Finite Group Databasebeta

← #484031

Subgroups — #484031

Order 512.

454 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33660C211yesmaximal, characteristic
2#18084C211yesmaximal, characteristic, complemented
3#18054C211yesmaximal, characteristic, complemented
4#12086C211yesmaximal, characteristic, complemented
5#12081C211yesmaximal, characteristic, complemented
6#12073C211yesmaximal, characteristic, complemented
7#12073C211yesmaximal, characteristic, complemented
8#12069C211yesmaximal, characteristic, complemented
9#12069C211yesmaximal, characteristic, complemented
10#12045C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic, complemented
12#12034C211yesmaximal, characteristic, complemented
13#11675C211yesmaximal, characteristic, complemented
14#11593C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#2979—21no—
18#2974—21no—
19C2×C4×D8C2211yescharacteristic
20#2232C411yescharacteristic
21#2232C411yescharacteristic
22#2224C411yescharacteristic
23#2224C411yescharacteristic
24#2206C411yescharacteristic
25#2206C411yescharacteristic
26#2184C411yescharacteristic
27#2184C411yescharacteristic
28C4×D16C2211yescharacteristic
29C4×D16C2211yescharacteristic
30C4×D16C2211yescharacteristic
31C4×D16C2211yescharacteristic
32#2162C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1589C2211yescharacteristic
35#1589C2211yescharacteristic
36#1589C2211yescharacteristic, complemented
37#1589C2211yescharacteristic, complemented
38#1586—21no—
39#1586—21no—
40#1586—21no—
41#1586—21no—
42C8×D8C2211yescharacteristic
43C8×D8C2211yescharacteristic
44C8×D8C2211yescharacteristic, complemented
45C8×D8C2211yescharacteristic, complemented
46#1568C2211yescharacteristic
47#1564C2211yescharacteristic
48#1490—21no—
49#1484—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic, complemented
52#1372C2211yescharacteristic, complemented
53#1371C2211yescharacteristic, complemented
54#1371C2211yescharacteristic, complemented
55#1331C2211yescharacteristic
56#1331C2211yescharacteristic
57#1331C2211yescharacteristic
58#1331C2211yescharacteristic
59#1331C2211yescharacteristic, complemented
60#1331C2211yescharacteristic, complemented
61#1331C2211yescharacteristic, complemented
62#1331C2211yescharacteristic, complemented
63#1312C2211yescharacteristic, complemented
64#1312C2211yescharacteristic, complemented
65#1311C2211yescharacteristic, complemented
66C8×C16C2211yescharacteristic, complemented
67C22×D8C2×C411yescharacteristic
68#532—21no—
69C2×C4×D4—21no—
70C2×D16C2×C411yescharacteristic
71C2×D16C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C2×M32—21no—
75#500C2×C411yescharacteristic
76#493C2×C411yescharacteristic
77#486—21no—
78#466—21no—
79#466C2×C411yescharacteristic
80#466C2×C411yescharacteristic
81C4×D8—21no—
82C4×D8—21no—
83C4×D8—21no—
84C4×D8C2×C411yescharacteristic
85C4×D8C2×C411yescharacteristic
86C4×D8C2311yescharacteristic
87C4×D8C2311yescharacteristic
88C4×D8C2311yescharacteristic
89C4×D8C2311yescharacteristic
90#436—21no—
91#436—21no—
92#436—21no—
93#436—21no—
94#436—41no—
95#435—41no—
96#435—41no—
97C8×D4—21no—
98C8×D4—21no—
99C8×D4—21no—
100C8×D4—21no—
101C8×D4—41no—
102#432—21no—
103#431—21no—
104#431D411yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#102—21no—
107#408—41no—
108#408—41no—
109#407—41no—
110C2×C22⋊C8—41no—
111C4×M16D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#366C2×C411yescharacteristic
114#366C2×C411yescharacteristic
115#366C2×C411yescharacteristic
116#366C2×C411yescharacteristic
117C4⋊C16C2×C411yescharacteristic
118C4⋊C16C2×C411yescharacteristic
119#357C2×C411yescharacteristic
120#357C2×C411yescharacteristic
121#357C2×C411yescharacteristic
122#357C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16C2×C411yescharacteristic
128C22⋊C16C2×C411yescharacteristic
129#346—21no—
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335C2311yescharacteristic
135#335C2311yescharacteristic
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#334—21no—
139#334—21no—
140D4⋊C8—21no—
141D4⋊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
152C2×#41—41no—
153C22×D4—21no—
154C23×C4—41no—
155C2×D8—22no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—41no—
159C2×D8C22×C411yescharacteristic
160C2×D8C22×C411yescharacteristic
161C2×D8C22×C411yescharacteristic
162C2×D8C22×C411yescharacteristic
