Finite Group Databasebeta

← #7196659

Subgroups — #7196659

Order 512.

2094 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3335C212yesmaximal, complemented
2#59823C211yesmaximal, characteristic, complemented
3#59815C211yesmaximal, characteristic
4#30705C214yesmaximal, complemented
5#30703C214yesmaximal, complemented
6#28205C211yesmaximal, characteristic, complemented
7#28202C211yesmaximal, characteristic, complemented
8#28141C212yesmaximal, complemented
9#26130C214yesmaximal, complemented
10#26128C214yesmaximal, complemented
11#24120C212yesmaximal, complemented
12#24118C212yesmaximal, complemented
13#23799C211yesmaximal, characteristic, complemented
14#22260C211yesmaximal, characteristic, complemented
15#22246C211yesmaximal, characteristic, complemented
16C2×#554C2212yes—
17C2×#550—21no—
18C2×#547C2212yes—
19#3335—216no—
20C2×#498C2211yescharacteristic
21C2×#481C2214yes—
22C2×#483C2214yes—
23C2×#475C2212yes—
24C2×#476C2212yes—
25C2×#465C2214yes—
26C2×C4×SD16C2212yes—
27C2×C4×SD16C2212yes—
28C2×C8×D4C2212yes—
29C22×#106C2214yes—
30#2732—22no—
31#2682—22no—
32#2552—21no—
33#2542C2212yes—
34#2541C2212yes—
35#2538—22no—
36#2538—21no—
37#2538C2212yes—
38#2537C2212yes—
39#2454—22no—
40#2445—22no—
41#2417C2214yes—
42#2416C2214yes—
43#2407—22no—
44#2366—21no—
45D4×C4⋊C4C2212yes—
46D4×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×#127C2211yescharacteristic
52#2096C2212yes—
53#2094C2212yes—
54#2084C2212yes—
55#2084C2212yes—
56#2077C2214yes—
57#2043C2212yes—
58#2041C2212yes—
59#2035C2214yes—
60#2034C2214yes—
61#2014C2212yes—
62#2013C2212yes—
63#2000C2214yes—
64#2000C2214yes—
65#2000C2214yes—
66#1990C2211yescharacteristic
67#1990C2211yescharacteristic
68#1954C2212yes—
69#1927C2214yes—
70#1926C2214yes—
71#1920C2214yes—
72#1919C2214yes—
73#1893C2214yes—
74#1892C2214yes—
75#1874C2214yes—
76#1872C2214yes—
77#1869C2214yes—
78#1869C2214yes—
79#1853C2214yes—
80#1853C2214yes—
81#1834C2211yescharacteristic
82#1831C2211yescharacteristic
83#1803C2212yes—
84#1802C2212yes—
85#1744C2211yescharacteristic
86C22×#41—22no—
87C22×SD16C2312yes—
88C22×SD16C2312yes—
89C23×C8C2314yes—
90#554—28no—
91#554—44no—
92#550—48no—
93#547—28no—
94#547—44no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127—22no—
99C2×#127D411yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#125—22no—
102C2×#123—24no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2314yes—
107C2×#121—22no—
108C2×#121—24no—
109C2×#121C2314yes—
110C2×#121C2314yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2314yes—
125#498D418yescomplemented
126#483—216no—
127#481—216no—
128#476—28no—
129#475—28no—
130#465—216no—
131C4×SD16—28no—
132C4×SD16—44no—
133C8×D4—28no—
134C2×#106—216no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154#399—24no—
155#398—24no—
156#396—24no—
157#396—24no—
158#396—24no—
159#394—24no—
160#394—24no—
161#392—24no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—21no—
169#390—21no—
170#390D412yes—
171#390D412yes—
172#390D412yes—
173#390C2312yes—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2312yes—
183#386—24no—
184#386—24no—
185#385—24no—
186#385—24no—
187#384—22no—
188#384D411yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#380—22no—
194#380—24no—
195#380C2314yes—
196#380C2314yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2312yes—
212#336C2312yes—
213C2×#41—48no—
214C2×#41—48no—
215C2×#41—84no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8C2412yes—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2×D412yes—
224C22×D4C2×D412yes—
225C22×D4C2×D412yes—
226C22×D4C2412yes—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4C2414yes—
231C2×SD16—24no—
232C2×SD16—42no—
233C2×SD16—42no—
234C2×SD16—42no—
235C2×SD16—42no—
236C2×SD16—42no—
237C22×C8—28no—
238C22×C8—44no—
239C22×C8—44no—
240C22×C8C2414yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—42no—
