Finite Group Databasebeta

← #226181

Subgroups — #226181

Order 512.

576 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#22746C211yesmaximal, characteristic, complemented
2#22669C211yesmaximal, characteristic
3#18040C211yesmaximal, characteristic
4M16×SD16C211yesmaximal, characteristic
5C2×#1588C211yesmaximal, 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#8659C211yesmaximal, characteristic, complemented
15#8296C211yesmaximal, characteristic, complemented
16Q8×M16C2211yescharacteristic
17Q8×M16C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
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#1837—21no—
34#1837C2211yescharacteristic
35#1811—21no—
36C4×#108—21no—
37C4×#108C2211yescharacteristic
38#1592—21no—
39#1590—21no—
40#1588—21no—
41#1588C411yescharacteristic
42#1588C411yescharacteristic
43#1588C411yescharacteristic
44#1588C411yescharacteristic
45#1588C411yescharacteristic
46#1588C411yescharacteristic
47#1588C411yescharacteristic
48#1588C411yescharacteristic
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—41no—
84#446—41no—
85C8×Q8—21no—
86C8×Q8—21no—
87C8×Q8—21no—
88C8×Q8—21no—
89C8×Q8C2×C411yescharacteristic
90C8×Q8C2×C411yescharacteristic
91C8×Q8C2×C411yescharacteristic
92C8×Q8C2×C411yescharacteristic
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×C4×C8—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—41no—
303#97—41no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
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—22no—
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—21no—
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—41no—
391C2×C8—41no—
392C2×C8—42no—
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—41no—
403C2×C8—82no—
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—42no—
446Q8#43012yescomplemented
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—42no—
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—41no—
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—22no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—21no—
491C8—21no—
492C8—22no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C8—81no—
506C8—81no—
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—22no—
522C4—21no—
523C4—21no—
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#2266811yescharacteristic
540C2#2267711yescharacteristic
541C2#2269411yescharacteristic
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. #22746C2semidirect
6. #9625C2semidirect
7. #9624C2semidirect
8. #8909C2semidirect
10. #8907C2semidirect
12. #8828C2semidirect
14. #8659C2semidirect
15. #8296C2semidirect
446. Q8#430semidirect

Maximal subgroups

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

subgroupindex
#227462
#226692
#180402
M16×SD162
C2×#15882
#96252
#96242
#89092
#89082
#89072
#89062
#88282
#86642
#86592
#82962