Finite Group Databasebeta

← #7618117

Subgroups — #7618117

Order 512.

1627 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45380C212yesmaximal, complemented
2#32662C212yesmaximal, complemented
3#32469C211yesmaximal, characteristic, complemented
4#31064C211yesmaximal, characteristic
5#29546C212yesmaximal, complemented
6#28846C214yesmaximal, complemented
7#28188C212yesmaximal, complemented
8#28094C212yesmaximal, complemented
9#27112C214yesmaximal, complemented
10#24909C214yesmaximal, complemented
11#22870C214yesmaximal, complemented
12#21635C211yesmaximal, characteristic, complemented
13#21290C212yesmaximal, complemented
14C2×#447C2214yes—
15C2×C8×Q8C2211yescharacteristic, complemented
16C2×#442C2212yes—
17C2×C4×D8C2212yes—
18C2×C8×D4C2211yescharacteristic
19C22×#102C2212yescomplemented
20#2760—22no—
21#2746—24no—
22#2736—22no—
23#2600—22no—
24#2580C2214yescomplemented
25#2494C2214yescomplemented
26#2486C2212yes—
27#2483—22no—
28#2420C2214yes—
29#2418—24no—
30#2408C2212yes—
31#2398C2211yescharacteristic
32#2361C2214yes—
33D4×C22⋊C4C2214yescomplemented
34#2319C2214yescomplemented
35#2092C2214yescomplemented
36#2087C2212yescomplemented
37#2087C2212yescomplemented
38#2085C2214yescomplemented
39#2077C2214yescomplemented
40#2067C2214yescomplemented
41#2054C2212yes—
42#2054C2212yes—
43#2034C2214yes—
44#2033C2214yes—
45#1985C2211yescharacteristic, complemented
46#1972C2212yes—
47#1970C2212yes—
48#1955C2211yescharacteristic
49#1944C2212yescomplemented
50#1941C2212yescomplemented
51#1936C2214yescomplemented
52#1934C2214yescomplemented
53#1934C2214yescomplemented
54#1925C2214yescomplemented
55#1924C2214yescomplemented
56#1921C2214yescomplemented
57#1893C2214yes—
58#1879C2214yes—
59#1879C2214yes—
60#1862C2214yescomplemented
61#1862C2214yescomplemented
62#1852C2214yescomplemented
63#1835C2212yescomplemented
64#1832C2211yescharacteristic, complemented
65#1823C2212yescomplemented
66#1808C2212yescomplemented
67#1804C2214yescomplemented
68#1786C2212yescomplemented
69#1776C2212yescomplemented
70C4×#102C2214yescomplemented
71C8×C22⋊C4C2214yescomplemented
72#1726C2211yescharacteristic, complemented
73C23×D4C2312yes—
74C22×D8C2312yes—
75C22×D8C2312yes—
76C23×C8C2312yes—
77C2×#123—22no—
78C2×#121C2314yes—
79C2×#120—22no—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—24no—
82C2×C4×D4C2314yes—
83C2×C4×D4C2314yes—
84C2×C4×D4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—24no—
87#447—216no—
88C8×Q8—42no—
89#442—28no—
90C4×D8—44no—
91C8×D4—42no—
92C2×#107C2314yes—
93C2×#107C2312yes—
94C2×#107C2312yes—
95C2×#106C2312yes—
96C2×#106C2312yes—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2311yescharacteristic
100C2×#102—44no—
101C2×#102C2314yes—
102C2×#102C2314yes—
103C2×#102C2314yes—
104C2×#102C2314yes—
105C2×#102C2314yes—
106C2×#102C2314yes—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2312yes—
109C2×#97C2312yes—
110C2×#97C2311yescharacteristic
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2312yes—
114#401—24no—
115#400D412yes—
116#399—24no—
117#398—22no—
118#397—24no—
119#397—22no—
120#397—24no—
121#397D412yes—
122#394—22no—
123#394—24no—
124#394D412yes—
125#393—24no—
126#393D412yes—
127#392—22no—
128#390C2314yes—
129#390C2314yes—
130#390C2314yes—
131#389C2314yes—
132#389C2314yes—
133#389C2314yes—
134#388—24no—
135#388—24no—
136#388—24no—
137#388—24no—
138#387—24no—
139#386—24no—
140#386—24no—
141#386—24no—
142#386—24no—
143#385—24no—
144#385—24no—
145#385—24no—
146#382C2314yes—
147#381—22no—
148#381—24no—
149#381—24no—
150#380—24no—
151#380C2312yes—
152#380C2312yes—
153#380C2314yes—
154#379—24no—
155#379—22no—
156C4×C4⋊C4C2314yes—
157C4×C22⋊C4C2314yes—
158#376C2311yescharacteristic
159#340C2312yes—
160#340C2312yes—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#340C2314yes—
165#340C2314yes—
166#336C2314yes—
167#336C2314yes—
168C25—22no—
169C22×Q8C2×D411yescharacteristic
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—22no—
173C22×D4—42no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4C2×D411yescharacteristic
177C22×D4C2414yes—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4C2412yes—
182C2×D8—28no—
183C2×D8—42no—
184C2×D8—42no—
185C2×D8—42no—
186C2×D8#4118yes—
187C22×C8—24no—
188C22×C8—44no—
189C22×C8C2412yes—
190C22×C8C2412yes—
191C22×C8C2411yescharacteristic
192C22×C8C2412yes—
193C22×C8C2411yescharacteristic
194#123—84no—
195#121—28no—
196#121—44no—
197#120—84no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×D4—28no—
