Finite Group Databasebeta

← #4378522

Subgroups — #4378522

Order 512.

2688 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic
3#16895C212yesmaximal, complemented
4#16395C212yesmaximal, complemented
5#16380C214yesmaximal, complemented
6#15983C212yesmaximal, complemented
7#15945C212yesmaximal, complemented
8#15814C212yesmaximal, complemented
9#15261C212yesmaximal
10#15122C212yesmaximal, complemented
11#14868C212yesmaximal, complemented
12#14842C212yesmaximal, complemented
13#14776C212yesmaximal, complemented
14#14375C214yesmaximal, complemented
15#13940C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#123C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2486—28no—
22C2×#398C2212yes—
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#394C2214yes—
27#2328—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#387C2214yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#382C2211yescharacteristic
38C2×#381C2212yes—
39C2×#381C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44#1445C2212yes—
45#1445C2212yes—
46#1444C2212yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2212yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2212yes—
66#1438C2214yes—
67#1437C2212yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2212yes—
71#1434C2212yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C2×#127—24no—
86C2×#127—28no—
87C2×#123—28no—
88C2×#123—48no—
89C2×C4×D4—24no—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
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⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#398—28no—
126#397—28no—
127#397—44no—
128#397—44no—
129#394—48no—
130#390—24no—
131#390—24no—
132#389—28no—
133#387—48no—
134#386—216no—
135#386—28no—
136#386—28no—
137#386—28no—
138#386—28no—
139#382—24no—
140#381—28no—
141#381—42no—
142#380—28no—
143#380—24no—
144C4×C22⋊C4—42no—
145#376—24no—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25C2411yescharacteristic
171C25C2412yes—
172C25C2411yescharacteristic
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4C2×D414yes—
192C22×D4#4118yes—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223#127—232no—
224#123—432no—
225C4×D4—416no—
226C4×D4—416no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
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×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C2×#4114yes—
346C24C2×#4114yes—
347C24C2511yescharacteristic
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—216no—
351C2×D4—216no—
352C2×D4—28no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—88no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—88no—
367C2×D4—88no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—28no—
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—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)14yes—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—84no—
464C22⋊C4—84no—
465C22⋊C4—88no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—84no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C42—28no—
474C42—28no—
475C42—44no—
476C42—84no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—28no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23#52514yes—
530C23C23×D411yescharacteristic
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23#58514yes—
536C23#58514yes—
537D4—416no—
538D4—416no—
539D4—816no—
540D4—816no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—416no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
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×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—88no—
605C2×C4—816no—
606C2×C4—816no—
607C2×C4#53414yes—
608C2×C4#53514yes—
609C22—24no—
610C22—28no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—48no—
615C22—48no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—48no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22C2×#52511yescharacteristic
635C22C2×#53411yescharacteristic
636C22C2×#53511yescharacteristic
637C22#348512yes—
638C22#349114yes—
639C22#350012yes—
640C22#351912yes—
641C22#351912yes—
642C22#352114yes—
643C22#353312yes—
644C22#353312yes—
645C22#353914yes—
646C22#354012yes—
647C22#355112yes—
648C22#356214yes—
649C4—24no—
650C4—24no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—84no—
660C4—84no—
661C2—44no—
662C2—82no—
663C2—84no—
664C2#3507611yescharacteristic
665C2#3601012yes—
666C2#3695511yescharacteristic
667C2#3741911yescharacteristic
668C2#4265212yes—
669C2#4706514yes—
670C2#5037614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2486C2semidirect
3. #16895C2semidirect
4. #16395C2semidirect
5. #16380C2semidirect
6. #15983C2semidirect
7. #15945C2semidirect
8. #15814C2semidirect
10. #15122C2semidirect
11. #14868C2semidirect
12. #14842C2semidirect
13. #14776C2semidirect
14. #14375C2semidirect
15. #13940C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23282
#168952
#163952
#163802
#159832
#159452
#158142
#152612
#151222
#148682
#148422
#147762
#143752
#139402