Finite Group Databasebeta

← #7590477

Subgroups — #7590477

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32703C214yesmaximal, complemented
2#29385C214yesmaximal, complemented
3#29230C211yesmaximal, characteristic, complemented
4#28249C211yesmaximal, characteristic, complemented
5#28181C212yesmaximal, complemented
6#28180C212yesmaximal, complemented
7#27834C212yesmaximal, complemented
8#27833C212yesmaximal, complemented
9#27692C212yesmaximal, complemented
10#27691C212yesmaximal, complemented
11#23043C212yesmaximal, complemented
12#23042C212yesmaximal, complemented
13#22694C211yesmaximal, characteristic, complemented
14#22694C211yesmaximal, characteristic, complemented
15#21548C211yesmaximal, characteristic, complemented
16#21545C211yesmaximal, characteristic, complemented
17#20828C211yesmaximal, characteristic, complemented
18#3067C2212yes—
19#3067C2212yes—
20#3054C2211yescharacteristic
21#3050C2211yescharacteristic
22#2957C2212yes—
23#2957C2212yes—
24#2956C2214yes—
25C2×#443C2211yescharacteristic
26C2×#443C2211yescharacteristic
27#2930C2212yes—
28#2929C2212yes—
29#2921C2212yes—
30#2916C2212yes—
31#2916C2212yes—
32C2×#405C2211yescharacteristic
33#2103C2214yes—
34#2103C2214yes—
35#2102C2214yes—
36#2091C2214yes—
37#2080C2214yes—
38#2080C2214yes—
39#2050C2214yes—
40#2010C2214yes—
41#1975C2211yescharacteristic
42#1974C2211yescharacteristic
43#1953C2212yes—
44#1953C2212yes—
45#1953C2212yes—
46#1953C2212yes—
47#1946C2211yescharacteristic
48#1945C2211yescharacteristic
49#1938C2212yes—
50#1938C2212yes—
51#1932C2214yes—
52#1932C2214yes—
53#1897C2212yes—
54#1897C2212yes—
55#1891C2214yes—
56#1891C2214yes—
57#1881C2214yes—
58#1881C2214yes—
59#1861C2214yes—
60#1861C2214yes—
61#1858C2214yes—
62#1858C2214yes—
63#1848C2211yescharacteristic
64#1848C2211yescharacteristic
65#1841C2214yes—
66#1838C2212yes—
67#1833C2212yes—
68#1833C2212yes—
69#1814C2212yes—
70#1814C2212yes—
71#1813C2212yes—
72#1813C2212yes—
73#1811C2212yes—
74#1811C2211yescharacteristic
75#1810C2212yes—
76#1810C2211yescharacteristic
77#1778C2212yes—
78C4×#108C2212yes—
79#1772C2211yescharacteristic
80#1772C2211yescharacteristic
81#1765C2212yes—
82#1764C2212yes—
83#575C2314yes—
84C2×#135C2311yescharacteristic
85C2×#135C2311yescharacteristic
86#568C2314yes—
87#568C2314yes—
88C2×#131C2312yes—
89C2×#131C2312yes—
90C22×M16C2311yescharacteristic
91C23×C8C2311yescharacteristic
92#525C2312yes—
93#525C2312yes—
94C2×#117C2311yescharacteristic
95#473—22no—
96#473—22no—
97#472—22no—
98#472—22no—
99#471—24no—
100#471—24no—
101#470—21no—
102#469—21no—
103#468—22no—
104#467—21no—
105#466—21no—
106#465—22no—
107#444—44no—
108#444—42no—
109#444—42no—
110#443—21no—
111#443—21no—
112#443—21no—
113#443—21no—
114#443—21no—
115#443—21no—
116#443—21no—
117#443—21no—
118#443—44no—
119#443—42no—
120#443—42no—
121#436—42no—
122#435—42no—
123#435—42no—
124C8×D4—42no—
125#433—44no—
126#430—42no—
127#430—42no—
128#430C2314yes—
129#430C2314yes—
130#430C2314yes—
131C2×#108—42no—
132C2×#108—42no—
133C2×#108C2311yescharacteristic
134C2×#108C2312yes—
135C2×#108C2312yes—
136C2×#108C2311yescharacteristic
137C2×#108C2311yescharacteristic
138C2×#108C2312yes—
139C2×#108C2312yes—
140C2×#108C2311yescharacteristic
141#427C2311yescharacteristic
142#427C2311yescharacteristic
143#424C2312yes—
144#424C2312yes—
