Finite Group Databasebeta

← #6555325

Subgroups — #6555325

Order 512.

1637 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic, complemented
4#29955C212yesmaximal, complemented
5#29259C211yesmaximal, characteristic, complemented
6#29159C212yesmaximal, complemented
7#28512C211yesmaximal, characteristic, complemented
8#27685C212yesmaximal, complemented
9#26531C212yesmaximal, complemented
10#26337C214yesmaximal, complemented
11#26091C212yesmaximal, complemented
12#25951C214yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14#23730C212yesmaximal, complemented
15#22696C211yesmaximal, characteristic, complemented
16#22657C212yesmaximal, complemented
17#22417C212yesmaximal
18C2×#547C2211yescharacteristic
19C2×#475C2214yes—
20C2×C8×D4C2212yes—
21C2×#427C2214yes—
22C2×#416C2214yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2571C2212yes—
26#2563—21no—
27#2538—21no—
28#2497—22no—
29#2492—22no—
30#2439C2214yes—
31#2374—22no—
32#2369—21no—
33#2365C2211yescharacteristic
34#2363—21no—
35#2362C2212yes—
36#2355—21no—
37#2354C2211yescharacteristic
38#2354C2211yescharacteristic
39#2354C2212yes—
40#2345—21no—
41#2341C2212yes—
42#2335—22no—
43#2324—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2297C2212yes—
48#2293C2211yescharacteristic
49#2285C2212yes—
50#2274—21no—
51C42×D4C2211yescharacteristic
52#2100C2211yescharacteristic
53#2099C2212yes—
54#2097C2214yes—
55#2096C2212yes—
56#2096C2212yes—
57#2095C2212yes—
58#2094C2211yescharacteristic
59#2093C2212yes—
60#2092C2214yes—
61#2088C2212yes—
62#2068C2214yes—
63#2065C2214yes—
64#2065C2212yes—
65#2058C2211yescharacteristic
66#2057C2212yes—
67#2024C2212yes—
68#2023C2211yescharacteristic
69#2023C2212yes—
70#1981C2212yes—
71#1956C2212yes—
72#1949C2214yes—
73#1947C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1919C2214yes—
77#1892C2214yes—
78#1873C2214yes—
79#1865C2214yes—
80#1863C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1850C2214yes—
84#1844C2212yes—
85#1831C2211yescharacteristic
86#1807C2214yes—
87#1805C2214yes—
88#1802C2212yes—
89#1797C2212yes—
90#1788C2214yes—
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#547—24no—
98#547Q818yescomplemented
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×#121—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D411yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—21no—
117C2×#117C2312yes—
118C2×#117C2314yes—
119C22×C4⋊C4—21no—
120C22×C42C2312yes—
121#475—48no—
122C8×D4—28no—
123#427—48no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2312yes—
128C2×#106C2314yes—
129C2×#106C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133#416—48no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C4×C8C2312yes—
143#400—24no—
144#396—24no—
145#393—24no—
146#391—21no—
147#391—22no—
148#391—24no—
149#391C2312yes—
150#391C2312yes—
151#390—22no—
152#390—21no—
153#390—22no—
154#390—22no—
155#390C2311yescharacteristic
156#390C2312yes—
157#390C2311yescharacteristic
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389C2312yes—
165#389C2311yescharacteristic
166#389C2311yescharacteristic
167#389C2314yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—24no—
172#388—21no—
173#388—21no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—24no—
179#386—22no—
180#386—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#382—22no—
189#382C2314yes—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—21no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—24no—
235C2×#41—44no—
236C2×#41—44no—
237C2×#41C2×Q818yes—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×D4—22no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C22×C8—28no—
250C22×C8—28no—
251C22×C8—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8C2×Q814yes—
261#123—216no—
262C22⋊Q8—24no—
263C22⋊Q8—84no—
264C22⋊Q8C2×Q818yes—
265#121—28no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×D4—24no—
270C4×D4—22no—
271C4×D4—42no—
272C4×D4C2×Q814yes—
273C4×D4C2×Q814yes—
274C4×D4#4114yes—
275#117—44no—
276#117—44no—
277#117—44no—
278#117—44no—
279#117—44no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—44no—
344#107—44no—
345#106—24no—
346#106—28no—
347#106—42no—
348#106—44no—
349C4⋊C8—24no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—28no—
353#103—44no—
354#103—44no—
355#103—44no—
356#102—24no—
357#102—42no—
358#102—44no—
359C22⋊C8—28no—
360C4×C8—24no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24C22×D412yes—
381#41—416no—
382#41—88no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8C22×Q814yes—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4C22⋊Q814yes—
402C2×D4C22×Q812yes—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×Q814yes—
435C22×C4C2×#4114yes—
436C22×C4C2511yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—416no—
446C2×C8H(5,𝔽2)14yes—
447C4⋊C4—24no—
448C4⋊C4—28no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4C22⋊Q814yes—
479C4⋊C4C22×Q812yes—
480C22⋊C4—28no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C22⋊C4C22⋊Q818yes—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—22no—
500C42—42no—
501C42—42no—
502C42—41no—
503C42—44no—
504C42—42no—
505C42—44no—
506C42—82no—
507C42—84no—
508C42—82no—
509C42—82no—
510C42#13514yes—
511C42C22×Q812yes—
512C42C2×#4112yes—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×C22⋊Q814yes—
527C23C23×D411yescharacteristic
528Q8—28no—
529Q8—44no—
530Q8—84no—
531Q8—164no—
532D4—28no—
533D4—24no—
534D4—42no—
535D4—88no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—88no—
585C2×C4C2×C22⋊Q812yes—
586C2×C4C2×C22⋊Q814yes—
587C2×C4C2×C22⋊Q812yes—
588C2×C4C2×#13512yes—
589C2×C4C2×#13512yes—
590C2×C4C2×#13512yes—
591C2×C4#57512yes—
592C2×C4#57712yes—
593C2×C4C23×Q811yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C2×H(5,𝔽2)12yes—
596C2×C4#58512yes—
597C8—24no—
598C8—42no—
599C8—44no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C22×C22⋊Q811yescharacteristic
615C22#348511yescharacteristic
616C22#349811yescharacteristic
617C22C22×#13511yescharacteristic
618C22C2×#57511yescharacteristic
619C22C2×#57711yescharacteristic
620C22#358611yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—44no—
631C4—81no—
632C4—82no—
633C4—81no—
634C4—84no—
635C4—82no—
636C4—82no—
637C4#307614yes—
638C4#350112yes—
639C4#351312yes—
640C2—44no—
641C2—82no—
642C2#3583611yescharacteristic
643C2C2×#307611yescharacteristic
644C2#6049711yescharacteristic
645C2#6170111yescharacteristic
646C2#6173111yescharacteristic
647C2#6241411yescharacteristic
648C2#6245211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #29980C2semidirect
4. #29955C2semidirect
5. #29259C2semidirect
6. #29159C2semidirect
7. #28512C2semidirect
8. #27685C2semidirect
9. #26531C2semidirect
10. #26337C2semidirect
11. #26091C2semidirect
12. #25951C2semidirect
13. #23775C2semidirect
14. #23730C2semidirect
15. #22696C2semidirect
16. #22657C2semidirect
98. #547Q8semidirect

Maximal subgroups

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

subgroupindex
#361942
#343162
#299802
#299552
#292592
#291592
#285122
#276852
#265312
#263372
#260912
#259512
#237752
#237302
#226962
#226572
#224172