Finite Group Databasebeta

← #6565915

Subgroups — #6565915

Order 512.

1583 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#34337C211yesmaximal, characteristic, complemented
3#30041C212yesmaximal
4#29983C211yesmaximal, characteristic, complemented
5#29555C212yesmaximal
6#29390C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28157C211yesmaximal, characteristic
9#27554C214yesmaximal, complemented
10#27066C214yesmaximal, complemented
11#24992C212yesmaximal, complemented
12#24187C212yesmaximal, complemented
13#23816C212yesmaximal, complemented
14#23678C211yesmaximal, characteristic
15#23094C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17#22767C212yesmaximal, complemented
18C2×#557C2211yescharacteristic
19C2×#556—21no—
20C2×#461C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2858C2212yes—
24#2547—21no—
25#2545—22no—
26#2538—21no—
27#2538C2212yes—
28#2512—22no—
29#2512C2214yes—
30#2370—21no—
31#2370—22no—
32#2365C2211yescharacteristic
33#2365C2212yes—
34#2354—21no—
35#2354C2212yes—
36Q8×C4⋊C4—21no—
37Q8×C4⋊C4C2211yescharacteristic
38Q8×C4⋊C4C2212yes—
39Q8×C4⋊C4C2211yescharacteristic
40#2330—22no—
41#2330—21no—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2295C2211yescharacteristic
47#2295C2212yes—
48#2276—21no—
49C42×Q8C2211yescharacteristic
50#2100C2214yes—
51#2099C2212yes—
52#2096C2212yes—
53#2084C2211yescharacteristic
54#2072C2211yescharacteristic
55#2072C2212yes—
56#2071C2214yes—
57#2065C2212yes—
58#2058C2212yes—
59#2055C2212yes—
60#2043C2214yes—
61#2041C2211yescharacteristic
62#2024C2212yes—
63#2024C2212yes—
64#2024C2212yes—
65#2014C2211yescharacteristic
66#2014C2212yes—
67#2013C2214yes—
68#1984C2212yes—
69#1968C2214yes—
70#1956C2212yes—
71#1948C2212yes—
72#1934C2214yes—
73#1927C2214yes—
74#1926C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1916C2214yes—
78#1880C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1880C2214yes—
82#1865C2214yes—
83#1839C2212yes—
84#1834C2212yes—
85#1831C2211yescharacteristic
86#1803C2212yes—
87#1803C2214yes—
88#1802C2212yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2214yes—
92C4×#103C2214yes—
93C4×#102C2212yes—
94#1744C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97#557—24no—
98#557—24no—
99#556—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#126—24no—
107C2×#125—21no—
108C2×#125—22no—
109C2×#124—22no—
110C2×#124—22no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127#461—28no—
128C4×SD16—28no—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#397—24no—
145#396—24no—
146#396—24no—
147#391—22no—
148#391C2311yescharacteristic
149#391C2312yes—
150#391C2311yescharacteristic
151#391C2312yes—
152#390—21no—
153#390—21no—
154#390—22no—
155#390—22no—
156#390C2312yes—
157#390C2314yes—
158#390C2314yes—
159#389—21no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2312yes—
175#388—24no—
176#388—24no—
177#387—22no—
178#387—21no—
179#387—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#384—21no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#380—22no—
194#380—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215#376—21no—
216#376—22no—
217C43C2311yescharacteristic
218C43C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#336C2312yes—
227#336C2314yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2412yes—
236C22×Q8C2411yescharacteristic
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2412yes—
240C23×C4—22no—
241C2×SD16—24no—
242C2×SD16—28no—
243C2×SD16—42no—
244C2×SD16—42no—
245C2×SD16—44no—
246C2×SD16—44no—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256C4⋊Q8#4114yes—
257#127—42no—
258#126—416no—
259#125—216no—
260#125—28no—
261#124—48no—
262#124—84no—
263C22⋊Q8—48no—
264C4×Q8—28no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—42no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×Q8SD1618yescomplemented
277C4×Q8#4114yes—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—44no—
281C4×D4—44no—
282#117—48no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#106—24no—
348#106—44no—
349C4⋊C8—24no—
350C4⋊C8—42no—
351#103—24no—
352#103—28no—
353#103—44no—
354#103—44no—
355#103—44no—
356#102—24no—
357#102—42no—
358#102—44no—
359#102—42no—
360C4×C8—24no—
361C4×C8—44no—
362#95—21no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—24no—
375#95—22no—
376#95—21no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8C2×SD1614yes—
393C2×Q8C2×#4114yes—
394C2×Q8C2×#4112yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2511yescharacteristic
429SD16—48no—
430SD16—88no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4C2×SD1614yes—
477C4⋊C4C2×SD1618yes—
478C4⋊C4C2×#4112yes—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C42—28no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—42no—
497C42—42no—
498C42—41no—
499C42—44no—
500C42—48no—
501C42—42no—
502C42—82no—
503C42—82no—
504C42—82no—
505C42C2×SD1614yes—
506C42C2×SD1618yes—
507C42C2×#4112yes—
508C42C2×#4112yes—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23C23×D411yescharacteristic
517Q8—28no—
518Q8—24no—
519Q8—28no—
520Q8—44no—
521Q8—42no—
522Q8—88no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—84no—
528D4—84no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—88no—
575C2×C4#52512yes—
576C2×C4#52512yes—
577C2×C4C22×SD1614yes—
578C2×C4C22×SD1612yes—
579C2×C4C22×SD1612yes—
580C2×C4C22×SD1614yes—
581C2×C4C22×SD1612yes—
582C2×C4#57512yes—
583C2×C4#57812yes—
584C2×C4C22×#4111yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4#58512yes—
587C2×C4#58512yes—
588C8—42no—
589C8—42no—
590C8—44no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22C2×#52511yescharacteristic
599C22#348711yescharacteristic
600C22#350011yescharacteristic
601C22C23×SD1611yescharacteristic
602C22C2×#57511yescharacteristic
603C22C2×#57811yescharacteristic
604C22#358511yescharacteristic
605C4—24no—
606C4—22no—
607C4—24no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—82no—
618C4—82no—
619C4—81no—
620C4—84no—
621C4—81no—
622C4#310314yes—
623C4#349312yes—
624C4#352112yes—
625C2—162no—
626C2#3601911yescharacteristic
627C2C2×#310311yescharacteristic
628C2#6060011yescharacteristic
629C2#6188111yescharacteristic
630C2#6190511yescharacteristic
631C2#6252611yescharacteristic
632C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34337C2semidirect
4. #29983C2semidirect
6. #29390C2semidirect
7. #29141C2semidirect
9. #27554C2semidirect
10. #27066C2semidirect
11. #24992C2semidirect
12. #24187C2semidirect
13. #23816C2semidirect
15. #23094C2semidirect
16. #22828C2semidirect
17. #22767C2semidirect
276. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365942
#343372
#300412
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