Finite Group Databasebeta

← #4335655

Subgroups — #4335655

Order 512.

2902 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2598C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic, complemented
3#16696C212yesmaximal, complemented
4#16653C212yesmaximal, complemented
5#16652C212yesmaximal, complemented
6#16566C212yesmaximal, complemented
7#15916C212yesmaximal, complemented
8#15561C212yesmaximal
9#15538C214yesmaximal, complemented
10#14902C212yesmaximal, complemented
11#14888C212yesmaximal, complemented
12#14873C212yesmaximal, complemented
13#14865C212yesmaximal, complemented
14#14553C214yesmaximal, complemented
15#13957C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2598—28no—
21C2×#400C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#394C2211yescharacteristic
24C2×#394C2214yes—
25C2×#393C2214yes—
26C2×#393C2212yes—
27C2×#393C2212yes—
28C2×#392C2211yescharacteristic
29#2310—28no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#388C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#385C2212yes—
39C2×#383C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#376C2211yescharacteristic
44#1446C2214yes—
45#1446C2212yes—
46#1446C2212yes—
47#1445C2214yes—
48#1444C2212yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2212yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2214yes—
65#1438C2214yes—
66#1437C2214yes—
67#1437C2212yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2212yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—44no—
80C23×Q8—42no—
81C23×Q8—42no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C2×C4×D4—24no—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×#117—24no—
93C2×#117—42no—
94C2×#117—42no—
95C2×#117—44no—
96C2×#117D418yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121#400—24no—
122#399—24no—
123#394—24no—
124#394—48no—
125#393—216no—
126#393—44no—
127#393—44no—
128#392—24no—
129#391—28no—
130#391—28no—
131#391—28no—
132#391—28no—
133#388—28no—
134#386—28no—
135#386—28no—
136#386—48no—
137#385—28no—
138#383—24no—
139#382—42no—
140#381—28no—
141#381—28no—
142#376—42no—
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×Q8—232no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—28no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—44no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—432no—
186C22×Q8—432no—
187C22×Q8—416no—
188C22×Q8—416no—
189C22×Q8#4118yes—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—28no—
194C22×D4—22no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—42no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C4×D4—416no—
229#117—416no—
230#117—416no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
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⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24C22×D412yes—
340C24H(5,𝔽2)14yes—
341C242−1+414yes—
342C24C2511yescharacteristic
343C2×Q8—216no—
344C2×Q8—216no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—416no—
348C2×Q8—416no—
349C2×Q8—4128no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—416no—
363C2×D4—48no—
364C2×D4—84no—
365C2×D4—88no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
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—22no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C2×#4118yes—
449C22×C4C2×#4114yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C42—28no—
473C42—28no—
474C42—48no—
475C42—48no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—44no—
490C23—44no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23#53514yes—
515C23C23×D411yescharacteristic
516C23C22×#4112yes—
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×2−1+412yes—
521C23#58512yes—
522C23#58514yes—
523Q8—432no—
524Q8—432no—
525D4—416no—
526D4—816no—
527C2×C4—232no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—432no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
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—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—816no—
599C2×C4#52514yes—
600C2×C4#53414yes—
601C22—28no—
602C22—24no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—48no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—48no—
611C22—48no—
612C22—44no—
613C22—44no—
614C22—82no—
615C22—84no—
616C22—82no—
617C22—84no—
618C22—82no—
619C22—82no—
620C22C2×#52511yescharacteristic
621C22C2×#53411yescharacteristic
622C22C2×#53511yescharacteristic
623C22#348912yes—
624C22#349012yes—
625C22#349012yes—
626C22#350012yes—
627C22#351614yes—
628C22#352414yes—
629C22#353514yes—
630C22#353811yescharacteristic
631C22#354012yes—
632C22#354011yescharacteristic
633C22#354112yes—
634C22#354514yes—
635C22#355112yes—
636C4—24no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—48no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—84no—
645C4—84no—
646C4—82no—
647C4—84no—
648C4—82no—
649C2—44no—
650C2—84no—
651C2#3505011yescharacteristic
652C2#3598512yes—
653C2#3737411yescharacteristic
654C2#3741911yescharacteristic
655C2#4229612yes—
656C2#4717112yes—
657C2#4766412yes—
658C2#5182814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2310C2semidirect
3. #16696C2semidirect
4. #16653C2semidirect
5. #16652C2semidirect
6. #16566C2semidirect
7. #15916C2semidirect
9. #15538C2semidirect
10. #14902C2semidirect
11. #14888C2semidirect
12. #14873C2semidirect
13. #14865C2semidirect
14. #14553C2semidirect
15. #13957C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25982
C2×#23102
#166962
#166532
#166522
#165662
#159162
#155612
#155382
#149022
#148882
#148732
#148652
#145532
#139572