Finite Group Databasebeta

← #211848

Subgroups — #211848

Order 512.

669 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22280C211yesmaximal, characteristic, complemented
2#20567C211yesmaximal, characteristic, complemented
3#19133C211yesmaximal, characteristic
4#17935C211yesmaximal, characteristic, complemented
5#17928C211yesmaximal, characteristic, complemented
6#9750C211yesmaximal, characteristic, complemented
7#9629C211yesmaximal, characteristic, complemented
8#9582C211yesmaximal, characteristic, complemented
9#9581C211yesmaximal, characteristic, complemented
10#9233C211yesmaximal, characteristic, complemented
11#9233C211yesmaximal, characteristic, complemented
12#8664C211yesmaximal, characteristic, complemented
13#8270C211yesmaximal, characteristic, complemented
14#8257C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic, complemented
16#3147C2211yescharacteristic
17#3100C2211yescharacteristic
18#3100C2211yescharacteristic
19#2931C2211yescharacteristic
20#2931C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1962C2211yescharacteristic
25#1962C2211yescharacteristic
26#1876—21no—
27#1876—21no—
28#1876—21no—
29#1876—21no—
30#1837C2211yescharacteristic
31C8×C4⋊C4C2211yescharacteristic
32C8×C4⋊C4C2211yescharacteristic
33#1767C2211yescharacteristic
34#1753—21no—
35#1753—21no—
36C4×C22⋊C8—21no—
37#1744C2211yescharacteristic
38#1728—21no—
39#1689C412yescomplemented
40#1687C412yescomplemented
41#1685C412yescomplemented
42#1685C412yescomplemented
43#1594—22no—
44#1589—22no—
45#1586—22no—
46#1581—22no—
47#1571C2211yescharacteristic
48#1570C2211yescharacteristic
49#1564C2211yescharacteristic
50#1564C2211yescharacteristic
51#1540C2211yescharacteristic
52#1540C2211yescharacteristic
53#1502C2211yescharacteristic
54#1502C2211yescharacteristic
55#1486C2211yescharacteristic
56#1486C2211yescharacteristic
57#1480C2211yescharacteristic
58#1480C2211yescharacteristic
59#1452C2211yescharacteristic
60#1450C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1301C2211yescharacteristic
64#1301C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C2×#135—21no—
68C2×#135—21no—
69C2×#131—21no—
70C2×#131—21no—
71C22×M16—21no—
72C22×M16—21no—
73#532C2311yescharacteristic
74#532C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C22×C42—21no—
78#495—21no—
79#495—21no—
80#494C2×C411yescharacteristic, complemented
81#493C2×C411yescharacteristic, complemented
82#492C2×C411yescharacteristic, complemented
83#492C2×C411yescharacteristic, complemented
84#480—21no—
85#480—21no—
86#474C2×C412yes—
87#474C2×C412yes—
88#459C2×C412yes—
89#459C2×C412yes—
90#447—21no—
91#447—21no—
92C4×D8—22no—
93C4×D8—22no—
94#436—21no—
95#436—21no—
96#436—21no—
97#436—21no—
98#436—21no—
99#436—21no—
100#436—21no—
101#436—21no—
102#436C2×C412yes—
103#436C2×C412yes—
104#436D412yes—
105#436D412yes—
106C8×D4—21no—
107C8×D4—21no—
108C8×D4—21no—
109C8×D4—21no—
110#431C2311yescharacteristic
111#431C2311yescharacteristic
112#431C2311yescharacteristic
113#431C2311yescharacteristic
114#423C2311yescharacteristic
115#423C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×#104—21no—
119C2×#104—21no—
120C2×#104—21no—
121C2×#104—21no—
122#408—21no—
123#408—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×#97—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C43C2311yescharacteristic
136#341—21no—
137#341—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#335—21no—
147#335C2×C412yescomplemented
148#334—22no—
149#334—21no—
150#334C2×C412yescomplemented
151#332C2×C412yescomplemented
152#332C2×C412yescomplemented
153#331C2×C412yescomplemented
154#331C2×C412yescomplemented
155#327—22no—
156#327—22no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161#322—21no—
162#322—21no—
163#322—21no—
164C82—21no—
165C2×#41—21no—
166C2×#41—21no—
167C22×D4#4111yescharacteristic
168C23×C4—21no—
169C23×C4—22no—
170#135—81no—
171#135—81no—
172#135—81no—
173#135—81no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×SD16—21no—
177C2×SD16—21no—
178C2×SD16—21no—
179C2×SD16—21no—
180C2×D8—22no—
181C2×D8—22no—
182C2×D8—21no—
183C2×D8—21no—
184#131—81no—
185#131—81no—
186#131—81no—
187#131—81no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—41no—
195C2×M16—41no—
196C2×M16—81no—
197C2×M16—81no—
198C2×M16—81no—
199C2×M16—81no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—41no—
203C22×C8—41no—
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C4⋊Q8C22×C411yescharacteristic
209C4⋊Q8C22×C411yescharacteristic
210#127C22×C411yescharacteristic
211#127C22×C411yescharacteristic
