Finite Group Databasebeta

← #225762

Subgroups — #225762

Order 512.

613 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22719C211yesmaximal, characteristic, complemented
2#22669C211yesmaximal, characteristic, complemented
3#18044C211yesmaximal, characteristic
4M16×D8C211yesmaximal, characteristic
5C2×#1594C211yesmaximal, characteristic
6#9626C211yesmaximal, characteristic, complemented
7#9623C211yesmaximal, characteristic, complemented
8#8908C211yesmaximal, characteristic, complemented
9#8908C211yesmaximal, characteristic, complemented
10#8906C211yesmaximal, characteristic, complemented
11#8906C211yesmaximal, characteristic, complemented
12#8828C211yesmaximal, characteristic, complemented
13#8664C211yesmaximal, characteristic, complemented
14#8662C211yesmaximal, characteristic, complemented
15#8297C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17D4×M16C2211yescharacteristic
18D4×M16C2211yescharacteristic
19D4×M16C2211yescharacteristic
20D4×M16C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×#430—21no—
24#1950C2211yescharacteristic
25#1950C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1855—21no—
29#1855—21no—
30#1855—21no—
31#1855—21no—
32#1848C2211yescharacteristic
33#1837C2211yescharacteristic
34#1811—21no—
35#1811—21no—
36C4×#108C2211yescharacteristic
37#1738—21no—
38#1594—21no—
39#1594C411yescharacteristic
40#1594C411yescharacteristic
41#1594C411yescharacteristic
42#1594C411yescharacteristic
43#1594C411yescharacteristic
44#1594C411yescharacteristic
45#1594C411yescharacteristic
46#1594C411yescharacteristic
47#1589—21no—
48#1586—21no—
49#1586—21no—
50#1581—21no—
51#1581—21no—
52#1580—21no—
53C8×D8—21no—
54#1567C2211yescharacteristic
55#1567C2211yescharacteristic
56#1566C2211yescharacteristic
57#1566C2211yescharacteristic
58#1564C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1486C2211yescharacteristic
61#1486C2211yescharacteristic
62#1486C2211yescharacteristic
63#1486C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C22×D8Q811yescharacteristic
76C22×M16—21no—
77C22×M16—21no—
78C22×M16—21no—
79C22×M16—21no—
80C22×M16—21no—
81C22×M16—21no—
82C2×C4×D4C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84C4×D8C2×C411yescharacteristic
85C4×D8C2×C411yescharacteristic
86C4×D8C2×C411yescharacteristic
87C4×D8C2×C411yescharacteristic
88C4×D8D411yescharacteristic
89C4×D8D411yescharacteristic
90C4×D8D411yescharacteristic
91C4×D8D411yescharacteristic
92#436—21no—
93#436—21no—
94#436—21no—
95#436—21no—
96#436—21no—
97#436—21no—
98#436—21no—
99#436—21no—
100#436C2×C411yescharacteristic
101#436C2×C411yescharacteristic
102#436C2×C411yescharacteristic
103#436C2×C411yescharacteristic
104#436C2×C411yescharacteristic
105#436C2×C411yescharacteristic
106#436C2×C411yescharacteristic
107#436C2×C411yescharacteristic
108#435—41no—
109#435—41no—
110#435—41no—
111#435—41no—
112C8×D4—21no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—21no—
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#430—44no—
119C2×#108—21no—
120C2×#108—21no—
121C2×#108—21no—
122C2×#108—21no—
123C2×#108—21no—
124C2×#108—21no—
125C2×#108D411yescharacteristic
126C2×#108D411yescharacteristic
127C2×#107Q811yescharacteristic
128#423C2311yescharacteristic
129#423C2311yescharacteristic
130#423C2311yescharacteristic
131#423C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×#102Q811yescharacteristic
135C2×#102Q811yescharacteristic
136#407—21no—
137#407—21no—
138#407—21no—
139#407—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C4×M16C2311yescharacteristic