201C4×D4—44no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—88no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2414yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—88no—
232C2×C22⋊C4C2×D414yes—
233C2×C22⋊C4C2×D412yes—
234C2×C22⋊C4C2×D414yes—
235C2×C42—24no—
236C2×C42C2412yes—
237C2×C42C2412yes—
238C2×C42C2411yescharacteristic
239#107—44no—
240#107—42no—
241#107—42no—
242#106—24no—
243#106—42no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—42no—
247#102—28no—
248#102—28no—
249#102—28no—
250#102—28no—
251#102—44no—
252#102—44no—
253C22⋊C8—28no—
254C22⋊C8—42no—
255#97—24no—
256#97—41no—
257C4×C8—22no—
258C4×C8—24no—
259C4×C8—42no—
260#95—24no—
261#95—22no—
262#95—24no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95C2×D412yes—
270#95C2×D412yes—
271#95C2×D414yes—
272#95C2×D414yes—
273C24—24no—
274C24—48no—
275C24—44no—
276C24—82no—
277C24—84no—
278C24—82no—
279C24C22×D412yes—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—41no—
289C2×D4—48no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—82no—
294C2×D4—88no—
295C2×D4—84no—
296C2×D4—84no—
297C2×D4C2×#4118yes—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4C22×D412yes—
315C22×C4C22×D412yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D414yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C2511yescharacteristic
321D8—48no—
322D8—48no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—84no—
333C2×C8C2×#4114yes—
334C2×C8H(5,𝔽2)12yes—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—84no—
345C4⋊C4—82no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C22⋊C4—28no—
349C22⋊C4—44no—
350C22⋊C4—42no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—48no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C22⋊C4—84no—
357C22⋊C4—84no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C22⋊C4—84no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—22no—
364C42—21no—
365C42—22no—
366C42—22no—
367C42—21no—
368C42—22no—
369C42—84no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—28no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—816no—
382C23—88no—
383C23—164no—
384C23—168no—
385C23—164no—
386C23#57614yes—
387C23C23×D411yescharacteristic
388Q8—44no—
389D4—28no—
390D4—28no—
391D4—44no—
392D4—44no—
393D4—44no—
394D4—88no—
395D4—816no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—168no—
423C2×C4—168no—
424C2×C4—164no—
425C2×C4C22×#4112yes—
426C2×C4C2×H(5,𝔽2)11yescharacteristic
427C2×C4#58511yescharacteristic
428C2×C4#58512yes—
429C8—24no—
430C8—22no—
431C8—41no—
432C8—42no—
433C8—42no—
434C22—28no—
435C22—24no—
436C22—44no—
437C22—42no—
438C22—42no—
439C22—48no—
440C22—48no—
441C22—48no—
442C22—84no—
443C22—84no—
444C22—84no—
445C22—84no—
446C22—1616no—
447C22D4×#4112yes—
448C22#349611yescharacteristic
449C22C2×#57612yes—
450C22#358412yes—
451C4—44no—
452C4—42no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—42no—
457C4—84no—
458C4—84no—
459C4—84no—
460C4#348111yescharacteristic
461C4#349112yes—
462C4#354511yescharacteristic
463C2—44no—
464C2—88no—
465C2#5983611yescharacteristic
466C2#6148914yes—
467C2#6227011yescharacteristic
468C2#6227311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#31064C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2398 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #45380C2semidirect
2. #32662C2semidirect
3. #32469C2semidirect
5. #29546C2semidirect
6. #28846C2semidirect
7. #28188C2semidirect
8. #28094C2semidirect
9. #27112C2semidirect
10. #24909C2semidirect
11. #22870C2semidirect
12. #21635C2semidirect
13. #21290C2semidirect
15. C2×C8×Q8C22semidirect
19. C22×#102C22semidirect
24. #2580C22semidirect
25. #2494C22semidirect
33. D4×C22⋊C4C22semidirect
34. #2319C22semidirect
35. #2092C22semidirect
36. #2087C22semidirect
37. #2087C22semidirect
38. #2085C22semidirect
39. #2077C22semidirect
40. #2067C22semidirect
45. #1985C22semidirect
49. #1944C22semidirect
50. #1941C22semidirect
51. #1936C22semidirect
52. #1934C22semidirect
53. #1934C22semidirect
54. #1925C22semidirect
55. #1924C22semidirect
56. #1921C22semidirect
60. #1862C22semidirect
61. #1862C22semidirect
62. #1852C22semidirect
63. #1835C22semidirect
64. #1832C22semidirect
65. #1823C22semidirect
66. #1808C22semidirect
67. #1804C22semidirect
68. #1786C22semidirect
69. #1776C22semidirect
70. C4×#102C22semidirect
71. C8×C22⋊C4C22semidirect
72. #1726C22semidirect

Maximal subgroups

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

subgroupindex
#453802
#326622
#324692
#310642
#295462
#288462
#281882
#280942
#271122
#249092
#228702
#216352
#212902