Finite Group Databasebeta

← #6457915

Subgroups — #6457915

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35629C211yesmaximal, characteristic, complemented
2#35557C211yesmaximal, characteristic, complemented
3#28029C214yesmaximal, complemented
4#28028C214yesmaximal, complemented
5#28019C214yesmaximal, complemented
6#28018C214yesmaximal, complemented
7#23631C211yesmaximal, characteristic, complemented
8#23630C211yesmaximal, characteristic, complemented
9#23611C211yesmaximal, characteristic, complemented
10#23610C211yesmaximal, characteristic, complemented
11#21230C211yesmaximal, characteristic
12#21229C211yesmaximal, characteristic
13#21228C211yesmaximal, characteristic
14#21157C211yesmaximal, characteristic, complemented
15#21068C211yesmaximal, characteristic, complemented
16#21067C211yesmaximal, characteristic, complemented
17C2×#1847C211yesmaximal, characteristic, complemented
18#20455C211yesmaximal, characteristic, complemented
19#20436C211yesmaximal, characteristic, complemented
20C22×#135C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#405C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2514C2214yes—
37#2513C2214yes—
38#2512C2214yes—
39#2511C2214yes—
40#2510—24no—
41#2497—22no—
42#2496—22no—
43#2495—22no—
44#2494—22no—
45#2481—24no—
46#2316C2211yescharacteristic
47#2316C2211yescharacteristic
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2311C2211yescharacteristic
52#2297C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2274C2211yescharacteristic
58#2274C2211yescharacteristic
59#2074C2214yes—
60#2073C2214yes—
61#2072C2214yes—
62#2071C2214yes—
63#2068C2214yes—
64#2067C2214yes—
65#2066C2214yes—
66#2065C2214yes—
67#1973C2212yes—
68#1973C2212yes—
69#1973C2212yes—
70#1973C2212yes—
71#1972C2212yes—
72#1972C2212yes—
73#1971C2212yes—
74#1971C2212yes—
75#1970C2212yes—
76#1969C2212yes—
77#1968C2212yes—
78#1968C2212yes—
79#1944C2214yes—
80#1941C2214yes—
81#1928C2214yes—
82#1927C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1847C2214yes—
86#1847C2214yes—
87#1776C2214yes—
88C4×#107C2214yes—
89#1771C2214yes—
90#1770C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C22×#41C2311yescharacteristic
99C2×#135—24no—
100C2×#135—24no—
101C2×#135—24no—
102C22×Q16C2311yescharacteristic
103C22×SD16C2311yescharacteristic
104C22×SD16C2311yescharacteristic
105C22×D8C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C22×M16C2311yescharacteristic
108C23×C8C2311yescharacteristic
109C23×C8C2311yescharacteristic
110C2×#126—24no—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C4⋊C4—22no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#427—24no—
122C2×#107—24no—
123C2×#107—42no—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×#106—24no—
129C2×#106—42no—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134#424—24no—
135C2×C4⋊C8C2314yes—
136#417—42no—
137#417—42no—
138#417—42no—
139#417—42no—
140#416—24no—
141#416—24no—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154#405—24no—
155#405—24no—
156C2×#97—24no—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2312yes—
163#401—24no—
164#400—24no—
165#400—24no—
166#400—24no—
167#397—24no—
168#397—24no—
169#397—24no—
170#396—24no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#389—22no—
184#389—22no—
185#389D412yes—
186#389D412yes—
187#389C2314yes—
188#389C2314yes—
189#388—24no—
190#388—24no—
191#388—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#386—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#385D414yes—
202#384—21no—
203#384—21no—
204#384C2314yes—
205#383—24no—
206#382—21no—
207#382—21no—
208#382C2314yes—
209#380—22no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—22no—
