Finite Group Databasebeta

← #6800345

Subgroups — #6800345

Order 512.

1371 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33148C211yesmaximal, characteristic, complemented
4#32319C214yesmaximal, complemented
5#31463C214yesmaximal, complemented
6#29973C212yesmaximal, complemented
7#28790C212yesmaximal, complemented
8#27964C212yesmaximal, complemented
9#27961C211yesmaximal, characteristic, complemented
10#27728C212yesmaximal, complemented
11#27651C212yesmaximal
12#27645C211yesmaximal, characteristic
13#26507C212yesmaximal, complemented
14#26107C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22076C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C2×#481C2212yes—
19C2×#480C2211yescharacteristic
20#2857C2211yescharacteristic
21#2851C2212yes—
22#2589—22no—
23#2571—22no—
24#2571—21no—
25#2571—21no—
26#2571C2214yes—
27#2571C2212yes—
28#2571C2212yes—
29#2563C2211yescharacteristic
30#2552C2211yescharacteristic
31#2509C2212yes—
32#2497—21no—
33#2487—21no—
34#2454C2212yes—
35#2450—22no—
36C2×#400—22no—
37#2369C2211yescharacteristic
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2211yescharacteristic
40#2345—21no—
41C4×#121C2212yes—
42C4×#121C2211yescharacteristic
43C2×#389—21no—
44C2×#389C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382—21no—
47C2×#382C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2101C2214yes—
53#2100C2214yes—
54#2088C2214yes—
55#2086C2214yes—
56#2085C2214yes—
57#2084C2212yes—
58#2073C2214yes—
59#2071C2212yes—
60#2059C2214yes—
61#1991C2214yes—
62#1990C2212yes—
63#1987C2212yes—
64#1949C2214yes—
65#1948C2212yes—
66#1948C2212yes—
67#1941C2214yes—
68#1939C2212yes—
69#1934C2214yes—
70#1925C2212yes—
71#1920C2214yes—
72#1920C2214yes—
73#1918C2212yes—
74#1916C2214yes—
75#1874C2211yescharacteristic
76#1864C2214yes—
77#1853C2212yes—
78#1852C2211yescharacteristic
79#1835C2214yes—
80#1834C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1824C2212yes—
84#1823C2212yes—
85#1823C2211yescharacteristic
86#1804C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125C2312yes—
97C2×C22⋊Q8—21no—
98C2×#121C2311yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4—22no—
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2312yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115#481—28no—
116#480—24no—
117C2×#107C2312yes—
118C2×#107C2312yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#102C2314yes—
124C2×#102C2312yes—
125C2×#102C2312yes—
126C2×C22⋊C8C2311yescharacteristic
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2311yescharacteristic
129C2×C4×C8C2312yes—
130C2×C4×C8C2314yes—
131#400—22no—
132#400—24no—
133#400—24no—
134#400—24no—
135#400—24no—
136#400—22no—
137#399—22no—
138#398—22no—
139#396—22no—
140#393—22no—
141#391—21no—
142#391—22no—
143#390C2312yes—
144#390C2311yescharacteristic
145#389—22no—
146#389—22no—
147#389—24no—
148#389—22no—
149#389—21no—
150#389—21no—
151#389—22no—
152#389—22no—
153#389—21no—
154#389C2314yes—
155#389C2314yes—
156#389C2312yes—
157#389C2312yes—
158#389C2311yescharacteristic
159#389C2312yes—
160#389C2312yes—
161#388—21no—
162#388—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#386—21no—
167#386—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#384—21no—
173#384—21no—
174#384—21no—
175#384C2312yes—
176#384C2312yes—
177#384C2311yescharacteristic
178#382—22no—
179#382—22no—
180#382—24no—
181#382—22no—
182#382—22no—
183#382C2312yes—
184#382C2314yes—
185#382C2314yes—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—21no—
190#380—22no—
191#380C2311yescharacteristic
192#380C2312yes—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—21no—
208C2×#95—21no—
209C2×#95—22no—
210C2×#95—22no—
211C2×#95—22no—
212C2×#95D412yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2314yes—
230C22×Q8—21no—
231C22×D4—21no—
232C22×D4C2411yescharacteristic
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2411yescharacteristic
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—24no—
247#125—44no—
248C22⋊Q8—82no—
249#121—22no—
250#121#4114yes—
251C4×D4—24no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4—84no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—84no—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C42—24no—
309C2×C42—21no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—24no—
324#107—42no—
325#106—28no—
326#106—44no—
327C4⋊C8—24no—
328C4⋊C8—44no—
329#102—24no—
330#102—28no—
331#102—42no—
332C22⋊C8—22no—
333C22⋊C8—24no—
334C22⋊C8—22no—
335C4×C8—24no—
336C4×C8—44no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—21no—
346#95—22no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—21no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24—21no—
361C24C22×D411yescharacteristic
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4C2×#4112yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4#13514yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4112yes—
419C22×C42−1+412yes—
420C22×C4C2511yescharacteristic
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—82no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4C2×#4112yes—
467C22⋊C4—24no—
468C22⋊C4—42no—
469C22⋊C4—44no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—81no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—24no—
480C42—24no—
481C42—42no—
482C42—42no—
483C42—44no—
484C42—44no—
485C42—42no—
486C42—82no—
487C422−1+414yes—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23#54812yes—
500C23C2×#13512yes—
501C23#57712yes—
502C23C23×D411yescharacteristic
503Q8—162no—
504D4—24no—
505D4—48no—
506D4—42no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—88no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—164no—
563C2×C4#53612yes—
564C2×C4#53612yes—
565C2×C4#54812yes—
566C2×C4C2×#13512yes—
567C2×C4C2×#13512yes—
568C2×C4#57512yes—
569C2×C4C22×#4111yescharacteristic
570C2×C4C2×2−1+412yes—
571C2×C4C2×2−1+411yescharacteristic
572C2×C4#58512yes—
573C8—44no—
574C8—42no—
575C8—42no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22#336014yes—
590C22C2×#53611yescharacteristic
591C22C2×#54811yescharacteristic
592C22#348511yescharacteristic
593C22C22×#13511yescharacteristic
594C22C2×#57511yescharacteristic
595C22C2×#57711yescharacteristic
596C22#358511yescharacteristic
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—81no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—84no—
611C4—81no—
612C4—82no—
613C4#323214yes—
614C4#351512yes—
615C4#356312yes—
616C2—24no—
617C2—82no—
618C2#3912211yescharacteristic
619C2C2×#323211yescharacteristic
620C2#6087311yescharacteristic
621C2C2×#336011yescharacteristic
622C2#6132411yescharacteristic
623C2#6170011yescharacteristic
624C2#6173911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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×#2571C2semidirect
3. #33148C2semidirect
4. #32319C2semidirect
5. #31463C2semidirect
6. #29973C2semidirect
7. #28790C2semidirect
8. #27964C2semidirect
9. #27961C2semidirect
10. #27728C2semidirect
13. #26507C2semidirect
14. #26107C2semidirect
15. #22625C2semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25712
#331482
#323192
#314632
#299732
#287902
#279642
#279612
#277282
#276512
#276452
#265072
#261072
#226252
#220762
C4×#4802