Finite Group Databasebeta

← #135380

Subgroups — #135380

Order 512.

366 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 94 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#20818C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic
3#10822C211yesmaximal, characteristic
4#7259C212yesmaximal
5#7259C212yesmaximal
6C22×C4×C8C2211yescharacteristic
7#2283—21no—
8C2×C22⋊C16C2212yes—
9#1846C411yescharacteristic
10#1846C411yescharacteristic
11#1835—21no—
12#1808C2211yescharacteristic
13#1808C2211yescharacteristic
14#1807—21no—
15C4×#107C411yescharacteristic
16C4×#107C411yescharacteristic
17C4×#106—21no—
18C2×#341—21no—
19C2×#341C2212yes—
20#1728—21no—
21#1374—21no—
22#1374—21no—
23C22×M16—22no—
24C23×C8D411yescharacteristic
25C23×C8C2311yescharacteristic
26C22×C42D411yescharacteristic
27C22×C16—22no—
28C2×#107C2×C411yescharacteristic
29C2×#107C2×C411yescharacteristic
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#106—21no—
33C2×#106—21no—
34C2×C4×C8—24no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#383—41no—
40#382—21no—
41#382—21no—
42#381—41no—
43#380—21no—
44#380—21no—
45C4×C4⋊C4—21no—
46C4×C4⋊C4—21no—
47C4×C22⋊C4—41no—
48#376—41no—
49C22⋊C16—24no—
50#341—24no—
51#341—24no—
52#340—21no—
53#340—21no—
54#340—21no—
55#340—21no—
56#336—41no—
57#336—41no—
58C23×C4SD1611yescharacteristic
59C23×C4SD1611yescharacteristic
60C23×C4C2×D411yescharacteristic
61C2×M16—24no—
62C2×M16—42no—
63C2×M16—42no—
64C2×M16—42no—
65C2×M16—42no—
66C2×M16—42no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8—42no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22×C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8#4114yes—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—41no—
87C2×C42—41no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42D811yescharacteristic
91C2×C42D811yescharacteristic
92C2×C16—42no—
93C2×C16—42no—
94C2×C16—42no—
95#107—41no—
96#107—41no—
97#107—41no—
98#107—41no—
99#106—42no—
100#106—42no—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—22no—
104C4×C8—22no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113C24C2×SD1611yescharacteristic
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4#10211yescharacteristic
129C22×C4#10211yescharacteristic
130C22×C4#10211yescharacteristic
131C22×C4#10211yescharacteristic
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4#12312yes—
134C22×C4C2×D811yescharacteristic
135C22×C4#13712yes—
136M16—44no—
137M16—44no—
138M16—82no—
139M16—82no—
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—44no—
150C2×C8—44no—
151C2×C8—44no—
152C2×C8—82no—
153C2×C8—82no—
154C2×C8#10412yes—
155C2×C8#11712yes—
156C2×C8#12312yes—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C22⋊C4—82no—
166C22⋊C4—82no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—22no—
172C42—22no—
173C42—82no—
174C42—82no—
175C16—82no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23C2×#10211yescharacteristic
183C23#48312yes—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4#35912yes—
216C2×C4#38111yescharacteristic
217C2×C4#41011yescharacteristic
218C2×C4#41011yescharacteristic
219C2×C4#41911yescharacteristic
220C2×C4C2×#10411yescharacteristic
221C2×C4#42111yescharacteristic
222C2×C4#48012yes—
223C8—22no—
224C8—22no—
225C8—22no—
226C8—41no—
227C8—84no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#150211yescharacteristic
236C22#151711yescharacteristic
237C22#182311yescharacteristic
238C22C2×#35911yescharacteristic
239C22#214311yescharacteristic
240C22#214811yescharacteristic
241C22#214811yescharacteristic
242C4—22no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—161no—
248C4—161no—
249C4—161no—
250C4—161no—
251C4#150611yescharacteristic
252C4#151511yescharacteristic
253C4#182011yescharacteristic
254C4#182611yescharacteristic
255C2—42no—
256C2#871611yescharacteristic
257C2#1174911yescharacteristic
258C2#1175011yescharacteristic
259C2#1186111yescharacteristic
260C2#1186611yescharacteristic
261C2#1286911yescharacteristic
262C2#1287211yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#208182
#128442
#108222
#72592