Finite Group Databasebeta

← #149721

Subgroups — #149721

Order 512.

304 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 68 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#22232C211yesmaximal, characteristic
2#11479C211yesmaximal, characteristic
3#11479C211yesmaximal, characteristic
4#7372C212yesmaximal, complemented
5#7359C212yesmaximal, complemented
6C2×#490C411yescharacteristic
7#2330C411yescharacteristic
8#2328C2211yescharacteristic
9#1991C2211yescharacteristic
10#1957C411yescharacteristic
11#1926—21no—
12#1925—21no—
13#1864—21no—
14#1862—21no—
15#1847C2211yescharacteristic
16#1746C411yescharacteristic
17#1459—21no—
18#1459—21no—
19#1324C2212yes—
20#1324C2212yes—
21#1318—21no—
22#1318—21no—
23C2×#127C2×C411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×C4×D4D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27#490—22no—
28#490C814yescomplemented
29C2×#107—21no—
30C2×#106—21no—
31C2×#102—21no—
32C2×#102—21no—
33C2×#102—21no—
34C2×#102—21no—
35C2×C22⋊C8—41no—
36C2×C22⋊C8—41no—
37C2×#97C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39#390D411yescharacteristic
40#390D411yescharacteristic
41#389—21no—
42#389—41no—
43#387—41no—
44#386—41no—
45#386—41no—
46#384C2×C411yescharacteristic
47#382C2311yescharacteristic
48C4×C4⋊C4—21no—
49#376C2×C411yescharacteristic
50#340—21no—
51#340—21no—
52#336—21no—
53#336—21no—
54#324—22no—
55#324—22no—
56#323—41no—
57#323—41no—
58C22×D4—41no—
59C22×D4C22⋊C411yescharacteristic
60C22×D4C22⋊C411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—81no—
66C22×C8—81no—
67#127—41no—
68#127C2×C812yes—
69#127M1612yes—
70#125—21no—
71#125—41no—
72#125C2×C812yescomplemented
73C4×D4—81no—
74C4×D4—81no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4C2×D411yescharacteristic
81C2×C4⋊C4C2×D411yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C42—41no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22×C411yescharacteristic
90#107—81no—
91#106—81no—
92#102—24no—
93#102—24no—
94#102—81no—
95#102—81no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98C22⋊C8—81no—
99C22⋊C8—81no—
100#97—21no—
101#97C2×C812yescomplemented
102C4×C8—41no—
103#95—21no—
104#95—21no—
105#95—41no—
106#95—41no—
107#95—41no—
108C24—21no—
109C24—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—41no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—161no—
119C2×D4—161no—
120C2×D4C22⋊C812yes—
121C2×D4C22⋊C812yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—161no—
133C22×C4—161no—
134C22×C4#9911yescharacteristic
135C22×C4#9911yescharacteristic
136C22×C4C2×C22⋊C411yescharacteristic
137C2×C8—22no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—82no—
144C2×C8—82no—
145C2×C8—162no—
146C2×C8—162no—
147C4⋊C4—22no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—161no—
156C4⋊C4—161no—
157C22⋊C4—81no—
158C22⋊C4—81no—
159C22⋊C4—161no—
160C22⋊C4—161no—
161C42—21no—
162C42—41no—
163C42—41no—
164C42—161no—
165C42C22×C811yescharacteristic
166C42C2×M1611yescharacteristic
167C23—22no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—81no—
174C23—81no—
175C23—81no—
176C23—81no—
177C23C2×#9911yescharacteristic
178D4—42no—
179D4—42no—
180D4—81no—
181D4—81no—
182D4—162no—
183D4—162no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
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—42no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—161no—
210C2×C4—161no—
211C2×C4C2×C22⋊C811yescharacteristic
212C2×C4#41311yescharacteristic
213C8—42no—
214C8—81no—
215C8—162no—
216C8—162no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—82no—
222C22—82no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—161no—
228C22—161no—
229C22#145911yescharacteristic
230C22#212711yescharacteristic
231C22#213511yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—81no—
239C4—82no—
240C4—81no—
241C4—81no—
242C4—161no—
243C2—21no—
244C2—21no—
245C2—82no—
246C2—82no—
247C2#1148111yescharacteristic
248C2#1360511yescharacteristic
249C2#1360711yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC42C22×C8
Frattini, cosocle#382C23
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2328 ◁ 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. #7359C2semidirect
28. #490C8semidirect
72. #125C2×C8semidirect
101. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#222322
#114792
#73722
#73592