Finite Group Databasebeta

← #6354845

Subgroups — #6354845

Order 512.

1745 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic, complemented
2C2×#3227C211yesmaximal, characteristic, complemented
3#60803C211yesmaximal, characteristic
4#60236C211yesmaximal, characteristic, complemented
5#19513C218yesmaximal, complemented
6#19427C214yesmaximal, complemented
7#19391C214yesmaximal, complemented
8#19196C214yesmaximal, complemented
9#19191C214yesmaximal, complemented
10#17803C211yesmaximal, characteristic, complemented
11#17746C211yesmaximal, characteristic, complemented
12#17717C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16#3227—22no—
17#3227—22no—
18#3227—22no—
19#3227—22no—
20#3227C2212yes—
21#3227C2212yes—
22#3227C2214yes—
23#3227C2214yescomplemented
24#3227C2214yescomplemented
25#3214C2214yes—
26#3213—22no—
27#3207—22no—
28#3206C2212yes—
29#3205C2212yes—
30C2×#498—21no—
31C2×#498C2212yescomplemented
32#3151—21no—
33#3141—21no—
34C2×#486—21no—
35C2×#486C2212yescomplemented
36C2×#481—22no—
37C2×#481—22no—
38C2×#476—22no—
39C2×#476C2214yescomplemented
40#3036C2214yes—
41#3006—22no—
42#2981C2214yescomplemented
43#2979—22no—
44#2975C2212yescomplemented
45#2974—22no—
46#2972C2212yescomplemented
47#2935—22no—
48#2931C2214yes—
49#2920C2211yescharacteristic
50#2920C2211yescharacteristic
51C2×#431C2211yescharacteristic
52C2×#431C2211yescharacteristic
53#2880—22no—
54#1697C22116yescomplemented
55#1691C22116yescomplemented
56#1677C22116yescomplemented
57#1671C22116yescomplemented
58#1572C2214yescomplemented
59#1571C2212yescomplemented
60#1570C2212yescomplemented
61#1564C2214yescomplemented
62#1558C2214yescomplemented
63#1553C2214yescomplemented
64#1549C2212yescomplemented
65#1548C2214yescomplemented
66#1548C2214yescomplemented
67#1545C2212yescomplemented
68#1540C2214yes—
69C22×#41—21no—
70C22×#41—21no—
71C23×Q8—21no—
72C2×AGL(1,ℤ/8)—24no—
73C2×#131—24no—
74#532—21no—
75#532—21no—
76#532—21no—
77#532—21no—
78#532—21no—
79#532—21no—
80#532—21no—
81#532—21no—
82#532—21no—
83#532—21no—
84#532D412yes—
85#532D412yes—
86#532C2312yes—
87#532C2312yes—
88#532C2312yes—
89#532C2312yes—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127D411yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#124—21no—
107C2×#124—21no—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121D412yes—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C42C2311yescharacteristic
119#498—24no—
120#498—24no—
121#498—24no—
122#498C2314yes—
123#498C2314yes—
124#496—24no—
125#493—22no—
126#492—22no—
127#489—24no—
128#487—22no—
129#486—22no—
130#486—24no—
131#486—24no—
132#486—24no—
133#486C2314yes—
134#486C2314yes—
135#481—28no—
136#481—28no—
137#481—24no—
138#481—24no—
139#481—28no—
140#476—28no—
141#476—24no—
142#476—24no—
143#476—28no—
144#476C23116yes—
145#463—28no—
146#461C2318yes—
147#459C2318yes—
148#449—28no—
149#448—24no—
150#447—24no—
151#436—24no—
152#436—24no—
153#436C2318yes—
154#435—28no—
155C8×D4—24no—
156C8×D4—24no—
157C8×D4C2318yes—
158#432—21no—
159#432—21no—
160#432C2312yes—
161#432C2312yes—
162#431—21no—
163#431—21no—
164#431—21no—
165#431—21no—
166#431D412yes—
167#431D412yes—
168#431C2312yes—
169#431C2312yes—
170#431C2312yes—
171#431C2312yes—
172#431C2311yescharacteristic
173#431C2311yescharacteristic
174#431C2312yes—
175C2×#106—24no—
176C2×#106—24no—
177#423C2312yes—
178#423C2312yes—
179C2×C4⋊C8D412yes—
180C2×C4⋊C8C2312yes—
181#417—24no—
182C2×#102—24no—
183C2×#102—24no—
184#408—24no—
185#408—28no—
186#407—21no—
187#407—21no—
