Finite Group Databasebeta

← #483863

Subgroups — #483863

Order 512.

451 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33665C211yesmaximal, characteristic
2#17973C211yesmaximal, characteristic, complemented
3#17854C211yesmaximal, characteristic, complemented
4#12074C211yesmaximal, characteristic, complemented
5#12073C211yesmaximal, characteristic, complemented
6#12070C211yesmaximal, characteristic, complemented
7#12069C211yesmaximal, characteristic, complemented
8#12045C211yesmaximal, characteristic, complemented
9#12043C211yesmaximal, characteristic, complemented
10C8×SD32C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic, complemented
12#12036C211yesmaximal, characteristic, complemented
13#11657C211yesmaximal, characteristic, complemented
14#11647C211yesmaximal, characteristic, complemented
15#11412C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#2993—21no—
18#2972—21no—
19C2×C4×D8C2211yescharacteristic
20#2236C411yescharacteristic
21#2233C411yescharacteristic
22#2228C411yescharacteristic
23#2228C411yescharacteristic
24#2210C411yescharacteristic
25#2210C411yescharacteristic
26#2187C411yescharacteristic
27#2184C411yescharacteristic
28C4×SD32C2211yescharacteristic
29C4×SD32C2211yescharacteristic
30C4×D16C2211yescharacteristic
31C4×D16C2211yescharacteristic
32#2162C2211yescharacteristic
33#2162C2211yescharacteristic
34#1592—21no—
35#1591C2211yescharacteristic, complemented
36#1589C2211yescharacteristic, complemented
37#1580—21no—
38#1580—21no—
39SD16⋊C8—21no—
40C8×Q16C2211yescharacteristic, complemented
41C8×D8C2211yescharacteristic
42C8×D8C2211yescharacteristic
43C8×D8C2211yescharacteristic
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic, complemented
46#1567C2211yescharacteristic
47#1564C2211yescharacteristic
48#1495—21no—
49#1478—21no—
50#1450C2211yescharacteristic
51#1372C2211yescharacteristic, complemented
52#1372C2211yescharacteristic, complemented
53#1371C2211yescharacteristic, complemented
54#1371C2211yescharacteristic, complemented
55#1332C2211yescharacteristic, complemented
56#1332C2211yescharacteristic, complemented
57#1331C2211yescharacteristic
58#1331C2211yescharacteristic
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic, complemented
62#1331C2211yescharacteristic, complemented
63#1312C2211yescharacteristic, complemented
64#1312C2211yescharacteristic, complemented
65#1311C2211yescharacteristic, complemented
66C8×C16C2211yescharacteristic, complemented
67C22×D8C2×C411yescharacteristic
68#529—21no—
69C2×C4×D4—21no—
70C2×SD32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74#499C2×C411yescharacteristic
75#495C2×C411yescharacteristic
76#488—21no—
77#467C2×C411yescharacteristic
78#466C2×C411yescharacteristic
79#465—21no—
80#446—21no—
81C8×Q8—21no—
82C4×Q16C2311yescharacteristic
83C4×SD16—21no—
84C4×D8—21no—
85C4×D8—21no—
86C4×D8C2×C411yescharacteristic
87C4×D8C2×C411yescharacteristic
88C4×D8C2311yescharacteristic
89C4×D8C2311yescharacteristic
90C4×D8C2311yescharacteristic
91#436—21no—
92#436—41no—
93#435—41no—
94C8×D4—21no—
95C8×D4—21no—
96C8×D4—21no—
97C8×D4—21no—
98C8×D4—21no—
99C8×D4—41no—
100C8×D4—41no—
101#433—41no—
102#431—21no—
103#431D411yescharacteristic
104#431D411yescharacteristic
105C2×#107C2×C411yescharacteristic
106#424—41no—
107#423—21no—
108C2×#102—21no—
109C2×C22⋊C8—41no—
110C2×C22⋊C8—41no—
111C2×C4×C8C2311yescharacteristic
112#367C2×C411yescharacteristic
113#367C2×C411yescharacteristic
114#366C2×C411yescharacteristic
115#366C2×C411yescharacteristic
116C4⋊C16—21no—
117C4⋊C16C2×C411yescharacteristic
118C4⋊C16C2×C411yescharacteristic
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#357C2×C411yescharacteristic
122#357C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16—21no—
128C22⋊C16C2×C411yescharacteristic
129C22⋊C16C2×C411yescharacteristic
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335—21no—
135#335C2311yescharacteristic
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#335C2311yescharacteristic
139#334—21no—
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142D4⋊C8—21no—
143D4⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C22×D4—21no—
153C23×C4—41no—
154C2×Q16C22×C411yescharacteristic
155C2×SD16—21no—
156C2×D8—22no—
157C2×D8—22no—
158C2×D8—21no—
159C2×D8C22×C411yescharacteristic
160C2×D8C22×C411yescharacteristic
