Finite Group Databasebeta

← #211879

Subgroups — #211879

Order 512.

596 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21628C211yesmaximal, characteristic
2#21627C211yesmaximal, characteristic
3#19126C211yesmaximal, characteristic
4#17977C211yesmaximal, characteristic, complemented
5#17316C211yesmaximal, characteristic
6#9718C211yesmaximal, characteristic, complemented
7#9624C211yesmaximal, characteristic, complemented
8#9589C211yesmaximal, characteristic, complemented
9#9589C211yesmaximal, characteristic, complemented
10#9207C211yesmaximal, characteristic
11#9207C211yesmaximal, characteristic
12#8667C211yesmaximal, characteristic
13#8291C211yesmaximal, characteristic, complemented
14#8256C211yesmaximal, characteristic, complemented
15#8252C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3076C2211yescharacteristic
18#3076C2211yescharacteristic
19C8×#41C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2960C2211yescharacteristic
22#2960C2211yescharacteristic
23C4×#131C2211yescharacteristic
24#1986C2211yescharacteristic
25#1986C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1922—21no—
29#1922—21no—
30#1866—21no—
31#1866—21no—
32#1806C2211yescharacteristic
33#1806C2211yescharacteristic
34C4×#107C2211yescharacteristic
35#1773—21no—
36#1772—21no—
37#1770C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39#1765—21no—
40#1764—21no—
41#1684C411yescharacteristic, complemented
42#1684C411yescharacteristic, complemented
43#1683C411yescharacteristic, complemented
44#1683C411yescharacteristic, complemented
45#1676C411yescharacteristic, complemented
46#1675C411yescharacteristic, complemented
47#1670C411yescharacteristic, complemented
48#1669C411yescharacteristic, complemented
49#1584—21no—
50#1584—21no—
51#1584—21no—
52#1584—21no—
53#1567C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1542C2211yescharacteristic
56#1542C2211yescharacteristic
57#1503C2211yescharacteristic
58#1503C2211yescharacteristic
59#1493C2211yescharacteristic
60#1493C2211yescharacteristic
61#1450—21no—
62#1450C2211yescharacteristic
63C8×M16—21no—
64C8×M16—21no—
65C8×M16C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C2×#322—21no—
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C2×#131D411yescharacteristic
76C2×#131D411yescharacteristic
77#533C2311yescharacteristic
78#533C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#500C2×C411yescharacteristic
81#494C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#487C2×C411yescharacteristic
84#479—21no—
85#479—21no—
86#478—21no—
87#478—21no—
88#477C2×C411yescharacteristic
89#477C2×C411yescharacteristic
90#476C2×C411yescharacteristic
91#476C2×C411yescharacteristic
92#475C2×C411yescharacteristic
93#475C2×C411yescharacteristic
94#474C2×C411yescharacteristic
95#474C2×C411yescharacteristic
96#467—21no—
97#467—21no—
98#446—21no—
99#446—21no—
100#446—21no—
101#446—21no—
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8—21no—
105C8×Q8—21no—
106C8×Q8—21no—
107C8×Q8C2×C411yescharacteristic
108C8×Q8C2×C411yescharacteristic
109C4×Q16—21no—
110C4×Q16—21no—
111C8×D4—21no—
112C8×D4—21no—
113C8×D4—21no—
114C8×D4—22no—
115C8×D4C2×C411yescharacteristic
116C8×D4C2×C411yescharacteristic
117#431—21no—
118#431—21no—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#108—21no—
122C2×#108—21no—
123#427—21no—
124#427—21no—
125C2×#107—21no—
126#424—21no—
127#424—21no—
128#423C2311yescharacteristic
129#423C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×#104—21no—
133C2×#104—21no—
134#416—21no—
135#416—21no—
136#405—21no—
137#405—21no—
138#405—21no—
139#405—21no—
140#405D411yescharacteristic
141#405D411yescharacteristic
142C4×M16—21no—
143C4×M16C2311yescharacteristic
144C2×#97—21no—
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154#341—21no—
155#341—21no—
156#340—21no—
157#339—21no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#336—21no—
164#336—21no—
165#335—21no—
166#335—21no—
167#335C2×C411yescharacteristic
168#335C2×C411yescharacteristic
169#335C2×C411yescharacteristic
170#335C2×C411yescharacteristic
171#333C2×C411yescharacteristic
172#333C2×C411yescharacteristic
173#333C2×C411yescharacteristic
174#333C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#332C2×C411yescharacteristic
179#328—22no—
180#328—22no—
181Q8⋊C8—21no—
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184Q8⋊C8—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192C82—21no—
193C82—21no—
194C82—21no—
195C2×#41C2×D411yescharacteristic
196C2×Q16—21no—
197C2×Q16—21no—
198#131—22no—
199#131—22no—
200#131—22no—
201#131Q1614yescomplemented
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16C2×D411yescharacteristic
207C2×M16C2×D411yescharacteristic
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8#4111yescharacteristic