188#407—24no—
189C2×C22⋊C8—21no—
190C2×C22⋊C8—24no—
191C2×C22⋊C8D412yes—
192C4×M16—21no—
193C4×M16—21no—
194C2×C4×C8—21no—
195C2×C4×C8C2312yes—
196#335C2318yes—
197#334C2318yes—
198#333C23116yes—
199#332C2318yes—
200#331C2318yes—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—22no—
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—41no—
208C2×#41—41no—
209C2×#41—84no—
210C2×#41—84no—
211C22×Q8—21no—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—41no—
217C22×Q8—41no—
218C22×Q8—41no—
219C22×Q8—84no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—41no—
227C22×D4C2×D412yes—
228C23×C4—22no—
229C23×C4—41no—
230C23×C4—41no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4#4111yescharacteristic
234AGL(1,ℤ/8)—88no—
235C2×SD16—28no—
236C2×D8—28no—
237#131—88no—
238C2×M16—22no—
239C2×M16—22no—
240C2×M16—28no—
241C22×C8—22no—
242C22×C8—28no—
243C22×C8C2×D414yes—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—41no—
259C4⋊Q8—41no—
260C4⋊Q8—41no—
261C4⋊Q8—41no—
262C4⋊Q8—41no—
263C4⋊Q8—41no—
264C4⋊Q8C2×D412yes—
265C4⋊Q8C2×D412yes—
266C4⋊Q8C2414yes—
267C4⋊Q8C2412yes—
268#127—21no—
269#127—21no—
270#127—21no—
271#127—21no—
272#127—21no—
273#127C2×D412yes—
274#127C2×D412yes—
275#127C2412yes—
276#124—22no—
277#124—22no—
278#124—22no—
279#124—22no—
280#124—22no—
281#124—22no—
282#124—41no—
283#124—41no—
284#121—22no—
285#121—22no—
286#121—22no—
287#121—22no—
288#121—22no—
289#121—22no—
290#121—22no—
291#121—22no—
292#121—22no—
293#121—22no—
294#121—41no—
295#121—41no—
296#121—41no—
297#121—41no—
298#121C2×D414yes—
299#121C2×D414yes—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—41no—
311C4×D4—41no—
312C4×D4—41no—
313C4×D4—41no—
314C4×D4C2×D414yes—
315C4×D4C2414yes—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—41no—
326C2×C4⋊C4—41no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4—41no—
329C2×C4⋊C4—41no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4C2×D412yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—41no—
335C2×C22⋊C4—41no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42#4111yescharacteristic
343C2×C42#4111yescharacteristic
344C2×C42#4111yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#106—28no—
349#106—28no—
350#106—216no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8C2×D412yes—
355C4⋊C8C2×D414yes—
356C4⋊C8C2×D412yes—
357C4⋊C8C2414yes—
358C4⋊C8C2414yes—
359C4⋊C8C2414yes—
360#103—28no—
361#102—216no—
362#102—28no—
363#102—28no—
364#102—28no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—28no—
372C22⋊C8—28no—
373C22⋊C8C2×D412yes—
374C22⋊C8C2×D412yes—
375C22⋊C8C2×D414yes—
376#97—22no—
377#97—22no—
378C4×C8—22no—
379C4×C8—22no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8C2×D414yes—
383C4×C8C2412yes—
384C4×C8C2412yes—
385C24—22no—
386C24—41no—
387C24#12111yescharacteristic
388#41—48no—
389#41—48no—
390#41—84no—
391#41—84no—
392#41—84no—
393#41—84no—
394#41—84no—
395#41—84no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—41no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—84no—
407C2×Q8—84no—
408C2×Q8—84no—
409C2×Q8—84no—
410C2×Q8—84no—
411C2×Q8—84no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—28no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—44no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4C22×D414yes—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4#12111yescharacteristic
452C22×C4#12111yescharacteristic
453C22×C4#12111yescharacteristic
454C22×C4#12111yescharacteristic
455C22×C4#12111yescharacteristic
