Finite Group Databasebeta

← #7622410

Subgroups — #7622410

Order 512.

1373 conjugacy classes of subgroups; only the 428 normal subgroups are shown.

normal subgroups only

This group has 1373 conjugacy classes of subgroups — too many to draw. Only its 428 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
1#32705C213yesmaximal, complemented
2#31077C211yesmaximal, characteristic, complemented
3#29384C216yesmaximal, complemented
4#29233C216yesmaximal, complemented
5#23994C216yesmaximal, complemented
6#22898C216yesmaximal, complemented
7#21417C213yesmaximal, complemented
8#3444C2213yes—
9#3025C2212yescomplemented
10#3018C2216yes—
11#2956C2216yes—
12#2920C2213yescomplemented
13#2901C2216yescomplemented
14C2×#376C2211yescharacteristic, complemented
15#2102C2216yescomplemented
16#2091C2212yescomplemented
17#2050C2216yes—
18#2031C22112yescomplemented
19#2010C2216yes—
20#1995C2213yescomplemented
21#1995C2213yescomplemented
22#1963C2213yes—
23#1963C2213yes—
24#1932C22112yescomplemented
25#1906C22112yes—
26#1896C22112yescomplemented
27#1891C2216yescomplemented
28#1884C2216yes—
29#1861C22112yescomplemented
30#1841C2216yescomplemented
31#1827C2216yescomplemented
32#1759C22112yescomplemented
33#585C2312yes—
34#575C2316yes—
35#568C2316yes—
36#533—23no—
37#533C2313yes—
38#532—23no—
39#532C2313yes—
40#529—26no—
41#529—43no—
42#529—43no—
43#525—43no—
44#525C2316yes—
45C4×#41—43no—
46C4×#41C2316yes—
47C22×C42C2313yes—
48#458—42no—
49B(4,𝔽2)—42no—
50#456—26no—
51#455—26no—
52#454—26no—
53#453—26no—
54#444—46no—
55#443—46no—
56#432—43no—
57#431—43no—
58#430C2316yes—
59#423C2313yes—
60#423C2313yes—
61#421—46no—
62#421C23112yes—
63C2×#104—46no—
64C2×#104C23112yes—
65C2×#104C23112yes—
66#413C2316yes—
67#410C2316yes—
68#407C2316yes—
69C4×M16C2316yes—
70#376—41no—
71#376C2311yescharacteristic
72#376C2313yes—
73C2×#95—41no—
74#344C23112yes—
75#339C23112yes—
76#339C23112yes—
77#338C2312yes—
78H(3,ℤ/4)C2316yes—
792−1+4—26no—
80H(5,𝔽2)—26no—
81H(5,𝔽2)—42no—
82C2×#41—43no—
83C2×#41—46no—
84C2×#41C2416yes—
85C23×C4—23no—
86C23×C4C2411yescharacteristic
87#137—212no—
88AGL(1,ℤ/8)—212no—
89#135—46no—
90#131—412no—
91C2×M16—46no—
92C2×M16C2416yes—
93C2×M16C2416yes—
94C4⋊Q8—23no—
95C4⋊Q8#4113yescomplemented
96#127#4113yescomplemented
97#126—46no—
98#125—23no—
99#125—43no—
100#124—43no—
101#124#4116yescomplemented
102#123—26no—
103#123—46no—
104C22⋊Q8—26no—
105C22⋊Q8—43no—
106C22⋊Q8—43no—
107#121—26no—
108#121—43no—
109#121—43no—
110#120—26no—
111C4×Q8—26no—
112C4×Q8—43no—
113C4×Q8—43no—
114C4×D4—26no—
115C4×D4—46no—
116C4×D4—43no—
117C4×D4—43no—
118C4×D4—46no—
119C4×D4—46no—
120#117—26no—
121#117—46no—
122#117—43no—
123#117C2416yes—
124C2×C42—23no—
125C2×C42—23no—
126C2×C42—23no—
127C2×C42—23no—
128C2×C42—23no—
129C2×C42—26no—
130C2×C42C2413yes—
131C2×C42C2413yes—
132#108—46no—
133C4⋊C8—43no—
134C4⋊C8—43no—
135#104—212no—
136#104—212no—
