Finite Group Databasebeta

← #120301

Subgroups — #120301

Order 512.

456 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 126 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#20809C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic
3#8096C212yesmaximal
4#8052C212yesmaximal
5#7980C211yesmaximal, characteristic
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C2×#431C2211yescharacteristic
9#2283C2212yes—
10C4×#117C2211yescharacteristic
11#1843C412yescomplemented
12#1832C412yescomplemented
13#1798—22no—
14C4×C4⋊C8—22no—
15#1574—24no—
16#1568—22no—
17#1303—24no—
18#1297—24no—
19#1289—22no—
20C2×#117—21no—
21C22×C42C2311yescharacteristic
22#432—22no—
23#432—22no—
24#431—22no—
25#431—22no—
26#431—22no—
27#431—22no—
28#431—22no—
29#431—22no—
30#423D414yes—
31C2×C4⋊C8—24no—
32C2×C4⋊C8C2×C412yes—
33C2×C4⋊C8C2×C412yes—
34#407—22no—
35C2×C22⋊C8—21no—
36C2×C22⋊C8D412yes—
37C4×M16—22no—
38C2×C4×C8—21no—
39C2×C4×C8D412yes—
40#383Q814yes—
41#382—22no—
42#381—22no—
43#380—22no—
44C4×C4⋊C4—21no—
45C4×C4⋊C4—22no—
46C4×C4⋊C4Q812yes—
47C4×C22⋊C4—21no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4D412yes—
50#376—22no—
51#376C2×C411yescharacteristic
52#376C2×C411yescharacteristic
53C43—21no—
54#336—24no—
55#335—24no—
56#334—24no—
57C23×C4C2×D411yescharacteristic
58C23×C4C2×D411yescharacteristic
59C23×C4#4111yescharacteristic
60C2×M16—24no—
61C22×C8—22no—
62C22×C8C22⋊C414yes—
63#117—44no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—42no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—42no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—42no—
73C2×C42C22×C411yescharacteristic
74C2×C42C2×D412yes—
75C2×C42C2×D411yescharacteristic
76C2×C42C2×Q812yes—
77C2×C42C2×Q811yescharacteristic
78C2×C42#4112yes—
79C2×C42#4112yes—
80C2×C42#4111yescharacteristic
81C4⋊C8—24no—
82C4⋊C8—24no—
83C4⋊C8—24no—
84C4⋊C8—24no—
85C4⋊C8—48no—
86C22⋊C8—24no—
87C22⋊C8—42no—
88C22⋊C8—42no—
89C22⋊C8—42no—
90#97—42no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—41no—
94C4×C8—41no—
95C4×C8—42no—
96#95—21no—
97#95—21no—
98#95—24no—
99#95—24no—
100#95—24no—
101#95—42no—
102#95C4⋊C412yes—
103#95C4⋊C412yes—
104C24#12111yescharacteristic
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—44no—
111C22×C4C2×C22⋊C411yescharacteristic
112C22×C4C2×C4⋊C411yescharacteristic
113C22×C4#11711yescharacteristic
114C22×C4#12112yes—
115C22×C4#12112yes—
116C22×C4#12111yescharacteristic
117C22×C4C22⋊Q811yescharacteristic
118C22×C4C22⋊Q812yes—
119C22×C4C22⋊Q812yes—
120C22×C4C22⋊Q812yes—
121C22×C4C22⋊Q811yescharacteristic
122C22×C4#12612yes—
123M16—84no—
124C2×C8—24no—
125C2×C8—44no—
126C2×C8—44no—
127C2×C8—42no—
128C2×C8—44no—
129C4⋊C4—44no—
130C4⋊C4—84no—
131C22⋊C4—44no—
132C22⋊C4—84no—
133C42—24no—
134C42—44no—
135C42—44no—
136C42—42no—
137C42—84no—
138C42C4×D412yes—
139C42C4×Q812yes—
140C42#12011yescharacteristic
141C42#12311yescharacteristic
142C42#12312yes—
143C42#12411yescharacteristic
144C42#12512yes—
145C42#12512yes—
146C42C4⋊Q812yes—
147C42C4⋊Q811yescharacteristic
148C23—22no—
149C23—41no—
150C23—41no—
151C23#38011yescharacteristic
152C23#39612yes—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
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—44no—
184C8—84no—
185C8—82no—
186C22—21no—
187C22—21no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—82no—
192C22#144111yescharacteristic
193C22#181412yes—
194C4—21no—
195C4—22no—
196C4—22no—
197C4—21no—
198C4—22no—
199C4—21no—
200C4—21no—
201C4—44no—
202C4—82no—
203C4—164no—
204C4#188412yes—
205C4#205312yes—
206C2—22no—
207C2—42no—
208C2#1121211yescharacteristic
209C2#1123611yescharacteristic
210C2#1124011yescharacteristic
subgroupG/H
centerC2×C4#394
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
11. #1843C4semidirect
12. #1832C4semidirect

Maximal subgroups

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

subgroupindex
#208092
#178192
#80962
#80522
#79802