143C4×M16C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#341—21no—
148#341—21no—
149#341—21no—
150#341—21no—
151#341—21no—
152#341—21no—
153#341D411yescharacteristic
154#341D411yescharacteristic
155#335—21no—
156#335—21no—
157#335—21no—
158#335—41no—
159#334—21no—
160#334—21no—
161#334—21no—
162#334—41no—
163#334C2×C411yescharacteristic
164#334C2×C411yescharacteristic
165#334C2×C411yescharacteristic
166#334C2×C411yescharacteristic
167D4⋊C8—21no—
168D4⋊C8—21no—
169D4⋊C8—21no—
170D4⋊C8—21no—
171D4⋊C8—21no—
172D4⋊C8—21no—
173D4⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8C2×C411yescharacteristic
176D4⋊C8C2×C411yescharacteristic
177D4⋊C8C2×C411yescharacteristic
178D4⋊C8C2×C411yescharacteristic
179D4⋊C8C2×C411yescharacteristic
180D4⋊C8C2×C411yescharacteristic
181D4⋊C8C2×C411yescharacteristic
182D4⋊C8C2×C411yescharacteristic
183#322—21no—
184#322—21no—
185#322—21no—
186#322C2×C411yescharacteristic
187#322C2×C411yescharacteristic
188#322C2×C411yescharacteristic
189#322C2×C411yescharacteristic
190C82—21no—
191C22×D4C2×Q811yescharacteristic
192C22×D4C2×Q811yescharacteristic
193C23×C4—21no—
194C23×C4—21no—
195C2×D8—42no—
196C2×D8C4⋊C411yescharacteristic
197C2×D8C4⋊C411yescharacteristic
198C2×D8C4⋊C411yescharacteristic
199C2×D8C4⋊C411yescharacteristic
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×M16—21no—
212C2×M16—42no—
213C2×M16—42no—
214C2×M16—41no—
215C2×M16—41no—
216C2×M16—41no—
217C2×M16—41no—
218C2×M16—42no—
219C2×M16—42no—
220C2×M16—42no—
221C2×M16—42no—
222C2×M16—81no—
223C2×M16—81no—
224C2×M16—81no—
225C2×M16—81no—
226C2×M16—81no—
227C2×M16—81no—
228C2×M16C2×D411yescharacteristic
229C2×M16C2×D411yescharacteristic
230C2×M16C2×D411yescharacteristic
231C2×M16C2×D411yescharacteristic
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—41no—
236C22×C8—41no—
237C22×C8—41no—
238C22×C8—41no—
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×Q811yescharacteristic
242C22×C8C2×Q811yescharacteristic
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4C22×C411yescharacteristic
248C4×D4C22×C411yescharacteristic
249C4×D4C22×C411yescharacteristic
250C4×D4C22×C411yescharacteristic
251C4×D4C2×D411yescharacteristic
252C4×D4C2×D411yescharacteristic
253C4×D4C2×D411yescharacteristic
254C4×D4C2×D411yescharacteristic
255C2×C4⋊C4C2×Q811yescharacteristic
256C2×C4⋊C4C2×Q811yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C42C2411yescharacteristic
260#108—22no—
261#108—41no—
262#108—42no—
263#108—42no—
264#108—42no—
265#108—81no—
266#108—81no—
267#108—81no—
268#107C4⋊C411yescharacteristic
269#107C4⋊C411yescharacteristic
270#107C4⋊C411yescharacteristic
271#107C4⋊C411yescharacteristic
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8C22×C411yescharacteristic
285C4⋊C8C22×C411yescharacteristic
286C4⋊C8C22×C411yescharacteristic
287C4⋊C8C22×C411yescharacteristic
288#102C4⋊C411yescharacteristic
289#102C4⋊C411yescharacteristic
290#102C4⋊C411yescharacteristic
291#102C4⋊C411yescharacteristic
292#102C4⋊C411yescharacteristic
293#102C4⋊C411yescharacteristic
294#102C4⋊C411yescharacteristic
295#102C4⋊C411yescharacteristic
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310C22⋊C8—41no—
311C22⋊C8—41no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314#97—21no—
315#97—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8C22×C411yescharacteristic
322C4×C8C22×C411yescharacteristic
323C4×C8C22×C411yescharacteristic
324C4×C8C22×C411yescharacteristic
325C4×C8C22×C411yescharacteristic
