Finite Group Databasebeta

← #6964602

Subgroups — #6964602

Order 512.

1509 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#35176C211yesmaximal, characteristic
3#32745C212yesmaximal, complemented
4#31568C214yesmaximal, complemented
5#30620C214yesmaximal, complemented
6#30199C212yesmaximal, complemented
7#29983C211yesmaximal, characteristic, complemented
8#29664C212yesmaximal, complemented
9#29141C211yesmaximal, characteristic, complemented
10#28161C211yesmaximal, characteristic, complemented
11#28151C212yesmaximal
12#26108C212yesmaximal, complemented
13#25504C212yesmaximal, complemented
14#25394C212yesmaximal
15#24732C212yesmaximal, complemented
16#23679C211yesmaximal, characteristic, complemented
17#22700C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#470C2214yes—
20C2×#462C2212yes—
21C2×#462C2214yes—
22C2×C4×Q16C2212yes—
23C2×#436C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2855C2212yes—
28#2667—22no—
29#2552—21no—
30#2552C2211yescharacteristic
31#2546C2211yescharacteristic
32#2542—21no—
33#2542C2211yescharacteristic
34#2538C2212yes—
35#2538C2211yescharacteristic
36#2522—22no—
37#2507—21no—
38#2454—21no—
39#2441—22no—
40#2429—22no—
41#2429C2212yes—
42#2423C2214yes—
43#2423C2212yes—
44#2365—21no—
45#2354—21no—
46Q8×C4⋊C4C2212yes—
47#2330—21no—
48#2319—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51#2295C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2099C2212yes—
54#2099C2214yes—
55#2096C2212yes—
56#2088C2214yes—
57#2083C2211yescharacteristic
58#2075C2214yes—
59#2072C2212yes—
60#2072C2211yescharacteristic
61#2068C2212yes—
62#2068C2214yes—
63#2058C2212yes—
64#2056C2212yes—
65#2043C2211yescharacteristic
66#2043C2214yes—
67#2035C2214yes—
68#2024C2212yes—
69#2024C2212yes—
70#2024C2212yes—
71#2016C2212yes—
72#2016C2211yescharacteristic
73#2001C2214yes—
74#2001C2214yes—
75#1956C2212yes—
76#1948C2212yes—
77#1939C2212yes—
78#1894C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1865C2214yes—
84#1851C2214yes—
85#1834C2212yes—
86#1831C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1803C2214yes—
90#1803C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1728C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97C22×M16C2314yes—
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125—21no—
106C2×#125—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2314yes—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2312yes—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C22×C42C2312yes—
123#470—48no—
124#462—44no—
125#462—48no—
126C4×Q16—44no—
127#436—44no—
128C2×#107C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C4×C8C2312yes—
144#399—22no—
145#399—22no—
146#399—24no—
147#398—22no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—24no—
152#396—22no—
153#396—22no—
154#396—24no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2311yescharacteristic
159#391C2312yes—
160#391C2311yescharacteristic
161#390—21no—
162#390—21no—
163#390—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#389C2314yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—24no—
185#387—22no—
186#387—22no—
187#387—21no—
188#387—21no—
189#387—22no—
190#387—24no—
191#385—22no—
192#384—21no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2312yes—
201#381—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206#380C2314yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376—21no—
221C43C2311yescharacteristic
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2314yes—
234C22×Q8—22no—
235C22×Q8C2411yescharacteristic
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C22×D4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C2×Q16—24no—
243C2×Q16—28no—
244C2×Q16—42no—
245C2×Q16—42no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×M16—44no—
249C2×M16—44no—
250C2×M16—44no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261#125—48no—
262#125—44no—
263C22⋊Q8—28no—
264C22⋊Q8—44no—
265C22⋊Q8—48no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—84no—
270C4×Q8—28no—
271C4×Q8—22no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—22no—
282#117—48no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—42no—
347#106—44no—
348C4⋊C8—24no—
349C4⋊C8—42no—
350C4⋊C8—44no—
351#103—24no—
352#103—44no—
353#103—42no—
354#103—42no—
355#103—42no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360C22⋊C8—44no—
361C4×C8—24no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24C22×D412yes—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8H(5,𝔽2)14yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2511yescharacteristic
432Q16—48no—
433Q16—88no—
434M16—88no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4#13714yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—48no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—84no—
494C42—22no—
495C42—22no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42—41no—
502C42—48no—
503C42—44no—
504C42—42no—
505C42—42no—
506C42—82no—
507C42#13714yes—
508C42H(5,𝔽2)12yes—
509C422−1+412yes—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—44no—
524Q8—44no—
525Q8—42no—
526Q8—88no—
527D4—44no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4#53712yes—
579C2×C4#53712yes—
580C2×C4C2×#13712yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13714yes—
583C2×C4C2×#13712yes—
584C2×C4C2×#13712yes—
585C2×C4#57812yes—
586C2×C4C2×H(5,𝔽2)11yescharacteristic
587C2×C4C2×H(5,𝔽2)12yes—
588C2×C4C2×2−1+411yescharacteristic
589C2×C4#58512yes—
590C8—42no—
591C8—44no—
592C8—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22C2×#53711yescharacteristic
601C22#348411yescharacteristic
602C22#350011yescharacteristic
603C22C22×#13711yescharacteristic
604C22C22×#13711yescharacteristic
605C22C2×#57811yescharacteristic
606C22#358511yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—24no—
612C4—41no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—82no—
617C4—81no—
618C4—82no—
619C4—81no—
620C4—84no—
621C4—81no—
622C4—81no—
623C4—82no—
624C4#325514yes—
625C4#354212yes—
626C4#355912yes—
627C2—82no—
628C2#4242411yescharacteristic
629C2C2×#325511yescharacteristic
630C2#6093011yescharacteristic
631C2#6158811yescharacteristic
632C2#6160211yescharacteristic
633C2#6253611yescharacteristic
634C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #32745C2semidirect
4. #31568C2semidirect
5. #30620C2semidirect
6. #30199C2semidirect
7. #29983C2semidirect
8. #29664C2semidirect
9. #29141C2semidirect
10. #28161C2semidirect
12. #26108C2semidirect
13. #25504C2semidirect
15. #24732C2semidirect
16. #23679C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#351762
#327452
#315682
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