Finite Group Databasebeta

← #128385

Subgroups — #128385

Order 512.

450 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C2×#1771C211yesmaximal, characteristic
2#12605C212yesmaximal
3#11440C212yesmaximal
4#7184C212yesmaximal
5C22×C4⋊C8C2211yescharacteristic
6C22×#97C411yescharacteristic
7C22×#97C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic
10#2112C2214yes—
11C2×#347—22no—
12#1771—22no—
13#1771—22no—
14C2×#336C411yescharacteristic
15C2×#336C2211yescharacteristic
16#1375—22no—
17#1322C418yescomplemented
18C23×C8C2×C411yescharacteristic
19C23×C8C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2311yescharacteristic
22C22×C4⋊C4—21no—
23C22×C42C2×C411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C22×C42C2×C411yescharacteristic
26C2×M32—24no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8—22no—
29C2×C4⋊C8C2×C414yes—
30C2×#97—21no—
31C2×#97—21no—
32C2×#97—22no—
33C2×#97—22no—
34C2×#97—42no—
35C2×#97D412yes—
36C2×#97D412yes—
37C2×C4×C8—21no—
38C2×C4×C8—21no—
39C2×C4×C8—42no—
40C4×C4⋊C4—22no—
41C4×C4⋊C4—22no—
42C2×#95—21no—
43C22⋊C16—24no—
44C22⋊C16C2×C418yes—
45#347—44no—
46#336—22no—
47#336—22no—
48C23×C4—21no—
49C23×C4—21no—
50C23×C4M1611yescharacteristic
51C23×C4M1611yescharacteristic
52C23×C4C22×C411yescharacteristic
53C22×C8—21no—
54C22×C8—21no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C4212yes—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8M1612yes—
71C22×C8C22×C412yes—
72C22×C8C2×D412yes—
73C22×C8#4112yes—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—44no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—41no—
83C2×C42—41no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42C4212yes—
87C2×C42C22⋊C412yes—
88C2×C42M1612yes—
89C2×C42M1612yes—
90M32—44no—
91M32—44no—
92C2×C16—28no—
93C4⋊C8—28no—
94#97—24no—
95#97—24no—
96#97—24no—
97#97—42no—
98C4×C8—42no—
99C4×C8—42no—
100#95—24no—
101C24C2×M1611yescharacteristic
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—24no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4#9712yes—
121C22×C4#9712yes—
122C22×C4C2×C4211yescharacteristic
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4#11711yescharacteristic
125C22×C4C2×M1611yescharacteristic
126C22×C4C2×M1611yescharacteristic
127C22×C4C2×M1611yescharacteristic
128C2×C8—24no—
129C2×C8—24no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8#9912yes—
149C2×C8#10112yes—
150C2×C8C4×D414yes—
151C4⋊C4—48no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—22no—
156C42—22no—
157C42—22no—
158C42—42no—
159C42—42no—
160C42—41no—
161C42—41no—
162C16—48no—
163C23—22no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23C2×#9711yescharacteristic
168C23#40711yescharacteristic
169C23#43211yescharacteristic
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—48no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—84no—
196C2×C4C4×C22⋊C411yescharacteristic
197C2×C4C4×M1611yescharacteristic
198C2×C4#40711yescharacteristic
199C2×C4C2×#9911yescharacteristic
200C2×C4#41011yescharacteristic
201C2×C4C2×#10111yescharacteristic
202C2×C4#41311yescharacteristic
203C2×C4#43211yescharacteristic
204C8—22no—
205C8—22no—
206C8—24no—
207C8—42no—
208C8—42no—
209C22—24no—
210C22—24no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22#146511yescharacteristic
218C22#146711yescharacteristic
219C22#174911yescharacteristic
220C22#211012yes—
221C22#216512yes—
222C4—21no—
223C4—21no—
224C4—44no—
225C4—42no—
226C4—42no—
227C4—82no—
228C4#146311yescharacteristic
229C4#146411yescharacteristic
230C4C4×#9911yescharacteristic
231C4C4×#10111yescharacteristic
232C2—21no—
233C2—22no—
234C2—22no—
235C2—21no—
236C2#815311yescharacteristic
237C2#1260611yescharacteristic
238C2#1260711yescharacteristic
subgroupG/H
centerC2×C4C4×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC23×C8C23
socleC22#1749
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1322C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17712
#126052
#114402
#71842