Finite Group Databasebeta

← #556857

Subgroups — #556857

Order 512.

3006 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2#14671C214yesmaximal, complemented
3#14349C211yesmaximal, characteristic, complemented
4#14347C211yesmaximal, characteristic, complemented
5#14341C211yesmaximal, characteristic
6#14338C211yesmaximal, characteristic
7#14293C214yesmaximal, complemented
8#14279C214yesmaximal, complemented
9#14158C214yesmaximal, complemented
10#13882C212yesmaximal, complemented
11#13878C212yesmaximal, complemented
12#13873C212yesmaximal
13#13868C212yesmaximal
14C4×#381C211yesmaximal, characteristic, complemented
15#13756C211yesmaximal, characteristic, complemented
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C2×#388C4116yescomplemented
23C2×#388C2214yes—
24C2×#388C2212yes—
25C2×#388C2212yes—
26C2×#388C2212yes—
27C2×#388C2212yes—
28C2×#387C2214yescomplemented
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#379C2212yes—
33C2×#379C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C22×#95C2211yescharacteristic
36C22×#95C2211yescharacteristic
37#1445C2214yescomplemented
38#1444C2214yescomplemented
39#1443C2214yescomplemented
40#1443C2214yescomplemented
41#1442C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2214yescomplemented
44#1441C2214yescomplemented
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59C4⋊#95C2212yescomplemented
60C4⋊#95C2212yescomplemented
61#1434C2212yescomplemented
62#1434C2212yescomplemented
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C4×#95C2212yescomplemented
73C4×#95C2212yescomplemented
74C4×#95C2212yescomplemented
75C4×#95C2212yescomplemented
76C26C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#117—28no—
84C2×#117—44no—
85C2×#117—44no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2×C418yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—28no—
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⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42—44no—
114C22×C42C2×C418yes—
115C22×C42C2314yes—
116C22×C42C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#388—48no—
122#388C2×C4116yes—
123#388C2×C4116yes—
124#388C2×C4116yes—
125#388C2×C4116yes—
126#387—48no—
127#381—28no—
128#380—44no—
129#380—44no—
130#379—44no—
131#379—44no—
132C4×C22⋊C4—44no—
133C2×#95C2×C418yes—
134C2×#95C2×C418yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C25—24no—
163C25—24no—
164C25—22no—
165C25—22no—
166C25—42no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25C22×C414yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C22×C414yes—
193C23×C4C22×C414yes—
194C23×C4C22×C414yes—
195C23×C4C22×C414yes—
196C23×C4C22×C414yes—
197C23×C4C22×C414yes—
198C23×C4#4118yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212#117—432no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—416no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C22×C418yes—
226C2×C4⋊C4C22×C418yes—
227C2×C22⋊C4—232no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
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—416no—
246C2×C22⋊C4—416no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C418yes—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—416no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42C22×C414yes—
282C2×C42C22×C414yes—
283C2×C42C22×C414yes—
284C2×C42C22×C414yes—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—42no—
293#95—42no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—42no—
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—28no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—28no—
328C24—28no—
329C24—44no—
330C24—44no—
331C24—44no—
332C24—44no—
333C24—416no—
334C24—416no—
335C24—48no—
336C24—48no—
337C24—88no—
338C24—88no—
339C24—84no—
340C24—84no—
341C24—84no—
342C24—84no—
343C24C23×C414yes—
344C24C23×C412yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)14yes—
347C24H(5,𝔽2)12yes—
348C242−1+412yes—
349C24C2511yescharacteristic
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
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—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—44no—
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—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4#117116yes—
433C22×C4C23×C414yes—
434C22×C4C23×C412yes—
435C22×C4C23×C412yes—
436C22×C4C23×C414yes—
437C22×C4C23×C414yes—
438C22×C4C23×C412yes—
439C22×C4C23×C412yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4H(5,𝔽2)14yes—
445C4⋊C4—216no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C42—44no—
465C42—44no—
466C42—44no—
467C42—44no—
468C42—416no—
469C42—88no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—416no—
490C23—416no—
491C23—416no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—88no—
509C23—88no—
510C23—88no—
511C23—88no—
512C23—832no—
513C23—832no—
514C23C2×#11714yes—
515C23#51814yes—
516C23#52014yes—
517C23#53818yes—
518C23#53918yes—
519C23C24×C411yescharacteristic
520C23C22×#4112yes—
521C23C2×H(5,𝔽2)12yes—
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×2−1+411yescharacteristic
525C23#58512yes—
526C23#58512yes—
527C23#58512yes—
528C2×C4—232no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
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—416no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
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—88no—
599C2×C4—88no—
600C2×C4C2×#11714yes—
601C2×C4C2×#11714yes—
602C2×C4#51814yes—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—216no—
610C22—216no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—832no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22#231818yes—
624C22C22×#11711yescharacteristic
625C22C2×#51811yescharacteristic
626C22C2×#51811yescharacteristic
627C22C2×#52011yescharacteristic
628C22#342811yescharacteristic
629C22#342811yescharacteristic
630C22#342811yescharacteristic
631C22C2×#53812yes—
632C22C2×#53912yes—
633C22#345514yes—
634C22#346014yes—
635C22#353314yes—
636C22#353414yes—
637C22#355812yes—
638C22#356012yes—
639C22#356112yes—
640C22#356412yes—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—84no—
651C4—84no—
652C4—84no—
653C4—84no—
654C2—28no—
655C2—84no—
656C2#3404811yescharacteristic
657C2C2×#231811yescharacteristic
658C2#3405811yescharacteristic
659C2#3435711yescharacteristic
660C2#3435811yescharacteristic
661C2#3437711yescharacteristic
662C2#3438111yescharacteristic
663C2#3749614yes—
664C2#3750614yes—
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. #14671C2semidirect
3. #14349C2semidirect
4. #14347C2semidirect
7. #14293C2semidirect
8. #14279C2semidirect
9. #14158C2semidirect
10. #13882C2semidirect
11. #13878C2semidirect
14. C4×#381C2semidirect
15. #13756C2semidirect
22. C2×#388C4semidirect
28. C2×#387C22semidirect
37. #1445C22semidirect
38. #1444C22semidirect
39. #1443C22semidirect
40. #1443C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. #1434C22semidirect
62. #1434C22semidirect
72. C4×#95C22semidirect
73. C4×#95C22semidirect
74. C4×#95C22semidirect
75. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
#146712
#143492
#143472
#143412
#143382
#142932
#142792
#141582
#138822
#138782
#138732
#138682
C4×#3812
#137562