456C22×C4#12111yescharacteristic
457C22×C4#12111yescharacteristic
458C22×C4#12111yescharacteristic
459C22×C4#12111yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4111yescharacteristic
467C22×C4C2×#4111yescharacteristic
468C22×C4C2×#4111yescharacteristic
469SD16—88no—
470D8—88no—
471M16—44no—
472M16—44no—
473M16—88no—
474C2×C8—28no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—44no—
479C2×C8—88no—
480C2×C8C22×D414yes—
481C2×C8C22×D414yes—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4C22×D414yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C42#12111yescharacteristic
507C42#12111yescharacteristic
508C42#12111yescharacteristic
509C42#12111yescharacteristic
510C42#12111yescharacteristic
511C42#12111yescharacteristic
512C42C22×D411yescharacteristic
513C42C22×D411yescharacteristic
514C42C22×D411yescharacteristic
515C42C22×D411yescharacteristic
516C42C22×D411yescharacteristic
517C42C22×D411yescharacteristic
518C42C2×#4111yescharacteristic
519C42C2×#4111yescharacteristic
520C42C2×#4111yescharacteristic
521C42C2511yescharacteristic
522C23—22no—
523C23—22no—
524C23—24no—
525C23—41no—
526C23—44no—
527C23—42no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23C2×#12111yescharacteristic
534C23C2×#12111yescharacteristic
535C23C2×#12111yescharacteristic
536Q8—44no—
537Q8—44no—
538Q8—82no—
539Q8—84no—
540Q8—84no—
541Q8—84no—
542Q8—82no—
543Q8—84no—
544D4—28no—
545D4—48no—
546D4—44no—
547D4—44no—
548D4—44no—
549D4—44no—
550D4—44no—
551D4—82no—
552D4—82no—
553D4—82no—
554D4—82no—
555D4—84no—
556D4—84no—
557D4—84no—
558D4—82no—
559D4—82no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12111yescharacteristic
590C2×C4C2×#12111yescharacteristic
591C2×C4C2×#12111yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4C2×#12111yescharacteristic
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×#12111yescharacteristic
597C2×C4#57714yes—
598C2×C4#57714yes—
599C2×C4C23×D411yescharacteristic
600C2×C4C23×D411yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C8—24no—
603C8—24no—
604C8—44no—
605C8—44no—
606C8—44no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—164no—
622C22—164no—
623C22#304212yes—
624C22#305712yes—
625C22C22×#12111yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—44no—
634C4—82no—
635C4—84no—
636C4#304212yes—
637C4#305712yes—
638C4#306812yes—
639C4C2×#57712yes—
640C4C2×#57712yes—
641C2—22no—
642C2—22no—
643C2—41no—
644C2—84no—
645C2—162no—
646C2C2×#304211yescharacteristic
647C2C2×#305711yescharacteristic
648C2C2×#306811yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60803C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3227C2semidirect
2. C2×#3227C2semidirect
4. #60236C2semidirect
5. #19513C2semidirect
6. #19427C2semidirect
7. #19391C2semidirect
8. #19196C2semidirect
9. #19191C2semidirect
10. #17803C2semidirect
11. #17746C2semidirect
12. #17717C2semidirect
23. #3227C22semidirect
24. #3227C22semidirect
31. C2×#498C22semidirect
35. C2×#486C22semidirect
39. C2×#476C22semidirect
42. #2981C22semidirect
44. #2975C22semidirect
46. #2972C22semidirect
54. #1697C22semidirect
55. #1691C22semidirect
56. #1677C22semidirect
57. #1671C22semidirect
58. #1572C22semidirect
59. #1571C22semidirect
60. #1570C22semidirect
61. #1564C22semidirect
62. #1558C22semidirect
63. #1553C22semidirect
64. #1549C22semidirect
65. #1548C22semidirect
66. #1548C22semidirect
67. #1545C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#608032
#602362
#195132
#194272
#193912
#191962
#191912
#178032
#177462
#177172