Finite Group Databasebeta

← #559868

Subgroups — #559868

Order 512.

3554 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#381C211yesmaximal, characteristic
2#14680C212yesmaximal
3#14674C212yesmaximal, complemented
4#14508C212yesmaximal, complemented
5#14507C212yesmaximal, complemented
6#14368C214yesmaximal, complemented
7#14367C214yesmaximal, complemented
8#13908C212yesmaximal, complemented
9#13903C211yesmaximal, characteristic
10#13883C212yesmaximal, complemented
11#13878C212yesmaximal, complemented
12#13874C212yesmaximal, complemented
13#13873C212yesmaximal
14#13868C211yesmaximal, characteristic
15#13828C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C2×#388C2214yes—
21C2×#381C4116yescomplemented
22C2×#381C2211yescharacteristic
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25C2×#381C2212yes—
26C2×#381C2212yes—
27C2×#381C2211yescharacteristic
28C2×#381C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#380C2214yes—
31C2×#380C2214yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#379C2212yes—
37C2×#379C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C22×#95C2212yes—
41C22×#95C2212yes—
42#1444C2214yescomplemented
43#1444C2214yescomplemented
44#1443C2214yescomplemented
45#1443C2214yescomplemented
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1440C2214yes—
51#1439C2212yes—
52#1439C2212yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2212yes—
56#1437C2212yes—
57#1437C2214yes—
58#1437C2212yes—
59#1437C2212yes—
60#1437C2212yes—
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1434C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C26C2311yescharacteristic
77C24×C4—28no—
78C24×C4—44no—
79C24×C4—42no—
80C24×C4—42no—
81C24×C4C2×C418yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C2×#117—44no—
90C2×#117—42no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—44no—
97C22×C22⋊C4—42no—
98C22×C22⋊C4—42no—
99C22×C22⋊C4C2×C418yes—
100C22×C22⋊C4C2×C418yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2314yes—
119#388—48no—
120#381—44no—
121#381—42no—
122#381C2×C4116yes—
123#381C2×C4116yes—
124#381C2×C418yes—
125#381C2×C418yes—
126#381C2×C418yes—
127#381C2×C418yes—
128#380—216no—
129#380—44no—
130#380—44no—
131#380—44no—
132#380—42no—
133#380—48no—
134#379—44no—
135#379—44no—
136C4×C22⋊C4—44no—
137C4×C22⋊C4—42no—
138C2×#95C2×C4116yes—
139C2×#95C2×C4116yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25—24no—
169C25—24no—
170C25—22no—
171C25—22no—
172C25—24no—
173C25—24no—
174C25—44no—
175C25C22×C414yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—28no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—232no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—416no—
195C23×C4—416no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—41no—
203C23×C4—41no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—432no—
209C23×C4—432no—
210C23×C4C22×C414yes—
211C23×C4C22×C418yes—
212C23×C4C22×C418yes—
213C23×C4C22×C414yes—
214C23×C4C22×C418yes—
215C23×C4C22×C414yes—
216C23×C4C22×C414yes—
217C23×C4#4118yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230#117—416no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—416no—
262C2×C22⋊C4—416no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
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⋊C4C22×C418yes—
282C2×C22⋊C4C22×C414yes—
283C2×C22⋊C4C22×C414yes—
284C2×C22⋊C4C22×C414yes—
285C2×C22⋊C4C22×C414yes—
286C2×C22⋊C4C22×C414yes—
287C2×C22⋊C4C22×C414yes—
288C2×C42—44no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95C22×C418yes—
316#95C22×C418yes—
317#95C22×C418yes—
318#95C22×C418yes—
319#95C22×C418yes—
320#95C22×C418yes—
321#95C22×C418yes—
322#95C22×C418yes—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—216no—
327C24—216no—
328C24—28no—
329C24—28no—
330C24—28no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—216no—
344C24—48no—
345C24—416no—
346C24—416no—
347C24—416no—
348C24—416no—
349C24—48no—
350C24—48no—
351C24—44no—
352C24—48no—
353C24—44no—
354C24C23×C412yes—
355C24C23×C412yes—
356C24C23×C412yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)14yes—
359C24H(5,𝔽2)12yes—
360C24H(5,𝔽2)12yes—
361C24H(5,𝔽2)14yes—
362C242−1+414yes—
363C24C2511yescharacteristic
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—24no—
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—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—4128no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4#117116yescomplemented
456C22×C4C23×C414yes—
457C22×C4C23×C414yes—
458C22×C4C23×C412yes—
459C22×C4C23×C412yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C414yes—
463C22×C4C23×C414yes—
464C22×C4C2×#4118yes—
465C4⋊C4—48no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C22⋊C4—216no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—48no—
486C42—84no—
487C42—88no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—28no—
499C23—232no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—432no—
517C23—432no—
518C23—416no—
519C23—432no—
520C23—432no—
521C23—464no—
522C23—416no—
523C23—44no—
524C23—44no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23C2×#11714yes—
538C23#51814yes—
539C23#51814yes—
540C23#52018yes—
541C23#56014yes—
542C23#56218yes—
543C23#56314yes—
544C23C24×C411yescharacteristic
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+412yes—
551C23#58514yes—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—416no—
574C2×C4—432no—
575C2×C4—432no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
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—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—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—88no—
623C2×C4—816no—
624C2×C4C2×#11718yes—
625C22—24no—
626C22—28no—
627C22—24no—
628C22—28no—
629C22—28no—
630C22—28no—
631C22—216no—
632C22—28no—
633C22—24no—
634C22—24no—
635C22—28no—
636C22—464no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—48no—
649C22#237618yes—
650C22C22×#11711yescharacteristic
651C22C2×#51811yescharacteristic
652C22C2×#51811yescharacteristic
653C22C2×#52012yes—
654C22#342812yes—
655C22#349012yes—
656C22#349012yes—
657C22#349414yes—
658C22#349514yes—
659C22C2×#56011yescharacteristic
660C22C2×#56212yes—
661C22C2×#56311yescharacteristic
662C22#353312yes—
663C22#353312yes—
664C22#353414yes—
665C22#353514yes—
666C4—28no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—84no—
675C4—84no—
676C4—84no—
677C4—84no—
678C2—28no—
679C2—44no—
680C2—44no—
681C2#3417611yescharacteristic
682C2#3418012yes—
683C2C2×#237611yescharacteristic
684C2#3435711yescharacteristic
685C2#3437612yes—
686C2#4547312yes—
687C2#4547412yes—
688C2#4547514yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14674C2semidirect
4. #14508C2semidirect
5. #14507C2semidirect
6. #14368C2semidirect
7. #14367C2semidirect
8. #13908C2semidirect
10. #13883C2semidirect
11. #13878C2semidirect
12. #13874C2semidirect
15. #13828C2semidirect
21. C2×#381C4semidirect
42. #1444C22semidirect
43. #1444C22semidirect
44. #1443C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
455. C22×C4#117semidirect

Maximal subgroups

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

subgroupindex
C22×#3812
#146802
#146742
#145082
#145072
#143682
#143672
#139082
#139032
#138832
#138782
#138742
#138732
#138682
#138282