Finite Group Databasebeta

← #415045

Subgroups — #415045

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1847C211yesmaximal, characteristic, complemented
5C2×#1749C211yesmaximal, characteristic
6#14682C211yesmaximal, characteristic
7#11220C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10843C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2857—24no—
21#2857—24no—
22C2×#391—21no—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380—21no—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32#1993—24no—
33#1993—24no—
34#1847—24no—
35#1749—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45#1434—21no—
46#1434—21no—
47C22⋊#95—21no—
48C22⋊#95—21no—
49C23×Q8—21no—
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D414yes—
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—44no—
63C2×C22⋊Q8—44no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—42no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—42no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8Q818yescomplemented
96C2×#97—22no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97D414yes—
102#391—28no—
103#389—28no—
104#389—24no—
105#389—24no—
106#384—22no—
107#384D414yes—
108#382—22no—
109#382—22no—
110#382—22no—
111#382—22no—
112#380—28no—
113#380—24no—
114#380—24no—
115#380—42no—
116#380—42no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—42no—
134#340—42no—
135#336—24no—
136C25C2×D411yescharacteristic
137C22×Q8—24no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×Q814yes—
179C23×C4C2411yescharacteristic
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C2×D414yes—
191C22×C8C2×Q814yes—
192C22×C8C2×Q814yes—
193C22×C8#4114yes—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196C4⋊Q8—44no—
197C4⋊Q8—44no—
198C22⋊Q8—432no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270#107—48no—
271#106—48no—
272C22⋊C8—216no—
273#97—28no—
274#97—44no—
275#95—28no—
276#95—28no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—42no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24C4⋊Q812yes—
300C24AGL(1,ℤ/8)12yes—
301C24#13712yes—
302C24C22×D411yescharacteristic
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4#12012yes—
409C22×C4#12012yes—
410C22×C4#12112yes—
411C22×C4#12112yes—
412C22×C4#12312yes—
413C22×C4#12312yes—
414C22×C4C4⋊Q812yes—
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4#13712yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×Q812yes—
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+412yes—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8C22⋊Q818yescomplemented
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—48no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C22⋊C4—416no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—44no—
469C42—44no—
470C42#12114yes—
471C42#12114yes—
472C42#12314yes—
473C42#12414yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23#50114yes—
509C23C2×#12011yescharacteristic
510C23C2×#12111yescharacteristic
511C23C2×#12311yescharacteristic
512C23C2×C4⋊Q811yescharacteristic
513C23#53211yescharacteristic
514C23#53511yescharacteristic
515C23#53611yescharacteristic
516C23C2×AGL(1,ℤ/8)11yescharacteristic
517C23C2×AGL(1,ℤ/8)11yescharacteristic
518C23C2×#13711yescharacteristic
519C23C2×#13711yescharacteristic
520C23#57511yescharacteristic
521C23#57511yescharacteristic
522C23#57511yescharacteristic
523C23#57511yescharacteristic
524Q8—816no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4#39414yes—
607C2×C4#39414yes—
608C2×C4C2×#12111yescharacteristic
609C2×C4C2×#12111yescharacteristic
610C2×C4C2×C22⋊Q812yes—
611C2×C4C2×#12311yescharacteristic
612C2×C4C2×#12411yescharacteristic
613C2×C4#53312yes—
614C2×C4#53712yes—
615C2×C4#54212yes—
616C2×C4#54612yes—
617C2×C4D4×Q812yes—
618C2×C4D4×Q812yes—
619C2×C4#55012yes—
620C2×C4#55212yes—
621C2×C4#55412yes—
622C2×C4#55412yes—
623C2×C4#55612yes—
624C8—28no—
625C8—44no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22C2×#39411yescharacteristic
642C22#243111yescharacteristic
643C22#243411yescharacteristic
644C22#249911yescharacteristic
645C22#250211yescharacteristic
646C22#262011yescharacteristic
647C22#285711yescharacteristic
648C22#300311yescharacteristic
649C22#300411yescharacteristic
650C22#300511yescharacteristic
651C22#300511yescharacteristic
652C22#305511yescharacteristic
653C22#305511yescharacteristic
654C22#305611yescharacteristic
655C22#305611yescharacteristic
656C22#309411yescharacteristic
657C22#309411yescharacteristic
658C22#309511yescharacteristic
659C22#309511yescharacteristic
660C22C2×#50111yescharacteristic
661C22#316211yescharacteristic
662C22#316511yescharacteristic
663C22#316611yescharacteristic
664C22#317111yescharacteristic
665C22#317211yescharacteristic
666C22#318011yescharacteristic
667C22#318111yescharacteristic
668C22#321311yescharacteristic
669C22#321411yescharacteristic
670C22#321511yescharacteristic
671C22#321611yescharacteristic
672C22#323811yescharacteristic
673C22#323911yescharacteristic
674C22#324011yescharacteristic
675C22#324011yescharacteristic
676C4—24no—
677C4—24no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—84no—
682C4—84no—
683C4—161no—
684C4—161no—
685C4—161no—
686C4—161no—
687C4#241712yes—
688C4#242312yes—
689C4#249212yes—
690C4#253812yes—
691C2—44no—
692C2#1518211yescharacteristic
693C2#2348011yescharacteristic
694C2#2348111yescharacteristic
695C2#2560211yescharacteristic
696C2#2560311yescharacteristic
697C2#2581611yescharacteristic
698C2#2581711yescharacteristic
699C2#2788211yescharacteristic
700C2#2788311yescharacteristic
701C2#2795511yescharacteristic
702C2#2795611yescharacteristic
703C2#2985311yescharacteristic
704C2#2985411yescharacteristic
705C2#3321711yescharacteristic
706C2#3321811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
2. C2×#1993C2semidirect
3. C2×#1993C2semidirect
4. C2×#1847C2semidirect
7. #11220C2semidirect
8. #11219C2semidirect
11. #10843C2semidirect
95. C2×C22⋊C8Q8semidirect
440. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#19932
C2×#18472
C2×#17492
#146822
#112202
#112192
#110962
#110952
#108432