Finite Group Databasebeta

← #220220

Subgroups — #220220

Order 512.

1129 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#576C211yesmaximal, characteristic
2#17858C211yesmaximal, characteristic, complemented
3#17849C211yesmaximal, characteristic, complemented
4#17438C212yesmaximal, complemented
5#9668C211yesmaximal, characteristic, complemented
6#9659C211yesmaximal, characteristic, complemented
7#9095C212yesmaximal, complemented
8#8954C212yesmaximal, complemented
9#8604C211yesmaximal, characteristic, complemented
10#8318C212yesmaximal, complemented
11#8185C211yesmaximal, characteristic, complemented
12C2×#576C411yescharacteristic
13C4×H(5,𝔽2)C2212yes—
14#2988C2212yescomplemented
15C8×#41C2212yescomplemented
16#2951—21no—
17#2951C411yescharacteristic
18#2948—21no—
19#2948C411yescharacteristic
20#2946—22no—
21#2946C412yes—
22C4×AGL(1,ℤ/8)—22no—
23C4×AGL(1,ℤ/8)C2212yes—
24C4×#135—21no—
25C4×#135C2211yescharacteristic
26C2×C4×D8—21no—
27C2×C4×D8C2211yescharacteristic
28#2913C411yescharacteristic
29#2897C412yes—
30C4×#131C2211yescharacteristic
31#1954C2212yescomplemented
32#1580—21no—
33#1580—21no—
34#1579—21no—
35#1579—21no—
36SD16⋊C8—22no—
37C8×SD16—22no—
38#1564C2211yescharacteristic, complemented
39C2×#334C2211yescharacteristic, complemented
40#1478—22no—
41#1478C2212yescomplemented
42#1477—22no—
43#1477C2212yes—
44#1477C2212yescomplemented
45#1476—22no—
46#1476C2211yescharacteristic
47#1476C2211yescharacteristic
48#1476C2212yescomplemented
49C2×D4⋊C8—22no—
50C2×D4⋊C8C2212yescomplemented
51#1460C2212yescomplemented
52#1450C2211yescharacteristic, complemented
53C2×#322C2211yescharacteristic, complemented
54#1283C2212yescomplemented
55#1278C2211yescharacteristic, complemented
56#1278C2211yescharacteristic, complemented
57C2×H(5,𝔽2)C2×C412yes—
58#576—24no—
59#576C818yescomplemented
60C2×AGL(1,ℤ/8)—22no—
61C2×AGL(1,ℤ/8)C2×C412yes—
62C2×#135—21no—
63C2×#135C2×C411yescharacteristic
64C22×D8—21no—
65C22×D8C2×C411yescharacteristic
66C2×#131C2×C411yescharacteristic
67C2×#131C2×C411yescharacteristic
68#520—22no—
69#520—22no—
70#520C2×C412yes—
71#520C2×C412yes—
72#518—22no—
73#518—22no—
74#518—42no—
75#518C2×C412yes—
76C4×#41—22no—
77C4×#41—22no—
78C4×#41C2312yes—
79C4×#41C2311yescharacteristic
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—42no—
83C2×C4×D4C2312yes—
84C8×Q8—22no—
85C8×Q8—22no—
86#442—22no—
87#442—22no—
88#442—22no—
89#440—22no—
90#440—22no—
91#440—22no—
92C4×Q16—21no—
93C4×Q16D411yescharacteristic
94C4×Q16D411yescharacteristic
95C4×SD16—22no—
96C4×SD16—22no—
97C4×SD16D412yes—
98C4×SD16D412yes—
99C4×D8—21no—
100C4×D8—22no—
101C4×D8—21no—
102C4×D8—21no—
103C4×D8—22no—
104C4×D8—21no—
105C4×D8—21no—
106C4×D8—21no—
107C4×D8—21no—
108C4×D8D411yescharacteristic
109C4×D8D411yescharacteristic
110C8×D4—22no—
111C8×D4—22no—
112C8×D4—42no—
113C8×D4—42no—
114#431—22no—
115#431—22no—
116#431C2312yes—
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#428—22no—
120#428C2×C412yes—
121#427—21no—
122#427C2×C411yescharacteristic
123C2×#107—21no—
124C2×#107C2×C411yescharacteristic
125C2×C4⋊C8—22no—
126C2×C4⋊C8C2312yes—
127#418—22no—
128#418C2×C412yes—
129#417—22no—
130#417C2×C412yes—
131#416—22no—
132#416C2×C412yes—
133C2×#102—22no—
134C2×#102C2×C412yes—
135C2×C22⋊C8—42no—
136#405—21no—
137#405—21no—
138#405C2×C411yescharacteristic
139#405C2×C411yescharacteristic
140C4×M16—21no—
141C4×M16C2311yescharacteristic
142C2×C4×C8—21no—
143C2×C4×C8—22no—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#335—21no—
148#334—21no—
149#334—21no—
