Finite Group Databasebeta

← #6506541

Subgroups — #6506541

Order 512.

1848 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35032C211yesmaximal, characteristic
2C2×#2439C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal, complemented
4#29335C211yesmaximal, characteristic, complemented
5#29334C211yesmaximal, characteristic, complemented
6#27983C211yesmaximal, characteristic, complemented
7#27982C211yesmaximal, characteristic, complemented
8#27714C214yesmaximal, complemented
9#27712C214yesmaximal, complemented
10#27269C214yesmaximal, complemented
11#27265C214yesmaximal, complemented
12#25908C212yesmaximal, complemented
13#21763C212yesmaximal, complemented
14#20908C212yesmaximal, complemented
15#20526C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525C2212yes—
18C22×#117C2211yescharacteristic
19C2×#489C2214yescomplemented
20C2×#416C2214yes—
21C22×C22⋊C8C2212yes—
22#2857C2211yescharacteristic
23#2856C2211yescharacteristic
24#2850C2212yes—
25#2617—21no—
26#2617—21no—
27#2602—22no—
28#2602—22no—
29#2500C2212yes—
30#2499C2212yes—
31#2439—22no—
32#2439—22no—
33#2439C2214yes—
34#2439C2214yes—
35#2432C2212yes—
36#2431C2212yes—
37C2×#396—21no—
38C2×#393—21no—
39#2325—21no—
40#2325—21no—
41C2×#391—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#382C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#2101C2214yescomplemented
49#2100C2212yescomplemented
50#2099C2212yescomplemented
51#2074C2214yescomplemented
52#2073C2214yescomplemented
53#2043C2212yescomplemented
54#2042C2214yescomplemented
55#2041C2212yescomplemented
56#2024C2214yescomplemented
57#2023C2214yescomplemented
58#1993C2214yescomplemented
59#1961C2212yescomplemented
60#1960C2212yescomplemented
61#1936C2214yescomplemented
62#1936C2214yescomplemented
63#1935C2214yescomplemented
64#1935C2214yescomplemented
65#1893C2214yes—
66#1892C2214yes—
67#1875C2214yes—
68#1874C2212yes—
69#1874C2212yes—
70#1874C2212yes—
71#1874C2212yes—
72#1873C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1872C2212yes—
77#1836C2212yescomplemented
78#1836C2212yescomplemented
79#1825C2214yes—
80#1788C2214yes—
81#1752C2214yes—
82#1752C2214yes—
83#1749C2212yes—
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
86#1735C2214yes—
87C22×#41C2312yes—
88C23×Q8—21no—
89C24×C4C2311yescharacteristic
90C23×C8C2314yes—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2314yes—
95C2×#127C2311yescharacteristic
96C2×#124C2312yes—
97#525—28no—
98#525—28no—
99C2×#123—22no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#120—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×#117—22no—
110C2×#117—44no—
111C2×#117C2312yes—
112C2×#117C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4D411yescharacteristic
123C22×C22⋊C4D411yescharacteristic
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#489—48no—
128#416—48no—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8—28no—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2312yes—
142C2×#97C2312yes—
143C2×#97C2314yes—
144#397—22no—
145#397—22no—
146#396—24no—
147#396—24no—
148#394—22no—
149#394—22no—
150#394—22no—
151#394—22no—
152#393—24no—
153#393—24no—
154#392—22no—
155#392—22no—
156#391—24no—
157#391—24no—
158#391C2312yes—
159#391C2312yes—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—24no—
179#387—24no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#382—21no—
189#382—21no—
190#382C2312yes—
191#382C2312yes—
192#382C2312yes—
193#380—21no—
194#380—21no—
195#380—22no—
196#380C2314yes—
197C4×C22⋊C4D412yes—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4C2312yes—
200C4×C22⋊C4C2312yes—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C25C2×D411yescharacteristic
220C2×#41—24no—
221C2×#41—24no—
222C2×#41—44no—
223C2×#41—44no—
224C22×Q8—24no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8—82no—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4#4118yes—
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C22×C8—28no—
252C22×C8—44no—
253C22×C8—44no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—28no—
257C4⋊Q8—44no—
258C4⋊Q8—42no—
259C4⋊Q8—48no—
260#127—42no—
261#124—44no—
262#123—48no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267#121—24no—
268#121—24no—
269#121—24no—
270#121—24no—
271#121—44no—
272#121—44no—
273#120—48no—
274C4×D4—48no—
275C4×D4—48no—
276#117—24no—
277#117—24no—
278#117—28no—
279#117—44no—
280#117—48no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—44no—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#103—28no—
343#103—44no—
344#103—44no—
345#103—44no—
346#102—28no—
347#102—44no—
348#102—44no—
349#102—44no—
350C22⋊C8—28no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353#97—42no—
354#97—42no—
355#97—44no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24#12414yes—
382C24C22×D412yes—
383C24C22×D411yescharacteristic
384#41—416no—
385#41—88no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—84no—
393C2×Q8—816no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4—82no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—416no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4#12414yes—
457C22×C4#12418yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2511yescharacteristic
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—88no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4H(5,𝔽2)14yes—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—42no—
525C42—48no—
526C42—44no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—28no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23C2×#12414yes—
552C23C2×#12412yes—
553C23#57512yes—
554C23#57512yes—
555C23C23×D411yescharacteristic
556Q8—44no—
557Q8—44no—
558Q8—816no—
559D4—44no—
560D4—44no—
561D4—88no—
562D4—88no—
563D4—88no—
564D4—162no—
565D4—162no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—88no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4C2×#12414yes—
624C2×C4C2×#12412yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4#58512yes—
628C8—44no—
629C8—44no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—88no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22#313514yes—
652C22C22×#12411yescharacteristic
653C22#348411yescharacteristic
654C22#348511yescharacteristic
655C22C2×#57511yescharacteristic
656C22C2×#57511yescharacteristic
657C22#358511yescharacteristic
658C22#358611yescharacteristic
659C4—24no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4#350514yes—
675C2—24no—
676C2—44no—
677C2—162no—
678C2#3537711yescharacteristic
679C2C2×#313511yescharacteristic
680C2#6062111yescharacteristic
681C2#6157611yescharacteristic
682C2#6157811yescharacteristic
683C2#6171011yescharacteristic
684C2#6171211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2439C2semidirect
3. #29336C2semidirect
4. #29335C2semidirect
5. #29334C2semidirect
6. #27983C2semidirect
7. #27982C2semidirect
8. #27714C2semidirect
9. #27712C2semidirect
10. #27269C2semidirect
11. #27265C2semidirect
12. #25908C2semidirect
13. #21763C2semidirect
14. #20908C2semidirect
15. #20526C2semidirect
19. C2×#489C22semidirect
48. #2101C22semidirect
49. #2100C22semidirect
50. #2099C22semidirect
51. #2074C22semidirect
52. #2073C22semidirect
53. #2043C22semidirect
54. #2042C22semidirect
55. #2041C22semidirect
56. #2024C22semidirect
57. #2023C22semidirect
58. #1993C22semidirect
59. #1961C22semidirect
60. #1960C22semidirect
61. #1936C22semidirect
62. #1936C22semidirect
63. #1935C22semidirect
64. #1935C22semidirect
77. #1836C22semidirect
78. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350322
C2×#24392
#293362
#293352
#293342
#279832
#279822
#277142
#277122
#272692
#272652
#259082
#217632
#209082
#205262