Finite Group Databasebeta

← #6407839

Subgroups — #6407839

Order 512.

1689 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#560C211yesmaximal, characteristic
2#23378C211yesmaximal, characteristic, complemented
3#23375C211yesmaximal, characteristic, complemented
4#23357C211yesmaximal, characteristic, complemented
5#23329C213yesmaximal, complemented
6#23327C213yesmaximal, complemented
7#23320C213yesmaximal, complemented
8#23308C213yesmaximal, complemented
9#21681C211yesmaximal, characteristic, complemented
10#21666C211yesmaximal, characteristic, complemented
11C8×#126C211yesmaximal, characteristic, complemented
12#21565C213yesmaximal, complemented
13#21490C213yesmaximal, complemented
14#21488C213yesmaximal, complemented
15#21441C213yesmaximal, complemented
16C23×M16C2211yescharacteristic
17C2×#560C411yescharacteristic
18C2×#436C2213yes—
19C2×#436C2213yes—
20C2×#436C2213yescomplemented
21C2×#435C2213yes—
22C2×C4×M16C2213yescomplemented
23#2383C411yescharacteristic
24#2383C411yescharacteristic
25#2380C411yescharacteristic
26#2377C413yes—
27#2377C413yes—
28#2376C413yes—
29#2375C413yes—
30C4×#127C2211yescharacteristic
31C4×#126C2211yescharacteristic
32C4×#126C2211yescharacteristic
33C4×#123C2213yes—
34C4×#121C2213yes—
35C4×#121C2213yes—
36C4×#120C2213yes—
37#1955C2213yes—
38#1954C2213yes—
39#1954C2213yes—
40#1954C2213yescomplemented
41#1940C2213yescomplemented
42#1939C2213yescomplemented
43#1924C2213yes—
44#1924C2213yes—
45#1924C2213yescomplemented
46#1923C2213yes—
47#1923C2213yes—
48#1923C2213yes—
49#1923C2213yes—
50#1923C2213yescomplemented
51#1917C2213yescomplemented
52#1916C2213yescomplemented
53#1916C2213yescomplemented
54#1916C2213yescomplemented
55#1870C2213yes—
56#1870C2213yes—
57#1870C2213yes—
58#1870C2213yes—
59#1870C2213yes—
60#1870C2213yescomplemented
61#1870C2213yescomplemented
62#1869C2213yes—
63#1851C2213yes—
64#1851C2213yes—
65#1851C2213yescomplemented
66#1850C2213yes—
67#1842C2211yescharacteristic, complemented
68#1839C2211yescharacteristic, complemented
69#1839C2211yescharacteristic, complemented
70#1817C2213yescomplemented
71#1797C2213yescomplemented
72#1797C2213yescomplemented
73#1779C2213yes—
74#1771C2213yescomplemented
75C8×C4⋊C4C2213yescomplemented
76#1753C2213yes—
77#1753C2213yes—
78#1753C2213yescomplemented
79C8×C22⋊C4C2213yes—
80C42×C8C2211yescharacteristic, complemented
81C24×C4C2311yescharacteristic
82C22×M16—46no—
83C22×M16C2313yes—
84C22×M16C2311yescharacteristic
85C22×M16C2311yescharacteristic
86C22×M16C2313yes—
87C22×M16C2311yescharacteristic
88C22×M16C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91#560—28no—
92C2×#127C2×C411yescharacteristic
93C2×#126C2×C411yescharacteristic
94C2×#126C2×C411yescharacteristic
95C2×#123C2×C413yes—
96C2×#121C2×C413yes—
97C2×#121C2×C413yes—
98C2×#120C2×C413yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×C4×D4C2313yes—
102C2×C4×D4C2313yes—
103C22×C42C2313yes—
