Finite Group Databasebeta

← #498121

Subgroups — #498121

Order 512.

757 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33521C211yesmaximal, characteristic
2C2×#2180C211yesmaximal, characteristic, complemented
3C22×#360C211yesmaximal, characteristic
4#23028C211yesmaximal, characteristic, complemented
5#22452C211yesmaximal, characteristic
6#13250C211yesmaximal, characteristic, complemented
7#13153C211yesmaximal, characteristic, complemented
8#13143C211yesmaximal, characteristic, complemented
9#13066C211yesmaximal, characteristic, complemented
10#13039C211yesmaximal, characteristic, complemented
11#13038C211yesmaximal, characteristic, complemented
12#12955C211yesmaximal, characteristic, complemented
13#12929C211yesmaximal, characteristic, complemented
14#12650C211yesmaximal, characteristic, complemented
15#12616C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C2×#510C2211yescharacteristic
18C2×#510C2211yescharacteristic
19C22×M32C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#444C2211yescharacteristic
22C2×#444C2211yescharacteristic
23C2×#441C2211yescharacteristic, complemented
24C22×#106C2211yescharacteristic
25#2880—21no—
26#2360—21no—
27#2189C412yescomplemented
28#2189C412yescomplemented
29#2188C412yescomplemented
30#2187C412yescomplemented
31#2180—24no—
32C2×#365C2211yescharacteristic, complemented
33C2×#365C2211yescharacteristic, complemented
34C2×#362C2211yescharacteristic, complemented
35C2×#362C2211yescharacteristic, complemented
36C2×#360—24no—
37C2×#360C2211yescharacteristic
38C2×#360C2211yescharacteristic
39C2×#360C2211yescharacteristic
40C2×#360C2211yescharacteristic
41#2139C2211yescharacteristic
42#2139C2211yescharacteristic
43#2112C2211yescharacteristic
44C2×#347C2211yescharacteristic, complemented
45C2×#347C2211yescharacteristic, complemented
46#1975C2211yescharacteristic
47#1975C2211yescharacteristic
48#1971C2211yescharacteristic, complemented
49#1943C2211yescharacteristic, complemented
50#1927—21no—
51#1876—21no—
52#1876—21no—
53#1872—21no—
54#1865—21no—
55#1810C2211yescharacteristic
56#1789—21no—
57#1776C2211yescharacteristic, complemented
58#1752C2211yescharacteristic
59#1387C2211yescharacteristic, complemented
60#1387C2211yescharacteristic, complemented
61#1383C2211yescharacteristic, complemented
62#1383C2211yescharacteristic, complemented
63#1375C2211yescharacteristic, complemented
64#1375C2211yescharacteristic, complemented
65C22×#41—21no—
66C2×#135—22no—
67C2×#135—22no—
68C2×#135—21no—
69C2×#135—21no—
70C2×#135—41no—
71C2×#135—41no—
72C2×#135C2×C411yescharacteristic
73C2×#135C2×C411yescharacteristic
74C2×#135C2311yescharacteristic
75C2×#135C2311yescharacteristic
76C22×Q16C2×C411yescharacteristic
77C22×Q16C2311yescharacteristic
78C22×SD16—21no—
79C22×D8C2×C411yescharacteristic
80C2×#131—21no—
81C2×#131—21no—
82C23×C8C2311yescharacteristic
83C2×C22⋊Q8—21no—
84C2×C4×Q8—21no—
85C22×C4⋊C4—21no—
86#510—22no—
87#510—22no—
88#510—41no—
89#510—41no—
90C2×Q32C2×C412yescomplemented
91C2×Q32C2×C412yescomplemented
92C2×SD32C2×C412yescomplemented
93C2×SD32C2×C412yescomplemented
94C2×M32—21no—
95C2×M32—22no—
96C2×M32—41no—
97C2×M32C2311yescharacteristic
98C2×M32C2311yescharacteristic
99C2×M32C2311yescharacteristic
100C2×M32C2311yescharacteristic
101C22×C16—21no—
102#467—21no—
103#467—21no—
104#467C2×C412yes—
105#467C2×C412yes—
106#444—22no—
107#444—22no—
108#444—41no—
109#444—41no—
110#441—22no—
111#441—41no—
112C2×#108C2311yescharacteristic
113C2×#108C2311yescharacteristic
114C2×#107C2311yescharacteristic
115C2×#106—22no—
116C2×#106—21no—
117C2×#106—41no—
118C2×#106C2311yescharacteristic
119C2×#106C2311yescharacteristic
120C2×#104—21no—
121C2×#104—21no—
122C2×#104—21no—
