Finite Group Databasebeta

← #2737659

Subgroups — #2737659

Order 512.

2100 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2421C211yesmaximal, characteristic
3#17225C214yesmaximal, complemented
4#17180C214yesmaximal
5#16901C212yesmaximal
6#16812C212yesmaximal
7#16806C212yesmaximal, complemented
8#15242C212yesmaximal, complemented
9#15220C212yesmaximal
10#15186C212yesmaximal, complemented
11#15182C212yesmaximal
12#15149C211yesmaximal, characteristic, complemented
13#15143C211yesmaximal, characteristic, complemented
14#15125C211yesmaximal, characteristic
15#14626C212yesmaximal, complemented
16#13935C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2568—28no—
22#2421—28no—
23C2×#400C2214yes—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#397C2212yes—
27C2×#397C2211yescharacteristic
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#393C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#384C2212yes—
42C2×#382C2212yes—
43C2×#382C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1446C2212yes—
50#1445C2214yes—
51#1445C2211yescharacteristic
52#1445C2214yes—
53#1445C2214yes—
54#1444C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2212yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2211yescharacteristic
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2212yes—
71#1439C2211yescharacteristic
72#1439C2214yes—
73#1439C2214yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—44no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—44no—
92C2×C4×Q8—44no—
93C2×C4×Q8D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#400—216no—
122#400—28no—
123#398—44no—
124#397—28no—
125#397—24no—
126#395—28no—
127#395—44no—
128#393—44no—
129#391—28no—
130#391—24no—
131#391—28no—
132#391—28no—
133#391—44no—
134#389—24no—
135#389—44no—
136#389—44no—
137#388—28no—
138#387—28no—
139#384—44no—
140#382—28no—
141#382—24no—
142#380—24no—
143#380—44no—
144C4×C4⋊C4—24no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—44no—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C25C2411yescharacteristic
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8C2×D414yes—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4⋊Q8—416no—
215C22⋊Q8—416no—
216C4×Q8—416no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
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⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D418yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—44no—
286C2×C42C2×D414yes—
287C2×C42C2×D418yes—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95#4114yes—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—22no—
331C24—21no—
332C24—22no—
333C242−1+412yes—
334C24C2511yescharacteristic
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—88no—
344C2×Q8—88no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
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—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4112yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+414yes—
442C22×C42−1+414yes—
443C22×C42−1+412yes—
444C4⋊C4—28no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—88no—
466C42—28no—
467C42—48no—
468C42—44no—
469C42—48no—
470C42—48no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23C23×D411yescharacteristic
495C23C22×#4111yescharacteristic
496C23C2×H(5,𝔽2)12yes—
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×2−1+412yes—
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+411yescharacteristic
502C23C2×2−1+412yes—
503C23#58512yes—
504C23#58511yescharacteristic
505Q8—416no—
506Q8—816no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—88no—
596C2×C4#53414yes—
597C2×C4#53614yes—
598C2×C4#53714yes—
599C2×C4#55014yes—
600C2×C4#56218yes—
601C2×C4#56318yes—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—82no—
612C22—82no—
613C22C2×#53411yescharacteristic
614C22C2×#53611yescharacteristic
615C22C2×#53711yescharacteristic
616C22C2×#55011yescharacteristic
617C22#348711yescharacteristic
618C22#348911yescharacteristic
619C22#349911yescharacteristic
620C22#350912yes—
621C22#351412yes—
622C22#351811yescharacteristic
623C22#352011yescharacteristic
624C22C2×#56212yes—
625C22C2×#56312yes—
626C22#354112yes—
627C22#354611yescharacteristic
628C22#355211yescharacteristic
629C22#355212yes—
630C22#355212yes—
631C22#355512yes—
632C22#355812yes—
633C22#356112yes—
634C22#356312yes—
635C22#356512yes—
636C4—24no—
637C4—44no—
638C4—44no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—81no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4#281418yes—
652C2—82no—
653C2C2×#281411yescharacteristic
654C2#3913511yescharacteristic
655C2#3913611yescharacteristic
656C2#3974411yescharacteristic
657C2#4287811yescharacteristic
658C2#4288811yescharacteristic
659C2#4291211yescharacteristic
660C2#4562612yes—
661C2#4605112yes—
662C2#5297512yes—
663C2#5300212yes—
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×#2568C2semidirect
3. #17225C2semidirect
7. #16806C2semidirect
8. #15242C2semidirect
10. #15186C2semidirect
12. #15149C2semidirect
13. #15143C2semidirect
15. #14626C2semidirect
16. #13935C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#24212
#172252
#171802
#169012
#168122
#168062
#152422
#152202
#151862
#151822
#151492
#151432
#151252
#146262
#139352
C4×#3972