Finite Group Databasebeta

← #6446022

Subgroups — #6446022

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#25071C214yesmaximal, complemented
4#25070C214yesmaximal, complemented
5#25069C214yesmaximal, complemented
6#25068C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8#23737C211yesmaximal, characteristic
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#21301C211yesmaximal, characteristic, complemented
12#21172C211yesmaximal, characteristic, complemented
13#21068C211yesmaximal, characteristic
14#21067C211yesmaximal, characteristic
15#21038C211yesmaximal, characteristic, complemented
16#21037C211yesmaximal, characteristic, complemented
17C2×#1847C211yesmaximal, characteristic, complemented
18#20648C211yesmaximal, characteristic
19#20411C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2851—22no—
35#2628—22no—
36#2581C2214yes—
37#2580C2214yes—
38#2564—24no—
39#2360—24no—
40#2355C2214yes—
41Q8×C4⋊C4C2214yes—
42#2330—22no—
43#2317C2211yescharacteristic
44#2317C2211yescharacteristic
45#2316C2211yescharacteristic
46#2316C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49C4×C4⋊Q8—22no—
50C2×#384C2211yescharacteristic
51C2×#382C2211yescharacteristic
52#2274C2211yescharacteristic
53#2274C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2097C2212yes—
56#2097C2212yes—
57#2096C2212yes—
58#2095C2212yes—
59#2094C2212yes—
60#2093C2212yes—
61#2092C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2085C2214yes—
65#2084C2214yes—
66#2083C2214yes—
67#1985C2214yes—
68#1985C2214yes—
69#1949C2214yes—
70#1949C2214yes—
71#1928C2214yes—
72#1927C2214yes—
73#1926C2214yes—
74#1925C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1919C2214yes—
78#1918C2214yes—
79#1847C2214yes—
80#1847C2214yes—
81#1803C2214yes—
82#1802C2214yes—
83#1763C2212yes—
84#1763C2212yes—
85#1762C2212yes—
86#1762C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102#533—24no—
103#533—24no—
104C2×C4⋊Q8D412yes—
105C2×#125D412yes—
106C2×C22⋊Q8D414yes—
107C4×#41—24no—
108C4×#41—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127#428—42no—
128#428—42no—
129C2×#107—24no—
130C2×#107—42no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#106—24no—
137C2×#106—42no—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143#423—24no—
144C2×C4⋊C8C2314yes—
145#417—42no—
146#417—42no—
147#416—24no—
148#416Q818yescomplemented
149C2×#103C2311yescharacteristic
150C2×#103C2311yescharacteristic
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#102C2311yescharacteristic
155C2×#102C2311yescharacteristic
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#97—24no—
160C2×#97C2312yes—
161C2×#97C2312yes—
162#400—24no—
163#399—24no—
164#395—24no—
165#393—24no—
166#391—24no—
167#391C2312yes—
168#391C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389—24no—
178#389C2314yes—
179#389C2314yes—
180#387—24no—
181#387—24no—
182#387—24no—
183#387—24no—
184#386—22no—
185#386—22no—
186#385—24no—
187#385—24no—
188#385—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2314yes—
193#384C2312yes—
194#384C2312yes—
195#382—21no—
196#382—21no—
197#382C2312yes—
198#382C2312yes—
199#382C2314yes—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214#376—21no—
215C2×#95—22no—
216C43—21no—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#336C2314yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41C2×Q814yes—
237C22×Q8C2×D414yes—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2411yescharacteristic
249C2×M16—22no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—22no—
254C2×M16—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8C2×Q814yes—
260C22×C8#4114yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2411yescharacteristic
265C22×C8C2411yescharacteristic
266C4⋊Q8—28no—
267#125—28no—
268C22⋊Q8—216no—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—88no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—48no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117—22no—
282#117—22no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—44no—
287#117C2×Q814yes—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42#4114yes—
341C2×C42#4114yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—24no—
346#107—24no—
347#107—44no—
348#107—42no—
349#107—42no—
350#106—24no—
351#106—24no—
352#106—44no—
353#106—42no—
354#106—42no—
355C4⋊C8—28no—
356#103—22no—
357#103—22no—
358#103—42no—
359#103—42no—
360#103C2×Q818yes—
361#102—22no—
362#102—22no—
363#102—42no—
364#102—42no—
365#102C2×Q818yes—
366#97—24no—
367#97—24no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380C24—21no—
381C24C22×D411yescharacteristic
382#41—28no—
383#41—28no—
384#41—44no—
385#41—44no—
386C2×Q8—28no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8C22×Q812yes—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4C22×Q812yes—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4#12318yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×Q812yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C42−1+412yes—
441C22×C4C2511yescharacteristic
442M16—44no—
443M16—44no—
444C2×C8—28no—
445C2×C8—28no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8C22×Q814yes—
459C2×C8C2×#4112yes—
460C2×C8C2×#4112yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4C22×Q814yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—44no—
507C42—44no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—84no—
512C42—84no—
513C42#12314yes—
514C42#12314yes—
515C42C2×#4114yes—
516C42C2×#4114yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23C2×#12312yes—
525C23D4×Q812yes—
526C23#55012yes—
527C23C23×D411yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—416no—
531Q8—41no—
532Q8—41no—
533D4—22no—
534D4—22no—
535D4—48no—
536D4—41no—
537D4—41no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—88no—
572C2×C4—162no—
573C2×C4—162no—
574C2×C4—162no—
575C2×C4—162no—
576C2×C4C2×#12314yes—
577C2×C4C2×#12314yes—
578C2×C4C2×#12312yes—
579C2×C4#53314yes—
580C2×C4#54214yes—
581C2×C4D4×Q812yes—
582C2×C4#55012yes—
583C2×C4#55214yes—
584C2×C4#55414yes—
585C2×C4C23×Q811yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C8—22no—
592C8—22no—
593C8—24no—
594C8—42no—
595C8—42no—
596C22—24no—
597C22—24no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—44no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22#242314yes—
606C22C22×#12311yescharacteristic
607C22C2×D4×Q811yescharacteristic
608C22C2×#55011yescharacteristic
609C22#358512yes—
610C22#358612yes—
611C4—24no—
612C4—24no—
613C4—44no—
614C4—44no—
615C4—42no—
616C4—44no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—81no—
623C4—82no—
624C4#242314yes—
625C4C2×#53311yescharacteristic
626C4C2×#54211yescharacteristic
627C4C2×#55211yescharacteristic
628C4C2×#55411yescharacteristic
629C2—24no—
630C2—82no—
631C2C2×#242311yescharacteristic
632C2#6051712yes—
633C2#6138012yes—
634C2#6141711yescharacteristic
635C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #25071C2semidirect
4. #25070C2semidirect
5. #25069C2semidirect
6. #25068C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
11. #21301C2semidirect
12. #21172C2semidirect
15. #21038C2semidirect
16. #21037C2semidirect
17. C2×#1847C2semidirect
148. #416Q8semidirect

Maximal subgroups

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

subgroupindex
#353342
#342142
#250712
#250702
#250692
#250682
#237382
#237372
#236612
#236602
#213012
#211722
#210682
#210672
#210382
#210372
C2×#18472
#206482
#204112