Finite Group Databasebeta

← #6987149

Subgroups — #6987149

Order 512.

1393 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36460C211yesmaximal, characteristic
2#35286C211yesmaximal, characteristic, complemented
3#32841C212yesmaximal, complemented
4#31126C214yesmaximal, complemented
5#31088C214yesmaximal, complemented
6#30379C212yesmaximal, complemented
7#30082C212yesmaximal, complemented
8#29920C211yesmaximal, characteristic
9#28830C212yesmaximal, complemented
10#28513C212yesmaximal, complemented
11#27906C211yesmaximal, characteristic
12#27750C212yesmaximal, complemented
13#27681C211yesmaximal, characteristic, complemented
14#26463C212yesmaximal, complemented
15#26043C212yesmaximal, complemented
16#23756C211yesmaximal, characteristic, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#481C2214yes—
19C2×#479C2214yes—
20C2×#478C2214yes—
21#2860C2212yes—
22#2859C2212yes—
23#2859C2212yes—
24#2856C2211yescharacteristic
25#2647—22no—
26#2571—21no—
27#2571C2212yes—
28#2571C2211yescharacteristic
29#2559—22no—
30#2552—21no—
31#2552C2212yes—
32#2548C2211yescharacteristic
33#2526—22no—
34#2512—21no—
35#2500—21no—
36#2442—22no—
37#2442C2214yes—
38#2426C2212yes—
39#2420C2212yes—
40#2366C2211yescharacteristic
41#2363—21no—
42#2357—21no—
43Q8×C4⋊C4C2211yescharacteristic
44#2317—21no—
45#2316—21no—
46C4×#124C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2100C2212yes—
52#2099C2211yescharacteristic
53#2096C2211yescharacteristic
54#2095C2212yes—
55#2094C2212yes—
56#2094C2212yes—
57#2093C2212yes—
58#2088C2214yes—
59#2085C2214yes—
60#2085C2214yes—
61#2084C2211yescharacteristic
62#2077C2212yes—
63#2073C2214yes—
64#2072C2211yescharacteristic
65#2071C2212yes—
66#2065C2214yes—
67#2059C2214yes—
68#2042C2214yes—
69#2035C2212yes—
70#2033C2214yes—
71#2025C2214yes—
72#1982C2212yes—
73#1958C2214yes—
74#1957C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1941C2214yes—
78#1939C2212yes—
79#1934C2214yes—
80#1923C2214yes—
81#1872C2212yes—
82#1865C2214yes—
83#1862C2214yes—
84#1852C2212yes—
85#1846C2214yes—
86#1844C2212yes—
87#1839C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#126—22no—
97C2×#125—22no—
98C2×#125C2314yes—
99C2×#124C2312yes—
100C2×C22⋊Q8C2311yescharacteristic
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117D411yescharacteristic
110C2×#117D411yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115#481—48no—
116#479—48no—
117#478—48no—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106C2312yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#106C2312yes—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2314yes—
137#400—24no—
138#400—24no—
139#400—22no—
140#398—24no—
141#398—24no—
142#397—22no—
143#396—22no—
144#394—22no—
145#394—22no—
146#394—22no—
147#391C2311yescharacteristic
148#390—21no—
149#390—21no—
150#390—21no—
151#390—22no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2314yes—
171#389C2311yescharacteristic
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#387—21no—
179#386—22no—
180#386—22no—
181#385—21no—
182#385—24no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384—21no—
187#384C2312yes—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#382—22no—
193#382—21no—
194#382—22no—
195#382—21no—
196#382C2314yes—
197#382C2312yes—
198#381—22no—
199#380—21no—
200#380—24no—
201#380C2311yescharacteristic
202#380C2312yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217#376D412yes—
218C43C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#336C2312yes—
227#336C2314yes—
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2311yescharacteristic
231C22×Q8C2411yescharacteristic
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244#127—24no—
245#126—84no—
246#125—44no—
247#125—44no—
248#125—84no—
249#124—24no—
250#124—42no—
251C22⋊Q8—22no—
252C22⋊Q8—41no—
253#121—44no—
254#121—44no—
255#121—82no—
256C4×Q8—24no—
257C4×Q8—44no—
258C4×D4—44no—
259C4×D4—44no—
260#117—24no—
261#117—42no—
262#117—44no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2412yes—
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107—44no—
329#107—44no—
330#106—24no—
331#106—28no—
332#106—42no—
333#106—44no—
334C4⋊C8—28no—
335C4⋊C8—24no—
336#103—24no—
337#103—24no—
338#103—44no—
339#102—28no—
340#102—44no—
341#102—44no—
342#102—44no—
343C22⋊C8—24no—
344C22⋊C8—42no—
345C4×C8—24no—
346C4×C8—44no—
347#95—21no—
348#95—21no—
349#95—22no—
350#95—22no—
351#95—21no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—22no—
367C24C22×D411yescharacteristic
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q82−1+412yes—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4—82no—
384C2×D4—82no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4#13514yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2511yescharacteristic
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8H(5,𝔽2)14yes—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—81no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—42no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—81no—
479C42—22no—
480C42—22no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—44no—
491C42—42no—
492C42—82no—
493C422−1+414yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C2×#13512yes—
508C23C23×D411yescharacteristic
509Q8—24no—
510Q8—42no—
511D4—44no—
512D4—44no—
513D4—84no—
514D4—162no—
515D4—164no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—164no—
566C2×C4—164no—
567C2×C4#53612yes—
568C2×C4#53612yes—
569C2×C4C2×#13512yes—
570C2×C4C2×#13512yes—
571C2×C4#57512yes—
572C2×C4#57712yes—
573C2×C4C2×H(5,𝔽2)12yes—
574C2×C4C2×2−1+411yescharacteristic
575C2×C4C2×2−1+412yes—
576C2×C4#58511yescharacteristic
577C8—24no—
578C8—42no—
579C8—44no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22C2×#53611yescharacteristic
595C22#348911yescharacteristic
596C22#349811yescharacteristic
597C22C22×#13511yescharacteristic
598C22C2×#57511yescharacteristic
599C22C2×#57711yescharacteristic
600C22#358511yescharacteristic
601C4—22no—
602C4—22no—
603C4—22no—
604C4—42no—
605C4—41no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4—81no—
612C4—81no—
613C4—82no—
614C4—82no—
615C4—84no—
616C4—81no—
617C4#323214yes—
618C4#355212yes—
619C4#355912yes—
620C2—42no—
621C2—162no—
622C2#4291411yescharacteristic
623C2C2×#323211yescharacteristic
624C2#6087311yescharacteristic
625C2#6210211yescharacteristic
626C2#6213511yescharacteristic
627C2#6241411yescharacteristic
628C2#6245511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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. #35286C2semidirect
3. #32841C2semidirect
4. #31126C2semidirect
5. #31088C2semidirect
6. #30379C2semidirect
7. #30082C2semidirect
9. #28830C2semidirect
10. #28513C2semidirect
12. #27750C2semidirect
13. #27681C2semidirect
14. #26463C2semidirect
15. #26043C2semidirect
16. #23756C2semidirect
17. #22639C2semidirect

Maximal subgroups

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

subgroupindex
#364602
#352862
#328412
#311262
#310882
#303792
#300822
#299202
#288302
#285132
#279062
#277502
#276812
#264632
#260432
#237562
#226392