Finite Group Databasebeta

← #6572340

Subgroups — #6572340

Order 512.

1571 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
1#36460C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30150C212yesmaximal, complemented
4#29661C211yesmaximal, characteristic
5#29494C212yesmaximal, complemented
6#28164C212yesmaximal, complemented
7#27973C211yesmaximal, characteristic
8#27647C211yesmaximal, characteristic, complemented
9#26933C214yesmaximal, complemented
10#26661C214yesmaximal, complemented
11#26113C212yesmaximal, complemented
12#24733C212yesmaximal, complemented
13#23817C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic, complemented
15#23095C212yesmaximal, complemented
16#22771C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#483C2212yes—
20C2×#464C2214yes—
21C2×#460C2212yes—
22C2×C4×Q16C2212yes—
23#2860C2212yes—
24#2856C2211yescharacteristic
25#2571C2211yescharacteristic
26#2559—22no—
27#2548C2211yescharacteristic
28#2526—22no—
29#2512—21no—
30#2500—21no—
31#2426C2212yes—
32#2420C2212yes—
33#2366C2211yescharacteristic
34#2363—21no—
35#2360—22no—
36#2360C2214yes—
37#2357—21no—
38#2355—21no—
39#2355C2212yes—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2211yescharacteristic
43#2330—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#124C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2088C2214yes—
54#2084C2212yes—
55#2072C2212yes—
56#2071C2212yes—
57#2071C2212yes—
58#2067C2214yes—
59#2057C2211yescharacteristic
60#2043C2211yescharacteristic
61#2042C2214yes—
62#2036C2212yes—
63#2024C2211yescharacteristic
64#2023C2212yes—
65#2016C2211yescharacteristic
66#2013C2212yes—
67#2001C2212yes—
68#2000C2214yes—
69#1968C2214yes—
70#1957C2212yes—
71#1948C2212yes—
72#1939C2212yes—
73#1933C2214yes—
74#1926C2214yes—
75#1923C2214yes—
76#1921C2214yes—
77#1880C2214yes—
78#1878C2214yes—
79#1877C2214yes—
80#1874C2214yes—
81#1865C2214yes—
82#1834C2212yes—
83#1831C2212yes—
84#1823C2212yes—
85#1805C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89C8×C4⋊C4C2212yes—
90C4×#103C2212yes—
91C4×#103C2214yes—
92C4×#102C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×Q16C2312yes—
96C22×SD16C2314yes—
97#533—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#126—22no—
102C2×#125—21no—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2311yescharacteristic
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117D411yescharacteristic
120C2×#117D411yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125#483—44no—
126#464—48no—
127#460—28no—
128C4×Q16—28no—
129C2×#107C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2314yes—
146#400—22no—
147#397—22no—
148#396—22no—
149#394—22no—
150#394—22no—
151#394—22no—
152#391—22no—
153#391C2314yes—
154#391C2311yescharacteristic
155#390—21no—
156#390—21no—
157#390—21no—
158#390—22no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389—24no—
170#389—22no—
171#389C2314yes—
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—21no—
182#387—24no—
183#387—24no—
184#386—22no—
185#386—22no—
186#385—21no—
187#385—24no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#382—21no—
196#382—22no—
197#382C2312yes—
198#381—22no—
199#380C2311yescharacteristic
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—24no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376D412yes—
220C43C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8C2412yes—
235C22×Q8C2411yescharacteristic
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2412yes—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×Q16—24no—
246C2×Q16—24no—
247C2×Q16—42no—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256#127—24no—
257#126—84no—
258#125—28no—
259#124—24no—
260#124—42no—
261C22⋊Q8—28no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—22no—
265C22⋊Q8—41no—
266#121—44no—
267#121—44no—
268#121—82no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—84no—
276C4×Q8#4118yes—
277C4×D4—44no—
278C4×D4—44no—
279#117—28no—
280#117—24no—
281#117—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—24no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—24no—
353#103—28no—
354#103—24no—
355#103—42no—
356#103—42no—
357#103—44no—
358#102—28no—
359#102—44no—
360#102—44no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C4×C8—24no—
364C4×C8—44no—
365#95—21no—
366#95—21no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—22no—
383C24C22×D411yescharacteristic
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8C2×#4114yes—
396C2×Q8H(5,𝔽2)12yes—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—82no—
409C2×D4—82no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4#13514yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444Q16—48no—
445SD16—88no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—24no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—84no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4#13714yes—
487C4⋊C4C2×#4114yes—
488C22⋊C4—28no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—81no—
503C42—22no—
504C42—22no—
505C42—28no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—82no—
517C42—84no—
518C42—84no—
519C42C2×#4114yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23C2×#13512yes—
534C23C23×D411yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—24no—
538Q8—48no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542D4—44no—
543D4—44no—
544D4—84no—
545D4—162no—
546D4—164no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—164no—
596C2×C4—164no—
597C2×C4#52512yes—
598C2×C4#52512yes—
599C2×C4C2×#13512yes—
600C2×C4C2×#13512yes—
601C2×C4C2×#13714yes—
602C2×C4C2×#13712yes—
603C2×C4#57712yes—
604C2×C4C22×#4112yes—
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4#58512yes—
607C2×C4#58511yescharacteristic
608C8—44no—
609C8—42no—
610C8—42no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C2×#52511yescharacteristic
626C22#348811yescharacteristic
627C22#350011yescharacteristic
628C22C22×#13511yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57711yescharacteristic
631C22#358511yescharacteristic
632C4—24no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—41no—
641C4—44no—
642C4—42no—
643C4—84no—
644C4—81no—
645C4—82no—
646C4—81no—
647C4—81no—
648C4#310814yes—
649C4#349412yes—
650C4#351212yes—
651C2—42no—
652C2—162no—
653C2#3608311yescharacteristic
654C2C2×#310811yescharacteristic
655C2#6058811yescharacteristic
656C2#6195311yescharacteristic
657C2#6200211yescharacteristic
658C2#6253111yescharacteristic
659C2#6258211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #30150C2semidirect
5. #29494C2semidirect
6. #28164C2semidirect
8. #27647C2semidirect
9. #26933C2semidirect
10. #26661C2semidirect
11. #26113C2semidirect
12. #24733C2semidirect
13. #23817C2semidirect
14. #23571C2semidirect
15. #23095C2semidirect
16. #22771C2semidirect
17. #22632C2semidirect

Maximal subgroups

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

subgroupindex
#364602
#342142
#301502
#296612
#294942
#281642
#279732
#276472
#269332
#266612
#261132
#247332
#238172
#235712
#230952
#227712
#226322