123C2×#104—21no—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—21no—
127#408—21no—
128#408—21no—
129#408—41no—
130#408—41no—
131#407—21no—
132#407—41no—
133C2×C22⋊C8—41no—
134C2×C22⋊C8D411yescharacteristic
135C2×#97C2311yescharacteristic
136C2×#97C2311yescharacteristic
137#391—21no—
138#389—21no—
139#389—41no—
140#387—41no—
141#385—41no—
142C4×C4⋊C4—21no—
143C4×C22⋊C4—41no—
144C4×C22⋊C4D411yescharacteristic
145#365—22no—
146#365—41no—
147#362—22no—
148#362—41no—
149#360—22no—
150#360—22no—
151#360—22no—
152#360D414yes—
153#358C2×C412yescomplemented
154#358C2×C412yescomplemented
155#357C2×C412yescomplemented
156#357C2×C412yescomplemented
157C22⋊C16C2×C412yes—
158C22⋊C16C2×C412yes—
159#347—22no—
160#347—41no—
161#341—21no—
162#340—21no—
163#340—21no—
164#336D411yescharacteristic
165#336D411yescharacteristic
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—41no—
173C2×#41—41no—
174C2×#41—81no—
175C2×#41—81no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×D4—21no—
179C22×D4—41no—
180C23×C4—41no—
181C23×C4—41no—
182C23×C4C2×D411yescharacteristic
183#135—22no—
184#135—22no—
185#135—21no—
186#135—21no—
187#135—21no—
188#135—21no—
189#135—21no—
190#135—21no—
191#135—41no—
192#135—41no—
193#135—42no—
194#135—42no—
195#135—42no—
196#135—42no—
197#135C2×D412yes—
198#135#4112yes—
199C2×Q16—22no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16—41no—
207C2×Q16C22×C411yescharacteristic
208C2×Q16C22×C412yes—
209C2×Q16C22×C411yescharacteristic
210C2×SD16—21no—
211C2×SD16—21no—
212C2×SD16—21no—
213C2×SD16—21no—
214C2×SD16—42no—
215C2×SD16—42no—
216C2×D8—22no—
217C2×D8—21no—
218C2×D8—41no—
219C2×D8C22×C411yescharacteristic
220C2×D8C22×C411yescharacteristic
221#131—42no—
222#131—42no—
223#131—81no—
224#131—81no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—41no—
228C2×M16—41no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—41no—
234C22×C8—41no—
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8#4111yescharacteristic
241C22×C8#4111yescharacteristic
242C22×C8C2411yescharacteristic
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C4×Q8—42no—
248C4×Q8—81no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—81no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264Q32—24no—
265Q32—42no—
266SD32—24no—
267SD32—42no—
268M32—21no—
269M32—21no—
270M32—21no—
271M32—21no—
272M32—21no—
273M32—21no—
274M32—21no—
275M32—42no—
276M32C2×D412yes—
277C2×C16—21no—
278C2×C16—21no—
279C2×C16—42no—
280C2×C16C22×C412yes—
281C2×C16C22×C412yes—
282#108—21no—
283#108—21no—
284#108—21no—
285#108—21no—
286#107—21no—
287#107C22×C412yes—
288#106—21no—
289#106—21no—
290#106—21no—
291#106—42no—
292#106C2×D412yes—
293#104—42no—
294#104—42no—
295#104—81no—
296#104—81no—
297#103—22no—
298#103—22no—
299#103—42no—
300#103—81no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309#97—21no—
310#97—21no—
311#97—21no—
312#97—21no—
313#95—21no—
314#95—41no—
315#95—41no—
316#95—41no—
317#95—41no—
318C24—41no—
319C24AGL(1,ℤ/8)11yescharacteristic
320#41—22no—
321#41—22no—
322#41—42no—
323#41—42no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—82no—
331#41—82no—
332#41—81no—
333#41—81no—
334#41—82no—
335#41—82no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—42no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—81no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—42no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—81no—
356C2×D4—82no—
357C2×D4—82no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—161no—
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4AGL(1,ℤ/8)11yescharacteristic
381C22×C4AGL(1,ℤ/8)11yescharacteristic
382C22×C4AGL(1,ℤ/8)11yescharacteristic
383C22×C4C22×D411yescharacteristic
384Q16—22no—
385Q16—21no—
386Q16—21no—
387Q16—22no—
