Finite Group Databasebeta

← #530773

Subgroups — #530773

Order 512.

3298 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2#14682C211yesmaximal, characteristic, complemented
3#14471C212yesmaximal, complemented
4#14440C212yesmaximal, complemented
5#14391C214yesmaximal, complemented
6#14284C212yesmaximal, complemented
7#13988C214yesmaximal, complemented
8#13956C212yesmaximal
9#13943C212yesmaximal, complemented
10#13917C212yesmaximal, complemented
11#13896C212yesmaximal, complemented
12#13887C212yesmaximal, complemented
13#13814C212yesmaximal, complemented
14#13791C212yesmaximal, complemented
15C4×#380C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2297C4116yescomplemented
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#388C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#381C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2214yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2212yes—
43#1446C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C22×#41—28no—
77C22×#41—42no—
78C22×#41—42no—
79C22×#41—44no—
80C22×#41C2×C418yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2314yes—
88C24×C4C2311yescharacteristic
89C2×#117—44no—
90C2×#117C2×C418yescomplemented
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C42—42no—
109C22×C42—42no—
110C22×C42C2×C418yescomplemented
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#391C2×C418yescomplemented
120#391C2×C418yescomplemented
121#390C2×C418yes—
122#390C2×C418yes—
123#388—48no—
124#386—216no—
125#386—48no—
126#386C2×C4116yescomplemented
127#386C2×C4116yescomplemented
128#385—216no—
129#385—48no—
130#382—42no—
131#382—42no—
132#381—44no—
133#380—28no—
134#380—44no—
135C4×C4⋊C4—42no—
136C4×C4⋊C4—42no—
137C4×C22⋊C4—44no—
138C4×C22⋊C4—48no—
139C4×C22⋊C4C2×C4116yescomplemented
140C4×C22⋊C4C2×C4116yescomplemented
141C2×#95—44no—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C2×#41—232no—
171C2×#41—416no—
172C2×#41—416no—
173C2×#41—432no—
174C2×#41—432no—
175C22×Q8—24no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8—42no—
179C22×Q8C22×C414yes—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4C22×C418yes—
188C22×D4C22×C414yes—
189C22×D4#4118yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4—41no—
209C23×C4—41no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C22×C414yes—
216C23×C4C22×C418yes—
217C23×C4C22×C414yes—
218C23×C4C22×C414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#117—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4#4118yescomplemented
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C418yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—416no—
286C2×C42—44no—
287C2×C42—41no—
288C2×C42—41no—
289C2×C42—41no—
290C2×C42—41no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42C22×C414yes—
299C2×C42C22×C414yes—
300C2×C42C22×C418yes—
301C2×C42C22×C414yes—
302C2×C42C22×C414yes—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C22×C418yes—
326#95C22×C418yes—
327#95C22×C418yes—
328#95C22×C418yes—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—24no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—24no—
346C24—22no—
347C24C23×C414yes—
348C24C23×C412yes—
349C24C2×#4118yes—
350C24H(5,𝔽2)14yes—
351C24C2511yescharacteristic
352#41—4128no—
353C2×Q8—28no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×D4—216no—
359C2×D4—216no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4#117116yes—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—216no—
414C22×C4—28no—
415C22×C4—28no—
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—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4C23×C414yes—
457C22×C4C23×C412yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—88no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—216no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C42—28no—
492C42—28no—
493C42—44no—
494C42—44no—
495C42—48no—
496C42—84no—
497C42—84no—
498C42—88no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23C2×#11718yes—
544C23#51814yes—
545C23C24×C411yescharacteristic
546C23C22×#4112yes—
547C23C22×#4112yes—
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23#58512yes—
556Q8—416no—
557D4—416no—
558D4—432no—
559C2×C4—232no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—416no—
600C2×C4—432no—
601C2×C4—416no—
602C2×C4—42no—
603C2×C4—42no—
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—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—832no—
636C2×C4C2×#11714yes—
637C2×C4C4×#4114yes—
638C2×C4#51814yes—
639C2×C4#52014yes—
640C2×C4#52014yes—
641C22—28no—
642C22—24no—
643C22—24no—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—42no—
648C22—44no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—44no—
653C22—48no—
654C22—44no—
655C22—48no—
656C22—44no—
657C22—42no—
658C22—48no—
659C22—44no—
660C22—48no—
661C22—44no—
662C22—44no—
663C22C22×#11711yescharacteristic
664C22C2×C4×#4111yescharacteristic
665C22C2×#51811yescharacteristic
666C22C2×#51811yescharacteristic
667C22C2×#52011yescharacteristic
668C22C2×#52011yescharacteristic
669C22#342811yescharacteristic
670C22#345912yes—
671C22#346012yes—
672C22#348112yes—
673C22#348212yes—
674C22#349012yes—
675C22#349012yes—
676C22#351412yes—
677C22#351512yes—
678C22#351912yes—
679C22#352212yes—
680C22#353611yescharacteristic
681C22#353611yescharacteristic
682C22#353711yescharacteristic
683C22#353811yescharacteristic
684C22#355812yes—
685C22#355912yes—
686C4—24no—
687C4—24no—
688C4—24no—
689C4—24no—
690C4—24no—
691C4—42no—
692C4—44no—
693C4—48no—
694C4—44no—
695C4—44no—
696C4—84no—
697C4—84no—
698C4—84no—
699C2—48no—
700C2—44no—
701C2#3408211yescharacteristic
702C2#3412711yescharacteristic
703C2#3417611yescharacteristic
704C2#3419111yescharacteristic
705C2#3423611yescharacteristic
706C2#3436011yescharacteristic
707C2#3439811yescharacteristic
708C2#3876012yes—
709C2#3876212yes—
710C2#3880012yes—
711C2#3880212yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14682C2semidirect
3. #14471C2semidirect
4. #14440C2semidirect
5. #14391C2semidirect
6. #14284C2semidirect
7. #13988C2semidirect
9. #13943C2semidirect
10. #13917C2semidirect
11. #13896C2semidirect
12. #13887C2semidirect
13. #13814C2semidirect
14. #13791C2semidirect
15. C4×#380C2semidirect
19. #2297C4semidirect
90. C2×#117C2×C4semidirect
110. C22×C42C2×C4semidirect
119. #391C2×C4semidirect
120. #391C2×C4semidirect
126. #386C2×C4semidirect
127. #386C2×C4semidirect
139. C4×C22⋊C4C2×C4semidirect
140. C4×C22⋊C4C2×C4semidirect
251. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
#146822
#144712
#144402
#143912
#142842
#139882
#139562
#139432
#139172
#138962
#138872
#138142
#137912
C4×#3802