104#436—46no—
105#436—46no—
106#436—46no—
107#435—46no—
108C2×C4⋊C8C2313yes—
109C2×C4⋊C8C2313yes—
110C2×C4⋊C8C2313yes—
111C2×C4⋊C8C2313yes—
112C2×C4⋊C8C2313yes—
113C2×C4⋊C8C2313yes—
114C2×C22⋊C8C2313yes—
115C2×C22⋊C8C2313yes—
116C2×C22⋊C8C2313yes—
117C2×C22⋊C8C2313yes—
118C2×C22⋊C8C2313yes—
119C2×C22⋊C8C2313yes—
120C2×C22⋊C8C2313yes—
121C2×C22⋊C8C2313yes—
122C2×C22⋊C8C2313yes—
123C2×C22⋊C8C2313yes—
124C2×C22⋊C8C2313yes—
125C2×C22⋊C8C2313yes—
126C4×M16—46no—
127C2×#97C2313yes—
128C2×#97C2313yes—
129C2×C4×C8C2313yes—
130C2×C4×C8C2313yes—
131C2×C4×C8C2313yes—
132C2×C4×C8C2313yes—
133#390C2×C413yes—
134#390C2×C413yes—
135#390C2×C413yes—
136#390C2×C413yes—
137#389C2×C413yes—
138#389C2×C413yes—
139#388C2×C413yes—
140#388C2×C413yes—
141#388C2×C413yes—
142#388C2×C413yes—
143#388C2×C413yes—
144#388C2×C413yes—
145#388C2×C413yes—
146#388C2×C413yes—
147#387C2×C413yes—
148#387C2×C413yes—
149#387C2×C413yes—
150#387C2×C413yes—
151#386C2×C413yes—
152#386C2×C413yes—
153#386C2×C413yes—
154#386C2×C413yes—
155#386C2×C413yes—
156#386C2×C413yes—
157#386C2×C413yes—
158#386C2×C413yes—
159#385C2×C413yes—
160#385C2×C413yes—
161#385C2×C413yes—
162#385C2×C413yes—
163#384C2×C411yescharacteristic
164#383C2×C411yescharacteristic
165#383C2×C411yescharacteristic
166#381C2×C413yes—
167#380C2×C413yes—
168#380C2×C413yes—
169#379C2×C413yes—
170C4×C4⋊C4C2313yes—
171C4×C4⋊C4C2313yes—
172C4×C22⋊C4C2313yes—
173C4×C22⋊C4C2313yes—
174C4×C22⋊C4C2313yes—
175C4×C22⋊C4C2313yes—
176C43C2311yescharacteristic
177#336C2313yes—
178#336C2313yes—
179#336C2313yes—
180#336C2313yes—
181#336C2313yes—
182#336C2313yes—
183#336C2313yes—
184#336C2313yes—
185#336C2313yes—
186#336C2313yes—
187#336C2313yes—
188#336C2313yes—
189C25C22×C411yescharacteristic
190C22×D4C22×C413yes—
191C22×D4C22×C413yes—
192C22×D4C22×C413yes—
193C22×D4C22×C413yes—
194C23×C4—43no—
195C23×C4—43no—
196C23×C4C22×C413yes—
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C413yes—
201C23×C4C2413yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C2×M16—26no—
206C2×M16—26no—
207C2×M16—26no—
208C2×M16—26no—
209C2×M16—26no—
210C2×M16—43no—
211C2×M16—43no—
212C2×M16—43no—
213C2×M16—43no—
214C2×M16—43no—
215C2×M16—416no—
216C22×C8—26no—
217C22×C8—43no—
218C22×C8C2411yescharacteristic
219C22×C8C2411yescharacteristic
220C22×C8C2411yescharacteristic
221C22×C8C2413yes—
222C22×C8C2413yes—
223C22×C8C2413yes—
224C22×C8C2413yes—
225C22×C8C2411yescharacteristic
226#127—24no—
227#126—24no—
228#126M1618yescomplemented
229#123—212no—
230#121—212no—
231#121—212no—
232#120—212no—
233C4×D4—43no—
234C4×D4—43no—
235C4×D4—43no—
236C4×D4—43no—
237C4×D4—43no—
238C4×D4—43no—
239C4×D4—43no—
240C4×D4—43no—
241C2×C4⋊C4C22×C413yes—
242C2×C4⋊C4C22×C413yes—
243C2×C4⋊C4C22×C413yes—
244C2×C4⋊C4C22×C413yes—
245C2×C4⋊C4C22×C413yes—
246C2×C4⋊C4C22×C413yes—
247C2×C4⋊C4C22×C413yes—
248C2×C4⋊C4C22×C413yes—
249C2×C22⋊C4C22×C413yes—
250C2×C22⋊C4C22×C413yes—
251C2×C22⋊C4C22×C413yes—
