Finite Group Databasebeta

← #394426

Subgroups — #394426

Order 512.

2404 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3C2×#2024C212yesmaximal, complemented
4C2×#1787C211yesmaximal, characteristic
5#13922C211yesmaximal, characteristic
6#11067C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#11050C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8—21no—
16C23×C4⋊C4C2211yescharacteristic
17C22×#103—28no—
18C22×#103—24no—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24#2440—24no—
25#2440—24no—
26C2×#398—21no—
27C2×#395—21no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#385—21no—
33C2×#384C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#2024—28no—
38#1787—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1446—21no—
46#1446—21no—
47C22⋊#95—21no—
48C22⋊#95—21no—
49C23×Q8—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×Q8Q814yescomplemented
61C23×Q8C2312yes—
62C24×C4—21no—
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C23×C8—22no—
66C23×C8—24no—
67C23×C8—22no—
68C23×C8—22no—
69C23×C8—42no—
70C23×C8C2312yes—
71C23×C8C2312yes—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—42no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—44no—
78C2×C22⋊Q8—44no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4D414yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C2×#103—232no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—42no—
111C2×#103—42no—
112C2×#103D418yes—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116#398—28no—
117#395—28no—
118#391—24no—
119#391—24no—
120#391—42no—
121#391—42no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#385—28no—
127#384—22no—
128#384—22no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380D414yes—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—21no—
143C2×#95D411yescharacteristic
144C2×#95D411yescharacteristic
145#340—24no—
146#340—24no—
147#340—42no—
148#340—42no—
149#336—24no—
150#336—42no—
151C25C2×D411yescharacteristic
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—28no—
158C22×Q8—216no—
159C22×Q8—28no—
160C22×Q8—216no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—44no—
172C22×Q8—44no—
173C22×Q8—44no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—48no—
178C22×Q8—48no—
179C22×Q8—48no—
180C22×Q8—48no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—48no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8—82no—
191C22×Q8C2×D414yes—
192C22×Q8C2×Q814yes—
193C22×Q8#4114yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—82no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2×Q812yes—
222C23×C4#4112yes—
223C23×C4#4112yes—
224C23×C4#4112yes—
225C23×C4C2411yescharacteristic
226C22×C8—28no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—48no—
241C22×C8—48no—
242C22×C8—44no—
243C22×C8—44no—
244C22×C8C2×D414yes—
245C4⋊Q8—48no—
246C4⋊Q8—44no—
247C4⋊Q8—44no—
248C22⋊Q8—432no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D412yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4SD1614yes—
300C2×C22⋊C4Q1614yes—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309#103—216no—
310#103—216no—
311#103—48no—
312#103—48no—
313C22⋊C8—44no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—42no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24#12012yes—
329C24C22⋊Q812yes—
330C24#12412yes—
331C24C2×SD1612yes—
332C24C2×Q1612yes—
333C24AGL(1,ℤ/8)12yes—
334C24#13712yes—
335C24C22×D411yescharacteristic
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—232no—
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—416no—
353C2×Q8—432no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—416no—
357C2×Q8—416no—
358C2×Q8—416no—
359C2×Q8—416no—
360C2×Q8—432no—
361C2×Q8—48no—
362C2×Q8—48no—
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×Q8—88no—
377C2×Q8—84no—
378C2×Q8—88no—
379C2×Q8C22⋊Q818yescomplemented
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
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—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—84no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4#12012yes—
457C22×C4C22⋊Q814yes—
458C22×C4C22⋊Q814yes—
459C22×C4C22⋊Q812yes—
460C22×C4#12414yes—
461C22×C4#12414yes—
462C22×C4#12412yes—
463C22×C4C2×SD1612yes—
464C22×C4C2×SD1612yes—
465C22×C4C2×Q1612yes—
466C22×C4C2×Q1612yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×Q811yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4C2×#4111yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—416no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—84no—
513C4⋊C4#12014yes—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—48no—
517C22⋊C4—84no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—42no—
523C42—42no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23#39314yes—
541C23#45014yes—
542C23#45114yes—
543C23#47514yes—
544C23#47714yes—
545C23#48714yes—
546C23#48914yes—
547C23C2×#12011yescharacteristic
548C23C2×C22⋊Q811yescharacteristic
549C23C2×#12411yescharacteristic
550C23D4212yes—
551C23#54612yes—
552C23C22×SD1611yescharacteristic
553C23C22×Q1611yescharacteristic
554C23C2×AGL(1,ℤ/8)11yescharacteristic
555C23C2×#13711yescharacteristic
556C23#57712yes—
557C23#57812yes—
558Q8—28no—
559Q8—28no—
560Q8—432no—
561Q8—44no—
562Q8—44no—
563Q8—44no—
564Q8—44no—
565Q8—44no—
566Q8—44no—
567Q8—816no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—432no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—816no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4#39314yes—
635C2×C4C2×#12011yescharacteristic
636C2×C4C2×C22⋊Q811yescharacteristic
637C2×C4C2×C22⋊Q812yes—
638C2×C4C2×C22⋊Q811yescharacteristic
639C2×C4#52511yescharacteristic
640C2×C4#52512yes—
641C2×C4C2×#12411yescharacteristic
642C2×C4C2×#12411yescharacteristic
643C2×C4#53411yescharacteristic
644C2×C4#53511yescharacteristic
645C2×C4#53812yes—
646C2×C4#54312yes—
647C2×C4D4212yes—
648C2×C4#54612yes—
649C2×C4#54612yes—
650C2×C4#54612yes—
651C2×C4#55112yes—
652C2×C4#55112yes—
653C8—44no—
654C8—82no—
655C22—24no—
656C22—24no—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—24no—
661C22—28no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22#202418yescomplemented
666C22C2×#39311yescharacteristic
667C22#240311yescharacteristic
668C22#240411yescharacteristic
669C22#243712yes—
670C22#247212yes—
671C22C2×#45011yescharacteristic
672C22C2×#45111yescharacteristic
673C22#300911yescharacteristic
674C22#301111yescharacteristic
675C22C2×#47511yescharacteristic
676C22C2×#47711yescharacteristic
677C22#308212yes—
678C22C2×#48711yescharacteristic
679C22C2×#48911yescharacteristic
680C22#313712yes—
681C22#326512yes—
682C22#327012yes—
683C22D4×SD1612yes—
684C22D4×Q1612yes—
685C22#329812yes—
686C22#329912yes—
687C22#330812yes—
688C22#331612yes—
689C4—24no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—48no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—162no—
700C4#243511yescharacteristic
701C4#243712yes—
702C4#247212yes—
703C4#256111yescharacteristic
704C4#262611yescharacteristic
705C4#284811yescharacteristic
706C2—28no—
707C2#1471711yescharacteristic
708C2C2×#202411yescharacteristic
709C2#2345711yescharacteristic
710C2#2384911yescharacteristic
711C2#2385611yescharacteristic
712C2#2396111yescharacteristic
713C2#2397011yescharacteristic
714C2#2595512yes—
715C2#2596312yes—
716C2#2725812yes—
717C2#2726612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
3. C2×#2024C2semidirect
6. #11067C2semidirect
7. #11067C2semidirect
8. #11050C2semidirect
60. C23×Q8Q8semidirect
379. C2×Q8C22⋊Q8semidirect
665. C22#2024semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#24402
C2×#20242
C2×#17872
#139222
#110672
#110502
#107302
#107212
#107142