Finite Group Databasebeta

← #490861

Subgroups — #490861

Order 512.

464 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33660C211yesmaximal, characteristic
2#33627C211yesmaximal, characteristic, complemented
3#19646C211yesmaximal, characteristic, complemented
4#18054C211yesmaximal, characteristic, complemented
5#12501C211yesmaximal, characteristic, complemented
6#12497C211yesmaximal, characteristic, complemented
7#12485C211yesmaximal, characteristic, complemented
8#12483C211yesmaximal, characteristic, complemented
9#12449C211yesmaximal, characteristic, complemented
10#12445C211yesmaximal, characteristic, complemented
11#12418C211yesmaximal, characteristic, complemented
12#12416C211yesmaximal, characteristic, complemented
13#12018C211yesmaximal, characteristic, complemented
14#11985C211yesmaximal, characteristic, complemented
15#11973C211yesmaximal, characteristic, complemented
16#3210—21no—
17#3182—21no—
18#3169C2211yescharacteristic
19#3160C2211yescharacteristic
20#2974—21no—
21C2×C4×D8C2211yescharacteristic
22#2232—21no—
23#2228—21no—
24#2228—21no—
25#2226C2211yescharacteristic
26#2226C2211yescharacteristic
27#2224C2211yescharacteristic
28#2224C2211yescharacteristic
29#2224C2211yescharacteristic
30#2224C2211yescharacteristic
31#2206C2211yescharacteristic
32#2206C2211yescharacteristic
33#2184—21no—
34C4×D16—21no—
35C4×D16—21no—
36#2162C2211yescharacteristic
37#2159—21no—
38#2150C2211yescharacteristic
39#2140—21no—
40#1720—21no—
41#1714—21no—
42#1713C2211yescharacteristic, complemented
43#1710C2211yescharacteristic, complemented
44#1706C2211yescharacteristic, complemented
45#1706C2211yescharacteristic, complemented
46#1589C2211yescharacteristic
47#1589C2211yescharacteristic
48#1586—21no—
49#1586—21no—
50C8×D8C2211yescharacteristic
51C8×D8C2211yescharacteristic
52#1568C2211yescharacteristic
53#1490—21no—
54C8×M16C2211yescharacteristic
55#1370C2211yescharacteristic, complemented
56#1370C2211yescharacteristic, complemented
57#1367C2211yescharacteristic, complemented
58#1367C2211yescharacteristic, complemented
59#1365C2211yescharacteristic, complemented
60#1365C2211yescharacteristic, complemented
61#1364C2211yescharacteristic, complemented
62#1364C2211yescharacteristic, complemented
63#1364C2211yescharacteristic, complemented
64#1364C2211yescharacteristic, complemented
65#1361C2211yescharacteristic, complemented
66#1361C2211yescharacteristic, complemented
67C22×D8D411yescharacteristic
68#533—21no—
69#532—21no—
70C2×C4×D4—21no—
71C2×D16—21no—
72C2×D16—21no—
73C2×M32—21no—
74#500—21no—
75#500C2311yescharacteristic
76#500C2311yescharacteristic
77#500C2311yescharacteristic
78#499—21no—
79#499—21no—
80#498—21no—
81#494—21no—
82#493C2311yescharacteristic
83#492—21no—
84#492—21no—
85#487—21no—
86#487—21no—
87#486—21no—
88#486—21no—
89#486—21no—
90#468—42no—
91#466—21no—
92#466—21no—
93#464—41no—
94#464—41no—
95#452—41no—
96#452—41no—
97#440—42no—
98C4×D8—21no—
99C4×D8—21no—
100C4×D8—21no—
101C4×D8—21no—
102C4×D8D411yescharacteristic
103C4×D8D411yescharacteristic
104C4×D8C2311yescharacteristic
105C4×D8C2311yescharacteristic
106#436—21no—
107#436—21no—
108#435—41no—
109#435—41no—
110C8×D4—21no—
111C8×D4—21no—
112#432—21no—
113#431D411yescharacteristic
114#427—21no—
115#427—21no—
116C2×#107D411yescharacteristic
117C2×#102—21no—
118#407—41no—
119C2×C22⋊C8—41no—
120C4×M16D411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#367—21no—
123#367—21no—
124#366—21no—
125#366—21no—
126#366—21no—
127#366—21no—
128C4⋊C16—21no—
129C4⋊C16C2311yescharacteristic
130C4⋊C16C2311yescharacteristic
131C4⋊C16C2311yescharacteristic
132C4⋊C16C2311yescharacteristic
133#357—21no—
134#357—21no—
135#357—21no—
136#357—21no—
137#357—21no—
138#357—21no—
139#357—21no—
140#357—21no—
141C22⋊C16—21no—
142C4×C16—21no—
143#335C2311yescharacteristic
144#335C2311yescharacteristic
145#334—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148D4⋊C8—21no—
149D4⋊C8—21no—
150#322D411yescharacteristic
151#322D411yescharacteristic
152C82C2311yescharacteristic
153C82C2311yescharacteristic
154C2×#41—41no—
155C22×D4—21no—
156C23×C4—41no—
157C2×Q16—41no—
158C2×Q16—41no—
159C2×SD16—42no—
160C2×SD16—41no—
161C2×SD16—41no—
162C2×D8—22no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—41no—
167C2×D8C2×D411yescharacteristic
168C2×D8C2×D411yescharacteristic
169C2×M16—21no—
170C2×M16—41no—
171C2×M16—41no—
172C22×C8—41no—
173C22×C8—41no—
174C22×C8C2×D411yescharacteristic
175C22×C8#4111yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179#127—21no—
180#125—41no—