388Q16—21no—
389Q16—42no—
390Q16—41no—
391Q16—41no—
392Q16C4×D412yes—
393SD16—22no—
394SD16—22no—
395SD16—42no—
396SD16—42no—
397SD16—41no—
398SD16—41no—
399D8—21no—
400D8—21no—
401D8—21no—
402D8—42no—
403D8C4×D412yes—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—82no—
409M16—82no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
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—41no—
427C2×C8—41no—
428C2×C8—82no—
429C2×C8—82no—
430C2×C8C4×D411yescharacteristic
431C2×C8C4×D411yescharacteristic
432C2×C8C23×C411yescharacteristic
433C2×C8C22×D411yescharacteristic
434C2×C8C2×#4111yescharacteristic
435C2×C8C2×#4111yescharacteristic
436C2×C8H(5,𝔽2)11yescharacteristic
437C2×C8H(5,𝔽2)11yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—161no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—82no—
457C42—161no—
458C16—22no—
459C16—22no—
460C16—22no—
461C16—41no—
462C23—22no—
463C23—21no—
464C23—21no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—82no—
469C23—82no—
470C23—161no—
471C23#44211yescharacteristic
472C23#44211yescharacteristic
473C23C2×AGL(1,ℤ/8)11yescharacteristic
474Q8—22no—
475Q8—41no—
476Q8—41no—
477Q8—42no—
478Q8—42no—
479Q8—41no—
480Q8—82no—
481Q8—81no—
482Q8—81no—
483D4—22no—
484D4—42no—
485D4—42no—
486D4—41no—
487D4—41no—
488D4—41no—
489D4—82no—
490D4—82no—
491D4—82no—
492D4—81no—
493D4—81no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—164no—
532C2×C4#44211yescharacteristic
533C2×C4#44211yescharacteristic
534C2×C4C2×C4×D411yescharacteristic
535C2×C4D4211yescharacteristic
536C2×C4#54611yescharacteristic
537C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
538C2×C4#57611yescharacteristic
539C2×C4#57711yescharacteristic
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—81no—
545C8—81no—
546C8C2×C4×D411yescharacteristic
547C8#51811yescharacteristic
548C8#54611yescharacteristic
549C8#55111yescharacteristic
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—164no—
566C22#218012yescomplemented
567C22C2×#44211yescharacteristic
568C22#326411yescharacteristic
569C22#330611yescharacteristic
570C4—21no—
571C4—21no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—82no—
576C4—81no—
577C4—82no—
578C4—82no—
579C4—161no—
580C4#235811yescharacteristic
581C4#294611yescharacteristic
582C4#326511yescharacteristic
583C4#331811yescharacteristic
584C2—21no—
585C2—21no—
586C2—22no—
587C2—41no—
588C2—82no—
589C2—161no—
590C2#2285811yescharacteristic
591C2C2×#218011yescharacteristic
592C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2180C2semidirect
4. #23028C2semidirect
6. #13250C2semidirect
7. #13153C2semidirect
8. #13143C2semidirect
9. #13066C2semidirect
10. #13039C2semidirect
11. #13038C2semidirect
12. #12955C2semidirect
13. #12929C2semidirect
14. #12650C2semidirect
15. #12616C2semidirect
23. C2×#441C22semidirect
27. #2189C4semidirect
28. #2189C4semidirect
29. #2188C4semidirect
30. #2187C4semidirect
32. C2×#365C22semidirect
33. C2×#365C22semidirect
34. C2×#362C22semidirect
35. C2×#362C22semidirect
44. C2×#347C22semidirect
45. C2×#347C22semidirect
48. #1971C22semidirect
49. #1943C22semidirect
57. #1776C22semidirect
59. #1387C22semidirect
60. #1387C22semidirect
61. #1383C22semidirect
62. #1383C22semidirect
63. #1375C22semidirect
64. #1375C22semidirect
90. C2×Q32C2×C4semidirect
91. C2×Q32C2×C4semidirect
92. C2×SD32C2×C4semidirect
93. C2×SD32C2×C4semidirect
153. #358C2×C4semidirect
154. #358C2×C4semidirect
155. #357C2×C4semidirect
156. #357C2×C4semidirect
566. C22#2180semidirect

Maximal subgroups

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

subgroupindex
#335212
C2×#21802
C22×#3602
#230282
#224522
#132502
#131532
#131432
#130662
#130392
#130382
#129552
#129292
#126502
#126162