Finite Group Databasebeta

← #225763

Subgroups — #225763

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22747C211yesmaximal, characteristic, complemented
2#22669C211yesmaximal, characteristic
3#18040C211yesmaximal, characteristic
4M16×SD16C211yesmaximal, characteristic
5C2×#1592C211yesmaximal, characteristic
6#9625C211yesmaximal, characteristic, complemented
7#9624C211yesmaximal, characteristic, complemented
8#8909C211yesmaximal, characteristic, complemented
9#8908C211yesmaximal, characteristic
10#8907C211yesmaximal, characteristic, complemented
11#8906C211yesmaximal, characteristic
12#8828C211yesmaximal, characteristic, complemented
13#8664C211yesmaximal, characteristic
14#8663C211yesmaximal, characteristic, complemented
15#8297C211yesmaximal, characteristic, complemented
16Q8×M16C2211yescharacteristic
17Q8×M16C2211yescharacteristic
18C2×#446C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20D4×M16C2211yescharacteristic
21D4×M16C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×#430—21no—
24#1950—21no—
25#1950—21no—
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1855—21no—
31#1855—21no—
32#1848C2211yescharacteristic
33#1838—21no—
34#1837C2211yescharacteristic
35#1811—21no—
36#1778—21no—
37C4×#108C2211yescharacteristic
38#1592—21no—
39#1592C411yescharacteristic
40#1592C411yescharacteristic
41#1592C411yescharacteristic
42#1592C411yescharacteristic
43#1592C411yescharacteristic
44#1592C411yescharacteristic
45#1592C411yescharacteristic
46#1592C411yescharacteristic
47#1590—21no—
48#1588—21no—
49#1585—21no—
50#1583—21no—
51#1582—21no—
52SD16⋊C8—21no—
53C8×SD16—21no—
54#1567C2211yescharacteristic
55#1567C2211yescharacteristic
56#1566C2211yescharacteristic
57#1566C2211yescharacteristic
58#1564C2211yescharacteristic
59C2×#335C2211yescharacteristic
60#1487C2211yescharacteristic
61#1487C2211yescharacteristic
62#1486C2211yescharacteristic
63#1486C2211yescharacteristic
64C2×Q8⋊C8C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C22×SD16Q811yescharacteristic
76C22×M16—21no—
77C22×M16—21no—
78C22×M16—21no—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81#446—21no—
82#446—21no—
83#446—21no—
84#446—21no—
85#446—41no—
86#446—41no—
87#446C2×C411yescharacteristic
88#446C2×C411yescharacteristic
89#446C2×C411yescharacteristic
90#446C2×C411yescharacteristic
91C8×Q8—21no—
92C8×Q8—21no—
93C4×SD16C2×C411yescharacteristic
94C4×SD16C2×C411yescharacteristic
95C4×SD16C2×C411yescharacteristic
96C4×SD16C2×C411yescharacteristic
97C4×SD16D411yescharacteristic
98C4×SD16D411yescharacteristic
99C4×SD16D411yescharacteristic
100C4×SD16D411yescharacteristic
101#436—21no—
102#436—21no—
103#436—21no—
104#436—21no—
105#436C2×C411yescharacteristic
106#436C2×C411yescharacteristic
107#436C2×C411yescharacteristic
108#436C2×C411yescharacteristic
109#435—41no—
110#435—41no—
111C8×D4—21no—
112C8×D4—21no—
113#431C2311yescharacteristic
114#431C2311yescharacteristic
115#430—44no—
116C2×#108—21no—
117C2×#108—21no—
118C2×#108—21no—
119C2×#108—21no—
120C2×#108—21no—
121C2×#108—21no—
122C2×#108D411yescharacteristic
123C2×#108D411yescharacteristic
124C2×#106Q811yescharacteristic
125#423—21no—
126#423—21no—
127#423C2311yescharacteristic
128#423C2311yescharacteristic
129#423C2311yescharacteristic
130#423C2311yescharacteristic
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×#103Q811yescharacteristic
135C2×#102Q811yescharacteristic
136#407—21no—
137#407—21no—
138C2×C22⋊C8—21no—
139C4×M16—21no—
140C4×M16—21no—
141C4×M16C2311yescharacteristic
142C4×M16C2311yescharacteristic
143C2×#97—21no—
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#335C2×C411yescharacteristic
160#335C2×C411yescharacteristic
161#335C2×C411yescharacteristic
162#335C2×C411yescharacteristic
163#334—21no—
164#334—21no—
165#334—21no—
166#334—41no—
167Q8⋊C8—21no—
168Q8⋊C8—21no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171Q8⋊C8C2×C411yescharacteristic
172Q8⋊C8C2×C411yescharacteristic
173Q8⋊C8C2×C411yescharacteristic
174Q8⋊C8C2×C411yescharacteristic
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
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×Q8C2×Q811yescharacteristic
192C22×D4C2×Q811yescharacteristic
193C23×C4—21no—
194C2×SD16—42no—
195C2×SD16C4⋊C411yescharacteristic
196C2×SD16C4⋊C411yescharacteristic
197C2×SD16C4⋊C411yescharacteristic
198C2×SD16C4⋊C411yescharacteristic
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—42no—
208C2×M16—42no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16—42no—
212C2×M16—42no—
213C2×M16—81no—
214C2×M16—81no—
215C2×M16C2×D411yescharacteristic
216C2×M16C2×D411yescharacteristic
217C2×M16C2×D411yescharacteristic
218C2×M16C2×D411yescharacteristic
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—41no—
223C22×C8—41no—
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×Q811yescharacteristic
227C22×C8C2×Q811yescharacteristic
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8C22×C411yescharacteristic
