Finite Group Databasebeta

← #6489074

Subgroups — #6489074

Order 512.

1644 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35314C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#32964C212yesmaximal
4#32915C212yesmaximal
5#30132C212yesmaximal, complemented
6#30131C212yesmaximal, complemented
7#29129C211yesmaximal, characteristic, complemented
8#29126C211yesmaximal, characteristic, complemented
9#28385C211yesmaximal, characteristic, complemented
10#28373C211yesmaximal, characteristic, complemented
11#28291C212yesmaximal, complemented
12#28261C212yesmaximal, complemented
13#27685C212yesmaximal
14#24514C214yesmaximal
15#24212C212yesmaximal
16#22164C214yesmaximal, complemented
17#20813C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519C2212yes—
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#462C2214yes—
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×#451C2214yes—
26C2×#433C2212yes—
27#2864C2211yescharacteristic
28#2863C2212yes—
29#2862C2211yescharacteristic
30#2628C2211yescharacteristic
31#2625C2212yes—
32#2621C2211yescharacteristic
33#2524—22no—
34#2523—22no—
35#2520—22no—
36#2517—22no—
37#2365—22no—
38#2365—22no—
39#2344—22no—
40#2344—22no—
41#2331C2211yescharacteristic
42#2327C2211yescharacteristic
43#2327C2212yes—
44#2327C2212yes—
45#2320C2212yes—
46#2310C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52#2285C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2100C2212yes—
55#2099C2212yes—
56#2096C2212yes—
57#2095C2212yes—
58#2068C2214yes—
59#2066C2214yes—
60#2056C2214yes—
61#2056C2214yes—
62#2043C2212yes—
63#2043C2212yes—
64#2016C2212yes—
65#2016C2212yes—
66#1981C2214yes—
67#1971C2214yes—
68#1969C2214yes—
69#1949C2214yes—
70#1935C2214yes—
71#1933C2214yes—
72#1878C2214yes—
73#1878C2214yes—
74#1873C2214yes—
75#1865C2214yes—
76#1863C2214yes—
77#1854C2214yes—
78#1843C2212yes—
79#1831C2212yes—
80#1805C2214yes—
81#1805C2212yes—
82#1805C2212yes—
83#1805C2212yes—
84#1805C2212yes—
85#1798C2212yes—
86#1797C2212yes—
87#1787C2214yes—
88#1768C2212yes—
89#1767C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C23×Q8C2312yes—
92C22×Q16C2314yes—
93C22×Q16C2314yes—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C22⋊Q8C2312yes—
102#519—44no—
103#519—44no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×#117—22no—
112C2×#117—22no—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—22no—
117C22×C42C2311yescharacteristic
118#477—44no—
119#475—44no—
120#462—216no—
121#462—44no—
122#462—44no—
123#451—48no—
124#433—44no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#400—21no—
144#399—21no—
145#398—21no—
146#398—21no—
147#398—22no—
148#397—21no—
149#397—21no—
150#397—22no—
151#397—22no—
152#395—21no—
153#393—21no—
154#391—24no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#387—22no—
186#387—22no—
187#387—22no—
188#385—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#383—22no—
192#383—22no—
193#382—22no—
194#382—22no—
195#382C2311yescharacteristic
196#382C2311yescharacteristic
197#382C2312yes—
198#381—24no—
199#380—22no—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4D412yes—
213#376C2311yescharacteristic
214#376C2311yescharacteristic
215C43C2311yescharacteristic
216C43C2311yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2312yes—
222#336C2312yes—
223#336C2314yes—
224C22×Q8—24no—
225C22×Q8—24no—
226C22×Q8—22no—
227C22×Q8—21no—
228C22×Q8—24no—
229C22×Q8—24no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C23×C4—22no—
236C23×C4C2×D412yes—
237C23×C4C2411yescharacteristic
238C2×Q16—28no—
239C2×Q16—28no—
240C2×Q16—44no—
241C2×Q16—44no—
242C2×Q16—44no—
243C2×Q16—44no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—216no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8#4118yes—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C4×Q8—28no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8—42no—
264C4×Q8—48no—
265C4×Q8—48no—
266#117—22no—
267#117—22no—
268#117—24no—
269#117—48no—
270#117—48no—
271#117—42no—
272#117—42no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4C2×D411yescharacteristic
304C2×C22⋊C4C2×D411yescharacteristic
305C2×C22⋊C4C2×D412yes—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#107—44no—
326#106—44no—
327C4⋊C8—28no—
328C4⋊C8—24no—
329C4⋊C8—24no—
330C4⋊C8—44no—
331#103—28no—
332#103—28no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#103—44no—
338#103—44no—
339C22⋊C8—28no—
340#97—42no—
341C4×C8—42no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24C22×D411yescharacteristic
370C2×Q8—28no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—82no—
393C2×Q8C2×#4114yes—
394C2×Q8C2×#4114yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422C22×C4C2511yescharacteristic
423Q16—88no—
424Q16—88no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—216no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—21no—
474C42—21no—
475C42—42no—
476C42—42no—
477C42—44no—
478C42—44no—
479C42—44no—
480C42—44no—
481C42—44no—
482C42—44no—
483C42—82no—
484C42C2×#4114yes—
485C42H(5,𝔽2)14yes—
486C23—22no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23#53712yes—
491C23C23×D411yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—24no—
495Q8—24no—
496Q8—416no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—44no—
502Q8—44no—
503Q8—162no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—48no—
527C2×C4—416no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4#52514yes—
549C2×C4#53512yes—
550C2×C4#53512yes—
551C2×C4#53712yes—
552C2×C4#57814yes—
553C2×C4C22×#4112yes—
554C2×C4C2×H(5,𝔽2)11yescharacteristic
555C2×C4C2×H(5,𝔽2)12yes—
556C2×C4C2×2−1+411yescharacteristic
557C8—44no—
558C8—44no—
559C22—22no—
560C22—24no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22C2×#52511yescharacteristic
565C22C2×#53511yescharacteristic
566C22C2×#53711yescharacteristic
567C22C2×#57812yes—
568C22#358612yes—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—24no—
573C4—24no—
574C4—41no—
575C4—41no—
576C4—44no—
577C4—44no—
578C4—42no—
579C4—42no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4#351912yes—
587C4#352012yes—
588C2—42no—
589C2#3523611yescharacteristic
590C2#6058312yes—
591C2#6082111yescharacteristic
592C2#6082811yescharacteristic
593C2#6092512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #30132C2semidirect
6. #30131C2semidirect
7. #29129C2semidirect
8. #29126C2semidirect
9. #28385C2semidirect
10. #28373C2semidirect
11. #28291C2semidirect
12. #28261C2semidirect
16. #22164C2semidirect

Maximal subgroups

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

subgroupindex
#353142
#342092
#329642
#329152
#301322
#301312
#291292
#291262
#283852
#283732
#282912
#282612
#276852
#245142
#242122
#221642
#208132