Finite Group Databasebeta

← #484023

Subgroups — #484023

Order 512.

458 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33665C211yesmaximal, characteristic
2#18069C211yesmaximal, characteristic, complemented
3M16×D8C211yesmaximal, characteristic, complemented
4#12086C211yesmaximal, characteristic, complemented
5#12082C211yesmaximal, characteristic, complemented
6#12075C211yesmaximal, characteristic, complemented
7#12073C211yesmaximal, characteristic, complemented
8#12070C211yesmaximal, characteristic, complemented
9#12069C211yesmaximal, characteristic, complemented
10#12043C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic, complemented
12#12036C211yesmaximal, characteristic, complemented
13#11674C211yesmaximal, characteristic, complemented
14#11593C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#2969—21no—
18C2×C4×D8C2211yescharacteristic
19D4×M16—21no—
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
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic, complemented
35#1589C2211yescharacteristic
36#1589C2211yescharacteristic
37#1589C2211yescharacteristic, complemented
38#1583—21no—
39#1582—21no—
40#1581—21no—
41#1581—21no—
42C8×Q16C2211yescharacteristic, complemented
43C8×D8C2211yescharacteristic
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic, complemented
46#1567C2211yescharacteristic
47#1564C2211yescharacteristic
48#1486—21no—
49#1482—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
66C8×C16C2211yescharacteristic, complemented
67C22×D8C2×C411yescharacteristic
68C22×M16—41no—
69#529—21no—
70C2×C4×D4—21no—
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C2×D16C2×C411yescharacteristic
75C2×M32—21no—
76#499C2×C411yescharacteristic
77#495C2×C411yescharacteristic
78#488—21no—
79#467C2×C411yescharacteristic
80#466C2×C411yescharacteristic
81#465—21no—
82#446—21no—
83C8×Q8—21no—
84C4×Q16C2311yescharacteristic
85C4×SD16—21no—
86C4×D8—21no—
87C4×D8—21no—
88C4×D8C2×C411yescharacteristic
89C4×D8C2×C411yescharacteristic
90C4×D8C2311yescharacteristic
91C4×D8C2311yescharacteristic
92C4×D8C2311yescharacteristic
93#436—21no—
94#436—21no—
95#436—21no—
96#436—41no—
97#435—41no—
98#435—41no—
99#435—41no—
100C8×D4—21no—
101C8×D4—21no—
102C8×D4—21no—
103#433—41no—
104#431—21no—
105#431D411yescharacteristic
106C2×#107C2×C411yescharacteristic
107#423—21no—
108C2×#102—21no—
109#407—41no—
110#405—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
129#346—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—41no—
163C2×M16—41no—
164C2×M16—41no—
165C2×M16—41no—
166C2×M16—84no—
167C2×M16D811yescharacteristic
168C2×M16D811yescharacteristic
169C22×C8—41no—
170C22×C8—41no—
171C22×C8—41no—
172C22×C8C22×C411yescharacteristic
173C22×C8#4111yescharacteristic
174#126—41no—
175#125—21no—
176#124—21no—
177#123—41no—
178C22⋊Q8—21no—
179#121—21no—
180C4×Q8—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—
189#117—41no—
190C2×C4⋊C4—21no—
191C2×C22⋊C4—41no—
192C2×C42C2×D411yescharacteristic
193SD32—22no—
194SD32—41no—
195D16—22no—
196D16—41no—
197M32—42no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16C22×C411yescharacteristic
201C2×C16C22×C411yescharacteristic
202C2×C16C22×C411yescharacteristic
203C2×C16C22×C411yescharacteristic
204#107—21no—
205#107C22×C411yescharacteristic
206#107C22×C411yescharacteristic
207#107C22×C411yescharacteristic
208#107C22×C411yescharacteristic
209#106—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218#103—21no—
219#103—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223#102—21no—
224#102—21no—
225#102—21no—
226C22⋊C8—21no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234#97—41no—
235#97D811yescharacteristic
236#97D811yescharacteristic
237C4×C8—41no—
238C4×C8—41no—
239C4×C8C22×C411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8C2×D411yescharacteristic
245C4×C8#4111yescharacteristic
246C4×C8C2411yescharacteristic
247C24—41no—
248C2×Q8—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—81no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4C4×D411yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4#13511yescharacteristic
267Q16—21no—
268Q16—21no—
269SD16—81no—
270D8—21no—
271D8—21no—
272D8—21no—
273D8—21no—
274D8—21no—
275D8—42no—
276D8#13112yes—
277M16—22no—
278M16—41no—
279M16—82no—
280M16—82no—
281M16—82no—
282M16—82no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—82no—
295C2×C8C4×D411yescharacteristic
296C2×C8C4×D411yescharacteristic
297C2×C8C2×D811yescharacteristic
298C2×C8C2×D811yescharacteristic
299C2×C8C2×D811yescharacteristic
300C2×C8C2×D811yescharacteristic
301C2×C8C23×C411yescharacteristic
302C2×C8C2×#4111yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42—41no—
322C42C4×D411yescharacteristic
323C42C2×D811yescharacteristic
324C42#13511yescharacteristic
325C42C22×D411yescharacteristic
326C16—22no—
327C16—21no—
328C16—21no—
329C16—21no—
330C16—21no—
331C16—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—82no—
336C23—82no—
337C23—161no—
338C23C4×D811yescharacteristic
339Q8—41no—
340Q8—41no—
341D4—22no—
342D4—41no—
343D4—41no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—82no—
348D4—161no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4C4×D811yescharacteristic
369C2×C4C4×D811yescharacteristic
370C2×C4C4×D811yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C22×D811yescharacteristic
373C2×C4C2×#13511yescharacteristic
374C8—21no—
375C8—21no—
376C8—21no—
377C8—21no—
378C8—21no—
379C8—21no—
380C8—41no—
381C8—41no—
382C8—81no—
383C8—81no—
384C8—81no—
385C8—81no—
386C8#50912yes—
387C8C2×#13111yescharacteristic
388C8#56811yescharacteristic
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—164no—
402C22C2×C4×D811yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—81no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4#293311yescharacteristic
413C4#295611yescharacteristic
414C4C2×#50911yescharacteristic
415C4#341411yescharacteristic
416C2—41no—
417C2—82no—
418C2—161no—
419C2—161no—
420C2#1799411yescharacteristic
421C2C4×#50911yescharacteristic
422C2#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 ◁ C4×M16 ◁ 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. #18069C2semidirect
3. M16×D8C2semidirect
4. #12086C2semidirect
5. #12082C2semidirect
6. #12075C2semidirect
7. #12073C2semidirect
8. #12070C2semidirect
9. #12069C2semidirect
10. #12043C2semidirect
11. C8×D16C2semidirect
12. #12036C2semidirect
13. #11674C2semidirect
14. #11593C2semidirect
15. #11409C2semidirect
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. C8×C16C22semidirect

Maximal subgroups

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

subgroupindex
#336652
#180692
M16×D82
#120862
#120822
#120752
#120732
#120702
#120692
#120432
C8×D162
#120362
#116742
#115932
#114092