8—28no—
613Q8—28no—
614Q8—432no—
615Q8—44no—
616Q8—44no—
617Q8—44no—
618Q8—44no—
619Q8—44no—
620Q8—44no—
621D4—42no—
622D4—44no—
623D4—42no—
624D4—816no—
625D4—81no—
626D4—82no—
627D4—81no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—432no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—816no—
675C2×C4—82no—
676C2×C4—82no—
677C2×C4—82no—
678C2×C4—82no—
679C2×C4—81no—
680C2×C4—81no—
681C2×C4—81no—
682C2×C4—81no—
683C2×C4—82no—
684C2×C4—82no—
685C2×C4#39014yes—
686C2×C4C2×C4×D411yescharacteristic
687C2×C4C2×C4×D411yescharacteristic
688C2×C4C2×C4×D412yes—
689C2×C4#51812yes—
690C2×C4C2×#12011yescharacteristic
691C2×C4#52512yes—
692C2×C4#52511yescharacteristic
693C2×C4C2×#12411yescharacteristic
694C2×C4C2×#12711yescharacteristic
695C2×C4#53411yescharacteristic
696C2×C4#53511yescharacteristic
697C2×C4#53812yes—
698C2×C4D4212yes—
699C2×C4D4212yes—
700C2×C4#54612yes—
701C2×C4#54612yes—
702C2×C4#54612yes—
703C2×C4#55112yes—
704C8—44no—
705C8—81no—
706C8—81no—
707C22—28no—
708C22—24no—
709C22—24no—
710C22—24no—
711C22—28no—
712C22—28no—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—28no—
717C22—82no—
718C22—82no—
719C22—84no—
720C22—84no—
721C22—82no—
722C22—162no—
723C22—161no—
724C22—161no—
725C22—161no—
726C22—161no—
727C22—162no—
728C22#198318yescomplemented
729C22C2×#39011yescharacteristic
730C22D4×C22⋊C412yes—
731C22#241011yescharacteristic
732C22#241111yescharacteristic
733C22#242412yes—
734C22C2×#41611yescharacteristic
735C22C2×#41711yescharacteristic
736C22#289812yes—
737C22C2×#46311yescharacteristic
738C22C2×#46311yescharacteristic
739C22C2×#46411yescharacteristic
740C22C2×#46411yescharacteristic
741C22#304211yescharacteristic
742C22#304411yescharacteristic
743C22#304612yes—
744C22#326512yes—
745C22#327012yes—
746C22#328212yes—
747C22#328512yes—
748C22#330012yes—
749C22#330112yes—
750C22#330812yes—
751C22#331612yes—
752C4—24no—
753C4—44no—
754C4—44no—
755C4—44no—
756C4—48no—
757C4—84no—
758C4—82no—
759C4—82no—
760C4C4×#12011yescharacteristic
761C4#232111yescharacteristic
762C4#235812yes—
763C4#243311yescharacteristic
764C4#247212yes—
765C4#253911yescharacteristic
766C2—28no—
767C2—162no—
768C2#1399511yescharacteristic
769C2C2×#198311yescharacteristic
770C2#2126211yescharacteristic
771C2#2241612yes—
772C2#2242112yes—
773C2#2435911yescharacteristic
774C2#2436111yescharacteristic
775C2#2448111yescharacteristic
776C2#2448311yescharacteristic
777C2#2508612yes—
778C2#2509012yes—
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
1. C23×#103C2semidirect
3. C2×#1983C2semidirect
6. C2×#1850C2semidirect
7. C2×#1787C2semidirect
8. C22⋊C4×C4⋊C4C2semidirect
9. #11077C2semidirect
10. #11009C2semidirect
11. #10932C2semidirect
29. #2436C4semidirect
80. C2×#124C2×C4semidirect
119. C2×#103D4semidirect
132. #391C2×C4semidirect
728. C22#1983semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#24362
C2×#19832
C2×#18932
C2×#18742
C2×#18502
C2×#17872
C22⋊C4×C4⋊C42
#110772
#110092
#109322