Finite Group Databasebeta

← #226154

Subgroups — #226154

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22782C211yesmaximal, characteristic
2#22686C211yesmaximal, characteristic
3#17863C211yesmaximal, characteristic
4#17830C211yesmaximal, characteristic, complemented
5#17826C211yesmaximal, characteristic, complemented
6#9632C211yesmaximal, characteristic, complemented
7#9631C211yesmaximal, characteristic, complemented
8#8904C211yesmaximal, characteristic, complemented
9#8904C211yesmaximal, characteristic, complemented
10#8903C211yesmaximal, characteristic, complemented
11#8902C211yesmaximal, characteristic, complemented
12#8828C211yesmaximal, characteristic, complemented
13#8668C211yesmaximal, characteristic, complemented
14#8667C211yesmaximal, characteristic, complemented
15#8287C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C2×#444—21no—
19C4×#135C2211yescharacteristic
20#2925C2211yescharacteristic
21#2906C2211yescharacteristic
22#2906C2211yescharacteristic
23#2906C2211yescharacteristic
24#1946—21no—
25#1938—21no—
26#1922—21no—
27#1876—21no—
28#1866—21no—
29#1866—21no—
30#1848C2211yescharacteristic
31#1837C2211yescharacteristic
32#1806C2211yescharacteristic
33#1806C2211yescharacteristic
34#1806C2211yescharacteristic
35#1806C2211yescharacteristic
36C4×#107C2211yescharacteristic
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39#1765—21no—
40#1584C411yescharacteristic, complemented
41#1584C411yescharacteristic, complemented
42#1583C411yescharacteristic, complemented
43#1583C411yescharacteristic, complemented
44#1582C411yescharacteristic, complemented
45#1582C411yescharacteristic, complemented
46#1581C411yescharacteristic, complemented
47#1581C411yescharacteristic, complemented
48#1574—21no—
49#1573—21no—
50#1571C2211yescharacteristic
51#1570—21no—
52#1570—21no—
53#1570C2211yescharacteristic
54#1570C2211yescharacteristic
55#1570C2211yescharacteristic
56#1569—21no—
57#1567C2211yescharacteristic
58#1567C2211yescharacteristic
59#1566—21no—
60#1566C2211yescharacteristic
61#1566C2211yescharacteristic
62#1565—21no—
63#1564C2211yescharacteristic
64C2×#334—21no—
65C2×#334C2211yescharacteristic
66#1476C2211yescharacteristic
67#1476C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1303C2211yescharacteristic
72#1303C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C2×#135D411yescharacteristic
76C2×#131—21no—
77C2×#131—21no—
78C2×#131—21no—
79C4×#41C2311yescharacteristic
80C4×#41C2311yescharacteristic
81#446C2×C411yescharacteristic
82#446C2×C411yescharacteristic
83C8×Q8C2×C411yescharacteristic
84C8×Q8C2×C411yescharacteristic
85#444—44no—
86C4×Q16C2×C411yescharacteristic
87C4×Q16Q811yescharacteristic
88C4×SD16C2×C411yescharacteristic
89C4×SD16C2×C411yescharacteristic
90C4×SD16Q811yescharacteristic
91C4×SD16Q811yescharacteristic
92C4×D8C2×C411yescharacteristic
93C4×D8Q811yescharacteristic
94#436C2×C411yescharacteristic
95#436C2×C411yescharacteristic
96#435—21no—
97#435—21no—
98C8×D4—21no—
99C8×D4—21no—
100C8×D4C2×C411yescharacteristic
101C8×D4C2×C411yescharacteristic
102#433—41no—
103#432—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109C2×#108—21no—
110C2×#108—21no—
111#427D411yescharacteristic
112C2×#107—21no—
113#424—21no—
114#424—21no—
115#424—21no—
116#424—41no—
117#424—41no—
118#424—41no—
119#423—21no—
120#423—21no—
121#423—21no—
122#423C2311yescharacteristic
123#423C2311yescharacteristic
124#423C2311yescharacteristic
125#423C2311yescharacteristic
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×#104—21no—
