Finite Group Databasebeta

← #2737545

Subgroups — #2737545

Order 512.

2032 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2489C211yesmaximal, characteristic
3#17203C214yesmaximal, complemented
4#17110C214yesmaximal
5#17046C212yesmaximal
6#16960C212yesmaximal, complemented
7#16518C212yesmaximal
8#15236C212yesmaximal
9#15228C212yesmaximal, complemented
10#15199C211yesmaximal, characteristic, complemented
11#15192C211yesmaximal, characteristic, complemented
12#15189C212yesmaximal, complemented
13#15135C211yesmaximal, characteristic
14#15131C212yesmaximal
15#14677C212yesmaximal, complemented
16#13918C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20#2571—28no—
21#2489—28no—
22C2×#400C2212yes—
23C2×#400C2214yes—
24C2×#399C2212yes—
25C2×#397C2211yescharacteristic
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#382C2212yes—
42C2×#382C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1445C2212yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1443C2212yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2211yescharacteristic
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1440C2212yes—
74#1440C2214yes—
75#1440C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#125—24no—
88C2×#125—44no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—44no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#400—216no—
121#400—28no—
122#399—44no—
123#397—28no—
124#397—24no—
125#396—48no—
126#391—24no—
127#391—24no—
128#391—44no—
129#391—44no—
130#389—28no—
131#389—28no—
132#389—44no—
133#389—44no—
134#388—28no—
135#384—24no—
136#384—44no—
137#382—28no—
138#382—28no—
139#382—28no—
140#382—28no—
141#380—24no—
142#380—44no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4D418yes—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—44no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208#125—416no—
209C22⋊Q8—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
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—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C42—24no—
275C2×C42—24no—
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×C42C2×D414yes—
286C2×C42C2×D418yes—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—28no—
304#95—24no—
305#95—42no—
306#95—42no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95C2×D418yes—
320#95#4114yes—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24H(5,𝔽2)12yes—
332C24C2511yescharacteristic
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—48no—
337C2×Q8—48no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4C22×D412yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+414yes—
434C22×C42−1+412yes—
435C22×C42−1+414yes—
436C22×C42−1+412yes—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—84no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C4⋊C4—88no—
453C4⋊C4—88no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—44no—
460C22⋊C4—88no—
461C42—28no—
462C42—216no—
463C42—44no—
464C42—48no—
465C42—48no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23C23×D411yescharacteristic
490C23C22×#4111yescharacteristic
491C23C2×H(5,𝔽2)12yes—
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×2−1+412yes—
495C23C2×2−1+411yescharacteristic
496C23C2×2−1+411yescharacteristic
497C23C2×2−1+412yes—
498C23#58512yes—
499C23#58511yescharacteristic
500Q8—88no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—88no—
590C2×C4#53514yes—
591C2×C4#53614yes—
592C2×C4#53714yes—
593C2×C4#54714yes—
594C2×C4#56218yes—
595C2×C4#56318yes—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—81no—
602C22—82no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22C2×#53511yescharacteristic
608C22C2×#53611yescharacteristic
609C22C2×#53711yescharacteristic
610C22C2×#54711yescharacteristic
611C22#348711yescharacteristic
612C22#348811yescharacteristic
613C22#349911yescharacteristic
614C22#351112yes—
615C22#351512yes—
616C22#351811yescharacteristic
617C22#351911yescharacteristic
618C22C2×#56212yes—
619C22C2×#56312yes—
620C22#354112yes—
621C22#354411yescharacteristic
622C22#354412yes—
623C22#354711yescharacteristic
624C22#355212yes—
625C22#355512yes—
626C22#355812yes—
627C22#356212yes—
628C22#356312yes—
629C22#356312yes—
630C4—24no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—82no—
637C4—84no—
638C4—81no—
639C4—81no—
640C4—82no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4#282018yes—
646C2—82no—
647C2C2×#282011yescharacteristic
648C2#3891911yescharacteristic
649C2#3892211yescharacteristic
650C2#3948811yescharacteristic
651C2#4287811yescharacteristic
652C2#4289011yescharacteristic
653C2#4296411yescharacteristic
654C2#4560312yes—
655C2#4611712yes—
656C2#5302412yes—
657C2#5303812yes—
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×#2571C2semidirect
3. #17203C2semidirect
6. #16960C2semidirect
9. #15228C2semidirect
10. #15199C2semidirect
11. #15192C2semidirect
12. #15189C2semidirect
15. #14677C2semidirect
16. #13918C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#24892
#172032
#171102
#170462
#169602
#165182
#152362
#152282
#151992
#151922
#151892
#151352
#151312
#146772
#139182
C4×#3972