145#423C2312yes—
146#423C2312yes—
147#421C2314yes—
148#421C2314yes—
149C2×#104C2312yes—
150C2×#104C2312yes—
151C2×#104C2312yes—
152C2×#104C2312yes—
153#416C2312yes—
154#416C2312yes—
155#413C2314yes—
156#413C2314yes—
157#408C2312yes—
158#408C2312yes—
159#407C2314yes—
160#407C2312yes—
161#407C2312yes—
162#405—21no—
163#405—21no—
164#405—21no—
165#405—21no—
166#405C2312yes—
167#405C2312yes—
168C4×M16C2314yes—
169C2×#97C2311yescharacteristic
170C2×#97C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173#344C2314yes—
174#344C2314yes—
175#344C2314yes—
176#344C2314yes—
177#341C2314yes—
178#341C2312yes—
179#341C2314yes—
180#341C2312yes—
181#339C2312yes—
182#339C2312yes—
183#339C2312yes—
184#339C2312yes—
185#338C2314yes—
186H(3,ℤ/4)C2314yes—
187C2×#41C2412yes—
188C2×#41C2412yes—
189C23×C4C2411yescharacteristic
190#137—24no—
191#137—24no—
192AGL(1,ℤ/8)—24no—
193AGL(1,ℤ/8)—24no—
194#135—21no—
195#135—21no—
196#135—21no—
197#135—21no—
198#135—21no—
199#135—21no—
200#135—21no—
201#135—21no—
202#135—44no—
203C2×Q16#4111yescharacteristic
204C2×Q16#4111yescharacteristic
205C2×SD16#4112yes—
206C2×SD16#4112yes—
207C2×D8#4111yescharacteristic
208C2×D8#4111yescharacteristic
209#131—22no—
210#131—22no—
211#131—22no—
212#131—22no—
213#131—22no—
214#131—22no—
215#131—22no—
216#131—22no—
217#131—44no—
218#131—44no—
219#131—44no—
220#131—44no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—22no—
228C2×M16—22no—
229C2×M16—44no—
230C2×M16—44no—
231C2×M16C2412yes—
232C2×M16C2412yes—
233C2×M16C2412yes—
234C2×M16C2412yes—
235C2×M16C2414yes—
236C2×M16C2414yes—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8C2411yescharacteristic
246C22×C8C2411yescharacteristic
247C22×C8C2411yescharacteristic
248C22×C8C2411yescharacteristic
249#123—22no—
250#123—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253#121—22no—
254#121—22no—
255#120—22no—
256#120—22no—
257C4×D4—42no—
258C4×D4—42no—
259#117—21no—
260#117—21no—
261#117—22no—
262#117C2412yes—
263#117C2412yes—
264C2×C4⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268#108—21no—
269#108—21no—
270#108—21no—
271#108—21no—
272#108—22no—
273#108—22no—
274#108—22no—
275#108—22no—
276#108—21no—
277#108—21no—
278#108—22no—
279#108—22no—
280#108—21no—
281#108—21no—
282#108—22no—
283#108—22no—
284#108—44no—
285#108—44no—
286#108—44no—
287#107—21no—
288#107—21no—
289#106—21no—
290#106—21no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295#104—22no—
296#104—22no—
297#104—22no—
298#104—22no—
299#104—22no—
300#104—24no—
301#104—22no—
302#104—44no—
303#104—44no—
304#103—22no—
305#103—22no—
306#102—22no—
307#102—22no—
308#101—24no—
309#101—24no—
310#100—24no—
311#100—24no—
312C22⋊C8—44no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317#97—21no—
318#97—21no—
319#97—22no—
320#97—22no—
321#97—21no—
322#97—21no—
323#97—24no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—24no—
329C4×C8#4111yescharacteristic
330C4×C8#4111yescharacteristic
331C4×C8#4111yescharacteristic
332C4×C8#4111yescharacteristic
333C242−1+411yescharacteristic
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342C2×Q8C2×#4112yes—
343C2×Q8C2×#4112yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4C2×#4112yes—
349C2×D4C2×#4112yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C42−1+411yescharacteristic
