Finite Group Databasebeta

← #6392656

Subgroups — #6392656

Order 512.

2426 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3C2×#2352C211yesmaximal, characteristic
4#22709C218yesmaximal, complemented
5#22659C218yesmaximal, complemented
6#21986C212yesmaximal, complemented
7#21942C212yesmaximal, complemented
8#21940C212yesmaximal, complemented
9#21365C212yesmaximal, complemented
10#20846C212yesmaximal, complemented
11C2×#1853C212yesmaximal, complemented
12C2×#518—21no—
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#2973C4116yescomplemented
17C2×#435C2218yes—
18C2×#432C2212yes—
19C2×#428C2214yescomplemented
20C22×#106C2214yescomplemented
21C2×#407C2212yes—
22C22×C22⋊C8C2212yes—
23#2352—24no—
24#2352C2218yes—
25#2349—24no—
26D4×C4⋊C4C2214yes—
27D4×C4⋊C4C2214yes—
28#2316—21no—
29#2315—21no—
30#2315—21no—
31C2×#388—21no—
32C2×#387—21no—
33C2×#386—21no—
34#2285C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4—21no—
42#1949C2218yescomplemented
43#1948C2214yescomplemented
44#1947C2214yescomplemented
45#1854C2218yescomplemented
46#1853C2214yes—
47#1853C2214yes—
48#1853C2218yescomplemented
49#1853C2214yescomplemented
50#1852C2214yes—
51#1852C2214yes—
52#1852C2214yescomplemented
53#1852C2214yescomplemented
54#1852C2214yescomplemented
55#1847C2214yescomplemented
56#1836C2214yescomplemented
57#1809C2218yescomplemented
58#1808C2218yescomplemented
59#1776C2218yescomplemented
60#1736C2214yescomplemented
61C2×#340C2214yescomplemented
62C23×D4D411yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C22×M16C2314yes—
67C23×C8C2314yes—
68#518—48no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4C2×C418yes—
72C2×C4×D4D414yes—
73C2×C4×D4Q818yes—
74C2×C4×D4C2314yes—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—44no—
78C2×#117C2314yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—22no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4D411yescharacteristic
98C22×C22⋊C4D411yescharacteristic
99C22×C42—21no—
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102#435C2×C4164yescomplemented
103#432C2×C4116yescomplemented
104#428—48no—
105C2×#107C2318yes—
106C2×#107C2318yes—
107C2×#106—48no—
108C2×#106C2318yes—
109C2×#106C2318yes—
110C2×C4⋊C8C2314yes—
111#407C2×C4116yescomplemented
112C2×C22⋊C8C2×C4116yescomplemented
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×#97C2314yes—
117#389—24no—
118#389—24no—
119#388—28no—
120#387—28no—
121#386—28no—
122#385—24no—
123#385—24no—
124#385—24no—
125#385—24no—
126#384C2312yes—
127#384C2312yes—
128#382—22no—
129#382—22no—
130#382C2314yes—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380C2314yes—
139#380C2314yes—
140#380C2314yes—
141#380C2314yes—
142#380C2314yes—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4C2314yes—
146C4×C4⋊C4C2314yes—
147C4×C22⋊C4—28no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95—21no—
153C2×#95—21no—
154C2×#95—21no—
155#340C2318yes—
156#340C2318yes—
157#340C2318yes—
158#340C2318yes—
159C25C2×D411yescharacteristic
160C25C2×D411yescharacteristic
161C22×D4—24no—
162C22×D4—24no—
163C22×D4C4⋊C418yes—
164C22×D4C2×D414yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—82no—
181C23×C4C22×C414yes—
182C23×C4C22×C414yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×Q814yes—
190C23×C4C2×Q814yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16C22×C4116yescomplemented
197C22×C8—44no—
198C22×C8—44no—
199C22×C8C22×C4116yes—
200C22×C8C2418yes—
201C4×D4—416no—
202C4×D4—416no—
203C4×D4C4⋊C4132yes—
204C4×D4C22×C4116yes—
205C4×D4C2×Q8116yes—
206#117—28no—
207#117—44no—
208#117—48no—
209#117—48no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C4⋊C418yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2414yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—82no—
254C2×C22⋊C4C4⋊C418yes—
255C2×C22⋊C4C4⋊C418yes—
256C2×C22⋊C4C2×D414yes—
257C2×C22⋊C4C2×D414yes—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—44no—
262C2×C42—82no—
263C2×C42C22×C414yes—
264C2×C42C2×D412yes—
265C2×C42C2×Q814yes—
266C2×C42C2412yes—
267#107—48no—
268#107—48no—
269#106—216no—
270#106—48no—
271C4⋊C8C22×C4116yescomplemented
272C22⋊C8C22×C4116yes—
273C22⋊C8C22×C4116yes—