150#334—21no—
151Q8⋊C8—22no—
152Q8⋊C8—22no—
153Q8⋊C8—22no—
154Q8⋊C8—21no—
155Q8⋊C8D412yes—
156D4⋊C8—22no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—21no—
160D4⋊C8—22no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8D412yes—
165#323—42no—
166#322—21no—
167#322—21no—
168#322—21no—
169C82—21no—
170H(5,𝔽2)—24no—
171H(5,𝔽2)—24no—
172H(5,𝔽2)—42no—
173H(5,𝔽2)C2×C818yescomplemented
174C2×#41—22no—
175C2×#41—22no—
176C2×#41C22×C412yes—
177C2×#41C22×C411yescharacteristic
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—42no—
181C22×D4C22×C412yes—
182C23×C4—22no—
183C23×C4—42no—
184AGL(1,ℤ/8)—24no—
185AGL(1,ℤ/8)—216no—
186AGL(1,ℤ/8)C2×C818yescomplemented
187#135—28no—
188#135C2×C814yes—
189#135M1614yes—
190C2×Q16C22⋊C411yescharacteristic
191C2×SD16—22no—
192C2×SD16C22⋊C412yes—
193C2×D8—22no—
194C2×D8—28no—
195C2×D8—21no—
196C2×D8—21no—
197C2×D8—22no—
198C2×D8C22⋊C411yescharacteristic
199C2×D8C2×C814yescomplemented
200#131—22no—
201#131—22no—
202#131—41no—
203#131C2×C814yescomplemented
204C2×M16—21no—
205C2×M16—21no—
206C2×M16C22×C411yescharacteristic
207C2×M16C22×C411yescharacteristic
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8C2×D412yes—
220C4×Q8C2×D412yes—
221C4×Q8C2×D411yescharacteristic
222C4×Q8#4111yescharacteristic
223C4×D4—21no—
224C4×D4—22no—
225C4×D4—21no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4C2×D412yes—
248C4×D4C2×D412yes—
249C4×D4C2×D411yescharacteristic
250C4×D4#4111yescharacteristic
251#117—21no—
252#117—22no—
253#117—22no—
254#117—22no—
255#117—42no—
256#117—42no—
257#117C22×C412yes—
258#117C22×C411yescharacteristic
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22×C412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—42no—
271C2×C42C2411yescharacteristic
272#107—21no—
273#107—21no—
274#107—22no—
275#107C22⋊C411yescharacteristic
276#107C22⋊C411yescharacteristic
277#106—22no—
278#106C22⋊C412yes—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—21no—
283C4⋊C8—42no—
284C4⋊C8—42no—
285C4⋊C8C2×D412yes—
286#103—22no—
287#103C22⋊C412yes—
288#103C22⋊C412yes—
289#102—22no—
290#102—22no—
291#102—22no—
292#102—22no—
293#102—22no—
294#102C22⋊C412yes—
295#102C22⋊C412yes—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—22no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301C22⋊C8—84no—
302#97—21no—
303#97—21no—
304#97—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—22no—
313C4×C8—21no—
314C4×C8—42no—
315C4×C8—42no—
316C4×C8C2×D412yes—
317C4×C8C2×D411yescharacteristic
318C4×C8C2×D411yescharacteristic
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C24—22no—
322C24—42no—
323#41—21no—
324#41—22no—
325#41—21no—
326#41—22no—
327#41—28no—
328#41—28no—
329#41—44no—
330#41—44no—
331#41C22×C814yes—
332#41C22×C812yes—
333#41C2×M1614yes—
334#41#13112yes—
335C2×Q8C2×C22⋊C412yes—
336C2×Q8C4×D411yescharacteristic
337C2×D4—21no—
338C2×D4—22no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4C2×C22⋊C412yes—
353C2×D4C4×D411yescharacteristic
354C2×D4C22×C814yes—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—162no—
372C22×C4—162no—
373C22×C4C23×C411yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376Q16C22⋊C814yescomplemented
377SD16—28no—
378SD16C22⋊C818yescomplemented
379D8—28no—
380D8—24no—
381D8—24no—
382D8C22⋊C814yescomplemented
383M16—21no—
384M16—24no—
385M16—21no—
386M16—21no—
387M16—42no—
388M16C22×C812yes—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—24no—
393C2×C8—21no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—81no—
403C2×C8—84no—
