Finite Group Databasebeta

← #4344620

Subgroups — #4344620

Order 512.

2978 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic
3#16986C212yesmaximal, complemented
4#16823C212yesmaximal, complemented
5#16694C212yesmaximal, complemented
6#16303C212yesmaximal, complemented
7#15779C214yesmaximal, complemented
8#15568C212yesmaximal, complemented
9#15406C212yesmaximal, complemented
10#14881C212yesmaximal, complemented
11#14874C212yesmaximal, complemented
12#14763C212yesmaximal, complemented
13#14758C212yesmaximal, complemented
14#14407C214yesmaximal
15#13784C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2580—28no—
21#2409—28no—
22C2×#399C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#395C2212yes—
25C2×#395C2212yes—
26C2×#395C2212yes—
27C2×#395C2214yes—
28C2×#393C2211yescharacteristic
29C2×#393C2211yescharacteristic
30C2×#393C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#382C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2214yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1446C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C4×Q8—24no—
92C2×C4×Q8—42no—
93C2×C4×Q8—42no—
94C2×C4×D4—42no—
95C2×C4×D4—42no—
96C2×C4×D4D418yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122#399—24no—
123#399—24no—
124#395—28no—
125#395—44no—
126#395—44no—
127#395—48no—
128#393—24no—
129#393—24no—
130#393—28no—
131#391—28no—
132#390—28no—
133#389—28no—
134#389—48no—
135#386—28no—
136#386Q8116yescomplemented
137#385—28no—
138#385—28no—
139#385—28no—
140#385—48no—
141#382—24no—
142#380—24no—
143#380—48no—
144C4×C22⋊C4—42no—
145C2×#95—42no—
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
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×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—41no—
183C22×Q8—41no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—416no—
190C22×Q8—432no—
191C22×Q8—432no—
192C22×Q8—432no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—42no—
196C22×D4C2×D414yes—
197C22×D4C2×Q818yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×Q818yescomplemented
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C22⋊Q8—432no—
230C4×Q8—416no—
231C4×D4—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×Q8116yes—
290C2×C22⋊C4C2×Q818yescomplemented
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—44no—
298C2×C42C2×D414yes—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
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—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×Q8116yescomplemented
325C24—24no—
326C24—21no—
327C24—24no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C22×Q814yes—
340C24C22×Q814yes—
341C24C2511yescharacteristic
342C2×Q8—216no—
343C2×Q8—48no—
344C2×Q8—44no—
345C2×Q8—416no—
346C2×Q8—416no—
347C2×Q8—416no—
348C2×Q8—48no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—416no—
354C2×Q8—44no—
355C2×Q8—4128no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4C4⋊Q8116yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—88no—
440C22×C4—84no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×Q818yes—
447C22×C4C22×Q818yes—
448C22×C4C22×Q814yes—
449C22×C4H(5,𝔽2)18yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+414yes—
453C4⋊C4—44no—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C42—28no—
476C42—48no—
477C42—84no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—44no—
502C23—44no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23C2×C4⋊Q814yes—
509C23C2×C4⋊Q814yes—
510C23C23×D411yescharacteristic
511C23C23×Q812yes—
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)14yes—
515C23C2×2−1+412yes—
516C23#58512yes—
517C23#58512yes—
518Q8—432no—
519Q8—416no—
520Q8—432no—
521D4—416no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—416no—
548C2×C4—432no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—88no—
595C2×C4—816no—
596C2×C4C2×C4⋊Q814yes—
597C2×C4#53414yes—
598C22—28no—
599C22—28no—
600C22—24no—
601C22—24no—
602C22—48no—
603C22—48no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22C22×C4⋊Q811yescharacteristic
615C22C2×#53411yescharacteristic
616C22#344911yescharacteristic
617C22#348412yes—
618C22#348912yes—
619C22#350114yes—
620C22#350112yes—
621C22#350112yes—
622C22#350314yes—
623C22#353212yes—
624C22#353212yes—
625C22#353811yescharacteristic
626C22#354014yes—
627C22#354011yescharacteristic
628C22#354112yes—
629C22#355414yes—
630C4—24no—
631C4—24no—
632C4—48no—
633C4—42no—
634C4—48no—
635C4—42no—
636C4—48no—
637C4—42no—
638C4—44no—
639C4—82no—
640C4—84no—
641C4—82no—
642C4—82no—
643C2—44no—
644C2—44no—
645C2#3501111yescharacteristic
646C2#3592812yes—
647C2#3670211yescharacteristic
648C2#3737411yescharacteristic
649C2#4060712yes—
650C2#4701812yes—
651C2#4702312yes—
652C2#4796514yes—
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. #16986C2semidirect
4. #16823C2semidirect
5. #16694C2semidirect
6. #16303C2semidirect
7. #15779C2semidirect
8. #15568C2semidirect
9. #15406C2semidirect
10. #14881C2semidirect
11. #14874C2semidirect
12. #14763C2semidirect
13. #14758C2semidirect
136. #386Q8semidirect
215. C23×C4C2×Q8semidirect
290. C2×C22⋊C4C2×Q8semidirect
324. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#24092
#169862
#168232
#166942
#163032
#157792
#155682
#154062
#148812
#148742
#147632
#147582
#144072
#137842