252C2×C22⋊C4C22×C413yes—
253C2×C22⋊C4C22×C413yes—
254C2×C22⋊C4C22×C413yes—
255C2×C22⋊C4C22×C413yes—
256C2×C22⋊C4C22×C413yes—
257C2×C22⋊C4C22×C413yes—
258C2×C22⋊C4C22×C413yes—
259C2×C22⋊C4C22×C413yes—
260C2×C22⋊C4C22×C413yes—
261C2×C22⋊C4C22×C413yes—
262C2×C22⋊C4C22×C413yes—
263C2×C22⋊C4C22×C413yes—
264C2×C22⋊C4C22×C413yes—
265C2×C42—43no—
266C2×C42—43no—
267C2×C42C22×C413yes—
268C2×C42C22×C413yes—
269C2×C42C22×C413yes—
270C2×C42C22×C411yescharacteristic
271C2×C42C2413yes—
272C2×C42C2413yes—
273C2×C42C2413yes—
274C4⋊C8—43no—
275C4⋊C8—43no—
276C4⋊C8—43no—
277C4⋊C8—43no—
278C4⋊C8—43no—
279C4⋊C8—43no—
280C22⋊C8—43no—
281C22⋊C8—43no—
282C22⋊C8—43no—
283C22⋊C8—43no—
284C22⋊C8—43no—
285C22⋊C8—43no—
286C22⋊C8—43no—
287C22⋊C8—43no—
288C22⋊C8—43no—
289C22⋊C8—43no—
290C22⋊C8—43no—
291C22⋊C8—43no—
292#97—26no—
293#97—43no—
294C4×C8—26no—
295C4×C8—26no—
296C4×C8—43no—
297C4×C8—43no—
298#95C22×C413yes—
299#95C22×C413yes—
300#95C22×C413yes—
301#95C22×C413yes—
302#95C22×C413yes—
303#95C22×C413yes—
304#95C22×C413yes—
305#95C22×C413yes—
306#95C22×C413yes—
307#95C22×C413yes—
308#95C22×C413yes—
309#95C22×C413yes—
310C24—22no—
311C24—43no—
312C24—43no—
313C24C23×C411yescharacteristic
314C24C23×C411yescharacteristic
315C24C23×C413yes—
316C24C23×C411yescharacteristic
317C2×D4—26no—
318C2×D4—26no—
319C2×D4—26no—
320C2×D4—26no—
321C2×D4—43no—
322C2×D4—43no—
323C2×D4—43no—
324C2×D4—43no—
325C2×D4—43no—
326C2×D4—43no—
327C2×D4—43no—
328C2×D4—43no—
329C22×C4—23no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—23no—
339C22×C4—23no—
340C22×C4—23no—
341C22×C4—26no—
342C22×C4—43no—
343C22×C4—43no—
344C22×C4—43no—
345C22×C4—43no—
346C22×C4—43no—
347C22×C4—43no—
348C22×C4—43no—
349C22×C4—43no—
350C22×C4—44no—
351C22×C4—412no—
352C22×C4C23×C413yes—
353C22×C4C23×C413yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C23×C411yescharacteristic
356C22×C4C23×C411yescharacteristic
357C22×C4C23×C413yes—
358C22×C4C23×C413yes—
359C22×C4C23×C413yes—
360C22×C4C23×C413yes—
361C22×C4C23×C413yes—
362C22×C4C2511yescharacteristic
363M16—412no—
364M16—44no—
365M16—44no—
366M16—412no—
367M16—44no—
368M16—44no—
369C2×C8—23no—
370C2×C8—23no—
371C2×C8—23no—
372C2×C8—23no—
373C2×C8—23no—
374C2×C8—23no—
375C2×C8—23no—
376C2×C8—23no—
377C2×C8—23no—
378C2×C8—23no—
379C2×C8—23no—
380C2×C8—23no—
381C2×C8—23no—
382C2×C8—23no—
383C2×C8—23no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8H(5,𝔽2)16yes—
387C4⋊C4—26no—
388C4⋊C4—43no—
389C4⋊C4—43no—
390C4⋊C4—43no—
391C4⋊C4—43no—
392C4⋊C4—43no—
393C4⋊C4—43no—
394C4⋊C4C2×M16112yescomplemented
395C22⋊C4—26no—
396C22⋊C4—26no—
397C22⋊C4—26no—
398C22⋊C4—43no—
399C22⋊C4—43no—
400C22⋊C4—43no—
401C22⋊C4—43no—
402C22⋊C4—43no—
403C22⋊C4—43no—
404C22⋊C4—43no—
405C22⋊C4—43no—
406C22⋊C4—43no—
407C22⋊C4—43no—
408C22⋊C4—43no—
409C22⋊C4—43no—
410C22⋊C4C2×M16112yes—
411C42—23no—
412C42—23no—
413C42—23no—
414C42—23no—
415C42—23no—
416C42—23no—
417C42—43no—
