Finite Group Databasebeta

← #644948

Subgroups — #644948

Order 512.

2476 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#16961C211yesmaximal, characteristic, complemented
4#16960C211yesmaximal, characteristic, complemented
5#16692C211yesmaximal, characteristic, complemented
6#16691C211yesmaximal, characteristic, complemented
7#16395C212yesmaximal, complemented
8#16392C212yesmaximal, complemented
9#15122C212yesmaximal, complemented
10#15120C212yesmaximal, complemented
11#15023C214yesmaximal, complemented
12#14964C214yesmaximal, complemented
13#14355C212yesmaximal, complemented
14#14290C212yesmaximal
15#14249C212yesmaximal, complemented
16#14231C212yesmaximal, complemented
17C4×#381C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2854—28no—
22C2×#399C2212yes—
23C2×#397C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#386C2212yes—
33#2283—28no—
34C2×#383C2212yes—
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2212yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1438C2212yes—
72#1438C2212yes—
73#1437C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78#1434C2212yes—
79#1434C2214yes—
80#1434C2211yescharacteristic
81#1434C2211yescharacteristic
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#123—28no—
91C2×#123—48no—
92C2×#121—24no—
93C2×#121—28no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42—28no—
119C22×C42D418yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2312yes—
127#399—28no—
128#397—28no—
129#390—24no—
130#390—24no—
131#389—24no—
132#389—24no—
133#388—28no—
134#388—28no—
135#388—48no—
136#387—48no—
137#386—28no—
138#383—28no—
139#382—24no—
140#382—24no—
141#381—28no—
142#381—28no—
143#380—24no—
144#380—24no—
145#380—24no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—28no—
150#376—28no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—42no—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4#4118yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219#123—432no—
220#121—416no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
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×C42—232no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—42no—
313#95—44no—
314#95—42no—
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#4118yes—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24C22×D412yes—
343C24C2×#4114yes—
344C24C2511yescharacteristic
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—88no—
352C2×D4—88no—
353C2×D4—88no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4#123116yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)14yes—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—416no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—88no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C42—28no—
467C42—28no—
468C42—28no—
469C42—28no—
470C42—48no—
471C42—48no—
472C42—88no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—28no—
480C23—28no—
481C23—24no—
482C23—24no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23C2×#12314yes—
513C23C23×D411yescharacteristic
514C23C22×#4112yes—
515C23C22×#4112yes—
516C23C2×H(5,𝔽2)12yes—
517C23#58512yes—
518C23#58512yes—
519C23#58512yes—
520C23#58511yescharacteristic
521C23#58511yescharacteristic
522D4—88no—
523D4—816no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—416no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
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—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4C2×#12314yes—
599C2×C4C2×#12314yes—
600C2×C4#53214yes—
601C2×C4#53514yes—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—48no—
607C22—48no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—82no—
612C22—82no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22C22×#12311yescharacteristic
623C22C2×#53211yescharacteristic
624C22C2×#53511yescharacteristic
625C22#344911yescharacteristic
626C22#344911yescharacteristic
627C22#344911yescharacteristic
628C22#344911yescharacteristic
629C22#346014yes—
630C22#347514yes—
631C22#349114yes—
632C22#349412yes—
633C22#349412yes—
634C22#352112yes—
635C22#352412yes—
636C22#356214yes—
637C22#356712yes—
638C22#356812yes—
639C4—24no—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—48no—
644C4—42no—
645C4—42no—
646C4—82no—
647C4—82no—
648C4—84no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C2—44no—
655C2—162no—
656C2#3462411yescharacteristic
657C2#3481011yescharacteristic
658C2#3509711yescharacteristic
659C2#3521211yescharacteristic
660C2#3526211yescharacteristic
661C2#3754111yescharacteristic
662C2#3755911yescharacteristic
663C2#3800014yes—
664C2#3800314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #16961C2semidirect
4. #16960C2semidirect
5. #16692C2semidirect
6. #16691C2semidirect
7. #16395C2semidirect
8. #16392C2semidirect
9. #15122C2semidirect
10. #15120C2semidirect
11. #15023C2semidirect
12. #14964C2semidirect
13. #14355C2semidirect
15. #14249C2semidirect
16. #14231C2semidirect
17. C4×#381C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#22832
#169612
#169602
#166922
#166912
#163952
#163922
#151222
#151202
#150232
#149642
#143552
#142902
#142492
#142312
C4×#3812