Finite Group Databasebeta

← #6450677

Subgroups — #6450677

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic, complemented
2#34188C211yesmaximal, characteristic
3#26101C214yesmaximal, complemented
4#26100C214yesmaximal, complemented
5#26099C214yesmaximal, complemented
6#26098C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8#23737C211yesmaximal, characteristic
9#23611C211yesmaximal, characteristic
10#23610C211yesmaximal, characteristic
11#21174C211yesmaximal, characteristic, complemented
12#21173C211yesmaximal, characteristic, complemented
13#21171C211yesmaximal, characteristic, complemented
14#21144C211yesmaximal, characteristic, complemented
15#21038C211yesmaximal, characteristic, complemented
16#21037C211yesmaximal, characteristic, complemented
17C2×#1847C211yesmaximal, characteristic, complemented
18#20649C211yesmaximal, characteristic
19#20411C211yesmaximal, characteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2864—22no—
36#2647—22no—
37#2570C2214yes—
38#2570C2214yes—
39#2569—24no—
40#2355C2214yes—
41#2354C2214yes—
42#2352—24no—
43#2317C2211yescharacteristic
44#2317C2211yescharacteristic
45#2316C2211yescharacteristic
46#2316C2211yescharacteristic
47#2314—22no—
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50#2285—22no—
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#382C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yes—
58#2097C2212yes—
59#2096C2212yes—
60#2095C2212yes—
61#2094C2212yes—
62#2093C2212yes—
63#2092C2212yes—
64#2092C2212yes—
65#2068C2214yes—
66#2067C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#1949C2214yes—
70#1949C2214yes—
71#1948C2214yes—
72#1948C2214yes—
73#1947C2214yes—
74#1947C2214yes—
75#1940C2214yes—
76#1940C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1847C2214yes—
82#1847C2214yes—
83#1805C2214yes—
84#1804C2214yes—
85#1763C2212yes—
86#1763C2212yes—
87#1762C2212yes—
88#1762C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C4×#41—24no—
103C4×#41Q818yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4C2312yes—
107C2×#117—24no—
108C2×#117—21no—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4—22no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121#428—42no—
122#428—42no—
123C2×#107—24no—
124C2×#107—42no—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106—24no—
131C2×#106—42no—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137#424—24no—
138#424—24no—
139#423—24no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143#417—42no—
144#417—42no—
145#416—24no—
146#416—24no—
147C2×#103C2312yes—
148C2×#103C2311yescharacteristic
149C2×#103C2311yescharacteristic
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#102C2312yes—
153C2×#102C2311yescharacteristic
154C2×#102C2311yescharacteristic
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#97—24no—
158C2×#97C2312yes—
159C2×#97C2312yes—
160#400—24no—
161#400—24no—
162#398—24no—
163#398—24no—
164#391C2312yes—
165#391C2312yes—
166#390C2312yes—
167#390C2312yes—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#388—24no—
175#388—24no—
176#387—24no—
177#387—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384D412yes—
189#384C2312yes—
190#384C2312yes—
191#384C2314yes—
192#382—24no—
193#382—21no—
194#382—21no—
195#382D412yes—
196#382C2314yes—
197#382C2312yes—
198#382C2312yes—
199#380—22no—
200#380—22no—
201#380—24no—
202#380—22no—
203#380—22no—
204#380D414yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—24no—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216#376—21no—
217C2×#95—22no—
218C43—21no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
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×Q8C2411yescharacteristic
238C22×D4—21no—
239C22×D4C2411yescharacteristic
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2411yescharacteristic
248C2×M16—22no—
249C2×M16—22no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8#4114yes—
259C22×C8#4114yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2411yescharacteristic
263C22×C8C2411yescharacteristic
