Finite Group Databasebeta

← #508752

Subgroups — #508752

Order 512.

394 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 124 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#33263C211yesmaximal, characteristic
2#21736C211yesmaximal, characteristic
3#13543C212yesmaximal
4#13541C212yesmaximal
5#13508C212yesmaximal
6#13454C212yesmaximal
7#13118C214yesmaximal
8#12896C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10#2961—21no—
11#2859—21no—
12#2159C2214yes—
13#1994—21no—
14#1946—21no—
15#1945C2212yes—
16#1944—21no—
17#1944C2212yes—
18#1943—21no—
19#1943C2212yes—
20#1849C2211yescharacteristic
21#1845C2211yescharacteristic
22#1844C2211yescharacteristic
23#1834—21no—
24#1773C2212yes—
25#1772—21no—
26C4×#97C2211yescharacteristic
27#1387C2214yes—
28#1386C2214yes—
29#1383C2214yes—
30#1377C2212yes—
31#1376C2212yes—
32#1375C2212yes—
33#533—21no—
34C2×#125—41no—
35C2×M32C2314yes—
36#501—22no—
37#501—22no—
38#446—42no—
39#446—42no—
40C2×#108—21no—
41C2×#108C2312yes—
42#427—21no—
43#427C2312yes—
44C2×#107—21no—
45C2×#107C2312yes—
46C2×#106—21no—
47C2×#106C2312yes—
48#424—22no—
49#423—21no—
50#405—22no—
51C2×#97D411yescharacteristic
52C2×#97D411yescharacteristic
53C2×#97C2311yescharacteristic
54C2×#97C2311yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#389—22no—
58#389—22no—
59#389—41no—
60#389—41no—
61#384—21no—
62#384—21no—
63#382—41no—
64C43D411yescharacteristic
65#367—24no—
66#366—24no—
67#340—22no—
68#340—22no—
69#339—22no—
70#339—22no—
71C2×M16—22no—
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8C2411yescharacteristic
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78#125—22no—
79#125—84no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C4×Q8—41no—
83C4×Q8—41no—
84#117—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42C2×D411yescharacteristic
98M32—24no—
99M32—24no—
100C2×C16—24no—
101#108—24no—
102#108—42no—
103#107—21no—
104#107—21no—
105#107—24no—
106#107—42no—
107#107#4112yes—
108#107#4112yes—
109#106—22no—
110#106—22no—
111#106—24no—
112#106—42no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—44no—
116#97—22no—
117#97—22no—
118#97—22no—
119#97—41no—
120#97—42no—
121C4×C8—22no—
122C4×C8—42no—
123C4×C8Q1614yes—
124#95—42no—
125#95—42no—
126C2×Q8—41no—
127C2×Q8—41no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134C22×C4C22×D411yescharacteristic
135M16—42no—
136M16—42no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—42no—
144C2×C8C2×Q1612yes—
145C2×C8C2×Q1612yes—
146C2×C8AGL(1,ℤ/8)12yes—
147C2×C8C2×#4112yes—
148C2×C82−1+412yes—
149C2×C82−1+412yes—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C22⋊C4—42no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—42no—
173C42—42no—
174C42—42no—
175C42C2×Q1612yes—
176C16—44no—
177C23C2×AGL(1,ℤ/8)11yescharacteristic
178Q8—161no—
179Q8—161no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4#52511yescharacteristic
199C2×C4#53611yescharacteristic
200C2×C4#53611yescharacteristic
201C2×C4C22×Q1611yescharacteristic
202C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C4#57511yescharacteristic
204C8—22no—
205C8—22no—
206C8—22no—
207C8—82no—
208C8#54114yes—
209C22—21no—
210C22—21no—
211C22#310911yescharacteristic
212C22#323011yescharacteristic
213C22#323511yescharacteristic
214C4—24no—
215C4—41no—
216C4—41no—
217C4—81no—
218C4—82no—
219C4—82no—
220C4—162no—
221C4—162no—
222C4#263411yescharacteristic
223C4#310411yescharacteristic
224C4#323011yescharacteristic
225C4#323411yescharacteristic
226C2—21no—
227C2—21no—
228C2#3009511yescharacteristic
229C2#3368811yescharacteristic
230C2#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

Maximal subgroups

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

subgroupindex
#332632
#217362
#135432
#135412
#135082
#134542
#131182
#128962