Finite Group Databasebeta

← #140959

Subgroups — #140959

Order 512.

365 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 82 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#33213C211yesmaximal, characteristic
2#11724C212yesmaximal
3#11723C212yesmaximal
4#10815C211yesmaximal, characteristic
5#8157C211yesmaximal, characteristic
6C2×#498C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2628C2212yes—
9#1993C412yes—
10#1943—21no—
11#1941—21no—
12#1846C411yescharacteristic
13#1809—21no—
14#1807—21no—
15#1742C2211yescharacteristic
16#1742C2211yescharacteristic
17C2×#341—21no—
18C2×#338—21no—
19#1458C2212yes—
20#1452—21no—
21#1354—24no—
22#1353—24no—
23#1281—21no—
24C22×M16—21no—
25C22×M16—21no—
26C22×M16D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C4⋊Q8C2×C412yes—
29C2×#117C2311yescharacteristic
30#498—42no—
31#431—22no—
32C2×#107—21no—
33C2×#106—21no—
34C2×#106—21no—
35C2×#106—21no—
36#405—42no—
37C2×#97C2×C411yescharacteristic
38C2×#97D411yescharacteristic
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#400—22no—
42#399—22no—
43#395—22no—
44#393—22no—
45#389—22no—
46#382C2×C412yes—
47#380—22no—
48#343—22no—
49#343—22no—
50#341—42no—
51#340—22no—
52#340—22no—
53#338—24no—
54#324D414yes—
55#323—22no—
56#322—42no—
57C22×Q8—22no—
58C23×C4C2×D411yescharacteristic
59C2×M16—21no—
60C2×M16—21no—
61C2×M16—22no—
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4C2×D412yes—
94C2×C22⋊C4—42no—
95C2×C22⋊C4#4112yes—
96C2×C42C22×C411yescharacteristic
97C2×C42C2×D411yescharacteristic
98#107—42no—
99#106—42no—
100#106—42no—
101#106—42no—
102C4⋊C8—42no—
103C22⋊C8—24no—
104#97—41no—
105#97—41no—
106C4×C8—22no—
107C4×C8—22no—
108C4×C8—42no—
109#95—22no—
110#95—22no—
111#95—42no—
112#95—42no—
113C24#12411yescharacteristic
114C2×Q8—82no—
115C2×Q8—162no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4C2×C22⋊C411yescharacteristic
123C22×C4C4×D412yes—
124C22×C4#12112yes—
125C22×C4#12711yescharacteristic
126C22×C4#13711yescharacteristic
127C22×C4#13711yescharacteristic
128C22×C4#13711yescharacteristic
129C22×C4#13711yescharacteristic
130M16—42no—
131M16—42no—
132M16—42no—
133M16—42no—
134M16—42no—
135M16—42no—
136M16—81no—
137M16—81no—
138M16—81no—
139M16—81no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—44no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—81no—
156C2×C8—81no—
157C2×C8—81no—
158C2×C8—81no—
159C2×C8#9912yes—
160C2×C8#10412yes—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4—162no—
168C22⋊C4—42no—
169C22⋊C4—162no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C23—21no—
175C23—21no—
176C23—42no—
177C23#39011yescharacteristic
178C23#49111yescharacteristic
179C23#49111yescharacteristic
180Q8—164no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4C2×#9911yescharacteristic
202C2×C4#41011yescharacteristic
203C2×C4#41911yescharacteristic
204C2×C4#41911yescharacteristic
205C2×C4C2×#10411yescharacteristic
206C2×C4#42111yescharacteristic
207C2×C4#49711yescharacteristic
208C2×C4#49711yescharacteristic
209C8—22no—
210C8—22no—
211C8—41no—
212C8—41no—
213C8—82no—
214C8—82no—
215C8—84no—
216C22—22no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—81no—
223C22#196011yescharacteristic
224C22#196411yescharacteristic
225C22#198011yescharacteristic
226C4—21no—
227C4—21no—
228C4—42no—
229C4—42no—
230C4—162no—
231C4—162no—
232C4#150611yescharacteristic
233C4#150811yescharacteristic
234C4#151111yescharacteristic
235C4#151511yescharacteristic
236C2—21no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2#968311yescharacteristic
241C2#1323212yes—
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332132
#117242
#117232
#108152
#81572