Finite Group Databasebeta

← #6556191

Subgroups — #6556191

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36419C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30203C211yesmaximal, characteristic, complemented
4#30188C212yesmaximal, complemented
5#29346C211yesmaximal, characteristic, complemented
6#29107C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#27976C212yesmaximal, complemented
9#27750C212yesmaximal, complemented
10#27730C212yesmaximal, complemented
11#27696C214yesmaximal, complemented
12#27261C214yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23627C212yesmaximal, complemented
15#22490C212yesmaximal, complemented
16#22418C212yesmaximal
17#20555C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#550C2211yescharacteristic
20C2×#487C2212yes—
21C2×C8×D4C2212yes—
22C2×#416C2214yes—
23#2868C2211yescharacteristic
24#2858C2211yescharacteristic
25#2857C2212yes—
26#2850C2212yes—
27#2571C2212yes—
28#2568C2211yescharacteristic
29#2563—21no—
30#2559—22no—
31#2559—22no—
32#2548C2211yescharacteristic
33#2544—21no—
34#2544—21no—
35#2538—21no—
36#2500C2212yes—
37#2497—22no—
38#2492—22no—
39#2439C2214yes—
40#2432C2212yes—
41#2369—21no—
42#2367—21no—
43#2367—21no—
44#2355—21no—
45#2354C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48#2297C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2100C2212yes—
53#2074C2214yes—
54#2073C2214yes—
55#2072C2211yescharacteristic
56#2072C2211yescharacteristic
57#2072C2212yes—
58#2072C2212yes—
59#2071C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2043C2212yes—
63#2042C2214yes—
64#2024C2211yescharacteristic
65#2024C2211yescharacteristic
66#2023C2214yes—
67#2023C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1982C2212yes—
72#1958C2214yes—
73#1957C2212yes—
74#1934C2214yes—
75#1934C2214yes—
76#1933C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1875C2214yes—
80#1873C2214yes—
81#1835C2214yes—
82#1825C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1788C2214yes—
86C8×C4⋊C4C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#103C2212yes—
89C4×#103C2212yes—
90C8×C22⋊C4C2214yes—
91C4×C22⋊C8C2212yes—
92#1744C2211yescharacteristic
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#556—28no—
98#550—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127—21no—
106C2×#127C2311yescharacteristic
107C2×#126—22no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#124—21no—
111C2×#124C2312yes—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×#121—22no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2314yes—
126C22×C42C2312yes—
127#487—44no—
128C8×D4—28no—
129C2×C4⋊C8C2312yes—
130#416—48no—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#400—24no—
147#396—24no—
148#394—22no—
149#394—22no—
150#394—22no—
151#394—22no—
152#394—22no—
153#393—24no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391—24no—
159#391C2311yescharacteristic
160#391C2311yescharacteristic
161#391C2312yes—
162#390—22no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2312yes—
176#389C2314yes—
177#388—24no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—24no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#382C2312yes—
197#382C2314yes—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233C2×#41—24no—
234C2×#41—28no—
235C2×#41—42no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2411yescharacteristic
241C22×Q8C2411yescharacteristic
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C22×C8—28no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—41no—
264C4⋊Q8—44no—
265C4⋊Q8—82no—
266#127—24no—
267#127—44no—
268#126—48no—
269#124—44no—
270#124—44no—
271#124—44no—
272#124—44no—
273C22⋊Q8—28no—
274C22⋊Q8—84no—
275#121—28no—
276#121—48no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—44no—
284C4×D4#4118yes—
285#117—28no—
286#117—44no—
287#117—44no—
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—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
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—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C4⋊C8—24no—
348#103—24no—
349#103—28no—
350#103—42no—
351#103—42no—
352#103—42no—
353#103—44no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358C22⋊C8—28no—
359C4×C8—24no—
360C4×C8—44no—
361C4×C8—42no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C24C22×D412yes—
381#41—416no—
382#41—88no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8H(5,𝔽2)12yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4#12414yes—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—82no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4118yes—
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—88no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—84no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4#12414yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4#12418yes—
506C42—21no—
507C42—24no—
508C42—22no—
509C42—21no—
510C42—42no—
511C42—42no—
512C42—41no—
513C42—41no—
514C42—44no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42—84no—
519C42#13514yes—
520C42#13514yes—
521C42C2×#4114yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C2×#12414yes—
536C23C23×D411yescharacteristic
537Q8—24no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—84no—
542Q8—84no—
543Q8—164no—
544D4—28no—
545D4—44no—
546D4—44no—
547D4—88no—
548D4—84no—
549D4—162no—
550D4—164no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4C2×#12414yes—
599C2×C4C2×#12412yes—
600C2×C4C2×#12412yes—
601C2×C4C2×#13512yes—
602C2×C4C2×#13512yes—
603C2×C4C2×#13512yes—
604C2×C4C2×#13512yes—
605C2×C4C2×#13512yes—
606C2×C4C2×#13512yes—
607C2×C4C22×#4112yes—
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4#58512yes—
610C2×C4#58511yescharacteristic
611C8—44no—
612C8—42no—
613C8—42no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×#12411yescharacteristic
629C22#348511yescharacteristic
630C22#348811yescharacteristic
631C22C22×#13511yescharacteristic
632C22C22×#13511yescharacteristic
633C22#358511yescharacteristic
634C22#358611yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—24no—
640C4—22no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—84no—
645C4—82no—
646C4—82no—
647C4—81no—
648C4—82no—
649C4—81no—
650C4—81no—
651C4#313114yes—
652C4#351312yes—
653C4#352112yes—
654C2—44no—
655C2—162no—
656C2#3584711yescharacteristic
657C2C2×#313111yescharacteristic
658C2#6062511yescharacteristic
659C2#6170011yescharacteristic
660C2#6170111yescharacteristic
661C2#6195311yescharacteristic
662C2#6195411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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
2. #36194C2semidirect
3. #30203C2semidirect
4. #30188C2semidirect
5. #29346C2semidirect
6. #29107C2semidirect
7. #28717C2semidirect
8. #27976C2semidirect
9. #27750C2semidirect
10. #27730C2semidirect
11. #27696C2semidirect
12. #27261C2semidirect
13. #23816C2semidirect
14. #23627C2semidirect
15. #22490C2semidirect
17. #20555C2semidirect

Maximal subgroups

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

subgroupindex
#364192
#361942
#302032
#301882
#293462
#291072
#287172
#279762
#277502
#277302
#276962
#272612
#238162
#236272
#224902
#224182
#205552