Finite Group Databasebeta

← #129247

Subgroups — #129247

Order 512.

840 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2905C211yesmaximal, characteristic
2#12681C212yesmaximal
3#11443C214yesmaximal, complemented
4C23×M16C411yescharacteristic
5C23×M16C2211yescharacteristic
6C22×#117C411yescharacteristic
7#2905—28no—
8C2×#424—22no—
9C2×#424C412yes—
10C2×#424C2212yes—
11C2×#423—22no—
12C2×#349—22no—
13C2×#349—22no—
14#2112C2214yes—
15#1323—28no—
16C24×C4C2×C411yescharacteristic
17C22×M16—21no—
18C22×M16—24no—
19C22×M16—24no—
20C22×M16—42no—
21C22×M16—42no—
22C22×M16—42no—
23C22×M16—42no—
24C22×M16C2×C412yes—
25C22×M16D412yes—
26C22×M16D412yes—
27C23×C8—21no—
28C23×C8C2×C411yescharacteristic
29C23×C8C2311yescharacteristic
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—24no—
33C2×#117—24no—
34C2×#117C812yes—
35C2×#117C2×C412yes—
36C22×C4⋊C4C812yes—
37C22×C22⋊C4—21no—
38C22×C42—21no—
39C2×M32—24no—
40#424—216no—
41#424—24no—
42#424—24no—
43#424—24no—
44#424D418yes—
45#423—216no—
46C2×C4⋊C8—24no—
47C2×C4⋊C8—24no—
48#349—44no—
49#349—44no—
50C22⋊C16—28no—
51C25M1611yescharacteristic
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C411yescharacteristic
61C23×C4C2×C811yescharacteristic
62C23×C4C2×C811yescharacteristic
63C23×C4M1611yescharacteristic
64C23×C4C22×C411yescharacteristic
65C2×M16—24no—
66C2×M16—24no—
67C2×M16—24no—
68C2×M16—22no—
69C2×M16—24no—
70C2×M16—24no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—416no—
74C2×M16—416no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16C22⋊C414yes—
80C22×C8—24no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8C22⋊C412yes—
90C22×C8C2×D412yes—
91C22×C8C2×D414yes—
92#117—28no—
93#117—24no—
94#117—24no—
95#117—22no—
96#117—22no—
97#117—28no—
98#117C22⋊C414yes—
99#117M1614yes—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4C2×C814yes—
103C2×C22⋊C4—28no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C42—24no—
107C2×C42—24no—
108C2×C42—22no—
109C2×C42—22no—
110M32—44no—
111M32—44no—
112C2×C16—44no—
113C4⋊C8—216no—
114C4⋊C8—216no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—82no—
119C24C22⋊C811yescharacteristic
120C24C22⋊C811yescharacteristic
121C24C2×M1611yescharacteristic
122C22×C4—28no—
123C22×C4—28no—
124C22×C4—28no—
125C22×C4—28no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—28no—
149C22×C4—84no—
150C22×C4C22⋊C811yescharacteristic
151C22×C4C22⋊C811yescharacteristic
152C22×C4#9912yes—
153C22×C4#9912yes—
154C22×C4#10112yes—
155C22×C4#10112yes—
156C22×C4C2×C22⋊C412yes—
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4C22×C811yescharacteristic
159C22×C4C2×M1612yes—
160M16—48no—
161M16—48no—
162M16—44no—
163M16—48no—
164M16—44no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—24no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—44no—
172C2×C8—44no—
173C2×C8—48no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8#12014yes—
177C4⋊C4—28no—
178C4⋊C4—24no—
179C4⋊C4C22⋊C818yes—
180C22⋊C4—28no—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C42—28no—
184C42—24no—
185C42—24no—
186C16—84no—
187C23—28no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—84no—
198C23—84no—
199C23—84no—
200C23#32412yes—
201C23#32412yes—
202C23C2×C22⋊C811yescharacteristic
203C23#40712yes—
204C2×C4—28no—
205C2×C4—28no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—28no—
226C2×C4—48no—
227C2×C4—88no—
228C2×C4#32412yes—
229C2×C4#32412yes—
230C2×C4#37911yescharacteristic
231C2×C4C2×C22⋊C812yes—
232C2×C4#40711yescharacteristic
233C2×C4C2×#9911yescharacteristic
234C2×C4C2×#9911yescharacteristic
235C2×C4C2×#10111yescharacteristic
236C2×C4C2×#10111yescharacteristic
237C8—24no—
238C8—42no—
239C8—44no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—24no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—88no—
254C22C2×#32411yescharacteristic
255C22C2×#32411yescharacteristic
256C22#177911yescharacteristic
257C4—22no—
258C4—22no—
259C4—22no—
260C4—48no—
261C4#146311yescharacteristic
262C4#146311yescharacteristic
263C4#178111yescharacteristic
264C4#178411yescharacteristic
265C2—21no—
266C2—21no—
267C2—42no—
268C2—42no—
269C2—42no—
270C2#829911yescharacteristic
271C2#1268612yes—
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1779
prime radicalC25M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11443C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29052
#126812
#114432