Finite Group Databasebeta

← #506357

Subgroups — #506357

Order 512.

537 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2231C212yesmaximal, complemented
2#21723C211yesmaximal, characteristic, complemented
3#21708C211yesmaximal, characteristic
4#13510C212yesmaximal, complemented
5#13410C212yesmaximal
6#13143C214yesmaximal, complemented
7#12881C211yesmaximal, characteristic, complemented
8#12821C212yesmaximal, complemented
9C2×#499C2211yescharacteristic
10#3150C2211yescharacteristic
11C2×#443—21no—
12C2×#443C2212yes—
13#2931—21no—
14#2330—21no—
15#2231—28no—
16C2×#365C2214yes—
17C2×#364C2212yes—
18C2×#360C2214yes—
19#2139C2214yes—
20#1992—21no—
21#1975—21no—
22#1975C2212yes—
23#1963—21no—
24#1944C2212yes—
25#1942—21no—
26#1849C2211yescharacteristic
27#1845C2211yescharacteristic
28#1776C2212yes—
29C4×#107—21no—
30#1726C2211yescharacteristic
31#1387C2214yes—
32#1377C2212yes—
33#1375C2212yes—
34C2×#135—21no—
35C2×#135C2312yes—
36C2×#131—22no—
37#532—21no—
38C2×#125—21no—
39C2×M32C2314yes—
40#499—22no—
41#499D414yescomplemented
42#497—22no—
43#496—22no—
44#443—24no—
45#443—24no—
46#443—44no—
47#436—41no—
48#436—41no—
49C8×D4—41no—
50C8×D4—41no—
51C2×#108—21no—
52C2×#108C2312yes—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107C2312yes—
56C2×#106C2312yes—
57#423—21no—
58C2×#104—22no—
59C2×#104—22no—
60#408—22no—
61C2×#97D411yescharacteristic
62C2×#97C2311yescharacteristic
63C2×#97C2311yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#389—22no—
68#389—42no—
69#387—42no—
70#384—21no—
71C4×C4⋊C4—22no—
72#376D411yescharacteristic
73#365—28no—
74#364—24no—
75#360—28no—
76#358—24no—
77#357—24no—
78#340—22no—
79#340—22no—
80C2×#41—22no—
81#135—22no—
82#135—22no—
83#135—22no—
84#135—22no—
85#135—42no—
86#135—42no—
87C2×Q16—21no—
88C2×Q16#4112yes—
89C2×SD16—22no—
90C2×SD16—22no—
91C2×D8—21no—
92C2×D8#4112yes—
93#131—44no—
94#131—44no—
95C2×M16—22no—
96C2×M16—42no—
97C22×C8—42no—
98C22×C8C2×D411yescharacteristic
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C22×C8C2411yescharacteristic
102C4⋊Q8—21no—
103#127—21no—
104#125—24no—
105#125—42no—
106#124—22no—
107#121—41no—
108#121—41no—
109C4×D4—41no—
110C4×D4—41no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C42—42no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120M32—24no—
121M32—24no—
122C2×C16—24no—
123#108—22no—
124#108—22no—
125#108—42no—
126#107—24no—
127#107—24no—
128#107—41no—
129#107—41no—
130#106—22no—
131#106C2×D414yes—
132C4⋊C8—41no—
133C4⋊C8—41no—
134#104—44no—
135#104—44no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#97—22no—
139#97—22no—
140#97—41no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8SD1614yes—
144C4×C8C2×D412yes—
145C4×C8#4112yes—
146#95—21no—
147#95—21no—
148#95—42no—
149#95—42no—
150#41—42no—
151#41—42no—
152#41—42no—
153#41—42no—
154C2×Q8—22no—
155C2×D4—22no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—42no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4AGL(1,ℤ/8)11yescharacteristic
165C22×C4AGL(1,ℤ/8)11yescharacteristic
166C22×C4C22×D411yescharacteristic
167Q16—22no—
168Q16—22no—
169Q16—42no—
170SD16—44no—
171SD16—42no—
172SD16—42no—
173D8—22no—
174D8—22no—
175D8—42no—
176M16—42no—
177M16—42no—
178M16—84no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—42no—
185C2×C8—84no—
186C2×C8#12112yes—
187C2×C8C2×SD1612yes—
188C2×C8C2×SD1612yes—
189C2×C8AGL(1,ℤ/8)11yescharacteristic
190C2×C8AGL(1,ℤ/8)11yescharacteristic
191C2×C8C22×D411yescharacteristic
192C2×C8C2×#4112yes—
193C2×C8C2×#4111yescharacteristic
194C2×C82−1+412yes—
195C4⋊C4—24no—
196C4⋊C4—48no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—162no—
203C4⋊C4—162no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—162no—
211C42#12112yes—
212C42C2×SD1612yes—
213C16—44no—
214C23—42no—
215C23C2×AGL(1,ℤ/8)11yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220D4—82no—
221D4—82no—
222D4—82no—
223D4—82no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4C2×#12111yescharacteristic
244C2×C4#52511yescharacteristic
245C2×C4#53611yescharacteristic
246C2×C4C22×SD1611yescharacteristic
247C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
248C2×C4#57511yescharacteristic
249C8—22no—
250C8—41no—
251C8—82no—
252C8#46012yes—
253C8#46012yes—
254C8#54812yes—
255C8#55212yes—
256C22—21no—
257C22—21no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22#303611yescharacteristic
262C22#310911yescharacteristic
263C22#322711yescharacteristic
264C4—42no—
265C4—42no—
266C4—41no—
267C4—41no—
268C4—84no—
269C4—82no—
270C4—162no—
271C4#253611yescharacteristic
272C4C2×#46011yescharacteristic
273C4#310111yescharacteristic
274C4#323511yescharacteristic
275C2—21no—
276C2—21no—
277C2—162no—
278C2#2820211yescharacteristic
279C2#3369211yescharacteristic
280C2#3369311yescharacteristic
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radical#3150C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2231C2semidirect
2. #21723C2semidirect
4. #13510C2semidirect
6. #13143C2semidirect
7. #12881C2semidirect
8. #12821C2semidirect
41. #499D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22312
#217232
#217082
#135102
#134102
#131432
#128812
#128212