Finite Group Databasebeta

← #6556197

Subgroups — #6556197

Order 512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30187C212yesmaximal, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#27980C212yesmaximal, complemented
9#27757C212yesmaximal, complemented
10#27736C212yesmaximal, complemented
11#27714C214yesmaximal, complemented
12#27265C214yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14#23627C212yesmaximal, complemented
15#22490C212yesmaximal, complemented
16#22417C212yesmaximal
17#20558C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#489C2212yes—
21C2×C8×D4C2212yes—
22C2×#416C2214yes—
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2571C2212yes—
28#2563—21no—
29#2553—22no—
30#2553—22no—
31#2546C2211yescharacteristic
32#2542C2211yescharacteristic
33#2538—21no—
34#2499C2212yes—
35#2497—22no—
36#2492—22no—
37#2439C2214yes—
38#2432C2212yes—
39#2427—21no—
40#2427—21no—
41#2369—21no—
42#2360—21no—
43#2360—21no—
44#2355—21no—
45#2354C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48#2297C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yes—
52#2100C2214yes—
53#2074C2214yes—
54#2073C2214yes—
55#2072C2211yescharacteristic
56#2072C2212yes—
57#2072C2212yes—
58#2072C2212yes—
59#2072C2212yes—
60#2071C2211yescharacteristic
61#2071C2212yes—
62#2043C2214yes—
63#2042C2212yes—
64#2024C2211yescharacteristic
65#2024C2214yes—
66#2024C2212yes—
67#2023C2211yescharacteristic
68#1993C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1981C2212yes—
72#1960C2214yes—
73#1956C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1935C2214yes—
77#1892C2214yes—
78#1892C2214yes—
79#1873C2214yes—
80#1873C2214yes—
81#1836C2214yes—
82#1825C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1752C2214yes—
91#1749C2212yes—
92#1744C2211yescharacteristic
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#556—28no—
98#547—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#126—22no—
107C2×#124—21no—
108C2×#124C2311yescharacteristic
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×#121—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2314yes—
125C22×C4⋊C4—21no—
126C22×C42C2312yes—
127#489—44no—
128C8×D4—28no—
129C2×C4⋊C8C2312yes—
130#416—48no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2314yes—
142C2×#97C2314yes—
143C2×#97C2312yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#399—22no—
147#398—22no—
148#397—22no—
149#397—22no—
150#396—24no—
151#394—22no—
152#394—22no—
153#393—24no—
154#391—21no—
155#391—22no—
156#391—21no—
157#391—21no—
158#391—24no—
159#391C2311yescharacteristic
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#390—22no—
164#390C2311yescharacteristic
165#390C2312yes—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2314yes—
175#388—24no—
176#388—21no—
177#388—21no—
178#387—24no—
179#387—22no—
180#387—22no—
181#386—22no—
182#386—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—21no—
193#382—21no—
194#382C2314yes—
195#382C2311yescharacteristic
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
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×Q8C2412yes—
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8—28no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264#126—48no—
265#124—22no—
266#124—41no—
267#124—44no—
268#123—48no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—84no—
274#121—44no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—42no—
280C4×D4—42no—
281C4×D4#4118yes—
282#117—28no—
283#117—44no—
284#117—44no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
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—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C4⋊C8—24no—
346#103—42no—
347#103—44no—
348#103—42no—
349#103—44no—
350#103—44no—
351#103—44no—
352#103—44no—
353#102—24no—
354#102—28no—
355#102—42no—
356C22⋊C8—28no—
357#97—44no—
358#97—42no—
359C4×C8—24no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
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—21no—
380C24C22×D412yes—
381#41—416no—
382#41—88no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4#12414yes—
409C2×D4H(5,𝔽2)12yes—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—82no—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4118yes—
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—88no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4#12414yes—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4#12418yes—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—24no—
507C42—22no—
508C42—21no—
509C42—42no—
510C42—42no—
511C42—44no—
512C42—42no—
513C42—44no—
514C42—84no—
515C42C2×#4114yes—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×#12414yes—
530C23C23×D411yescharacteristic
531Q8—42no—
532Q8—44no—
533Q8—42no—
534Q8—44no—
535Q8—84no—
536Q8—84no—
537Q8—164no—
538D4—28no—
539D4—24no—
540D4—42no—
541D4—88no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—88no—
591C2×C4—84no—
592C2×C4C2×#12414yes—
593C2×C4C2×#12412yes—
594C2×C4C2×#12412yes—
595C2×C4#57512yes—
596C2×C4#57512yes—
597C2×C4#57712yes—
598C2×C4#57812yes—
599C2×C4C22×#4112yes—
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4#58512yes—
602C2×C4#58511yescharacteristic
603C8—44no—
604C8—42no—
605C8—42no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22C22×#12411yescharacteristic
621C22#348511yescharacteristic
622C22#348811yescharacteristic
623C22C2×#57511yescharacteristic
624C22C2×#57511yescharacteristic
625C22C2×#57711yescharacteristic
626C22C2×#57811yescharacteristic
627C4—24no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4#313514yes—
644C4#351312yes—
645C4#352112yes—
646C2—44no—
647C2—82no—
648C2#3584711yescharacteristic
649C2C2×#313511yescharacteristic
650C2#6062311yescharacteristic
651C2#6173011yescharacteristic
652C2#6173111yescharacteristic
653C2#6198511yescharacteristic
654C2#6198611yescharacteristic
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
2. #36194C2semidirect
3. #30205C2semidirect
4. #30187C2semidirect
5. #29355C2semidirect
6. #29113C2semidirect
7. #28726C2semidirect
8. #27980C2semidirect
9. #27757C2semidirect
10. #27736C2semidirect
11. #27714C2semidirect
12. #27265C2semidirect
13. #23817C2semidirect
14. #23627C2semidirect
15. #22490C2semidirect
17. #20558C2semidirect

Maximal subgroups

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

subgroupindex
#364062
#361942
#302052
#301872
#293552
#291132
#287262
#279802
#277572
#277362
#277142
#272652
#238172
#236272
#224902
#224172
#205582