Finite Group Databasebeta

← #398596

Subgroups — #398596

Order 512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#1933C212yesmaximal, complemented
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11267C211yesmaximal, characteristic, complemented
8#11200C212yesmaximal
9#11050C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10880C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C22⋊Q8C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2212yes—
20#2857—28no—
21C2×#391—21no—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30#2024—24no—
31#1933—28no—
32C4×C22⋊C8—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C22⋊Q8—28no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—44no—
64C2×C22⋊Q8—44no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42—24no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—42no—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C4×C8—24no—
96C2×C4×C8—24no—
97C2×C4×C8—42no—
98#391—28no—
99#391—24no—
100#391—22no—
101#391—22no—
102#389—28no—
103#389—24no—
104#389D418yes—
105#384—24no—
106#380—28no—
107#380—24no—
108#380—24no—
109#380—24no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—42no—
127#336—22no—
128#336D414yes—
129C25C2×D411yescharacteristic
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8#4112yes—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4SD1614yes—
175C23×C4Q1614yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4114yes—
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C2×D414yes—
196C22×C8#4114yes—
197C4⋊Q8—48no—
198C4⋊Q8—48no—
199C22⋊Q8—48no—
200C22⋊Q8—48no—
201C22⋊Q8—432no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
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—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4#4112yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—48no—
260C2×C42—48no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42C2×D414yes—
264C2×C42C2×D412yes—
265C2×C42#4112yes—
266#103—216no—
267#103—48no—
268C22⋊C8—216no—
269C4×C8—48no—
270#95—28no—
271#95—28no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—42no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24#12412yes—
294C24C2×SD1612yes—
295C24C2×Q1612yes—
296C24C22×D411yescharacteristic
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—42no—
305C2×Q8—41no—
306C2×Q8—42no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×Q8—84no—
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—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
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—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4#12012yes—
415C22×C4#12012yes—
416C22×C4#12112yes—
417C22×C4#12112yes—
418C22×C4#12412yes—
419C22×C4C2×SD1612yes—
420C22×C4C2×SD1612yes—
421C22×C4C2×Q1612yes—
422C22×C4C2×Q1612yes—
423C22×C4#13512yes—
424C22×C4#13512yes—
425C22×C4AGL(1,ℤ/8)12yes—
426C22×C4#13712yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—416no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—84no—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—44no—
478C42—44no—
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—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23#48714yes—
514C23C2×#12011yescharacteristic
515C23C2×#12111yescharacteristic
516C23#52511yescharacteristic
517C23C2×#12411yescharacteristic
518C23#53211yescharacteristic
519C23#53411yescharacteristic
520C23#53611yescharacteristic
521C23C22×SD1611yescharacteristic
522C23C22×Q1611yescharacteristic
523C23C2×#13511yescharacteristic
524C23C2×#13511yescharacteristic
525C23C2×AGL(1,ℤ/8)11yescharacteristic
526C23C2×#13711yescharacteristic
527C23#57511yescharacteristic
528C23#57511yescharacteristic
529Q8—44no—
530Q8—44no—
531Q8—816no—
532Q8—82no—
533Q8—82no—
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—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4#45014yes—
614C2×C4#45114yes—
615C2×C4#46514yes—
616C2×C4#46714yes—
617C2×C4#52512yes—
618C2×C4#52512yes—
619C2×C4#52912yes—
620C2×C4#53212yes—
621C2×C4#53912yes—
622C2×C4#54212yes—
623C2×C4D4212yes—
624C2×C4#54612yes—
625C2×C4#54711yescharacteristic
626C2×C4#54812yes—
627C2×C4#54812yes—
628C2×C4#54811yescharacteristic
629C2×C4#55112yes—
630C2×C4#55112yes—
631C2×C4#55211yescharacteristic
632C2×C4#55611yescharacteristic
633C8—44no—
634C8—44no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22#238811yescharacteristic
651C22#241011yescharacteristic
652C22#241511yescharacteristic
653C22#243611yescharacteristic
654C22#256111yescharacteristic
655C22#266311yescharacteristic
656C22#285011yescharacteristic
657C22C2×#45011yescharacteristic
658C22C2×#45111yescharacteristic
659C22#300211yescharacteristic
660C22#300211yescharacteristic
661C22C2×#46511yescharacteristic
662C22C2×#46711yescharacteristic
663C22#305511yescharacteristic
664C22#305611yescharacteristic
665C22#310311yescharacteristic
666C22#310411yescharacteristic
667C22#310611yescharacteristic
668C22#310711yescharacteristic
669C22C2×#48711yescharacteristic
670C22#313111yescharacteristic
671C22#314011yescharacteristic
672C22#314111yescharacteristic
673C22#316711yescharacteristic
674C22#317011yescharacteristic
675C22#317111yescharacteristic
676C22#317211yescharacteristic
677C22#318711yescharacteristic
678C22#318811yescharacteristic
679C22#318911yescharacteristic
680C22#319011yescharacteristic
681C22#322911yescharacteristic
682C22#323011yescharacteristic
683C22#323311yescharacteristic
684C22#323311yescharacteristic
685C4—24no—
686C4—24no—
687C4—24no—
688C4—24no—
689C4—42no—
690C4—84no—
691C4—84no—
692C4—82no—
693C4—82no—
694C4—161no—
695C4—161no—
696C4#246112yes—
697C4#246312yes—
698C4#247712yes—
699C4#256012yes—
700C2—44no—
701C2#1476511yescharacteristic
702C2#2357011yescharacteristic
703C2#2357111yescharacteristic
704C2#2437911yescharacteristic
705C2#2438011yescharacteristic
706C2#2470211yescharacteristic
707C2#2470311yescharacteristic
708C2#2590211yescharacteristic
709C2#2590611yescharacteristic
710C2#2910511yescharacteristic
711C2#2910711yescharacteristic
712C2#3035811yescharacteristic
713C2#3035911yescharacteristic
714C2#3304311yescharacteristic
715C2#3305711yescharacteristic
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×#2024C2semidirect
3. C2×#1933C2semidirect
6. #11384C2semidirect
7. #11267C2semidirect
9. #11050C2semidirect
11. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20242
C2×#19332
C2×C4×C22⋊C82
#147022
#113842
#112672
#112002
#110502
#109702
#108802