Finite Group Databasebeta

← #135589

Subgroups — #135589

Order 512.

252 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 58 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#21915C211yesmaximal, characteristic
2#8743C211yesmaximal, characteristic
3#8742C211yesmaximal, characteristic
4#7327C211yesmaximal, characteristic
5#7327C211yesmaximal, characteristic
6#7325C211yesmaximal, characteristic
7#7325C211yesmaximal, characteristic
8#2296C2211yescharacteristic
9#1917—21no—
10#1917—21no—
11#1916—21no—
12#1916—21no—
13#1818C2211yescharacteristic
14#1818C2211yescharacteristic
15#1798C411yescharacteristic
16#1798C411yescharacteristic
17#1749C411yescharacteristic
18C4×C22⋊C8C411yescharacteristic
19#1464C2211yescharacteristic
20#1464C2211yescharacteristic
21#1464C2211yescharacteristic
22#1464C2211yescharacteristic
23#1285—21no—
24#1285—21no—
25#1285—21no—
26#1285—21no—
27C2×#117C2×C411yescharacteristic
28C2×#117C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C22⋊C8C2×C411yescharacteristic
37C2×C22⋊C8C2×C411yescharacteristic
38C2×C22⋊C8C2×C411yescharacteristic
39C2×C22⋊C8C2×C411yescharacteristic
40C2×#97—21no—
41C2×C4×C8—21no—
42#389—21no—
43#389—21no—
44#385—21no—
45#385—21no—
46C4×C4⋊C4D411yescharacteristic
47C4×C4⋊C4D411yescharacteristic
48C4×C4⋊C4D411yescharacteristic
49C4×C4⋊C4D411yescharacteristic
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#324—21no—
59#324—21no—
60#324—21no—
61#324—21no—
62#323—21no—
63#323—21no—
64#323—21no—
65#323—21no—
66C23×C4—21no—
67C23×C4C22×C411yescharacteristic
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72#117—41no—
73#117—41no—
74#117—81no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4#4111yescharacteristic
81C2×C4⋊C4#4111yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4#4111yescharacteristic
84C2×C22⋊C4#4111yescharacteristic
85C2×C42—21no—
86C2×C42—81no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22⋊C411yescharacteristic
91C2×C42C2×D411yescharacteristic
92C2×C42C2×D411yescharacteristic
93C4⋊C8—41no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—81no—
98C4⋊C8—81no—
99C22⋊C8—41no—
100C22⋊C8—41no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—41no—
106C22⋊C8—41no—
107#97—41no—
108#97—41no—
109#97—41no—
110#97—41no—
111#97—81no—
112C4×C8—81no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117C24#11711yescharacteristic
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—81no—
128C22×C4—81no—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4#11711yescharacteristic
131C22×C4#12311yescharacteristic
132C22×C4#12311yescharacteristic
133C22×C4#12311yescharacteristic
134C22×C4#12311yescharacteristic
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C22⋊C4—81no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—41no—
166C42—41no—
167C42—41no—
168C42—41no—
169C42—81no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23#38111yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—84no—
204C2×C4#41011yescharacteristic
205C2×C4#41011yescharacteristic
206C8—81no—
207C8—81no—
208C8—81no—
209C8—81no—
210C8—161no—
211C8—161no—
212C8—161no—
213C8—161no—
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—81no—
221C22#182011yescharacteristic
222C22#182911yescharacteristic
223C22#183011yescharacteristic
224C4—21no—
225C4—21no—
226C4—42no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—81no—
234C4—81no—
235C2—21no—
236C2—21no—
237C2—81no—
238C2#875511yescharacteristic
239C2#1286411yescharacteristic
240C2#1286511yescharacteristic
subgroupG/H
center, socleC22#1820
commutatorC23×C4C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#117
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2296 ◁ 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
#219152
#87432
#87422
#73272
#73252