264C22×C8C2412yes—
265C4×Q8—24no—
266C4×Q8C2×Q818yes—
267C4×D4—216no—
268C4×D4—24no—
269C4×D4C2×Q818yes—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—28no—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—88no—
281#117C2×Q814yes—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—84no—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—28no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×Q814yes—
334C2×C42#4114yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338#107—24no—
339#107—24no—
340#107—44no—
341#107—42no—
342#107—42no—
343#106—24no—
344#106—24no—
345#106—44no—
346#106—42no—
347#106—42no—
348C4⋊C8—28no—
349C4⋊C8—28no—
350C4⋊C8—28no—
351#103—24no—
352#103—22no—
353#103—22no—
354#103—42no—
355#103—42no—
356#102—24no—
357#102—22no—
358#102—22no—
359#102—42no—
360#102—42no—
361#97—24no—
362#97—24no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95C2×D414yes—
376#95C2×D414yes—
377C24—21no—
378C24C22×D411yescharacteristic
379#41—28no—
380#41—28no—
381#41—44no—
382#41—44no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8C22×Q812yes—
389C2×D4—24no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4C22×Q812yes—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4C22⋊Q818yes—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×Q812yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C42−1+412yes—
435C22×C4C2511yescharacteristic
436M16—44no—
437M16—44no—
438C2×C8—28no—
439C2×C8—28no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8C2×#4114yes—
453C2×C8C2×#4112yes—
454C2×C8C2×#4112yes—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4C22×Q814yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—28no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C42—24no—
495C42—28no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—42no—
501C42—42no—
502C42—82no—
503C42—82no—
504C42—84no—
505C42—84no—
506C42C22⋊Q814yes—
507C42C22⋊Q814yes—
508C42C22×Q814yes—
509C42C2×#4114yes—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23C2×C22⋊Q812yes—
518C23#54712yes—
519C23#55012yes—
520C23C23×D411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525D4—22no—
526D4—22no—
527D4—48no—
528D4—41no—
529D4—41no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—48no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—162no—
565C2×C4—162no—
566C2×C4—162no—
567C2×C4—162no—
568C2×C4C2×C22⋊Q814yes—
569C2×C4C2×C22⋊Q814yes—
570C2×C4C2×C22⋊Q812yes—
571C2×C4#53314yes—
572C2×C4#54414yes—
573C2×C4#54712yes—
574C2×C4#55012yes—
575C2×C4#55214yes—
576C2×C4#55314yes—
577C2×C4C23×Q811yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)11yescharacteristic
582C2×C4C2×2−1+411yescharacteristic
583C8—22no—
584C8—22no—
585C8—24no—
586C8—42no—
587C8—42no—
588C22—24no—
589C22—24no—
590C22—22no—
591C22—22no—
592C22—44no—
593C22—42no—
594C22—82no—
595C22—81no—
596C22—81no—
597C22#244114yes—
598C22C22×C22⋊Q811yescharacteristic
599C22C2×#54711yescharacteristic
600C22C2×#55011yescharacteristic
601C22#358512yes—
602C22#358612yes—
603C4—24no—
604C4—24no—
605C4—44no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—82no—
610C4—81no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4#244114yes—
617C4C2×#53311yescharacteristic
618C4C2×#54411yescharacteristic
619C4C2×#55211yescharacteristic
620C4C2×#55311yescharacteristic
621C2—24no—
622C2—82no—
623C2C2×#244111yescharacteristic
624C2#6049212yes—
625C2#6129412yes—
626C2#6141711yescharacteristic
627C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35339C2semidirect
3. #26101C2semidirect
4. #26100C2semidirect
5. #26099C2semidirect
6. #26098C2semidirect
11. #21174C2semidirect
12. #21173C2semidirect
13. #21171C2semidirect
14. #21144C2semidirect
15. #21038C2semidirect
16. #21037C2semidirect
17. C2×#1847C2semidirect

Maximal subgroups

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

subgroupindex
#353392
#341882
#261012
#261002
#260992
#260982
#237382
#237372
#236112
#236102
#211742
#211732
#211712
#211442
#210382
#210372
C2×#18472
#206492
#204112