Finite Group Databasebeta

← #559867

Subgroups — #559867

Order 512.

3398 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2280C211yesmaximal, characteristic
2#14680C212yesmaximal
3#14674C212yesmaximal, complemented
4#14370C214yesmaximal, complemented
5#14369C214yesmaximal, complemented
6#14366C212yesmaximal
7#14365C212yesmaximal
8#13957C211yesmaximal, characteristic, complemented
9#13944C212yesmaximal, complemented
10#13938C211yesmaximal, characteristic, complemented
11#13919C212yesmaximal, complemented
12#13915C212yesmaximal, complemented
13#13898C212yesmaximal, complemented
14#13891C212yesmaximal, complemented
15#13828C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#388C2214yes—
20C2×#388C2214yes—
21C2×#387C2214yescomplemented
22C2×#383C2211yescharacteristic, complemented
23C2×#382C2211yescharacteristic, complemented
24#2280C4116yescomplemented
25C2×#381C2211yescharacteristic
26C2×#381C2211yescharacteristic
27C2×#380C2214yes—
28C2×#380C2214yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#379C2214yes—
34C2×#379C2211yescharacteristic
35C2×#379C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic, complemented
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×#376C2211yescharacteristic, complemented
40#1445C2214yescomplemented
41#1444C2214yescomplemented
42#1444C2214yescomplemented
43#1442C2214yescomplemented
44#1442C2214yescomplemented
45#1441C2212yescomplemented
46#1441C2212yescomplemented
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1437C2212yes—
55#1437C2212yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2212yescomplemented
64#1436C2212yescomplemented
65C4⋊#95C2212yescomplemented
66#1434C2212yescomplemented
67#1434C2212yescomplemented
68#1434C2212yescomplemented
69#1434C2212yescomplemented
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2212yescomplemented
76C26C2311yescharacteristic
77C24×C4—28no—
78C24×C4—44no—
79C24×C4—42no—
80C24×C4—42no—
81C24×C4C2×C418yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C2×#117—42no—
87C2×#117—42no—
88C2×#117—44no—
89C2×#117C2×C418yes—
90C2×#117C2×C418yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119#388—48no—
120#388—48no—
121#387—48no—
122#383—42no—
123#382—42no—
124#381C2×C418yes—
125#381C2×C418yes—
126#380—216no—
127#380—44no—
128#380—44no—
129#380—48no—
130#380C2×C4116yes—
131#380C2×C4116yes—
132#379—48no—
133#379C2×C418yes—
134#379C2×C418yes—
135C4×C4⋊C4—42no—
136C4×C22⋊C4C2×C4116yes—
137C4×C22⋊C4C2×C4116yes—
138#376—42no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C25—24no—
166C25—24no—
167C25—24no—
168C25—41no—
169C25—41no—
170C25—42no—
171C25—44no—
172C25C22×C414yes—
173C25C2411yescharacteristic
174C25C2412yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—232no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—416no—
186C23×C4—416no—
187C23×C4—44no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—432no—
194C23×C4—432no—
195C23×C4C22×C414yes—
196C23×C4C22×C418yes—
197C23×C4C22×C418yes—
198C23×C4C22×C414yes—
199C23×C4#4118yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213#117—416no—
214#117—416no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C22×C414yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—41no—
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⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C414yes—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C414yes—
271C2×C22⋊C4C22×C414yes—
272C2×C22⋊C4C22×C418yes—
273C2×C22⋊C4C22×C418yes—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42—44no—
281C2×C42—42no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42C22×C414yes—
287C2×C42C22×C414yes—
288C2×C42C22×C418yes—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95C22×C418yes—
312#95C22×C418yes—
313#95C22×C418yes—
314#95C22×C418yes—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—28no—
330C24—216no—
331C24—44no—
332C24—44no—
333C24—48no—
334C24—416no—
335C24—416no—
336C24—48no—
337C24—44no—
338C24—44no—
339C24—48no—
340C24—48no—
341C24—416no—
342C24—416no—
343C24—48no—
344C24—48no—
345C24C23×C412yes—
346C24C23×C414yes—
347C24C2×#4114yes—
348C24H(5,𝔽2)14yes—
349C242−1+414yes—
350C24C2511yescharacteristic
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—216no—
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—48no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—4128no—
429C22×C4#117116yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C23×C412yes—
434C22×C4C23×C414yes—
435C22×C4C23×C414yes—
436C22×C4C23×C414yes—
437C22×C4C23×C412yes—
438C22×C4C23×C412yes—
439C22×C4C2×#4118yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)14yes—
443C4⋊C4—216no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—84no—
450C4⋊C4—88no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C42—28no—
466C42—28no—
467C42—48no—
468C42—48no—
469C42—84no—
470C42—88no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—416no—
494C23—432no—
495C23—416no—
496C23—432no—
497C23—416no—
498C23—464no—
499C23—432no—
500C23—432no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23C2×#11714yes—
517C23#52018yes—
518C23C24×C411yescharacteristic
519C23C22×#4112yes—
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)12yes—
524C23C2×2−1+412yes—
525C23#58514yes—
526C2×C4—232no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—416no—
554C2×C4—432no—
555C2×C4—432no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
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×C4C2×#11718yes—
599C2×C4#51814yes—
600C2×C4#51814yes—
601C22—28no—
602C22—28no—
603C22—24no—
604C22—28no—
605C22—28no—
606C22—464no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—44no—
618C22—48no—
619C22—44no—
620C22—48no—
621C22C22×#11711yescharacteristic
622C22C2×#51811yescharacteristic
623C22C2×#51811yescharacteristic
624C22C2×#52012yes—
625C22#342812yes—
626C22#349012yes—
627C22#349012yes—
628C22#349414yes—
629C22#349514yes—
630C22#352912yes—
631C22#352912yes—
632C22#353612yes—
633C22#353711yescharacteristic
634C22#353811yescharacteristic
635C22#354114yes—
636C22#354214yes—
637C4—24no—
638C4—24no—
639C4—28no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—84no—
647C4—84no—
648C4—84no—
649C4—84no—
650C2—48no—
651C2—44no—
652C2#3417611yescharacteristic
653C2#3418012yes—
654C2#3434211yescharacteristic
655C2#3436011yescharacteristic
656C2#3440412yes—
657C2#4547014yes—
658C2#4547112yes—
659C2#4547212yes—
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. #14370C2semidirect
5. #14369C2semidirect
8. #13957C2semidirect
9. #13944C2semidirect
10. #13938C2semidirect
11. #13919C2semidirect
12. #13915C2semidirect
13. #13898C2semidirect
14. #13891C2semidirect
15. #13828C2semidirect
21. C2×#387C22semidirect
22. C2×#383C22semidirect
23. C2×#382C22semidirect
24. #2280C4semidirect
36. C2×C4×C4⋊C4C22semidirect
39. C2×#376C22semidirect
40. #1445C22semidirect
41. #1444C22semidirect
42. #1444C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
69. #1434C22semidirect
75. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22802
#146802
#146742
#143702
#143692
#143662
#143652
#139572
#139442
#139382
#139192
#139152
#138982
#138912
#138282