Finite Group Databasebeta

← #128317

Subgroups — #128317

Order 512.

554 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 142 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#20377C211yesmaximal, characteristic
2C4×#349C212yesmaximal, complemented
3C2×#1320C212yesmaximal
4#7191C212yesmaximal, complemented
5C22×#117C411yescharacteristic
6C2×#407C411yescharacteristic
7C2×#407C2211yescharacteristic
8C2×C4×M16C411yescharacteristic
9C2×C4×M16C2211yescharacteristic
10C2×#349C2214yes—
11C4×M32—22no—
12#1750—22no—
13#1750—22no—
14#1728C411yescharacteristic
15#1728C2211yescharacteristic
16#1377—22no—
17#1320C418yescomplemented
18C24×C4C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C22×M16C2×C411yescharacteristic
22C22×M16C2311yescharacteristic
23C2×#117—21no—
24C2×#117—21no—
25C2×#117—22no—
26C2×#117—42no—
27C2×#117—42no—
28C22×C4⋊C4C812yes—
29C22×C22⋊C4C812yes—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C2×M32—24no—
33#407—22no—
34#407—22no—
35#407C814yes—
36#407C2×C414yes—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C4×M16—42no—
40C4×M16—42no—
41C2×#97—21no—
42C2×#97D412yes—
43C2×C4×C8—21no—
44C2×C4×C8D412yes—
45#349—24no—
46#349C2×C418yes—
47#346—42no—
48C4×C16—42no—
49#336—22no—
50#336—22no—
51C25C2×C811yescharacteristic
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×C4C4212yes—
61C23×C4C2×C811yescharacteristic
62C23×C4C2×C812yes—
63C23×C4C2×C812yes—
64C23×C4C2×C812yes—
65C23×C4C22×C411yescharacteristic
66C2×M16—21no—
67C2×M16—21no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16C4212yes—
77C2×M16C2×C812yes—
78C2×M16C2×C812yes—
79C2×M16M1612yes—
80C2×M16C22×C412yes—
81C2×M16#4112yes—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8C22⋊C412yes—
85C22×C8C2×D412yes—
86#117—48no—
87#117—42no—
88#117—42no—
89#117—42no—
90#117—42no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—44no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—44no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—41no—
102C2×C42—41no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22⋊C411yescharacteristic
105C2×C42M1612yes—
106M32—28no—
107M32—44no—
108C2×C16—44no—
109C22⋊C8—28no—
110C22⋊C8—28no—
111#97—42no—
112#97—42no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—42no—
116C24—21no—
117C24—21no—
118C24—41no—
119C24—41no—
120C24—82no—
121C24C4×C812yes—
122C24C22×C811yescharacteristic
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
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—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—82no—
153C22×C4—82no—
154C22×C4C4×C812yes—
155C22×C4C22⋊C812yes—
156C22×C4C22⋊C812yes—
157C22×C4C2×C4211yescharacteristic
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4#11711yescharacteristic
160C22×C4C22×C811yescharacteristic
161C22×C4C2×M1611yescharacteristic
162C22×C4C2×M1611yescharacteristic
163M16—42no—
164M16—42no—
165M16—42no—
166M16—42no—
167M16—42no—
168M16—42no—
169M16—42no—
170M16—42no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8C22⋊C814yes—
182C2×C8C4×D414yes—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C42—24no—
188C42—21no—
189C42—21no—
190C42—42no—
191C42—41no—
192C42—41no—
193C42#9912yes—
194C42#10012yes—
195C16—48no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—42no—
200C23—42no—
201C23—44no—
202C23—42no—
203C23—41no—
204C23—41no—
205C23—82no—
206C23—82no—
207C23—84no—
208C23—84no—
209C23C2×C4×C811yescharacteristic
210C23C2×C22⋊C811yescharacteristic
211C23#43111yescharacteristic
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—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—84no—
239C2×C4—84no—
240C2×C4#32314yes—
241C2×C4C4×C22⋊C411yescharacteristic
242C2×C4C4×M1611yescharacteristic
243C2×C4C2×C22⋊C811yescharacteristic
244C2×C4C2×#9911yescharacteristic
245C2×C4#41011yescharacteristic
246C2×C4C2×#10011yescharacteristic
247C2×C4#41311yescharacteristic
248C2×C4#43111yescharacteristic
249C8—24no—
250C8—24no—
251C8—42no—
252C8—42no—
253C22—22no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—42no—
259C22—44no—
260C22—42no—
261C22—42no—
262C22—82no—
263C22—84no—
264C22—84no—
265C22—82no—
266C22C2×#32311yescharacteristic
267C22#145811yescharacteristic
268C22C4×C22⋊C811yescharacteristic
269C4—24no—
270C4—21no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—82no—
275C4#145911yescharacteristic
276C4#146011yescharacteristic
277C4C4×#9911yescharacteristic
278C4C4×#10011yescharacteristic
279C2—21no—
280C2—21no—
281C2—41no—
282C2—41no—
283C2—82no—
284C2C4×#32311yescharacteristic
285C2#1143612yes—
subgroupG/H
centerC2×C4C2×#99
commutatorC23C2×C4×C8
Frattini, cosocleC22×M16C23
socleC22C2×#323
prime radicalC25C2×C8
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#349C2semidirect
4. #7191C2semidirect
17. #1320C4semidirect

Maximal subgroups

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

subgroupindex
#203772
C4×#3492
C2×#13202
#71912