Finite Group Databasebeta

← #4430409

Subgroups — #4430409

Order 512.

3885 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic, complemented
2C2×#2613C211yesmaximal, characteristic
3#16367C214yesmaximal, complemented
4#16340C214yesmaximal, complemented
5#15916C212yesmaximal, complemented
6#15669C212yesmaximal, complemented
7#15284C212yesmaximal, complemented
8#15283C212yesmaximal, complemented
9#14922C214yesmaximal, complemented
10#14754C214yesmaximal, complemented
11#14255C212yesmaximal, complemented
12#13867C212yesmaximal, complemented
13C22⋊C42C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#123C2211yescharacteristic
16C22×#120C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19#2842—28no—
20#2613—28no—
21C2×#399C2212yes—
22C2×#399C2212yes—
23C2×#397C2212yescomplemented
24C2×#397C2212yescomplemented
25C2×#396C2214yescomplemented
26C2×#396C2214yescomplemented
27C2×#394C2214yes—
28C2×#394C2214yescomplemented
29C2×#392C2214yes—
30C2×#390C2212yescomplemented
31C2×#388C2214yescomplemented
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yescomplemented
35C2×#382C2212yescomplemented
36C2×#381C2211yescharacteristic
37C2×#379C2212yes—
38C2×#379C2211yescharacteristic
39C2×#379C2214yes—
40C2×#379C2211yescharacteristic
41C2×C4×C22⋊C4C2214yescomplemented
42#1445C2212yescomplemented
43#1443C2214yescomplemented
44#1442C2214yescomplemented
45#1442C2214yescomplemented
46#1441C2214yescomplemented
47#1440C2214yes—
48#1440C2212yescomplemented
49#1440C2214yescomplemented
50#1440C2212yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yescomplemented
59#1438C2212yescomplemented
60#1438C2212yescomplemented
61#1437C2212yes—
62#1437C2212yes—
63#1437C2212yescomplemented
64#1437C2212yescomplemented
65#1436C2214yescomplemented
66C22⋊#95C2214yes—
67C22⋊#95C2214yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×D4—24no—
72C23×D4—28no—
73C23×D4—44no—
74C23×D4—44no—
75C23×D4C2314yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2314yes—
81C2×#123—24no—
82C2×#123—44no—
83C2×#120—24no—
84C2×#120—28no—
85C2×#120—24no—
86C2×#120—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—48no—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117#399—28no—
118#399—28no—
119#397—44no—
120#397—44no—
121#396—216no—
122#396—216no—
123#394—216no—
124#394—48no—
125#392—216no—
126#390—44no—
127#388—48no—
128#386—48no—
129#386—48no—
130#386—48no—
131#382—44no—
132#381—24no—
133#379—28no—
134#379—24no—
135#379—48no—
136#379D418yes—
137C4×C22⋊C4—48no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25—22no—
156C25—24no—
157C25—24no—
158C25—44no—
159C25—42no—
160C25—42no—
161C25—42no—
162C25—44no—
163C25—44no—
164C25—44no—
165C25C2×D414yes—
166C25#4118yes—
167C25C2411yescharacteristic
168C25C2412yes—
169C25C2411yescharacteristic
170C25C2412yes—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—22no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—416no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—464no—
194C22×D4—416no—
195C22×D4—416no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216#123—416no—
217#120—416no—
218#120—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—432no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
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×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⋊C4C2×D414yes—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D4116yes—
291C2×C42—28no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—42no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—44no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—28no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—48no—
338C24—44no—
339C24—44no—
340C24—416no—
341C24—416no—
342C24—416no—
343C24—48no—
344C24—416no—
345C24—44no—
346C24—416no—
347C24—44no—
348C24—84no—
349C24—84no—
350C24—88no—
351C24—84no—
352C24—88no—
353C24—84no—
354C24—84no—
355C24—88no—
356C24—84no—
357C24—84no—
358C24—84no—
359C24—88no—
360C24—88no—
361C24—84no—
362C24—84no—
363C24#124116yes—
364C24C22×D412yes—
365C24C22×D414yes—
366C24C2×#4114yes—
367C24C2×#4114yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)14yes—
370C24C2511yescharacteristic
371C2×D4—232no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—4128no—
379C2×D4—416no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—416no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—88no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—88no—
390C2×D4—84no—
391C2×D4—88no—
392C2×D4—88no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
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—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—416no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)18yes—
456C4⋊C4—48no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C22⋊C4—216no—
462C22⋊C4—416no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—48no—
483C42—84no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—24no—
504C23—28no—
505C23—24no—
506C23—28no—
507C23—24no—
508C23—416no—
509C23—432no—
510C23—416no—
511C23—44no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—41no—
542C23—44no—
543C23—41no—
544C23—41no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—42no—
555C23—816no—
556C23—816no—
557C23—816no—
558C23—832no—
559C23—832no—
560C23—816no—
561C23—832no—
562C23—832no—
563C23—816no—
564C23C2×#12414yes—
565C23C2×#12418yes—
566C23C23×D411yescharacteristic
567C23C22×#4112yes—
568C23C2×H(5,𝔽2)14yes—
569C23C2×H(5,𝔽2)12yes—
570C23C2×H(5,𝔽2)12yes—
571C23#58514yes—
572D4—464no—
573D4—816no—
574D4—88no—
575D4—816no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—416no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—832no—
630C22—24no—
631C22—28no—
632C22—28no—
633C22—28no—
634C22—28no—
635C22—48no—
636C22—44no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—44no—
649C22—44no—
650C22—832no—
651C22—832no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—81no—
657C22—82no—
658C22—82no—
659C22—84no—
660C22—84no—
661C22—82no—
662C22C22×#12411yescharacteristic
663C22#345914yes—
664C22#348411yescharacteristic
665C22#348411yescharacteristic
666C22#349014yes—
667C22#349012yes—
668C22#349112yes—
669C22#349614yes—
670C22#353312yes—
671C22#353414yes—
672C22#353712yes—
673C22#354312yes—
674C22#354614yes—
675C22#355112yes—
676C4—44no—
677C4—44no—
678C4—48no—
679C4—44no—
680C4—81no—
681C4—84no—
682C4—82no—
683C4—84no—
684C4—84no—
685C2—48no—
686C2—44no—
687C2—84no—
688C2—161no—
689C2#3499312yes—
690C2#3536711yescharacteristic
691C2#4091714yes—
692C2#4092814yes—
693C2#4784612yes—
694C2#4956412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
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×#2842C2semidirect
3. #16367C2semidirect
4. #16340C2semidirect
5. #15916C2semidirect
6. #15669C2semidirect
7. #15284C2semidirect
8. #15283C2semidirect
9. #14922C2semidirect
10. #14754C2semidirect
11. #14255C2semidirect
12. #13867C2semidirect
13. C22⋊C42C2semidirect
23. C2×#397C22semidirect
24. C2×#397C22semidirect
25. C2×#396C22semidirect
26. C2×#396C22semidirect
28. C2×#394C22semidirect
30. C2×#390C22semidirect
31. C2×#388C22semidirect
34. C2×#386C22semidirect
35. C2×#382C22semidirect
41. C2×C4×C22⋊C4C22semidirect
42. #1445C22semidirect
43. #1443C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
58. #1438C22semidirect
59. #1438C22semidirect
60. #1438C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. #1436C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
C2×#26132
#163672
#163402
#159162
#156692
#152842
#152832
#149222
#147542
#142552
#138672
C22⋊C422