Finite Group Databasebeta

← #152159

Subgroups — #152159

Order 512.

648 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C22×#508C211yesmaximal, characteristic
2C2×#1416C212yesmaximal, complemented
3C2×#1415C212yesmaximal, complemented
4C2×C22⋊C32C211yesmaximal, characteristic, complemented
5#12834C211yesmaximal, characteristic, complemented
6C22×#135—21no—
7C2×#508—24no—
8C2×#508—24no—
9C2×#508—22no—
10C2×#508C414yescomplemented
11C22×Q32C2211yescharacteristic
12C22×SD32—21no—
13C22×D16C2211yescharacteristic
14C23×C16C2211yescharacteristic
15C22×C32C2212yes—
16C2×#366C2212yes—
17#1808—21no—
18#1416—24no—
19#1415—24no—
20C22⋊C32—22no—
21#1380—21no—
22C22×#41—41no—
23C2×#135—28no—
24C2×#135—24no—
25C2×#135—44no—
26C2×#135—42no—
27C22×Q16—21no—
28C22×SD16—41no—
29C22×D8—21no—
30C23×C8D411yescharacteristic
31#508—216no—
32#508—28no—
33#508—28no—
34C2×Q32—22no—
35C2×Q32—21no—
36C2×Q32C2×C412yes—
37C2×Q32D414yes—
38C2×SD32—28no—
39C2×SD32—24no—
40C2×D16—22no—
41C2×D16—21no—
42C2×D16C2×C412yes—
43C2×D16D414yes—
44C22×C16—22no—
45C22×C16—22no—
46C22×C16C2×C412yes—
47C22×C16D411yescharacteristic
48C22×C16D412yes—
49C22×C16C2311yescharacteristic
50C2×#107—22no—
51#380—41no—
52C2×C32—22no—
53C2×C32—22no—
54C2×C32—22no—
55#366—22no—
56#366—22no—
57#340—41no—
58C2×#41—48no—
59C2×#41—44no—
60C2×#41—84no—
61C2×#41—82no—
62C22×Q8—41no—
63C22×D4—41no—
64C22×D4—81no—
65C23×C4—81no—
66C23×C4D811yescharacteristic
67#135—216no—
68#135—48no—
69#135—416no—
70C2×Q16—22no—
71C2×Q16—24no—
72C2×Q16—41no—
73C2×Q16—42no—
74C2×SD16—48no—
75C2×SD16—44no—
76C2×D8—22no—
77C2×D8—24no—
78C2×D8—41no—
79C2×D8—42no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8C22⋊C412yes—
83C22×C8D811yescharacteristic
84C22×C8SD1612yes—
85C22×C8C2×D411yescharacteristic
86C2×C4⋊C4—42no—
87C2×C22⋊C4—81no—
88Q32—22no—
89Q32—24no—
90Q32C22⋊C414yescomplemented
91SD32—216no—
92D16—22no—
93D16—24no—
94D16C22⋊C414yescomplemented
95C2×C16—22no—
96C2×C16—22no—
97C2×C16—22no—
98C2×C16—24no—
99C2×C16C22⋊C412yes—
100C2×C16C22×C411yescharacteristic
101C2×C16C2×D412yes—
102C2×C16C2×D412yes—
103C2×C16C2×D411yescharacteristic
104#107—42no—
105#107—42no—
106#95—81no—
107C32—42no—
108C24—81no—
109C24D1611yescharacteristic
110#41—416no—
111#41—88no—
112#41—816no—
113C2×Q8—42no—
114C2×Q8—44no—
115C2×Q8—81no—
116C2×Q8—82no—
117C2×D4—42no—
118C2×D4—44no—
119C2×D4—88no—
120C2×D4—84no—
121C2×D4—81no—
122C2×D4—82no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—88no—
126C22×C4—81no—
127C22×C4—84no—
128C22×C4—82no—
129C22×C4#10212yes—
130C22×C4D1611yescharacteristic
131C22×C4SD3212yes—
132C22×C4C2×D811yescharacteristic
133Q16—24no—
134Q16—42no—
135Q16—44no—
136SD16—416no—
137D8—24no—
138D8—42no—
139D8—44no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—24no—
144C2×C8#10212yes—
145C2×C8C2×C22⋊C411yescharacteristic
146C2×C8#12012yes—
147C2×C8C2×SD1611yescharacteristic
148C2×C8AGL(1,ℤ/8)12yes—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C22⋊C4—162no—
152C16—22no—
153C16C2×C22⋊C411yescharacteristic
154C16C2×C22⋊C411yescharacteristic
155C16#12012yes—
156C23—22no—
157C23—22no—
158C23—21no—
159C23—21no—
160C23—82no—
161C23—84no—
162C23—162no—
163C23—161no—
164C23—161no—
165C23#35712yes—
166C23C2×D1611yescharacteristic
167Q8—44no—
168Q8—82no—
169Q8—84no—
170D4—44no—
171D4—816no—
172D4—82no—
173D4—84no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—816no—
179C2×C4—82no—
180C2×C4—84no—
181C2×C4—162no—
182C2×C4—162no—
183C2×C4—162no—
184C2×C4#35712yes—
185C2×C4C2×#10211yescharacteristic
186C2×C4#44712yes—
187C2×C4C2×SD3211yescharacteristic
188C2×C4#50912yes—
189C8—22no—
190C8#37911yescharacteristic
191C8#41811yescharacteristic
192C8#45012yes—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—84no—
199C22—162no—
200C22—162no—
201C22—164no—
202C22—164no—
203C22C2×#35711yescharacteristic
204C22#218412yes—
205C22#226212yes—
206C22#226311yescharacteristic
207C22#226411yescharacteristic
208C4—22no—
209C4—84no—
210C4—322no—
211C4#178611yescharacteristic
212C4#214011yescharacteristic
213C4#218712yes—
214C2—42no—
215C2—164no—
216C2#1273111yescharacteristic
217C2#1361511yescharacteristic
218C2#1361611yescharacteristic
219C2#1366312yes—
220C2#1366612yes—
subgroupG/H
center, socleC23C2×D16
commutatorC2×C16C22×C4
Frattini, cosocleC22×C16C23
prime radicalC22×#508C2
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1416C2semidirect
3. C2×#1415C2semidirect
4. C2×C22⋊C32C2semidirect
5. #12834C2semidirect
10. C2×#508C4semidirect
90. Q32C22⋊C4semidirect
94. D16C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5082
C2×#14162
C2×#14152
C2×C22⋊C322
#128342