Finite Group Databasebeta

← #128165

Subgroups — #128165

Order 512.

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

normal subgroups only

This group has 758 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×#1463C212yesmaximal
2#14342C211yesmaximal, characteristic
3#7326C214yesmaximal, complemented
4C23×C4⋊C4C2211yescharacteristic
5C2×#407—22no—
6C2×#381C411yescharacteristic
7C2×#381C411yescharacteristic
8#1463—22no—
9#1463—22no—
10#1463C414yescomplemented
11#1463C2214yes—
12C2×#324—22no—
13C2×#324C414yescomplemented
14#1445—22no—
15#1445C412yes—
16#1445C2212yes—
17C22⋊#95—22no—
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×M16—42no—
21C22×C4⋊C4—42no—
22C22×C4⋊C4—42no—
23C22×C4⋊C4C2×C411yescharacteristic
24C22×C4⋊C4C2×C412yes—
25C22×C4⋊C4C2×C412yes—
26C22×C4⋊C4D412yes—
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4Q811yescharacteristic
30C22×C4⋊C4Q811yescharacteristic
31C22×C4⋊C4C2311yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34#407—24no—
35#407—24no—
36#407—42no—
37#407—42no—
38C2×C22⋊C8—24no—
39C2×C22⋊C8—42no—
40#381—42no—
41#381—42no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—24no—
45C2×#95—24no—
46C2×#95C2×C412yes—
47C2×#95C2×C412yes—
48#324—24no—
49#324—24no—
50#324—24no—
51#324C2×C418yes—
52C25C22⋊C411yescharacteristic
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—44no—
62C23×C4—84no—
63C23×C4C4211yescharacteristic
64C23×C4C4211yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C412yes—
69C23×C4C4⋊C411yescharacteristic
70C23×C4C4⋊C411yescharacteristic
71C23×C4C4⋊C412yes—
72C23×C4C4⋊C412yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D412yes—
75C2×M16—44no—
76C2×M16—44no—
77C2×M16—88no—
78C22×C8—44no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4C4212yes—
94C2×C4⋊C4C22⋊C412yes—
95C2×C4⋊C4C22⋊C412yes—
96C2×C4⋊C4C4⋊C412yes—
97C2×C4⋊C4C4⋊C412yes—
98C2×C4⋊C4C4⋊C411yescharacteristic
99C2×C4⋊C4C4⋊C411yescharacteristic
100C2×C4⋊C4C22×C412yes—
101C2×C4⋊C4C2×D411yescharacteristic
102C2×C4⋊C4C2×Q811yescharacteristic
103C2×C22⋊C4—44no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C22⋊C8—28no—
108C22⋊C8—44no—
109C22⋊C8—44no—
110C22⋊C8—44no—
111#95—42no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—84no—
116#95—84no—
117C24—24no—
118C24—42no—
119C24—42no—
120C24#9511yescharacteristic
121C24#9511yescharacteristic
122C24#9912yes—
123C24#9912yes—
124C24C2×C22⋊C411yescharacteristic
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—48no—
152C22×C4—48no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—84no—
171C22×C4—84no—
172C22×C4—84no—
173C22×C4#9511yescharacteristic
174C22×C4#9511yescharacteristic
175C22×C4#9512yes—
176C22×C4#9512yes—
177C22×C4C2×C4211yescharacteristic
178C22×C4C2×C22⋊C412yes—
179C22×C4C2×C4⋊C411yescharacteristic
180C22×C4C2×C4⋊C412yes—
181M16—88no—
182M16—88no—
183C2×C8—48no—
184C2×C8—88no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C22⋊C4—84no—
196C22⋊C4—84no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—24no—
201C23—24no—
202C23—24no—
203C23—24no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—48no—
212C23—48no—
213C23#34212yes—
214C23#34212yes—
215C23#34414yes—
216C23C2×#9511yescharacteristic
217C23C2×#9911yescharacteristic
218C23C2×#9911yescharacteristic
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C8—88no—
266C22—24no—
267C22—24no—
268C22—21no—
269C22—22no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—44no—
280C22—48no—
281C22—48no—
282C22—44no—
283C22#130614yes—
284C22C2×#34211yescharacteristic
285C22C2×#34412yes—
286C22#212712yes—
287C22#213312yes—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—162no—
297C4—162no—
298C2—22no—
299C2—22no—
300C2—22no—
301C2—24no—
302C2—42no—
303C2C2×#130611yescharacteristic
304C2#1258211yescharacteristic
305C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7326C2semidirect
10. #1463C4semidirect
13. C2×#324C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14632
#143422
#73262