Finite Group Databasebeta

← #227609

Subgroups — #227609

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#18480C211yesmaximal, characteristic
2#18025C211yesmaximal, characteristic, complemented
3#18018C211yesmaximal, characteristic, complemented
4#17522C211yesmaximal, characteristic, complemented
5#17522C211yesmaximal, characteristic, complemented
6#9860C211yesmaximal, characteristic, complemented
7#9588C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9213C211yesmaximal, characteristic, complemented
10#9213C211yesmaximal, characteristic, complemented
11#9069C211yesmaximal, characteristic, complemented
12#8901C211yesmaximal, characteristic, complemented
13#8901C211yesmaximal, characteristic, complemented
14#8189C211yesmaximal, characteristic, complemented
15#8188C211yesmaximal, characteristic, complemented
16#3481C2211yescharacteristic
17#3147C2211yescharacteristic
18#3106C2211yescharacteristic
19#3106C2211yescharacteristic
20#2991C2211yescharacteristic
21#2991C2211yescharacteristic
22#2979C2211yescharacteristic, complemented
23#2922C2211yescharacteristic
24#2922C2211yescharacteristic
25#1837C2211yescharacteristic, complemented
26#1632C411yescharacteristic
27#1632C411yescharacteristic
28#1630C411yescharacteristic
29#1629C411yescharacteristic
30#1622C411yescharacteristic
31#1622C411yescharacteristic
32#1620C411yescharacteristic
33#1619C411yescharacteristic
34#1594—21no—
35#1589—21no—
36#1588—21no—
37#1586—21no—
38#1583—21no—
39#1581—21no—
40SD16⋊C8—21no—
41C8×SD16—21no—
42#1574C2211yescharacteristic, complemented
43#1574C2211yescharacteristic, complemented
44#1543C2211yescharacteristic
45#1525C2211yescharacteristic, complemented
46#1495—21no—
47#1495—21no—
48#1492C2211yescharacteristic
49#1492C2211yescharacteristic
50#1492C2211yescharacteristic, complemented
51#1492C2211yescharacteristic, complemented
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
75C2×#135—41no—
76#559C2×C411yescharacteristic
77#554C2×C411yescharacteristic
78#554C2×C411yescharacteristic
79#553—21no—
80#553—21no—
81#551—41no—
82#546—21no—
83#546—21no—
84D42C2×C411yescharacteristic
85#532C2311yescharacteristic
86#532C2311yescharacteristic
87#529—21no—
88#529—21no—
89#525—41no—
90C4×#41C2311yescharacteristic
91C4×#41C2311yescharacteristic
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94#495—41no—
95#494C2×C411yescharacteristic
96#493C2×C411yescharacteristic
97#492C2×C411yescharacteristic
98#492C2×C411yescharacteristic
99#482—21no—
100#482—21no—
101#477C2×C411yescharacteristic
102#477C2×C411yescharacteristic
103#476C2×C411yescharacteristic
104#476C2×C411yescharacteristic
105#460C2×C411yescharacteristic
106#460C2×C411yescharacteristic
107#459C2×C411yescharacteristic
108#459C2×C411yescharacteristic
109#449—21no—
110#449—21no—
111C8×Q8—21no—
112C8×Q8—21no—
113C4×SD16—21no—
114C4×SD16—21no—
115C4×D8—21no—
116C4×D8—21no—
117#436—21no—
118#436—21no—
119#436—21no—
120#436—21no—
121#436—41no—
122#436—41no—
123C8×D4—21no—
124C8×D4—21no—
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
137#424—21no—
138#424—21no—
139#423—21no—
140#423—21no—
141#423C2311yescharacteristic
142#423C2311yescharacteristic
143#408—21no—
144#408—21no—
145#408—41no—
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151#341—41no—
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#329—41no—
161Q8⋊C8—21no—
162Q8⋊C8—21no—
163Q8⋊C8C2×C411yescharacteristic
164Q8⋊C8C2×C411yescharacteristic
165Q8⋊C8C2×C411yescharacteristic
166Q8⋊C8C2×C411yescharacteristic
167Q8⋊C8D411yescharacteristic
168Q8⋊C8D411yescharacteristic
169Q8⋊C8D411yescharacteristic
170Q8⋊C8D411yescharacteristic
171D4⋊C8—21no—
172D4⋊C8—21no—
173D4⋊C8—21no—
174D4⋊C8—21no—
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—21no—
200C2×#41#4111yescharacteristic
201C2×#41#4111yescharacteristic
202C22×D4—21no—
203C22×D4—21no—
204C23×C4—41no—
205#135—82no—
206#135—82no—
207C2×Q16—41no—
208C2×SD16—21no—
209C2×SD16—21no—
210C2×SD16—41no—
211C2×SD16—41no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—41no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—41no—
220C2×M16—41no—
221C2×M16—41no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—41no—
226C4⋊Q8C22×C411yescharacteristic
227C4⋊Q8C22×C411yescharacteristic
228#127C22×C411yescharacteristic
229#127C22×C411yescharacteristic
230#126—41no—
231#126—41no—
232#125—21no—
233#125—21no—
234#124—21no—
235#124—21no—
236#124—21no—
237#124—21no—
238#123—41no—
239#123—41no—
240#123—41no—
241#123—41no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—41no—
245C22⋊Q8—41no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251#121—21no—
252#121—41no—
