Finite Group Databasebeta

← #4375669

Subgroups — #4375669

Order 512.

2986 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2322C211yesmaximal, characteristic
3#16815C212yesmaximal, complemented
4#16663C212yesmaximal, complemented
5#16325C214yesmaximal, complemented
6#15966C212yesmaximal
7#15930C212yesmaximal, complemented
8#15622C212yesmaximal, complemented
9#15463C212yesmaximal
10#15001C212yesmaximal, complemented
11#14901C212yesmaximal, complemented
12#14874C212yesmaximal, complemented
13#14720C212yesmaximal, complemented
14#14541C214yesmaximal, complemented
15#13959C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2409—28no—
22C2×#397C2214yes—
23C2×#395C2212yes—
24C2×#395C2214yes—
25C2×#395C2212yes—
26C2×#395C2212yes—
27C2×#393C2212yes—
28#2322—28no—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#388C2212yes—
32C2×#387C2212yes—
33C2×#387C2214yes—
34C2×#386C2211yescharacteristic
35C2×#386C2212yes—
36C2×#385C2211yescharacteristic
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#383C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×#376C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1445C2214yes—
50#1445C2212yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2212yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1437C2212yes—
69#1437C2212yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—24no—
90C2×#123—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—48no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C2×C4×D4D418yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#397—216no—
123#395—28no—
124#395—44no—
125#395—44no—
126#395—48no—
127#393—28no—
128#391—28no—
129#389—28no—
130#388—28no—
131#387—28no—
132#387—48no—
133#386—24no—
134#386—28no—
135#385—28no—
136#385—24no—
137#385—28no—
138#385—48no—
139#385—48no—
140#383—42no—
141#381—24no—
142#380—24no—
143C4×C22⋊C4—24no—
144#376—42no—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—416no—
186C22×Q8—432no—
187C22×Q8—432no—
188C22×Q8—432no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4C2×D414yes—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—28no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229#123—232no—
230C22⋊Q8—432no—
231C4×D4—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—416no—
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—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C2×D414yes—
285C2×C22⋊C4C2×D414yes—
286C2×C22⋊C4#4118yes—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42C2×D414yes—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—24no—
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—42no—
311#95—42no—
312#95—42no—
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—22no—
323C24—24no—
324C24—24no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24C22×D412yes—
333C24C22×D412yes—
334C24C2×#4114yes—
335C242−1+414yes—
336C24C2511yescharacteristic
337C2×Q8—216no—
338C2×Q8—416no—
339C2×Q8—416no—
340C2×Q8—416no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—416no—
345C2×Q8—4128no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—48no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)18yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+418yes—
446C4⋊C4—216no—
447C4⋊C4—44no—
448C4⋊C4—416no—
449C4⋊C4—416no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—84no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—28no—
457C22⋊C4—216no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—84no—
466C42—28no—
467C42—48no—
468C42—84no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23#52514yes—
498C23#53614yes—
499C23C23×D411yescharacteristic
500C23C22×#4112yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)14yes—
504C23C2×2−1+412yes—
505C23C2×2−1+414yes—
506Q8—432no—
507Q8—432no—
508D4—416no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—432no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
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—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
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—88no—
583C2×C4—88no—
584C2×C4#53414yes—
585C22—24no—
586C22—24no—
587C22—28no—
588C22—28no—
589C22—42no—
590C22—42no—
591C22—48no—
592C22—48no—
593C22—44no—
594C22—42no—
595C22—42no—
596C22—48no—
597C22—48no—
598C22—44no—
599C22C2×#52511yescharacteristic
600C22C2×#53411yescharacteristic
601C22C2×#53611yescharacteristic
602C22#348412yes—
603C22#349912yes—
604C22#351114yes—
605C22#352012yes—
606C22#352012yes—
607C22#352014yes—
608C22#353212yes—
609C22#353212yes—
610C22#353812yes—
611C22#353814yes—
612C22#355012yes—
613C22#355214yes—
614C4—24no—
615C4—42no—
616C4—44no—
617C4—42no—
618C4—42no—
619C4—48no—
620C4—42no—
621C4—48no—
622C4—44no—
623C4—84no—
624C4—82no—
625C4—82no—
626C4—84no—
627C2—44no—
628C2—44no—
629C2#3508411yescharacteristic
630C2#3574012yes—
631C2#3671411yescharacteristic
632C2#3738611yescharacteristic
633C2#4049512yes—
634C2#4712614yes—
635C2#4852514yes—
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. #16815C2semidirect
4. #16663C2semidirect
5. #16325C2semidirect
7. #15930C2semidirect
8. #15622C2semidirect
10. #15001C2semidirect
11. #14901C2semidirect
12. #14874C2semidirect
13. #14720C2semidirect
14. #14541C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23222
#168152
#166632
#163252
#159662
#159302
#156222
#154632
#150012
#149012
#148742
#147202
#145412
#139592