Finite Group Databasebeta

← #7623203

Subgroups — #7623203

Order 512.

2444 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3416C214yesmaximal, complemented
2#60648C211yesmaximal, characteristic, complemented
3#60647C211yesmaximal, characteristic, complemented
4#33722C2112yesmaximal, complemented
5#33718C213yesmaximal, complemented
6#33717C213yesmaximal, complemented
7#33713C213yesmaximal, complemented
8C2×#2248C213yesmaximal, complemented
9C4×#504C211yesmaximal, characteristic, complemented
10C2×#578—21no—
11C2×#578C2212yes—
12C2×#576—21no—
13C2×#576C2212yes—
14#3499—21no—
15#3498—21no—
16#3416—232no—
17C2×#510C22112yescomplemented
18C2×#509C22112yescomplemented
19C2×#508C22112yescomplemented
20C22×SD32C22112yescomplemented
21C2×#504C2212yes—
22#3156—23no—
23#3156C2216yescomplemented
24#3155—23no—
25#3155C2216yes—
26#3152—23no—
27#3151—23no—
28#3147C2213yes—
29#3147C2213yescomplemented
30C2×#494C2213yescomplemented
31C2×#493C2213yes—
32C4×#131C2211yescharacteristic
33#2251C22112yescomplemented
34#2250C22112yescomplemented
35#2249C22112yescomplemented
36#2248C2213yescomplemented
37#2248C2216yescomplemented
38#2248C22112yescomplemented
39#2248C2211yescharacteristic, complemented
40#2247C2213yescomplemented
41#2246C2213yescomplemented
42#2108C2216yescomplemented
43C4×M32C2213yescomplemented
44C2×C4×C16C2213yescomplemented
45C2×2−1+4—22no—
46C2×H(5,𝔽2)—22no—
47#578—216no—
48#578—48no—
49#576—216no—
50#576—48no—
51C2×#137—26no—
52C2×#137—26no—
53C2×AGL(1,ℤ/8)—26no—
54C2×AGL(1,ℤ/8)—26no—
55C2×#135—23no—
56C2×#135—23no—
57C2×#135C2316yes—
58C2×#135C2316yes—
59C22×Q16—23no—
60C22×Q16C2316yes—
61C22×D8—23no—
62C22×D8C2316yes—
63C2×#131D411yescharacteristic
64C2×#131C2311yescharacteristic
65#550—42no—
66D4×Q8—42no—
67#547—46no—
68D42—46no—
69#536—26no—
70#535—26no—
71#532—23no—
72#532—23no—
73C2×C4⋊Q8—23no—
74C2×#127—23no—
75C4×#41D411yescharacteristic
76#510—248no—
77#509—248no—
78#508—248no—
79C2×Q32C23112yes—
80C2×SD32—248no—
81C2×SD32C23112yes—
82C2×SD32C23124yes—
83C2×SD32C2314yes—
84C2×D16C23112yes—
85#504D4116yescomplemented
86C2×M32C2316yes—
87C22×C16C2316yes—
88#497—212no—
89#496—212no—
90#495—23no—
91#495—23no—
92#495C2316yes—
93#495C2316yes—
94#494—23no—
95#494—23no—
96#494C2313yes—
97#494C2316yes—
98#494C2311yescharacteristic
99#493—23no—
100#493—23no—
101#493C2313yes—
102#493C2316yes—
103#493C2311yescharacteristic
104#492—23no—
105#492—23no—
106#405D413yes—
107#405C2313yes—
108C4×M16D413yes—
109C2×C4×C8C2313yes—
110#346C2316yes—
111C4×C16C2311yescharacteristic
112C4×C16C2313yes—
113C4×C16C2316yes—
1142−1+4—48no—
1152−1+4—48no—
116H(5,𝔽2)—48no—
117H(5,𝔽2)—48no—
118C2×#41—26no—
119C2×#41—26no—
120C2×#41—42no—
121C2×#41—46no—
122C2×#41C2×D411yescharacteristic
123C22×Q8—26no—
124C22×Q8—42no—
125C22×D4—26no—
126C22×D4—46no—
127C22×D4—46no—
128#137—424no—
129#137—424no—
130AGL(1,ℤ/8)—424no—
131AGL(1,ℤ/8)—424no—
132#135—224no—
133#135—224no—
134#135—412no—
135#135—412no—
136C2×Q16—224no—
137C2×Q16—23no—
138C2×Q16—26no—
139C2×Q16—26no—
140C2×Q16—21no—
141C2×Q16—412no—
142C2×Q16C2416yes—
143C2×Q16C2412yes—
144C2×SD16—26no—
145C2×SD16—26no—
146C2×SD16—26no—
147C2×SD16—26no—
148C2×D8—224no—
149C2×D8—23no—
150C2×D8—26no—
151C2×D8—26no—
152C2×D8—21no—
153C2×D8—412no—
154C2×D8C2416yes—
155C2×D8C2412yes—
156#131D818yes—
157#131C2×D418yes—
158C2×M16C2×D413yes—
159C2×M16C2×D413yes—
160C22×C8C2×D413yes—
161C22×C8C2413yes—
162C4⋊Q8—23no—
163C4⋊Q8—26no—
164C4⋊Q8—21no—
165C4⋊Q8—46no—
166#127—23no—
167#127—26no—
168#127—21no—
169#127—46no—
170#124—26no—
171#124—26no—
172#123—412no—
173C22⋊Q8—412no—
174#121—46no—