404C2×C8—82no—
405C2×C8—162no—
406C2×C8C2×C22⋊C411yescharacteristic
407C2×C8C2×C22⋊C411yescharacteristic
408C2×C8C2×C22⋊C411yescharacteristic
409C2×C8C2×C22⋊C411yescharacteristic
410C2×C8C22×C812yes—
411C2×C8C2×M1612yes—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4C2×C22⋊C412yes—
421C4⋊C4C2×C22⋊C412yes—
422C4⋊C4C4×D411yescharacteristic
423C4⋊C4C4×D411yescharacteristic
424C22⋊C4—21no—
425C22⋊C4—22no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C42—21no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—42no—
443C42—42no—
444C42C22×D411yescharacteristic
445C42C22×D411yescharacteristic
446C42C22×D411yescharacteristic
447C42C2×#4111yescharacteristic
448C23—21no—
449C23—22no—
450C23—24no—
451C23—48no—
452C23—42no—
453C23—42no—
454C23—82no—
455C23—82no—
456C23—162no—
457C23—162no—
458C23#51811yescharacteristic
459Q8—22no—
460Q8—21no—
461Q8C2×C22⋊C814yes—
462Q8C8×D412yes—
463D4—22no—
464D4—24no—
465D4—22no—
466D4—24no—
467D4—24no—
468D4—22no—
469D4—24no—
470D4—21no—
471D4—48no—
472D4—48no—
473D4—42no—
474D4—42no—
475D4C2×C22⋊C814yes—
476D4C8×D412yes—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—48no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—162no—
500C2×C4—162no—
501C2×C4—162no—
502C2×C4—162no—
503C2×C4—162no—
504C2×C4C22×C22⋊C411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4D4211yescharacteristic
507C2×C4D4211yescharacteristic
508C2×C4#54611yescharacteristic
509C2×C4#54611yescharacteristic
510C2×C4C23×C811yescharacteristic
511C2×C4C22×M1611yescharacteristic
512C2×C4C2×#13111yescharacteristic
513C2×C4#56811yescharacteristic
514C8—22no—
515C8—21no—
516C8—21no—
517C8—42no—
518C8—41no—
519C8—42no—
520C8—81no—
521C8—81no—
522C8—82no—
523C8—82no—
524C8C2×C22⋊C812yes—
525C8C2×C22⋊C812yes—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—24no—
530C22—42no—
531C22—41no—
532C22—41no—
533C22—42no—
534C22—44no—
535C22—84no—
536C22—82no—
537C22—82no—
538C22—162no—
539C22—168no—
540C22D4×C22⋊C411yescharacteristic
541C22#297211yescharacteristic
542C22#297411yescharacteristic
543C4—22no—
544C4—24no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—42no—
549C4—41no—
550C4—42no—
551C4—41no—
552C4—82no—
553C4C22×C22⋊C811yescharacteristic
554C4#288011yescharacteristic
555C4C2×C8×D411yescharacteristic
556C4D4×M1611yescharacteristic
557C4#328812yes—
558C2—21no—
559C2—21no—
560C2—44no—
561C2—42no—
562C2—84no—
563C2D4×C22⋊C811yescharacteristic
564C2#2246111yescharacteristic
565C2#2246311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC22D4×C22⋊C4
prime radicalC2×#576C4
LCSG ▷ C2×C4 ▷ C2 ▷ 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. #17858C2semidirect
3. #17849C2semidirect
4. #17438C2semidirect
5. #9668C2semidirect
6. #9659C2semidirect
7. #9095C2semidirect
8. #8954C2semidirect
9. #8604C2semidirect
10. #8318C2semidirect
11. #8185C2semidirect
14. #2988C22semidirect
15. C8×#41C22semidirect
31. #1954C22semidirect
38. #1564C22semidirect
39. C2×#334C22semidirect
41. #1478C22semidirect
44. #1477C22semidirect
48. #1476C22semidirect
50. C2×D4⋊C8C22semidirect
51. #1460C22semidirect
52. #1450C22semidirect
53. C2×#322C22semidirect
54. #1283C22semidirect
55. #1278C22semidirect
56. #1278C22semidirect
59. #576C8semidirect
173. H(5,𝔽2)C2×C8semidirect
186. AGL(1,ℤ/8)C2×C8semidirect
199. C2×D8C2×C8semidirect
203. #131C2×C8semidirect
376. Q16C22⋊C8semidirect
378. SD16C22⋊C8semidirect
382. D8C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5762
#178582
#178492
#174382
#96682
#96592
#90952
#89542
#86042
#83182
#81852