Finite Group Databasebeta

← #99538

Subgroups — #99538

Order 512.

450 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 104 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#23369C211yesmaximal, characteristic, complemented
2C4×#490C211yesmaximal, characteristic
3#9642C212yesmaximal
4#7568C211yesmaximal, characteristic, complemented
5#7030C212yesmaximal, complemented
6C2×#490C411yescharacteristic
7C4×#127C2211yescharacteristic
8C4×#125C2211yescharacteristic
9#1992C411yescharacteristic
10#1961C411yescharacteristic
11#1939—22no—
12#1936—22no—
13#1916—22no—
14#1916—22no—
15#1836C411yescharacteristic
16#1831C2212yes—
17C4×#102—22no—
18#1744C2212yes—
19C4×#97C2211yescharacteristic
20C2×D4⋊C8—22no—
21C2×D4⋊C8—22no—
22#1271—22no—
23C2×#127C2×C411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×C4×D4—22no—
26C2×C4×D4—21no—
27#490—22no—
28#490C814yescomplemented
29C2×C4⋊C8—22no—
30C2×C4⋊C8D412yes—
31C2×C4⋊C8D412yes—
32C2×#102—22no—
33C2×#102—22no—
34C2×#97—21no—
35C2×#97C2×C411yescharacteristic
36C2×#97C2×C411yescharacteristic
37C2×C4×C8—22no—
38C2×C4×C8—21no—
39C2×C4×C8—22no—
40#390—21no—
41#390—22no—
42#389D412yes—
43#387—22no—
44#384C2×C411yescharacteristic
45#382C2×C411yescharacteristic
46C4×C4⋊C4—22no—
47C4×C4⋊C4D412yes—
48C43C2311yescharacteristic
49#340—22no—
50#340—22no—
51#336—22no—
52#336—22no—
53#336—22no—
54#336—22no—
55#336—22no—
56#336—22no—
57D4⋊C8—44no—
58D4⋊C8—44no—
59C22×D4—21no—
60C22×D4—22no—
61C23×C4—41no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—42no—
67#127—21no—
68#127—22no—
69#127C2×C812yes—
70#125—22no—
71#125C2×C812yes—
72#125M1612yes—
73C4×D4—42no—
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78C4×D4—42no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4C22⋊C412yes—
84C2×C4⋊C4C22⋊C412yes—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—41no—
87C2×C42—22no—
88C2×C42C22×C411yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92C2×C42#4112yes—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—82no—
96#102—28no—
97#102—44no—
98#97—21no—
99#97—22no—
100#97—42no—
101#97C2×C812yes—
102C4×C8—24no—
103C4×C8—41no—
104C4×C8—41no—
105C4×C8—42no—
106C4×C8—82no—
107#95—22no—
108#95—22no—
109#95—22no—
110C24—41no—
111C2×D4—22no—
112C2×D4—24no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—41no—
120C2×D4—42no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—41no—
124C22×C4—82no—
125C22×C4—82no—
126C22×C4—81no—
127C22×C4—81no—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C4×D412yes—
130C22×C4#12011yescharacteristic
131C22×C4#12112yes—
132C22×C4#12311yescharacteristic
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—44no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—82no—
147C2×C8—82no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—82no—
156C4⋊C4C22⋊C814yes—
157C22⋊C4—82no—
158C22⋊C4—82no—
159C22⋊C4—82no—
160C42—22no—
161C42—22no—
162C42—21no—
163C42—21no—
164C42—41no—
165C42—42no—
166C42—42no—
167C42—42no—
168C42C22×C811yescharacteristic
169C42C2×M1611yescharacteristic
170C42#13112yes—
171C23—42no—
172C23—82no—
173C23—82no—
174C23—81no—
175C23—81no—
176C23#38611yescharacteristic
177D4—44no—
178D4—44no—
179D4—44no—
180D4—82no—
181D4—82no—
182D4—82no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—81no—
200C2×C4—161no—
201C2×C4—162no—
202C2×C4—161no—
203C2×C4C2×C22⋊C811yescharacteristic
204C2×C4#40811yescharacteristic
205C2×C4C8×D412yes—
206C2×C4#43512yes—
207C2×C4#45312yes—
208C2×C4#45411yescharacteristic
209C2×C4#45411yescharacteristic
210C2×C4#45412yes—
211C8—42no—
212C8—42no—
213C8—84no—
214C8—82no—
215C8—82no—
216C22—82no—
217C22—84no—
218C22—162no—
219C22—161no—
220C22—161no—
221C22#186911yescharacteristic
222C22#188112yes—
223C22#188312yes—
224C22#188411yescharacteristic
225C22#188811yescharacteristic
226C4—21no—
227C4—21no—
228C4—22no—
229C4—22no—
230C4—42no—
231C4—41no—
232C4—44no—
233C4—82no—
234C4—161no—
235C4#165611yescharacteristic
236C4#165712yes—
237C4#165811yescharacteristic
238C2—162no—
239C2#895412yes—
240C2#896011yescharacteristic
241C2#899712yes—
242C2#900211yescharacteristic
243C2#900811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC23#386
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23369C2semidirect
4. #7568C2semidirect
5. #7030C2semidirect
28. #490C8semidirect

Maximal subgroups

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

subgroupindex
#233692
C4×#4902
#96422
#75682
#70302