Finite Group Databasebeta

← #1494985

Subgroups — #1494985

Order 512.

3226 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2809C211yesmaximal, characteristic, complemented
2C2×#2431C211yesmaximal, characteristic
3#15820C212yesmaximal, complemented
4#15674C212yesmaximal, complemented
5#15479C214yesmaximal, complemented
6#15428C214yesmaximal, complemented
7#15140C212yesmaximal, complemented
8#15007C214yesmaximal, complemented
9#14492C212yesmaximal, complemented
10#14229C212yesmaximal, complemented
11#14196C214yesmaximal, complemented
12#13937C212yesmaximal, complemented
13#13826C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#121C2211yescharacteristic
18#2809—28no—
19#2431—28no—
20C2×#400C2212yes—
21C2×#398C2214yes—
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#392C2214yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2214yes—
32C2×#390C2214yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#387C2214yes—
39C2×#386C2214yes—
40C2×#380C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43#1445C2212yes—
44#1442C2212yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2212yes—
55#1438C2214yes—
56#1438C2214yes—
57#1437C2212yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1434C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C4×#95C2212yes—
68C4×#95C2214yes—
69C22×#41—24no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C23×D4C2314yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×#127—24no—
83C2×#127—44no—
84C2×#124—24no—
85C2×#124—28no—
86C2×#121—24no—
87C2×#121—42no—
88C2×#121—42no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116#400—28no—
117#398—216no—
118#397—28no—
119#397—44no—
120#394—216no—
121#394—48no—
122#392—216no—
123#391—28no—
124#391—28no—
125#391D418yes—
126#390—24no—
127#390—48no—
128#390—48no—
129#390—44no—
130#390—44no—
131#390—44no—
132#389—28no—
133#388—48no—
134#387—216no—
135#386—216no—
136#380—44no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—44no—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C25C2412yes—
158C25C2411yescharacteristic
159C25C2412yes—
160C2×#41—416no—
161C2×#41—432no—
162C2×#41—416no—
163C2×#41—432no—
164C2×#41—416no—
165C22×Q8—42no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8C2×D414yes—
169C22×Q8#4118yes—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221#127—416no—
222#124—416no—
223#121—416no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C2×D414yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
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—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
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⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—44no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C2×D414yes—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95C2×D4116yes—
318#95#4118yescomplemented
319C24—21no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24H(5,𝔽2)12yes—
346C24H(5,𝔽2)14yes—
347C24C2511yescharacteristic
348#41—4128no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8#124116yes—
354C2×D4—28no—
355C2×D4—216no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—44no—
374C2×D4—416no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—84no—
382C2×D4—88no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
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—22no—
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—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C2×#4118yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—44no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C22⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C42—44no—
477C42—44no—
478C42—84no—
479C42—84no—
480C42—88no—
481C42—84no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—44no—
542C23C23×D411yescharacteristic
543C23C22×#4112yes—
544C23C22×#4112yes—
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×H(5,𝔽2)11yescharacteristic
549C23#58514yes—
550Q8—416no—
551D4—416no—
552D4—432no—
553D4—832no—
554D4—88no—
555D4—816no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—416no—
578C2×C4—432no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
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×C4C2×#12414yes—
616C2×C4C2×#12418yes—
617C2×C4#55018yes—
618C22—42no—
619C22—48no—
620C22—48no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—44no—
626C22—48no—
627C22—82no—
628C22—84no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22—82no—
641C22—84no—
642C22—82no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22C22×#12411yescharacteristic
647C22C2×#55012yes—
648C22#348014yes—
649C22#348311yescharacteristic
650C22#348411yescharacteristic
651C22#348411yescharacteristic
652C22#349012yes—
653C22#349012yes—
654C22#350012yes—
655C22#351314yes—
656C22#351714yes—
657C22#351912yes—
658C22#353312yes—
659C22#353414yes—
660C22#354911yescharacteristic
661C22#355112yes—
662C4—24no—
663C4—28no—
664C4—28no—
665C4—44no—
666C4—48no—
667C4—42no—
668C4—84no—
669C4—84no—
670C4—82no—
671C4—84no—
672C2—82no—
673C2—84no—
674C2—162no—
675C2#3536711yescharacteristic
676C2#3658312yes—
677C2#4300612yes—
678C2#4406614yes—
679C2#5193012yes—
680C2#5197811yescharacteristic
681C2#5905611yescharacteristic
682C2#5979812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2809C2semidirect
3. #15820C2semidirect
4. #15674C2semidirect
5. #15479C2semidirect
6. #15428C2semidirect
7. #15140C2semidirect
8. #15007C2semidirect
9. #14492C2semidirect
10. #14229C2semidirect
11. #14196C2semidirect
12. #13937C2semidirect
13. #13826C2semidirect
318. #95#41semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
C2×#24312
#158202
#156742
#154792
#154282
#151402
#150072
#144922
#142292
#141962
#139372
#138262