Finite Group Databasebeta

← #629417

Subgroups — #629417

Order 512.

2592 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic, complemented
2C2×#2621C211yesmaximal, characteristic
3C2×#2628C211yesmaximal, characteristic
4C2×#2327C211yesmaximal, characteristic, complemented
5#15740C214yesmaximal, complemented
6#15739C214yesmaximal
7#15738C214yesmaximal, complemented
8#15737C214yesmaximal
9#15186C212yesmaximal, complemented
10#15179C212yesmaximal
11#14707C211yesmaximal, characteristic, complemented
12#14282C211yesmaximal, characteristic, complemented
13#13969C212yesmaximal
14#13935C212yesmaximal, complemented
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21#2862—28no—
22#2628—28no—
23#2621—28no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#397C2212yes—
29C2×#397C2212yes—
30C2×#395C2212yes—
31C2×#393C2212yes—
32#2327—28no—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2214yes—
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#387C2214yes—
41C2×#387C2214yes—
42C2×#382C2214yes—
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49C2×C43C2211yescharacteristic
50#1446C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1437C2212yes—
65#1437C2214yes—
66#1437C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2214yes—
71#1436C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77C4×#95C2211yescharacteristic
78C4×#95C2211yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C24×C4C2311yescharacteristic
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8D418yes—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—44no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—44no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124#400—28no—
125#399—28no—
126#398—28no—
127#398—28no—
128#397—28no—
129#397—28no—
130#395—28no—
131#393—28no—
132#391—28no—
133#391—28no—
134#391—28no—
135#391—28no—
136#391—216no—
137#389—28no—
138#389—44no—
139#387—216no—
140#387—216no—
141#382—216no—
142#382—24no—
143#382D418yes—
144#380—28no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147#376—24no—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C43—24no—
178C25C2411yescharacteristic
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—44no—
189C22×Q8—44no—
190C22×Q8C2×D418yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4#4114yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4⋊Q8—232no—
213C4⋊Q8—416no—
214C22⋊Q8—432no—
215C4×Q8—432no—
216#117—416no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—28no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42#4114yes—
297C2×C42#4114yes—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C2×D418yes—
326#95C2×D418yes—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24C2×#4112yes—
338C24C2511yescharacteristic
339C2×Q8—216no—
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—416no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—88no—
349C2×Q8—84no—
350C2×Q8—84no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4C22×D418yes—
430C22×C4C22×D418yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+414yes—
446C22×C42−1+414yes—
447C22×C42−1+412yes—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—88no—
466C42—216no—
467C42—28no—
468C42—416no—
469C42—48no—
470C42—44no—
471C42—44no—
472C42—44no—
473C42—44no—
474C42#121116yes—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23C23×D411yescharacteristic
499C23C23×D411yescharacteristic
500C23C22×#4111yescharacteristic
501C23C22×#4111yescharacteristic
502C23C22×#4111yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23C2×2−1+412yes—
506C23C2×2−1+412yes—
507C23#58512yes—
508C23#58512yes—
509Q8—432no—
510Q8—816no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—416no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4C2×#12114yes—
594C2×C4C2×#12114yes—
595C2×C4C2×#12114yes—
596C2×C4#52914yes—
597C2×C4#54214yes—
598C2×C4#54614yes—
599C2×C4#54614yes—
600C2×C4#54714yes—
601C2×C4#54814yes—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—82no—
612C22—82no—
613C22C22×#12111yescharacteristic
614C22C2×#52911yescharacteristic
615C22C2×#54211yescharacteristic
616C22#346112yes—
617C22#346112yes—
618C22C2×#54611yescharacteristic
619C22C2×#54611yescharacteristic
620C22C2×#54711yescharacteristic
621C22C2×#54811yescharacteristic
622C22#348612yes—
623C22#348712yes—
624C22#349512yes—
625C22#349512yes—
626C22#349912yes—
627C22#350012yes—
628C22#350911yescharacteristic
629C22#350911yescharacteristic
630C22#351011yescharacteristic
631C22#351011yescharacteristic
632C22#355412yes—
633C22#355512yes—
634C22#356112yes—
635C22#356312yes—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—42no—
641C4—48no—
642C4—44no—
643C4—48no—
644C4—82no—
645C4—81no—
646C4—81no—
647C4—84no—
648C4—84no—
649C4#242718yes—
650C2—82no—
651C2C2×#242711yescharacteristic
652C2#3531411yescharacteristic
653C2#3532411yescharacteristic
654C2#3566711yescharacteristic
655C2#3567711yescharacteristic
656C2#3739611yescharacteristic
657C2#3739711yescharacteristic
658C2#3774912yes—
659C2#3776512yes—
660C2#3941812yes—
661C2#3942712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2862C2semidirect
4. C2×#2327C2semidirect
5. #15740C2semidirect
7. #15738C2semidirect
9. #15186C2semidirect
11. #14707C2semidirect
12. #14282C2semidirect
14. #13935C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#26212
C2×#26282
C2×#23272
#157402
#157392
#157382
#157372
#151862
#151792
#147072
#142822
#139692
#139352
#137612