Finite Group Databasebeta

← #6451266

Subgroups — #6451266

Order 512.

1669 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35557C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic, complemented
3#27510C214yesmaximal, complemented
4#27505C214yesmaximal, complemented
5#27501C214yesmaximal, complemented
6#27490C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8C2×#2096C211yesmaximal, characteristic, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23547C211yesmaximal, characteristic
11#21234C211yesmaximal, characteristic
12#21171C211yesmaximal, characteristic, complemented
13#21068C211yesmaximal, characteristic
14#21064C211yesmaximal, characteristic, complemented
15#21058C211yesmaximal, characteristic
16#21027C211yesmaximal, characteristic, complemented
17#20930C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20454C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C2×#405C2211yescharacteristic
35#2512C2214yes—
36#2511C2214yes—
37#2497—22no—
38#2496—22no—
39#2481—24no—
40#2356C2214yes—
41#2355C2214yes—
42#2343—24no—
43#2320—22no—
44#2317C2211yescharacteristic
45#2316C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2211yescharacteristic
49#2297C2211yescharacteristic
50#2296C2211yescharacteristic
51#2295—22no—
52C2×#391C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2274C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yes—
58#2096C2214yes—
59#2096C2212yes—
60#2096C2214yes—
61#2094C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2084C2214yes—
65#2056C2214yes—
66#2055C2214yes—
67#1973C2212yes—
68#1971C2212yes—
69#1969C2212yes—
70#1968C2212yes—
71#1948C2214yes—
72#1947C2214yes—
73#1928C2212yes—
74#1928C2214yes—
75#1927C2212yes—
76#1927C2214yes—
77#1927C2214yes—
78#1926C2212yes—
79#1926C2214yes—
80#1925C2212yes—
81#1917C2214yes—
82#1916C2214yes—
83#1880C2214yes—
84#1878C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1776C2214yes—
88C4×#106C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#137—24no—
100C2×#137—24no—
101C2×#137—42no—
102C22×Q16C2312yes—
103C22×SD16C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C2×#126—24no—
109#519—24no—
110#519—24no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2314yes—
113C2×#117—22no—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#428—42no—
125#428—42no—
126#427—24no—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106—24no—
131C2×#106—42no—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138#423—24no—
139C2×C4⋊C8C2314yes—
140#419—24no—
141#417—42no—
142#416—24no—
143#416—24no—
144C2×#103—24no—
145C2×#103—42no—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2311yescharacteristic
150C2×#103C2311yescharacteristic
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#102C2311yescharacteristic
155C2×#102C2311yescharacteristic
156C2×#102C2312yes—
157C2×#102C2312yes—
158#405—24no—
159C2×#97C2312yes—
160C2×C4×C8C2312yes—
161#400—24no—
162#400—24no—
163#397—24no—
164#396—24no—
165#391—22no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#391C2314yes—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389—22no—
179#389D412yes—
180#389D412yes—
181#389C2314yes—
182#388—24no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—24no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385D414yes—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2314yes—
198#383—24no—
199#382—24no—
200#382C2312yes—
201#381—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218#376—21no—
219#376—24no—
220C2×#95—22no—
221C43—21no—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8#4114yes—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257#137—416no—
258C2×Q16—24no—
259C2×Q16—24no—
260C2×Q16—42no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16#4114yes—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C22×C8C2412yes—
281#126—88no—
282C4×Q8—216no—
283C4×Q8—28no—
284C4×Q8—28no—
285#117—216no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117#4114yes—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—24no—
347#107—42no—
348#107—42no—
349#106—24no—
350#106—24no—
351#106—24no—
352#106—44no—
353#106—42no—
354#106—42no—
355C4⋊C8—28no—
356#103—28no—
357#103—24no—
358#103—24no—
359#103—22no—
360#103—22no—
361#103—24no—
362#103—44no—
363#103—42no—
364#102—22no—
365#102—22no—
366#102—24no—
367#102—42no—
368#97—24no—
369C4×C8—24no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95C2×D414yes—
385#95C2×D414yes—
386C24—21no—
387C24C22×D411yescharacteristic
388#41—44no—
389#41—44no—
390#41—44no—
391#41—44no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8C2×#4114yes—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456Q16—48no—
457SD16—48no—
458M16—28no—
459M16—44no—
460C2×C8—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—44no—
474C2×C8C2×#4114yes—
475C2×C8C2×#4112yes—
476C2×C8C2×#4112yes—
477C4⋊C4—216no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—48no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4C2×#4114yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—216no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—82no—
526C42—84no—
527C42—84no—
528C23—22no—
529C23—22no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23#52512yes—
536C23#54712yes—
537C23#55012yes—
538C23C23×D411yescharacteristic
539Q8—24no—
540Q8—24no—
541Q8—42no—
542Q8—42no—
543Q8—48no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—84no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—84no—
589C2×C4—162no—
590C2×C4—162no—
591C2×C4—162no—
592C2×C4#52512yes—
593C2×C4#52914yes—
594C2×C4#54214yes—
595C2×C4#54712yes—
596C2×C4#55012yes—
597C2×C4#55114yes—
598C2×C4#55214yes—
599C2×C4#57714yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C8—22no—
607C8—22no—
608C8—24no—
609C8—42no—
610C8—42no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—22no—
615C22—82no—
616C22—81no—
617C22—82no—
618C22—81no—
619C22—81no—
620C22#247914yes—
621C22C2×#52511yescharacteristic
622C22C2×#54711yescharacteristic
623C22C2×#55011yescharacteristic
624C22C2×#57712yes—
625C22#358612yes—
626C4—24no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—44no—
632C4—81no—
633C4—82no—
634C4—81no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4#247914yes—
640C4C2×#52911yescharacteristic
641C4C2×#54211yescharacteristic
642C4C2×#55111yescharacteristic
643C4C2×#55211yescharacteristic
644C2—24no—
645C2—161no—
646C2C2×#247911yescharacteristic
647C2#6059912yes—
648C2#6127112yes—
649C2#6141011yescharacteristic
650C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34254C2semidirect
3. #27510C2semidirect
4. #27505C2semidirect
5. #27501C2semidirect
6. #27490C2semidirect
8. C2×#2096C2semidirect
9. #23661C2semidirect
12. #21171C2semidirect
14. #21064C2semidirect
16. #21027C2semidirect
18. #20880C2semidirect
19. #20454C2semidirect

Maximal subgroups

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

subgroupindex
#355572
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#275102
#275052
#275012
#274902
#237382
C2×#20962
#236612
#235472
#212342
#211712
#210682
#210642
#210582
#210272
#209302
#208802
#204542