Finite Group Databasebeta

← #120583

Subgroups — #120583

Order 512.

538 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 136 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#21353C211yesmaximal, characteristic
2#20451C211yesmaximal, characteristic
3#8092C212yesmaximal
4#7941C212yesmaximal
5#7861C211yesmaximal, characteristic
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C2×#427—21no—
9C22×#97C2211yescharacteristic
10#2283C2211yescharacteristic
11C4×#117C2212yes—
12#1940C412yescomplemented
13#1940C412yescomplemented
14#1850—21no—
15#1850—21no—
16#1776—24no—
17#1768—22no—
18#1735—21no—
19#1303—24no—
20#1284—24no—
21#1275—22no—
22C23×C8D411yescharacteristic
23C23×C8D411yescharacteristic
24C2×#117—22no—
25C22×C42C2311yescharacteristic
26#432—22no—
27#432—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31#431Q814yes—
32#427—28no—
33C2×#107—22no—
34C2×#106—22no—
35#423—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8C2×C411yescharacteristic
40#407—22no—
41C2×C22⋊C8—21no—
42C4×M16—22no—
43C2×#97—22no—
44C2×#97—22no—
45C2×#97C2×C412yes—
46C2×#97D412yes—
47C2×C4×C8—21no—
48#383Q812yes—
49#382—21no—
50#381D412yes—
51#380—21no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4C2×C412yes—
54C4×C4⋊C4D414yes—
55C4×C22⋊C4—21no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58#376—21no—
59#376—22no—
60C43—22no—
61#340—24no—
62#336—22no—
63#336—22no—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4#4111yescharacteristic
67C2×M16—42no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—24no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—42no—
76C22×C8C22⋊C412yes—
77C22×C8C22⋊C412yes—
78#117—28no—
79#117—44no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—42no—
84C2×C42—22no—
85C2×C42—24no—
86C2×C42—42no—
87C2×C42—42no—
88C2×C42C22×C411yescharacteristic
89C2×C42C2×D412yes—
90C2×C42C2×D411yescharacteristic
91C2×C42C2×Q812yes—
92C2×C42C2×Q811yescharacteristic
93C2×C42#4112yes—
94C2×C42#4112yes—
95C2×C42#4111yescharacteristic
96#107—28no—
97#106—28no—
98C4⋊C8—22no—
99C4⋊C8—22no—
100C4⋊C8—24no—
101C4⋊C8—44no—
102C4⋊C8—44no—
103C4⋊C8—42no—
104C22⋊C8—24no—
105C22⋊C8—41no—
106C22⋊C8—42no—
107C22⋊C8—41no—
108#97—24no—
109#97—24no—
110#97—42no—
111C4×C8—24no—
112C4×C8—42no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95—24no—
117#95—42no—
118#95C4⋊C414yes—
119C24#12111yescharacteristic
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—42no—
124C22×C4—44no—
125C22×C4—81no—
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4C2×C4⋊C411yescharacteristic
128C22×C4#11711yescharacteristic
129C22×C4#12011yescharacteristic
130C22×C4#12112yes—
131C22×C4#12111yescharacteristic
132C22×C4C22⋊Q811yescharacteristic
133C22×C4C22⋊Q812yes—
134C22×C4C22⋊Q811yescharacteristic
135C22×C4#12311yescharacteristic
136C22×C4#12312yes—
137C22×C4#12411yescharacteristic
138C22×C4#12512yes—
139C22×C4C4⋊Q811yescharacteristic
140M16—84no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—24no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—44no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8#10414yes—
157C4⋊C4—28no—
158C4⋊C4—44no—
159C4⋊C4—82no—
160C22⋊C4—44no—
161C22⋊C4—48no—
162C42—48no—
163C42—42no—
164C42—42no—
165C42—84no—
166C42—84no—
167C42C4×D412yes—
168C42C4×Q812yes—
169C42#12112yes—
170C42C22⋊Q812yes—
171C42C22⋊Q812yes—
172C42#12512yes—
173C42#12612yes—
174C42C4⋊Q812yes—
175C23—22no—
176C23—22no—
177C23—41no—
178C23#38011yescharacteristic
179C23#39411yescharacteristic
180C23#39811yescharacteristic
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—48no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—84no—
201C2×C4—84no—
202C2×C4#38611yescharacteristic
203C2×C4#38712yes—
204C2×C4#38912yes—
205C2×C4#39111yescharacteristic
206C2×C4#39511yescharacteristic
207C2×C4#39612yes—
208C2×C4#39812yes—
209C2×C4#40011yescharacteristic
210C2×C4C2×#10412yes—
211C2×C4#42112yes—
212C8—24no—
213C8—42no—
214C8—84no—
215C8—81no—
216C8—81no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—44no—
225C22#144111yescharacteristic
226C22#180612yes—
227C22#205312yes—
228C4—21no—
229C4—21no—
230C4—22no—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—84no—
235C4—88no—
236C4#189611yescharacteristic
237C4#189711yescharacteristic
238C4#198612yes—
239C2—22no—
240C2—22no—
241C2—41no—
242C2#1122611yescharacteristic
243C2#1123611yescharacteristic
244C2#1124511yescharacteristic
subgroupG/H
centerC2×C4#391
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1441
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1940C4semidirect
13. #1940C4semidirect

Maximal subgroups

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

subgroupindex
#213532
#204512
#80922
#79412
#78612