Finite Group Databasebeta

← #158398

Subgroups — #158398

Order 512.

3840 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#395C212yesmaximal
2C22×#391C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4C2×#1444C212yesmaximal
5C2×#1441C212yesmaximal
6#7557C212yesmaximal
7#7557C214yesmaximal
8#7554C211yesmaximal, characteristic
9C24×Q8C2211yescharacteristic
10C23×C4⋊C4C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2214yes—
15C23×C42C2211yescharacteristic
16C2×#395—28no—
17C2×#395—28no—
18C2×#395—28no—
19C2×#391—24no—
20C2×#391—28no—
21C2×#389—24no—
22C2×#389—28no—
23C22×#95C2212yes—
24C22×#95C2212yes—
25C22×#95C2214yes—
26C22×#95C2214yes—
27C22×#95C2214yes—
28C22×#95C2212yes—
29C22×#95C2212yes—
30C22×#95C2212yes—
31C22×#95C2212yes—
32#1444—28no—
33#1441—28no—
34C23×Q8—24no—
35C23×Q8—22no—
36C23×Q8—24no—
37C23×Q8—24no—
38C23×Q8—24no—
39C23×Q8—22no—
40C23×Q8—42no—
41C23×Q8—42no—
42C23×Q8D414yes—
43C24×C4C2312yes—
44C24×C4C2312yes—
45C24×C4C2312yes—
46C24×C4C2311yescharacteristic
47C24×C4C2314yes—
48C24×C4C2312yes—
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—28no—
56C22×C4⋊C4—28no—
57C22×C4⋊C4—28no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4—44no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4D418yes—
77C22×C4⋊C4Q814yescomplemented
78C22×C42—24no—
79C22×C42—24no—
80C22×C42—42no—
81C22×C42D414yes—
82#395—264no—
83#391—232no—
84#389—232no—
85C2×#95—28no—
86C2×#95—28no—
87C2×#95—28no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—28no—
101C2×#95—28no—
102C2×#95—28no—
103C2×#95—28no—
104C2×#95—42no—
105C2×#95—42no—
106C2×#95—44no—
107C2×#95—44no—
108C2×#95D418yes—
109C2×#95D418yes—
110C2×#95Q818yescomplemented
111C2×#95Q818yescomplemented
112C25C2411yescharacteristic
113C22×Q8—28no—
114C22×Q8—216no—
115C22×Q8—28no—
116C22×Q8—216no—
117C22×Q8—216no—
118C22×Q8—28no—
119C22×Q8—48no—
120C22×Q8—416no—
121C22×Q8—416no—
122C22×Q8—416no—
123C22×Q8—416no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—48no—
127C22×Q8—44no—
128C22×Q8—48no—
129C22×Q8—44no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D418yes—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D414yes—
210C23×C4C2×Q814yes—
211C23×C4C2×Q818yes—
212C23×C4C2×Q818yes—
213C23×C4C2×Q814yes—
214C23×C4C2×Q812yes—
215C23×C4C2×Q814yes—
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4#4112yes—
219C23×C4#4112yes—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—232no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—416no—
232C2×C4⋊C4—416no—
233C2×C4⋊C4—416no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—48no—
247C2×C42—28no—
248C2×C42—28no—
249C2×C42—44no—
250C2×C42—416no—
251#95—216no—
252#95—216no—
253#95—232no—
254#95—216no—
255#95—216no—
256#95—216no—
257#95—48no—
258#95—416no—
259#95—416no—
260C24C22×D411yescharacteristic
261C24C22×D411yescharacteristic
262C24C22×D412yes—
263C24C22×D412yes—
264C24C22×D412yes—
265C24C22×Q811yescharacteristic
266C24C22×Q812yes—
267C24C22×Q812yes—
268C24C2×#4111yescharacteristic
269C24C2×#4112yes—
270C24C2×#4111yescharacteristic
271C24C2×#4112yes—
272C24H(5,𝔽2)12yes—
273C24H(5,𝔽2)12yes—
274C24H(5,𝔽2)11yescharacteristic
275C242−1+412yes—
276C242−1+411yescharacteristic
277C242−1+412yes—
278C242−1+412yes—
279C2×Q8—232no—
280C2×Q8—464no—
281C2×Q8—416no—
282C2×Q8—416no—
283C2×Q8—416no—
284C2×Q8—464no—
285C2×Q8—416no—
286C2×Q8—416no—
287C2×Q8—416no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4#12018yes—
465C22×C4#12014yes—
466C22×C4#12014yes—
467C22×C4#12118yes—
468C22×C4#12118yes—
469C22×C4#12114yes—
470C22×C4C22⋊Q818yes—
471C22×C4C22⋊Q818yes—
472C22×C4C22⋊Q814yes—
473C22×C4#12714yes—
474C22×C4C4⋊Q814yes—
475C22×C4C4⋊Q8116yes—
476C22×C4C4⋊Q814yes—
477C22×C4C4⋊Q818yes—
478C4⋊C4—232no—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—464no—
482C4⋊C4—432no—
483C42—416no—
484C23C2×#12011yescharacteristic
485C23C2×#12011yescharacteristic
486C23C2×#12012yes—
487C23C2×#12111yescharacteristic
488C23C2×#12112yes—
489C23C2×#12112yes—
490C23C2×C22⋊Q811yescharacteristic
491C23C2×C22⋊Q812yes—
492C23C2×C22⋊Q812yes—
493C23#52511yescharacteristic
494C23#52512yes—
495C23#52511yescharacteristic
496C23#52512yes—
497C23C2×#12711yescharacteristic
498C23C2×C4⋊Q811yescharacteristic
