Finite Group Databasebeta

← #521668

Subgroups — #521668

Order 512.

3802 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#381C211yesmaximal, characteristic
2#14681C211yesmaximal, characteristic
3#14508C214yesmaximal, complemented
4#14507C214yesmaximal, complemented
5#14349C211yesmaximal, characteristic, complemented
6#14347C211yesmaximal, characteristic, complemented
7#13992C214yesmaximal, complemented
8#13972C214yesmaximal, complemented
9#13883C212yesmaximal, complemented
10#13883C212yesmaximal, complemented
11#13877C212yesmaximal
12C22⋊C42C212yesmaximal
13#13796C211yesmaximal, characteristic, complemented
14#13778C211yesmaximal, characteristic, complemented
15C2×C22⋊#95C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C2×#385C2214yes—
21C2×#381C4116yescomplemented
22C2×#381C2212yes—
23C2×#381C2212yes—
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#381C2212yes—
27C2×#381C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#379C2212yes—
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2214yes—
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1443C2214yescomplemented
41#1443C2214yescomplemented
42#1443C2214yescomplemented
43#1443C2214yescomplemented
44#1442C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1439C2214yes—
49#1439C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yescomplemented
53#1436C2214yescomplemented
54#1436C2214yescomplemented
55#1436C2214yescomplemented
56#1434C2214yescomplemented
57#1434C2214yescomplemented
58#1434C2214yescomplemented
59#1434C2214yescomplemented
60C22⋊#95C2211yescharacteristic
61C22⋊#95C2212yes—
62C22⋊#95C2211yescharacteristic
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C26C2311yescharacteristic
77C24×C4—28no—
78C24×C4—44no—
79C24×C4—44no—
80C24×C4C2×C418yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2314yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4C2×C418yes—
100C22×C22⋊C4C2×C418yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2314yes—
113C22×C42C2314yes—
114#385—48no—
115#381—44no—
116#381—44no—
117#381C2×C4116yes—
118#381C2×C4116yes—
119#381C2×C4116yescomplemented
120#381C2×C4116yescomplemented
121#380—28no—
122#380—44no—
123#379—28no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—44no—
126C4×C22⋊C4—48no—
127C2×#95—44no—
128C2×#95—44no—
129C2×#95C2×C418yescomplemented
130C2×#95C2×C418yescomplemented
131C2×#95C2×C418yescomplemented
132C2×#95C2×C418yescomplemented
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C25—24no—
166C25—22no—
167C25—24no—
168C25—24no—
169C25—22no—
170C25—42no—
171C25—42no—
172C25C22×C414yes—
173C25#4118yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C23×C4—232no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—416no—
199C23×C4—416no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—464no—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4C22×C414yes—
227C23×C4C22×C414yes—
228C23×C4C22×C414yes—
229C23×C4C22×C414yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2414yes—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—416no—
268C2×C22⋊C4—416no—
269C2×C22⋊C4—432no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4#4118yes—
280C2×C42—28no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286#95—28no—
287#95—28no—
288#95—42no—
289#95—44no—
290#95—42no—
291#95—44no—
292#95—42no—
293#95—42no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95C22×C418yes—
311#95C22×C418yes—
312#95C22×C418yes—
313#95C22×C418yes—
314#95C22×C418yes—
315#95C22×C418yes—
316#95C22×C418yes—
317#95C22×C418yes—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—28no—
328C24—28no—
329C24—28no—
330C24—28no—
331C24—232no—
332C24—216no—
333C24—216no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—44no—
342C24—44no—
343C24—416no—
344C24—416no—
345C24—416no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24#117116yes—
351C24C23×C412yes—
352C24C23×C412yes—
353C24C23×C412yes—
354C24C2×#4114yes—
355C24C2×#4114yes—
356C24C2×#4114yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)14yes—
359C24H(5,𝔽2)14yes—
360C24H(5,𝔽2)12yes—
361C242−1+412yes—
362C24C2511yescharacteristic
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—4128no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4—84no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—88no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4C23×C414yes—
469C22×C4C23×C414yes—
470C22×C4C23×C414yes—
471C22×C4C23×C414yes—
472C22×C4C23×C412yes—
473C22×C4C23×C412yes—
474C22×C4C23×C412yes—
475C22×C4C23×C412yes—
476C22×C4C2×#4118yes—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—88no—
490C42—88no—
491C42—88no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—232no—
518C23—232no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—28no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—416no—
530C23—416no—
531C23—416no—
532C23—464no—
533C23—464no—
534C23—416no—
535C23—416no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23C2×#11714yes—
543C23C2×#11714yes—
544C23C2×#11714yes—
545C23C4×#4114yes—
546C23#51814yes—
547C23#51814yes—
548C23#52014yes—
549C23#538116yes—
550C23#55118yes—
551C23#55218yes—
552C23C24×C411yescharacteristic
553C23C22×#4112yes—
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×2−1+411yescharacteristic
559C23#58512yes—
560C23#58512yes—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—416no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
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—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—88no—
632C2×C4—88no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4—832no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—216no—
644C22—216no—
645C22—216no—
646C22—216no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—464no—
656C22—42no—
657C22—42no—
658C22—48no—
659C22—48no—
660C22—42no—
661C22—42no—
662C22#231218yes—
663C22#234318yes—
664C22C22×#11711yescharacteristic
665C22C2×C4×#4111yescharacteristic
666C22C2×#51811yescharacteristic
667C22C2×#51811yescharacteristic
668C22C2×#52011yescharacteristic
669C22#342811yescharacteristic
670C22#342811yescharacteristic
671C22C2×#53814yes—
672C22C2×#55112yes—
673C22C2×#55212yes—
674C22#349114yes—
675C22#351314yes—
676C22#351314yes—
677C22#353412yes—
678C22#353412yes—
679C22#353512yes—
680C22#353512yes—
681C4—44no—
682C4—44no—
683C4—48no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—84no—
691C4—84no—
692C4—84no—
693C2—28no—
694C2—28no—
695C2—44no—
696C2C2×#231211yescharacteristic
697C2C2×#234311yescharacteristic
698C2#3417711yescharacteristic
699C2#3423811yescharacteristic
700C2#3423811yescharacteristic
701C2#3437611yescharacteristic
702C2#3437611yescharacteristic
703C2#3805314yes—
704C2#3806514yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14508C2semidirect
4. #14507C2semidirect
5. #14349C2semidirect
6. #14347C2semidirect
7. #13992C2semidirect
8. #13972C2semidirect
9. #13883C2semidirect
10. #13883C2semidirect
13. #13796C2semidirect
14. #13778C2semidirect
21. C2×#381C4semidirect
40. #1443C22semidirect
41. #1443C22semidirect
42. #1443C22semidirect
43. #1443C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
52. #1436C22semidirect
53. #1436C22semidirect
54. #1436C22semidirect
55. #1436C22semidirect
56. #1434C22semidirect
57. #1434C22semidirect
58. #1434C22semidirect
59. #1434C22semidirect
119. #381C2×C4semidirect
120. #381C2×C4semidirect
129. C2×#95C2×C4semidirect
130. C2×#95C2×C4semidirect
131. C2×#95C2×C4semidirect
132. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3812
#146812
#145082
#145072
#143492
#143472
#139922
#139722
#138832
#138772
C22⋊C422
#137962
#137782
C2×C22⋊#952