Finite Group Databasebeta

← #226164

Subgroups — #226164

Order 512.

589 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#22688C211yesmaximal, characteristic
3#17868C211yesmaximal, characteristic
4#17832C211yesmaximal, characteristic, complemented
5#17828C211yesmaximal, characteristic, complemented
6#9630C211yesmaximal, characteristic, complemented
7#9629C211yesmaximal, characteristic, complemented
8#8905C211yesmaximal, characteristic, complemented
9#8904C211yesmaximal, characteristic, complemented
10#8902C211yesmaximal, characteristic, complemented
11#8902C211yesmaximal, characteristic, complemented
12#8828C211yesmaximal, characteristic, complemented
13#8669C211yesmaximal, characteristic, complemented
14#8666C211yesmaximal, characteristic, complemented
15#8289C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C2×#443—21no—
19C4×#135C2211yescharacteristic
20#2925C2211yescharacteristic
21#2906C2211yescharacteristic
22#2906C2211yescharacteristic
23#2906C2211yescharacteristic
24#1945—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×#106C2211yescharacteristic
37#1770C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39#1765—21no—
40#1588C411yescharacteristic, complemented
41#1588C411yescharacteristic, complemented
42#1587C411yescharacteristic, complemented
43#1587C411yescharacteristic, complemented
44#1586C411yescharacteristic, complemented
45#1586C411yescharacteristic, complemented
46#1585C411yescharacteristic, complemented
47#1585C411yescharacteristic, complemented
48#1574—21no—
49#1573—21no—
50#1571—21no—
51#1571—21no—
52#1571C2211yescharacteristic
53#1571C2211yescharacteristic
54#1571C2211yescharacteristic
55#1570C2211yescharacteristic
56#1569—21no—
57#1567—21no—
58#1567C2211yescharacteristic
59#1567C2211yescharacteristic
60#1566C2211yescharacteristic
61#1566C2211yescharacteristic
62#1565—21no—
63#1564C2211yescharacteristic
64C2×#335—21no—
65C2×#335C2211yescharacteristic
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#443—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×#106—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×C4×C8—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—21no—
162#335—21no—
163#335—21no—
164#335—21no—
165#335—41no—
166#335C2×C411yescharacteristic
167#335C2×C411yescharacteristic
168#335D411yescharacteristic
169#335D411yescharacteristic
170#335D411yescharacteristic
171#335D411yescharacteristic
172#334—21no—
173#334—21no—
174#334—21no—
175#334—21no—
176#334—41no—
177#334C2×C411yescharacteristic
178#334C2×C411yescharacteristic
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#107C4⋊C411yescharacteristic
252#107C4⋊C411yescharacteristic
253#106—42no—
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—81no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—41no—
314C4×C8—42no—
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—21no—
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—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
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—22no—
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—42no—
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#57611yescharacteristic
488C2×C4#57811yescharacteristic
489C8—22no—
490C8—21no—
491C8—21no—
492C8—21no—
493C8—21no—
494C8—21no—
495C8—21no—
496C8—21no—
497C8—21no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—81no—
508C8—81no—
509C8—81no—
510C8—81no—
511C22—21no—
512C22—21no—
513C22—22no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22D4×C4⋊C411yescharacteristic
520C22#334311yescharacteristic
521C22#334611yescharacteristic
522C4—22no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—82no—
532C4—161no—
533C4C2×#42811yescharacteristic
534C4#291311yescharacteristic
535C4#295511yescharacteristic
536C4#295611yescharacteristic
537C4#326411yescharacteristic
538C4#327011yescharacteristic
539C2—21no—
540C2—21no—
541C2—42no—
542C2—81no—
543C2#2266111yescharacteristic
544C2#2268711yescharacteristic
545C2#2269411yescharacteristic
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. #17832C2semidirect
5. #17828C2semidirect
6. #9630C2semidirect
7. #9629C2semidirect
8. #8905C2semidirect
9. #8904C2semidirect
10. #8902C2semidirect
11. #8902C2semidirect
12. #8828C2semidirect
13. #8669C2semidirect
14. #8666C2semidirect
15. #8289C2semidirect
40. #1588C4semidirect
41. #1588C4semidirect
42. #1587C4semidirect
43. #1587C4semidirect
44. #1586C4semidirect
45. #1586C4semidirect
46. #1585C4semidirect
47. #1585C4semidirect

Maximal subgroups

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

subgroupindex
#227822
#226882
#178682
#178322
#178282
#96302
#96292
#89052
#89042
#89022
#88282
#86692
#86662
#82892