175#121—46no—
176#121—412no—
177#120—412no—
178C4×Q8D812yes—
179C4×D4—46no—
180C4×D4—46no—
181C4×D4D816yes—
182#117C2×D413yes—
183C2×C4⋊C4—46no—
184C2×C42C2×D413yes—
185Q32—224no—
186SD32—224no—
187SD32—248no—
188SD32—28no—
189D16—224no—
190M32C2×D4124yescomplemented
191C2×C16C2×D4124yescomplemented
192C2×C16C2412yes—
193C2×C16C2416yes—
194#97C2×D416yes—
195C4×C8C2×D413yes—
196C4×C8C2×D413yes—
197C4×C8C2411yescharacteristic
198C4×C8C2413yes—
199C24—46no—
200#41—412no—
201#41—412no—
202#41—412no—
203#41—412no—
204#41—88no—
205#41—824no—
206#41#12714yes—
207#41C2×D814yes—
208C2×Q8—22no—
209C2×Q8—26no—
210C2×Q8—46no—
211C2×Q8—412no—
212C2×Q8—412no—
213C2×Q8—88no—
214C2×Q8C2×D811yescharacteristic
215C2×D4—22no—
216C2×D4—26no—
217C2×D4—46no—
218C2×D4—46no—
219C2×D4—412no—
220C2×D4—412no—
221C2×D4—42no—
222C2×D4—46no—
223C2×D4—46no—
224C2×D4—412no—
225C2×D4—824no—
226C2×D4—1612no—
227C2×D4C2×D813yes—
228C22×C4—46no—
229C22×C4—46no—
230C22×C4—46no—
231C22×C4C2×D816yes—
232C22×C4C22×D413yes—
233Q16—212no—
234Q16—24no—
235Q16—412no—
236Q16—46no—
237Q16—412no—
238Q16—42no—
239SD16—412no—
240SD16—412no—
241SD16—412no—
242SD16—412no—
243D8—24no—
244D8—212no—
245D8—412no—
246D8—46no—
247D8—412no—
248D8—42no—
249M16#127112yes—
250M16C2×D8112yes—
251C2×C8C2×D8112yes—
252C2×C8C22×D413yes—
253C2×C8C22×D4112yes—
254C2×C8C22×D411yescharacteristic
255C2×C8C22×D413yes—
256C2×C8C2511yescharacteristic
257C4⋊C4—46no—
258C4⋊C4—412no—
259C4⋊C4—42no—
260C4⋊C4C2×D816yes—
261C22⋊C4—412no—
262C22⋊C4—412no—
263C22⋊C4C2×D816yes—
264C42C2×D816yes—
265C42C22×D411yescharacteristic
266C42C22×D413yes—
267C16C22×D414yes—
268C16C22×D4112yes—
269C23—42no—
270C23—46no—
271C23—812no—
272C23—1612no—
273C23C22×D813yes—
274Q8—42no—
275Q8—46no—
276Q8—42no—
277Q8—46no—
278Q8—812no—
279Q8#49314yes—
280D4—46no—
281D4—42no—
282D4—46no—
283D4—42no—
284D4—812no—
285D4—812no—
286D4—812no—
287D4—84no—
288D4—812no—
289D4—1612no—
290D4#493112yes—
291C2×C4—212no—
292C2×C4—42no—
293C2×C4—46no—
294C2×C4—812no—
295C2×C4—812no—
296C2×C4—812no—
297C2×C4C2×#12716yes—
298C2×C4C22×D816yes—
299C2×C4C22×D813yes—
300C2×C4C22×D816yes—
301C2×C4C22×D812yes—
302C2×C4C23×D411yescharacteristic
303C8C2×#12716yes—
304C8C22×D812yes—
305C8C22×D816yes—
306C8C23×D412yes—
307C22—26no—
308C22—812no—
309C22—84no—
310C22—82no—
311C22—86no—
312C22—3212no—
313C22C2×#49316yes—
314C22C23×D811yescharacteristic
315C4—26no—
316C4—82no—
317C4—86no—
318C4C2×#49316yes—
319C4#341614yescomplemented
320C4C22×#12711yescharacteristic
321C4C23×D811yescharacteristic
322C2—26no—
323C2—162no—
324C2—166no—
325C2C22×#49311yescharacteristic
326C2C2×#341612yes—
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3416C2semidirect
2. #60648C2semidirect
3. #60647C2semidirect
4. #33722C2semidirect
5. #33718C2semidirect
6. #33717C2semidirect
7. #33713C2semidirect
8. C2×#2248C2semidirect
9. C4×#504C2semidirect
17. C2×#510C22semidirect
18. C2×#509C22semidirect
19. C2×#508C22semidirect
20. C22×SD32C22semidirect
23. #3156C22semidirect
29. #3147C22semidirect
30. C2×#494C22semidirect
33. #2251C22semidirect
34. #2250C22semidirect
35. #2249C22semidirect
36. #2248C22semidirect
37. #2248C22semidirect
38. #2248C22semidirect
39. #2248C22semidirect
40. #2247C22semidirect
41. #2246C22semidirect
42. #2108C22semidirect
43. C4×M32C22semidirect
44. C2×C4×C16C22semidirect
85. #504D4semidirect
190. M32C2×D4semidirect
191. C2×C16C2×D4semidirect
319. C4#3416semidirect

Maximal subgroups

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

subgroupindex
C2×#34162
#606482
#606472
#337222
#337182
#337172
#337132
C2×#22482
C4×#5042