Finite Group Databasebeta

← #6436952

Subgroups — #6436952

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35292C211yesmaximal, characteristic
2#34060C211yesmaximal, characteristic, complemented
3#29942C212yesmaximal, complemented
4#29938C212yesmaximal, complemented
5#29925C212yesmaximal, complemented
6#29921C212yesmaximal, complemented
7#29916C214yesmaximal, complemented
8#29899C214yesmaximal, complemented
9#23771C214yesmaximal, complemented
10#23766C214yesmaximal, complemented
11#21395C211yesmaximal, characteristic, complemented
12#21390C211yesmaximal, characteristic, complemented
13#21364C211yesmaximal, characteristic, complemented
14#21359C211yesmaximal, characteristic, complemented
15#20721C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2857C2212yes—
21#2856C2212yes—
22#2624C2212yes—
23#2615C2212yes—
24#2592—21no—
25#2580—21no—
26#2540—21no—
27#2535—21no—
28#2414—22no—
29#2407—22no—
30#2324C2214yes—
31#2320—21no—
32#2319—21no—
33#2318—22no—
34#2317C2212yes—
35#2314C2212yes—
36#2314C2212yes—
37#2311C2212yes—
38#2295—21no—
39#2293—21no—
40#2290—22no—
41#2285C2212yes—
42#2285C2211yescharacteristic
43#2283C2212yes—
44C4×#117C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2101C2214yes—
47#2099C2214yes—
48#2094C2214yes—
49#2092C2214yes—
50#2085C2214yes—
51#2084C2214yes—
52#2077C2214yes—
53#2077C2214yes—
54#2077C2214yes—
55#2075C2214yes—
56#2073C2214yes—
57#2071C2214yes—
58#2034C2214yes—
59#2034C2214yes—
60#2034C2214yes—
61#2033C2214yes—
62#1987C2211yescharacteristic
63#1987C2211yescharacteristic
64#1987C2212yes—
65#1982C2211yescharacteristic
66#1982C2211yescharacteristic
67#1982C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1875C2214yes—
73#1874C2214yes—
74#1854C2214yes—
75#1853C2214yes—
76#1818C2212yes—
77#1797C2212yes—
78#1797C2212yes—
79#1780C2212yes—
80#1776C2214yes—
81C4×#106C2212yes—
82C4×#106C2212yes—
83#1762C2212yes—
84#1762C2212yes—
85C4×#102C2214yes—
86#1750C2214yes—
87C22×#41—21no—
88C22×#41C2311yescharacteristic
89#532—24no—
90#532—24no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#124—21no—
94C2×C22⋊Q8—22no—
95C2×#121—21no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121C2312yes—
99C2×#121C2314yes—
100C4×#41—24no—
101C4×#41—24no—
102C2×C4×Q8D411yescharacteristic
103C2×C4×Q8D411yescharacteristic
104C2×C4×D4—21no—
105C2×C4×D4D412yes—
106C2×C4×D4D411yescharacteristic
107C2×C4×D4D412yes—
108C2×C4×D4D411yescharacteristic
109C2×C4×D4C2312yes—
110C2×#117—21no—
111C2×#117—22no—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C42—21no—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#432—24no—
120#431—24no—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138#399—22no—
139#397—22no—
140#394—24no—
141#393—22no—
142#392—24no—
143#392—22no—
144#391—22no—
145#391—22no—
146#390—22no—
147#390—24no—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#388—24no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#386—22no—
165#386—22no—
166#386—24no—
167#385—24no—
168#385—22no—
169#385—22no—
170#384C2312yes—
171#384C2312yes—
172#384C2312yes—
173#383—22no—
174#382—22no—
175#382—21no—
176#382C2312yes—
177#382C2312yes—
178#381—22no—
179#380—22no—
180#380—22no—
181#380—21no—
182#380C2312yes—
183#380C2312yes—
184#380C2312yes—
185#380C2312yes—
186#380C2314yes—
187#380C2312yes—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—22no—
202#376—21no—
203#376—22no—
204C2×#95—21no—
205C43—21no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C2×#41—22no—
217C2×#41—22no—
218C2×#41—42no—
219C2×#41—42no—
220C2×#41—44no—
221C2×#41—84no—
222C22×Q8—21no—
223C22×Q8C2×D411yescharacteristic
224C22×D4—24no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C2×D412yes—
229C22×D4C2×D411yescharacteristic
