Finite Group Databasebeta

← #6855865

Subgroups — #6855865

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33107C211yesmaximal, characteristic
4#32241C214yesmaximal, complemented
5#30819C214yesmaximal, complemented
6#29761C212yesmaximal, complemented
7#28790C212yesmaximal, complemented
8#27973C211yesmaximal, characteristic
9#27770C212yesmaximal, complemented
10#27680C211yesmaximal, characteristic, complemented
11#26467C212yesmaximal, complemented
12#26027C212yesmaximal, complemented
13#25907C212yesmaximal, complemented
14#24305C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22076C212yesmaximal, complemented
17C4×#484C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C2×#488C2214yes—
20C2×#484C2211yescharacteristic
21C2×#481C2212yes—
22C2×#478C2214yes—
23#2856C2211yescharacteristic
24#2850C2212yes—
25#2597—22no—
26#2571C2211yescharacteristic
27#2552—22no—
28#2552—21no—
29#2552—21no—
30#2552C2212yes—
31#2552C2214yes—
32#2552C2212yes—
33#2548C2211yescharacteristic
34#2513—21no—
35#2513C2212yes—
36#2504—21no—
37#2447C2212yes—
38#2444—22no—
39C2×#398—22no—
40#2366C2211yescharacteristic
41#2361—21no—
42#2357—21no—
43#2355C2211yescharacteristic
44C4×C22⋊Q8C2211yescharacteristic
45C4×#121C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384—21no—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2212yes—
53#2096C2212yes—
54#2085C2214yes—
55#2071C2214yes—
56#2067C2214yes—
57#2066C2214yes—
58#2059C2214yes—
59#2042C2214yes—
60#2026C2214yes—
61#2025C2214yes—
62#2025C2214yes—
63#1991C2214yes—
64#1989C2214yes—
65#1982C2212yes—
66#1957C2212yes—
67#1948C2212yes—
68#1948C2212yes—
69#1941C2214yes—
70#1939C2212yes—
71#1934C2214yes—
72#1928C2212yes—
73#1926C2214yes—
74#1921C2212yes—
75#1920C2214yes—
76#1916C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1864C2214yes—
83#1853C2212yes—
84#1852C2211yescharacteristic
85#1834C2212yes—
86#1824C2211yescharacteristic
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1797C2212yes—
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#125—22no—
97C2×#125—22no—
98C2×#125—48no—
99C2×#125C2314yes—
100C2×#125C2314yes—
101C2×#124C2312yes—
102C2×C22⋊Q8C2311yescharacteristic
103C2×#121—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#488—48no—
124#484—24no—
125#481—28no—
126#478—48no—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2311yescharacteristic
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#398—24no—
147#398—24no—
148#398—24no—
149#398—24no—
150#397—22no—
151#397—22no—
152#397—22no—
153#397—22no—
154#394—22no—
155#391C2312yes—
156#391C2311yescharacteristic
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#389C2311yescharacteristic
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—21no—
181#387—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#385—21no—
187#385—22no—
188#384—22no—
189#384—22no—
190#384—24no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#382—21no—
197#382—21no—
198#382C2312yes—
199#381—22no—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8C2411yescharacteristic
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2411yescharacteristic
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250#127—24no—
251#125—28no—
252#125—28no—
253#125—44no—
254#125—44no—
255#125—48no—
256#125—48no—
257#124—44no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260#121—24no—
261#121—24no—
262#121—44no—
263#121—82no—
264C4×Q8—24no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—84no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—84no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4#4114yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—24no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—24no—
340#107—24no—
341#106—28no—
342#106—44no—
343C4⋊C8—24no—
344C4⋊C8—44no—
345#103—24no—
346#103—24no—
347#103—44no—
348#102—28no—
349#102—28no—
350#102—44no—
351#102—44no—
352C22⋊C8—24no—
353C22⋊C8—22no—
354C22⋊C8—22no—
355C4×C8—24no—
356C4×C8—44no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24—22no—
377C24C22×D411yescharacteristic
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—81no—
394C2×D4—81no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4#13514yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4112yes—
439C22×C42−1+412yes—
440C22×C4C2511yescharacteristic
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—81no—
485C4⋊C4C2×#4114yes—
486C4⋊C4H(5,𝔽2)14yes—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—42no—
505C42—42no—
506C42—44no—
507C42—42no—
508C42—44no—
509C42—42no—
510C42—82no—
511C422−1+414yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23#54812yes—
524C23C2×#13512yes—
525C23#57712yes—
526C23C23×D411yescharacteristic
527Q8—42no—
528Q8—42no—
529D4—44no—
530D4—44no—
531D4—88no—
532D4—84no—
533D4—162no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—88no—
588C2×C4—164no—
589C2×C4#53612yes—
590C2×C4#53612yes—
591C2×C4#54812yes—
592C2×C4C2×#13512yes—
593C2×C4C2×#13512yes—
594C2×C4#57512yes—
595C2×C4C22×#4111yescharacteristic
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4C2×2−1+412yes—
598C2×C4C2×2−1+411yescharacteristic
599C8—44no—
600C8—42no—
601C8—42no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22#336514yes—
616C22C2×#53611yescharacteristic
617C22C2×#54811yescharacteristic
618C22#348411yescharacteristic
619C22C22×#13511yescharacteristic
620C22C2×#57511yescharacteristic
621C22C2×#57711yescharacteristic
622C22#358511yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—41no—
633C4—81no—
634C4—82no—
635C4—84no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4#323214yes—
640C4#351112yes—
641C4#355212yes—
642C2—24no—
643C2—162no—
644C2#3972011yescharacteristic
645C2C2×#323211yescharacteristic
646C2#6087311yescharacteristic
647C2C2×#336511yescharacteristic
648C2#6131811yescharacteristic
649C2#6156611yescharacteristic
650C2#6160411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
4. #32241C2semidirect
5. #30819C2semidirect
6. #29761C2semidirect
7. #28790C2semidirect
9. #27770C2semidirect
10. #27680C2semidirect
11. #26467C2semidirect
12. #26027C2semidirect
13. #25907C2semidirect
14. #24305C2semidirect
15. #22625C2semidirect
16. #22076C2semidirect
17. C4×#484C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#25522
#331072
#322412
#308192
#297612
#287902
#279732
#277702
#276802
#264672
#260272
#259072
#243052
#226252
#220762
C4×#4842