Finite Group Databasebeta

← #4381115

Subgroups — #4381115

Order 512.

3212 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2539C211yesmaximal, characteristic
2C2×#2482C211yesmaximal, characteristic, complemented
3C2×#2624C211yesmaximal, characteristic, complemented
4C2×#2312C211yesmaximal, characteristic, complemented
5#16255C214yesmaximal, complemented
6#15696C214yesmaximal, complemented
7#15689C214yesmaximal, complemented
8#15423C214yesmaximal, complemented
9#15351C214yesmaximal, complemented
10#14761C214yesmaximal, complemented
11#14659C211yesmaximal, characteristic, complemented
12#14297C211yesmaximal, characteristic, complemented
13#13939C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#121C2211yescharacteristic
16C22×#120C2212yes—
17C22×#117C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2624—28no—
20#2539—28no—
21#2482—28no—
22C2×#399C2212yes—
23C2×#399C2212yes—
24C2×#399C2212yescomplemented
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#397C2212yescomplemented
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#392C2212yes—
31#2312—28no—
32C2×#390C2214yes—
33C2×#390C2212yescomplemented
34C2×#389C2212yescomplemented
35C2×#389C2214yescomplemented
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#388C2214yescomplemented
40C2×#385C2212yes—
41C2×#381C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1445C2212yescomplemented
49#1444C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1438C2214yes—
61#1438C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1438C2212yescomplemented
65#1437C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68C4⋊#95C2212yescomplemented
69C4⋊#95C2212yescomplemented
70#1434C2214yescomplemented
71C26C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2314yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×#123—28no—
78C2×#123—48no—
79C2×#121—24no—
80C2×#121—24no—
81C2×#121—24no—
82C2×#120—28no—
83C2×#120—48no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4D418yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2314yes—
120#399—28no—
121#399—28no—
122#399—28no—
123#397—28no—
124#397—28no—
125#397—48no—
126#394—216no—
127#394—48no—
128#392—28no—
129#390—28no—
130#390—216no—
131#389—28no—
132#389—48no—
133#388—216no—
134#388—216no—
135#388—216no—
136#388—48no—
137#385—28no—
138#381—24no—
139#380—24no—
140#380D418yescomplemented
141#379—28no—
142#379—28no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—28no—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C25—22no—
163C25—22no—
164C25—42no—
165C25—42no—
166C25—44no—
167C25—42no—
168C25—42no—
169C25C2×D414yes—
170C25C2412yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—28no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4#4114yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209#123—432no—
210#121—416no—
211#120—432no—
212#117—416no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D418yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—232no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
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—44no—
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⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×D418yes—
292C2×C22⋊C4C2×D418yes—
293C2×C22⋊C4C2×D418yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42—42no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95C2×D418yes—
320C24—24no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—28no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—44no—
349C24—44no—
350C24—48no—
351C24—84no—
352C24—82no—
353C24—82no—
354C24—84no—
355C24—84no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24—88no—
360C24C22×D414yes—
361C24C22×D412yes—
362C24C22×D412yes—
363C24C2×#4112yes—
364C24C2511yescharacteristic
365C2×D4—416no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—416no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—84no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—88no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
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—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
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—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4#121116yes—
440C22×C4C22×D418yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)18yes—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—84no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C22⋊C4—216no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C42—48no—
477C42—84no—
478C42—88no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—416no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—88no—
540C23—88no—
541C23—832no—
542C23—816no—
543C23—816no—
544C23—816no—
545C23—816no—
546C23C2×#12114yes—
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C22×#4111yescharacteristic
550C23C2×H(5,𝔽2)14yes—
551C23#58512yes—
552C23#58512yes—
553C23#58512yes—
554C23#58512yes—
555D4—88no—
556D4—832no—
557D4—816no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—416no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4C2×#12114yes—
617C2×C4C2×#12114yes—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—48no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—44no—
631C22—832no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—82no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22—84no—
645C22—84no—
646C22—82no—
647C22C22×#12111yescharacteristic
648C22#344912yes—
649C22#348412yes—
650C22#348511yescharacteristic
651C22#348512yes—
652C22#348512yes—
653C22#348711yescharacteristic
654C22#349111yescharacteristic
655C22#349111yescharacteristic
656C22#349312yes—
657C22#349812yes—
658C22#350812yes—
659C22#353314yes—
660C22#354014yes—
661C22#356014yes—
662C22#356214yes—
663C4—24no—
664C4—24no—
665C4—48no—
666C4—48no—
667C4—44no—
668C4—82no—
669C4—84no—
670C4—84no—
671C4—82no—
672C4—82no—
673C2—44no—
674C2—84no—
675C2—162no—
676C2#3515011yescharacteristic
677C2#3544711yescharacteristic
678C2#3552811yescharacteristic
679C2#4059614yes—
680C2#4721212yes—
681C2#4721312yes—
682C2#5018712yes—
683C2#5018812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2539C2
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×#2482C2semidirect
3. C2×#2624C2semidirect
4. C2×#2312C2semidirect
5. #16255C2semidirect
6. #15696C2semidirect
7. #15689C2semidirect
8. #15423C2semidirect
9. #15351C2semidirect
10. #14761C2semidirect
11. #14659C2semidirect
12. #14297C2semidirect
13. #13939C2semidirect
24. C2×#399C22semidirect
27. C2×#397C22semidirect
33. C2×#390C22semidirect
34. C2×#389C22semidirect
35. C2×#389C22semidirect
39. C2×#388C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
64. #1438C22semidirect
65. #1437C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect
70. #1434C22semidirect
140. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25392
C2×#24822
C2×#26242
C2×#23122
#162552
#156962
#156892
#154232
#153512
#147612
#146592
#142972
#139392