Finite Group Databasebeta

← #227586

Subgroups — #227586

Order 512.

679 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#18480C211yesmaximal, characteristic
2#18022C211yesmaximal, characteristic, complemented
3#18015C211yesmaximal, characteristic, complemented
4#17517C211yesmaximal, characteristic, complemented
5#17517C211yesmaximal, characteristic, complemented
6#9860C211yesmaximal, characteristic, complemented
7#9588C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9211C211yesmaximal, characteristic, complemented
10#9211C211yesmaximal, characteristic, complemented
11#9069C211yesmaximal, characteristic, complemented
12#8900C211yesmaximal, characteristic, complemented
13#8900C211yesmaximal, characteristic, complemented
14#8186C211yesmaximal, characteristic, complemented
15#8185C211yesmaximal, characteristic, complemented
16#3481C2211yescharacteristic
17#3147C2211yescharacteristic
18#3106C2211yescharacteristic
19#3106C2211yescharacteristic
20#2990C2211yescharacteristic
21#2990C2211yescharacteristic
22#2979C2211yescharacteristic, complemented
23C4×#131C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1837C2211yescharacteristic, complemented
26#1632C411yescharacteristic
27#1632C411yescharacteristic
28#1630C411yescharacteristic
29#1629C411yescharacteristic
30#1622C411yescharacteristic
31#1622C411yescharacteristic
32#1620C411yescharacteristic
33#1619C411yescharacteristic
34#1592—21no—
35#1590—21no—
36#1586—21no—
37#1585—21no—
38#1582—21no—
39#1581—21no—
40#1580—21no—
41C8×D8—21no—
42#1574C2211yescharacteristic, complemented
43#1574C2211yescharacteristic, complemented
44#1543C2211yescharacteristic
45#1525C2211yescharacteristic, complemented
46#1494—21no—
47#1494—21no—
48#1492C2211yescharacteristic
49#1492C2211yescharacteristic
50#1492C2211yescharacteristic, complemented
51#1492C2211yescharacteristic, complemented
52#1490—21no—
53#1490—21no—
54#1489C2211yescharacteristic
55#1489C2211yescharacteristic
56#1484C2211yescharacteristic
57#1484C2211yescharacteristic
58#1483—21no—
59#1483—21no—
60#1476C2211yescharacteristic
61#1476C2211yescharacteristic
62#1476C2211yescharacteristic
63#1476C2211yescharacteristic
64C2×D4⋊C8—21no—
65C2×D4⋊C8—21no—
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—
76C2×#131—21no—
77C2×#131—21no—
78#559C2×C411yescharacteristic
79#554C2×C411yescharacteristic
80#554C2×C411yescharacteristic
81#553—21no—
82#553—21no—
83#551—41no—
84#546—21no—
85#546—21no—
86D42C2×C411yescharacteristic
87#532C2311yescharacteristic
88#532C2311yescharacteristic
89#529—21no—
90#529—21no—
91#525—41no—
92C4×#41C2311yescharacteristic
93C4×#41C2311yescharacteristic
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96#495—41no—
97#494C2×C411yescharacteristic
98#493C2×C411yescharacteristic
99#492C2×C411yescharacteristic
100#492C2×C411yescharacteristic
101#482—21no—
102#482—21no—
103#477C2×C411yescharacteristic
104#477C2×C411yescharacteristic
105#476C2×C411yescharacteristic
106#476C2×C411yescharacteristic
107#460C2×C411yescharacteristic
108#460C2×C411yescharacteristic
109#459C2×C411yescharacteristic
110#459C2×C411yescharacteristic
111#449—21no—
112#449—21no—
113#446—21no—
114#446—21no—
115C4×SD16—21no—
116C4×SD16—21no—
117C4×D8—21no—
118C4×D8—21no—
119#436—21no—
120#436—21no—
121#435—41no—
122#435—41no—
123C8×D4—21no—
124C8×D4—21no—
125C8×D4—21no—
126C8×D4—21no—
127#432—21no—
128#432—21no—
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#431C2311yescharacteristic
133#423C2311yescharacteristic
134#423C2311yescharacteristic
135C2×C4⋊C8—21no—
136C2×C4⋊C8—21no—
137#408—41no—
138#405—21no—
139#405—21no—
140#405—41no—
141#405—41no—
142C4×M16—21no—
143C4×M16—21no—
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8—21no—
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—
215#131—24no—
216#131—42no—
217#131—42no—
218#131—42no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—41no—
224C2×M16—41no—
225C2×M16—41no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—41no—
230C22×C8—41no—
231C22×C8—41no—
232C22×C8—41no—
233C22×C8—41no—
234C4⋊Q8C22×C411yescharacteristic
235C4⋊Q8C22×C411yescharacteristic
236#127C22×C411yescharacteristic
237#127C22×C411yescharacteristic
238#126—41no—
239#126—41no—
240#125—21no—
241#125—21no—
242#124—21no—
243#124—21no—
244#124—21no—
245#124—21no—
246#123—41no—
247#123—41no—
248#123—41no—
249#123—41no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—41no—
253C22⋊Q8—41no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#121—41no—
261#121—41no—
262#120—41no—
263#120—41no—
264C4×Q8C22×C411yescharacteristic
265C4×Q8C22×C411yescharacteristic
266C4×Q8C2×D411yescharacteristic
