Finite Group Databasebeta

← #6506477

Subgroups — #6506477

Order 512.

2254 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37307C211yesmaximal, characteristic, complemented
2#35666C211yesmaximal, characteristic
3#29117C212yesmaximal, complemented
4#29107C212yesmaximal, complemented
5#27283C214yesmaximal, complemented
6#27274C214yesmaximal, complemented
7#27258C214yesmaximal, complemented
8#27255C214yesmaximal, complemented
9#25868C212yesmaximal, complemented
10#25856C212yesmaximal, complemented
11#21771C212yesmaximal, complemented
12#20603C212yesmaximal
13#20530C211yesmaximal, characteristic, complemented
14C22×2−1+4C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#491C2212yes—
19C2×#488C2212yes—
20C2×#487C2214yes—
21C2×#417C2212yes—
22C2×#419C2212yes—
23C2×#416C2212yes—
24C22×#103C2212yes—
25C2×#408C2212yes—
26#2857C2212yes—
27#2806—22no—
28#2748—22no—
29#2732—22no—
30#2683—22no—
31#2629C2212yes—
32#2441C2214yes—
33#2440C2214yes—
34#2434—22no—
35#2434C2214yes—
36#2432—22no—
37#2432C2214yes—
38#2406—22no—
39#2406—22no—
40#2311C2212yes—
41#2310C2211yescharacteristic
42#2297C2211yescharacteristic
43#2297C2211yescharacteristic
44#2072C2214yes—
45#2072C2212yes—
46#2071C2212yes—
47#2043C2214yes—
48#2043C2214yes—
49#2043C2214yes—
50#2042C2214yes—
51#2026C2214yes—
52#2024C2212yes—
53#2023C2212yes—
54#1959C2212yes—
55#1956C2214yes—
56#1956C2212yes—
57#1935C2214yes—
58#1934C2214yes—
59#1933C2214yes—
60#1933C2214yes—
61#1894C2214yes—
62#1892C2214yes—
63#1873C2214yes—
64#1872C2214yes—
65#1825C2214yes—
66#1788C2214yes—
67#1788C2214yes—
68#1787C2214yes—
69#1787C2214yes—
70#1787C2214yes—
71#1753C2214yes—
72C8×C22⋊C4C2214yes—
73#1750C2212yes—
74C4×C22⋊C8C2212yes—
75C2×2−1+4—28no—
76C2×2−1+4—44no—
77C2×2−1+4—42no—
78C2×2−1+4—42no—
79C22×#41—22no—
80C22×#41—21no—
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C22×M16C2312yes—
89C23×C8C2312yes—
90#536—28no—
91#536—42no—
92#536—42no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×#125C2312yes—
100C2×#124—21no—
101C2×#124C2312yes—
102C2×#123—22no—
103C2×#123—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8C2314yes—
108C2×#117C2311yescharacteristic
109C2×#117C2312yes—
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115#491—44no—
116#488—44no—
117#487—48no—
118#419—44no—
119#417—44no—
120#416—44no—
121C2×#103—44no—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134#408—28no—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2312yes—
138C2×C4×C8C2314yes—
139C2×C4×C8C2312yes—
140#399—24no—
141#398—24no—
142#398—24no—
143#396—24no—
144#396—24no—
145#395—24no—
146#395—24no—
147#395—24no—
148#394—24no—
149#394—24no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391D412yes—
154#391C2314yes—
155#391C2312yes—
156#390—22no—
157#390D412yes—
158#390C2312yes—
159#389—22no—
160#389—24no—
161#389C2314yes—
162#389C2314yes—
163#387—24no—
164#386—22no—
165#386—22no—
166#386—21no—
167#386—21no—
168#386D412yes—
169#386D412yes—
170#385—24no—
171#385—24no—
172#385—24no—
173#385—24no—
174#385—24no—
175#384C2312yes—
176#384C2312yes—
177#381—22no—
178#380—22no—
179#380C2314yes—
180#380C2314yes—
181C4×C22⋊C4—22no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2312yes—
189#336C2314yes—
1902−1+4—432no—
1912−1+4—432no—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—24no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—41no—
201C2×#41—41no—
202C2×#41—42no—
203C2×#41—42no—
204C2×#41—42no—
205C2×#41—48no—
206C2×#41—44no—
207C2×#41—48no—
208C2×#41—44no—
209C2×#41—48no—
210C2×#41—44no—
211C2×#41#4118yes—
212C22×Q8—24no—
213C22×Q8—21no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8—41no—
226C22×Q8—41no—
227C22×Q8—42no—
228C22×Q8—41no—
229C22×Q8—41no—
230C22×Q8—44no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2412yes—
235C22×Q8C2412yes—
236C22×Q8C2411yescharacteristic
237C22×D4—21no—
238C22×D4—22no—
239C22×D4C2×D412yes—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—42no—
254C2×M16—42no—