499C23C2×C4⋊Q811yescharacteristic
500C23C2×C4⋊Q812yes—
501C23C2×C4⋊Q814yes—
502C23#53314yes—
503C23#53411yescharacteristic
504C23#53412yes—
505C23#53511yescharacteristic
506C23#53611yescharacteristic
507C23#53611yescharacteristic
508C23#53611yescharacteristic
509C23#53612yes—
510C23#53612yes—
511C23#53614yes—
512C23#53711yescharacteristic
513C23#53712yes—
514C23#53712yes—
515C23#53812yes—
516C23#53912yes—
517C23#54114yes—
518C23#54214yes—
519C23#54312yes—
520C23#54412yes—
521C23#54412yes—
522C23#54414yes—
523C23D4214yes—
524C23D4214yes—
525C23#54614yes—
526C23#54612yes—
527C23#54614yes—
528C23#54714yes—
529C23#54712yes—
530C23#54812yes—
531C23#54812yes—
532C23D4×Q811yescharacteristic
533C23D4×Q812yes—
534C23D4×Q814yes—
535C23D4×Q814yes—
536C23D4×Q812yes—
537C23D4×Q814yes—
538C23D4×Q812yes—
539C23#55011yescharacteristic
540C23#55212yes—
541C23#55412yes—
542C23#55414yes—
543C23#55411yescharacteristic
544C23#55412yes—
545C23#55611yescharacteristic
546C23#55612yes—
547C23#55712yes—
548C23#55712yes—
549C23#55712yes—
550C23#55712yes—
551C23Q8212yes—
552C23#56212yes—
553C23#56314yes—
554C23#56312yes—
555Q8—464no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—216no—
566C2×C4—216no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—216no—
570C2×C4—216no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—216no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—216no—
581C2×C4—216no—
582C2×C4—216no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4#39218yes—
645C2×C4#39518yescomplemented
646C22C2×#39211yescharacteristic
647C22C2×#39511yescharacteristic
648C22#239911yescharacteristic
649C22#240311yescharacteristic
650C22#240411yescharacteristic
651C22#240611yescharacteristic
652C22#240712yes—
653C22#240912yes—
654C22#241014yes—
655C22#241512yes—
656C22#241712yes—
657C22#242414yes—
658C22#242914yes—
659C22#243311yescharacteristic
660C22#243411yescharacteristic
661C22#243411yescharacteristic
662C22#243411yescharacteristic
663C22#243414yes—
664C22#244014yes—
665C22#244014yes—
666C22#244914yes—
667C22#245114yes—
668C22#246214yes—
669C22#246514yes—
670C22#253611yescharacteristic
671C22#253811yescharacteristic
672C22#253812yes—
673C22#253911yescharacteristic
674C22#254011yescharacteristic
675C22#254211yescharacteristic
676C22#254614yes—
677C22#254611yescharacteristic
678C22#254611yescharacteristic
679C22#254612yes—
680C22#254914yes—
681C22#255014yes—
682C22#255114yes—
683C22#256711yescharacteristic
684C22#258112yes—
685C22#258111yescharacteristic
686C22#258311yescharacteristic
687C22#258412yes—
688C22#258411yescharacteristic
689C22#261611yescharacteristic
690C22#261611yescharacteristic
691C22#261711yescharacteristic
692C22#262711yescharacteristic
693C22#263211yescharacteristic
694C22#263411yescharacteristic
695C22#265811yescharacteristic
696C22#265911yescharacteristic
697C22#266011yescharacteristic
698C22#267214yes—
699C22#267414yes—
700C22#267714yes—
701C22#268314yes—
702C22#268914yes—
703C22#270414yes—
704C22#271314yes—
705C22#271314yes—
706C22#272914yes—
707C22#273212yes—
708C22#273412yes—
709C22#277414yes—
710C22#280512yes—
711C22#280612yes—
712C22#281711yescharacteristic
713C22#282011yescharacteristic
714C22#284711yescharacteristic
715C22#284811yescharacteristic
716C22#286811yescharacteristic
717C4—28no—
718C4—28no—
719C4—48no—
720C4—48no—
721C4—44no—
722C4—416no—
723C4—84no—
724C4—84no—
725C2#1471611yescharacteristic
726C2#1472011yescharacteristic
727C2#1488011yescharacteristic
728C2#1488111yescharacteristic
729C2#1514311yescharacteristic
730C2#1515811yescharacteristic
731C2#1517311yescharacteristic
732C2#1517711yescharacteristic
733C2#1519011yescharacteristic
734C2#1527614yes—
735C2#1531814yes—
736C2#1536014yes—
737C2#1562314yes—
738C2#1625411yescharacteristic
739C2#1651011yescharacteristic
740C2#1660811yescharacteristic
741C2#1681311yescharacteristic
742C2#1699311yescharacteristic
743C2#1707811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
77. C22×C4⋊C4Q8semidirect
110. C2×#95Q8semidirect
111. C2×#95Q8semidirect
645. C2×C4#395semidirect

Maximal subgroups

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

subgroupindex
C22×#3952
C22×#3912
C22×#3892
C2×#14442
C2×#14412
#75572
#75542