231C4×Q8C22×C411yescharacteristic
232C4×Q8C2×D411yescharacteristic
233C4×Q8C2×D411yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236C4×D4C22×C411yescharacteristic
237C4×D4C22×C411yescharacteristic
238C4×D4C2×D411yescharacteristic
239C4×D4C2×D411yescharacteristic
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×Q811yescharacteristic
242C2×C4⋊C4C2×Q811yescharacteristic
243C2×C22⋊C4—21no—
244C2×C42—21no—
245C2×C42C2411yescharacteristic
246#108—22no—
247#108—41no—
248#108—42no—
249#108—42no—
250#108—42no—
251#108—42no—
252#108—42no—
253#108—81no—
254#106C4⋊C411yescharacteristic
255#106C4⋊C411yescharacteristic
256#106C4⋊C411yescharacteristic
257#106C4⋊C411yescharacteristic
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8—41no—
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C22×C411yescharacteristic
281C4⋊C8C22×C411yescharacteristic
282#103C4⋊C411yescharacteristic
283#103C4⋊C411yescharacteristic
284#103C4⋊C411yescharacteristic
285#103C4⋊C411yescharacteristic
286#102C4⋊C411yescharacteristic
287#102C4⋊C411yescharacteristic
288#102C4⋊C411yescharacteristic
289#102C4⋊C411yescharacteristic
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300#97—21no—
301#97—21no—
302#97—21no—
303#97—21no—
304#97—21no—
305#97—21no—
306#97—41no—
307#97—41no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—41no—
314C4×C8—41no—
315C4×C8C22×C411yescharacteristic
316C4×C8C22×C411yescharacteristic
317C4×C8C22×C411yescharacteristic
318C4×C8C22×C411yescharacteristic
319C4×C8C22×C411yescharacteristic
320C4×C8C22×C411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8#4111yescharacteristic
324C4×C8#4111yescharacteristic
325C24—21no—
326C2×Q8—21no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8C2×C4⋊C411yescharacteristic
330C2×Q8C2×C4⋊C411yescharacteristic
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4C2×C4⋊C411yescharacteristic
335C2×D4C2×C4⋊C411yescharacteristic
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C42−1+411yescharacteristic
350SD16—22no—
351SD16—22no—
352SD16—41no—
353SD16—41no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—21no—
358M16—21no—
359M16—21no—
360M16—21no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16—42no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—82no—
374M16—82no—
375M16AGL(1,ℤ/8)12yes—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—82no—
403C2×C8—81no—
404C2×C8—82no—
405C2×C8C2×C4⋊C411yescharacteristic
406C2×C8C2×C4⋊C411yescharacteristic
407C2×C8C2×C4⋊C411yescharacteristic
408C2×C8C2×C4⋊C411yescharacteristic
409C2×C8C4×D411yescharacteristic
410C2×C8C4×D411yescharacteristic
411C2×C8C4×D411yescharacteristic
412C2×C8C4×D411yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4C2×C4⋊C411yescharacteristic
420C4⋊C4C2×C4⋊C411yescharacteristic
421C4⋊C4C2×C4⋊C411yescharacteristic
422C4⋊C4C2×C4⋊C411yescharacteristic
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—21no—
426C22⋊C4—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42C23×C411yescharacteristic
432C42C22×D411yescharacteristic
433C42C2×#4111yescharacteristic
434C42H(5,𝔽2)11yescharacteristic
435C23—21no—
436C23—21no—
437C23—41no—
438C23—41no—
439C23—81no—
440C23—81no—
441C23D4×Q811yescharacteristic
442Q8—21no—
443Q8—21no—
444Q8—21no—
445Q8—21no—
446Q8—81no—
447D4—21no—
448D4—21no—
449D4—21no—
450D4—21no—
451D4—81no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4C22×C4⋊C411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4#52011yescharacteristic
476C2×C4D4211yescharacteristic
477C2×C4#54711yescharacteristic
478C2×C4#55411yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57811yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—22no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—81no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C8—81no—
506C8#43012yescomplemented
507C22—21no—
508C22—21no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—161no—
516C22—161no—
517C22D4×C4⋊C411yescharacteristic
518C22#338611yescharacteristic
519C22#338811yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4C2×#43011yescharacteristic
530C4#291611yescharacteristic
531C4C4×AGL(1,ℤ/8)11yescharacteristic
532C4#294711yescharacteristic
533C4#326511yescharacteristic
534C4#326911yescharacteristic
535C2—21no—
536C2—21no—
537C2—81no—
538C2—81no—
539C2#2266911yescharacteristic
540C2#2267711yescharacteristic
541C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC22×D4C2×Q8
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. #22747C2semidirect
6. #9625C2semidirect
7. #9624C2semidirect
8. #8909C2semidirect
10. #8907C2semidirect
12. #8828C2semidirect
14. #8663C2semidirect
15. #8297C2semidirect
506. C8#430semidirect

Maximal subgroups

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

subgroupindex
#227472
#226692
#180402
M16×SD162
C2×#15922
#96252
#96242
#89092
#89082
#89072
#89062
#88282
#86642
#86632
#82972