267C4×Q8C2×D411yescharacteristic
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—41no—
283C4×D4—41no—
284C4×D4C22×C411yescharacteristic
285C4×D4C22×C411yescharacteristic
286C4×D4C2×D411yescharacteristic
287C4×D4C2×D411yescharacteristic
288#117—21no—
289#117—21no—
290#117—41no—
291#117#4111yescharacteristic
292#117#4111yescharacteristic
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42C2411yescharacteristic
300#107—21no—
301#107—21no—
302#106—21no—
303#106—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—41no—
309C4⋊C8—41no—
310C4⋊C8C22×C411yescharacteristic
311C4⋊C8C22×C411yescharacteristic
312C4⋊C8C22×C411yescharacteristic
313C4⋊C8C22×C411yescharacteristic
314C4⋊C8C2×D411yescharacteristic
315C4⋊C8C2×D411yescharacteristic
316C4⋊C8C2×D411yescharacteristic
317C4⋊C8C2×D411yescharacteristic
318#103—21no—
319#103—21no—
320#102—21no—
321#102—21no—
322#102—21no—
323#102—21no—
324#102—21no—
325#102—21no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C22⋊C8—41no—
331C22⋊C8—41no—
332C22⋊C8—41no—
333C22⋊C8—41no—
334#97—21no—
335#97—21no—
336#97—41no—
337#97—41no—
338#97—41no—
339#97—41no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—41no—
350C4×C8—41no—
351C4×C8C22×C411yescharacteristic
352C4×C8C22×C411yescharacteristic
353C4×C8C2×D411yescharacteristic
354C4×C8C2×D411yescharacteristic
355C4×C8C2×D411yescharacteristic
356C4×C8C2×D411yescharacteristic
357C24—41no—
358#41—22no—
359#41—22no—
360#41—41no—
361#41—41no—
362#41—81no—
363#41—81no—
364#41—81no—
365#41—81no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8C4×D411yescharacteristic
369C2×Q8C4×D411yescharacteristic
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4C4×D411yescharacteristic
387C2×D4C4×D411yescharacteristic
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—161no—
400C22×C4—161no—
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404Q16—82no—
405SD16—81no—
406SD16—81no—
407SD16—82no—
408SD16—82no—
409D8—42no—
410D8—42no—
411D8—82no—
412M16—22no—
413M16—22no—
414M16—44no—
415M16—41no—
416M16—41no—
417M16—44no—
418M16—81no—
419M16—81no—
420M16—161no—
421C2×C8—22no—
422C2×C8—22no—
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—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—44no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—44no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—161no—
448C2×C8—161no—
449C2×C8—161no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4C4×D411yescharacteristic
463C4⋊C4C4×D411yescharacteristic
464C4⋊C4C4×D411yescharacteristic
465C4⋊C4C4×D411yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42C23×C411yescharacteristic
489C42C22×D411yescharacteristic
490C42C22×D411yescharacteristic
491C42H(5,𝔽2)11yescharacteristic
492C23—21no—
493C23—21no—
494C23—41no—
495C23—41no—
496C23—82no—
497C23—82no—
498C23—161no—
499C23#55111yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—21no—
503Q8—21no—
504Q8—81no—
505Q8—81no—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—42no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—81no—
519D4—81no—
520D4—82no—
521D4—82no—
522D4—161no—
523D4—161no—
524D4#44312yes—
525D4#44412yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—161no—
555C2×C4—161no—
556C2×C4—161no—
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4#51811yescharacteristic
560C2×C4D4211yescharacteristic
561C2×C4#54611yescharacteristic
562C2×C4#54611yescharacteristic
563C8—22no—
564C8—22no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—42no—
570C8—42no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8—41no—
577C8—41no—
578C8—41no—
579C8—81no—
580C8—81no—
581C22—21no—
582C22—22no—
583C22—21no—
584C22—22no—
585C22—21no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—164no—
597C22#235811yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—81no—
608C4—81no—
609C4C2×#44311yescharacteristic
610C4C2×#44411yescharacteristic
611C4#295611yescharacteristic
612C4#295611yescharacteristic
613C4#328811yescharacteristic
614C4#328811yescharacteristic
615C2—41no—
616C2—42no—
617C2—42no—
618C2—161no—
619C2—161no—
620C2#2288611yescharacteristic
621C2#2288811yescharacteristic
622C2#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. #18022C2semidirect
3. #18015C2semidirect
4. #17517C2semidirect
5. #17517C2semidirect
6. #9860C2semidirect
7. #9588C2semidirect
8. #9586C2semidirect
9. #9211C2semidirect
10. #9211C2semidirect
11. #9069C2semidirect
12. #8900C2semidirect
13. #8900C2semidirect
14. #8186C2semidirect
15. #8185C2semidirect
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
#180222
#180152
#175172
#98602
#95882
#95862
#92112
#90692
#89002
#81862
#81852