Finite Group Databasebeta

← #377673

Subgroups — #377673

Order 512.

1019 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2#21147C211yesmaximal, characteristic, complemented
3C2×#1938C211yesmaximal, characteristic
4C2×#1938C211yesmaximal, characteristic
5#21052C211yesmaximal, characteristic
6C2×#1773C211yesmaximal, characteristic, complemented
7C2×#1767C211yesmaximal, characteristic, complemented
8#11233C211yesmaximal, characteristic, complemented
9#11221C211yesmaximal, characteristic, complemented
10#11125C211yesmaximal, characteristic
11#10934C211yesmaximal, characteristic
12#10923C211yesmaximal, characteristic
13#10887C211yesmaximal, characteristic, complemented
14#10878C211yesmaximal, characteristic, complemented
15#10819C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×#97C2211yescharacteristic
32#2297C2211yescharacteristic
33#2296C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#2068C412yescomplemented
36#2067C412yescomplemented
37#2066C412yescomplemented
38#2065C412yescomplemented
39#1940—22no—
40#1939—22no—
41#1938—24no—
42#1938—24no—
43#1924—22no—
44#1923—22no—
45#1773—24no—
46#1767—24no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1728C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#108—22no—
81C2×#108—22no—
82C2×#108—22no—
83#427—22no—
84#427—22no—
85C2×#107C2×C412yes—
86C2×#106C2×C412yes—
87#424—22no—
88#424—22no—
89#424—22no—
90#424—41no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2×C412yes—
99C2×C4⋊C8C2×C412yes—
100C2×C4⋊C8D412yes—
101C2×C4⋊C8D412yes—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—22no—
106C2×#104—22no—
107C2×#104—41no—
108#416—22no—
109#416—22no—
110#416—22no—
111#416—22no—
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C412yes—
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C412yes—
117C2×#102C2×C411yescharacteristic
118#408—22no—
119#408—22no—
120#408—22no—
121#408—41no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8—21no—
125#405—22no—
126#405—22no—
127#405—41no—
128#405—41no—
129C2×#97—22no—
130C2×#97—22no—
131C2×#97—21no—
132C2×#97—21no—
133C2×#97—22no—
134C2×#97—22no—
135C2×#97—22no—
136C2×#97D414yes—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139#391C2×C412yes—
140#390C2×C412yes—
141#389C2×C412yes—
142#389C2×C412yes—
143#386—21no—
144#386—21no—
145#385—21no—
146#385—21no—
147C4×C4⋊C4—22no—
148C4×C22⋊C4—22no—
149C43—22no—
150C43—22no—
151#341—22no—
152#341—22no—
153#341—22no—
154#340C2×C412yes—
155#340C2×C412yes—
156#339—22no—
157#339—22no—
158#339—22no—
159#339—22no—
160#339—41no—
161#339D414yes—
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C22×Q8C22×C411yescharacteristic
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4#4111yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4C2411yescharacteristic
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16C2×D412yes—
212C2×M16#4112yes—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8C22×C412yes—
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8#4112yes—
246C22×C8#4112yes—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117C2×D412yes—
258#117#4112yes—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C22×C412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42#4112yes—
294C2×C42#4112yes—
295#108—28no—
296#107—24no—
297#106—24no—
298C4⋊C8—24no—
299C4⋊C8—24no—
300C4⋊C8—24no—
301C4⋊C8—24no—
302C4⋊C8—44no—
303C4⋊C8—42no—
304C4⋊C8—42no—
305#104—44no—
306#104—44no—
307#103—22no—
308#103—22no—
309#103—24no—
310#102—22no—
311#102—22no—
312#102—24no—
313C22⋊C8—44no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316#97—28no—
317#97—42no—
318#97—42no—
319#97—44no—
320C4×C8—42no—
321C4×C8—42no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326C24—21no—
327C24C2×#4111yescharacteristic
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4C4×D412yes—
390C22×C4C4×D412yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C42−1+411yescharacteristic
403C22×C42−1+411yescharacteristic
404C22×C42−1+411yescharacteristic
405M16—22no—
406M16—22no—
407M16—22no—
408M16—22no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419M16—41no—
420M16—41no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8C4×D412yes—
464C2×C8C4×D412yes—
465C2×C8#12112yes—
466C2×C8#12112yes—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—44no—
475C4⋊C4—82no—
476C4⋊C4C4×D414yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—44no—
484C22⋊C4—82no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—48no—
497C42—48no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42#12114yes—
505C23—22no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23C2×C4×D411yescharacteristic
513C23#52511yescharacteristic
514C23#52911yescharacteristic
515C23#54711yescharacteristic
516C23#55211yescharacteristic
517C23#55311yescharacteristic
518C23#55611yescharacteristic
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4#44414yes—
568C2×C4C2×C4×D411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C4×#4111yescharacteristic
571C2×C4C4×#4111yescharacteristic
572C2×C4#51911yescharacteristic
573C2×C4#52011yescharacteristic
574C2×C4C2×#12111yescharacteristic
575C2×C4C2×#12111yescharacteristic
576C2×C4#52511yescharacteristic
577C2×C4#52911yescharacteristic
578C2×C4#53211yescharacteristic
579C2×C4#53211yescharacteristic
580C2×C4#53611yescharacteristic
581C2×C4#53711yescharacteristic
582C2×C4#54111yescharacteristic
583C2×C4#54211yescharacteristic
584C2×C4D4211yescharacteristic
585C2×C4#54711yescharacteristic
586C2×C4#54711yescharacteristic
587C2×C4#54811yescharacteristic
588C2×C4#54811yescharacteristic
589C2×C4#54811yescharacteristic
590C2×C4#55111yescharacteristic
591C2×C4#55211yescharacteristic
592C2×C4#55211yescharacteristic
593C2×C4#55311yescharacteristic
594C2×C4#55611yescharacteristic
595C2×C4#55611yescharacteristic
596C8—22no—
597C8—22no—
598C8—22no—
599C8—22no—
600C8—42no—
601C8—42no—
602C8—42no—
603C8—42no—
604C8—81no—
605C8—81no—
606C22—22no—
607C22—22no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—22no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—82no—
619C22C4×#12111yescharacteristic
620C22#236111yescharacteristic
621C22#236211yescharacteristic
622C22#241311yescharacteristic
623C22#246911yescharacteristic
624C22#252511yescharacteristic
625C22#260111yescharacteristic
626C22C2×#44412yes—
627C22#295612yes—
628C4—24no—
629C4—44no—
630C4—44no—
631C4—42no—
632C4—44no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—82no—
637C4C4×#12111yescharacteristic
638C4#232311yescharacteristic
639C4#236211yescharacteristic
640C4#237311yescharacteristic
641C4#241611yescharacteristic
642C4#246111yescharacteristic
643C4#251911yescharacteristic
644C4#254511yescharacteristic
645C2—21no—
646C2—21no—
647C2—21no—
648C2—21no—
649C2—161no—
650C2#1420011yescharacteristic
651C2#2154611yescharacteristic
652C2#2155111yescharacteristic
653C2#2301612yes—
654C2#2308112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21147C2semidirect
6. C2×#1773C2semidirect
7. C2×#1767C2semidirect
8. #11233C2semidirect
9. #11221C2semidirect
13. #10887C2semidirect
14. #10878C2semidirect
15. #10819C2semidirect
35. #2068C4semidirect
36. #2067C4semidirect
37. #2066C4semidirect
38. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
#211472
C2×#19382
#210522
C2×#17732
C2×#17672
#112332
#112212
#111252
#109342
#109232
#108872
#108782
#108192