Finite Group Databasebeta

← #229909

Subgroups — #229909

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23088C211yesmaximal, characteristic, complemented
2#22986C211yesmaximal, characteristic, complemented
3C4×#469C211yesmaximal, characteristic
4C2×#1581C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6#9665C211yesmaximal, characteristic, complemented
7#9597C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic, complemented
10#8944C211yesmaximal, characteristic, complemented
11#8943C211yesmaximal, characteristic, complemented
12#8847C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8289C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C2×#469C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×#435—21no—
20C2×C8×D4C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#1972C411yescharacteristic
27#1954—21no—
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1944C411yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1936C411yescharacteristic
34#1936C411yescharacteristic
35#1924—21no—
36#1924—21no—
37#1895C411yescharacteristic
38#1895C411yescharacteristic
39#1870—21no—
40#1870—21no—
41#1850—21no—
42#1809C411yescharacteristic
43C4×#106C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1581—24no—
52#1569—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16—21no—
72C22×M16C2×C411yescharacteristic
73C22×M16C2×C411yescharacteristic
74C23×C8—21no—
75C2×#121C2×C411yescharacteristic
76C2×#121C2×C411yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#469—24no—
83C4×D8—22no—
84C4×D8—22no—
85#436—22no—
86#436—41no—
87#435—44no—
88C8×D4—22no—
89C8×D4—41no—
90#432—22no—
91#432—41no—
92C2×#107C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95#423—22no—
96#423—41no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8D411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C4×M16—22no—
121C4×M16—22no—
122C2×#97—21no—
123C2×#97—21no—
124C2×#97C2×C411yescharacteristic
125C2×#97C2×C411yescharacteristic
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#388—21no—
138#388—21no—
139#386—21no—
140#386—21no—
141#380C2×C411yescharacteristic
142#380C2×C411yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336D411yescharacteristic
159#336D411yescharacteristic
160#335—22no—
161#335D414yescomplemented
162#334—22no—
163#334—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168#322—22no—
169#322—22no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C2×D8—22no—
180C2×D8—22no—
181C2×D8—41no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—41no—
186C2×M16—41no—
187C2×M16—41no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—81no—
193C2×M16—81no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—81no—
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211#121—22no—
212#121—22no—
213#121—41no—
214#121—41no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—81no—
226C4×D4—81no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42C2411yescharacteristic
255#107—22no—
256#106—22no—
257#106—41no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—81no—
267C4⋊C8C2×D412yes—
268C4⋊C8C2×D412yes—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—41no—
273#102—41no—
274#102M1614yescomplemented
275C22⋊C8—22no—
276C22⋊C8—41no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—81no—
284C22⋊C8—81no—
285#97—22no—
286#97—22no—
287#97—42no—
288#97—42no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—81no—
300C4×C8C2×D412yes—
301C4×C8#4112yes—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308C24—21no—
309C24—21no—
310C24C4×D411yescharacteristic
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—81no—
327C2×D4—81no—
328C2×D4C2×M1612yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369D8—44no—
370M16—44no—
371M16—44no—
372M16—84no—
373M16—84no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—84no—
413C2×C8C4×D412yes—
414C2×C8C4×D412yes—
415C2×C8C2×M1612yes—
416C2×C8AGL(1,ℤ/8)12yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4C2×M1612yes—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42C22×D411yescharacteristic
454C42C2×#4111yescharacteristic
455C42H(5,𝔽2)11yescharacteristic
456C422−1+411yescharacteristic
457C23—21no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23C2×C4×D411yescharacteristic
479D4—22no—
480D4—22no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—84no—
488D4—84no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4C2×C4×D411yescharacteristic
537C2×C4#52011yescharacteristic
538C2×C4D4211yescharacteristic
539C2×C4#54711yescharacteristic
540C2×C4#54811yescharacteristic
541C2×C4#55311yescharacteristic
542C2×C4C22×M1611yescharacteristic
543C2×C4#56811yescharacteristic
544C2×C4#56811yescharacteristic
545C2×C4#56811yescharacteristic
546C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
547C2×C4#57511yescharacteristic
548C2×C4#57611yescharacteristic
549C2×C4#57711yescharacteristic
550C8—22no—
551C8—22no—
552C8—22no—
553C8—22no—
554C8—42no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—42no—
559C8—42no—
560C8—41no—
561C8—81no—
562C8—81no—
563C8—81no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—41no—
569C22—42no—
570C22—41no—
571C22—42no—
572C22—41no—
573C22—41no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22#236111yescharacteristic
586C22#293211yescharacteristic
587C22D4×M1611yescharacteristic
588C22C4×AGL(1,ℤ/8)11yescharacteristic
589C22#294611yescharacteristic
590C22#295511yescharacteristic
591C22#295611yescharacteristic
592C4—21no—
593C4—21no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4#293011yescharacteristic
604C4D4×M1611yescharacteristic
605C4#298911yescharacteristic
606C4#299511yescharacteristic
607C4#326411yescharacteristic
608C4#327111yescharacteristic
609C4#335411yescharacteristic
610C4#335511yescharacteristic
611C2—81no—
612C2—82no—
613C2—161no—
614C2#1798311yescharacteristic
615C2#1798711yescharacteristic
616C2#1804811yescharacteristic
617C2#1805011yescharacteristic
618C2#2298811yescharacteristic
619C2#2302111yescharacteristic
620C2#2303711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#469C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23088C2semidirect
2. #22986C2semidirect
4. C2×#1581C2semidirect
5. C2×#1569C2semidirect
6. #9665C2semidirect
7. #9597C2semidirect
8. #9546C2semidirect
9. #9541C2semidirect
10. #8944C2semidirect
11. #8943C2semidirect
12. #8847C2semidirect
13. #8845C2semidirect
14. #8289C2semidirect
15. #8147C2semidirect
161. #335D4semidirect
274. #102M16semidirect

Maximal subgroups

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

subgroupindex
#230882
#229862
C4×#4692
C2×#15812
C2×#15692
#96652
#95972
#95462
#95412
#89442
#89432
#88472
#88452
#82892
#81472