255C22×C8—24no—
256C22×C8—42no—
257C22×C8—42no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—42no—
265C4⋊Q8—48no—
266C4⋊Q8—44no—
267#125—42no—
268#125—42no—
269#124—24no—
270#124—42no—
271#124—44no—
272#123—48no—
273#123—48no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—48no—
277C22⋊Q8—88no—
278C22⋊Q8—84no—
279#117—24no—
280#117—41no—
281#117—41no—
282#117—42no—
283#117—42no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4C2×D414yes—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D412yes—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—42no—
318C2×C42C2×D412yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321#103—28no—
322#103—28no—
323#103—28no—
324#103—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329#103—44no—
330#103—44no—
331#102—28no—
332#102—44no—
333C22⋊C8—28no—
334C22⋊C8—28no—
335#97—42no—
336C4×C8—44no—
337C4×C8—42no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95C2×D414yes—
347#95C2×D414yes—
348C24—21no—
349C24C22×D412yes—
350C24C22×D411yescharacteristic
351#41—216no—
352#41—48no—
353#41—48no—
354#41—48no—
355#41—48no—
356#41—48no—
357#41—48no—
358#41—48no—
359#41—432no—
360#41—464no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—216no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—48no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—432no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—82no—
393C2×Q8—82no—
394C2×Q8—84no—
395C2×Q8—84no—
396C2×Q8#12414yes—
397C2×Q8H(5,𝔽2)12yes—
398C2×Q8H(5,𝔽2)14yes—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—41no—
404C2×D4—42no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4#12418yes—
413C2×D4#12414yes—
414C2×D4H(5,𝔽2)12yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—82no—
448C22×C4—84no—
449C22×C4C22×D414yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2511yescharacteristic
458M16—84no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—84no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—42no—
487C22⋊C4—48no—
488C22⋊C4—42no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—42no—
500C42—82no—
501C23—21no—
502C23—22no—
503C23—21no—
504C23—22no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23C2×#12414yes—
515C23C2×#12412yes—
516C23C23×D411yescharacteristic
517Q8—24no—
518Q8—24no—
519Q8—24no—
520Q8—44no—
521Q8—416no—
522Q8—48no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—44no—
527Q8—44no—
528Q8—84no—
529Q8—164no—
530D4—24no—
531D4—48no—
532D4—44no—
533D4—48no—
534D4—416no—
535D4—42no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—416no—
559C2×C4—48no—
560C2×C4—416no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—84no—
578C2×C4—164no—
579C2×C4C2×#12412yes—
580C2×C4C2×#12414yes—
581C2×C4#57714yes—
582C2×C4#57814yes—
583C2×C4C22×#4112yes—
584C2×C4C2×H(5,𝔽2)12yes—
585C2×C4C2×H(5,𝔽2)11yescharacteristic
586C2×C4C2×H(5,𝔽2)11yescharacteristic
587C8—44no—
588C8—44no—
589C22—22no—
590C22—24no—
591C22—28no—
592C22—24no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22C22×#12411yescharacteristic
605C22#348411yescharacteristic
606C22#348411yescharacteristic
607C22C2×#57712yes—
608C22C2×#57812yes—
609C4—24no—
610C4—22no—
611C4—24no—
612C4—24no—
613C4—42no—
614C4—44no—
615C4—44no—
616C4—82no—
617C4—84no—
618C4—82no—
619C4—84no—
620C4#349012yes—
621C4#349012yes—
622C2—42no—
623C2—44no—
624C2—82no—
625C2#3536711yescharacteristic
626C2#6062312yes—
627C2#6163311yescharacteristic
628C2#6163411yescharacteristic
629C2#6166712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37307C2semidirect
3. #29117C2semidirect
4. #29107C2semidirect
5. #27283C2semidirect
6. #27274C2semidirect
7. #27258C2semidirect
8. #27255C2semidirect
9. #25868C2semidirect
10. #25856C2semidirect
11. #21771C2semidirect
13. #20530C2semidirect

Maximal subgroups

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

subgroupindex
#373072
#356662
#291172
#291072
#272832
#272742
#272582
#272552
#258682
#258562
#217712
#206032
#205302