Finite Group Databasebeta

← #158424

Subgroups — #158424

Order 512.

3476 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic
3C22×#391C211yesmaximal, characteristic
4C2×#1446C211yesmaximal, characteristic
5C2×#1445C211yesmaximal, characteristic
6C2×#1436C211yesmaximal, characteristic
7#7557C214yesmaximal
8#7557C214yesmaximal
9#7554C211yesmaximal, characteristic
10C24×Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C4⋊C4C2212yes—
15C23×C42C2211yescharacteristic
16C2×#398—24no—
17C2×#398—24no—
18C2×#398—24no—
19C2×#395—24no—
20C2×#395—24no—
21C2×#395—24no—
22C2×#391—24no—
23C2×#391—24no—
24C2×#391—24no—
25C22×#95C2211yescharacteristic
26C22×#95C2212yes—
27C22×#95C2214yes—
28C22×#95C2212yes—
29C22×#95C2212yes—
30C22×#95C2214yes—
31C22×#95C2214yes—
32C22×#95C2211yescharacteristic
33C22×#95C2211yescharacteristic
34C22×#95C2212yes—
35C22×#95C2214yes—
36#1446—24no—
37#1445—24no—
38#1436—24no—
39C23×Q8—22no—
40C23×Q8—22no—
41C23×Q8—22no—
42C23×Q8—22no—
43C23×Q8—22no—
44C23×Q8—28no—
45C23×Q8—22no—
46C23×Q8—44no—
47C23×Q8D414yes—
48C24×C4C2311yescharacteristic
49C24×C4C2312yes—
50C24×C4C2312yes—
51C24×C4C2312yes—
52C24×C4C2312yes—
53C24×C4C2312yes—
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—28no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4D414yes—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—44no—
84C22×C42D414yes—
85#398—232no—
86#395—232no—
87#391—232no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95—24no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—24no—
99C2×#95—28no—
100C2×#95—28no—
101C2×#95—28no—
102C2×#95—24no—
103C2×#95—24no—
104C2×#95—24no—
105C2×#95—22no—
106C2×#95—28no—
107C2×#95—22no—
108C2×#95—28no—
109C2×#95—22no—
110C2×#95—28no—
111C2×#95—28no—
112C2×#95—44no—
113C2×#95—44no—
114C2×#95—44no—
115C2×#95—44no—
116C2×#95—44no—
117C2×#95D414yes—
118C2×#95D414yes—
119C2×#95Q8116yescomplemented
120C25C2411yescharacteristic
121C22×Q8—216no—
122C22×Q8—216no—
123C22×Q8—28no—
124C22×Q8—28no—
125C22×Q8—28no—
126C22×Q8—216no—
127C22×Q8—48no—
128C22×Q8—416no—
129C22×Q8—416no—
130C22×Q8—416no—
131C22×Q8—416no—
132C22×Q8—48no—
133C22×Q8—48no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—44no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—21no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4C2×D412yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D412yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2×D412yes—
222C23×C4C2×Q818yes—
223C23×C4C2×Q818yes—
224C23×C4C2×Q818yes—
225C23×C4#4114yes—
226C23×C4#4118yes—
227C23×C4#4114yes—
228C23×C4#4112yes—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C42—28no—
254C2×C42—48no—
255C2×C42—48no—
256C2×C42—48no—
257#95—216no—
258#95—232no—
259#95—216no—
260#95—216no—
261#95—28no—
262#95—28no—
263#95—416no—
264#95—416no—
265#95—416no—
266#95—48no—
267#95—44no—
268C24C22×D411yescharacteristic
269C24C22×D411yescharacteristic
270C24C22×D411yescharacteristic
271C24C22×D411yescharacteristic
272C24C22×D411yescharacteristic
273C24C22×D411yescharacteristic
274C24C22×Q814yes—
275C24C2×#4111yescharacteristic
276C24C2×#4114yes—
277C24C2×#4112yes—
278C24C2×#4112yes—
279C24H(5,𝔽2)14yes—
280C24H(5,𝔽2)12yes—
281C242−1+414yes—
282C242−1+412yes—
283C2×Q8—232no—
284C2×Q8—432no—
285C2×Q8—432no—
286C2×Q8—416no—
287C2×Q8—432no—
288C2×Q8—432no—
289C2×Q8—432no—
290C2×Q8—432no—
291C2×Q8—416no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
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—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4#12014yes—
472C22×C4#12014yes—
473C22×C4#12014yes—
