Finite Group Databasebeta

← #507940

Subgroups — #507940

Order 512.

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

normal subgroups only

This group has 475 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#21281C212yesmaximal
2#13515C211yesmaximal, characteristic
3#13477C212yesmaximal
4#13450C211yesmaximal, characteristic
5#13133C212yesmaximal
6#13111C212yesmaximal
7#13015C212yesmaximal
8#12938C212yesmaximal
9#12880C211yesmaximal, characteristic
10C22×Q32C2211yescharacteristic
11C2×C4×Q16C2212yes—
12#2320—22no—
13C2×#367C2212yes—
14C2×#366C2211yescharacteristic
15C2×C4⋊C16C2211yescharacteristic
16C2×C4⋊C16C2211yescharacteristic
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×#358C2212yes—
20#1969C2212yes—
21#1969C2212yes—
22#1942C2212yes—
23#1942C2212yes—
24#1935—22no—
25#1935—22no—
26C4×#107C2212yes—
27#1763—22no—
28#1726C2211yescharacteristic
29#1380C2211yescharacteristic
30#1380C2212yes—
31#1380C2212yes—
32#1380C2211yescharacteristic
33#1374C2212yes—
34#1374C2212yes—
35C22×Q16C2312yes—
36C2×C4×Q8—22no—
37C2×Q32—24no—
38C2×Q32—41no—
39C22×C16C2311yescharacteristic
40C22×C16C2311yescharacteristic
41C22×C16C2312yes—
42C4×Q16—24no—
43C4×Q16—24no—
44C2×#107C2312yes—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#103—22no—
48C2×#103—22no—
49C2×#103—22no—
50C2×#97D411yescharacteristic
51C2×#97D412yes—
52C2×C4×C8C2312yes—
53C2×C4×C8C2311yescharacteristic
54#391—22no—
55#391—22no—
56#389—22no—
57#389—22no—
58#387—42no—
59#387—42no—
60#383—42no—
61#382—42no—
62C4×C4⋊C4—22no—
63#376D411yescharacteristic
64#367—24no—
65#366—41no—
66C4⋊C16—41no—
67C4⋊C16—41no—
68#358—24no—
69#358—24no—
70#358—42no—
71#340—22no—
72#340—22no—
73#340—22no—
74C22×Q8—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16#4114yes—
81C22×C8C2×D412yes—
82C22×C8C2×D411yescharacteristic
83C22×C8C2411yescharacteristic
84C4×Q8—28no—
85C4×Q8—44no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C42—42no—
93C2×C42—42no—
94C2×C42C2×D412yes—
95C2×C42C2×D411yescharacteristic
96Q32—44no—
97C2×C16—22no—
98C2×C16—22no—
99C2×C16—22no—
100C2×C16—22no—
101C2×C16—22no—
102C2×C16—21no—
103C2×C16—21no—
104#107—22no—
105#107—22no—
106#107—22no—
107#107—22no—
108#107—22no—
109#107#4114yes—
110#103—28no—
111#103—44no—
112#103—44no—
113#97—41no—
114#97—42no—
115C4×C8—22no—
116C4×C8—22no—
117C4×C8—21no—
118C4×C8—21no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95D811yescharacteristic
127#95D812yes—
128#95D811yescharacteristic
129C2×Q8—24no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C2×D811yescharacteristic
140C22×C4C2×D812yes—
141C22×C4C2×D812yes—
142C22×C4C2×D811yescharacteristic
143C22×C4C22×D411yescharacteristic
144Q16—24no—
145Q16—24no—
146Q16—42no—
147Q16—42no—
148C2×C8—22no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8C2×#4112yes—
154C2×C8C2×#4112yes—
155C2×C8H(5,𝔽2)11yescharacteristic
156C2×C82−1+411yescharacteristic
157C4⋊C4—24no—
158C4⋊C4—48no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—162no—
165C4⋊C4—162no—
166C42—22no—
167C42—21no—
168C42—21no—
169C42—22no—
170C42—48no—
171C42—162no—
172C16—41no—
173C16—41no—
174C16—42no—
175C23C22×D811yescharacteristic
176Q8—44no—
177Q8—44no—
178Q8—82no—
179Q8—82no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4#52512yes—
204C2×C4#53711yescharacteristic
205C2×C4#57511yescharacteristic
206C2×C4#57512yes—
207C8—42no—
208C8—41no—
209C8#55111yescharacteristic
210C8#55212yes—
211C8#55511yescharacteristic
212C22#310012yes—
213C22#324511yescharacteristic
214C22#341612yes—
215C22#341711yescharacteristic
216C22#341711yescharacteristic
217C4—41no—
218C4—41no—
219C4—42no—
220C4—42no—
221C4—84no—
222C4—84no—
223C4—162no—
224C4—162no—
225C4#260911yescharacteristic
226C4#311112yes—
227C4#325611yescharacteristic
228C2#2953211yescharacteristic
229C2#3368012yes—
230C2#3368112yes—
231C2#3382711yescharacteristic
232C2#3382911yescharacteristic
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
#212812
#135152
#134772
#134502
#131332
#131112
#130152
#129382
#128802