Finite Group Databasebeta

← #4868103

Subgroups — #4868103

Order 512.

2378 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2536C211yesmaximal, characteristic
3#17221C212yesmaximal, complemented
4#17140C212yesmaximal
5#16963C212yesmaximal, complemented
6#16676C212yesmaximal, complemented
7#16652C212yesmaximal
8#16629C212yesmaximal, complemented
9#16564C212yesmaximal, complemented
10#16561C214yesmaximal, complemented
11#16511C212yesmaximal, complemented
12#15265C214yesmaximal, complemented
13#15110C212yesmaximal, complemented
14#14807C212yesmaximal
15#14346C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2546—28no—
22#2536—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#398C2211yescharacteristic
26C2×#398C2212yes—
27C2×#398C2211yescharacteristic
28C2×#398C2212yes—
29C2×#395C2214yes—
30C2×#394C2211yescharacteristic
31C2×#394C2211yescharacteristic
32C2×#393C2214yes—
33C2×#391C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#383C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45#1446C2212yes—
46#1445C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1434C2212yes—
76#1434C2212yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—42no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—44no—
86C2×C4⋊Q8—44no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—48no—
89C2×C4×Q8—42no—
90C2×C4×Q8—42no—
91C2×C4×Q8D418yes—
92C2×C4×D4—24no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#399—44no—
124#399—44no—
125#398—24no—
126#398—24no—
127#398—28no—
128#398—28no—
129#395—48no—
130#394—24no—
131#394—24no—
132#393—48no—
133#391—28no—
134#389—28no—
135#389—28no—
136#388—28no—
137#388—28no—
138#385—48no—
139#385—48no—
140#383—42no—
141#382—24no—
142#382—42no—
143#380—28no—
144C4×C4⋊C4—24no—
145C2×#95C2314yes—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C25C2412yes—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—41no—
179C22×Q8—41no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8C2×D414yes—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—42no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4⋊Q8—416no—
210C4⋊Q8—432no—
211C22⋊Q8—432no—
212C4×Q8—416no—
213C4×D4—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C2×D414yes—
289C2×C42C2×D414yes—
290C2×C42C2×D414yes—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C242−1+414yes—
329C24C2511yescharacteristic
330C2×Q8—28no—
331C2×Q8—28no—
332C2×Q8—44no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×D4—48no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4—44no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
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—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
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—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C42−1+418yes—
429C4⋊C4—216no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—416no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4—84no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C42—28no—
454C42—28no—
455C42—28no—
456C42—48no—
457C23—24no—
458C23—28no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—44no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C23×D411yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×H(5,𝔽2)12yes—
485C23C2×2−1+414yes—
486C23C2×2−1+412yes—
487C23#58512yes—
488Q8—416no—
489Q8—816no—
490D4—416no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
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—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—816no—
566C2×C4#53414yes—
567C2×C4#53514yes—
568C2×C4#53514yes—
569C22—44no—
570C22—44no—
571C22—48no—
572C22—48no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—84no—
578C22—82no—
579C22—84no—
580C22C2×#53411yescharacteristic
581C22C2×#53511yescharacteristic
582C22C2×#53511yescharacteristic
583C22#348612yes—
584C22#348912yes—
585C22#353712yes—
586C22#353712yes—
587C22#353814yes—
588C22#353811yescharacteristic
589C22#354011yescharacteristic
590C22#354114yes—
591C22#354112yes—
592C22#354412yes—
593C22#354412yes—
594C22#355214yes—
595C22#355414yes—
596C4—24no—
597C4—24no—
598C4—24no—
599C4—48no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—84no—
608C4—84no—
609C4—84no—
610C2—84no—
611C2#3725811yescharacteristic
612C2#3737411yescharacteristic
613C2#3745011yescharacteristic
614C2#4280812yes—
615C2#4281512yes—
616C2#5375312yes—
617C2#5390012yes—
618C2#5670614yes—
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
1. C2×#2546C2semidirect
3. #17221C2semidirect
5. #16963C2semidirect
6. #16676C2semidirect
8. #16629C2semidirect
9. #16564C2semidirect
10. #16561C2semidirect
11. #16511C2semidirect
12. #15265C2semidirect
13. #15110C2semidirect
15. #14346C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#25362
#172212
#171402
#169632
#166762
#166522
#166292
#165642
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