474C22×C4#12114yes—
475C22×C4#12114yes—
476C22×C4#12114yes—
477C22×C4#12318yes—
478C22×C4#12414yes—
479C22×C4#12714yes—
480C22×C4C4⋊Q818yes—
481C22×C4C4⋊Q818yes—
482C22×C4C4⋊Q818yes—
483C4⋊C4—432no—
484C4⋊C4—432no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C42—416no—
488C23C2×#12011yescharacteristic
489C23C2×#12011yescharacteristic
490C23C2×#12011yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×#12111yescharacteristic
493C23C2×#12111yescharacteristic
494C23C2×#12312yes—
495C23#52512yes—
496C23#52512yes—
497C23#52512yes—
498C23C2×#12411yescharacteristic
499C23#52912yes—
500C23C2×#12711yescharacteristic
501C23C2×C4⋊Q812yes—
502C23C2×C4⋊Q812yes—
503C23C2×C4⋊Q812yes—
504C23#53211yescharacteristic
505C23#53314yes—
506C23#53314yes—
507C23#53412yes—
508C23#53412yes—
509C23#53512yes—
510C23#53511yescharacteristic
511C23#53612yes—
512C23#53611yescharacteristic
513C23#53612yes—
514C23#53612yes—
515C23#53712yes—
516C23#53914yes—
517C23#54114yes—
518C23#54212yes—
519C23#54214yes—
520C23#54214yes—
521C23#54314yes—
522C23#54414yes—
523C23#54614yes—
524C23#54614yes—
525C23#54614yes—
526C23#54612yes—
527C23#54612yes—
528C23#54612yes—
529C23#54712yes—
530C23#54812yes—
531C23#54812yes—
532C23#54812yes—
533C23D4×Q814yes—
534C23D4×Q814yes—
535C23D4×Q814yes—
536C23#55012yes—
537C23#55114yes—
538C23#55214yes—
539C23#55414yes—
540C23#55414yes—
541C23#55414yes—
542C23#55614yes—
543C23#55714yes—
544C23#55714yes—
545C23#56214yes—
546C23#56314yes—
547Q8—464no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—216no—
568C2×C4—216no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—81no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—84no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—84no—
637C2×C4—82no—
638C2×C4#39218yes—
639C22C2×#39211yescharacteristic
640C22#239912yes—
641C22#240412yes—
642C22#240512yes—
643C22#240712yes—
644C22#241112yes—
645C22#241212yes—
646C22#242314yes—
647C22#242612yes—
648C22#242712yes—
649C22#243111yescharacteristic
650C22#243211yescharacteristic
651C22#243412yes—
652C22#243412yes—
653C22#243412yes—
654C22#243511yescharacteristic
655C22#243611yescharacteristic
656C22#244614yes—
657C22#246214yes—
658C22#246514yes—
659C22#249214yes—
660C22#252014yes—
661C22#253412yes—
662C22#253912yes—
663C22#254012yes—
664C22#254612yes—
665C22#254612yes—
666C22#254612yes—
667C22#254914yes—
668C22#254914yes—
669C22#255014yes—
670C22#255014yes—
671C22#255614yes—
672C22#258312yes—
673C22#258412yes—
674C22#258814yes—
675C22#259912yes—
676C22#261311yescharacteristic
677C22#262011yescharacteristic
678C22#263712yes—
679C22#263812yes—
680C22#265811yescharacteristic
681C22#265912yes—
682C22#268714yes—
683C22#269014yes—
684C22#269214yes—
685C22#269214yes—
686C22#270014yes—
687C22#271314yes—
688C22#271614yes—
689C22#272014yes—
690C22#272914yes—
691C22#273214yes—
692C22#275514yes—
693C22#279914yes—
694C22#281412yes—
695C22#284911yescharacteristic
696C22#285011yescharacteristic
697C22#286811yescharacteristic
698C4—28no—
699C4—48no—
700C4—48no—
701C4—48no—
702C4—44no—
703C4—48no—
704C4—82no—
705C4—84no—
706C4—84no—
707C2#1487912yes—
708C2#1491111yescharacteristic
709C2#1509712yes—
710C2#1517312yes—
711C2#1519011yescharacteristic
712C2#1521511yescharacteristic
713C2#1546612yes—
714C2#1550014yes—
715C2#1567112yes—
716C2#1574014yes—
717C2#1578312yes—
718C2#1584912yes—
719C2#1660612yes—
720C2#1680612yes—
721C2#1681212yes—
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
119. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#3982
C22×#3952
C22×#3912
C2×#14462
C2×#14452
C2×#14362
#75572
#75542