Finite Group Databasebeta

← #149636

Subgroups — #149636

Order 512.

280 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 74 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#21991C211yesmaximal, characteristic
2#11449C211yesmaximal, characteristic
3#11448C211yesmaximal, characteristic
4#7372C212yesmaximal, complemented
5#7366C212yesmaximal, complemented
6C2×#433C411yescharacteristic
7#2317C2211yescharacteristic
8#2314C411yescharacteristic
9#1949—21no—
10#1947—21no—
11#1854—21no—
12#1852—21no—
13#1847C2211yescharacteristic
14#1845C411yescharacteristic
15#1836C411yescharacteristic
16#1835C2211yescharacteristic
17#1458—21no—
18C2×#324—21no—
19#1325—21no—
20#1324—21no—
21#1324C2212yes—
22#1324C2212yes—
23C2×#117C2×C411yescharacteristic
24C2×#117D411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26#433—22no—
27#433C814yescomplemented
28#431—41no—
29C2×#107—21no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#106—21no—
33C2×C4⋊C8—21no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—41no—
36C2×#97C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38#389—41no—
39#388—41no—
40#387—41no—
41#385—41no—
42#384C2×C411yescharacteristic
43#384C2×C411yescharacteristic
44#384D411yescharacteristic
45#382C2×C411yescharacteristic
46#382D411yescharacteristic
47#382C2311yescharacteristic
48#380—21no—
49C4×C4⋊C4—21no—
50#340—21no—
51#340—21no—
52#340—21no—
53#340—21no—
54#324—22no—
55#324—22no—
56#324—41no—
57#323—41no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—81no—
63#117—22no—
64#117—81no—
65#117C2×C812yes—
66#117M1612yes—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—81no—
75C2×C4⋊C4C22⋊C411yescharacteristic
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—41no—
82C2×C42—21no—
83C2×C42—41no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22×C411yescharacteristic
86#107—81no—
87#107—81no—
88#107—81no—
89#106—81no—
90C4⋊C8—24no—
91C4⋊C8—81no—
92C22⋊C8—24no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C22⋊C8—81no—
96#97—21no—
97#97C2×C812yescomplemented
98C4×C8—21no—
99C4×C8—81no—
100C4×C8C2×C812yescomplemented
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106C24#9911yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—161no—
120C22×C4C22⋊C812yes—
121C22×C4#9911yescharacteristic
122C22×C4#10112yes—
123C22×C4C2×C22⋊C411yescharacteristic
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—82no—
131C2×C8—82no—
132C2×C8—162no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—81no—
143C4⋊C4—161no—
144C4⋊C4—161no—
145C4⋊C4—161no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—81no—
149C22⋊C4—161no—
150C42—21no—
151C42—21no—
152C42—41no—
153C42—41no—
154C42—161no—
155C42C22⋊C812yes—
156C42C22×C811yescharacteristic
157C42C2×M1611yescharacteristic
158C23—21no—
159C23—21no—
160C23—41no—
161C23—41no—
162C23#32412yes—
163C23C2×#9911yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—82no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—161no—
193C2×C4#32412yes—
194C2×C4C2×C22⋊C811yescharacteristic
195C2×C4C2×#10111yescharacteristic
196C8—42no—
197C8—42no—
198C8—162no—
199C8—162no—
200C22—21no—
201C22—21no—
202C22—22no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—81no—
207C22C2×#32411yescharacteristic
208C22#212711yescharacteristic
209C22#213311yescharacteristic
210C4—21no—
211C4—22no—
212C4—21no—
213C4—42no—
214C4—41no—
215C4—41no—
216C4—81no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4—161no—
221C4—161no—
222C2—21no—
223C2—21no—
224C2—42no—
225C2#1144811yescharacteristic
226C2#1360511yescharacteristic
227C2#1360811yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC42C22×C8
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2317 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7372C2semidirect
5. #7366C2semidirect
27. #433C8semidirect
97. #97C2×C8semidirect
100. C4×C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#219912
#114492
#114482
#73722
#73662