Finite Group Databasebeta

← #4862584

Subgroups — #4862584

Order 512.

2464 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2638C211yesmaximal, characteristic
3#17108C214yesmaximal, complemented
4#16830C212yesmaximal, complemented
5#16603C212yesmaximal, complemented
6#16563C212yesmaximal, complemented
7#16522C212yesmaximal, complemented
8#16364C212yesmaximal, complemented
9#16303C212yesmaximal, complemented
10#16264C212yesmaximal, complemented
11#16254C212yesmaximal, complemented
12#15764C214yesmaximal
13#15080C212yesmaximal, complemented
14#14758C212yesmaximal, complemented
15#13960C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2638—28no—
22#2542—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#398C2211yescharacteristic
26C2×#398C2214yes—
27C2×#397C2211yescharacteristic
28C2×#397C2212yes—
29C2×#397C2214yes—
30C2×#395C2211yescharacteristic
31C2×#395C2212yes—
32C2×#395C2214yes—
33C2×#395C2212yes—
34C2×#394C2212yes—
35C2×#393C2211yescharacteristic
36C2×#390C2212yes—
37C2×#387C2212yes—
38C2×#387C2214yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#384C2212yes—
42C2×#381C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2212yes—
48#1446C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2212yes—
52#1444C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1434C2212yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—42no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—48no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8—42no—
92C2×C22⋊Q8—42no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C2×C4×D4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#399—28no—
123#399—28no—
124#398—24no—
125#398—48no—
126#397—24no—
127#397—28no—
128#397—48no—
129#395—24no—
130#395—48no—
131#395—44no—
132#395—44no—
133#394—28no—
134#393—24no—
135#390—28no—
136#387—28no—
137#387—48no—
138#386—44no—
139#385—28no—
140#384—28no—
141#381—24no—
142#380—42no—
143#380—48no—
144C4×C4⋊C4—24no—
145C2×#95—42no—
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—41no—
182C22×Q8—41no—
183C22×Q8—44no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4C2×D414yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C4⋊Q8—416no—
218C22⋊Q8—432no—
219C22⋊Q8—432no—
220C22⋊Q8—416no—
221C4×D4—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42C2×D414yes—
295#95—24no—
296#95—24no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322C24—24no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—24no—
327C24—21no—
328C24—24no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—24no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24C22×D412yes—
337C24C22×D412yes—
338C242−1+412yes—
339C242−1+412yes—
340C24C2511yescharacteristic
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—88no—
346C2×Q8—88no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4H(5,𝔽2)18yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+418yes—
442C22×C42−1+412yes—
443C4⋊C4—216no—
444C4⋊C4—44no—
445C4⋊C4—416no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—416no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—88no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—28no—
468C42—48no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—22no—
477C23—22no—
478C23—28no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—42no—
490C23—44no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—44no—
502C23#53614yes—
503C23C23×D411yescharacteristic
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)14yes—
508C23C2×2−1+411yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23C2×2−1+411yescharacteristic
511C23C2×2−1+414yes—
512Q8—816no—
513Q8—88no—
514D4—416no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4—816no—
589C2×C4#53414yes—
590C2×C4#53714yes—
591C22—28no—
592C22—24no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22C2×#53411yescharacteristic
610C22C2×#53611yescharacteristic
611C22C2×#53711yescharacteristic
612C22#348412yes—
613C22#349912yes—
614C22#353611yescharacteristic
615C22#353611yescharacteristic
616C22#353614yes—
617C22#353812yes—
618C22#353814yes—
619C22#353811yescharacteristic
620C22#353811yescharacteristic
621C22#354312yes—
622C22#354312yes—
623C22#354414yes—
624C22#355012yes—
625C22#355214yes—
626C4—24no—
627C4—24no—
628C4—42no—
629C4—48no—
630C4—42no—
631C4—42no—
632C4—48no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4—84no—
639C2—44no—
640C2—82no—
641C2#3723211yescharacteristic
642C2#3738611yescharacteristic
643C2#3744211yescharacteristic
644C2#4049512yes—
645C2#4242812yes—
646C2#5349212yes—
647C2#5485412yes—
648C2#5895212yes—
649C2#5898212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17108C2semidirect
4. #16830C2semidirect
5. #16603C2semidirect
6. #16563C2semidirect
7. #16522C2semidirect
8. #16364C2semidirect
9. #16303C2semidirect
10. #16264C2semidirect
11. #16254C2semidirect
13. #15080C2semidirect
14. #14758C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#26382
#171082
#168302
#166032
#165632
#165222
#163642
#163032
#162642
#162542
#157642
#150802
#147582
#139602