Finite Group Databasebeta

← #6495161

Subgroups — #6495161

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35343C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic, complemented
3#33297C211yesmaximal, characteristic, complemented
4#33295C211yesmaximal, characteristic, complemented
5#33238C211yesmaximal, characteristic, complemented
6#33234C211yesmaximal, characteristic, complemented
7#30217C212yesmaximal, complemented
8#30216C212yesmaximal, complemented
9#29151C214yesmaximal, complemented
10#28729C212yesmaximal, complemented
11#28728C212yesmaximal, complemented
12#27984C214yesmaximal, complemented
13#25915C212yesmaximal
14#25914C212yesmaximal, complemented
15#21868C212yesmaximal, complemented
16#21867C212yesmaximal, complemented
17#20467C211yesmaximal, characteristic, complemented
18C2×#544C2212yes—
19C2×#529C2212yes—
20C2×#491C2212yes—
21C2×#490C2212yes—
22C2×#487C2212yes—
23C2×#486C2212yes—
24C2×#432C2212yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2861C2212yes—
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2660C2212yes—
33#2660C2212yes—
34#2621C2211yescharacteristic
35#2620C2212yes—
36#2619C2211yescharacteristic
37#2602—22no—
38#2602—22no—
39#2601—22no—
40#2601—22no—
41#2552—22no—
42#2551—22no—
43#2440—22no—
44#2440—22no—
45#2328C2211yescharacteristic
46#2327C2211yescharacteristic
47#2326C2212yes—
48#2317C2212yes—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2074C2214yes—
53#2073C2214yes—
54#2072C2214yes—
55#2071C2214yes—
56#2026C2214yes—
57#2025C2214yes—
58#2024C2214yes—
59#2023C2214yes—
60#1993C2212yes—
61#1993C2212yes—
62#1992C2212yes—
63#1992C2212yes—
64#1992C2212yes—
65#1992C2212yes—
66#1991C2212yes—
67#1991C2212yes—
68#1959C2212yes—
69#1959C2212yes—
70#1959C2212yes—
71#1959C2212yes—
72#1959C2214yes—
73#1958C2214yes—
74#1875C2214yes—
75#1874C2214yes—
76#1873C2214yes—
77#1872C2214yes—
78#1836C2214yes—
79#1834C2214yes—
80#1805C2214yes—
81#1804C2214yes—
82#1803C2214yes—
83#1802C2214yes—
84#1768C2212yes—
85#1768C2212yes—
86#1750C2212yes—
87#1750C2212yes—
88#1745C2212yes—
89C4×#97C2211yescharacteristic
90C4×#97C2211yescharacteristic
91#544—44no—
92#544—44no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×#127C2311yescharacteristic
100#529—216no—
101C2×#126—22no—
102C2×#125—22no—
103C2×#125C2312yes—
104C2×#125C2312yes—
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×#124C2312yes—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×#121C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123#491—44no—
124#490—44no—
125#487—44no—
126#486—44no—
127#432—28no—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2312yes—
139C2×#97C2312yes—
140C2×#97C2312yes—
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#398—21no—
145#398—21no—
146#398—24no—
147#398—24no—
148#397—21no—
149#397—21no—
150#396—22no—
151#396—22no—
152#395—24no—
153#395—24no—
154#394—21no—
155#394—21no—
156#394—22no—
157#394—22no—
158#392—21no—
159#392—21no—
160#391C2312yes—
161#391C2312yes—
162#390—22no—
163#390—22no—
164#390C2312yes—
165#390C2312yes—
166#389—24no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#386—22no—
182#386—22no—
183#385—24no—
184#385—22no—
185#385—22no—
186#384—22no—
187#384—22no—
188#384—22no—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#382C2312yes—
198#382C2312yes—
199#382C2312yes—
200#382C2311yescharacteristic
201#382C2311yescharacteristic
202#380—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4D412yes—
208#376C2311yescharacteristic
209#376C2311yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8C2412yes—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C2412yes—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4C2×D412yes—
232C23×C4C2411yescharacteristic
233C22×C8C2414yes—
234C22×C8C2414yes—
235C4⋊Q8—24no—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—42no—
239C4⋊Q8—42no—
240C4⋊Q8—48no—
241C4⋊Q8—84no—
242C4⋊Q8—84no—
243#127—24no—
244#126—48no—
245#125—28no—
246#125—24no—
247#125—24no—
248#125—42no—
249#125—42no—
250#125—48no—
251#124—28no—
252#124—28no—
253#123—48no—
254C22⋊Q8—28no—
255C22⋊Q8—48no—
256C22⋊Q8—48no—
257#121—28no—
258C4×Q8—28no—
259C4×D4—28no—
260C4×D4—48no—
261#117—24no—
262#117—24no—
263#117—42no—
264#117—42no—
265#117—48no—
266C2×C4⋊C4—22no—
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—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—84no—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2412yes—
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⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4C2×D411yescharacteristic
313C2×C22⋊C4C2×D411yescharacteristic
314C2×C22⋊C4C2×D412yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42#4118yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C4⋊C8—28no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—44no—
334#102—44no—
335#102—44no—
336#102—44no—
337#102—44no—
338C22⋊C8—28no—
339#97—24no—
340#97—24no—
341#97—24no—
342#97—24no—
343C4×C8—24no—
344C4×C8—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24—22no—
361C24C22×D411yescharacteristic
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—84no—
395C22×C4#12418yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C2×#4114yes—
406C22×C4C2511yescharacteristic
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C42—22no—
464C42—22no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—44no—
470C42—44no—
471C42#12418yes—
472C42C2×#4114yes—
473C42C2×#4114yes—
474C42H(5,𝔽2)14yes—
475C422−1+414yes—
476C23—22no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23C2×#12412yes—
488C23C23×D411yescharacteristic
489Q8—48no—
490Q8—164no—
491Q8—164no—
492D4—48no—
493D4—88no—
494D4—162no—
495D4—162no—
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—21no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
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—88no—
533C2×C4—164no—
534C2×C4C2×#12414yes—
535C2×C4C2×#12412yes—
536C2×C4C2×#12414yes—
537C2×C4#53512yes—
538C2×C4#53512yes—
539C2×C4#53614yes—
540C2×C4C22×#4112yes—
541C2×C4C2×H(5,𝔽2)12yes—
542C2×C4C2×2−1+412yes—
543C8—44no—
544C8—44no—
545C22—24no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22C22×#12411yescharacteristic
558C22C2×#53511yescharacteristic
559C22C2×#53611yescharacteristic
560C22#358512yes—
561C22#358612yes—
562C4—22no—
563C4—22no—
564C4—24no—
565C4—24no—
566C4—24no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#351114yes—
579C2—42no—
580C2—162no—
581C2#3528911yescharacteristic
582C2#6062112yes—
583C2#6082711yescharacteristic
584C2#6082811yescharacteristic
585C2#6088712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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
2. #35275C2semidirect
3. #33297C2semidirect
4. #33295C2semidirect
5. #33238C2semidirect
6. #33234C2semidirect
7. #30217C2semidirect
8. #30216C2semidirect
9. #29151C2semidirect
10. #28729C2semidirect
11. #28728C2semidirect
12. #27984C2semidirect
14. #25914C2semidirect
15. #21868C2semidirect
16. #21867C2semidirect
17. #20467C2semidirect

Maximal subgroups

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

subgroupindex
#353432
#352752
#332972
#332952
#332382
#332342
#302172
#302162
#291512
#287292
#287282
#279842
#259152
#259142
#218682
#218672
#204672