Finite Group Databasebeta

← #4861918

Subgroups — #4861918

Order 512.

2728 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2633C211yesmaximal, characteristic
3#17201C212yesmaximal, complemented
4#17106C212yesmaximal, complemented
5#16892C212yesmaximal, complemented
6#16696C212yesmaximal, complemented
7#16608C212yesmaximal, complemented
8#16553C214yesmaximal, complemented
9#16309C214yesmaximal
10#16262C212yesmaximal, complemented
11#15579C212yesmaximal, complemented
12#15406C212yesmaximal, complemented
13#14874C212yesmaximal, complemented
14#14763C212yesmaximal, complemented
15#13940C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2633—28no—
21#2409—28no—
22C2×#399C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#398C2212yes—
25C2×#397C2211yescharacteristic
26C2×#397C2211yescharacteristic
27C2×#397C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32C2×#394C2212yes—
33C2×#393C2212yes—
34C2×#391C2212yes—
35C2×#389C2214yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#383C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2214yes—
46#1446C2212yes—
47#1446C2212yes—
48#1445C2214yes—
49#1445C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2212yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1440C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1437C2212yes—
72#1436C2212yes—
73#1436C2214yes—
74#1434C2212yes—
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⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#399—24no—
121#399—24no—
122#398—28no—
123#397—24no—
124#397—24no—
125#397—48no—
126#395—28no—
127#395—44no—
128#395—44no—
129#395—48no—
130#394—28no—
131#393—28no—
132#391—28no—
133#389—48no—
134#386—28no—
135#385—28no—
136#385—28no—
137#385—28no—
138#385—48no—
139#383—24no—
140#381—42no—
141#380—24no—
142C4×C22⋊C4—42no—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
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×#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—41no—
189C22×D4—41no—
190C22×D4—42no—
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×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218#125—416no—
219C22⋊Q8—432no—
220C4×D4—416no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C2×D414yes—
281C2×C22⋊C4C2×D414yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42C2×D414yes—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315C24—24no—
316C24—21no—
317C24—21no—
318C24—24no—
319C24—22no—
320C24—24no—
321C24—21no—
322C24—21no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24C22×D412yes—
327C24C22×D412yes—
328C242−1+414yes—
329C24C2511yescharacteristic
330C2×Q8—216no—
331C2×Q8—416no—
332C2×Q8—416no—
333C2×Q8—416no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—416no—
338C2×Q8—4128no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×D4—28no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4H(5,𝔽2)18yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+418yes—
430C4⋊C4—44no—
431C4⋊C4—416no—
432C4⋊C4—416no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—416no—
446C22⋊C4—416no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C42—28no—
454C42—84no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—42no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23#53614yes—
485C23C23×D411yescharacteristic
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×H(5,𝔽2)14yes—
489C23C2×2−1+412yes—
490C23C2×2−1+414yes—
491C23#58512yes—
492Q8—432no—
493Q8—432no—
494D4—416no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—416no—
514C2×C4—432no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
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—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—88no—
568C2×C4#53414yes—
569C22—28no—
570C22—24no—
571C22—42no—
572C22—48no—
573C22—44no—
574C22—42no—
575C22—48no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—84no—
582C22—84no—
583C22C2×#53411yescharacteristic
584C22C2×#53611yescharacteristic
585C22#344911yescharacteristic
586C22#348412yes—
587C22#349912yes—
588C22#353212yes—
589C22#353212yes—
590C22#353812yes—
591C22#353814yes—
592C22#354112yes—
593C22#354114yes—
594C22#354112yes—
595C22#355012yes—
596C22#355214yes—
597C22#355414yes—
598C4—24no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—48no—
603C4—42no—
604C4—48no—
605C4—84no—
606C4—82no—
607C4—84no—
608C4—82no—
609C4—82no—
610C4—82no—
611C2—44no—
612C2—82no—
613C2#3670211yescharacteristic
614C2#3738611yescharacteristic
615C2#3743811yescharacteristic
616C2#4049512yes—
617C2#4050312yes—
618C2#5798714yes—
619C2#5889614yes—
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. #17201C2semidirect
4. #17106C2semidirect
5. #16892C2semidirect
6. #16696C2semidirect
7. #16608C2semidirect
8. #16553C2semidirect
10. #16262C2semidirect
11. #15579C2semidirect
12. #15406C2semidirect
13. #14874C2semidirect
14. #14763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#26332
#172012
#171062
#168922
#166962
#166082
#165532
#163092
#162622
#155792
#154062
#148742
#147632
#139402