Finite Group Databasebeta

← #4866542

Subgroups — #4866542

Order 512.

2728 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic
3#17122C214yesmaximal, complemented
4#17005C212yesmaximal, complemented
5#16686C212yesmaximal, complemented
6#16608C212yesmaximal, complemented
7#16522C212yesmaximal, complemented
8#16324C214yesmaximal, complemented
9#16264C212yesmaximal, complemented
10#16262C212yesmaximal, complemented
11#16074C212yesmaximal
12#15342C212yesmaximal
13#15089C212yesmaximal, complemented
14#14874C212yesmaximal, complemented
15#14344C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2566—28no—
21#2409—28no—
22C2×#398C2214yes—
23C2×#397C2214yes—
24C2×#396C2212yes—
25C2×#396C2212yes—
26C2×#395C2212yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#395C2214yes—
30C2×#393C2212yes—
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#386C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#381C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#1446C2214yes—
46#1446C2212yes—
47#1446C2212yes—
48#1445C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1438C2212yes—
72#1434C2212yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#125—24no—
88C2×#125—42no—
89C2×#125—42no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C2×C4×D4D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#398—48no—
123#397—48no—
124#396—28no—
125#396—28no—
126#395—28no—
127#395—44no—
128#395—44no—
129#395—48no—
130#393—28no—
131#391—28no—
132#390—24no—
133#390—24no—
134#386—28no—
135#385—28no—
136#385—28no—
137#385—28no—
138#385—48no—
139#384—24no—
140#382—24no—
141#381—42no—
142#380—24no—
143#380—42no—
144C4×C4⋊C4—24no—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—28no—
176C22×Q8—28no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—416no—
184C22×Q8—432no—
185C22×Q8—432no—
186C22×Q8—432no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4C2×D414yes—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218#125—416no—
219C22⋊Q8—432no—
220C4×D4—416no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
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⋊C4—44no—
255C2×C4⋊C4C2×D414yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
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—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
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—24no—
290C2×C42C2×D414yes—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317C24—24no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24C22×D412yes—
331C24C22×D412yes—
332C242−1+414yes—
333C24C2511yescharacteristic
334C2×Q8—216no—
335C2×Q8—416no—
336C2×Q8—416no—
337C2×Q8—416no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—416no—
342C2×Q8—4128no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—48no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
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—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4H(5,𝔽2)18yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+414yes—
434C22×C42−1+414yes—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—416no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—88no—
446C4⋊C4—88no—
447C4⋊C4—84no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C42—28no—
460C42—44no—
461C42—44no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—28no—
469C23—28no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—44no—
489C23—41no—
490C23—41no—
491C23—44no—
492C23—44no—
493C23#53614yes—
494C23C23×D411yescharacteristic
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×H(5,𝔽2)11yescharacteristic
497C23C2×H(5,𝔽2)14yes—
498C23C2×2−1+412yes—
499C23C2×2−1+412yes—
500C23C2×2−1+412yes—
501C23#58512yes—
502Q8—432no—
503Q8—432no—
504D4—416no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—416no—
524C2×C4—432no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—88no—
578C2×C4#53414yes—
579C2×C4#53614yes—
580C22—24no—
581C22—28no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—48no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—84no—
596C22C2×#53411yescharacteristic
597C22C2×#53611yescharacteristic
598C22C2×#53611yescharacteristic
599C22#348412yes—
600C22#348912yes—
601C22#353212yes—
602C22#353212yes—
603C22#353812yes—
604C22#353812yes—
605C22#354114yes—
606C22#354712yes—
607C22#354714yes—
608C22#355011yescharacteristic
609C22#355111yescharacteristic
610C22#355214yes—
611C22#355214yes—
612C4—24no—
613C4—24no—
614C4—42no—
615C4—42no—
616C4—48no—
617C4—42no—
618C4—44no—
619C4—82no—
620C4—84no—
621C4—82no—
622C4—84no—
623C4—84no—
624C4—82no—
625C2—44no—
626C2—82no—
627C2#3670211yescharacteristic
628C2#3737611yescharacteristic
629C2#3745311yescharacteristic
630C2#4058912yes—
631C2#4248512yes—
632C2#5630414yes—
633C2#5888912yes—
634C2#5894312yes—
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. #17122C2semidirect
4. #17005C2semidirect
5. #16686C2semidirect
6. #16608C2semidirect
7. #16522C2semidirect
8. #16324C2semidirect
9. #16264C2semidirect
10. #16262C2semidirect
13. #15089C2semidirect
14. #14874C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#24092
#171222
#170052
#166862
#166082
#165222
#163242
#162642
#162622
#160742
#153422
#150892
#148742
#143442