Finite Group Databasebeta

← #6470784

Subgroups — #6470784

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3202C212yesmaximal
2#37386C211yesmaximal, characteristic
3#34276C211yesmaximal, characteristic
4#33114C212yesmaximal, complemented
5#32915C212yesmaximal, complemented
6#29766C212yesmaximal, complemented
7#29715C212yesmaximal, complemented
8#25226C214yesmaximal, complemented
9#25215C214yesmaximal, complemented
10#25095C214yesmaximal, complemented
11#25082C214yesmaximal, complemented
12#21936C211yesmaximal, characteristic
13#20913C212yesmaximal
14C2×#536C2212yes—
15C2×#536C2211yescharacteristic
16C2×#520C2211yescharacteristic
17C2×#519C2211yescharacteristic
18#3202—216no—
19C2×#485C2214yes—
20C2×#485C2214yes—
21C2×#470C2214yes—
22C2×#467C2214yes—
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×#451C2214yes—
26C2×#451C2214yes—
27C2×#419C2212yes—
28C22×#103C2212yes—
29C2×#408C2212yes—
30#2857C2212yes—
31#2734—21no—
32#2734—21no—
33#2724—21no—
34#2724—21no—
35#2717—24no—
36#2621C2212yes—
37#2499C2214yes—
38#2434C2214yes—
39#2362C2214yes—
40#2360C2214yes—
41#2359—24no—
42#2327—21no—
43#2326—21no—
44#2317C2211yescharacteristic
45C4×C4⋊Q8—21no—
46C4×#124—21no—
47#2297C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2085C2214yes—
51#2072C2214yes—
52#2068C2214yes—
53#2066C2214yes—
54#2056C2214yes—
55#2056C2214yes—
56#2043C2214yes—
57#2016C2214yes—
58#1981C2212yes—
59#1981C2212yes—
60#1948C2212yes—
61#1947C2212yes—
62#1928C2214yes—
63#1927C2214yes—
64#1878C2214yes—
65#1878C2214yes—
66#1873C2214yes—
67#1873C2214yes—
68#1872C2214yes—
69#1872C2214yes—
70#1851C2214yes—
71#1850C2214yes—
72#1805C2212yes—
73#1805C2212yes—
74#1798C2212yes—
75#1797C2212yes—
76#1736C2212yes—
77#1735C2212yes—
78C22×#41C2311yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2311yescharacteristic
81C22×Q16C2314yes—
82C22×Q16C2314yes—
83C22×M16C2312yes—
84C23×C8C2312yes—
85#536—24no—
86#536—28no—
87#536—28no—
88#536D418yes—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×#124—21no—
94C2×#124—22no—
95C2×#123—22no—
96C2×#123—22no—
97C2×#123—22no—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2314yes—
101#520—42no—
102#520—42no—
103#519—42no—
104#519—42no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×#117—22no—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#485—216no—
122#485—216no—
123#470—216no—
124#467—216no—
125#462—44no—
126#462—44no—
127#451—216no—
128#451—216no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135#419—44no—
136C2×#103—44no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145#408—28no—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148#398—22no—
149#398—22no—
150#397—22no—
151#397—22no—
152#397—24no—
153#397—24no—
154#395—24no—
155#395—24no—
156#391—21no—
157#391—21no—
158#391D411yescharacteristic
159#391D411yescharacteristic
160#391C2314yes—
161#391C2314yes—
162#391C2314yes—
163#390D411yescharacteristic
164#390D411yescharacteristic
165#389—22no—
166#389—22no—
167#389—24no—
168#389C2314yes—
169#388—24no—
170#388—24no—
171#387—24no—
172#387—24no—
173#387—24no—
174#387—24no—
175#386D414yes—
176#385—22no—
177#385—24no—
178#385—24no—
179#385—24no—
180#384—21no—
181#384C2312yes—
182#382—21no—
183#382C2312yes—
184#380—22no—
185#380—22no—
186C4×C4⋊C4—24no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—24no—
191C4×C22⋊C4C2314yes—
192#376—21no—
193C2×#95—22no—
194C43—21no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C2×#41—41no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—41no—
205C2×#41C2×D414yes—
206C2×#41#4114yes—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—41no—
213C22×Q8—41no—
214C22×Q8—41no—
215C22×Q8—41no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8C2×D414yes—
221C22×Q8C2×D411yescharacteristic
222C22×Q8C2414yes—
223C22×Q8C2412yes—
224C22×D4C2×D412yes—
225C22×D4C2×D411yescharacteristic
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4C2×D412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C2×Q16—28no—
234C2×Q16—28no—
235C2×Q16—44no—
236C2×Q16—44no—
237C2×Q16—44no—