181#124—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184#121—41no—
185#121—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196#117—41no—
197C2×C4⋊C4—21no—
198C2×C22⋊C4—41no—
199C2×C42C2×D411yescharacteristic
200D16—42no—
201D16—42no—
202M32—42no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—42no—
215#107C2×D411yescharacteristic
216#107C2×D411yescharacteristic
217#106—41no—
218#106—41no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8D811yescharacteristic
222C4⋊C8D811yescharacteristic
223#103—42no—
224#103—41no—
225#103—41no—
226#102—21no—
227#102—21no—
228#102—21no—
229#102—21no—
230#102—42no—
231#102—41no—
232#102—41no—
233#102—41no—
234#102—41no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8D811yescharacteristic
241C22⋊C8D811yescharacteristic
242#97—21no—
243#97—41no—
244C4×C8C2×D411yescharacteristic
245C4×C8C2×D411yescharacteristic
246C4×C8C2×D411yescharacteristic
247C4×C8C2×D411yescharacteristic
248C4×C8C2×D411yescharacteristic
249C4×C8C2×D411yescharacteristic
250C4×C8#4111yescharacteristic
251C4×C8C2411yescharacteristic
252C24—41no—
253#41—161no—
254#41—161no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—81no—
266C2×D4—81no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—81no—
273C22×C4—161no—
274C22×C4—161no—
275C22×C4#12111yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4AGL(1,ℤ/8)11yescharacteristic
278Q16—161no—
279Q16—161no—
280SD16—84no—
281SD16—161no—
282SD16—161no—
283D8—21no—
284D8—21no—
285D8—21no—
286D8—42no—
287D8—41no—
288D8—41no—
289D8—81no—
290D8#13712yes—
291M16—42no—
292M16—161no—
293M16—161no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—161no—
302C2×C8—161no—
303C2×C8#12111yescharacteristic
304C2×C8#12111yescharacteristic
305C2×C8C2×D811yescharacteristic
306C2×C8C2×D811yescharacteristic
307C2×C8C2×D811yescharacteristic
308C2×C8C2×D811yescharacteristic
309C2×C8C22×D411yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C4⋊C4—81no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—81no—
329C42—81no—
330C42#12111yescharacteristic
331C42C2×D811yescharacteristic
332C42AGL(1,ℤ/8)11yescharacteristic
333C42C22×D411yescharacteristic
334C16—41no—
335C16—41no—
336C16—41no—
337C16—41no—
338C23—41no—
339C23—41no—
340C23—82no—
341C23—82no—
342C23—81no—
343C23—161no—
344C23#45911yescharacteristic
345Q8—82no—
346Q8—81no—
347Q8—81no—
348Q8—81no—
349Q8—81no—
350D4—22no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—82no—
355D4—82no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—162no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—161no—
376C2×C4—161no—
377C2×C4—162no—
378C2×C4—161no—
379C2×C4#45911yescharacteristic
380C2×C4#45911yescharacteristic
381C2×C4#45911yescharacteristic
382C2×C4C2×#12111yescharacteristic
383C2×C4C22×D811yescharacteristic
384C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
385C8—22no—
386C8—21no—
387C8—21no—
388C8—21no—
389C8—21no—
390C8—21no—
391C8—21no—
392C8—21no—
393C8—41no—
394C8—81no—
395C8—81no—
396C8C2×#13711yescharacteristic
397C8#57511yescharacteristic
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—164no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22C2×#45911yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—81no—
417C4—81no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4#305611yescharacteristic
422C4#306911yescharacteristic
423C4#341511yescharacteristic
424C4#341511yescharacteristic
425C2—41no—
426C2—82no—
427C2—161no—
428C2—321no—
429C2#1916311yescharacteristic
430C2#3358511yescharacteristic
431C2#3359411yescharacteristic
subgroupG/H
center, socleC22C2×#459
commutator, Frattini, cosocleC4×C8C24
prime radical#33660C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33627C2semidirect
3. #19646C2semidirect
4. #18054C2semidirect
5. #12501C2semidirect
6. #12497C2semidirect
7. #12485C2semidirect
8. #12483C2semidirect
9. #12449C2semidirect
10. #12445C2semidirect
11. #12418C2semidirect
12. #12416C2semidirect
13. #12018C2semidirect
14. #11985C2semidirect
15. #11973C2semidirect
42. #1713C22semidirect
43. #1710C22semidirect
44. #1706C22semidirect
45. #1706C22semidirect
55. #1370C22semidirect
56. #1370C22semidirect
57. #1367C22semidirect
58. #1367C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1364C22semidirect
62. #1364C22semidirect
63. #1364C22semidirect
64. #1364C22semidirect
65. #1361C22semidirect
66. #1361C22semidirect

Maximal subgroups

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

subgroupindex
#336602
#336272
#196462
#180542
#125012
#124972
#124852
#124832
#124492
#124452
#124182
#124162
#120182
#119852
#119732