Finite Group Databasebeta

← #134263

Subgroups — #134263

Order 512.

394 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 114 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#20436C211yesmaximal, characteristic
2#11452C212yesmaximal
3#8601C212yesmaximal
4#7190C212yesmaximal
5C2×#424C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×#97C411yescharacteristic
9#2296C2211yescharacteristic
10#1771—22no—
11#1770—22no—
12#1729C2211yescharacteristic
13#1728C411yescharacteristic
14#1564—22no—
15#1458C2214yes—
16#1326—24no—
17#1275—22no—
18C22×M16C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C2×#117C2311yescharacteristic
24C22×C42C2×C411yescharacteristic
25#431—24no—
26#424—22no—
27#424—22no—
28C2×C4⋊C8—22no—
29#405—22no—
30#405—42no—
31#405C814yescomplemented
32C2×#97—21no—
33C2×#97—21no—
34C2×#97—21no—
35C2×#97—42no—
36C2×#97D412yes—
37C2×C4×C8—21no—
38C2×C4×C8D412yes—
39#389—22no—
40#385D412yes—
41#385Q812yes—
42C4×C4⋊C4—22no—
43#336—22no—
44#336—22no—
45#335—42no—
46#334—42no—
47#324—24no—
48#323—24no—
49C23×C4C4⋊C412yes—
50C23×C4C22×C411yescharacteristic
51C2×M16—21no—
52C2×M16—21no—
53C2×M16—21no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16C2×C812yes—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8C22⋊C412yes—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C2×C812yes—
70C22×C8M1612yes—
71C22×C8M1612yes—
72#117—21no—
73#117—21no—
74#117—41no—
75#117—41no—
76#117C2×C812yes—
77#117M1612yes—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4#4112yes—
82C2×C22⋊C4C4⋊C412yes—
83C2×C22⋊C4C2×D411yescharacteristic
84C2×C22⋊C4C2×Q811yescharacteristic
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—41no—
89C2×C42—41no—
90C2×C42C2×D412yes—
91C4⋊C8—28no—
92C4⋊C8—44no—
93C22⋊C8—44no—
94#97—24no—
95#97—42no—
96#97—42no—
97#97—42no—
98C4×C8—24no—
99C4×C8—22no—
100C4×C8—22no—
101C4×C8—44no—
102#95—22no—
103#95—22no—
104C24C2×C4⋊C411yescharacteristic
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—82no—
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4#11711yescharacteristic
115C22×C4#12112yes—
116C22×C4C22⋊Q812yes—
117C22×C4C22×C811yescharacteristic
118C22×C4C2×M1611yescharacteristic
119C22×C4C2×M1611yescharacteristic
120C22×C4C2×M1611yescharacteristic
121M16—42no—
122M16—42no—
123M16—42no—
124M16—42no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—24no—
128C2×C8—24no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—42no—
140C2×C8—84no—
141C2×C8C22⋊C812yes—
142C2×C8C22⋊C812yes—
143C2×C8#9912yes—
144C2×C8#10112yes—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C22⋊C4—22no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C22⋊C4C4⋊C814yes—
154C42—22no—
155C42—22no—
156C42—22no—
157C42—22no—
158C42—42no—
159C42—41no—
160C42—41no—
161C23—21no—
162C23—21no—
163C23—42no—
164C23#38011yescharacteristic
165C23C2×C4⋊C811yescharacteristic
166C23#42311yescharacteristic
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—84no—
188C2×C4C2×C22⋊C811yescharacteristic
189C2×C4#40711yescharacteristic
190C2×C4C2×#9911yescharacteristic
191C2×C4#41011yescharacteristic
192C2×C4C2×#10111yescharacteristic
193C2×C4#41311yescharacteristic
194C2×C4#43111yescharacteristic
195C2×C4#43211yescharacteristic
196C8—22no—
197C8—42no—
198C8—42no—
199C8—42no—
200C8—164no—
201C8#32414yes—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—42no—
208C22—42no—
209C22#179711yescharacteristic
210C22#179911yescharacteristic
211C22#180111yescharacteristic
212C4—21no—
213C4—21no—
214C4—44no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4C2×#32411yescharacteristic
219C4#145811yescharacteristic
220C4#146511yescharacteristic
221C4#146711yescharacteristic
222C2—21no—
223C2—21no—
224C2—41no—
225C2—41no—
226C2#855811yescharacteristic
227C2#1280612yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC24C2×C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #405C8semidirect

Maximal subgroups

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

subgroupindex
#204362
#114522
#86012
#71902