Finite Group Databasebeta

← #120545

Subgroups — #120545

Order 512.

433 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 124 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#21975C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4#7960C212yesmaximal
5#7944C212yesmaximal
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C2×#423C2211yescharacteristic
9#2283C2211yescharacteristic
10#2283C2212yes—
11#1939—22no—
12#1851—22no—
13#1843C412yescomplemented
14#1832C412yescomplemented
15#1574—24no—
16#1568—22no—
17#1289—22no—
18#1286—24no—
19#1284—24no—
20C22×M16—21no—
21C22×C42C2311yescharacteristic
22#432—22no—
23#432—22no—
24#431—22no—
25#431—22no—
26#431—22no—
27#431D414yes—
28#423—22no—
29#423—22no—
30#423—22no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—24no—
33C2×C4⋊C8C2×C412yes—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8C2×C411yescharacteristic
36#407—22no—
37C2×C22⋊C8D412yes—
38C4×M16—22no—
39C2×C4×C8D412yes—
40#383—21no—
41#383—22no—
42#382—22no—
43#382Q812yes—
44#381—22no—
45#381D412yes—
46#380—21no—
47#380—22no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4—22no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4—22no—
52#376C2×C412yes—
53#376Q814yes—
54#336—24no—
55#335—24no—
56#334—24no—
57C23×C4C2×D411yescharacteristic
58C23×C4C2×D411yescharacteristic
59C23×C4#4111yescharacteristic
60C2×M16—42no—
61C2×M16—42no—
62C2×M16—82no—
63C22×C8—22no—
64C22×C8C22⋊C412yes—
65C22×C8C22⋊C412yes—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—42no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—42no—
70C2×C42—22no—
71C2×C42—24no—
72C2×C42C22×C411yescharacteristic
73C2×C42C2×D412yes—
74C2×C42C2×D411yescharacteristic
75C2×C42C2×Q812yes—
76C2×C42C2×Q811yescharacteristic
77C2×C42#4112yes—
78C2×C42#4112yes—
79C2×C42#4111yescharacteristic
80C4⋊C8—22no—
81C4⋊C8—22no—
82C4⋊C8—24no—
83C4⋊C8—24no—
84C4⋊C8—41no—
85C4⋊C8—41no—
86C4⋊C8—42no—
87C4⋊C8—84no—
88C22⋊C8—24no—
89C22⋊C8—24no—
90C22⋊C8—42no—
91#97—42no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8—24no—
95#95—22no—
96#95—22no—
97#95—22no—
98#95—24no—
99#95—24no—
100#95—42no—
101#95—42no—
102#95C4⋊C414yes—
103C24#12111yescharacteristic
104C22×C4—24no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—42no—
108C22×C4—44no—
109C22×C4—81no—
110C22×C4C2×C22⋊C411yescharacteristic
111C22×C4C2×C4⋊C411yescharacteristic
112C22×C4#11711yescharacteristic
113C22×C4#12112yes—
114C22×C4#12112yes—
115C22×C4#12111yescharacteristic
116C22×C4C22⋊Q811yescharacteristic
117C22×C4C22⋊Q812yes—
118C22×C4C22⋊Q812yes—
119C22×C4C22⋊Q812yes—
120C22×C4C22⋊Q811yescharacteristic
121C22×C4#12612yes—
122M16—84no—
123M16—84no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—44no—
129C2×C8—42no—
130C2×C8—44no—
131C4⋊C4—44no—
132C4⋊C4—82no—
133C4⋊C4—82no—
134C22⋊C4—82no—
135C22⋊C4—84no—
136C42—42no—
137C42—42no—
138C42—84no—
139C42C4×D412yes—
140C42C4×Q812yes—
141C42#12011yescharacteristic
142C42#12311yescharacteristic
143C42#12312yes—
144C42#12411yescharacteristic
145C42#12512yes—
146C42#12512yes—
147C42C4⋊Q812yes—
148C42C4⋊Q811yescharacteristic
149C23—22no—
150C23—41no—
151C23—41no—
152C23#38011yescharacteristic
153C23#39612yes—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—84no—
172C2×C4—84no—
173C2×C4—84no—
174C2×C4#38611yescharacteristic
175C2×C4#38712yes—
176C2×C4#38912yes—
177C2×C4#39111yescharacteristic
178C2×C4#39411yescharacteristic
179C2×C4#39511yescharacteristic
180C2×C4#39812yes—
181C2×C4#39811yescharacteristic
182C2×C4#40011yescharacteristic
183C8—42no—
184C8—42no—
185C8—84no—
186C8—82no—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—82no—
193C22#144111yescharacteristic
194C22#181412yes—
195C4—21no—
196C4—22no—
197C4—21no—
198C4—22no—
199C4—21no—
200C4—22no—
201C4—21no—
202C4—21no—
203C4—84no—
204C4—164no—
205C4#190911yescharacteristic
206C4#191011yescharacteristic
207C2—22no—
208C2—42no—
209C2#1122511yescharacteristic
210C2#1123411yescharacteristic
211C2#1124011yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1843C4semidirect
14. #1832C4semidirect

Maximal subgroups

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

subgroupindex
#219752
#178192
#79802
#79602
#79442