Finite Group Databasebeta

← #395644

Subgroups — #395644

Order 512.

2306 conjugacy classes of subgroups.

The subgroup lattice has 789 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#391C211yesmaximal, characteristic
2C2×#2068C212yesmaximal, complemented
3C2×#1852C211yesmaximal, characteristic
4C2×#1797C211yesmaximal, characteristic
5C2×#1787C211yesmaximal, characteristic
6C2×#1787C211yesmaximal, characteristic
7#14697C211yesmaximal, characteristic
8#11124C212yesmaximal, complemented
9#11077C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#391—21no—
24C2×#391—21no—
25C2×#391—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#385—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C22×#95—21no—
34C22×#95—21no—
35#2068—28no—
36#1852—24no—
37#1797—24no—
38#1787—24no—
39#1787—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2211yescharacteristic
48C22⋊#95—21no—
49C23×Q8—24no—
50C23×Q8—24no—
51C23×Q8—21no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—22no—
57C23×Q8—42no—
58C23×Q8—42no—
59C23×Q8Q814yes—
60C23×Q8C2312yes—
61C24×C4—21no—
62C24×C4—21no—
63C24×C4D411yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4C2311yescharacteristic
66C23×C8C2312yes—
67C23×C8C2312yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—24no—
85C22×C42—42no—
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—42no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—42no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#103—42no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8D414yes—
114#391—28no—
115#391—28no—
116#391—24no—
117#391—24no—
118#391—24no—
119#391D418yes—
120#389—24no—
121#389—42no—
122#385—44no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380D414yes—
127C2×#95—28no—
128C2×#95—28no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141#340—24no—
142#340—24no—
143#340—42no—
144#336—24no—
145C25C2×D411yescharacteristic
146C22×Q8—24no—
147C22×Q8—216no—
148C22×Q8—24no—
149C22×Q8—216no—
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
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—48no—
165C22×Q8—48no—
166C22×Q8—48no—
167C22×Q8—48no—
168C22×Q8—48no—
169C22×Q8—48no—
170C22×Q8—48no—
171C22×Q8—48no—
172C22×Q8—48no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—48no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×Q8C2×D414yes—
181C22×Q8C2×Q814yes—
182C22×Q8#4114yes—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4—41no—
209C23×C4—41no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4Q1614yes—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2×Q812yes—
229C23×C4#4112yes—
230C23×C4#4112yes—
231C23×C4#4112yes—
232C23×C4C2411yescharacteristic
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
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×C8C2×D414yes—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—48no—
298C2×C42—48no—
299C2×C42—41no—
300C2×C42—41no—
301C2×C42C2×D414yes—
302#107—48no—
303C4⋊C8—48no—
304#103—48no—
305#103—48no—
306#103—48no—
307#103—48no—
308C22⋊C8—28no—
309C22⋊C8—44no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—42no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24#12112yes—
331C24#12112yes—
332C24C22⋊Q812yes—
333C24C2×Q1612yes—
334C24#13512yes—
335C24AGL(1,ℤ/8)12yes—
336C24#13712yes—
337C24C22×D411yescharacteristic
338C2×Q8—216no—
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—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—432no—
355C2×Q8—432no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—432no—
362C2×Q8—432no—
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—42no—
375C2×Q8—48no—
376C2×Q8C22⋊Q818yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
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—28no—
398C22×C4—216no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4#12012yes—
486C22×C4#12012yes—
487C22×C4#12114yes—
488C22×C4#12112yes—
489C22×C4#12112yes—
490C22×C4C22⋊Q814yes—
491C22×C4C22⋊Q814yes—
492C22×C4C22⋊Q812yes—
493C22×C4#12614yes—
494C22×C4C2×Q1612yes—
495C22×C4C2×Q1612yes—
496C22×C4#13712yes—
497C22×C4#13712yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×Q811yescharacteristic
502C22×C4C2×#4111yescharacteristic
503C22×C4C2×#4112yes—
504C22×C4C2×#4111yescharacteristic
505C22×C4C2×#4111yescharacteristic
506C22×C4H(5,𝔽2)12yes—
507C22×C4H(5,𝔽2)12yes—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—42no—
