Finite Group Databasebeta

← #6468120

Subgroups — #6468120

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29233C214yesmaximal, complemented
2#29230C211yesmaximal, characteristic, complemented
3#29154C211yesmaximal, characteristic, complemented
4#29154C211yesmaximal, characteristic, complemented
5#28249C211yesmaximal, characteristic
6#27834C212yesmaximal, complemented
7#27833C212yesmaximal, complemented
8#24986C212yesmaximal, complemented
9#24985C212yesmaximal, complemented
10#23834C214yesmaximal, complemented
11#21733C211yesmaximal, characteristic, complemented
12#21559C212yesmaximal, complemented
13#21558C212yesmaximal, complemented
14#20788C212yesmaximal, complemented
15#20787C212yesmaximal, complemented
16#20514C211yesmaximal, characteristic, complemented
17#20510C211yesmaximal, characteristic, complemented
18#3160C2211yescharacteristic
19#3067C2212yes—
20#3067C2212yes—
21#2961C2212yes—
22#2955C2214yes—
23C2×#444C2211yescharacteristic
24C2×#443C2211yescharacteristic
25C2×#443C2211yescharacteristic
26C2×#443C2211yescharacteristic
27#2943C2211yescharacteristic
28C4×#135C2211yescharacteristic
29#2875C2212yes—
30C4×#131C2212yes—
31#2103C2214yes—
32#2102C2214yes—
33#2080C2214yes—
34#2080C2214yes—
35#2053C2214yes—
36#2050C2214yes—
37#2031C2214yes—
38#2031C2214yes—
39#1995C2212yes—
40#1976C2212yes—
41#1976C2212yes—
42#1953C2212yes—
43#1953C2212yes—
44#1946C2212yes—
45#1945C2212yes—
46#1932C2214yes—
47#1932C2214yes—
48#1906C2214yes—
49#1906C2214yes—
50#1848C2211yescharacteristic
51#1841C2214yes—
52#1841C2214yes—
53#1833C2212yes—
54#1833C2212yes—
55#1833C2212yes—
56#1833C2212yes—
57#1827C2214yes—
58#1827C2214yes—
59#1814C2212yes—
60#1814C2212yes—
61#1813C2212yes—
62#1813C2212yes—
63#1812C2212yes—
64#1812C2212yes—
65#1778C2211yescharacteristic
66C4×#108C2211yescharacteristic
67#1773C2211yescharacteristic
68#1773C2212yes—
69#1772C2211yescharacteristic
70#1772C2211yescharacteristic
71#1772C2211yescharacteristic
72#1772C2212yes—
73#1765C2212yes—
74#1765C2212yes—
75#1765C2212yes—
76#1764C2212yes—
77#1759C2212yes—
78#1759C2212yes—
79#1757C2214yes—
80#1757C2214yes—
81#1747C2214yes—
82#1726C2211yescharacteristic
83#575C2314yes—
84C2×#135C2311yescharacteristic
85C2×#135C2311yescharacteristic
86C2×#131C2312yes—
87C2×#131C2312yes—
88C2×#131C2312yes—
89C2×#131C2312yes—
90#533C2312yes—
91C4×#41C2312yes—
92C4×#41C2312yes—
93#500—21no—
94#499—21no—
95#499—21no—
96#498—21no—
97#473—22no—
98#473—22no—
99#472—22no—
100#472—22no—
101#471—24no—
102#471—24no—
103#446—44no—
104#444—21no—
105#444—21no—
106#444—21no—
107#444—21no—
108#444—44no—
109#443—22no—
110#443—22no—
111#443—21no—
112#443—21no—
113#443—21no—
114#443—21no—
115#443—21no—
116#443—21no—
117#443—21no—
118#443—21no—
119#443—44no—
120#442—21no—
121#441—21no—
122#440—22no—
123C4×Q16—21no—
124C4×SD16—22no—
125C4×D8—21no—
126#430C2314yes—
127C2×#108C2311yescharacteristic
128C2×#108C2311yescharacteristic
129C2×#108C2312yes—
130C2×#108C2312yes—
131C2×#108C2311yescharacteristic
132C2×#108C2311yescharacteristic
133#427C2311yescharacteristic
134#427C2311yescharacteristic
135#427C2312yes—
136#427C2312yes—
137#424C2314yes—
138#423C2312yes—
139#421C2314yes—
140#421C2314yes—
141C2×#104C2312yes—
142C2×#104C2312yes—
143C2×#104C2312yes—
144C2×#104C2312yes—
145#416C2312yes—
146#416C2312yes—
147#413C2314yes—
148#413C2314yes—
149#410C2314yes—
150#410C2314yes—
151#405—22no—
