Finite Group Databasebeta

← #134243

Subgroups — #134243

Order 512.

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

normal subgroups only

This group has 518 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#20386C211yesmaximal, characteristic
2#12790C212yesmaximal, complemented
3C2×#1320C212yesmaximal
4#7194C212yesmaximal, complemented
5C23×M16C411yescharacteristic
6C2×#407C2211yescharacteristic
7C2×#405C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2280C411yescharacteristic
10C2×#364—22no—
11C2×#349C2214yes—
12#1753—22no—
13#1753—22no—
14#1729C411yescharacteristic
15#1729C2211yescharacteristic
16#1375—22no—
17#1320—24no—
18C24×C4C2×C411yescharacteristic
19C22×M16—21no—
20C22×M16—21no—
21C22×M16—22no—
22C22×M16—42no—
23C22×M16—42no—
24C22×M16C814yes—
25C22×M16C2×C411yescharacteristic
26C22×M16C2311yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C2×#117C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C2×M32—24no—
32#407—22no—
33#407D412yes—
34#407Q812yes—
35C2×C22⋊C8—22no—
36#405—42no—
37#405—42no—
38C2×#97—21no—
39C2×#97D412yes—
40C2×C4×C8—21no—
41C2×C4×C8D412yes—
42#381—21no—
43#380—22no—
44#379—21no—
45C4×C22⋊C4—22no—
46#364—44no—
47#349—24no—
48#349—24no—
49#336—22no—
50#336—22no—
51C25M1611yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—41no—
57C23×C4—41no—
58C23×C4—41no—
59C23×C4—41no—
60C23×C4C4⋊C412yes—
61C23×C4C2×C812yes—
62C23×C4M1611yescharacteristic
63C23×C4C22×C411yescharacteristic
64C2×M16—21no—
65C2×M16—24no—
66C2×M16—24no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—48no—
72C2×M16—48no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16C4⋊C412yes—
78C2×M16C2×D411yescharacteristic
79C2×M16C2×Q811yescharacteristic
80C2×M16#4112yes—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C2×C812yes—
91C22×C8C2×C812yes—
92C22×C8M1612yes—
93C22×C8C2×D412yes—
94#117—41no—
95#117—41no—
96#117—41no—
97#117—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4M1612yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—24no—
103C2×C42—21no—
104C2×C42C22⋊C412yes—
105M32—44no—
106M32—44no—
107C2×C16—44no—
108C22⋊C8—24no—
109C22⋊C8—24no—
110#97—42no—
111#97—42no—
112C4×C8—24no—
113C4×C8—42no—
114#95—24no—
115C24—21no—
116C24—21no—
117C24—41no—
118C24—41no—
119C24—82no—
120C24C4⋊C812yes—
121C24C2×M1611yescharacteristic
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4C4⋊C812yes—
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×C4⋊C411yescharacteristic
153C22×C4#11711yescharacteristic
154C22×C4C22×C811yescharacteristic
155C22×C4C2×M1611yescharacteristic
156C22×C4C2×M1611yescharacteristic
157M16—41no—
158M16—41no—
159M16—42no—
160M16—48no—
161M16—42no—
162M16—48no—
163M16—42no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—21no—
168C2×C8—24no—
169C2×C8—24no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8C22⋊C812yes—
179C2×C8C22⋊C812yes—
180C2×C8#9912yes—
181C2×C8#10012yes—
182C2×C8#12112yes—
183C2×C8C22⋊Q812yes—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—84no—
189C42—22no—
190C42—22no—
191C42—41no—
192C42—41no—
193C16—84no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—41no—
203C23—41no—
204C23—82no—
205C23—82no—
206C23—84no—
207C23—84no—
208C23#40711yescharacteristic
209C23C2×C4⋊C811yescharacteristic
210C23#43211yescharacteristic
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4#38011yescharacteristic
237C2×C4C2×C22⋊C811yescharacteristic
238C2×C4C2×#9911yescharacteristic
239C2×C4#41011yescharacteristic
240C2×C4C2×#10011yescharacteristic
241C2×C4#41311yescharacteristic
242C2×C4#42311yescharacteristic
243C2×C4#43111yescharacteristic
244C8—22no—
245C8—22no—
246C8—22no—
247C8—42no—
248C8—42no—
249C8#32314yes—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—41no—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—82no—
261C22—84no—
262C22—84no—
263C22—82no—
264C22#146511yescharacteristic
265C22#146611yescharacteristic
266C22#179711yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—82no—
271C4—82no—
272C4C2×#32311yescharacteristic
273C4#145811yescharacteristic
274C4#179911yescharacteristic
275C4#180011yescharacteristic
276C2—21no—
277C2—21no—
278C2—41no—
279C2—41no—
280C2—82no—
281C2#855711yescharacteristic
282C2#1153812yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22#1466
prime radicalC25M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12790C2semidirect
4. #7194C2semidirect

Maximal subgroups

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

subgroupindex
#203862
#127902
C2×#13202
#71942