Finite Group Databasebeta

← #6626297

Subgroups — #6626297

Order 512.

1743 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#32960C212yesmaximal
4#32954C211yesmaximal, characteristic
5#29303C212yesmaximal
6#29240C212yesmaximal
7#27948C211yesmaximal, characteristic
8#27681C211yesmaximal, characteristic
9#27314C214yesmaximal, complemented
10#26395C214yesmaximal, complemented
11#26043C212yesmaximal, complemented
12#24818C212yesmaximal, complemented
13#22970C212yesmaximal, complemented
14#22674C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16#21455C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#479C2214yes—
21C2×#475C2212yes—
22C2×#451C2211yescharacteristic
23C2×#441C2214yes—
24C2×C4×Q16C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2590—22no—
28#2568C2211yescharacteristic
29#2552C2211yescharacteristic
30#2511—21no—
31#2508C2212yes—
32#2505—22no—
33#2502—21no—
34#2453C2212yes—
35#2365C2211yescharacteristic
36#2360—21no—
37#2355—21no—
38Q8×C4⋊C4—22no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2214yes—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2212yes—
44Q8×C4⋊C4C2211yescharacteristic
45C4×C22⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47C2×#389—22no—
48C2×#389C2212yes—
49C2×#384—21no—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2212yes—
54#2095C2212yes—
55#2088C2214yes—
56#2085C2214yes—
57#2084C2214yes—
58#2066C2214yes—
59#2059C2214yes—
60#1990C2212yes—
61#1947C2212yes—
62#1944C2214yes—
63#1943C2214yes—
64#1939C2212yes—
65#1919C2214yes—
66#1917C2214yes—
67#1878C2212yes—
68#1878C2214yes—
69#1872C2212yes—
70#1872C2211yescharacteristic
71#1865C2214yes—
72#1865C2214yes—
73#1865C2214yes—
74#1865C2214yes—
75#1863C2214yes—
76#1863C2212yes—
77#1853C2212yes—
78#1853C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1852C2211yescharacteristic
82#1844C2212yes—
83#1805C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1787C2212yes—
89#1787C2211yescharacteristic
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8C2311yescharacteristic
101C2×C4×Q8—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—48no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C42—21no—
124C22×C42—22no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#479—48no—
128#475—28no—
129#451Q818yescomplemented
130#441—48no—
131C4×Q16—28no—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#400—22no—
153#400—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391C2312yes—
162#391C2311yescharacteristic
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—24no—
170#389—24no—
171#389—24no—
172#389—24no—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#389C2311yescharacteristic
180#387—22no—
181#387—21no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—21no—
191#384—22no—
192#384—22no—
193#384—24no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382—22no—
202#382C2312yes—
203#380—21no—
204#380C2311yescharacteristic
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—24no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2×Q814yes—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2411yescharacteristic
251C2×Q16—24no—
252C2×Q16—24no—
253C2×Q16C2×Q818yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258#125—44no—
259C22⋊Q8—22no—
260C22⋊Q8—44no—
261C22⋊Q8C2×Q814yes—
262C4×Q8—28no—
263C4×Q8—28no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—28no—
268C4×Q8—28no—
269C4×Q8—24no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—48no—
275C4×Q8—48no—
276C4×Q8#4118yes—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C42—28no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42—48no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—24no—
348#107—44no—
349#106—28no—
350#106—44no—
351#106—44no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#103—24no—
355#103—28no—
356#103—28no—
357#103—24no—
358#103—24no—
359#103—44no—
360#103C2×Q818yes—
361C22⋊C8—24no—
362C22⋊C8—22no—
363C22⋊C8C2×Q814yes—
364#97—44no—
365C4×C8—24no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—216no—
391C2×Q8—216no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8C22×Q812yes—
403C2×Q8C22×Q814yes—
404C2×Q8C2×#4114yes—
405C2×Q8H(5,𝔽2)14yes—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4#13714yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q812yes—
448C22×C42−1+412yes—
449C22×C4C2511yescharacteristic
450Q16—216no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8C22×Q814yes—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C22×Q812yes—
500C4⋊C4C2×#4114yes—
501C22⋊C4—24no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C42—28no—
505C42—28no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—24no—
510C42—24no—
511C42—42no—
512C42—42no—
513C42—48no—
514C42—48no—
515C42—44no—
516C42—42no—
517C42—42no—
518C42—82no—
519C42C2×#4114yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23D4×Q812yes—
525C23C2×#13712yes—
526C23#57712yes—
527C23C23×D411yescharacteristic
528Q8—216no—
529Q8—24no—
530Q8—24no—
531Q8—24no—
532Q8—24no—
533Q8—42no—
534Q8—84no—
535Q8#55418yes—
536C2×C4—216no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4D4×Q812yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4#57512yes—
599C2×C4C23×Q811yescharacteristic
600C2×C4C22×#4112yes—
601C2×C4C2×H(5,𝔽2)12yes—
602C2×C4C2×2−1+411yescharacteristic
603C8—24no—
604C8—42no—
605C8—44no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22#338714yes—
613C22C2×#52511yescharacteristic
614C22C2×D4×Q811yescharacteristic
615C22#348411yescharacteristic
616C22C22×#13711yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57711yescharacteristic
619C22#358611yescharacteristic
620C4—22no—
621C4—22no—
622C4—28no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—82no—
633C4—82no—
634C4—81no—
635C4—84no—
636C4—81no—
637C4#311714yes—
638C4C2×#55412yes—
639C4#349212yes—
640C2—24no—
641C2#3654611yescharacteristic
642C2C2×#311711yescharacteristic
643C2#6058511yescharacteristic
644C2C2×#338711yescharacteristic
645C2#6136911yescharacteristic
646C2#6158411yescharacteristic
647C2#6159611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #27314C2semidirect
10. #26395C2semidirect
11. #26043C2semidirect
12. #24818C2semidirect
13. #22970C2semidirect
14. #22674C2semidirect
129. #451Q8semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×Q8×C4⋊C42
#329602
#329542
#293032
#292402
#279482
#276812
#273142
#263952
#260432
#248182
#229702
#226742
#226292
#214552
C4×#4512