Finite Group Databasebeta

← #158402

Subgroups — #158402

Order 512.

2928 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#400C212yesmaximal
2C22×#389C211yesmaximal, characteristic
3C2×#1441C212yesmaximal
4C2×C4⋊#95C211yesmaximal, characteristic
5#7557C212yesmaximal
6#7557C212yesmaximal
7#7556C214yesmaximal
8#7554C211yesmaximal, characteristic
9C23×C4⋊C4C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#400—28no—
16C2×#400—28no—
17C2×#400—28no—
18C2×#389—24no—
19C2×#389—28no—
20C22×#95C2212yes—
21C22×#95C2212yes—
22C22×#95C2212yes—
23C22×#95C2214yes—
24C22×#95C2214yes—
25C22×#95C2212yes—
26C22×#95C2214yes—
27C22×#95C2214yes—
28C22×#95C2212yes—
29C22×#95C2212yes—
30#1441—28no—
31C4⋊#95—24no—
32C24×C4C2312yes—
33C24×C4C2312yes—
34C24×C4C2314yes—
35C24×C4C2312yes—
36C24×C4C2312yes—
37C24×C4C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C22×C4⋊C4—24no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—42no—
61C22×C4⋊C4—42no—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4D418yes—
64C22×C4⋊C4Q814yes—
65C22×C42—24no—
66C22×C42—24no—
67C22×C42—42no—
68C22×C42D414yes—
69#400—264no—
70#389—232no—
71C2×#95—28no—
72C2×#95—28no—
73C2×#95—28no—
74C2×#95—28no—
75C2×#95—28no—
76C2×#95—28no—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95—24no—
84C2×#95—28no—
85C2×#95—28no—
86C2×#95—28no—
87C2×#95—28no—
88C2×#95—24no—
89C2×#95—28no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—44no—
96C2×#95—44no—
97C2×#95—42no—
98C2×#95—42no—
99C2×#95D418yes—
100C25C2411yescharacteristic
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4C2×D414yes—
173C23×C4C2×D414yes—
174C23×C4C2×D418yes—
175C23×C4C2×D414yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D414yes—
179C23×C4C2×D414yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×Q814yes—
183C23×C4C2×Q812yes—
184C23×C4#4114yes—
185C23×C4#4114yes—
186C23×C4#4114yes—
187C23×C4#4112yes—
188C23×C4#4112yes—
189C23×C4#4114yes—
190C23×C4#4114yes—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—216no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—216no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—216no—
199C2×C4⋊C4—216no—
200C2×C4⋊C4—216no—
201C2×C4⋊C4—216no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—416no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C42—28no—
217C2×C42—28no—
218C2×C42—44no—
219C2×C42—416no—
220#95—216no—
221#95—232no—
222#95—232no—
223#95—216no—
224#95—416no—
225#95—416no—
226#95—48no—
227#95—48no—
228#95—48no—
229#95—48no—
230C24C22×D412yes—
231C24C22×D411yescharacteristic
232C24C22×D411yescharacteristic
233C24C22×D412yes—
234C24C22×Q811yescharacteristic
235C24C2×#4111yescharacteristic
236C24C2×#4111yescharacteristic
237C24C2×#4112yes—
238C24C2×#4112yes—
239C24C2×#4112yes—
240C24C2×#4112yes—
241C24C2×#4112yes—
242C24H(5,𝔽2)12yes—
243C24H(5,𝔽2)11yescharacteristic
244C242−1+412yes—
245C242−1+412yes—
246C242−1+411yescharacteristic
247C242−1+412yes—
248C242−1+412yes—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—24no—
301C22×C4—28no—
302C22×C4—24no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
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—44no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4#12018yes—
428C22×C4#12014yes—
429C22×C4#12114yes—
430C22×C4#12118yes—
431C22×C4C22⋊Q818yes—
432C22×C4C22⋊Q814yes—
433C22×C4#12314yes—
434C22×C4#12414yes—
435C22×C4#12414yes—
436C22×C4#12518yes—
437C22×C4#12618yes—
438C22×C4#12714yes—
439C4⋊C4—232no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—432no—
443C4⋊C4—416no—
444C42—416no—
445C23C2×#12011yescharacteristic
446C23C2×#12012yes—
447C23C2×#12111yescharacteristic
448C23C2×#12112yes—
449C23C2×C22⋊Q811yescharacteristic
450C23C2×C22⋊Q812yes—
451C23C2×#12311yescharacteristic
452C23#52511yescharacteristic
453C23#52511yescharacteristic
454C23#52512yes—
455C23C2×#12411yescharacteristic
456C23C2×#12411yescharacteristic
457C23C2×#12512yes—
458C23C2×#12612yes—
459C23#52912yes—
460C23#52914yes—
