Finite Group Databasebeta

← #4378651

Subgroups — #4378651

Order 512.

2604 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2322C211yesmaximal, characteristic
3#16603C212yesmaximal, complemented
4#16335C214yesmaximal, complemented
5#16262C212yesmaximal, complemented
6#15983C212yesmaximal
7#15916C212yesmaximal, complemented
8#15406C212yesmaximal, complemented
9#15342C212yesmaximal
10#15080C212yesmaximal, complemented
11#15001C212yesmaximal, complemented
12#14763C212yesmaximal, complemented
13#14720C212yesmaximal, complemented
14#14406C214yesmaximal, complemented
15#13940C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2542—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#397C2214yes—
26C2×#395C2212yes—
27C2×#395C2212yes—
28#2322—28no—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#388C2212yes—
33C2×#387C2212yes—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#386C2211yescharacteristic
37C2×#385C2211yescharacteristic
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2212yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1437C2212yes—
70#1436C2214yes—
71#1436C2212yes—
72#1436C2214yes—
73#1434C2214yes—
74#1434C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—42no—
86C2×#123—24no—
87C2×#123—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8—48no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C2×C4×D4D418yes—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126#399—28no—
127#399—28no—
128#397—48no—
129#395—44no—
130#395—44no—
131#390—28no—
132#389—28no—
133#389—48no—
134#388—28no—
135#387—216no—
136#387—28no—
137#387—48no—
138#386—24no—
139#385—28no—
140#385—24no—
141#385—48no—
142#384—28no—
143#381—24no—
144#381—42no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—42no—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4C2×D414yes—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—41no—
194C23×C4—41no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C4⋊Q8—416no—
216#123—232no—
217C22⋊Q8—432no—
218C22⋊Q8—432no—
219C4×D4—416no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4#4118yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C2×D414yes—
285C2×C22⋊C4C2×D414yes—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320C24—22no—
321C24—24no—
322C24—21no—
323C24—21no—
324C24—24no—
325C24—22no—
326C24—24no—
327C24—21no—
328C24—21no—
329C24—24no—
330C24—22no—
331C24C22×D412yes—
332C24C22×D412yes—
333C242−1+414yes—
334C24C2511yescharacteristic
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—88no—
339C2×Q8—88no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—44no—
343C2×D4—48no—
344C2×D4—48no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
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—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C4H(5,𝔽2)18yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+418yes—
433C4⋊C4—232no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—44no—
437C4⋊C4—416no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C22⋊C4—28no—
447C22⋊C4—216no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—44no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C42—28no—
458C42—48no—
459C42—84no—
460C42—84no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23#53614yes—
491C23C23×D411yescharacteristic
492C23C22×#4112yes—
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)14yes—
496C23C2×2−1+412yes—
497C23C2×2−1+414yes—
498Q8—816no—
499D4—416no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—416no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
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—88no—
571C2×C4—88no—
572C2×C4—88no—
573C2×C4#52514yes—
574C2×C4#53414yes—
575C22—24no—
576C22—28no—
577C22—42no—
578C22—48no—
579C22—44no—
580C22—42no—
581C22—48no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—84no—
588C22—84no—
589C22C2×#52511yescharacteristic
590C22C2×#53411yescharacteristic
591C22C2×#53611yescharacteristic
592C22#348412yes—
593C22#349214yes—
594C22#349912yes—
595C22#352012yes—
596C22#352012yes—
597C22#352214yes—
598C22#353812yes—
599C22#353814yes—
600C22#354312yes—
601C22#354312yes—
602C22#355012yes—
603C22#355214yes—
604C4—24no—
605C4—24no—
606C4—42no—
607C4—44no—
608C4—48no—
609C4—44no—
610C4—48no—
611C4—42no—
612C4—42no—
613C4—82no—
614C4—82no—
615C4—84no—
616C4—84no—
617C2—44no—
618C2—82no—
619C2#3508411yescharacteristic
620C2#3569312yes—
621C2#3723211yescharacteristic
622C2#3738611yescharacteristic
623C2#4049512yes—
624C2#4667714yes—
625C2#4862214yes—
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. #16603C2semidirect
4. #16335C2semidirect
5. #16262C2semidirect
7. #15916C2semidirect
8. #15406C2semidirect
10. #15080C2semidirect
11. #15001C2semidirect
12. #14763C2semidirect
13. #14720C2semidirect
14. #14406C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#23222
#166032
#163352
#162622
#159832
#159162
#154062
#153422
#150802
#150012
#147632
#147202
#144062
#139402