Finite Group Databasebeta

← #555996

Subgroups — #555996

Order 512.

3114 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic, complemented
3#14684C211yesmaximal, characteristic, complemented
4#14509C214yesmaximal, complemented
5#14481C214yesmaximal, complemented
6#14436C214yesmaximal, complemented
7#14433C214yesmaximal, complemented
8#14349C211yesmaximal, characteristic, complemented
9#14347C211yesmaximal, characteristic, complemented
10#13882C212yesmaximal, complemented
11#13877C212yesmaximal
12#13874C212yesmaximal, complemented
13#13869C212yesmaximal
14#13794C211yesmaximal, characteristic
15#13775C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C2×#388C4116yescomplemented
23C2×#388C2212yes—
24C2×#388C2212yes—
25C2×#388C2212yes—
26C2×#388C2212yes—
27C2×#387C2214yescomplemented
28C2×#385C2214yes—
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#379C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C22×#95C2211yescharacteristic
36C22×#95C2211yescharacteristic
37#1445C2214yescomplemented
38#1443C2214yescomplemented
39#1443C2214yescomplemented
40#1443C2214yescomplemented
41#1442C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2214yescomplemented
44#1441C2214yescomplemented
45#1440C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1437C2214yes—
50#1437C2214yes—
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53C4⋊#95C2212yescomplemented
54C4⋊#95C2212yescomplemented
55#1434C2214yescomplemented
56#1434C2212yescomplemented
57#1434C2212yescomplemented
58#1434C2212yescomplemented
59#1434C2212yescomplemented
60#1434C2212yescomplemented
61#1434C2212yescomplemented
62C22⋊#95C2212yes—
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2211yescharacteristic
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C26C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2314yes—
84C22×C4⋊C4—44no—
85C22×C4⋊C4C2×C418yescomplemented
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4C2×C418yes—
97C22×C22⋊C4C2×C418yes—
98C22×C22⋊C4C2×C418yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
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⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42—44no—
113C22×C42C2×C418yescomplemented
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#388C2×C4116yescomplemented
120#388C2×C4116yescomplemented
121#388C2×C4116yescomplemented
122#388C2×C4116yescomplemented
123#387—48no—
124#385—48no—
125#381—44no—
126#381—44no—
127#380—44no—
128#380—44no—
129#379—28no—
130C4×C22⋊C4—44no—
131C2×#95C2×C418yescomplemented
132C2×#95C2×C418yescomplemented
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C25—24no—
163C25—22no—
164C25—22no—
165C25—42no—
166C25—42no—
167C25—42no—
168C25—42no—
169C25C22×C414yes—
170C25#4118yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C22×C414yes—
201C23×C4C22×C414yes—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C22×C414yes—
205C23×C4C22×C414yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C22×C418yes—
229C2×C4⋊C4C22×C418yes—
230C2×C22⋊C4—232no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—416no—
252C2×C22⋊C4—416no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—432no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C418yes—
270C2×C22⋊C4C22×C418yes—
271C2×C42—416no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C22×C414yes—
280C2×C42C22×C414yes—
281C2×C42C22×C414yes—
282C2×C42C22×C414yes—
283#95—28no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—44no—
293#95—42no—
294#95—42no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95C22×C418yes—
309#95C22×C418yes—
310#95C22×C418yes—
311#95C22×C418yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—28no—
325C24—28no—
326C24—216no—
327C24—216no—
328C24—44no—
329C24—44no—
330C24—44no—
331C24—44no—
332C24—416no—
333C24—416no—
334C24—88no—
335C24—88no—
336C24—84no—
337C24—84no—
338C24—84no—
339C24—84no—
340C24#117116yes—
341C24C23×C414yes—
342C24C23×C412yes—
343C24C2×#4114yes—
344C24C2×#4114yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)14yes—
347C24H(5,𝔽2)12yes—
348C242−1+412yes—
349C24C2511yescharacteristic
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
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—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—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—88no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4C23×C414yes—
443C22×C4C23×C412yes—
444C22×C4C23×C412yes—
445C22×C4C23×C412yes—
446C22×C4C23×C412yes—
447C22×C4C23×C414yes—
448C22×C4C23×C414yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4H(5,𝔽2)14yes—
452C4⋊C4—216no—
453C4⋊C4—48no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—216no—
457C22⋊C4—216no—
458C22⋊C4—216no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C42—44no—
470C42—44no—
471C42—44no—
472C42—44no—
473C42—88no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—232no—
501C23—416no—
502C23—416no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—88no—
512C23—88no—
513C23—88no—
514C23—88no—
515C23—832no—
516C23—832no—
517C23C2×#11718yes—
518C23C2×#11714yes—
519C23#51814yes—
520C23#52014yes—
521C23#53818yes—
522C23#53918yes—
523C23C24×C411yescharacteristic
524C23C22×#4112yes—
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)12yes—
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23#58512yes—
530C23#58512yes—
531C23#58512yes—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
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×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—88no—
605C2×C4—88no—
606C2×C4—832no—
607C2×C4#51814yes—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—216no—
613C22—216no—
614C22—24no—
615C22—24no—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—48no—
624C22—48no—
625C22—48no—
626C22—48no—
627C22—832no—
628C22#231818yes—
629C22C22×#11711yescharacteristic
630C22C2×#51811yescharacteristic
631C22C2×#51811yescharacteristic
632C22C2×#52011yescharacteristic
633C22#342811yescharacteristic
634C22#342811yescharacteristic
635C22#342811yescharacteristic
636C22C2×#53812yes—
637C22C2×#53912yes—
638C22#349114yes—
639C22#350514yes—
640C22#353314yes—
641C22#353414yes—
642C22#355812yes—
643C22#356012yes—
644C22#356112yes—
645C22#356412yes—
646C4—24no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—84no—
656C4—84no—
657C4—84no—
658C4—84no—
659C2—28no—
660C2—48no—
661C2C2×#231811yescharacteristic
662C2#3417711yescharacteristic
663C2#3418511yescharacteristic
664C2#3435711yescharacteristic
665C2#3435811yescharacteristic
666C2#3437711yescharacteristic
667C2#3438111yescharacteristic
668C2#3698514yes—
669C2#3698614yes—
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
2. #14685C2semidirect
3. #14684C2semidirect
4. #14509C2semidirect
5. #14481C2semidirect
6. #14436C2semidirect
7. #14433C2semidirect
8. #14349C2semidirect
9. #14347C2semidirect
10. #13882C2semidirect
12. #13874C2semidirect
22. C2×#388C4semidirect
27. C2×#387C22semidirect
37. #1445C22semidirect
38. #1443C22semidirect
39. #1443C22semidirect
40. #1443C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect
53. C4⋊#95C22semidirect
54. C4⋊#95C22semidirect
55. #1434C22semidirect
56. #1434C22semidirect
57. #1434C22semidirect
58. #1434C22semidirect
59. #1434C22semidirect
60. #1434C22semidirect
61. #1434C22semidirect
85. C22×C4⋊C4C2×C4semidirect
113. C22×C42C2×C4semidirect
119. #388C2×C4semidirect
120. #388C2×C4semidirect
121. #388C2×C4semidirect
122. #388C2×C4semidirect
131. C2×#95C2×C4semidirect
132. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
#146852
#146842
#145092
#144812
#144362
#144332
#143492
#143472
#138822
#138772
#138742
#138692
#137942
#137752