246C4⋊Q8C2×D414yes—
247C4⋊Q8C2×D414yes—
248#127—42no—
249#127—48no—
250#127SD1618yescomplemented
251#125—48no—
252#123—416no—
253C22⋊Q8—28no—
254C22⋊Q8—416no—
255C22⋊Q8—44no—
256C22⋊Q8—48no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259#121—28no—
260#121—44no—
261#121—48no—
262#121—416no—
263#121—44no—
264#121—44no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—44no—
268C4×Q8—82no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—48no—
276C4×D4—48no—
277C4×D4—44no—
278C4×D4—82no—
279C4×D4—84no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#106—24no—
335#106—28no—
336#106—28no—
337#106—42no—
338#106C2×D4116yes—
339C4⋊C8—24no—
340#103—24no—
341#103—28no—
342#103—44no—
343#103—42no—
344#102—24no—
345#102—28no—
346#102—44no—
347#102—42no—
348C22⋊C8—28no—
349C22⋊C8—44no—
350C22⋊C8—44no—
351C4×C8—22no—
352C4×C8C2×D414yes—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361C24—22no—
362C24C22×D414yes—
363#41—832no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—81no—
375C2×Q8—81no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—48no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—42no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4C2×SD1618yes—
401C2×D4C2×SD1618yes—
402C2×D4C2×SD1618yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D414yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433SD16—48no—
434SD16—48no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—416no—
444C2×C8C22×D414yes—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4C22×D414yes—
476C4⋊C4C22×D414yes—
477C22⋊C4—28no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—44no—
495C42—44no—
496C42—44no—
497C42C2×SD1614yes—
498C42C22×D412yes—
499C42H(5,𝔽2)12yes—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23C22×SD1618yes—
514C23C23×D411yescharacteristic
515Q8—48no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—84no—
521D4—216no—
522D4—48no—
523D4—48no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—84no—
529D4—816no—
530D4—84no—
531D4—168no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—816no—
573C2×C4—88no—
574C2×C4—168no—
575C2×C4D4212yes—
576C2×C4D4212yes—
577C2×C4C22×SD1612yes—
578C2×C4C22×SD1612yes—
579C2×C4C22×SD1612yes—
580C2×C4#57712yes—
581C2×C4C23×D411yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4#58512yes—
584C2×C4#58512yes—
585C8—24no—
586C8—44no—
587C8—42no—
588C22—28no—
589C22—28no—
590C22—28no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22C2×D4211yescharacteristic
603C22#348511yescharacteristic
604C22#348511yescharacteristic
605C22C23×SD1611yescharacteristic
606C22C2×#57711yescharacteristic
607C22#358511yescharacteristic
608C22#358611yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—44no—
613C4—41no—
614C4—44no—
615C4—41no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4D4×SD1614yes—
623C4#348712yes—
624C4#353912yes—
625C2—48no—
626C2—162no—
627C2#4538711yescharacteristic
628C2C2×D4×SD1611yescharacteristic
629C2#6108011yescharacteristic
630C2#6169511yescharacteristic
631C2#6169711yescharacteristic
632C2#6180811yescharacteristic
633C2#6180911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3335C2semidirect
2. #59823C2semidirect
4. #30705C2semidirect
5. #30703C2semidirect
6. #28205C2semidirect
7. #28202C2semidirect
8. #28141C2semidirect
9. #26130C2semidirect
10. #26128C2semidirect
11. #24120C2semidirect
12. #24118C2semidirect
13. #23799C2semidirect
14. #22260C2semidirect
15. #22246C2semidirect
125. #498D4semidirect
250. #127SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#33352
#598232
#598152
#307052
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#282052
#282022
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#261302
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#237992
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#222462