Finite Group Databasebeta

← #4862605

Subgroups — #4862605

Order 512.

2346 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2638C211yesmaximal, characteristic
3#17160C212yesmaximal, complemented
4#17106C212yesmaximal, complemented
5#16603C212yesmaximal, complemented
6#16528C214yesmaximal, complemented
7#16527C214yesmaximal
8#16364C212yesmaximal, complemented
9#16262C212yesmaximal, complemented
10#16254C212yesmaximal, complemented
11#15790C212yesmaximal, complemented
12#15625C212yesmaximal, complemented
13#15102C212yesmaximal, complemented
14#14763C212yesmaximal, complemented
15#13940C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2638—28no—
22#2542—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#398C2211yescharacteristic
26C2×#397C2211yescharacteristic
27C2×#397C2212yes—
28C2×#397C2214yes—
29C2×#397C2214yes—
30C2×#395C2211yescharacteristic
31C2×#395C2214yes—
32C2×#395C2212yes—
33C2×#395C2212yes—
34C2×#394C2212yes—
35C2×#393C2211yescharacteristic
36C2×#390C2212yes—
37C2×#389C2214yes—
38C2×#387C2212yes—
39C2×#385C2212yes—
40C2×#384C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2212yes—
46#1446C2212yes—
47#1445C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2212yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1440C2212yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1437C2212yes—
71#1436C2212yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—42no—
84C2×C4⋊Q8—42no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—48no—
89C2×C22⋊Q8—48no—
90C2×C22⋊Q8—42no—
91C2×C22⋊Q8—42no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C2×C4×D4D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122#399—28no—
123#399—28no—
124#398—24no—
125#397—24no—
126#397—28no—
127#397—48no—
128#397—48no—
129#395—24no—
130#395—48no—
131#395—44no—
132#395—44no—
133#394—28no—
134#393—24no—
135#390—28no—
136#389—48no—
137#387—28no—
138#385—28no—
139#384—28no—
140#381—24no—
141#381—42no—
142C4×C4⋊C4—24no—
143C4×C22⋊C4—42no—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×Q8—24no—
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8—41no—
180C22×Q8—41no—
181C22×Q8—44no—
182C22×D4—24no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4C2×D414yes—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C4⋊Q8—416no—
209C22⋊Q8—432no—
210C22⋊Q8—432no—
211C22⋊Q8—416no—
212C4×D4—416no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—44no—
285C2×C42C2×D414yes—
286#95—24no—
287#95—24no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—24no—
314C24—24no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—24no—
320C24—24no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24C22×D412yes—
326C24C22×D412yes—
327C24C2511yescharacteristic
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—44no—
331C2×Q8—48no—
332C2×Q8—88no—
333C2×Q8—88no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×D4—28no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4—48no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
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—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
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—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4H(5,𝔽2)18yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C42−1+418yes—
420C22×C42−1+414yes—
421C4⋊C4—216no—
422C4⋊C4—44no—
423C4⋊C4—416no—
424C4⋊C4—416no—
425C4⋊C4—416no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—84no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—88no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C42—28no—
446C42—48no—
447C42—84no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23C23×D411yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×H(5,𝔽2)14yes—
482C23C2×2−1+412yes—
483C23C2×2−1+414yes—
484C23#58512yes—
485Q8—816no—
486Q8—88no—
487D4—416no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
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—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—88no—
558C2×C4—88no—
559C2×C4#53414yes—
560C2×C4#53614yes—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22C2×#53411yescharacteristic
576C22C2×#53611yescharacteristic
577C22#344911yescharacteristic
578C22#348412yes—
579C22#349912yes—
580C22#353514yes—
581C22#353812yes—
582C22#354112yes—
583C22#354112yes—
584C22#354312yes—
585C22#354312yes—
586C22#355012yes—
587C22#355014yes—
588C22#355214yes—
589C22#355914yes—
590C4—24no—
591C4—24no—
592C4—42no—
593C4—48no—
594C4—42no—
595C4—42no—
596C4—48no—
597C4—82no—
598C4—84no—
599C4—82no—
600C4—82no—
601C4—84no—
602C4—82no—
603C2—82no—
604C2—82no—
605C2#3723211yescharacteristic
606C2#3738611yescharacteristic
607C2#3743811yescharacteristic
608C2#4050412yes—
609C2#4242712yes—
610C2#5345114yes—
611C2#5586914yes—
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. #17160C2semidirect
4. #17106C2semidirect
5. #16603C2semidirect
6. #16528C2semidirect
8. #16364C2semidirect
9. #16262C2semidirect
10. #16254C2semidirect
11. #15790C2semidirect
12. #15625C2semidirect
13. #15102C2semidirect
14. #14763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#26382
#171602
#171062
#166032
#165282
#165272
#163642
#162622
#162542
#157902
#156252
#151022
#147632
#139402