361C22×C4C2511yescharacteristic
362Q16—21no—
363Q16—21no—
364Q16—21no—
365Q16—21no—
366Q16—44no—
367SD16—22no—
368SD16—22no—
369SD16—22no—
370SD16—22no—
371SD16—44no—
372SD16—44no—
373D8—21no—
374D8—21no—
375D8—21no—
376D8—21no—
377D8—44no—
378M16—22no—
379M16—22no—
380M16—24no—
381M16—24no—
382M16—22no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16—24no—
387M16—24no—
388M16—22no—
389M16—22no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—42no—
395M16—42no—
396M16—42no—
397M16—42no—
398M16—84no—
399M16—84no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—84no—
421C2×C8—84no—
422C2×C8C2×#4111yescharacteristic
423C2×C8C2×#4111yescharacteristic
424C2×C8C2×#4111yescharacteristic
425C2×C8C2×#4111yescharacteristic
426C2×C8C2×#4111yescharacteristic
427C2×C8C2×#4111yescharacteristic
428C2×C8C2×#4111yescharacteristic
429C2×C8C2×#4111yescharacteristic
430C2×C8H(5,𝔽2)11yescharacteristic
431C2×C8H(5,𝔽2)11yescharacteristic
432C2×C8H(5,𝔽2)11yescharacteristic
433C2×C8H(5,𝔽2)11yescharacteristic
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C42—21no—
449C42—21no—
450C42—22no—
451C42—22no—
452C42C2×#4111yescharacteristic
453C42C2×#4111yescharacteristic
454C42H(5,𝔽2)12yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23—81no—
461C23C2×2−1+411yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—22no—
465Q8—22no—
466D4—22no—
467D4—22no—
468D4—22no—
469D4—22no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—84no—
475D4—162no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4C22×#4111yescharacteristic
497C2×C4C22×#4111yescharacteristic
498C2×C4C22×#4111yescharacteristic
499C2×C4C2×H(5,𝔽2)11yescharacteristic
500C2×C4C2×H(5,𝔽2)11yescharacteristic
501C2×C4C2×H(5,𝔽2)11yescharacteristic
502C8—41no—
503C8—41no—
504C8—42no—
505C8—42no—
506C8—44no—
507C8—42no—
508C8—42no—
509C8—44no—
510C8#55311yescharacteristic
511C8#55311yescharacteristic
512C8#55311yescharacteristic
513C8#55311yescharacteristic
514C8#55311yescharacteristic
515C8#55311yescharacteristic
516C8#55311yescharacteristic
517C8#55311yescharacteristic
518C22—41no—
519C22—42no—
520C22—41no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22#349211yescharacteristic
529C22#352011yescharacteristic
530C22#352011yescharacteristic
531C4—41no—
532C4—41no—
533C4—42no—
534C4—41no—
535C4—42no—
536C4—42no—
537C4—82no—
538C4—82no—
539C4C2×#55311yescharacteristic
540C4C2×#55311yescharacteristic
541C4#348111yescharacteristic
542C4#354911yescharacteristic
543C2—21no—
544C2—21no—
545C2—21no—
546C2—81no—
547C2—82no—
548C2—82no—
549C2#5922011yescharacteristic
subgroupG/H
centerC4#3549
commutator, Frattini, cosocleC22×C4C25
socleC2#59220
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #32703C2semidirect
2. #29385C2semidirect
3. #29230C2semidirect
4. #28249C2semidirect
5. #28181C2semidirect
6. #28180C2semidirect
7. #27834C2semidirect
8. #27833C2semidirect
9. #27692C2semidirect
10. #27691C2semidirect
11. #23043C2semidirect
12. #23042C2semidirect
13. #22694C2semidirect
14. #22694C2semidirect
15. #21548C2semidirect
16. #21545C2semidirect
17. #20828C2semidirect

Maximal subgroups

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

subgroupindex
#327032
#293852
#292302
#282492
#281812
#281802
#278342
#278332
#276922
#276912
#230432
#230422
#226942
#215482
#215452
#208282