230C22×D4C2412yes—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2414yes—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244#127—22no—
245#127—22no—
246#124—44no—
247C22⋊Q8—84no—
248#121—24no—
249#121—24no—
250#121—44no—
251#121—44no—
252#121—44no—
253#121—84no—
254#121—84no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×D4—28no—
258C4×D4—28no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—44no—
264#117—22no—
265#117—22no—
266#117—24no—
267#117—24no—
268#117—42no—
269#117—42no—
270#117—84no—
271#117—82no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—21no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—82no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42#4114yes—
324C2×C42#4114yes—
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107—44no—
329#106—44no—
330#106—44no—
331#106—44no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—24no—
335#102—44no—
336#102—44no—
337#102—44no—
338#102—44no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8—44no—
342C22⋊C8—44no—
343#97—24no—
344C4×C8—24no—
345#95—22no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363C24—22no—
364C24C22×D412yes—
365C24C22×D411yescharacteristic
366#41—44no—
367#41—44no—
368#41—48no—
369#41—816no—
370C2×Q8—22no—
371C2×Q8—42no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4—84no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—84no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4#12314yes—
422C22×C4#12314yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2511yescharacteristic
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
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—22no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—84no—
481C42—24no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—44no—
486C42—42no—
487C42—44no—
488C42—42no—
489C42—82no—
490C42—81no—
491C42—81no—
492C42—82no—
493C42#12314yes—
494C42#12314yes—
495C42C2×#4112yes—
496C42C2×#4112yes—
497C42C2×#4112yes—
498C42C2×#4112yes—
499C23—22no—
500C23—22no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23C2×#12312yes—
516C23C23×D411yescharacteristic
517Q8—44no—
518Q8—84no—
519D4—44no—
520D4—48no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—84no—
525D4—88no—
526D4—84no—
527D4—168no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—84no—
570C2×C4—88no—
571C2×C4—88no—
572C2×C4C2×#12312yes—
573C2×C4C2×#12312yes—
574C2×C4C2×#12312yes—
575C2×C4C2×#12312yes—
576C2×C4C2×#12312yes—
577C2×C4#57712yes—
578C2×C4#57712yes—
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4#58512yes—
582C2×C4#58512yes—
583C8—42no—
584C8—42no—
585C8—44no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—84no—
599C22—84no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22C22×#12311yescharacteristic
604C22#344912yes—
605C22C2×#57711yescharacteristic
606C22C2×#57711yescharacteristic
607C22#358512yes—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—84no—
620C4—82no—
621C4#309914yes—
622C4#345511yescharacteristic
623C4#345511yescharacteristic
624C4#345512yes—
625C2—42no—
626C2—82no—
627C2—162no—
628C2#3463511yescharacteristic
629C2C2×#309911yescharacteristic
630C2#6051811yescharacteristic
631C2#6098912yes—
632C2#6099412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35292C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34060C2semidirect
3. #29942C2semidirect
4. #29938C2semidirect
5. #29925C2semidirect
6. #29921C2semidirect
7. #29916C2semidirect
8. #29899C2semidirect
9. #23771C2semidirect
10. #23766C2semidirect
11. #21395C2semidirect
12. #21390C2semidirect
13. #21364C2semidirect
14. #21359C2semidirect
15. #20721C2semidirect

Maximal subgroups

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

subgroupindex
#352922
#340602
#299422
#299382
#299252
#299212
#299162
#298992
#237712
#237662
#213952
#213902
#213642
#213592
#207212