214#380—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226C2×#95—22no—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C22×Q8C2411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259#135—416no—
260C2×Q16—22no—
261C2×Q16—22no—
262C2×Q16—22no—
263C2×SD16—22no—
264C2×SD16—22no—
265C2×SD16—22no—
266C2×SD16—22no—
267C2×SD16—22no—
268C2×SD16—22no—
269C2×D8—22no—
270C2×D8—22no—
271C2×D8—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8#4114yes—
281C22×C8#4114yes—
282C22×C8#4114yes—
283C22×C8#4114yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2411yescharacteristic
288C22×C8C2411yescharacteristic
289#126—88no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—84no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—24no—
349#107—24no—
350#107—24no—
351#107—44no—
352#106—24no—
353#106—24no—
354#106—24no—
355#106—44no—
356C4⋊C8—28no—
357#103—24no—
358#103—24no—
359#103—42no—
360#103—42no—
361#103—42no—
362#103—42no—
363#102—24no—
364#102—24no—
365#102—42no—
366#102—42no—
367#102—42no—
368#102—42no—
369#97—24no—
370#97—24no—
371#97—24no—
372#97—24no—
373C4×C8—24no—
374C4×C8—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—21no—
392C24—21no—
393C24C22×D411yescharacteristic
394#41—44no—
395#41—44no—
396#41—44no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401#41—44no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469Q16—44no—
470SD16—44no—
471SD16—44no—
472D8—44no—
473M16—44no—
474M16—44no—
475C2×C8—28no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8#126116yescomplemented
485C2×C8C2×#4114yes—
486C2×C8C2×#4112yes—
487C2×C8C2×#4112yes—
488C2×C8C2×#4112yes—
489C2×C8C2×#4112yes—
490C2×C8C2×#4112yes—
491C2×C8C2×#4112yes—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—84no—
523C22⋊C4—82no—
524C22⋊C4—84no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—42no—
537C42—42no—
538C42—84no—
539C42—84no—
540C23—22no—
541C23—22no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23#53212yes—
553C23#54712yes—
554C23#55012yes—
555C23C23×D411yescharacteristic
556Q8—41no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—84no—
573D4—84no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—162no—
610C2×C4—162no—
611C2×C4—162no—
612C2×C4—162no—
613C2×C4C2×#12614yes—
614C2×C4#53212yes—
615C2×C4#54214yes—
616C2×C4#54712yes—
617C2×C4#55012yes—
618C2×C4#55314yes—
619C2×C4#55614yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C8—24no—
627C8—42no—
628C8—42no—
629C8C2×#12614yes—
630C8C2×#12614yes—
631C22—24no—
632C22—24no—
633C22—22no—
634C22—22no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22#251214yes—
646C22C2×#53211yescharacteristic
647C22C2×#54711yescharacteristic
648C22C2×#55011yescharacteristic
649C22#358512yes—
650C22#358612yes—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#251214yes—
664C4C22×#12611yescharacteristic
665C4C2×#54211yescharacteristic
666C4C2×#55311yescharacteristic
667C4C2×#55611yescharacteristic
668C2—24no—
669C2—161no—
670C2—161no—
671C2C2×#251211yescharacteristic
672C2#6072112yes—
673C2#6126612yes—
674C2#6141711yescharacteristic
675C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#135C22
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
1. #35629C2semidirect
2. #35557C2semidirect
3. #28029C2semidirect
4. #28028C2semidirect
5. #28019C2semidirect
6. #28018C2semidirect
7. #23631C2semidirect
8. #23630C2semidirect
9. #23611C2semidirect
10. #23610C2semidirect
14. #21157C2semidirect
15. #21068C2semidirect
16. #21067C2semidirect
17. C2×#1847C2semidirect
18. #20455C2semidirect
19. #20436C2semidirect
484. C2×C8#126semidirect

Maximal subgroups

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

subgroupindex
#356292
#355572
#280292
#280282
#280192
#280182
#236312
#236302
#236112
#236102
#212302
#212292
#212282
#211572
#210682
#210672
C2×#18472
#204552
#204362