Finite Group Databasebeta

← #134262

Subgroups — #134262

Order 512.

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

normal subgroups only

This group has 408 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#20435C211yesmaximal, characteristic
2#11452C212yesmaximal
3#8601C212yesmaximal
4#7190C212yesmaximal
5C2×#424C411yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2296C2211yescharacteristic
10#1771—22no—
11C8×C4⋊C4—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—41no—
31#405—41no—
32#405C814yescomplemented
33C2×#97—21no—
34C2×#97D412yes—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—21no—
38C2×C4×C8—41no—
39C2×C4×C8—41no—
40C2×C4×C8D412yes—
41#389—22no—
42#385D412yes—
43#385Q812yes—
44C4×C4⋊C4—22no—
45#336—22no—
46#336—22no—
47#335—42no—
48#334—42no—
49#324—24no—
50#323—24no—
51C23×C4C4⋊C412yes—
52C23×C4C22×C411yescharacteristic
53C2×M16—21no—
54C2×M16—21no—
55C2×M16—21no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16C2×C812yes—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—22no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C22⋊C412yes—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C2×C812yes—
73C22×C8M1612yes—
74C22×C8M1612yes—
75#117—21no—
76#117—21no—
77#117—41no—
78#117—41no—
79#117C2×C812yes—
80#117M1612yes—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4#4112yes—
85C2×C22⋊C4C4⋊C412yes—
86C2×C22⋊C4C2×D411yescharacteristic
87C2×C22⋊C4C2×Q811yescharacteristic
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—41no—
92C2×C42—41no—
93C2×C42C2×D412yes—
94C4⋊C8—28no—
95C4⋊C8—44no—
96C22⋊C8—44no—
97#97—24no—
98#97—42no—
99C4×C8—24no—
100C4×C8—22no—
101C4×C8—22no—
102C4×C8—44no—
103C4×C8—42no—
104C4×C8—41no—
105C4×C8—41no—
106#95—22no—
107#95—22no—
108C24C2×C4⋊C411yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4#11711yescharacteristic
119C22×C4#12112yes—
120C22×C4C22⋊Q812yes—
121C22×C4C22×C811yescharacteristic
122C22×C4C2×M1611yescharacteristic
123C22×C4C2×M1611yescharacteristic
124C22×C4C2×M1611yescharacteristic
125M16—42no—
126M16—42no—
127M16—42no—
128M16—42no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—42no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—42no—
146C2×C8—84no—
147C2×C8C22⋊C812yes—
148C2×C8C22⋊C812yes—
149C2×C8#9911yescharacteristic
150C2×C8#9911yescharacteristic
151C2×C8#10112yes—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C22⋊C4—22no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C22⋊C4C4⋊C814yes—
161C42—22no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—42no—
166C42—41no—
167C42—41no—
168C23—21no—
169C23—21no—
170C23—42no—
171C23#38011yescharacteristic
172C23C2×C4⋊C811yescharacteristic
173C23#42311yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4C2×C22⋊C811yescharacteristic
197C2×C4#40711yescharacteristic
198C2×C4C2×#9911yescharacteristic
199C2×C4#41011yescharacteristic
200C2×C4C2×#10111yescharacteristic
201C2×C4#41311yescharacteristic
202C2×C4#43111yescharacteristic
203C2×C4#43211yescharacteristic
204C8—22no—
205C8—42no—
206C8—42no—
207C8—42no—
208C8—164no—
209C8#32412yes—
210C8#32412yes—
211C22—22no—
212C22—21no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—42no—
217C22—42no—
218C22#179711yescharacteristic
219C22#179911yescharacteristic
220C22#180111yescharacteristic
221C4—21no—
222C4—21no—
223C4—44no—
224C4—42no—
225C4—42no—
226C4—44no—
227C4C2×#32411yescharacteristic
228C4#145811yescharacteristic
229C4#146511yescharacteristic
230C4#146711yescharacteristic
231C2—21no—
232C2—21no—
233C2—41no—
234C2—41no—
235C2#855811yescharacteristic
236C2#1280512yes—
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
32. #405C8semidirect

Maximal subgroups

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

subgroupindex
#204352
#114522
#86012
#71902