Finite Group Databasebeta

← #227613

Subgroups — #227613

Order 512.

611 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#18485C211yesmaximal, characteristic
2#18024C211yesmaximal, characteristic, complemented
3#18019C211yesmaximal, characteristic, complemented
4#17522C211yesmaximal, characteristic
5#17522C211yesmaximal, characteristic
6#9858C211yesmaximal, characteristic, complemented
7#9588C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9214C211yesmaximal, characteristic, complemented
10#9214C211yesmaximal, characteristic, complemented
11#9070C211yesmaximal, characteristic, complemented
12#8899C211yesmaximal, characteristic, complemented
13#8899C211yesmaximal, characteristic, complemented
14#8189C211yesmaximal, characteristic, complemented
15#8188C211yesmaximal, characteristic, complemented
16#3481C2211yescharacteristic
17#3131C2211yescharacteristic
18#3107C2211yescharacteristic
19#3107C2211yescharacteristic
20#2991C2211yescharacteristic
21#2991C2211yescharacteristic
22#2986C2211yescharacteristic, complemented
23#2922C2211yescharacteristic
24#2922C2211yescharacteristic
25#1848C2211yescharacteristic, complemented
26#1644C411yescharacteristic
27#1644C411yescharacteristic
28#1643C411yescharacteristic
29#1643C411yescharacteristic
30#1640C411yescharacteristic
31#1640C411yescharacteristic
32#1639C411yescharacteristic
33#1639C411yescharacteristic
34#1592—21no—
35#1590—21no—
36#1588—21no—
37#1587—21no—
38#1584—21no—
39#1583—21no—
40#1579—21no—
41C8×Q16—21no—
42#1574C2211yescharacteristic, complemented
43#1574C2211yescharacteristic, complemented
44#1543C2211yescharacteristic
45#1526C2211yescharacteristic, complemented
46#1495—21no—
47#1495—21no—
48#1493C2211yescharacteristic, complemented
49#1493C2211yescharacteristic, complemented
50#1492C2211yescharacteristic
51#1492C2211yescharacteristic
52#1489C2211yescharacteristic
53#1489C2211yescharacteristic
54#1486—21no—
55#1486—21no—
56#1484C2211yescharacteristic
57#1484C2211yescharacteristic
58#1482—21no—
59#1482—21no—
60#1478—21no—
61#1478—21no—
62#1476C2211yescharacteristic
63#1476C2211yescharacteristic
64#1476C2211yescharacteristic
65#1476C2211yescharacteristic
66#1452C2211yescharacteristic, complemented
67#1450C2211yescharacteristic, complemented
68C2×#322C2211yescharacteristic, complemented
69#1299C2211yescharacteristic, complemented
70#1299C2211yescharacteristic, complemented
71#1278C2211yescharacteristic, complemented
72#1278C2211yescharacteristic, complemented
73#1278C2211yescharacteristic, complemented
74#1278C2211yescharacteristic, complemented
75#559C2×C411yescharacteristic
76#554C2×C411yescharacteristic
77#554C2×C411yescharacteristic
78#553—21no—
79#553—21no—
80#551—41no—
81#546—21no—
82#546—21no—
83D42C2×C411yescharacteristic
84#533C2311yescharacteristic
85#532C2311yescharacteristic
86#529—21no—
87#529—21no—
88#525—41no—
89C4×#41C2311yescharacteristic
90C4×#41C2311yescharacteristic
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93#488—41no—
94#487C2×C411yescharacteristic
95#487C2×C411yescharacteristic
96#486C2×C411yescharacteristic
97#486C2×C411yescharacteristic
98#485—21no—
99#485—21no—
100#475C2×C411yescharacteristic
101#475C2×C411yescharacteristic
102#474C2×C411yescharacteristic
103#474C2×C411yescharacteristic
104#462C2×C411yescharacteristic
105#462C2×C411yescharacteristic
106#461C2×C411yescharacteristic
107#461C2×C411yescharacteristic
108#452—21no—
109#452—21no—
110#446—21no—
111#446—21no—
112C8×Q8—21no—
113C8×Q8—21no—
114C8×Q8—21no—
115C8×Q8—21no—
116C4×Q16—21no—
117C4×Q16—21no—
118C4×SD16—21no—