137#104—212no—
138#104—212no—
139#104—412no—
140#101—26no—
141#100—26no—
142#99—46no—
143C22⋊C8—46no—
144#97—26no—
145C4×C8—26no—
146#95—43no—
147#95—41no—
148C24H(5,𝔽2)11yescharacteristic
149#41—212no—
150#41—212no—
151#41—412no—
152#41—86no—
153#41—82no—
154#41—812no—
155C2×Q8—43no—
156C2×Q8—46no—
157C2×Q8C2×#4116yes—
158C2×D4—26no—
159C2×D4—43no—
160C2×D4—46no—
161C2×D4—46no—
162C2×D4—86no—
163C2×D4C2×#4116yes—
164C22×C4—23no—
165C22×C4—23no—
166C22×C4—23no—
167C22×C4—23no—
168C22×C4—23no—
169C22×C4—23no—
170C22×C4—23no—
171C22×C4—23no—
172C22×C4—26no—
173C22×C4—46no—
174C22×C4—82no—
175C22×C4—86no—
176C22×C4H(5,𝔽2)11yescharacteristic
177C22×C4C2511yescharacteristic
178Q16—46no—
179SD16—412no—
180D8—46no—
181M16—212no—
182M16—212no—
183M16—86no—
184C2×C8—26no—
185C2×C8—26no—
186C2×C8—86no—
187C4⋊C4—26no—
188C4⋊C4—43no—
189C4⋊C4—46no—
190C4⋊C4—46no—
191C4⋊C4—43no—
192C4⋊C4—46no—
193C22⋊C4—26no—
194C22⋊C4—46no—
195C22⋊C4—46no—
196C22⋊C4—46no—
197C22⋊C4—46no—
198C42—23no—
199C42—23no—
200C42—23no—
201C42—23no—
202C42—23no—
203C42—23no—
204C42—26no—
205C42—46no—
206C42C2×#4113yes—
207C42C2×#4113yes—
208C42H(5,𝔽2)13yes—
209C42H(5,𝔽2)13yes—
210C23—23no—
211C23—26no—
212C23—42no—
213C23—82no—
214C23C2×H(5,𝔽2)11yescharacteristic
215Q8—212no—
216Q8—82no—
217Q8—86no—
218D4—212no—
219D4—412no—
220D4—86no—
221D4—86no—
222D4—812no—
223C2×C4—23no—
224C2×C4—23no—
225C2×C4—23no—
226C2×C4—23no—
227C2×C4—23no—
228C2×C4—26no—
229C2×C4—46no—
230C2×C4—46no—
231C2×C4—43no—
232C2×C4—46no—
233C2×C4—46no—
234C2×C4—41no—
235C2×C4—46no—
236C2×C4—83no—
237C2×C4—83no—
238C2×C4—83no—
239C2×C4—83no—
240C2×C4—86no—
241C2×C4—812no—
242C2×C4C22×#4113yes—
243C2×C4C2×H(5,𝔽2)13yes—
244C8—46no—
245C8—46no—
246C22—26no—
247C22—41no—
248C22—43no—
249C22—41no—
250C22—46no—
251C22—86no—
252C22—322no—
253C22#349013yes—
254C4—43no—
255C4—43no—
256C4—43no—
257C4—43no—
258C4—41no—
259C4—46no—
260C4—86no—
261C4#348113yes—
262C4#354911yescharacteristic
263C2—23no—
264C2—42no—
265C2—86no—
266C2#6004511yescharacteristic
subgroupG/H
centerC4#3549
commutator, Frattini, cosocleC22×C4C25
socleC2#60045
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #32705C2semidirect
2. #31077C2semidirect
3. #29384C2semidirect
4. #29233C2semidirect
5. #23994C2semidirect
6. #22898C2semidirect
7. #21417C2semidirect
9. #3025C22semidirect
12. #2920C22semidirect
13. #2901C22semidirect
14. C2×#376C22semidirect
15. #2102C22semidirect
16. #2091C22semidirect
18. #2031C22semidirect
20. #1995C22semidirect
21. #1995C22semidirect
24. #1932C22semidirect
26. #1896C22semidirect
27. #1891C22semidirect
29. #1861C22semidirect
30. #1841C22semidirect
31. #1827C22semidirect
32. #1759C22semidirect
95. C4⋊Q8#41semidirect
96. #127#41semidirect
101. #124#41semidirect

Maximal subgroups

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

subgroupindex
#327052
#310772
#293842
#292332
#239942
#228982
#214172