212#124—21no—
213#124—21no—
214#121—21no—
215#121—21no—
216#121—21no—
217#121—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—44no—
225C4×D4C22×C412yes—
226C4×D4C2×D412yes—
227C2×C4⋊C4—21no—
228C2×C4⋊C4#4111yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—81no—
236C2×C42—81no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2411yescharacteristic
240#107—22no—
241#107—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253C4⋊C8C2×D412yes—
254#104—81no—
255#104—81no—
256#104—81no—
257#104—81no—
258#102—22no—
259#102—22no—
260#102—22no—
261#102—22no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273C22⋊C8—41no—
274#97—81no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—81no—
280C4×C8—81no—
281C4×C8C22×C411yescharacteristic
282C4×C8C22×C411yescharacteristic
283C4×C8C2×D411yescharacteristic
284C4×C8C2×D411yescharacteristic
285C4×C8#4111yescharacteristic
286C4×C8#4111yescharacteristic
287C4×C8#4111yescharacteristic
288C4×C8#4111yescharacteristic
289#95—22no—
290#95#4112yes—
291C24—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4C4×D412yes—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—82no—
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331Q16—81no—
332Q16—81no—
333SD16—81no—
334SD16—81no—
335SD16—81no—
336SD16—81no—
337D8—81no—
338D8—81no—
339D8—82no—
340D8—82no—
341M16—41no—
342M16—41no—
343M16—41no—
344M16—41no—
345M16—82no—
346M16—82no—
347M16—82no—
348M16—82no—
349M16—82no—
350M16—82no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—82no—
372C2×C8—82no—
373C2×C8C4×D411yescharacteristic
374C2×C8C4×D411yescharacteristic
375C2×C8C4×D411yescharacteristic
376C2×C8C4×D411yescharacteristic
377C2×C8#12111yescharacteristic
378C2×C8#12111yescharacteristic
379C2×C8#12111yescharacteristic
380C2×C8#12111yescharacteristic
381C4⋊C4—24no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4C4×D412yes—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—44no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—42no—
400C42—42no—
401C42—41no—
402C42—41no—
403C42—82no—
404C42C23×C411yescharacteristic
405C42C22×D411yescharacteristic
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C23—22no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23#52511yescharacteristic
416Q8—41no—
417Q8—41no—
418Q8—41no—
419Q8—41no—
420D4—22no—
421D4—22no—
422D4—42no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—42no—
428D4—42no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—81no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—88no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4C2×C4×D411yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C4×#4111yescharacteristic
463C2×C4C2×#12111yescharacteristic
464C2×C4#52911yescharacteristic
465C2×C4#53211yescharacteristic
466C2×C4#57611yescharacteristic
467C2×C4#57611yescharacteristic
468C2×C4#57611yescharacteristic
469C2×C4#57711yescharacteristic
470C8—21no—
471C8—21no—
472C8—21no—
473C8—21no—
474C8—21no—
475C8—21no—
476C8—21no—
477C8—21no—
478C8—42no—
479C8—42no—
480C8—42no—
481C8—42no—
482C8—81no—
483C8—81no—
484C8—81no—
485C8—81no—
486C22—21no—
487C22—21no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22—82no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22C4×#12111yescharacteristic
500C22#311911yescharacteristic
501C22#312211yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4#294611yescharacteristic
512C4#294811yescharacteristic
513C4#295612yes—
514C4#304111yescharacteristic
515C4#304511yescharacteristic
516C2—21no—
517C2—21no—
518C2—82no—
519C2—161no—
520C2—161no—
521C2#2151311yescharacteristic
522C2#2155111yescharacteristic
523C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#19133C2
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. #22280C2semidirect
2. #20567C2semidirect
4. #17935C2semidirect
5. #17928C2semidirect
6. #9750C2semidirect
7. #9629C2semidirect
8. #9582C2semidirect
9. #9581C2semidirect
10. #9233C2semidirect
11. #9233C2semidirect
12. #8664C2semidirect
13. #8270C2semidirect
14. #8257C2semidirect
15. #8251C2semidirect
39. #1689C4semidirect
40. #1687C4semidirect
41. #1685C4semidirect
42. #1685C4semidirect
80. #494C2×C4semidirect
81. #493C2×C4semidirect
82. #492C2×C4semidirect
83. #492C2×C4semidirect
147. #335C2×C4semidirect
150. #334C2×C4semidirect
151. #332C2×C4semidirect
152. #332C2×C4semidirect
153. #331C2×C4semidirect
154. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222802
#205672
#191332
#179352
#179282
#97502
#96292
#95822
#95812
#92332
#86642
#82702
#82572
#82512