Finite Group Databasebeta

← #150039

Subgroups — #150039

Order 512.

342 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 58 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#11532C211yesmaximal, characteristic
3#7571C211yesmaximal, characteristic
4#7376C214yesmaximal
5#3502C411yescharacteristic
6#3078—21no—
7C4×C4⋊Q8C2211yescharacteristic
8#2060—21no—
9#2028—21no—
10#1994—21no—
11#1886C412yes—
12#1857C2212yes—
13#1801C411yescharacteristic
14C4×C4⋊C8—21no—
15#1467—21no—
16#1325—21no—
17#1319C2212yes—
18#1272—21no—
19#1272C2212yes—
20Q82—41no—
21#557—42no—
22#557—41no—
23#544—21no—
24#544—41no—
25#533—21no—
26#533—41no—
27C2×C4⋊Q8C2×C411yescharacteristic
28#519C2×C412yes—
29C2×C4×Q8C2×C411yescharacteristic
30C2×C4×Q8D411yescharacteristic
31#479—42no—
32#477—41no—
33#475—41no—
34#432—41no—
35#428—22no—
36#423—21no—
37C2×C4⋊C8—41no—
38#419—22no—
39C2×#101C2×C412yes—
40C2×C4×C8—41no—
41#391D411yescharacteristic
42#391C2311yescharacteristic
43#389—41no—
44#384D411yescharacteristic
45C4×C4⋊C4—41no—
46C43D411yescharacteristic
47#343—21no—
48#339—22no—
49#339—22no—
50#336—22no—
51#324—22no—
52#324—41no—
53C22×Q8C22⋊C411yescharacteristic
54C22×Q8C22×C411yescharacteristic
55C2×M16—22no—
56C22×C8—42no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—42no—
62C4⋊Q8—41no—
63C4⋊Q8—44no—
64C4⋊Q8—41no—
65#125—22no—
66#125—42no—
67#125—42no—
68#125—44no—
69C22⋊Q8—42no—
70C22⋊Q8—41no—
71C22⋊Q8—41no—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×Q8—41no—
75C4×Q8—41no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×Q8—42no—
79C4×Q8—42no—
80C4×Q8—42no—
81C4×Q8—42no—
82#117—22no—
83#117—42no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4C22⋊C411yescharacteristic
89C2×C4⋊C4C2×D411yescharacteristic
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22⋊C411yescharacteristic
96C2×C42C2×D411yescharacteristic
97#107—42no—
98#106—42no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—44no—
102C4⋊C8—81no—
103#103—42no—
104#103—42no—
105#101—24no—
106C22⋊C8—42no—
107#97—81no—
108C4×C8—82no—
109#95—41no—
110#95—41no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—42no—
120C2×Q8—41no—
121C2×Q8#13112yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4#9911yescharacteristic
129C22×C4#9911yescharacteristic
130C22×C4C2×C22⋊C411yescharacteristic
131M16—42no—
132M16—42no—
133C2×C8—42no—
134C2×C8—44no—
135C2×C8—82no—
136C2×C8—82no—
137C2×C8—82no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—44no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—41no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—41no—
152C4⋊C4—81no—
153C4⋊C4—82no—
154C22⋊C4—42no—
155C42—22no—
156C42—22no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—41no—
161C42—42no—
162C42—42no—
163C42—44no—
164C42—41no—
165C42—41no—
166C42—41no—
167C42—41no—
168C42#10112yes—
169C23C2×#9911yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—42no—
173Q8—81no—
174Q8—82no—
175Q8—161no—
176Q8—162no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4AΣL(1,𝔽16)11yescharacteristic
202C2×C4AΣL(1,𝔽16)11yescharacteristic
203C2×C4#40811yescharacteristic
204C2×C4C2×#10111yescharacteristic
205C8—82no—
206C8—84no—
207C8—162no—
208C22—21no—
209C22—21no—
210C22#147211yescharacteristic
211C22C2×AΣL(1,𝔽16)11yescharacteristic
212C22#212311yescharacteristic
213C4—24no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—82no—
224C4—82no—
225C4—81no—
226C4#140512yes—
227C2—21no—
228C2—21no—
229C2#1158011yescharacteristic
230C2#1356811yescharacteristic
231C2C2×#140511yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC22×Q8C22×C4
Frattini, cosocle#391C23
prime radicalC23C2×#99
LCSG ▷ C22×Q8 ▷ C2×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
#115322
#75712
#73762