461C23C2×#12711yescharacteristic
462C23#53212yes—
463C23#53214yes—
464C23#53411yescharacteristic
465C23#53611yescharacteristic
466C23#53611yescharacteristic
467C23#53611yescharacteristic
468C23#53611yescharacteristic
469C23#53612yes—
470C23#53614yes—
471C23#53711yescharacteristic
472C23#53712yes—
473C23#53712yes—
474C23#53812yes—
475C23#53912yes—
476C23#53912yes—
477C23#53912yes—
478C23#54112yes—
479C23#54114yes—
480C23#54114yes—
481C23#54114yes—
482C23#54212yes—
483C23#54412yes—
484C23#54614yes—
485C23#54612yes—
486C23#54614yes—
487C23#54714yes—
488C23#54712yes—
489C23#54714yes—
490C23#54712yes—
491C23#54712yes—
492C23#54812yes—
493C23#54812yes—
494C23#54814yes—
495C23#54814yes—
496C23D4×Q811yescharacteristic
497C23D4×Q812yes—
498C23#55011yescharacteristic
499C23#55212yes—
500C23#55212yes—
501C23#55214yes—
502C23#55214yes—
503C23#55312yes—
504C23#55314yes—
505C23#55411yescharacteristic
506C23#55612yes—
507C23#55612yes—
508C23#55611yescharacteristic
509C23#55612yes—
510C23#55612yes—
511C23#55712yes—
512C23#55712yes—
513C23#56314yes—
514C23#56312yes—
515C23#56412yes—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—216no—
525C2×C4—216no—
526C2×C4—24no—
527C2×C4—216no—
528C2×C4—216no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—82no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4#39218yes—
605C2×C4#40018yescomplemented
606C22C2×#39211yescharacteristic
607C22C2×#40011yescharacteristic
608C22#239811yescharacteristic
609C22#240511yescharacteristic
610C22#240611yescharacteristic
611C22#240712yes—
612C22#240912yes—
613C22#241512yes—
614C22#241712yes—
615C22#242514yes—
616C22#243014yes—
617C22#243214yes—
618C22#243611yescharacteristic
619C22#243611yescharacteristic
620C22#244214yes—
621C22#245314yes—
622C22#246914yes—
623C22#248114yes—
624C22#248411yescharacteristic
625C22#249614yes—
626C22#250111yescharacteristic
627C22#250311yescharacteristic
628C22#250614yes—
629C22#250612yes—
630C22#250912yes—
631C22#251212yes—
632C22#251412yes—
633C22#251914yes—
634C22#253314yes—
635C22#253611yescharacteristic
636C22#253811yescharacteristic
637C22#253911yescharacteristic
638C22#254011yescharacteristic
639C22#254211yescharacteristic
640C22#254514yes—
641C22#254711yescharacteristic
642C22#254711yescharacteristic
643C22#255814yes—
644C22#255914yes—
645C22#256211yescharacteristic
646C22#256811yescharacteristic
647C22#258111yescharacteristic
648C22#259111yescharacteristic
649C22#261611yescharacteristic
650C22#261811yescharacteristic
651C22#262511yescharacteristic
652C22#262711yescharacteristic
653C22#263411yescharacteristic
654C22#263411yescharacteristic
655C22#263911yescharacteristic
656C22#264211yescharacteristic
657C22#265811yescharacteristic
658C22#268914yes—
659C22#269014yes—
660C22#269814yes—
661C22#270214yes—
662C22#271314yes—
663C22#271914yes—
664C22#271914yes—
665C22#272514yes—
666C22#272514yes—
667C22#272612yes—
668C22#272712yes—
669C22#277614yes—
670C22#280512yes—
671C22#281012yes—
672C22#281711yescharacteristic
673C22#282211yescharacteristic
674C22#284811yescharacteristic
675C22#285411yescharacteristic
676C22#285811yescharacteristic
677C4—28no—
678C4—28no—
679C4—48no—
680C4—44no—
681C4—84no—
682C4—88no—
683C4—84no—
684C4—84no—
685C2#1482211yescharacteristic
686C2#1488411yescharacteristic
687C2#1492611yescharacteristic
688C2#1509011yescharacteristic
689C2#1515311yescharacteristic
690C2#1515611yescharacteristic
691C2#1517011yescharacteristic
692C2#1517711yescharacteristic
693C2#1524311yescharacteristic
694C2#1553014yes—
695C2#1567814yes—
696C2#1584714yes—
697C2#1594014yes—
698C2#1636611yescharacteristic
699C2#1663411yescharacteristic
700C2#1681311yescharacteristic
701C2#1696411yescharacteristic
702C2#1701011yescharacteristic
703C2#1707411yescharacteristic
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
605. C2×C4#400semidirect

Maximal subgroups

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

subgroupindex
C22×#4002
C22×#3892
C2×#14412
C2×C4⋊#952
#75572
#75562
#75542