152#405—22no—
153#405—22no—
154#405—22no—
155#405C2312yes—
156#405C2312yes—
157#405C2312yes—
158#405C2312yes—
159#405C2314yes—
160C4×M16—22no—
161C4×M16—22no—
162C4×M16C2312yes—
163C4×M16C2312yes—
164C2×#97—22no—
165C2×#97C2311yescharacteristic
166C2×#97C2311yescharacteristic
167C2×#97C2311yescharacteristic
168C2×C4×C8—22no—
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172#376C2311yescharacteristic
173#344C2314yes—
174#344C2314yes—
175#344C2314yes—
176#344C2314yes—
177#341C2312yes—
178#341C2312yes—
179#339C2312yes—
180#339C2312yes—
181#339C2314yes—
182#339C2314yes—
183#339C2312yes—
184#339C2312yes—
185#338C2314yes—
186H(3,ℤ/4)C2314yes—
187H(3,ℤ/4)C2314yes—
188H(3,ℤ/4)C2314yes—
189C2×#41C2412yes—
190C2×#41C2412yes—
191#137—24no—
192#137—24no—
193AGL(1,ℤ/8)—24no—
194AGL(1,ℤ/8)—24no—
195#135—21no—
196#135—21no—
197#135—21no—
198#135—21no—
199#135—21no—
200#135—21no—
201#135—21no—
202#135—21no—
203#135—44no—
204C2×Q16#4111yescharacteristic
205C2×Q16#4111yescharacteristic
206C2×SD16#4112yes—
207C2×SD16#4112yes—
208C2×D8#4111yescharacteristic
209C2×D8#4111yescharacteristic
210#131—22no—
211#131—22no—
212#131—22no—
213#131—22no—
214#131—24no—
215#131—22no—
216#131—22no—
217#131—22no—
218#131—22no—
219#131—22no—
220#131—22no—
221#131—22no—
222#131—24no—
223#131—22no—
224C2×M16—22no—
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16C2412yes—
229C2×M16C2412yes—
230C2×M16C2412yes—
231C2×M16C2412yes—
232C2×M16C2414yes—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8C2411yescharacteristic
238C22×C8C2411yescharacteristic
239C22×C8C2411yescharacteristic
240C22×C8C2411yescharacteristic
241C4⋊Q8—22no—
242#125—22no—
243C22⋊Q8—42no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—42no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—42no—
250C4×D4—42no—
251#117—22no—
252#117—22no—
253#117C2412yes—
254#117C2412yes—
255#117C2414yes—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260C2×C42C2411yescharacteristic
261#108—21no—
262#108—21no—
263#108—21no—
264#108—21no—
265#108—21no—
266#108—21no—
267#108—21no—
268#108—21no—
269#108—22no—
270#108—22no—
271#108—22no—
272#108—22no—
273#108—44no—
274#107—21no—
275#107—21no—
276#107—22no—
277#107—22no—
278#106—21no—
279#106—21no—
280#106—22no—
281#106—22no—
282C4⋊C8—42no—
283C4⋊C8—44no—
284#104—24no—
285#104—22no—
286#104—22no—
287#104—22no—
288#104—22no—
289#104—22no—
290#104—22no—
291#104—44no—
292#104—44no—
293#103—22no—
294#103—22no—
295#102—22no—
296#102—22no—
297#101—24no—
298#101—24no—
299#100—24no—
300#100—24no—
301#99—44no—
302#99—44no—
303#97—22no—
304#97—22no—
305#97—21no—
306#97—21no—
307#97—22no—
308#97—22no—
309#97—22no—
310#97—24no—
311#97—22no—
312#97—22no—
313#97#4111yescharacteristic
314#97#4111yescharacteristic
315#97#4111yescharacteristic
316#97#4111yescharacteristic
317C4×C8—22no—
318C4×C8—22no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—24no—
324C4×C8—22no—
325C4×C8—22no—
326C4×C8#4112yes—
327C4×C8#4112yes—
328C4×C8#4111yescharacteristic
329C4×C8#4111yescharacteristic
330C4×C8#4111yescharacteristic
331C4×C8#4111yescharacteristic
332#95—41no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341C2×Q8—42no—
342C2×Q8C2×#4112yes—
343C2×Q8C2×#4112yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4C2×#4112yes—
347C2×D4C2×#4112yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4C2511yescharacteristic
