Finite Group Databasebeta

← #141979

Subgroups — #141979

Order 512.

356 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 84 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#21023C211yesmaximal, characteristic
2#12791C211yesmaximal, characteristic
3#12607C211yesmaximal, characteristic
4#7193C212yesmaximal
5#7191C212yesmaximal
6C2×#423C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×#364—21no—
10C2×#349C2212yes—
11C2×#349C2212yes—
12C2×#347—21no—
13#1917—21no—
14#1917—21no—
15#1916—21no—
16#1916—21no—
17#1728C411yescharacteristic
18#1728C411yescharacteristic
19#1728C2211yescharacteristic
20#1728C2211yescharacteristic
21#1376—21no—
22#1375—21no—
23C22×M16C2×C411yescharacteristic
24C22×M16C2×C411yescharacteristic
25C22×M16C2×C411yescharacteristic
26C22×M16C2311yescharacteristic
27C22×C4⋊C4—21no—
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C2×M32—22no—
32C2×M32—22no—
33#423—42no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C4×M16—42no—
37C2×#97D411yescharacteristic
38C2×#97D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4—21no—
43C4×C4⋊C4—21no—
44C4×C4⋊C4—21no—
45C2×#95—21no—
46#364—42no—
47#349—22no—
48#349—24no—
49#349Q814yes—
50#347—42no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4M1611yescharacteristic
62C23×C4M1611yescharacteristic
63C23×C4C22×C411yescharacteristic
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—22no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16C2×Q812yes—
76C2×M16#4112yes—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—44no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42—41no—
95C2×C42—41no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C22⋊C411yescharacteristic
100C2×C42M1611yescharacteristic
101C2×C42M1611yescharacteristic
102M32—24no—
103M32—42no—
104M32—42no—
105M32—42no—
106C2×C16—42no—
107C2×C16—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110#97—22no—
111#97—41no—
112C4×C8—22no—
113C4×C8—41no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118C24C2×M1611yescharacteristic
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—81no—
137C22×C4—81no—
138C22×C4—81no—
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C4×Q812yes—
141C22×C4C2×M1611yescharacteristic
142C22×C4#13112yes—
143M16—42no—
144M16—42no—
145M16—44no—
146M16—82no—
147M16—82no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8#9912yes—
164C2×C8C22⋊Q812yes—
165C2×C8#12411yescharacteristic
166C2×C8C4⋊Q811yescharacteristic
167C4⋊C4—24no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—41no—
176C42—41no—
177C42—42no—
178C42—42no—
179C42—41no—
180C42—41no—
181C42—41no—
182C42—41no—
183C16—44no—
184C16—82no—
185C23—21no—
186C23—21no—
187C23—41no—
188C23—41no—
189C23#40711yescharacteristic
190C23#44612yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4#39111yescharacteristic
220C2×C4#40811yescharacteristic
221C2×C4C2×#9911yescharacteristic
222C2×C4#41011yescharacteristic
223C2×C4#41311yescharacteristic
224C2×C4#41311yescharacteristic
225C2×C4#44612yes—
226C8—22no—
227C8—22no—
228C8—21no—
229C8—21no—
230C8—41no—
231C8—81no—
232C8—81no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22#146411yescharacteristic
241C22#146511yescharacteristic
242C22#198511yescharacteristic
243C4—22no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—81no—
248C4—81no—
249C4—81no—
250C4#146811yescharacteristic
251C4#147311yescharacteristic
252C4#199611yescharacteristic
253C4#199811yescharacteristic
254C2—21no—
255C2—21no—
256C2—42no—
257C2#988111yescharacteristic
258C2#1152111yescharacteristic
259C2#1152211yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC22×M16C23
socleC22#1464
prime radicalC24C2×M16
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#210232
#127912
#126072
#71932
#71912