Finite Group Databasebeta

← #6485300

Subgroups — #6485300

Order 512.

1623 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic, complemented
2#34280C211yesmaximal, characteristic
3#33040C214yesmaximal, complemented
4#30113C212yesmaximal, complemented
5#30106C212yesmaximal, complemented
6#29369C212yesmaximal, complemented
7#29368C212yesmaximal, complemented
8#29221C211yesmaximal, characteristic, complemented
9#28755C211yesmaximal, characteristic, complemented
10#28516C211yesmaximal, characteristic, complemented
11#28438C211yesmaximal, characteristic, complemented
12#27902C212yesmaximal, complemented
13#25606C212yesmaximal
14#24163C214yesmaximal, complemented
15#22275C212yesmaximal, complemented
16#22231C212yesmaximal, complemented
17#20781C211yesmaximal, characteristic, complemented
18C2×#541C2212yes—
19C2×#540C2211yescharacteristic
20C2×#492C2212yes—
21C2×#491C2212yes—
22C2×#478C2214yes—
23C2×#461C2214yes—
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2642C2212yes—
29#2621C2212yes—
30#2506—22no—
31#2506—22no—
32#2496—22no—
33#2496—22no—
34#2374—22no—
35#2364—22no—
36#2363—22no—
37#2359—22no—
38#2330C2211yescharacteristic
39#2328C2211yescharacteristic
40#2327C2212yes—
41#2324C2212yes—
42#2314C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125C2211yescharacteristic
47C4×#121C2212yes—
48#2293C2212yes—
49#2275C2212yes—
50C4×#117C2211yescharacteristic
51#2100C2212yes—
52#2099C2212yes—
53#2094C2212yes—
54#2093C2212yes—
55#2043C2212yes—
56#2043C2212yes—
57#2036C2214yes—
58#2035C2214yes—
59#2013C2212yes—
60#2013C2212yes—
61#2000C2214yes—
62#2000C2214yes—
63#1991C2212yes—
64#1989C2214yes—
65#1981C2214yes—
66#1961C2212yes—
67#1960C2212yes—
68#1956C2212yes—
69#1928C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1927C2214yes—
73#1880C2214yes—
74#1880C2214yes—
75#1877C2214yes—
76#1877C2214yes—
77#1877C2214yes—
78#1877C2214yes—
79#1865C2214yes—
80#1863C2214yes—
81#1847C2212yes—
82#1839C2212yes—
83#1835C2212yes—
84#1832C2212yes—
85#1817C2212yes—
86#1798C2212yes—
87#1767C2211yescharacteristic
88C4×C4⋊C8C2211yescharacteristic
89#1750C2212yes—
90#1746C2212yes—
91C22×SD16C2314yes—
92C22×SD16C2314yes—
93#541—48no—
94#540—24no—
95#540—24no—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#126—22no—
102C2×#126—22no—
103C2×#125—22no—
104C2×#125—22no—
105C2×#125C2312yes—
106C2×#125C2311yescharacteristic
107C2×#123—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×#121—22no—
111C2×#121C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×#117—22no—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#492—28no—
124#491—28no—
125#478—48no—
126#461—48no—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#400—24no—
143#400—24no—
144#398—24no—
145#397—24no—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#390—22no—
151#390—22no—
152#390C2312yes—
153#390C2312yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#387—22no—
169#387—22no—
170#387—24no—
171#387—24no—
172#387—22no—
173#386—22no—
174#386—22no—
175#386—22no—
176#386—22no—
177#385—22no—
178#384—22no—
179#384—22no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#382C2311yescharacteristic
184#382C2312yes—
185#380—22no—
186#380—22no—
187#380C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203#376C2311yescharacteristic
204#376C2311yescharacteristic
205C2×#95—22no—
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217C22×Q8C2412yes—
218C22×Q8C2412yes—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2412yes—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4C2×D412yes—
226C23×C4C2411yescharacteristic
227C2×SD16—28no—
228C2×SD16—28no—
229C2×SD16—44no—
230C2×SD16—44no—
231C2×SD16—44no—
232C2×SD16—44no—
233C22×C8C2414yes—
234C22×C8C2414yes—
235C4⋊Q8—24no—
236C4⋊Q8—24no—
237C4⋊Q8—28no—
238C4⋊Q8—42no—
239C4⋊Q8—42no—
240#127—22no—
241#127#4114yes—
242#126—48no—
243#126—48no—
244#125—22no—
245#125—28no—
246#125—48no—
247#125—48no—
248#125#4114yes—
249#123—216no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252#121—216no—
253#121—24no—
254#121—24no—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—28no—
260C4×D4—44no—
261C4×D4—84no—
262C4×D4—84no—
263#117—22no—
264#117—22no—
265#117—24no—
266#117—84no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D411yescharacteristic
305C2×C22⋊C4C2×D411yescharacteristic
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—42no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—44no—
324#106—44no—
325C4⋊C8—24no—
326C4⋊C8—24no—
327#103—28no—
328#103—28no—
329#103—44no—
330#103—44no—
331#103—44no—
332#103—44no—
333#102—44no—
334#102—44no—
335C22⋊C8—44no—
336#97—24no—
337C4×C8—24no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D412yes—
354C24—22no—
355C24C22×D411yescharacteristic
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4C2×#4114yes—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C2511yescharacteristic
407SD16—88no—
408SD16—88no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4C2×#4114yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C42—24no—
456C42—24no—
457C42—21no—
458C42—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—44no—
466C42—44no—
467C42—82no—
468C42—82no—
469C42—82no—
470C42C2×#4112yes—
471C42C2×#4112yes—
472C42H(5,𝔽2)14yes—
473C23—22no—
474C23—21no—
475C23—24no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23C23×D411yescharacteristic
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8—44no—
490D4—28no—
491D4—44no—
492D4—88no—
493D4—88no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—84no—
535C2×C4—88no—
536C2×C4—164no—
537C2×C4—164no—
538C2×C4#52514yes—
539C2×C4#53512yes—
540C2×C4#53512yes—
541C2×C4#57714yes—
542C2×C4C22×#4111yescharacteristic
543C2×C4C22×#4111yescharacteristic
544C2×C4C2×H(5,𝔽2)12yes—
545C2×C4#58512yes—
546C8—44no—
547C8—44no—
548C22—42no—
549C22—41no—
550C22—44no—
551C22—41no—
552C22—41no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22C2×#52511yescharacteristic
561C22C2×#53511yescharacteristic
562C22#344911yescharacteristic
563C22C2×#57712yes—
564C22#358612yes—
565C4—24no—
566C4—22no—
567C4—22no—
568C4—24no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—44no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4#351412yes—
582C4#351712yes—
583C2—81no—
584C2—84no—
585C2#3519011yescharacteristic
586C2#6059112yes—
587C2#6082011yescharacteristic
588C2#6083111yescharacteristic
589C2#6098212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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
1. #35319C2semidirect
3. #33040C2semidirect
4. #30113C2semidirect
5. #30106C2semidirect
6. #29369C2semidirect
7. #29368C2semidirect
8. #29221C2semidirect
9. #28755C2semidirect
10. #28516C2semidirect
11. #28438C2semidirect
12. #27902C2semidirect
14. #24163C2semidirect
15. #22275C2semidirect
16. #22231C2semidirect
17. #20781C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#342802
#330402
#301132
#301062
#293692
#293682
#292212
#287552
#285162
#284382
#279022
#256062
#241632
#222752
#222312
#207812