Finite Group Databasebeta

← #125609

Subgroups — #125609

Order 512.

305 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 66 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#33059C211yesmaximal, characteristic
2#12849C211yesmaximal, characteristic
3#12791C211yesmaximal, characteristic
4#7130C212yesmaximal, complemented
5#7117C212yesmaximal, complemented
6C2×#491C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8#2857C411yescharacteristic
9#2851C411yescharacteristic
10C2×#364—21no—
11C2×#349—21no—
12C2×#349—21no—
13#1992C2211yescharacteristic
14#1960C2211yescharacteristic
15#1933—21no—
16#1933—21no—
17#1894—21no—
18#1894—21no—
19#1836C411yescharacteristic
20#1376—21no—
21#1375C2212yes—
22#1375C2212yes—
23C22×M16D411yescharacteristic
24C22×M16D411yescharacteristic
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×#125C2×C411yescharacteristic
27C2×C22⋊Q8—21no—
28C22×C42C2×C411yescharacteristic
29C2×M32—22no—
30C2×M32—22no—
31#491—42no—
32C2×#103—21no—
33C2×#103—21no—
34C2×#103—21no—
35C2×#103—21no—
36C4×M16—42no—
37C2×#97C2×C411yescharacteristic
38C2×#97C2311yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#391—21no—
42#389—21no—
43#389—41no—
44#387—41no—
45#385—41no—
46#384C2×C411yescharacteristic
47#382C2×C411yescharacteristic
48#380—21no—
49#380—41no—
50#364—42no—
51#349—42no—
52#349—42no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340—21no—
57C22×Q8—21no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—21no—
63C2×M16—21no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73#125—22no—
74#125—41no—
75C22⋊Q8—42no—
76C22⋊Q8—42no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42—41no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C22×C411yescharacteristic
92M32—42no—
93M32—42no—
94M32—42no—
95M32—42no—
96C2×C16—42no—
97C2×C16—42no—
98#103—42no—
99#103—42no—
100#103—42no—
101#103—41no—
102#103—41no—
103#97—22no—
104#97—41no—
105C4×C8—22no—
106C4×C8—41no—
107#95—41no—
108#95—42no—
109#95—41no—
110#95—41no—
111C24#10011yescharacteristic
112C2×Q8—41no—
113C2×Q8—41no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—41no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—81no—
133C22×C4#10011yescharacteristic
134C22×C4C2×C22⋊C411yescharacteristic
135M16—44no—
136M16—42no—
137M16—42no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8#9912yes—
148C2×C8#13512yes—
149C2×C8AGL(1,ℤ/8)11yescharacteristic
150C2×C8#13711yescharacteristic
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C22⋊C4—82no—
163C22⋊C4—82no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—42no—
168C42#13112yes—
169C16—82no—
170C16—82no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23C2×#10011yescharacteristic
177Q8—81no—
178Q8—81no—
179Q8—81no—
180Q8—81no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—84no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4#40811yescharacteristic
207C2×C4C2×#9911yescharacteristic
208C2×C4#41611yescharacteristic
209C2×C4#41711yescharacteristic
210C2×C4#42111yescharacteristic
211C2×C4#42111yescharacteristic
212C8—21no—
213C8—21no—
214C8—22no—
215C8—22no—
216C8—41no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22#147111yescharacteristic
225C22#153311yescharacteristic
226C22#155111yescharacteristic
227C4—42no—
228C4—42no—
229C4—41no—
230C4—161no—
231C4—161no—
232C4—161no—
233C4—161no—
234C4#148211yescharacteristic
235C4#149511yescharacteristic
236C4#151311yescharacteristic
237C4#151611yescharacteristic
238C2—21no—
239C2—21no—
240C2—81no—
241C2#773011yescharacteristic
242C2#1191611yescharacteristic
243C2#1191711yescharacteristic
subgroupG/H
center, socleC22#1533
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7130C2semidirect
5. #7117C2semidirect

Maximal subgroups

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

subgroupindex
#330592
#128492
#127912
#71302
#71172