274C22⋊C8C22×C4116yescomplemented
275#97C22×C4116yescomplemented
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285C24—22no—
286C24—22no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24C2×C4⋊C414yes—
296C24C2×C4⋊C414yes—
297C24AGL(1,ℤ/8)14yes—
298C24#13714yes—
299C24C22×D412yes—
300C24C22×D412yes—
301C24C22×D411yescharacteristic
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—232no—
305C2×D4C2×C4⋊C4116yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—48no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—84no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—816no—
362C22×C4C2×C4⋊C414yes—
363C22×C4C2×C4⋊C414yes—
364C22×C4C2×C4⋊C414yes—
365C22×C4C2×C4⋊C414yes—
366C22×C4C2×C4⋊C418yes—
367C22×C4C2×C4⋊C418yes—
368C22×C4C23×C412yes—
369C22×C4C23×C414yes—
370C22×C4C23×C414yes—
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×Q812yes—
377C22×C4C22×Q814yes—
378C22×C4C22×Q814yes—
379C22×C4C2511yescharacteristic
380M16—416no—
381C2×C8—28no—
382C2×C8—28no—
383C2×C8—416no—
384C2×C8C23×C4116yes—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4C2×C4⋊C4116yes—
405C4⋊C4H(5,𝔽2)14yes—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—84no—
412C22⋊C4C2×C4⋊C4116yes—
413C22⋊C4C2×C4⋊C4116yes—
414C42—48no—
415C42—48no—
416C42—84no—
417C42C2×C4⋊C418yes—
418C42C23×C414yes—
419C42C22×Q814yes—
420C23—21no—
421C23—22no—
422C23—21no—
423C23—22no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—28no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23#428116yescomplemented
448C23C22×C4⋊C414yes—
449C23C22×C4⋊C414yes—
450C23C22×C4⋊C412yes—
451C23C2×AGL(1,ℤ/8)12yes—
452C23C2×AGL(1,ℤ/8)14yes—
453C23C2×#13712yes—
454C23C2×#13714yes—
455C23C23×D411yescharacteristic
456D4—232no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—816no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4—164no—
503C2×C4—164no—
504C2×C4—164no—
505C2×C4—164no—
506C2×C4C22×C4⋊C412yes—
507C2×C4C22×C4⋊C414yes—
508C2×C4C22×C4⋊C414yes—
509C2×C4C22×C4⋊C414yes—
510C2×C4C22×C4⋊C414yes—
511C2×C4C24×C411yescharacteristic
512C2×C4C23×Q811yescharacteristic
513C2×C4C2×H(5,𝔽2)12yes—
514C2×C4#58512yes—
515C8—216no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—24no—
520C22—22no—
521C22—24no—
522C22—22no—
523C22—24no—
524C22—22no—
525C22—24no—
526C22—22no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—416no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—88no—
537C22C2×#42814yes—
538C22C2×#42818yes—
539C22C23×C4⋊C411yescharacteristic
540C22#348411yescharacteristic
541C22#348511yescharacteristic
542C22C22×AGL(1,ℤ/8)11yescharacteristic
543C22C22×#13711yescharacteristic
544C22#358511yescharacteristic
545C22#358611yescharacteristic
546C4—22no—
547C4—24no—
548C4—24no—
549C4—24no—
550C4—24no—
551C4—44no—
552C4—82no—
553C4—84no—
554C4—82no—
555C4—82no—
556C4—84no—
557C4C4×H(5,𝔽2)12yes—
558C4#350112yes—
559C2—24no—
560C2—28no—
561C2—44no—
562C2#3424811yescharacteristic
563C2C22×#42811yescharacteristic
564C2#6010911yescharacteristic
565C2#6157011yescharacteristic
566C2#6157211yescharacteristic
567C2#6170511yescharacteristic
568C2#6170711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22709C2semidirect
5. #22659C2semidirect
6. #21986C2semidirect
7. #21942C2semidirect
8. #21940C2semidirect
9. #21365C2semidirect
10. #20846C2semidirect
11. C2×#1853C2semidirect
16. #2973C4semidirect
19. C2×#428C22semidirect
20. C22×#106C22semidirect
42. #1949C22semidirect
43. #1948C22semidirect
44. #1947C22semidirect
45. #1854C22semidirect
48. #1853C22semidirect
49. #1853C22semidirect
52. #1852C22semidirect
53. #1852C22semidirect
54. #1852C22semidirect
55. #1847C22semidirect
56. #1836C22semidirect
57. #1809C22semidirect
58. #1808C22semidirect
59. #1776C22semidirect
60. #1736C22semidirect
61. C2×#340C22semidirect
102. #435C2×C4semidirect
103. #432C2×C4semidirect
111. #407C2×C4semidirect
112. C2×C22⋊C8C2×C4semidirect
196. C2×M16C22×C4semidirect
271. C4⋊C8C22×C4semidirect
274. C22⋊C8C22×C4semidirect
275. #97C22×C4semidirect
447. C23#428semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
#341812
C2×#23522
#227092
#226592
#219862
#219422
#219402
#213652
#208462
C2×#18532