238C2×Q16—44no—
239C2×M16—24no—
240C2×M16—42no—
241C2×M16—42no—
242C22×C8—24no—
243C22×C8—42no—
244C22×C8—42no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247C4⋊Q8—28no—
248C4⋊Q8—42no—
249C4⋊Q8—42no—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—48no—
253#124—28no—
254#124—48no—
255#123—216no—
256#123—48no—
257#123—48no—
258C22⋊Q8—28no—
259C22⋊Q8—28no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8C2×D4116yes—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—42no—
267C4×Q8—42no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×D4—44no—
272C4×D4—48no—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—24no—
277#117—24no—
278#117—24no—
279#117—41no—
280#117—41no—
281#117—48no—
282#117—42no—
283#117C2×D414yes—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—41no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D414yes—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—41no—
328C2×C42—41no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2412yes—
333#107—28no—
334#106—28no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339#103—28no—
340#103—28no—
341#103—28no—
342#103—28no—
343#103—28no—
344#103—28no—
345#103—44no—
346#103—44no—
347C22⋊C8—28no—
348C22⋊C8—28no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24C22×D412yes—
363C24C22×D411yescharacteristic
364#41—44no—
365#41—44no—
366#41—44no—
367#41—44no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—44no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×Q8—84no—
394C2×Q8—84no—
395C2×Q8#12114yes—
396C2×Q8C22×D414yes—
397C2×D4—41no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4#12118yes—
402C2×D4#12114yes—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—84no—
430C22×C4—82no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4114yes—
441C22×C4C2511yescharacteristic
442Q16—88no—
443Q16—88no—
444M16—84no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—84no—
452C4⋊C4—216no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4C22×D414yes—
476C4⋊C4H(5,𝔽2)12yes—
477C4⋊C4H(5,𝔽2)12yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—216no—
481C22⋊C4—28no—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—44no—
497C42—44no—
498C42—44no—
499C42—44no—
500C42—44no—
501C42—82no—
502C23—24no—
503C23—22no—
504C23—21no—
505C23—22no—
506C23—21no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23C2×#12114yes—
511C23C2×#12112yes—
512C23C23×D411yescharacteristic
513Q8—24no—
514Q8—24no—
515Q8—44no—
516Q8—48no—
517Q8—42no—
518Q8—42no—
519Q8—44no—
520Q8—44no—
521Q8—44no—
522Q8—44no—
523Q8—88no—
524D4—44no—
525D4—48no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—84no—
567C2×C4—88no—
568C2×C4—164no—
569C2×C4—162no—
570C2×C4C2×#12112yes—
571C2×C4C2×#12114yes—
572C2×C4#57814yes—
573C2×C4C23×D411yescharacteristic
574C2×C4C22×#4111yescharacteristic
575C2×C4C2×H(5,𝔽2)11yescharacteristic
576C2×C4C2×H(5,𝔽2)11yescharacteristic
577C2×C4#58511yescharacteristic
578C2×C4#58511yescharacteristic
579C8—44no—
580C8—44no—
581C22—28no—
582C22—24no—
583C22—22no—
584C22—24no—
585C22—42no—
586C22—44no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—41no—
591C22C22×#12111yescharacteristic
592C22#348411yescharacteristic
593C22#348511yescharacteristic
594C22C2×#57812yes—
595C22#358612yes—
596C4—24no—
597C4—22no—
598C4—24no—
599C4—44no—
600C4—42no—
601C4—44no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—84no—
607C4—84no—
608C4—82no—
609C4#345911yescharacteristic
610C4#345911yescharacteristic
611C4#349611yescharacteristic
612C4#349611yescharacteristic
613C2—42no—
614C2—44no—
615C2#3498411yescharacteristic
616C2#6043212yes—
617C2#6166511yescharacteristic
618C2#6167411yescharacteristic
619C2#6175712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33114C2semidirect
5. #32915C2semidirect
6. #29766C2semidirect
7. #29715C2semidirect
8. #25226C2semidirect
9. #25215C2semidirect
10. #25095C2semidirect
11. #25082C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32022
#373862
#342762
#331142
#329152
#297662
#297152
#252262
#252152
#250952
#250822
#219362
#209132