163C2×M16—21no—
164C2×M16—41no—
165C2×M16—41no—
166C2×M16—41no—
167C2×M16—41no—
168C22×C8—41no—
169C22×C8—41no—
170C22×C8—41no—
171C22×C8—41no—
172C22×C8C22×C411yescharacteristic
173C22×C8#4111yescharacteristic
174C4⋊Q8—21no—
175#127—21no—
176#124—41no—
177#121—21no—
178#121—21no—
179#121—41no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—21no—
190C2×C22⋊C4—41no—
191C2×C42C2×D411yescharacteristic
192D16—22no—
193D16—22no—
194D16—41no—
195D16—41no—
196M32—42no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C22×C411yescharacteristic
200C2×C16C22×C411yescharacteristic
201C2×C16C22×C411yescharacteristic
202C2×C16C22×C411yescharacteristic
203#107—21no—
204#107—21no—
205#107C22×C411yescharacteristic
206#107C22×C411yescharacteristic
207#107C22×C411yescharacteristic
208#107C22×C411yescharacteristic
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8D811yescharacteristic
214C4⋊C8D811yescharacteristic
215#102—21no—
216#102—21no—
217#102—21no—
218#102—21no—
219#102—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C22⋊C8D811yescharacteristic
234C22⋊C8D811yescharacteristic
235#97—21no—
236#97—41no—
237C4×C8—41no—
238C4×C8C22×C411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246C24—41no—
247#41—161no—
248#41—161no—
249C2×Q8—41no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—161no—
269C22×C4—161no—
270C22×C4C4×D411yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4#13511yescharacteristic
273D8—21no—
274D8—21no—
275D8—21no—
276D8—21no—
277D8—21no—
278D8—21no—
279D8—21no—
280D8—42no—
281D8—81no—
282D8#13112yes—
283M16—42no—
284M16—161no—
285M16—161no—
286M16—161no—
287M16—161no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—161no—
298C2×C8—161no—
299C2×C8—161no—
300C2×C8—161no—
301C2×C8C4×D411yescharacteristic
302C2×C8C4×D411yescharacteristic
303C2×C8C2×D811yescharacteristic
304C2×C8C2×D811yescharacteristic
305C2×C8C2×D811yescharacteristic
306C2×C8C2×D811yescharacteristic
307C2×C8C23×C411yescharacteristic
308C2×C8C2×#4111yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42C4×D411yescharacteristic
324C42C2×D811yescharacteristic
325C42#13511yescharacteristic
326C42C22×D411yescharacteristic
327C16—22no—
328C16—21no—
329C16—21no—
330C16—21no—
331C16—21no—
332C16—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—82no—
338C23—82no—
339C23—161no—
340C23C4×D811yescharacteristic
341Q8—161no—
342D4—22no—
343D4—41no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—82no—
351D4—82no—
352D4—161no—
353D4—161no—
354D4—161no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—161no—
373C2×C4—161no—
374C2×C4—161no—
375C2×C4C4×D811yescharacteristic
376C2×C4C4×D811yescharacteristic
377C2×C4C4×D811yescharacteristic
378C2×C4C2×C4×D411yescharacteristic
379C2×C4C22×D811yescharacteristic
380C2×C4C2×#13511yescharacteristic
381C8—22no—
382C8—21no—
383C8—21no—
384C8—21no—
385C8—21no—
386C8—21no—
387C8—21no—
388C8—21no—
389C8—41no—
390C8—81no—
391C8—81no—
392C8—81no—
393C8—81no—
394C8C2×#13111yescharacteristic
395C8#56811yescharacteristic
396C22—21no—
397C22—21no—
398C22—21no—
399C22—42no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—164no—
412C22C2×C4×D811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4#293311yescharacteristic
422C4#295611yescharacteristic
423C4#341511yescharacteristic
424C4#341511yescharacteristic
425C2—41no—
426C2—82no—
427C2—161no—
428C2—161no—
429C2—161no—
430C2#1799411yescharacteristic
431C2#3345611yescharacteristic
432C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#33660C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #431 ◁ 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. #18084C2semidirect
3. #18054C2semidirect
4. #12086C2semidirect
5. #12081C2semidirect
6. #12073C2semidirect
7. #12073C2semidirect
8. #12069C2semidirect
9. #12069C2semidirect
10. #12045C2semidirect
11. C8×D16C2semidirect
12. #12034C2semidirect
13. #11675C2semidirect
14. #11593C2semidirect
15. #11409C2semidirect
36. #1589C22semidirect
37. #1589C22semidirect
44. C8×D8C22semidirect
45. C8×D8C22semidirect
51. #1372C22semidirect
52. #1372C22semidirect
53. #1371C22semidirect
54. #1371C22semidirect
59. #1331C22semidirect
60. #1331C22semidirect
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
#336602
#180842
#180542
#120862
#120812
#120732
#120692
#120452
C8×D162
#120342
#116752
#115932
#114092