Finite Group Databasebeta

← #128166

Subgroups — #128166

Order 512.

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

normal subgroups only

This group has 658 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
1#17357C212yesmaximal
2#14343C211yesmaximal, characteristic
3#7326C214yesmaximal, complemented
4C22×#117C2211yescharacteristic
5#2920—22no—
6C2×#380C411yescharacteristic
7C2×#380C411yescharacteristic
8#1464—22no—
9#1464C414yescomplemented
10#1463C2214yes—
11#1462—22no—
12#1458—22no—
13#1458C414yescomplemented
14#1445C412yes—
15#1445C2212yes—
16#1440—22no—
17C4⋊#95—22no—
18C24×C4C2×C411yescharacteristic
19C2×#117—42no—
20C2×#117—42no—
21C2×#117C2×C412yes—
22C2×#117C2×C412yes—
23C2×#117D411yescharacteristic
24C2×#117D411yescharacteristic
25C2×#117Q811yescharacteristic
26C2×#117Q811yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D412yes—
34C22×C42D412yes—
35#432—24no—
36#432—42no—
37#431—24no—
38#431—42no—
39#423—42no—
40#407—24no—
41C4×M16—42no—
42#380—42no—
43#380—42no—
44C2×#95—24no—
45C2×#95—24no—
46C2×#95C2×C412yes—
47C2×#95C2×C412yes—
48#324—24no—
49#324C2×C418yes—
50#323—24no—
51#323—24no—
52C25C22⋊C411yescharacteristic
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—44no—
58C23×C4C4211yescharacteristic
59C23×C4C4211yescharacteristic
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C4⋊C411yescharacteristic
63C23×C4C4⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D412yes—
66C2×M16—44no—
67#117—42no—
68#117—42no—
69#117—42no—
70#117—42no—
71#117—42no—
72#117—42no—
73#117—42no—
74#117—42no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4C4212yes—
84C2×C4⋊C4C4⋊C411yescharacteristic
85C2×C4⋊C4C4⋊C411yescharacteristic
86C2×C4⋊C4C22×C412yes—
87C2×C4⋊C4C2×D411yescharacteristic
88C2×C4⋊C4C2×Q811yescharacteristic
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—82no—
94C2×C22⋊C4—82no—
95C2×C22⋊C4C22⋊C412yes—
96C2×C22⋊C4C22⋊C412yes—
97C2×C22⋊C4C4⋊C412yes—
98C2×C22⋊C4C4⋊C412yes—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42C22⋊C412yes—
104C2×C42C22⋊C412yes—
105C2×C42C4⋊C412yes—
106C2×C42C4⋊C412yes—
107C4⋊C8—44no—
108C4⋊C8—44no—
109C22⋊C8—28no—
110C22⋊C8—44no—
111#97—44no—
112C4×C8—44no—
113#95—42no—
114#95—42no—
115#95—84no—
116#95—84no—
117C24—21no—
118C24—21no—
119C24—42no—
120C24—41no—
121C24—41no—
122C24—41no—
123C24—41no—
124C24#9511yescharacteristic
125C24#9511yescharacteristic
126C24C2×C22⋊C411yescharacteristic
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—48no—
144C22×C4—48no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4#9511yescharacteristic
160C22×C4#9512yes—
161C22×C4#9512yes—
162C22×C4#9511yescharacteristic
163C22×C4#9912yes—
164C22×C4#9912yes—
165C22×C4C2×C4211yescharacteristic
166C22×C4C2×C22⋊C412yes—
167C22×C4C2×C4⋊C411yescharacteristic
168C22×C4C2×C4⋊C412yes—
169M16—84no—
170M16—84no—
171C2×C8—48no—
172C4⋊C4—42no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—84no—
185C22⋊C4—84no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42—22no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—21no—
201C23—22no—
202C23—22no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23C2×#9511yescharacteristic
220C23C2×#9911yescharacteristic
221C23C2×#9911yescharacteristic
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4#34212yes—
260C2×C4#34212yes—
261C2×C4#34414yes—
262C8—88no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22C2×#34211yescharacteristic
286C22C2×#34412yes—
287C22#212811yescharacteristic
288C22#212911yescharacteristic
289C22#213411yescharacteristic
290C22#213511yescharacteristic
291C4—22no—
292C4—42no—
293C4—42no—
294C4—42no—
295C4—42no—
296C4—41no—
297C4—41no—
298C4—162no—
299C4—162no—
300C4#130614yes—
301C2—21no—
302C2—21no—
303C2—21no—
304C2—21no—
305C2—22no—
306C2—81no—
307C2—81no—
308C2C2×#130611yescharacteristic
309C2#1258311yescharacteristic
310C2#1258711yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7326C2semidirect
9. #1464C4semidirect
13. #1458C4semidirect

Maximal subgroups

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

subgroupindex
#173572
#143432
#73262