Finite Group Databasebeta

← #582897

Subgroups — #582897

Order 512.

2820 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#14666C212yesmaximal
3#14655C212yesmaximal, complemented
4#14649C214yesmaximal, complemented
5#14599C212yesmaximal, complemented
6#14516C212yesmaximal
7#14457C214yesmaximal, complemented
8#14261C212yesmaximal, complemented
9#14138C212yesmaximal, complemented
10#14021C212yesmaximal, complemented
11#13997C212yesmaximal, complemented
12#13993C212yesmaximal, complemented
13#13922C212yesmaximal, complemented
14#13871C211yesmaximal, characteristic
15#13847C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2409C4116yescomplemented
21C2×#395C2212yes—
22C2×#393C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yes—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×#376C2211yescharacteristic
39#1446C2214yes—
40#1446C2214yes—
41#1445C2211yescharacteristic
42#1445C2211yescharacteristic
43#1445C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1440C2212yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2211yescharacteristic
59#1438C2211yescharacteristic
60#1438C2212yes—
61#1437C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C23×Q8—42no—
77C23×Q8—42no—
78C23×Q8—44no—
79C23×Q8—44no—
80C23×Q8C2×C418yes—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×C22⋊Q8—44no—
88C2×C22⋊Q8—44no—
89C2×C22⋊Q8C2×C4116yes—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4C2×C418yes—
93C2×#117—44no—
94C2×#117—42no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2314yes—
118#395C2×C4116yes—
119#393C2×C4116yes—
120#391—44no—
121#391—44no—
122#391—44no—
123#391—44no—
124#391C2×C4116yes—
125#389—44no—
126#386C2×C4116yes—
127#385C2×C4116yes—
128#380—44no—
129#380—42no—
130#380C2×C418yes—
131C4×C4⋊C4—42no—
132C4×C4⋊C4—44no—
133C4×C22⋊C4—44no—
134C4×C22⋊C4—44no—
135#376—42no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2311yescharacteristic
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—416no—
179C22×Q8—432no—
180C22×Q8—416no—
181C22×Q8—432no—
182C22×Q8—416no—
183C22×Q8C22×C418yes—
184C22×Q8C22×C418yes—
185C22×D4—24no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—42no—
189C22×D4C22×C414yes—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C22×C418yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C22⋊Q8—432no—
219C4×D4—416no—
220#117—416no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C22×C4116yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4C22×C414yes—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C4116yes—
271C2×C22⋊C4C22×C418yes—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42C22×C414yes—
286#95—28no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—41no—
294#95—41no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—41no—
300#95—41no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C22×C418yes—
314#95C22×C4116yes—
315#95C22×C4116yes—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24C23×C412yes—
332C24C23×C412yes—
333C242−1+414yes—
334C24C2511yescharacteristic
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—216no—
338C2×Q8—48no—
339C2×Q8—416no—
340C2×Q8—4128no—
341C2×Q8—416no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×D4—28no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—48no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
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—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—416no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4C23×C412yes—
438C22×C4C23×C412yes—
439C22×C4C23×C418yes—
440C22×C4C23×C412yes—
441C22×C4C23×C418yes—
442C22×C4C23×C412yes—
443C22×C4C23×C412yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C42−1+414yes—
449C4⋊C4—216no—
450C4⋊C4—44no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—88no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—416no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C22⋊C4—84no—
467C42—28no—
468C42—48no—
469C42—48no—
470C42—48no—
471C42—88no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23#51914yes—
514C23C24×C411yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)12yes—
519C23C2×2−1+412yes—
520C23C2×2−1+412yes—
521C23#58512yes—
522C23#58512yes—
523Q8—432no—
524Q8—432no—
525D4—416no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—432no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4#51814yes—
606C2×C4#51818yes—
607C2×C4#56014yes—
608C2×C4#56018yes—
609C2×C4H(3,𝔽4)14yes—
610C22—24no—
611C22—28no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×#51811yescharacteristic
629C22C2×#51812yes—
630C22C2×#51911yescharacteristic
631C22#342811yescharacteristic
632C22C4×2−1+412yes—
633C22C2×#56012yes—
634C22C2×#56011yescharacteristic
635C22C2×H(3,𝔽4)11yescharacteristic
636C22#353812yes—
637C22#354012yes—
638C22#354112yes—
639C22#354112yes—
640C22#354112yes—
641C22#354112yes—
642C22#354112yes—
643C22#355011yescharacteristic
644C22#355111yescharacteristic
645C22#355212yes—
646C22#355412yes—
647C22#356112yes—
648C22#356212yes—
649C4—42no—
650C4—42no—
651C4—48no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—48no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—82no—
660C4—84no—
661C4#237518yes—
662C2—44no—
663C2—82no—
664C2C2×#237511yescharacteristic
665C2#3437011yescharacteristic
666C2#3441311yescharacteristic
667C2#3447412yes—
668C2#3447512yes—
669C2#5313612yes—
670C2#5313712yes—
671C2#5314412yes—
672C2#5315112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14655C2semidirect
4. #14649C2semidirect
5. #14599C2semidirect
7. #14457C2semidirect
8. #14261C2semidirect
9. #14138C2semidirect
10. #14021C2semidirect
11. #13997C2semidirect
12. #13993C2semidirect
13. #13922C2semidirect
20. #2409C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#146662
#146552
#146492
#145992
#145162
#144572
#142612
#141382
#140212
#139972
#139932
#139222
#138712
#138472