Finite Group Databasebeta

← #211720

Subgroups — #211720

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21660C211yesmaximal, characteristic, complemented
2D4×#97C211yesmaximal, characteristic, complemented
3#19110C211yesmaximal, characteristic
4#17902C211yesmaximal, characteristic, complemented
5#17900C211yesmaximal, characteristic
6#9754C211yesmaximal, characteristic, complemented
7#9632C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9582C211yesmaximal, characteristic, complemented
10#9233C211yesmaximal, characteristic
11#9231C211yesmaximal, characteristic, complemented
12#8660C211yesmaximal, characteristic, complemented
13#8268C211yesmaximal, characteristic, complemented
14#8256C211yesmaximal, characteristic, complemented
15#8255C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#3101C2211yescharacteristic
18#3100C2211yescharacteristic
19#2960C2211yescharacteristic
20#2931C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C22×#97—21no—
24C42×D4C2211yescharacteristic
25#1994C2211yescharacteristic
26#1962C2211yescharacteristic
27#1938—21no—
28#1938—21no—
29#1837C2211yescharacteristic
30#1770C2211yescharacteristic
31#1770C2211yescharacteristic
32#1767C2211yescharacteristic
33#1765—21no—
34#1765—21no—
35#1752—21no—
36#1752—21no—
37#1749—21no—
38C4×#97C2211yescharacteristic
39#1683C412yescomplemented
40#1681C412yescomplemented
41#1679C412yescomplemented
42#1677C412yescomplemented
43#1582—22no—
44#1581—22no—
45C8×SD16—22no—
46C8×D8—22no—
47#1571C2211yescharacteristic
48#1570C2211yescharacteristic
49#1564C2211yescharacteristic
50#1564C2211yescharacteristic
51#1542C2211yescharacteristic
52#1540C2211yescharacteristic
53#1503C2211yescharacteristic
54#1502C2211yescharacteristic
55#1486C2211yescharacteristic
56#1486C2211yescharacteristic
57#1481C2211yescharacteristic
58#1480C2211yescharacteristic
59#1450C2211yescharacteristic
60#1450C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1301C2211yescharacteristic
64#1301C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C2×#131—21no—
68C23×C8—21no—
69C23×C8—21no—
70#533C2311yescharacteristic
71#532C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#488—21no—
76#488—21no—
77#487C2×C411yescharacteristic, complemented
78#487C2×C411yescharacteristic, complemented
79#486C2×C411yescharacteristic, complemented
80#486C2×C411yescharacteristic, complemented
81#483—21no—
82#480—21no—
83#476C2×C412yes—
84#474C2×C412yes—
85#461C2×C412yes—
86#459C2×C412yes—
87#450—21no—
88#447—21no—
89#446—21no—
90#446—21no—
91C8×Q8—21no—
92C8×Q8—21no—
93C4×SD16—22no—
94C4×D8—22no—
95#436—21no—
96#436—21no—
97C8×D4—21no—
98C8×D4—21no—
99C8×D4—21no—
100C8×D4—21no—
101C8×D4—21no—
102C8×D4—21no—
103C8×D4C2×C412yes—
104C8×D4C2×C412yes—
105C8×D4D412yes—
106C8×D4D412yes—
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#431C2311yescharacteristic
111#424—21no—
112#423C2311yescharacteristic
113#423C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×#104—21no—
117C2×#104—21no—
118#416—21no—
119#416—21no—
120#408—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123#405—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—44no—
128C2×#97D411yescharacteristic
129C2×#97D411yescharacteristic
130C2×#97D411yescharacteristic
131C2×#97D411yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—21no—
137C4×C22⋊C4—22no—
138C43C2311yescharacteristic
139#341—21no—
140#341—21no—
141#339—21no—
142#339—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#335—22no—
147#335—21no—
148#335C2×C412yescomplemented
149#334—22no—
150#334—21no—
151#334C2×C412yescomplemented
152#333C2×C412yescomplemented
153#332C2×C412yescomplemented
154#332C2×C412yescomplemented
155#331C2×C412yescomplemented
156#328—22no—
157#327—22no—
158Q8⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161D4⋊C8—22no—
162#322—21no—
163#322—21no—
164C82—21no—
165C82—21no—
166C2×#41—21no—
167C22×D4#4111yescharacteristic
168C23×C4—21no—
169C23×C4—22no—
170C2×SD16—22no—
171C2×D8—22no—
172#131—42no—
173#131—42no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—41no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—41no—
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C4⋊Q8C22×C411yescharacteristic
193C4⋊Q8C22×C411yescharacteristic
194C4⋊Q8C22×C411yescharacteristic
195#127C22×C411yescharacteristic
196#125—21no—
197#124—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200#121—21no—
201#121—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—44no—
209C4×D4C22×C412yes—
