Finite Group Databasebeta

← #140960

Subgroups — #140960

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#33215C211yesmaximal, characteristic
2#11724C212yesmaximal
3#11723C212yesmaximal
4#10815C211yesmaximal, characteristic
5#8157C211yesmaximal, characteristic
6C2×#500C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2628C2212yes—
9#1993C412yes—
10#1944—21no—
11#1942—21no—
12#1846C411yescharacteristic
13#1809—21no—
14#1808—21no—
15#1742C2212yes—
16C2×#341—21no—
17C2×#338—21no—
18#1458C2211yescharacteristic
19#1458C2211yescharacteristic
20#1452—21no—
21#1354—24no—
22#1353—24no—
23#1281—21no—
24C22×M16—22no—
25C22×M16D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8C2×C412yes—
28C2×#117C2311yescharacteristic
29#500—42no—
30#431—21no—
31#431—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—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#324D412yes—
55#324D412yes—
56#323—21no—
57#323—21no—
58#322—42no—
59C22×Q8—22no—
60C23×C4C2×D411yescharacteristic
61C2×M16—22no—
62C2×M16—22no—
63C2×M16—24no—
64C2×M16—41no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—41no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80#117—41no—
81#117—41no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4C2×D411yescharacteristic
87C2×C4⋊C4C2×D411yescharacteristic
88C2×C22⋊C4—42no—
89C2×C22⋊C4#4111yescharacteristic
90C2×C22⋊C4#4111yescharacteristic
91C2×C42C22×C411yescharacteristic
92C2×C42C2×D411yescharacteristic
93#107—42no—
94#107—42no—
95#107—42no—
96#106—42no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C22⋊C8—22no—
100C22⋊C8—22no—
101#97—41no—
102#97—41no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—41no—
106C4×C8—41no—
107#95—22no—
108#95—22no—
109#95—42no—
110#95—42no—
111C24#12411yescharacteristic
112C2×Q8—82no—
113C2×Q8—162no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C4×D411yescharacteristic
122C22×C4C4×D411yescharacteristic
123C22×C4#12111yescharacteristic
124C22×C4#12111yescharacteristic
125C22×C4#12711yescharacteristic
126C22×C4#13712yes—
127C22×C4#13712yes—
128M16—44no—
129M16—44no—
130M16—44no—
131M16—82no—
132M16—82no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—44no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—44no—
147C2×C8—82no—
148C2×C8—82no—
149C2×C8#9912yes—
150C2×C8#10412yes—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—82no—
158C4⋊C4—162no—
159C22⋊C4—41no—
160C22⋊C4—41no—
161C22⋊C4—162no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23#39011yescharacteristic
171C23#49112yes—
172Q8—164no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4C2×#9911yescharacteristic
194C2×C4#41011yescharacteristic
195C2×C4#41911yescharacteristic
196C2×C4#41911yescharacteristic
197C2×C4C2×#10411yescharacteristic
198C2×C4#42111yescharacteristic
199C2×C4#49712yes—
200C8—22no—
201C8—22no—
202C8—41no—
203C8—41no—
204C8—84no—
205C8—82no—
206C8—82no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—41no—
212C22—41no—
213C22—81no—
214C22#196011yescharacteristic
215C22#196411yescharacteristic
216C22#198011yescharacteristic
217C4—22no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—162no—
223C4—162no—
224C4#150611yescharacteristic
225C4#150811yescharacteristic
226C4#151111yescharacteristic
227C4#151511yescharacteristic
228C2—21no—
229C2—21no—
230C2—42no—
231C2#968311yescharacteristic
232C2#1323311yescharacteristic
233C2#1323411yescharacteristic
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
#332152
#117242
#117232
#108152
#81572