253#121—41no—
254#120—41no—
255#120—41no—
256C4×Q8C22×C411yescharacteristic
257C4×Q8C22×C411yescharacteristic
258C4×Q8C2×D411yescharacteristic
259C4×Q8C2×D411yescharacteristic
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4C22×C411yescharacteristic
277C4×D4C22×C411yescharacteristic
278C4×D4C2×D411yescharacteristic
279C4×D4C2×D411yescharacteristic
280#117—21no—
281#117—21no—
282#117—41no—
283#117#4111yescharacteristic
284#117#4111yescharacteristic
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C2411yescharacteristic
292#107—21no—
293#107—21no—
294#106—21no—
295#106—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—41no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303C4⋊C8—41no—
304C4⋊C8—41no—
305C4⋊C8—41no—
306C4⋊C8C22×C411yescharacteristic
307C4⋊C8C22×C411yescharacteristic
308C4⋊C8C22×C411yescharacteristic
309C4⋊C8C22×C411yescharacteristic
310C4⋊C8C2×D411yescharacteristic
311C4⋊C8C2×D411yescharacteristic
312C4⋊C8C2×D411yescharacteristic
313C4⋊C8C2×D411yescharacteristic
314#103—21no—
315#103—21no—
316#102—21no—
317#102—21no—
318#102—21no—
319#102—21no—
320#102—21no—
321#102—21no—
322C22⋊C8—21no—
323C22⋊C8—21no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C22⋊C8—41no—
327C22⋊C8—41no—
328C22⋊C8—41no—
329C22⋊C8—41no—
330C22⋊C8—41no—
331C22⋊C8—41no—
332C22⋊C8—41no—
333C22⋊C8—41no—
334#97—21no—
335#97—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—41no—
346C4×C8—41no—
347C4×C8C22×C411yescharacteristic
348C4×C8C22×C411yescharacteristic
349C4×C8C2×D411yescharacteristic
350C4×C8C2×D411yescharacteristic
351C4×C8C2×D411yescharacteristic
352C4×C8C2×D411yescharacteristic
353C24—41no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—81no—
359#41—81no—
360#41—81no—
361#41—81no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8C4×D411yescharacteristic
365C2×Q8C4×D411yescharacteristic
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4C4×D411yescharacteristic
383C2×D4C4×D411yescharacteristic
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—161no—
396C22×C4—161no—
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400Q16—82no—
401SD16—42no—
402SD16—42no—
403SD16—82no—
404SD16—82no—
405D8—81no—
406D8—81no—
407D8—82no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—81no—
413M16—81no—
414M16—161no—
415M16—161no—
416M16—161no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—161no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4C4×D411yescharacteristic
455C4⋊C4C4×D411yescharacteristic
456C4⋊C4C4×D411yescharacteristic
457C4⋊C4C4×D411yescharacteristic
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42C23×C411yescharacteristic
481C42C22×D411yescharacteristic
482C42C22×D411yescharacteristic
483C42H(5,𝔽2)11yescharacteristic
484C23—21no—
485C23—21no—
486C23—41no—
487C23—41no—
488C23—82no—
489C23—82no—
490C23—161no—
491C23#55111yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—81no—
497Q8—81no—
498D4—21no—
499D4—21no—
500D4—21no—
501D4—21no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—81no—
511D4—81no—
512D4—81no—
513D4—81no—
514D4—82no—
515D4—82no—
516D4—161no—
517D4—161no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—161no—
547C2×C4—161no—
548C2×C4—161no—
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4#51811yescharacteristic
552C2×C4D4211yescharacteristic
553C2×C4#54611yescharacteristic
554C2×C4#54611yescharacteristic
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—81no—
570C8—81no—
571C8—81no—
572C8—81no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—164no—
589C22#235811yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4#295511yescharacteristic
602C4#295511yescharacteristic
603C4#295711yescharacteristic
604C4#295711yescharacteristic
605C4#328811yescharacteristic
606C4#328811yescharacteristic
607C2—41no—
608C2—81no—
609C2—81no—
610C2—161no—
611C2—161no—
612C2#2288711yescharacteristic
613C2#2288911yescharacteristic
614C2#2289111yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radical#18480C2
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. #18025C2semidirect
3. #18018C2semidirect
4. #17522C2semidirect
5. #17522C2semidirect
6. #9860C2semidirect
7. #9588C2semidirect
8. #9586C2semidirect
9. #9213C2semidirect
10. #9213C2semidirect
11. #9069C2semidirect
12. #8901C2semidirect
13. #8901C2semidirect
14. #8189C2semidirect
15. #8188C2semidirect
22. #2979C22semidirect
25. #1837C22semidirect
42. #1574C22semidirect
43. #1574C22semidirect
45. #1525C22semidirect
50. #1492C22semidirect
51. #1492C22semidirect
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
#184802
#180252
#180182
#175222
#98602
#95882
#95862
#92132
#90692
#89012
#81892
#81882