Finite Group Databasebeta

← #508748

Subgroups — #508748

Order 512.

401 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 132 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#33261C211yesmaximal, characteristic
2#21725C211yesmaximal, characteristic
3#13539C212yesmaximal
4#13539C212yesmaximal
5#13506C212yesmaximal
6#13452C212yesmaximal
7#13111C212yesmaximal
8#13110C212yesmaximal
9#12895C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11#2647—21no—
12#2331—21no—
13C2×#367C2212yes—
14C2×#367C2212yes—
15#1993—21no—
16#1944—21no—
17#1943—21no—
18#1943—21no—
19#1942C2212yes—
20#1942C2212yes—
21#1942C2212yes—
22#1846C2211yescharacteristic
23#1846C2211yescharacteristic
24#1845C2211yescharacteristic
25#1836—21no—
26#1776—21no—
27C4×#107C2212yes—
28#1726C2211yescharacteristic
29#1380C2212yes—
30#1380C2212yes—
31#1380C2212yes—
32#1380C2212yes—
33#1380C2212yes—
34#1380C2212yes—
35#1374C2211yescharacteristic
36#1374C2211yescharacteristic
37#1374C2212yes—
38#1374C2212yes—
39C2×C4⋊Q8—21no—
40C22×C16C2312yes—
41C22×C16C2312yes—
42#500—24no—
43C2×#107—21no—
44C2×#107—21no—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#107C2312yes—
48C2×#107C2312yes—
49C2×#106—21no—
50C2×#106—21no—
51C2×#97D411yescharacteristic
52C2×#97D411yescharacteristic
53C2×#97D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#400—42no—
58#398—42no—
59#391—42no—
60#389—22no—
61#389—22no—
62#389—22no—
63#387—42no—
64#384—21no—
65#382—21no—
66#382—21no—
67C4×C4⋊C4—22no—
68#376D411yescharacteristic
69#367—24no—
70#367—24no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75C22×Q8—41no—
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C22×C8C2411yescharacteristic
80C4⋊Q8—24no—
81C4⋊Q8—42no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C42—42no—
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C16—22no—
96C2×C16—22no—
97C2×C16—22no—
98C2×C16—22no—
99C2×C16—22no—
100C2×C16—22no—
101#107—24no—
102#107—22no—
103#107—22no—
104#107—22no—
105#107—24no—
106#107—22no—
107#107—22no—
108#107—42no—
109#107#4114yes—
110#106—42no—
111#106—42no—
112#97—22no—
113#97—22no—
114#97—41no—
115C4×C8—22no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—22no—
119#95—42no—
120#95—41no—
121#95—41no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95D812yes—
127#95D812yes—
128C2×Q8—44no—
129C2×Q8—162no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4C2×D811yescharacteristic
135C22×C4C2×D812yes—
136C22×C4C2×D812yes—
137C22×C4C2×D811yescharacteristic
138C22×C4C22×D411yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8#13712yes—
147C2×C8#13712yes—
148C2×C8C2×#4112yes—
149C2×C82−1+411yescharacteristic
150C2×C82−1+411yescharacteristic
151C2×C82−1+412yes—
152C4⋊C4—24no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—162no—
162C4⋊C4—162no—
163C4⋊C4—162no—
164C4⋊C4—162no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—162no—
172C16—42no—
173C16—42no—
174C23C22×D811yescharacteristic
175Q8—164no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—44no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
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×C4C2×#13711yescharacteristic
202C2×C4C2×#13711yescharacteristic
203C2×C4#57511yescharacteristic
204C8—22no—
205C8—22no—
206C8—41no—
207C8#54112yes—
208C8#54112yes—
209C22#310011yescharacteristic
210C22#322611yescharacteristic
211C22#322611yescharacteristic
212C22#341612yes—
213C22#341612yes—
214C4—42no—
215C4—42no—
216C4—41no—
217C4—41no—
218C4—84no—
219C4—162no—
220C4—162no—
221C4—162no—
222C4#263411yescharacteristic
223C4#311311yescharacteristic
224C4#323611yescharacteristic
225C4#323711yescharacteristic
226C2#3009111yescharacteristic
227C2#3368311yescharacteristic
228C2#3368611yescharacteristic
229C2#3381012yes—
230C2#3381012yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ 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
#332612
#217252
#135392
#135062
#134522
#131112
#131102
#128952