Finite Group Databasebeta

← #150898

Subgroups — #150898

Order 512.

354 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 64 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#24315C211yesmaximal, characteristic
2#12802C211yesmaximal, characteristic
3#7571C211yesmaximal, characteristic
4#7431C214yesmaximal
5#3502C411yescharacteristic
6#3078—21no—
7C4×C4⋊Q8C2211yescharacteristic
8C2×#356—21no—
9#2060C412yes—
10#2028C2212yes—
11#1994—21no—
12#1886—21no—
13#1857—21no—
14#1801—21no—
15#1801C411yescharacteristic
16C4×C4⋊C8—21no—
17#1394C2212yes—
18#1272—21no—
19#1272C2212yes—
20Q82—41no—
21#557—41no—
22#557—42no—
23#544—41no—
24#544C2×C412yes—
25#533—21no—
26#533—41no—
27C2×C4⋊Q8D411yescharacteristic
28#519—21no—
29C2×C4×Q8C2×C411yescharacteristic
30C2×C4×Q8D411yescharacteristic
31#479—42no—
32#477—41no—
33#475—41no—
34#428—22no—
35#423—21no—
36#423—41no—
37C2×C4⋊C8—41no—
38#419—22no—
39C2×#101—21no—
40C2×#101—22no—
41C2×C4×C8—41no—
42#391D411yescharacteristic
43#391C2311yescharacteristic
44#389—41no—
45#384C2×C411yescharacteristic
46C4×C4⋊C4—41no—
47C43D411yescharacteristic
48#356—42no—
49#343—41no—
50#343C2×C412yes—
51#339—22no—
52#339—22no—
53#336—22no—
54C22×Q8C22⋊C411yescharacteristic
55C22×Q8C2×D411yescharacteristic
56C2×M16—22no—
57C2×M16—42no—
58C22×C8—42no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—41no—
62C4⋊Q8—42no—
63C4⋊Q8—41no—
64C4⋊Q8—44no—
65C4⋊Q8—41no—
66C4⋊Q8SD1612yes—
67C4⋊Q8Q1612yes—
68#125—22no—
69#125—42no—
70#125—42no—
71#125—44no—
72C22⋊Q8—41no—
73C22⋊Q8—41no—
74C22⋊Q8—42no—
75C4×Q8—41no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—41no—
81C4×Q8—42no—
82C4×Q8—42no—
83C4×Q8—42no—
84C4×Q8—42no—
85#117—22no—
86#117—42no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4C22⋊C411yescharacteristic
92C2×C4⋊C4C22×C411yescharacteristic
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42C2×D411yescharacteristic
100#107—42no—
101#106—42no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—44no—
105C4⋊C8—81no—
106C4⋊C8—81no—
107#103—42no—
108#103—42no—
109#101—42no—
110#101—42no—
111#101—42no—
112C4×C8—82no—
113#95—41no—
114#95—41no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—42no—
124C2×Q8#10312yes—
125C2×Q8C2×SD1611yescharacteristic
126C2×Q8C2×Q1611yescharacteristic
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4#9911yescharacteristic
134C22×C4#9911yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136M16—42no—
137M16—42no—
138M16—82no—
139M16—82no—
140C2×C8—42no—
141C2×C8—44no—
142C2×C8—82no—
143C2×C8—82no—
144C2×C8—82no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C4⋊C4—41no—
159C4⋊C4—42no—
160C4⋊C4—81no—
161C4⋊C4—82no—
162C22⋊C4—42no—
163C42—22no—
164C42—22no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—41no—
169C42—42no—
170C42—42no—
171C42—44no—
172C42—41no—
173C42—41no—
174C42—41no—
175C42—41no—
176C42#10312yes—
177C23C2×#9911yescharacteristic
178Q8—42no—
179Q8—41no—
180Q8—41no—
181Q8—42no—
182Q8—81no—
183Q8—82no—
184Q8—161no—
185Q8—162no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4AΣL(1,𝔽16)11yescharacteristic
212C2×C4AΣL(1,𝔽16)11yescharacteristic
213C2×C4C2×#10311yescharacteristic
214C2×C4#42111yescharacteristic
215C8—82no—
216C8—84no—
217C8—162no—
218C22—21no—
219C22—21no—
220C22#152011yescharacteristic
221C22C2×AΣL(1,𝔽16)11yescharacteristic
222C22#212911yescharacteristic
223C4—24no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—41no—
233C4—82no—
234C4—81no—
235C4—81no—
236C4—82no—
237C4#140512yes—
238C2—21no—
239C2—21no—
240C2#1189811yescharacteristic
241C2#1356411yescharacteristic
242C2C2×#140511yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#391C23
prime radicalC23C2×#99
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#243152
#128022
#75712
#74312