161C2×D8C22×C411yescharacteristic
162C2×M16—41no—
163C2×M16—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8C22×C411yescharacteristic
170C22×C8#4111yescharacteristic
171#126—41no—
172#125—21no—
173#124—21no—
174#123—41no—
175C22⋊Q8—21no—
176#121—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186#117—41no—
187C2×C4⋊C4—21no—
188C2×C22⋊C4—41no—
189C2×C42C2×D411yescharacteristic
190SD32—22no—
191SD32—22no—
192D16—22no—
193D16—22no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199C2×C16C22×C411yescharacteristic
200#107—21no—
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#107C22×C411yescharacteristic
204#107C22×C411yescharacteristic
205#106—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8D811yescharacteristic
215C4⋊C8D811yescharacteristic
216#103—21no—
217#103—21no—
218#102—21no—
219#102—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223#102—21no—
224C22⋊C8—21no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8D811yescharacteristic
233C22⋊C8D811yescharacteristic
234#97—41no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8C22×C411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8#4111yescharacteristic
244C4×C8C2411yescharacteristic
245C24—41no—
246C2×Q8—21no—
247C2×D4—24no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—161no—
261C22×C4—161no—
262C22×C4C4×D411yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4#13511yescharacteristic
265Q16—21no—
266Q16—21no—
267SD16—42no—
268D8—24no—
269D8—21no—
270D8—21no—
271D8—21no—
272D8—21no—
273D8#13112yes—
274D8#13112yes—
275M16—161no—
276M16—161no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—161no—
287C2×C8—161no—
288C2×C8—161no—
289C2×C8—161no—
290C2×C8—161no—
291C2×C8C4×D411yescharacteristic
292C2×C8C4×D411yescharacteristic
293C2×C8C2×D811yescharacteristic
294C2×C8C2×D811yescharacteristic
295C2×C8C2×D811yescharacteristic
296C2×C8C2×D811yescharacteristic
297C2×C8C23×C411yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—41no—
318C42C4×D411yescharacteristic
319C42C2×D811yescharacteristic
320C42#13511yescharacteristic
321C42C22×D411yescharacteristic
322C16—21no—
323C16—21no—
324C16—21no—
325C16—21no—
326C16—41no—
327C16—41no—
328C23—44no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—161no—
333C23—161no—
334C23C4×D811yescharacteristic
335Q8—41no—
336Q8—41no—
337D4—22no—
338D4—22no—
339D4—44no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—41no—
344D4—161no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—161no—
362C2×C4—161no—
363C2×C4—161no—
364C2×C4C4×D811yescharacteristic
365C2×C4C4×D811yescharacteristic
366C2×C4C4×D811yescharacteristic
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C22×D811yescharacteristic
369C2×C4C2×#13511yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—21no—
375C8—21no—
376C8—21no—
377C8—41no—
378C8—41no—
379C8—81no—
380C8—81no—
381C8—81no—
382C8—81no—
383C8C2×#13111yescharacteristic
384C8C2×#13111yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—164no—
398C22C2×C4×D811yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4#292911yescharacteristic
409C4#295511yescharacteristic
410C4#341511yescharacteristic
411C4#341611yescharacteristic
412C2—41no—
413C2—82no—
414C2—82no—
415C2—161no—
416C2#1790011yescharacteristic
417C2#3345411yescharacteristic
418C2#3346511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#2184C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17973C2semidirect
3. #17854C2semidirect
4. #12074C2semidirect
5. #12073C2semidirect
6. #12070C2semidirect
7. #12069C2semidirect
8. #12045C2semidirect
9. #12043C2semidirect
10. C8×SD32C2semidirect
11. C8×D16C2semidirect
12. #12036C2semidirect
13. #11657C2semidirect
14. #11647C2semidirect
15. #11412C2semidirect
35. #1591C22semidirect
36. #1589C22semidirect
40. C8×Q16C22semidirect
45. C8×D8C22semidirect
51. #1372C22semidirect
52. #1372C22semidirect
53. #1371C22semidirect
54. #1371C22semidirect
55. #1332C22semidirect
56. #1332C22semidirect
61. #1331C22semidirect
62. #1331C22semidirect
63. #1312C22semidirect
64. #1312C22semidirect
65. #1311C22semidirect
66. C8×C16C22semidirect

Maximal subgroups

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

subgroupindex
#336652
#179732
#178542
#120742
#120732
#120702
#120692
#120452
#120432
C8×SD322
C8×D162
#120362
#116572
#116472
#114122