418C42—43no—
419C42—43no—
420C42C2×M1614yescomplemented
421C23—23no—
422C23—23no—
423C23—23no—
424C23—23no—
425C23—23no—
426C23—23no—
427C23—23no—
428C23—21no—
429C23—21no—
430C23—23no—
431C23—23no—
432C23—23no—
433C23—23no—
434C23—23no—
435C23—23no—
436C23—44no—
437C23—86no—
438C23—86no—
439C23C22×M1612yes—
440C23C24×C411yescharacteristic
441D4—86no—
442D4—86no—
443D4—86no—
444D4—86no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—23no—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—23no—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—23no—
463C2×C4—23no—
464C2×C4—23no—
465C2×C4—23no—
466C2×C4—23no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—43no—
474C2×C4—43no—
475C2×C4—43no—
476C2×C4—43no—
477C2×C4—43no—
478C2×C4—43no—
479C2×C4—43no—
480C2×C4—86no—
481C2×C4—86no—
482C2×C4C22×M1616yes—
483C2×C4C22×M1616yes—
484C2×C4C2×H(5,𝔽2)13yes—
485C2×C4#58513yes—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—43no—
490C8—43no—
491C8—43no—
492C8—43no—
493C8#56014yescomplemented
494C22—26no—
495C22—41no—
496C22—43no—
497C22—41no—
498C22—41no—
499C22—43no—
500C22—43no—
501C22—43no—
502C22—43no—
503C22—43no—
504C22—43no—
505C22—43no—
506C22—43no—
507C22—43no—
508C22—43no—
509C22—43no—
510C22—88no—
511C22C4×H(5,𝔽2)13yes—
512C22C23×M1611yescharacteristic
513C22#357313yes—
514C4—26no—
515C4—26no—
516C4—43no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—43no—
521C4—43no—
522C4—43no—
523C4—43no—
524C4C2×#56011yescharacteristic
525C4#352913yes—
526C2—42no—
527C2—83no—
528C2—81no—
529C2—81no—
530C2C4×#56011yescharacteristic
531C2#6032513yes—
532C2#6032713yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC2×#560C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23378C2semidirect
3. #23375C2semidirect
4. #23357C2semidirect
5. #23329C2semidirect
6. #23327C2semidirect
7. #23320C2semidirect
8. #23308C2semidirect
9. #21681C2semidirect
10. #21666C2semidirect
11. C8×#126C2semidirect
12. #21565C2semidirect
13. #21490C2semidirect
14. #21488C2semidirect
15. #21441C2semidirect
20. C2×#436C22semidirect
22. C2×C4×M16C22semidirect
40. #1954C22semidirect
41. #1940C22semidirect
42. #1939C22semidirect
45. #1924C22semidirect
50. #1923C22semidirect
51. #1917C22semidirect
52. #1916C22semidirect
53. #1916C22semidirect
54. #1916C22semidirect
60. #1870C22semidirect
61. #1870C22semidirect
65. #1851C22semidirect
67. #1842C22semidirect
68. #1839C22semidirect
69. #1839C22semidirect
70. #1817C22semidirect
71. #1797C22semidirect
72. #1797C22semidirect
74. #1771C22semidirect
75. C8×C4⋊C4C22semidirect
78. #1753C22semidirect
80. C42×C8C22semidirect
228. #126M16semidirect
394. C4⋊C4C2×M16semidirect
420. C42C2×M16semidirect
493. C8#560semidirect

Maximal subgroups

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

subgroupindex
C4×#5602
#233782
#233752
#233572
#233292
#233272
#233202
#233082
#216812
#216662
C8×#1262
#215652
#214902
#214882
#214412