219C22×C8#4111yescharacteristic
220C4⋊Q8C22×C411yescharacteristic
221C4⋊Q8C22×C411yescharacteristic
222C4⋊Q8C22×C411yescharacteristic
223C4⋊Q8C22×C411yescharacteristic
224#125—21no—
225#125—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C22×C411yescharacteristic
235C4×Q8#4111yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C22×C411yescharacteristic
239C4×D4#4111yescharacteristic
240#117—21no—
241#117—21no—
242#117—21no—
243#117C2×D411yescharacteristic
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C2411yescharacteristic
250#108—42no—
251#108—42no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—42no—
259#106—21no—
260#106—21no—
261#106—21no—
262#106—21no—
263C4⋊C8—22no—
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—41no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C22×C411yescharacteristic
281C4⋊C8C22×C411yescharacteristic
282C4⋊C8C22×C411yescharacteristic
283#104—42no—
284#104—42no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#102—21no—
297C22⋊C8—22no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8#4112yes—
302#97—21no—
303#97—21no—
304#97—21no—
305#97—41no—
306#97—41no—
307#97—42no—
308C4×C8—22no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—41no—
318C4×C8—41no—
319C4×C8—42no—
320C4×C8C22×C411yescharacteristic
321C4×C8C22×C411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8C2×D411yescharacteristic
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8#4111yescharacteristic
329C4×C8#4111yescharacteristic
330#95—21no—
331#95—21no—
332#41—21no—
333#41—21no—
334#41C2×Q1612yes—
335#41#13512yes—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8C4×D411yescharacteristic
341C2×D4—21no—
342C2×D4C4×D411yescharacteristic
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351Q16—81no—
352Q16—81no—
353M16—24no—
354M16—21no—
355M16—21no—
356M16—24no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16C2×Q1612yes—
362M16#13512yes—
363C2×C8—24no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—24no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—81no—
387C2×C8—81no—
388C2×C8—81no—
389C2×C8—82no—
390C2×C8C4×D411yescharacteristic
391C2×C8C4×D411yescharacteristic
392C2×C8C4×D411yescharacteristic
393C2×C8C4×D411yescharacteristic
394C2×C8#12111yescharacteristic
395C2×C8#12111yescharacteristic
396C2×C8#12111yescharacteristic
397C2×C8#12111yescharacteristic
398C2×C8C2×Q1612yes—
399C2×C8#13512yes—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
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⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4C4×D411yescharacteristic
419C4⋊C4C4×D411yescharacteristic
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42C23×C411yescharacteristic
435C42C22×D411yescharacteristic
436C42C2×#4111yescharacteristic
437C42C2×#4111yescharacteristic
438C23—21no—
439C23#53211yescharacteristic
440Q8—21no—
441Q8—41no—
442Q8—41no—
443Q8—41no—
444Q8—41no—
445Q8—81no—
446Q8—81no—
447Q8C4×Q1612yes—
448D4—22no—
449D4—22no—
450D4—21no—
451D4C4×Q1612yes—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4C4×#4111yescharacteristic
476C2×C4C2×#12111yescharacteristic
477C2×C4#52511yescharacteristic
478C2×C4#52911yescharacteristic
479C2×C4C22×Q1611yescharacteristic
480C2×C4C2×#13511yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—22no—
488C8—21no—
489C8—22no—
490C8—21no—
491C8—42no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—81no—
501C8—81no—
502C8#46712yes—
503C8#46712yes—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22C4×#12111yescharacteristic
510C22#316811yescharacteristic
511C22#317011yescharacteristic
512C4—21no—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—82no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—161no—
524C4C2×C4×Q1611yescharacteristic
525C4C4×#13511yescharacteristic
526C4#295611yescharacteristic
527C4#295711yescharacteristic
528C4C2×#46711yescharacteristic
529C4#305011yescharacteristic
530C2—21no—
531C2—21no—
532C2—42no—
533C2#2151911yescharacteristic
534C2C4×#46711yescharacteristic
535C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×#41C2×D4
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
4. #17977C2semidirect
6. #9718C2semidirect
7. #9624C2semidirect
8. #9589C2semidirect
9. #9589C2semidirect
13. #8291C2semidirect
14. #8256C2semidirect
15. #8252C2semidirect
41. #1684C4semidirect
42. #1684C4semidirect
43. #1683C4semidirect
44. #1683C4semidirect
45. #1676C4semidirect
46. #1675C4semidirect
47. #1670C4semidirect
48. #1669C4semidirect
201. #131Q16semidirect

Maximal subgroups

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

subgroupindex
#216282
#216272
#191262
#179772
#173162
#97182
#96242
#95892
#92072
#86672
#82912
#82562
#82522