356Q16—21no—
357Q16—21no—
358Q16—21no—
359Q16—21no—
360Q16—44no—
361SD16—22no—
362SD16—22no—
363SD16—22no—
364SD16—22no—
365SD16—44no—
366SD16—44no—
367D8—21no—
368D8—21no—
369D8—21no—
370D8—21no—
371D8—44no—
372M16—22no—
373M16—22no—
374M16—24no—
375M16—24no—
376M16—22no—
377M16—22no—
378M16—22no—
379M16—22no—
380M16—22no—
381M16—22no—
382M16—42no—
383M16—42no—
384M16—42no—
385M16—42no—
386M16—42no—
387M16—44no—
388M16—42no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—24no—
396C2×C8—21no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8—42no—
404C2×C8C2×#4111yescharacteristic
405C2×C8C2×#4111yescharacteristic
406C2×C8C2×#4111yescharacteristic
407C2×C8C2×#4111yescharacteristic
408C2×C8C2×#4111yescharacteristic
409C2×C8C2×#4111yescharacteristic
410C2×C8C2×#4111yescharacteristic
411C2×C8C2×#4111yescharacteristic
412C2×C8C2×#4111yescharacteristic
413C2×C8C2×#4111yescharacteristic
414C2×C8C2×#4111yescharacteristic
415C2×C8C2×#4111yescharacteristic
416C2×C8C2×#4111yescharacteristic
417C2×C8C2×#4112yes—
418C2×C8C2×#4111yescharacteristic
419C2×C8H(5,𝔽2)12yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—24no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—42no—
439C42—42no—
440C42C2×#4111yescharacteristic
441C42C2×#4112yes—
442C42C2×#4111yescharacteristic
443C42C2×#4111yescharacteristic
444C42C2×#4111yescharacteristic
445C23—22no—
446C23—22no—
447C23#58511yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—22no—
451Q8—22no—
452Q8—162no—
453D4—22no—
454D4—22no—
455D4—22no—
456D4—22no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—84no—
479C2×C4—162no—
480C2×C4C22×#4111yescharacteristic
481C2×C4C22×#4111yescharacteristic
482C2×C4C22×#4111yescharacteristic
483C2×C4C22×#4111yescharacteristic
484C2×C4C22×#4111yescharacteristic
485C2×C4C2×H(5,𝔽2)11yescharacteristic
486C8—42no—
487C8—42no—
488C8—42no—
489C8—42no—
490C8—41no—
491C8—44no—
492C8#52911yescharacteristic
493C8#52911yescharacteristic
494C8#52911yescharacteristic
495C8#52911yescharacteristic
496C8#52911yescharacteristic
497C8#52911yescharacteristic
498C8#52911yescharacteristic
499C8#52911yescharacteristic
500C8#55512yes—
501C8#55512yes—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22#346011yescharacteristic
508C22#347511yescharacteristic
509C22#347511yescharacteristic
510C4—41no—
511C4—41no—
512C4—42no—
513C4—41no—
514C4—41no—
515C4—42no—
516C4—42no—
517C4—82no—
518C4—82no—
519C4—84no—
520C4C2×#52911yescharacteristic
521C4C2×#52911yescharacteristic
522C4C2×#55511yescharacteristic
523C4#348111yescharacteristic
524C2—21no—
525C2—21no—
526C2—21no—
527C2—82no—
528C2—82no—
529C2#3494311yescharacteristic
subgroupG/H
centerC4C2×#555
commutator, Frattini, cosocleC22×C4C25
socleC2#34943
prime radical#28249C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29233C2semidirect
2. #29230C2semidirect
3. #29154C2semidirect
4. #29154C2semidirect
6. #27834C2semidirect
7. #27833C2semidirect
8. #24986C2semidirect
9. #24985C2semidirect
10. #23834C2semidirect
11. #21733C2semidirect
12. #21559C2semidirect
13. #21558C2semidirect
14. #20788C2semidirect
15. #20787C2semidirect
16. #20514C2semidirect
17. #20510C2semidirect

Maximal subgroups

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

subgroupindex
#292332
#292302
#291542
#282492
#278342
#278332
#249862
#249852
#238342
#217332
#215592
#215582
#207882
#207872
#205142
#205102