Finite Group Databasebeta

← #507604

Subgroups — #507604

Order 512.

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

normal subgroups only

This group has 443 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
1#21736C211yesmaximal, characteristic
2#21727C211yesmaximal, characteristic
3#13541C212yesmaximal
4#13454C212yesmaximal
5#13272C212yesmaximal
6#13181C212yesmaximal
7#13119C214yesmaximal
8#12882C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10C2×#498C2211yescharacteristic
11#2961—21no—
12C4×C4⋊Q8—21no—
13#2159C2214yes—
14C2×#364C2212yes—
15#1994—21no—
16#1990—21no—
17#1946C2212yes—
18#1945—21no—
19#1944—21no—
20#1944C2212yes—
21#1849C2211yescharacteristic
22#1845C2211yescharacteristic
23#1776—21no—
24#1776C2212yes—
25#1773—21no—
26#1772C2212yes—
27C4×#97C2211yescharacteristic
28#1387C2214yes—
29#1386C2214yes—
30#1383C2214yes—
31#1377C2212yes—
32#1375C2212yes—
33#533—21no—
34C2×C4⋊Q8—21no—
35C2×C4×Q8—41no—
36C2×M32C2314yes—
37#501—22no—
38#501—22no—
39#498—22no—
40#498Q814yescomplemented
41#446—42no—
42#446—42no—
43C2×#108—21no—
44C2×#108C2312yes—
45#427—21no—
46#427C2312yes—
47C2×#107—21no—
48C2×#107C2312yes—
49C2×#106—21no—
50C2×#106C2312yes—
51#424—22no—
52#423—21no—
53#405—22no—
54C2×#97D411yescharacteristic
55C2×#97D411yescharacteristic
56C2×#97C2311yescharacteristic
57C2×#97C2311yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8C2311yescharacteristic
60#391—41no—
61#389—22no—
62#389—41no—
63#384—21no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—41no—
66C43D411yescharacteristic
67#367—24no—
68#366—24no—
69#364—24no—
70#340—22no—
71#340—22no—
72#339—22no—
73#339—22no—
74C22×Q8—41no—
75C2×M16—22no—
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C22×C8C2411yescharacteristic
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—24no—
83C4⋊Q8—42no—
84#125—22no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×Q8—84no—
90#117—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—42no—
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D411yescharacteristic
103M32—24no—
104M32—24no—
105C2×C16—24no—
106#108—24no—
107#108—42no—
108#107—21no—
109#107—21no—
110#107—24no—
111#107—42no—
112#107#4112yes—
113#107#4112yes—
114#106—22no—
115#106—24no—
116#106—22no—
117#106—22no—
118#106C2×Q814yes—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—44no—
122#97—22no—
123#97—22no—
124#97—22no—
125#97—42no—
126#97Q1614yes—
127C4×C8—41no—
128C4×C8—42no—
129C4×C8C2×Q812yes—
130C4×C8#4112yes—
131#95—42no—
132#95—42no—
133C2×Q8—44no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—82no—
137C2×Q8—82no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4AGL(1,ℤ/8)11yescharacteristic
143C22×C4AGL(1,ℤ/8)11yescharacteristic
144C22×C4C22×D411yescharacteristic
145M16—42no—
146M16—42no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—42no—
153C2×C8C22⋊Q812yes—
154C2×C8C2×Q1612yes—
155C2×C8C2×Q1612yes—
156C2×C8AGL(1,ℤ/8)12yes—
157C2×C8C22×Q811yescharacteristic
158C2×C8C2×#4111yescharacteristic
159C2×C8C2×#4112yes—
160C2×C82−1+412yes—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—44no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—84no—
175C22⋊C4—42no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—42no—
181C42—42no—
182C42—42no—
183C42—84no—
184C42C22⋊Q812yes—
185C42C2×Q1612yes—
186C16—44no—
187C23C2×AGL(1,ℤ/8)11yescharacteristic
188Q8—88no—
189Q8—161no—
190Q8—161no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4C2×C22⋊Q811yescharacteristic
210C2×C4#52511yescharacteristic
211C2×C4#53611yescharacteristic
212C2×C4C22×Q1611yescharacteristic
213C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C4#57511yescharacteristic
215C8—22no—
216C8—41no—
217C8—82no—
218C8#47714yes—
219C8#55612yes—
220C8#55712yes—
221C22—21no—
222C22—21no—
223C22#307711yescharacteristic
224C22#310911yescharacteristic
225C22#323011yescharacteristic
226C4—24no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—84no—
231C4—82no—
232C4—82no—
233C4—162no—
234C4#258111yescharacteristic
235C4C2×#47711yescharacteristic
236C4#310411yescharacteristic
237C4#323511yescharacteristic
238C2—21no—
239C2—21no—
240C2#2927711yescharacteristic
241C2#3368811yescharacteristic
242C2#3368911yescharacteristic
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
40. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#217362
#217272
#135412
#134542
#132722
#131812
#131192
#128822