119C4×SD16—21no—
120#436—21no—
121#436—21no—
122#436—41no—
123#436—41no—
124#433—41no—
125#433—41no—
126#433—41no—
127#431—21no—
128#431—21no—
129#431—21no—
130#431—21no—
131#431—21no—
132#431—21no—
133#431C2311yescharacteristic
134#431C2311yescharacteristic
135#431C2311yescharacteristic
136#431C2311yescharacteristic
137C2×#108—41no—
138#424—21no—
139#424—21no—
140#423—21no—
141#423—21no—
142#423C2311yescharacteristic
143#423C2311yescharacteristic
144#416—41no—
145#408—21no—
146#408—21no—
147C4×M16C2311yescharacteristic
148C4×M16C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152#335—21no—
153#335—21no—
154#334—21no—
155#334—21no—
156#333C2×C411yescharacteristic
157#333C2×C411yescharacteristic
158#331C2×C411yescharacteristic
159#331C2×C411yescharacteristic
160#330—41no—
161Q8⋊C8—21no—
162Q8⋊C8—21no—
163Q8⋊C8—21no—
164Q8⋊C8—21no—
165Q8⋊C8—21no—
166Q8⋊C8—21no—
167Q8⋊C8C2×C411yescharacteristic
168Q8⋊C8C2×C411yescharacteristic
169Q8⋊C8C2×C411yescharacteristic
170Q8⋊C8C2×C411yescharacteristic
171Q8⋊C8D411yescharacteristic
172Q8⋊C8D411yescharacteristic
173Q8⋊C8D411yescharacteristic
174Q8⋊C8D411yescharacteristic
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8C2×C411yescharacteristic
186D4⋊C8C2×C411yescharacteristic
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8D411yescharacteristic
190D4⋊C8D411yescharacteristic
191D4⋊C8D411yescharacteristic
192D4⋊C8D411yescharacteristic
193#322—21no—
194#322—21no—
195#322—21no—
196#322—41no—
197C82—21no—
198C2×#41—21no—
199C2×#41#4111yescharacteristic
200C2×#41#4111yescharacteristic
201C22×D4—21no—
202C22×D4—21no—
203C23×C4—41no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×SD16—21no—
207C2×SD16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—41no—
213C2×M16—41no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C4⋊Q8C22×C411yescharacteristic
218C4⋊Q8C22×C411yescharacteristic
219C4⋊Q8C22×C411yescharacteristic
220#127C22×C411yescharacteristic
221#126—41no—
222#126—41no—
223#125—21no—
224#125—21no—
225#125—21no—
226#124—21no—
227#124—21no—
228#124—21no—
229#123—41no—
230#123—41no—
231#123—41no—
232#123—41no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—41no—
238C22⋊Q8—41no—
239#121—21no—
240#121—21no—
241#121—21no—
242#121—21no—
243#121—41no—
244#121—41no—
245#120—41no—
246#120—41no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C22×C411yescharacteristic
250C4×Q8C22×C411yescharacteristic
251C4×Q8C2×D411yescharacteristic
252C4×Q8C2×D411yescharacteristic
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4C22×C411yescharacteristic
268C4×D4C22×C411yescharacteristic
269C4×D4C2×D411yescharacteristic
270C4×D4C2×D411yescharacteristic
271#117—21no—
272#117—21no—
273#117—21no—
274#117—41no—
275#117#4111yescharacteristic
276#117#4111yescharacteristic
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C2411yescharacteristic
284#108—82no—
285#107—21no—
286#107—21no—
287#106—21no—
288#106—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—41no—
294C4⋊C8—41no—
295C4⋊C8—41no—
296C4⋊C8—41no—
297C4⋊C8—41no—
298C4⋊C8—41no—
299C4⋊C8—41no—
300C4⋊C8—41no—
301C4⋊C8C22×C411yescharacteristic
302C4⋊C8C22×C411yescharacteristic
303C4⋊C8C22×C411yescharacteristic
304C4⋊C8C22×C411yescharacteristic
305C4⋊C8C2×D411yescharacteristic
306C4⋊C8C2×D411yescharacteristic
307C4⋊C8C2×D411yescharacteristic
308C4⋊C8C2×D411yescharacteristic
309#103—21no—
310#103—21no—
311#103—21no—
312#103—21no—
313#103—21no—
314#103—21no—
315#103—41no—
316#103—41no—
317#102—21no—
318#102—21no—
319#102—41no—
320#102—41no—
321C22⋊C8—21no—
322C22⋊C8—21no—