512C2×C8—42no—
513C2×C8—42no—
514C2×C8—42no—
515C2×C8—42no—
516C2×C8—42no—
517C2×C8—42no—
518C2×C8—42no—
519C2×C8—42no—
520C2×C8—42no—
521C2×C8—42no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C42—24no—
540C42—24no—
541C42—44no—
542C42—44no—
543C42—42no—
544C42—42no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23#39614yes—
574C23#46214yes—
575C23#46314yes—
576C23#46714yes—
577C23#46814yes—
578C23#47714yes—
579C23#47914yes—
580C23C2×#12011yescharacteristic
581C23C2×#12111yescharacteristic
582C23C2×#12111yescharacteristic
583C23C2×C22⋊Q811yescharacteristic
584C23#52511yescharacteristic
585C23#53411yescharacteristic
586C23#53511yescharacteristic
587C23C22×Q1611yescharacteristic
588C23C2×#13511yescharacteristic
589C23C2×AGL(1,ℤ/8)11yescharacteristic
590C23C2×#13711yescharacteristic
591C23C2×#13711yescharacteristic
592C23C2×#13711yescharacteristic
593C23#57511yescharacteristic
594C23#57511yescharacteristic
595Q8—28no—
596Q8—28no—
597Q8—432no—
598Q8—44no—
599Q8—44no—
600Q8—44no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—416no—
631C2×C4—416no—
632C2×C4—432no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—816no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—82no—
667C2×C4—81no—
668C2×C4—82no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—81no—
674C2×C4—81no—
675C2×C4—81no—
676C2×C4—81no—
677C2×C4—81no—
678C2×C4—81no—
679C2×C4—81no—
680C2×C4—81no—
681C2×C4—81no—
682C2×C4—81no—
683C2×C4—81no—
684C2×C4—81no—
685C2×C4—81no—
686C2×C4—81no—
687C2×C4#39614yes—
688C2×C4#45114yes—
689C2×C4C2×#12111yescharacteristic
690C2×C4C2×C22⋊Q812yes—
691C2×C4C2×C22⋊Q811yescharacteristic
692C2×C4C2×C22⋊Q811yescharacteristic
693C2×C4#52512yes—
694C2×C4C2×#12611yescharacteristic
695C2×C4#53812yes—
696C2×C4#54312yes—
697C2×C4D4212yes—
698C2×C4D4212yes—
699C2×C4#54612yes—
700C2×C4#54612yes—
701C2×C4#54612yes—
702C2×C4#54612yes—
703C2×C4#55112yes—
704C2×C4#55112yes—
705C8—44no—
706C8—82no—
707C22—24no—
708C22—24no—
709C22—24no—
710C22—24no—
711C22—24no—
712C22—24no—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—24no—
717C22—24no—
718C22—24no—
719C22—24no—
720C22—24no—
721C22#206818yescomplemented
722C22C2×#39611yescharacteristic
723C22#240811yescharacteristic
724C22#241011yescharacteristic
725C22#241111yescharacteristic
726C22#253411yescharacteristic
727C22#262611yescharacteristic
728C22#284811yescharacteristic
729C22C2×#45111yescharacteristic
730C22#300211yescharacteristic
731C22#300411yescharacteristic
732C22#300611yescharacteristic
733C22C2×#46311yescharacteristic
734C22C2×#46211yescharacteristic
735C22#303911yescharacteristic
736C22#304011yescharacteristic
737C22C2×#46711yescharacteristic
738C22C2×#46811yescharacteristic
739C22#305611yescharacteristic
740C22#305611yescharacteristic
741C22C2×#47711yescharacteristic
742C22C2×#47911yescharacteristic
743C22#307811yescharacteristic
744C22#307811yescharacteristic
745C22#310411yescharacteristic
746C22#310811yescharacteristic
747C22#311211yescharacteristic
748C22#311711yescharacteristic
749C22#318911yescharacteristic
750C22#319011yescharacteristic
751C22#319311yescharacteristic
752C22#319611yescharacteristic
753C22#320911yescharacteristic
754C22#321111yescharacteristic
755C22#321311yescharacteristic
756C22#321811yescharacteristic
757C4—24no—
758C4—24no—
759C4—42no—
760C4—42no—
761C4—42no—
762C4—48no—
763C4—82no—
764C4—84no—
765C4—82no—
766C4—161no—
767C4—161no—
768C4#242412yes—
769C4#243712yes—
770C4#243712yes—
771C4#248012yes—
772C2—28no—
773C2#1472611yescharacteristic
774C2C2×#206811yescharacteristic
775C2#2361911yescharacteristic
776C2#2421211yescharacteristic
777C2#2421511yescharacteristic
778C2#2438011yescharacteristic
779C2#2438711yescharacteristic
780C2#2450211yescharacteristic
781C2#2450911yescharacteristic
782C2#2811811yescharacteristic
783C2#2812111yescharacteristic
784C2#2995311yescharacteristic
785C2#2995711yescharacteristic
786C2#3295711yescharacteristic
787C2#3296011yescharacteristic
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×#2068C2semidirect
8. #11124C2semidirect
9. #11077C2semidirect
10. #11067C2semidirect
721. C22#2068semidirect

Maximal subgroups

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

subgroupindex
C22×#3912
C2×#20682
C2×#18522
C2×#17972
C2×#17872
#146972
#111242
#110772
#110672
#107302