Finite Group Databasebeta

← #368210

Subgroups — #368210

Order 512.

1904 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2352C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#13872C211yesmaximal, characteristic
6#11268C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10731C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#107C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20#2352—24no—
21#2352—24no—
22C2×#388—21no—
23C2×#387—21no—
24C2×#386—21no—
25C2×#386—21no—
26C2×#382C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#1869C418yescomplemented
34#1852—24no—
35C2×#340—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1441—21no—
45#1439—21no—
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—42no—
56C23×C8—42no—
57C23×C8C2×C414yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C414yes—
65C2×C4×D4Q814yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—24no—
69C2×#117—42no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D414yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97C2×#107—24no—
98C2×#107—24no—
99C2×#107—42no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8C2×C414yes—
105C2×C22⋊C8C2×C418yescomplemented
106#388—28no—
107#387—28no—
108#386—28no—
109#386—44no—
110#382—22no—
111#382—22no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—24no—
118#380—42no—
119#380D414yes—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—28no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—42no—
137#340—42no—
138#340—42no—
139#336C2×C418yescomplemented
140C25C2×D411yescharacteristic
141C25C2×D411yescharacteristic
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C4⋊C414yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×Q812yes—
186C23×C4C2×Q812yes—
187C23×C4#4112yes—
188C23×C4#4112yes—
189C23×C4C2411yescharacteristic
190C22×C8—28no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—48no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8—48no—
207C22×C8C22×C414yes—
208C22×C8C22×C414yes—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—28no—
213#117—48no—
214#117—44no—
215#117—44no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
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—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C4⋊C414yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4#4114yes—
256C2×C4⋊C4#4114yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4C4⋊C414yes—
275C2×C22⋊C4C4⋊C414yes—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—44no—
283C2×C42—82no—
284C2×C42C22×C412yes—
285C2×C42C2×Q812yes—
286#107—48no—
287C22⋊C8—28no—
288C22⋊C8—48no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—82no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—21no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24C2×C4⋊C412yes—
313C24C2×C4⋊C412yes—
314C24#12012yes—
315C24#12312yes—
316C24#13512yes—
317C24#13512yes—
318C24AGL(1,ℤ/8)12yes—
319C24#13712yes—
320C24C22×D411yescharacteristic
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—48no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—84no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C2×C4⋊C412yes—
421C22×C4C2×C4⋊C412yes—
422C22×C4C2×C4⋊C412yes—
423C22×C4C2×C4⋊C412yes—
424C22×C4C4×D414yes—
425C22×C4C4×D414yes—
426C22×C4#12012yes—
427C22×C4C22⋊Q814yes—
428C22×C4C22⋊Q814yes—
429C22×C4#12312yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×Q811yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—416no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—28no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4#12314yes—
478C22⋊C4—28no—
479C22⋊C4—28no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—84no—
483C22⋊C4—88no—
484C22⋊C4—84no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—48no—
490C42—84no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23#38514yes—
531C23#42714yes—
532C23#42814yes—
533C23#44914yes—
534C23#45214yes—
535C23#48214yes—
536C23#48514yes—
537C23C22×C4⋊C411yescharacteristic
538C23C2×#12011yescharacteristic
539C23C2×#12311yescharacteristic
540C23#54612yes—
541C23#54612yes—
542C23C2×#13511yescharacteristic
543C23C2×#13511yescharacteristic
544C23C2×AGL(1,ℤ/8)11yescharacteristic
545C23C2×#13711yescharacteristic
546C23#57612yes—
547C23#57812yes—
548D4—416no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—816no—
600C2×C4—816no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4#38514yes—
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C4×#4112yes—
622C2×C4#51812yes—
623C2×C4C2×C22⋊Q811yescharacteristic
624C2×C4C2×C22⋊Q811yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4#52511yescharacteristic
627C2×C4#52912yes—
628C2×C4#53411yescharacteristic
629C2×C4#53511yescharacteristic
630C2×C4#53812yes—
631C2×C4#54612yes—
632C2×C4#54612yes—
633C2×C4#55312yes—
634C2×C4#55312yes—
635C2×C4#55412yes—
636C2×C4#55412yes—
637C8—44no—
638C8—44no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—42no—
654C22—82no—
655C22—82no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22#185418yes—
661C22C2×#38511yescharacteristic
662C22#234912yes—
663C22#240411yescharacteristic
664C22#240511yescharacteristic
665C22#244412yes—
666C22C2×#42711yescharacteristic
667C22C2×#42811yescharacteristic
668C22#291312yes—
669C22C2×#44911yescharacteristic
670C22C2×#45211yescharacteristic
671C22#300811yescharacteristic
672C22#301111yescharacteristic
673C22C2×#48211yescharacteristic
674C22C2×#48511yescharacteristic
675C22#309812yes—
676C22#329712yes—
677C22#329712yes—
678C22#330012yes—
679C22#330012yes—
680C22#330812yes—
681C22#331212yes—
682C22#331412yes—
683C22#331812yes—
684C4—24no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—44no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4—82no—
693C4—162no—
694C4C4×#12311yescharacteristic
695C4#232111yescharacteristic
696C4#236412yes—
697C4#245012yes—
698C4#248711yescharacteristic
699C4#257911yescharacteristic
700C2—28no—
701C2—82no—
702C2#1401611yescharacteristic
703C2C2×#185411yescharacteristic
704C2#2084711yescharacteristic
705C2#2270412yes—
706C2#2270912yes—
707C2#2396311yescharacteristic
708C2#2396711yescharacteristic
709C2#2403511yescharacteristic
710C2#2403911yescharacteristic
711C2#2620812yes—
712C2#2621012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1852C2semidirect
4. C22×#340C2semidirect
6. #11268C2semidirect
7. #11096C2semidirect
8. #10907C2semidirect
9. #10731C2semidirect
10. #10730C2semidirect
11. #10721C2semidirect
33. #1869C4semidirect
105. C2×C22⋊C8C2×C4semidirect
139. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23522
C2×#18692
C2×#18522
C22×#3402
#138722
#112682
#110962
#109072
#107312
#107302
#107212