323C22⋊C8—21no—
324C22⋊C8—21no—
325C22⋊C8—41no—
326C22⋊C8—41no—
327C22⋊C8—41no—
328C22⋊C8—41no—
329C22⋊C8—41no—
330C22⋊C8—41no—
331#97—21no—
332#97—21no—
333#97—41no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—41no—
344C4×C8—41no—
345C4×C8—41no—
346C4×C8C22×C411yescharacteristic
347C4×C8C22×C411yescharacteristic
348C4×C8C2×D411yescharacteristic
349C4×C8C2×D411yescharacteristic
350C4×C8C2×D411yescharacteristic
351C4×C8C2×D411yescharacteristic
352C24—41no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—81no—
358#41—81no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8C4×D411yescharacteristic
363C2×Q8C4×D411yescharacteristic
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4C4×D411yescharacteristic
377C2×D4C4×D411yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—161no—
389C22×C4—161no—
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393Q16—42no—
394Q16—42no—
395SD16—81no—
396SD16—81no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—81no—
402M16—81no—
403M16—161no—
404M16—161no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4C4×D411yescharacteristic
446C4⋊C4C4×D411yescharacteristic
447C4⋊C4C4×D411yescharacteristic
448C4⋊C4C4×D411yescharacteristic
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—41no—
471C42C23×C411yescharacteristic
472C42C22×D411yescharacteristic
473C42C22×D411yescharacteristic
474C42H(5,𝔽2)11yescharacteristic
475C23—21no—
476C23—21no—
477C23—41no—
478C23—82no—
479C23—82no—
480C23—161no—
481C23#55111yescharacteristic
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—21no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—81no—
491D4—21no—
492D4—21no—
493D4—21no—
494D4—21no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—81no—
500D4—81no—
501D4—81no—
502D4—82no—
503D4—82no—
504D4—161no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—161no—
533C2×C4—161no—
534C2×C4—161no—
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4#51811yescharacteristic
538C2×C4D4211yescharacteristic
539C2×C4#54611yescharacteristic
540C2×C4#54611yescharacteristic
541C8—41no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—81no—
556C8—81no—
557C8—81no—
558C8—81no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—164no—
572C22#235811yescharacteristic
573C4—21no—
574C4—21no—
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4#295511yescharacteristic
586C4#295511yescharacteristic
587C4#295711yescharacteristic
588C4#295711yescharacteristic
589C4#328811yescharacteristic
590C4#328911yescharacteristic
591C2—41no—
592C2—81no—
593C2—81no—
594C2—161no—
595C2#2288711yescharacteristic
596C2#2288911yescharacteristic
597C2#2289011yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radical#3481C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18024C2semidirect
3. #18019C2semidirect
6. #9858C2semidirect
7. #9588C2semidirect
8. #9586C2semidirect
9. #9214C2semidirect
10. #9214C2semidirect
11. #9070C2semidirect
12. #8899C2semidirect
13. #8899C2semidirect
14. #8189C2semidirect
15. #8188C2semidirect
22. #2986C22semidirect
25. #1848C22semidirect
42. #1574C22semidirect
43. #1574C22semidirect
45. #1526C22semidirect
48. #1493C22semidirect
49. #1493C22semidirect
66. #1452C22semidirect
67. #1450C22semidirect
68. C2×#322C22semidirect
69. #1299C22semidirect
70. #1299C22semidirect
71. #1278C22semidirect
72. #1278C22semidirect
73. #1278C22semidirect
74. #1278C22semidirect

Maximal subgroups

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

subgroupindex
#184852
#180242
#180192
#175222
#98582
#95882
#95862
#92142
#90702
#88992
#81892
#81882