Finite Group Databasebeta

← #128308

Subgroups — #128308

Order 512.

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

normal subgroups only

This group has 588 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#20385C211yesmaximal, characteristic
2#11425C212yesmaximal
3#8156C212yesmaximal, complemented
4#7188C212yesmaximal, complemented
5C24×C8C411yescharacteristic
6C2×#407C411yescharacteristic
7C2×#405C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2280C2211yescharacteristic
10#1752—22no—
11C8×C22⋊C4—22no—
12#1729C411yescharacteristic
13#1729C2211yescharacteristic
14#1458C2214yes—
15#1450—22no—
16#1317C414yes—
17#1316C414yes—
18#1281—22no—
19C24×C4C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C22×M16C2×C411yescharacteristic
22C23×C8—21no—
23C23×C8—21no—
24C23×C8—22no—
25C23×C8—41no—
26C23×C8—41no—
27C23×C8—42no—
28C23×C8C814yescomplemented
29C23×C8C2×C411yescharacteristic
30C23×C8C2×C411yescharacteristic
31C2×#117C2×C411yescharacteristic
32C2×#117C2311yescharacteristic
33#431—24no—
34#407—22no—
35#407C814yescomplemented
36C2×C22⋊C8—22no—
37#405—41no—
38#405—41no—
39#405—42no—
40C2×#97—21no—
41C2×#97D412yes—
42C2×C4×C8—21no—
43C2×C4×C8D412yes—
44#381C2×C412yes—
45#380—22no—
46#379C2×C412yes—
47C4×C22⋊C4—22no—
48#336—22no—
49#336—22no—
50#324—24no—
51#323C2×C418yes—
52#322—42no—
53C82—42no—
54C25C4211yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4C4211yescharacteristic
64C23×C4C2×C812yes—
65C23×C4C2×C812yes—
66C23×C4C22×C411yescharacteristic
67C2×M16—21no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16C2×C812yes—
73C2×M16C2×C812yes—
74C2×M16M1612yes—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—24no—
82C22×C8—24no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—44no—
86C22×C8—44no—
87C22×C8—44no—
88C22×C8—44no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—42no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C412yes—
96C22×C8C22⋊C411yescharacteristic
97C22×C8C2×C812yes—
98C22×C8C2×C812yes—
99C22×C8M1612yes—
100#117—41no—
101#117—41no—
102#117—41no—
103#117—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4#4112yes—
106C2×C22⋊C4—24no—
107C2×C22⋊C4C4212yes—
108C2×C22⋊C4C22×C412yes—
109C2×C42—21no—
110C2×C42C2×D412yes—
111C4⋊C8—44no—
112C22⋊C8—28no—
113C22⋊C8—28no—
114#97—42no—
115#97—42no—
116C4×C8—22no—
117C4×C8—22no—
118C4×C8—44no—
119C4×C8—42no—
120#95—24no—
121C24—21no—
122C24—21no—
123C24—41no—
124C24—41no—
125C24—44no—
126C24C4×C812yes—
127C24C2×C4211yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4C4×C812yes—
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4#11711yescharacteristic
158C22×C4C4×D414yes—
159C22×C4C22×C811yescharacteristic
160C22×C4C22×C811yescharacteristic
161C22×C4C2×M1611yescharacteristic
162C22×C4C2×M1611yescharacteristic
163M16—42no—
164M16—42no—
165M16—42no—
166M16—42no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—48no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—44no—
184C2×C8—44no—
185C2×C8—42no—
186C2×C8—48no—
187C2×C8C22⋊C814yes—
188C2×C8C22⋊C812yes—
189C2×C8C22⋊C812yes—
190C2×C8#9911yescharacteristic
191C2×C8#9911yescharacteristic
192C2×C8#10012yes—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—48no—
197C22⋊C4—42no—
198C42—22no—
199C42—22no—
200C42—41no—
201C42—41no—
202C23—24no—
203C23—22no—
204C23—21no—
205C23—21no—
206C23—42no—
207C23—44no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—44no—
212C23—44no—
213C23—48no—
214C23—44no—
215C23—41no—
216C23—41no—
217C23C4×C22⋊C411yescharacteristic
218C23C2×C4×C811yescharacteristic
219C23C4×M1611yescharacteristic
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4C2×C22⋊C811yescharacteristic
246C2×C4C2×C22⋊C811yescharacteristic
247C2×C4C2×#9911yescharacteristic
248C2×C4#41011yescharacteristic
249C2×C4C2×#10011yescharacteristic
250C2×C4#41311yescharacteristic
251C2×C4#43111yescharacteristic
252C2×C4#43111yescharacteristic
253C8—22no—
254C8—24no—
255C8—42no—
256C8—42no—
257C8—88no—
258C8#32312yes—
259C8#32312yes—
260C22—22no—
261C22—24no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—44no—
270C22—44no—
271C22—48no—
272C22—44no—
273C22—42no—
274C22—42no—
275C22C4×C22⋊C811yescharacteristic
276C22C4×#9911yescharacteristic
277C22C4×#10011yescharacteristic
278C4—21no—
279C4—21no—
280C4—44no—
281C4—42no—
282C4—82no—
283C4C2×#32311yescharacteristic
284C4#145811yescharacteristic
285C4#145911yescharacteristic
286C4#146011yescharacteristic
287C2—21no—
288C2—21no—
289C2—44no—
290C2—41no—
291C2—41no—
292C2C4×#32311yescharacteristic
293C2#1262312yes—
subgroupG/H
centerC2×C8#99
commutatorC23C2×C4×C8
Frattini, cosocleC2×#117C23
socleC22C4×#99
prime radicalC25C42
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ 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
3. #8156C2semidirect
4. #7188C2semidirect
28. C23×C8C8semidirect
35. #407C8semidirect

Maximal subgroups

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

subgroupindex
#203852
#114252
#81562
#71882