132C2×#104—21no—
133C2×#104—21no—
134C2×#104—21no—
135#416D411yescharacteristic
136#416D411yescharacteristic
137#408—21no—
138#405—21no—
139C4×M16—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C4×C4⋊C4C2311yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148#341—21no—
149#341—21no—
150#340—21no—
151#339—21no—
152#339—21no—
153#339—21no—
154#339—21no—
155#336—21no—
156#336—21no—
157#335—21no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335—41no—
162#335C2×C411yescharacteristic
163#335C2×C411yescharacteristic
164#334—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168#334—21no—
169#334—21no—
170#334—21no—
171#334—21no—
172#334—41no—
173#334C2×C411yescharacteristic
174#334C2×C411yescharacteristic
175#334D411yescharacteristic
176#334D411yescharacteristic
177#334D411yescharacteristic
178#334D411yescharacteristic
179Q8⋊C8C2×C411yescharacteristic
180Q8⋊C8C2×C411yescharacteristic
181Q8⋊C8C2×C411yescharacteristic
182Q8⋊C8C2×C411yescharacteristic
183D4⋊C8C2×C411yescharacteristic
184D4⋊C8C2×C411yescharacteristic
185D4⋊C8C2×C411yescharacteristic
186D4⋊C8C2×C411yescharacteristic
187#322C2×C411yescharacteristic
188#322C2×C411yescharacteristic
189#322C2×C411yescharacteristic
190#322C2×C411yescharacteristic
191C2×#41C2×D411yescharacteristic
192C2×#41C2×D411yescharacteristic
193#135—22no—
194#135—22no—
195C2×Q16C4⋊C411yescharacteristic
196C2×SD16C4⋊C411yescharacteristic
197C2×SD16C4⋊C411yescharacteristic
198C2×D8C4⋊C411yescharacteristic
199#131—42no—
200#131—42no—
201#131—42no—
202#131—81no—
203#131—81no—
204#131—81no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—41no—
212C2×M16—41no—
213C2×M16—41no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—41no—
220C22×C8—41no—
221C22×C8—41no—
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8#4111yescharacteristic
225C22×C8#4111yescharacteristic
226C4×Q8C22×C411yescharacteristic
227C4×Q8C22×C411yescharacteristic
228C4×Q8C2×Q811yescharacteristic
229C4×Q8C2×Q811yescharacteristic
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4C22×C411yescharacteristic
235C4×D4C22×C411yescharacteristic
236C4×D4C2×Q811yescharacteristic
237C4×D4C2×Q811yescharacteristic
238#117—21no—
239#117—21no—
240#117C2×D411yescharacteristic
241#117C2×D411yescharacteristic
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C2411yescharacteristic
249#108—42no—
250#108—81no—
251#107—42no—
252#107C4⋊C411yescharacteristic
253#107C4⋊C411yescharacteristic
254#106C4⋊C411yescharacteristic
255#106C4⋊C411yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
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—41no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8C22×C411yescharacteristic
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C2×D411yescharacteristic
281C4⋊C8C2×D411yescharacteristic
282C4⋊C8C2×D411yescharacteristic
283C4⋊C8C2×D411yescharacteristic
284#104—42no—
285#104—42no—
286#104—42no—
287#104—81no—
288#103C4⋊C411yescharacteristic
289#103C4⋊C411yescharacteristic
290#103C4⋊C411yescharacteristic
291#103C4⋊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—41no—
303C22⋊C8—41no—
304#97—21no—
305#97—21no—
306#97—41no—
307#97—42no—
308#97—81no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
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×C8C2×Q811yescharacteristic
324C4×C8C2×Q811yescharacteristic
325#95—21no—
326#95—21no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41AGL(1,ℤ/8)12yes—
334#41#13712yes—
335C2×Q8C2×C4⋊C411yescharacteristic
