Finite Group Databasebeta

← #647316

Subgroups — #647316

Order 512.

2498 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic
3#17044C214yesmaximal, complemented
4#16900C212yesmaximal, complemented
5#16820C212yesmaximal, complemented
6#15024C212yesmaximal, complemented
7#15019C211yesmaximal, characteristic
8#15015C211yesmaximal, characteristic
9#15014C211yesmaximal, characteristic
10#14936C211yesmaximal, characteristic
11#14935C212yesmaximal
12#14929C212yesmaximal, complemented
13#14728C212yesmaximal
14#14314C214yesmaximal, complemented
15#14043C212yesmaximal
16#13972C212yesmaximal
17C4×#380C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2851—28no—
23C2×#398C2214yes—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2214yes—
27C2×#387—28no—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#382C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1445C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2212yes—
64#1442C2214yes—
65#1442C2212yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77#1434C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2211yescharacteristic
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#125—24no—
93C2×#125—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42—44no—
118C22×C42D418yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#398—216no—
125#389—24no—
126#389—24no—
127#389—48no—
128#387—28no—
129#387—28no—
130#387—28no—
131#387—28no—
132#387—28no—
133#387—28no—
134#385—28no—
135#385—28no—
136#385—28no—
137#385—44no—
138#382—24no—
139#381—28no—
140#381—44no—
141#380—24no—
142#380—24no—
143#380—28no—
144C4×C4⋊C4—44no—
145C4×C22⋊C4—44no—
146C2×#95—24no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2411yescharacteristic
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#125—416no—
233C2×C4⋊C4—232no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4#4118yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4#4118yes—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—416no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42C2×D414yes—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24C22×D412yes—
333C24C2×#4114yes—
334C24H(5,𝔽2)12yes—
335C242−1+414yes—
336C242−1+412yes—
337C24C2511yescharacteristic
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C2×#4118yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C42−1+414yes—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—88no—
464C22⋊C4—216no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C42—28no—
469C42—28no—
470C42—44no—
471C42—44no—
472C42—88no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23#53214yes—
486C23#53614yes—
487C23#53714yes—
488C23#53918yes—
489C23#54118yes—
490C23C23×D411yescharacteristic
491C23C22×#4112yes—
492C23C22×#4112yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×2−1+412yes—
496C23C2×2−1+412yes—
497C23C2×2−1+411yescharacteristic
498C23#58512yes—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—416no—
526C2×C4—416no—
527C2×C4—416no—
528C2×C4—416no—
529C2×C4—416no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
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—88no—
582C2×C4—88no—
583C2×C4#52514yes—
584C2×C4#53414yes—
585C2×C4#53614yes—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—216no—
590C22—216no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22#266418yes—
595C22C2×#52511yescharacteristic
596C22C2×#53211yescharacteristic
597C22C2×#53411yescharacteristic
598C22C2×#53611yescharacteristic
599C22C2×#53611yescharacteristic
600C22C2×#53711yescharacteristic
601C22#344911yescharacteristic
602C22C2×#53912yes—
603C22C2×#54112yes—
604C22#348114yes—
605C22#349514yes—
606C22#351514yes—
607C22#352212yes—
608C22#352312yes—
609C22#353412yes—
610C22#353512yes—
611C22#353814yes—
612C4—24no—
613C4—24no—
614C4—24no—
615C4—42no—
616C4—42no—
617C4—48no—
618C4—42no—
619C4—44no—
620C4—44no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—84no—
626C4—82no—
627C4—82no—
628C2—28no—
629C2C2×#266411yescharacteristic
630C2#3495711yescharacteristic
631C2#3513911yescharacteristic
632C2#3524711yescharacteristic
633C2#3532511yescharacteristic
634C2#3745211yescharacteristic
635C2#3745311yescharacteristic
636C2#3850714yes—
637C2#3853614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17044C2semidirect
4. #16900C2semidirect
5. #16820C2semidirect
6. #15024C2semidirect
12. #14929C2semidirect
14. #14314C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C22×#3872
#170442
#169002
#168202
#150242
#150192
#150152
#150142
#149362
#149352
#149292
#147282
#143142
#140432
#139722
C4×#3802