Finite Group Databasebeta

← #6563268

Subgroups — #6563268

Order 512.

1573 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36505C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29944C212yesmaximal, complemented
4#29937C211yesmaximal, characteristic, complemented
5#27951C212yesmaximal, complemented
6#27950C211yesmaximal, characteristic, complemented
7#27652C212yesmaximal, complemented
8#27647C211yesmaximal, characteristic, complemented
9#26305C214yesmaximal, complemented
10#26303C212yesmaximal, complemented
11#25021C214yesmaximal, complemented
12#25017C212yesmaximal, complemented
13#23757C211yesmaximal, characteristic, complemented
14#23757C212yesmaximal, complemented
15#22727C212yesmaximal, complemented
16#22530C211yesmaximal, characteristic
17#22521C212yesmaximal, complemented
18C2×#557C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yes—
21C2×#483C2212yes—
22C2×C8×Q8C2212yes—
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2562C2212yes—
26#2547C2211yescharacteristic
27#2538C2211yescharacteristic
28#2536C2212yes—
29#2475—22no—
30#2460—22no—
31#2446—21no—
32#2432—21no—
33#2367—21no—
34#2362—21no—
35#2355—21no—
36#2355C2212yes—
37#2355C2211yescharacteristic
38#2354—21no—
39#2354C2212yes—
40#2352—22no—
41#2352C2214yes—
42#2314—22no—
43#2311—21no—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2211yescharacteristic
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2100C2212yes—
52#2100C2211yescharacteristic
53#2095C2211yescharacteristic
54#2084C2212yes—
55#2084C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2083C2211yescharacteristic
59#2077C2214yes—
60#2077C2214yes—
61#2076C2214yes—
62#2076C2212yes—
63#2075C2214yes—
64#2072C2211yescharacteristic
65#2072C2212yes—
66#2036C2214yes—
67#2036C2212yes—
68#1991C2214yes—
69#1990C2212yes—
70#1989C2214yes—
71#1988C2212yes—
72#1954C2212yes—
73#1954C2214yes—
74#1949C2214yes—
75#1949C2214yes—
76#1948C2212yes—
77#1948C2212yes—
78#1854C2214yes—
79#1854C2214yes—
80#1852C2214yes—
81#1846C2214yes—
82#1845C2212yes—
83#1825C2214yes—
84#1823C2212yes—
85#1805C2214yes—
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90C4×#107C2214yes—
91C4×#103C2212yes—
92C4×#103C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96#557—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—22no—
100C2×#125C2312yes—
101C2×#124—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105C4×#41—28no—
106C2×C4×Q8—21no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4D412yes—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—22no—
117C2×#117D412yes—
118C2×#117D411yescharacteristic
119C2×#117C2314yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#485—48no—
126#483—44no—
127C8×Q8—28no—
128C2×#107C2312yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#398—22no—
147#397—22no—
148#396—22no—
149#394—22no—
150#394—22no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391C2311yescharacteristic
156#390—22no—
157#390—21no—
158#389—21no—
159#389—21no—
160#389C2312yes—
161#389C2311yescharacteristic
162#388—22no—
163#388—22no—
164#388—24no—
165#388—24no—
166#388—22no—
167#388—22no—
168#387—24no—
169#387—24no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—21no—
175#386—24no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—21no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—21no—
188#382—22no—
189#382C2314yes—
190#382C2312yes—
191#380—22no—
192#380—22no—
193#380—21no—
194#380—21no—
195#380—22no—
196#380C2311yescharacteristic
197#380C2312yes—
198#380C2312yes—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—24no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—42no—
236C22×Q8—21no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×D4C2×D411yescharacteristic
240C22×D4C2×D412yes—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—44no—
253#125—28no—
254#125—48no—
255#124—44no—
256C22⋊Q8—22no—
257C22⋊Q8—24no—
258C22⋊Q8—41no—
259C22⋊Q8—42no—
260C22⋊Q8—44no—
261C4×Q8—24no—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—44no—
265C4×Q8#4118yes—
266C4×D4—24no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—24no—
270#117—24no—
271#117—28no—
272#117—44no—
273#117—44no—
274#117—44no—
275#117—84no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—28no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—44no—
343#107—42no—
344#107—44no—
345#106—44no—
346#106—44no—
347C4⋊C8—28no—
348C4⋊C8—24no—
349#103—28no—
350#103—24no—
351#103—24no—
352C22⋊C8—24no—
353C22⋊C8—28no—
354C22⋊C8—42no—
355C22⋊C8—44no—
356C4×C8—28no—
357C4×C8—24no—
358#95—21no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—21no—
364#95—24no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379#41—416no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8#12314yes—
389C2×Q8H(5,𝔽2)12yes—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4#13514yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2511yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C82−1+414yes—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4#12314yes—
482C4⋊C4#12318yes—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—81no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C42—24no—
497C42—22no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—44no—
502C42—42no—
503C42—44no—
504C42—42no—
505C42—82no—
506C42—84no—
507C42—84no—
508C42C2×#4118yes—
509C42C2×#4114yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23C2×#13512yes—
524C23C23×D411yescharacteristic
525Q8—28no—
526Q8—24no—
527Q8—42no—
528Q8—84no—
529D4—48no—
530D4—44no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—164no—
583C2×C4C2×#12314yes—
584C2×C4C2×#12312yes—
585C2×C4C2×#12314yes—
586C2×C4C2×#12312yes—
587C2×C4C2×#13512yes—
588C2×C4C2×#13512yes—
589C2×C4#57512yes—
590C2×C4#57812yes—
591C2×C4C22×#4112yes—
592C2×C4C2×H(5,𝔽2)11yescharacteristic
593C2×C4C2×2−1+412yes—
594C2×C4#58511yescharacteristic
595C8—24no—
596C8—44no—
597C8—42no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22C22×#12311yescharacteristic
613C22#348811yescharacteristic
614C22#349911yescharacteristic
615C22C22×#13511yescharacteristic
616C22C2×#57511yescharacteristic
617C22C2×#57811yescharacteristic
618C22#358511yescharacteristic
619C4—22no—
620C4—24no—
621C4—22no—
622C4—24no—
623C4—22no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—82no—
628C4—84no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—81no—
633C4—82no—
634C4—81no—
635C4#309114yes—
636C4#351612yes—
637C4#352212yes—
638C2—42no—
639C2—82no—
640C2#3599611yescharacteristic
641C2C2×#309111yescharacteristic
642C2#6052611yescharacteristic
643C2#6195311yescharacteristic
644C2#6199811yescharacteristic
645C2#6247311yescharacteristic
646C2#6251211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29944C2semidirect
4. #29937C2semidirect
5. #27951C2semidirect
6. #27950C2semidirect
7. #27652C2semidirect
8. #27647C2semidirect
9. #26305C2semidirect
10. #26303C2semidirect
11. #25021C2semidirect
12. #25017C2semidirect
13. #23757C2semidirect
14. #23757C2semidirect
15. #22727C2semidirect
17. #22521C2semidirect

Maximal subgroups

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

subgroupindex
#365052
#341882
#299442
#299372
#279512
#279502
#276522
#276472
#263052
#263032
#250212
#250172
#237572
#227272
#225302
#225212