Finite Group Databasebeta

← #484017

Subgroups — #484017

Order 512.

444 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33665C211yesmaximal, characteristic
2#18087C211yesmaximal, characteristic, complemented
3#18054C211yesmaximal, characteristic, complemented
4#12086C211yesmaximal, characteristic, complemented
5#12084C211yesmaximal, characteristic, complemented
6#12071C211yesmaximal, characteristic, complemented
7#12070C211yesmaximal, characteristic, complemented
8#12069C211yesmaximal, characteristic, complemented
9#12069C211yesmaximal, characteristic, complemented
10#12045C211yesmaximal, characteristic, complemented
11#12043C211yesmaximal, characteristic, complemented
12#12035C211yesmaximal, characteristic, complemented
13#11673C211yesmaximal, characteristic, complemented
14C2×#1331C211yesmaximal, characteristic, complemented
15#11408C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#2994—21no—
18#2974—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
33C2×C4×C16C2211yescharacteristic
34#1591C2211yescharacteristic, complemented
35#1589C2211yescharacteristic
36#1589C2211yescharacteristic
37#1589C2211yescharacteristic, complemented
38#1588—21no—
39#1586—21no—
40#1586—21no—
41#1585—21no—
42C8×Q16C2211yescharacteristic, complemented
43C8×D8C2211yescharacteristic
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic, complemented
46#1568C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1490—21no—
49#1483—21no—
50C8×M16C2211yescharacteristic
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
66#1311C2211yescharacteristic, complemented
67C22×D8C2×C411yescharacteristic
68#529—21no—
69C2×C4×D4—21no—
70C2×SD32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C22×C16—21no—
75#499C2×C411yescharacteristic
76#495C2×C411yescharacteristic
77#488—21no—
78#467C2×C411yescharacteristic
79#466C2×C411yescharacteristic
80#465—21no—
81#446—21no—
82C8×Q8—21no—
83C4×Q16C2311yescharacteristic
84C4×SD16—21no—
85C4×D8—21no—
86C4×D8—21no—
87C4×D8C2×C411yescharacteristic
88C4×D8C2×C411yescharacteristic
89C4×D8C2311yescharacteristic
90C4×D8C2311yescharacteristic
91C4×D8C2311yescharacteristic
92#436—21no—
93#436—21no—
94#436—21no—
95#435—41no—
96#435—41no—
97#435—41no—
98C8×D4—21no—
99C8×D4—21no—
100C8×D4—21no—
101C8×D4—41no—
102#433—41no—
103#432—21no—
104#431D411yescharacteristic
105C2×#107C2×C411yescharacteristic
106#424—41no—
107C2×C4⋊C8—21no—
108C2×#102—21no—
109#407—41no—
110C2×C22⋊C8—41no—
111C4×M16D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#367C2×C411yescharacteristic
114#367C2×C411yescharacteristic
115#366C2×C411yescharacteristic
116#366C2×C411yescharacteristic
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⋊C16C2×C411yescharacteristic
128C22⋊C16C2×C411yescharacteristic
129C4×C16—21no—
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—21no—
158C2×D8—41no—
159C2×D8C22×C411yescharacteristic
160C2×D8C22×C411yescharacteristic
161C2×D8C22×C411yescharacteristic
162C2×M16—21no—
163C2×M16—41no—
164C2×M16—41no—
165C2×M16—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8—41no—
170C22×C8C22×C411yescharacteristic
171C22×C8#4111yescharacteristic
172#126—41no—
173#125—21no—
174#124—21no—
175#123—41no—
176C22⋊Q8—21no—
177#121—21no—
178C4×Q8—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187#117—41no—
188C2×C4⋊C4—21no—
189C2×C22⋊C4—41no—
190C2×C42C2×D411yescharacteristic
191SD32—22no—
192SD32—41no—
193D16—22no—
194D16—41no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—42no—
198C2×C16C22×C411yescharacteristic
199C2×C16C22×C411yescharacteristic
200C2×C16C22×C411yescharacteristic
201C2×C16C22×C411yescharacteristic
202#107—21no—
203#107C22×C411yescharacteristic
204#107C22×C411yescharacteristic
205#107C22×C411yescharacteristic
206#107C22×C411yescharacteristic
207#106—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8D811yescharacteristic
216C4⋊C8D811yescharacteristic
217#103—21no—
218#103—21no—
219#102—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223#102—21no—
224#102—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—21no—
235#97—41no—
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—
247C2×Q8—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—81no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—161no—
262C22×C4—161no—
263C22×C4C4×D411yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4#13511yescharacteristic
266Q16—21no—
267Q16—21no—
268SD16—81no—
269D8—21no—
270D8—21no—
271D8—21no—
272D8—21no—
273D8—21no—
274D8—42no—
275D8#13112yes—
276M16—42no—
277M16—161no—
278M16—161no—
279M16—161no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—161no—
290C2×C8—161no—
291C2×C8—161no—
292C2×C8—161no—
293C2×C8C4×D411yescharacteristic
294C2×C8C4×D411yescharacteristic
295C2×C8C2×D811yescharacteristic
296C2×C8C2×D811yescharacteristic
297C2×C8C2×D811yescharacteristic
298C2×C8C2×D811yescharacteristic
299C2×C8C23×C411yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C42—41no—
320C42C4×D411yescharacteristic
321C42C2×D811yescharacteristic
322C42#13511yescharacteristic
323C42C22×D411yescharacteristic
324C16—22no—
325C16—21no—
326C16—21no—
327C16—21no—
328C16—21no—
329C16—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—82no—
334C23—82no—
335C23—161no—
336C23C4×D811yescharacteristic
337Q8—41no—
338Q8—41no—
339D4—22no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—41no—
344D4—41no—
345D4—82no—
346D4—82no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—161no—
364C2×C4—161no—
365C2×C4—161no—
366C2×C4C4×D811yescharacteristic
367C2×C4C4×D811yescharacteristic
368C2×C4C4×D811yescharacteristic
369C2×C4C2×C4×D411yescharacteristic
370C2×C4C22×D811yescharacteristic
371C2×C4C2×#13511yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8—21no—
376C8—21no—
377C8—21no—
378C8—21no—
379C8—41no—
380C8—41no—
381C8—81no—
382C8—81no—
383C8—81no—
384C8—81no—
385C8C2×#13111yescharacteristic
386C8#56811yescharacteristic
387C22—21no—
388C22—21no—
389C22—42no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—164no—
399C22#217912yes—
400C22C2×C4×D811yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—81no—
410C4#293311yescharacteristic
411C4#295611yescharacteristic
412C4#341511yescharacteristic
413C4#341611yescharacteristic
414C2—22no—
415C2—82no—
416C2—161no—
417C2—161no—
418C2#1799411yescharacteristic
419C2#3345411yescharacteristic
420C2C2×#217911yescharacteristic
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 ◁ 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. #18087C2semidirect
3. #18054C2semidirect
4. #12086C2semidirect
5. #12084C2semidirect
6. #12071C2semidirect
7. #12070C2semidirect
8. #12069C2semidirect
9. #12069C2semidirect
10. #12045C2semidirect
11. #12043C2semidirect
12. #12035C2semidirect
13. #11673C2semidirect
14. C2×#1331C2semidirect
15. #11408C2semidirect
34. #1591C22semidirect
37. #1589C22semidirect
42. 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. #1311C22semidirect

Maximal subgroups

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

subgroupindex
#336652
#180872
#180542
#120862
#120842
#120712
#120702
#120692
#120452
#120432
#120352
#116732
C2×#13312
#114082