336C2×Q8C2×C4⋊C411yescharacteristic
337C2×D4—21no—
338C2×D4—21no—
339C2×D4C2×C4⋊C411yescharacteristic
340C2×D4C2×C4⋊C411yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C42−1+411yescharacteristic
352Q16—22no—
353SD16—22no—
354SD16—22no—
355D8—22no—
356M16—42no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—81no—
366M16—82no—
367M16—82no—
368M16—82no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
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—21no—
383C2×C8—21no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—81no—
395C2×C8—81no—
396C2×C8—82no—
397C2×C8—82no—
398C2×C8—82no—
399C2×C8C2×C4⋊C411yescharacteristic
400C2×C8C2×C4⋊C411yescharacteristic
401C2×C8C2×C4⋊C411yescharacteristic
402C2×C8C2×C4⋊C411yescharacteristic
403C2×C8C4×D411yescharacteristic
404C2×C8C4×D411yescharacteristic
405C2×C8C4×D411yescharacteristic
406C2×C8C4×D411yescharacteristic
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4C2×C4⋊C411yescharacteristic
416C4⋊C4C2×C4⋊C411yescharacteristic
417C4⋊C4C2×C4⋊C411yescharacteristic
418C4⋊C4C2×C4⋊C411yescharacteristic
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42C23×C411yescharacteristic
436C42C22×D411yescharacteristic
437C42C22×Q811yescharacteristic
438C42H(5,𝔽2)11yescharacteristic
439C23—21no—
440C23—21no—
441C23#54711yescharacteristic
442Q8—21no—
443Q8—21no—
444Q8—21no—
445Q8#42812yes—
446D4—22no—
447D4—22no—
448D4—21no—
449D4—21no—
450D4—21no—
451D4—41no—
452D4—41no—
453D4#42812yes—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
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×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4C22×C4⋊C411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4#52011yescharacteristic
482C2×C4D4211yescharacteristic
483C2×C4D4×Q811yescharacteristic
484C2×C4#55411yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C2×C4C2×#13711yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57711yescharacteristic
489C8—21no—
490C8—21no—
491C8—21no—
492C8—21no—
493C8—21no—
494C8—21no—
495C8—21no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—81no—
507C8—81no—
508C8—81no—
509C8—81no—
510C8#44412yescomplemented
511C22—21no—
512C22—21no—
513C22—22no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22D4×C4⋊C411yescharacteristic
520C22#334411yescharacteristic
521C22#334511yescharacteristic
522C4—22no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—161no—
532C4—161no—
533C4C2×#42811yescharacteristic
534C4#291211yescharacteristic
535C4C2×#44411yescharacteristic
536C4#295711yescharacteristic
537C4#326511yescharacteristic
538C4#326611yescharacteristic
539C2—21no—
540C2—21no—
541C2—42no—
542C2—81no—
543C2#2265911yescharacteristic
544C2#2268611yescharacteristic
545C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC2×#135D4
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
4. #17830C2semidirect
5. #17826C2semidirect
6. #9632C2semidirect
7. #9631C2semidirect
8. #8904C2semidirect
9. #8904C2semidirect
10. #8903C2semidirect
11. #8902C2semidirect
12. #8828C2semidirect
13. #8668C2semidirect
14. #8667C2semidirect
15. #8287C2semidirect
40. #1584C4semidirect
41. #1584C4semidirect
42. #1583C4semidirect
43. #1583C4semidirect
44. #1582C4semidirect
45. #1582C4semidirect
46. #1581C4semidirect
47. #1581C4semidirect
510. C8#444semidirect

Maximal subgroups

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

subgroupindex
#227822
#226862
#178632
#178302
#178262
#96322
#96312
#89042
#89032
#89022
#88282
#86682
#86672
#82872