Finite Group Databasebeta

← #536341

Subgroups — #536341

Order 512.

2876 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2392C211yesmaximal, characteristic
2#14613C212yesmaximal, complemented
3#14536C214yesmaximal, complemented
4#14535C212yesmaximal, complemented
5#14238C212yesmaximal, complemented
6#14207C212yesmaximal
7#14198C214yesmaximal, complemented
8#13949C212yesmaximal, complemented
9#13939C211yesmaximal, characteristic, complemented
10#13938C211yesmaximal, characteristic, complemented
11#13899C212yesmaximal, complemented
12#13892C212yesmaximal, complemented
13#13820C212yesmaximal, complemented
14#13814C212yesmaximal, complemented
15#13784C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×#117C2211yescharacteristic
19#2392C4116yescomplemented
20C2×#398C2211yescharacteristic
21C2×#397C2211yescharacteristic
22C2×#396C2212yes—
23C2×#394C2211yescharacteristic
24C2×#394C2212yes—
25C2×#392C2211yescharacteristic
26C2×#390C2214yes—
27C2×#387C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#383C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2214yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2214yes—
40C2×C4×C22⋊C4C2212yes—
41C22×#95C2212yes—
42#1442C2214yes—
43#1441C2214yes—
44#1441C2212yes—
45#1441C2214yes—
46#1440C2212yes—
47#1440C2214yes—
48#1440C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2212yes—
52#1437C2212yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68#1434C2212yes—
69#1434C2212yes—
70#1434C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2312yes—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C2×C4×D4—48no—
84C2×C4×D4C2×C418yes—
85C2×C4×D4C2×C4116yescomplemented
86C2×#117—42no—
87C2×#117—42no—
88C2×#117C2×C418yescomplemented
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118#398C2×C418yescomplemented
119#397C2×C418yescomplemented
120#396C2×C4116yescomplemented
121#394C2×C418yes—
122#394C2×C4116yescomplemented
123#392C2×C418yes—
124#390—216no—
125#387—48no—
126#386—216no—
127#386—48no—
128#385—216no—
129#385—48no—
130#383—42no—
131#380—42no—
132C4×C4⋊C4—42no—
133C4×C4⋊C4—48no—
134C4×C4⋊C4C2×C418yescomplemented
135C4×C22⋊C4—42no—
136C4×C22⋊C4—44no—
137C4×C22⋊C4—48no—
138C4×C22⋊C4C2×C4116yescomplemented
139C2×#95—44no—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C25C2411yescharacteristic
166C25C2412yes—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—44no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4C22×C414yes—
175C22×D4C22×C418yes—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C22×C414yes—
192C23×C4C22×C418yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C4×D4—416no—
207C4×D4—432no—
208#117—416no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C22×C418yes—
230C2×C4⋊C4C22×C418yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C414yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4C22×C418yes—
261C2×C22⋊C4C22×C418yes—
262C2×C22⋊C4C22×C418yes—
263C2×C22⋊C4C22×C414yes—
264C2×C22⋊C4C22×C414yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4#41116yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—24no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—44no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42C22×C414yes—
285C2×C42C22×C414yes—
286C2×C42C22×C418yes—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95C22×C418yes—
309#95C22×C418yes—
310#95C22×C414yes—
311#95C22×C414yes—
312C24—24no—
313C24—22no—
314C24—24no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24C23×C412yes—
329C24C23×C414yes—
330C24C2×#4118yes—
331C24C2511yescharacteristic
332C2×D4—216no—
333C2×D4—216no—
334C2×D4—216no—
335C2×D4—48no—
336C2×D4—44no—
337C2×D4—416no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—48no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
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—24no—
398C22×C4—24no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4C23×C412yes—
414C22×C4C23×C414yes—
415C22×C4C23×C412yes—
416C22×C4C23×C412yes—
417C22×C4C23×C414yes—
418C22×C4C23×C412yes—
419C22×C4C23×C412yes—
420C22×C4C23×C412yes—
421C22×C4C23×C414yes—
422C22×C4C2×#4118yes—
423C22×C4C2×#4118yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C42−1+414yes—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C22⋊C4—216no—
439C22⋊C4—216no—
440C22⋊C4—216no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—416no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—84no—
451C42—28no—
452C42—28no—
453C42—48no—
454C42—416no—
455C42—84no—
456C42—88no—
457C42—84no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23C4×#4118yes—
497C23C24×C411yescharacteristic
498C23C22×#4114yes—
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×H(5,𝔽2)12yes—
503C23C2×2−1+412yes—
504C23#58512yes—
505D4—432no—
506D4—416no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—416no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
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—816no—
577C2×C4—832no—
578C2×C4#51814yes—
579C2×C4#51814yes—
580C2×C4#52018yes—
581C22—28no—
582C22—28no—
583C22—28no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—48no—
589C22—48no—
590C22—48no—
591C22—48no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—84no—
598C22—82no—
599C22—82no—
600C22C2×C4×#4112yes—
601C22C2×#51811yescharacteristic
602C22C2×#51811yescharacteristic
603C22C2×#52012yes—
604C22#342811yescharacteristic
605C22#348112yes—
606C22#348112yes—
607C22#350514yes—
608C22#350714yes—
609C22#351914yes—
610C22#352014yes—
611C22#353412yes—
612C22#353412yes—
613C22#354812yes—
614C22#354911yescharacteristic
615C22#355011yescharacteristic
616C4—24no—
617C4—24no—
618C4—28no—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—44no—
625C4—48no—
626C4—48no—
627C4—84no—
628C4—84no—
629C2—48no—
630C2—82no—
631C2#3415211yescharacteristic
632C2#3422412yes—
633C2#3423312yes—
634C2#3435811yescharacteristic
635C2#3436811yescharacteristic
636C2#4346114yes—
637C2#4346212yes—
638C2#4347812yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#392C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14613C2semidirect
3. #14536C2semidirect
4. #14535C2semidirect
5. #14238C2semidirect
7. #14198C2semidirect
8. #13949C2semidirect
9. #13939C2semidirect
10. #13938C2semidirect
11. #13899C2semidirect
12. #13892C2semidirect
13. #13820C2semidirect
14. #13814C2semidirect
15. #13784C2semidirect
19. #2392C4semidirect
85. C2×C4×D4C2×C4semidirect
88. C2×#117C2×C4semidirect
118. #398C2×C4semidirect
119. #397C2×C4semidirect
120. #396C2×C4semidirect
122. #394C2×C4semidirect
134. C4×C4⋊C4C2×C4semidirect
138. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23922
#146132
#145362
#145352
#142382
#142072
#141982
#139492
#139392
#139382
#138992
#138922
#138202
#138142
#137842