326C4×C8C22×C411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8#4111yescharacteristic
330C4×C8#4111yescharacteristic
331C24—21no—
332C24—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4C2×C4⋊C411yescharacteristic
340C2×D4C2×C4⋊C411yescharacteristic
341C2×D4C2×C4⋊C411yescharacteristic
342C2×D4C2×C4⋊C411yescharacteristic
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
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×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C42−1+411yescharacteristic
364D8—22no—
365D8—22no—
366D8—41no—
367D8—41no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—42no—
376M16—42no—
377M16—42no—
378M16—42no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—81no—
388M16—81no—
389M16—81no—
390M16—81no—
391M16—82no—
392M16—82no—
393M16—82no—
394M16—82no—
395M16AGL(1,ℤ/8)12yes—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—82no—
423C2×C8—82no—
424C2×C8—81no—
425C2×C8—82no—
426C2×C8—82no—
427C2×C8C2×C4⋊C411yescharacteristic
428C2×C8C2×C4⋊C411yescharacteristic
429C2×C8C2×C4⋊C411yescharacteristic
430C2×C8C2×C4⋊C411yescharacteristic
431C2×C8C4×D411yescharacteristic
432C2×C8C4×D411yescharacteristic
433C2×C8C4×D411yescharacteristic
434C2×C8C4×D411yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4C2×C4⋊C411yescharacteristic
438C4⋊C4C2×C4⋊C411yescharacteristic
439C4⋊C4C2×C4⋊C411yescharacteristic
440C4⋊C4C2×C4⋊C411yescharacteristic
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C42C23×C411yescharacteristic
450C42C22×D411yescharacteristic
451C42C2×#4111yescharacteristic
452C42H(5,𝔽2)11yescharacteristic
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23D4×Q811yescharacteristic
466D4—21no—
467D4—21no—
468D4—21no—
469D4—21no—
470D4—21no—
471D4—21no—
472D4—21no—
473D4—21no—
474D4—81no—
475D4—81no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4C22×C4⋊C411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4#52011yescharacteristic
506C2×C4D4211yescharacteristic
507C2×C4#54711yescharacteristic
508C2×C4#55411yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C2×C4#57511yescharacteristic
511C2×C4#57611yescharacteristic
512C2×C4#57711yescharacteristic
513C8—21no—
514C8—21no—
515C8—21no—
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—22no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—81no—
531C8—81no—
532C8—81no—
533C8—81no—
534C8—81no—
535C8—81no—
536C8#43012yescomplemented
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—161no—
552C22—161no—
553C22—161no—
554C22—161no—
555C22D4×C4⋊C411yescharacteristic
556C22#338411yescharacteristic
557C22#338911yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4C2×#43011yescharacteristic
566C4#291611yescharacteristic
567C4C4×AGL(1,ℤ/8)11yescharacteristic
568C4#294611yescharacteristic
569C4#326411yescharacteristic
570C4#326711yescharacteristic
571C2—21no—
572C2—21no—
573C2—81no—
574C2—81no—
575C2—81no—
576C2—81no—
577C2#2266911yescharacteristic
578C2#2267611yescharacteristic
579C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC22×D8Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22719C2semidirect
2. #22669C2semidirect
6. #9626C2semidirect
7. #9623C2semidirect
8. #8908C2semidirect
9. #8908C2semidirect
10. #8906C2semidirect
11. #8906C2semidirect
12. #8828C2semidirect
13. #8664C2semidirect
14. #8662C2semidirect
15. #8297C2semidirect
536. C8#430semidirect

Maximal subgroups

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

subgroupindex
#227192
#226692
#180442
M16×D82
C2×#15942
#96262
#96232
#89082
#89062
#88282
#86642
#86622
#82972