Finite Group Databasebeta

← #211280

Subgroups — #211280

Order 512.

731 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22219C211yesmaximal, characteristic
2#21616C211yesmaximal, characteristic, complemented
3C4×#460C211yesmaximal, characteristic
4#20569C211yesmaximal, characteristic, complemented
5C2×#1678C211yesmaximal, characteristic
6#9735C211yesmaximal, characteristic
7#9612C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9564C211yesmaximal, characteristic
10#9222C211yesmaximal, characteristic
11#9216C211yesmaximal, characteristic, complemented
12#8643C211yesmaximal, characteristic, complemented
13#8277C211yesmaximal, characteristic, complemented
14#8234C211yesmaximal, characteristic
15C4×#333C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C4×SD16—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1982C2211yescharacteristic
28#1957C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1926—21no—
32#1926—21no—
33#1926—21no—
34#1918—21no—
35#1880—21no—
36#1865—21no—
37#1864—21no—
38#1864—21no—
39#1844C2211yescharacteristic
40#1803C2211yescharacteristic
41#1803C2211yescharacteristic
42#1776—21no—
43C8×C4⋊C4—21no—
44C8×C4⋊C4—21no—
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1767C2211yescharacteristic
48C4×C4⋊C8—21no—
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51#1763—21no—
52C4×#102—21no—
53#1744—21no—
54#1744C2211yescharacteristic
55#1678C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78#486—22no—
79#486C2×C414yes—
80#477—22no—
81#477C2×C414yescomplemented
82#460—22no—
83#460C2×C414yes—
84C8×Q8—22no—
85C8×Q8—22no—
86C8×Q8—22no—
87C8×Q8C2×C414yescomplemented
88C4×SD16—44no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#389—21no—
126#389—21no—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C43—21no—
138C43C2311yescharacteristic
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#336D411yescharacteristic
155#336D411yescharacteristic
156#334—22no—
157#334C2×C414yescomplemented
158#333—22no—
159#333C2×C414yescomplemented
160#331—22no—
161#331C2×C414yes—
162C22×Q8—21no—
163C22×Q8C2×D411yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C23×C4—41no—
168C2×SD16—24no—
169C2×SD16—42no—
170C2×SD16—42no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C22×C412yes—
185#127—21no—
186#127—21no—
187#127—21no—
188#127C22×C412yes—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—82no—
195C4×Q8SD1614yes—
196C4×Q8C22×C412yes—
197C4×Q8#4112yes—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—81no—
201C4×D4—81no—
202C4×D4—82no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—24no—
236#106—42no—
237#106—42no—
238#106—81no—
239#106—81no—
240C4⋊C8—24no—
241C4⋊C8—24no—
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⋊C8#4112yes—
254#103—24no—
255#103—42no—
256#103—42no—
257#102—24no—
258#102—24no—
259#102—24no—
260#102—42no—
261#102—42no—
262#102—81no—
263#102—81no—
264#97—42no—
265C4×C8—24no—
266C4×C8—24no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—42no—
270C4×C8—81no—
271C4×C8—81no—
272C4×C8—81no—
273C4×C8C22×C412yes—
274C4×C8#4112yes—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280C24—41no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×Q8C4×D412yes—
289C2×Q8C2×SD1612yes—
290C2×Q8#13512yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—81no—
306C2×D4—81no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C2×#4111yescharacteristic
324SD16—48no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8C4×D412yes—
354C2×C8C4×D412yes—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—82no—
377C4⋊C4C4×D412yes—
378C4⋊C4C2×SD1612yes—
379C4⋊C4#13512yes—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C42—24no—
384C42—22no—
385C42—22no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—24no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—81no—
398C42—81no—
399C42—82no—
400C42C2×SD1612yes—
401C42#13512yes—
402C42C23×C411yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C23—42no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23C2×#12111yescharacteristic
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—82no—
418Q8—82no—
419Q8C4×SD1614yes—
420D4—42no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—82no—
425D4—82no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—81no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—164no—
464C2×C4#46512yes—
465C2×C4#46512yes—
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C4×#4111yescharacteristic
469C2×C4#52511yescharacteristic
470C2×C4#52911yescharacteristic
471C2×C4#53211yescharacteristic
472C2×C4C22×SD1611yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4C2×#13511yescharacteristic
475C2×C4C2×#13511yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57711yescharacteristic
479C2×C4#57811yescharacteristic
480C8—22no—
481C8—22no—
482C8—41no—
483C8—41no—
484C8—42no—
485C8—42no—
486C8—42no—
487C8—42no—
488C8—81no—
489C8—81no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—161no—
494C22—161no—
495C22—161no—
496C22—161no—
497C22C4×#12111yescharacteristic
498C22#304311yescharacteristic
499C22#304611yescharacteristic
500C22C2×#46511yescharacteristic
501C22#305011yescharacteristic
502C22#306811yescharacteristic
503C22#306811yescharacteristic
504C4—24no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—42no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—82no—
515C4—161no—
516C4—161no—
517C4C2×C4×SD1611yescharacteristic
518C4C4×#13511yescharacteristic
519C4#294711yescharacteristic
520C4#295211yescharacteristic
521C4#312111yescharacteristic
522C4#312511yescharacteristic
523C4#316711yescharacteristic
524C4#316811yescharacteristic
525C2—162no—
526C2#1939611yescharacteristic
527C2#1940111yescharacteristic
528C2#1941411yescharacteristic
529C2#1941911yescharacteristic
530C2#2151111yescharacteristic
531C2C4×#46511yescharacteristic
532C2#2155511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21616C2semidirect
4. #20569C2semidirect
7. #9612C2semidirect
8. #9567C2semidirect
11. #9216C2semidirect
12. #8643C2semidirect
13. #8277C2semidirect
15. C4×#333C2semidirect
55. #1678C4semidirect
81. #477C2×C4semidirect
87. C8×Q8C2×C4semidirect
157. #334C2×C4semidirect
159. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222192
#216162
C4×#4602
#205692
C2×#16782
#97352
#96122
#95672
#95642
#92222
#92162
#86432
#82772
#82342
C4×#3332