Finite Group Databasebeta

← #4866481

Subgroups — #4866481

Order 512.

2246 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2C2×#2621C211yesmaximal, characteristic, complemented
3#17157C214yesmaximal, complemented
4#17046C212yesmaximal, complemented
5#16736C214yesmaximal, complemented
6#16665C212yesmaximal, complemented
7#16602C212yesmaximal, complemented
8#16596C212yesmaximal, complemented
9#16563C212yesmaximal, complemented
10#16253C212yesmaximal
11#15945C212yesmaximal, complemented
12#15814C212yesmaximal, complemented
13#15001C212yesmaximal, complemented
14#14866C212yesmaximal, complemented
15#13938C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2621—28no—
21#2500—28no—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#394C2212yes—
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#387C2214yes—
37C2×#384C2211yescharacteristic
38C2×#383C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43#1446C2212yes—
44#1446C2212yes—
45#1444C2212yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1437C2214yes—
71#1437C2212yes—
72#1436C2212yes—
73#1436C2214yes—
74#1434C2214yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8—42no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—48no—
86C2×#121—24no—
87C2×#121—42no—
88C2×#121—42no—
89C2×#117—42no—
90C2×#117—42no—
91C2×#117D418yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
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⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#398—28no—
123#398—28no—
124#397—28no—
125#397—28no—
126#394—28no—
127#394—28no—
128#394—48no—
129#393—44no—
130#393—44no—
131#391—28no—
132#390—24no—
133#390—24no—
134#388—28no—
135#388—28no—
136#387—48no—
137#384—24no—
138#383—42no—
139#382—24no—
140#382—24no—
141C4×C4⋊C4—42no—
142C4×C22⋊C4—24no—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×D4—24no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—44no—
184C22×D4—44no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4C2×D414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C4⋊Q8—416no—
204C22⋊Q8—432no—
205#121—416no—
206#117—416no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
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—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D414yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42C2×D414yes—
287C2×C42C2×D414yes—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—22no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24C22×D412yes—
332C24C2511yescharacteristic
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—88no—
337C2×Q8—88no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—88no—
347C2×D4—88no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
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—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)14yes—
423C22×C42−1+414yes—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—416no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C42—28no—
450C42—28no—
451C42—44no—
452C42—44no—
453C42—84no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—41no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23C23×D411yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×H(5,𝔽2)12yes—
497C23C2×2−1+412yes—
498C23#58512yes—
499C23#58512yes—
500C23#58514yes—
501Q8—816no—
502D4—88no—
503D4—816no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
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—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#53414yes—
575C2×C4#53514yes—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—84no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—84no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—84no—
594C22—84no—
595C22C2×#53411yescharacteristic
596C22C2×#53511yescharacteristic
597C22#344911yescharacteristic
598C22#348912yes—
599C22#350012yes—
600C22#353412yes—
601C22#353412yes—
602C22#354012yes—
603C22#354114yes—
604C22#354712yes—
605C22#355011yescharacteristic
606C22#355111yescharacteristic
607C22#355112yes—
608C22#355114yes—
609C22#356514yes—
610C22#356614yes—
611C4—24no—
612C4—24no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—44no—
618C4—82no—
619C4—84no—
620C4—82no—
621C4—84no—
622C4—84no—
623C4—84no—
624C2—82no—
625C2—162no—
626C2#3737611yescharacteristic
627C2#3741911yescharacteristic
628C2#3744011yescharacteristic
629C2#4230712yes—
630C2#4291012yes—
631C2#5653012yes—
632C2#5849114yes—
633C2#5858612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2621C2semidirect
3. #17157C2semidirect
4. #17046C2semidirect
5. #16736C2semidirect
6. #16665C2semidirect
7. #16602C2semidirect
8. #16596C2semidirect
9. #16563C2semidirect
11. #15945C2semidirect
12. #15814C2semidirect
13. #15001C2semidirect
14. #14866C2semidirect
15. #13938C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#26212
#171572
#170462
#167362
#166652
#166022
#165962
#165632
#162532
#159452
#158142
#150012
#148662
#139382