210C4×D4C2×D412yes—
211#117—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4#4111yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225#107—22no—
226#106—22no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C412yes—
240C4⋊C8C2×D412yes—
241#104—42no—
242#104—42no—
243#103—21no—
244#103—21no—
245#103—21no—
246#103—21no—
247#103—22no—
248#102—22no—
249#102—22no—
250#102—21no—
251#102—21no—
252#102—21no—
253#102—21no—
254#102—22no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265#97—22no—
266#97—22no—
267#97—22no—
268#97—41no—
269#97—42no—
270#97—42no—
271#97—41no—
272#97—42no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—41no—
278C4×C8C22×C411yescharacteristic
279C4×C8C22×C411yescharacteristic
280C4×C8C2×D411yescharacteristic
281C4×C8C2×D411yescharacteristic
282C4×C8#4111yescharacteristic
283C4×C8#4111yescharacteristic
284C4×C8#4111yescharacteristic
285C4×C8#4111yescharacteristic
286#95—22no—
287#95#4112yes—
288C24—21no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×D4—24no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4C4×D412yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—44no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—44no—
315C22×C4—41no—
316C22×C4—82no—
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320SD16—44no—
321D8—44no—
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—82no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—44no—
344C2×C8—44no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—82no—
350C2×C8C4×D411yescharacteristic
351C2×C8C4×D411yescharacteristic
352C2×C8C4×D411yescharacteristic
353C2×C8C4×D411yescharacteristic
354C2×C8#12111yescharacteristic
355C2×C8#12111yescharacteristic
356C2×C8#12111yescharacteristic
357C2×C8#12111yescharacteristic
358C2×C8#13512yes—
359C2×C8AGL(1,ℤ/8)12yes—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4C4×D412yes—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—44no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—44no—
381C42—42no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—81no—
386C42#13512yes—
387C42C23×C411yescharacteristic
388C42C22×D411yescharacteristic
389C42C2×#4111yescharacteristic
390C42C2×#4111yescharacteristic
391C23—24no—
392C23—22no—
393C23—42no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23#52511yescharacteristic
398Q8—41no—
399Q8—41no—
400Q8—42no—
401Q8—81no—
402D4—24no—
403D4—22no—
404D4—41no—
405D4—41no—
406D4—42no—
407D4—81no—
408D4—81no—
409D4—81no—
410D4#44414yes—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—88no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C4×#4111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4#52911yescharacteristic
440C2×C4#53211yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4C2×#13511yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4#57511yescharacteristic
445C8—21no—
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—21no—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—42no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8#46312yes—
460C8#46312yes—
461C22—21no—
462C22—21no—
463C22—24no—
464C22—44no—
465C22—44no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22C4×#12111yescharacteristic
472C22#310211yescharacteristic
473C22#310511yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—82no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4C4×#13511yescharacteristic
485C4C4×AGL(1,ℤ/8)11yescharacteristic
486C4C2×#44412yes—
487C4C2×#46311yescharacteristic
488C4#303511yescharacteristic
489C2—21no—
490C2—21no—
491C2—44no—
492C2—161no—
493C2C4×#46311yescharacteristic
494C2#2154611yescharacteristic
495C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#3100C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21660C2semidirect
2. D4×#97C2semidirect
4. #17902C2semidirect
6. #9754C2semidirect
7. #9632C2semidirect
8. #9584C2semidirect
9. #9582C2semidirect
11. #9231C2semidirect
12. #8660C2semidirect
13. #8268C2semidirect
14. #8256C2semidirect
15. #8255C2semidirect
39. #1683C4semidirect
40. #1681C4semidirect
41. #1679C4semidirect
42. #1677C4semidirect
77. #487C2×C4semidirect
78. #487C2×C4semidirect
79. #486C2×C4semidirect
80. #486C2×C4semidirect
148. #335C2×C4semidirect
151. #334C2×C4semidirect
152. #333C2×C4semidirect
153. #332C2×C4semidirect
154. #332C2×C4semidirect
155. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#216602
D4×#972
#191102
#179022
#179002
#97542
#96322
#95842
#95822
#92332
#92312
#86602
#82682
#82562
#82552