Finite Group Databasebeta

← #211442

Subgroups — #211442

Order 512.

700 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#22148C211yesmaximal, characteristic, complemented
2#22147C211yesmaximal, characteristic
3C4×#486C211yesmaximal, characteristic
4#20516C211yesmaximal, characteristic, complemented
5C2×#1702C211yesmaximal, characteristic
6#9746C211yesmaximal, characteristic
7#9622C211yesmaximal, characteristic, complemented
8#9565C211yesmaximal, characteristic, complemented
9#9563C211yesmaximal, characteristic
10#9222C211yesmaximal, characteristic
11#9218C211yesmaximal, characteristic, complemented
12#8657C211yesmaximal, characteristic, complemented
13C4×#334C211yesmaximal, characteristic, complemented
14#8237C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic
16C2×#486C2211yescharacteristic
17C2×#462C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2295C2211yescharacteristic
24#1990C2211yescharacteristic
25#1982C2211yescharacteristic
26#1981C2211yescharacteristic
27#1969—21no—
28#1968—21no—
29#1957C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1934—21no—
33#1934—21no—
34#1934—21no—
35#1933—21no—
36#1878—21no—
37#1877—21no—
38#1864—21no—
39#1863—21no—
40#1834C2211yescharacteristic
41#1803C2211yescharacteristic
42#1803C2211yescharacteristic
43#1771—21no—
44#1770C2211yescharacteristic
45C8×C4⋊C4—21no—
46C8×C4⋊C4C2211yescharacteristic
47#1768—21no—
48#1767C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×#102—21no—
52C4×#102—21no—
53#1726—21no—
54C42×C8C2211yescharacteristic
55#1702C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—21no—
69C22×SD16—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77#486—22no—
78#486C2×C414yes—
79#462—22no—
80#462C2×C414yescomplemented
81#460—22no—
82#460C2×C414yes—
83#446—22no—
84#446C2×C414yescomplemented
85C8×Q8—22no—
86C8×Q8—22no—
87C2×#107—21no—
88C2×#106—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—21no—
123#391D411yescharacteristic
124#391D411yescharacteristic
125#390—21no—
126#390—21no—
127#390—21no—
128#389—21no—
129#389—21no—
130#389D411yescharacteristic
131#389D411yescharacteristic
132#387—21no—
133#387—21no—
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140#376—21no—
141C43C2311yescharacteristic
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#334—22no—
156#334C2×C414yescomplemented
157#332—22no—
158#332C2×C414yescomplemented
159#331—22no—
160#331C2×C414yes—
161C22×Q8—21no—
162C22×Q8C2×D411yescharacteristic
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C23×C4—41no—
167C2×Q16—24no—
168C2×Q16—81no—
169C2×Q16—81no—
170C2×SD16—24no—
171C2×SD16—81no—
172C2×SD16—81no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187#127—21no—
188#127—21no—
189#127—21no—
190#127C22×C412yes—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—81no—
194C4×Q8—81no—
195C4×Q8C22×C412yes—
196C4×Q8#4112yes—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—81no—
200C4×D4—81no—
201C4×D4—82no—
202C2×C4⋊C4—21no—
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—41no—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—41no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#107—81no—
231#106—81no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244C4⋊C8C22×C412yes—
245C4⋊C8#4112yes—
246#103—24no—
247#103—81no—
248#103—81no—
249#103—81no—
250#103—81no—
251#102—24no—
252#102—24no—
253#102—24no—
254#102—42no—
255#102—42no—
256#102—81no—
257#102—81no—
258#97—24no—
259#97—42no—
260#97—42no—
261C4×C8—24no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—41no—
268C4×C8—42no—
269C4×C8—81no—
270C4×C8—81no—
271C4×C8C22×C412yes—
272C4×C8#4112yes—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282C24—41no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8C4×D412yes—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—81no—
310C2×D4—81no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328Q16—84no—
329SD16—84no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
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×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8C4×D412yes—
358C2×C8C4×D412yes—
359C2×C8#13512yes—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—82no—
384C4⋊C4C4×D412yes—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C42—22no—
389C42—22no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—82no—
401C42#13512yes—
402C42#13712yes—
403C42C23×C411yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C42C2×#4111yescharacteristic
407C23—42no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23C2×#12111yescharacteristic
415Q8—22no—
416Q8—22no—
417Q8—41no—
418Q8—41no—
419Q8—42no—
420Q8—42no—
421Q8—81no—
422Q8—81no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—82no—
428D4—82no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—164no—
466C2×C4#46412yes—
467C2×C4#46412yes—
468C2×C4#47112yes—
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C4×#4111yescharacteristic
472C2×C4#52511yescharacteristic
473C2×C4#52911yescharacteristic
474C2×C4#53211yescharacteristic
475C2×C4C2×#13511yescharacteristic
476C2×C4C2×#13511yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C2×C4#57511yescharacteristic
479C2×C4#57611yescharacteristic
480C2×C4#57611yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57711yescharacteristic
483C8—22no—
484C8—22no—
485C8—22no—
486C8—41no—
487C8—42no—
488C8—42no—
489C8—42no—
490C8—42no—
491C8—81no—
492C8—81no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22C4×#12111yescharacteristic
501C22C2×#46411yescharacteristic
502C22#303511yescharacteristic
503C22#305811yescharacteristic
504C22#306011yescharacteristic
505C22C2×#47111yescharacteristic
506C22#306711yescharacteristic
507C4—22no—
508C4—22no—
509C4—22no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—81no—
517C4—82no—
518C4—161no—
519C4—161no—
520C4C4×#13511yescharacteristic
521C4C4×#13711yescharacteristic
522C4#294611yescharacteristic
523C4#294611yescharacteristic
524C4#310211yescharacteristic
525C4#310811yescharacteristic
526C4#317911yescharacteristic
527C4#317911yescharacteristic
528C2—162no—
529C2#1921411yescharacteristic
530C2#1921611yescharacteristic
531C2#1921911yescharacteristic
532C2#1922111yescharacteristic
533C2C4×#46411yescharacteristic
534C2#2153611yescharacteristic
535C2C4×#47111yescharacteristic
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
1. #22148C2semidirect
4. #20516C2semidirect
7. #9622C2semidirect
8. #9565C2semidirect
11. #9218C2semidirect
12. #8657C2semidirect
13. C4×#334C2semidirect
14. #8237C2semidirect
55. #1702C4semidirect
80. #462C2×C4semidirect
84. #446C2×C4semidirect
156. #334C2×C4semidirect
158. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#221482
#221472
C4×#4862
#205162
C2×#17022
#97462
#96222
#95652
#95632
#92222
#92182
#86572
C4×#3342
#82372
#82342