Finite Group Databasebeta

← #129277

Subgroups — #129277

Order 512.

824 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C22×#432C211yesmaximal, characteristic
2#12846C212yesmaximal, complemented
3#11440C212yesmaximal, complemented
4#11418C212yesmaximal
5C23×C42C411yescharacteristic
6C2×#432—24no—
7C2×#432—24no—
8C2×#432—24no—
9C22×C4⋊C8C411yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C22×C22⋊C8C411yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×#97C411yescharacteristic
14C22×#97C2211yescharacteristic
15#2112C2214yes—
16#1377—24no—
17#1322—24no—
18#1314—24no—
19C24×C4C811yescharacteristic
20C24×C4C811yescharacteristic
21C24×C4C2×C411yescharacteristic
22C23×C8C2×C411yescharacteristic
23C23×C8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8C2311yescharacteristic
26C22×C42—22no—
27C22×C42—22no—
28C22×C42—22no—
29C22×C42—22no—
30C22×C42—22no—
31C22×C42—22no—
32C22×C42C811yescharacteristic
33C22×C42C811yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C2×M32—24no—
37#432—232no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8D414yes—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8D414yes—
50C2×#97—22no—
51C2×#97—22no—
52C2×#97—22no—
53C2×#97—22no—
54C2×#97—22no—
55C2×#97D414yes—
56C22⋊C16—24no—
57C22⋊C16—24no—
58C25C2×C811yescharacteristic
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—44no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4—44no—
73C23×C4C22⋊C412yes—
74C23×C4C2×C811yescharacteristic
75C23×C4C2×C811yescharacteristic
76C23×C4C2×C811yescharacteristic
77C23×C4C2×C811yescharacteristic
78C23×C4C2×C811yescharacteristic
79C23×C4M1612yes—
80C23×C4C22×C411yescharacteristic
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8C22⋊C412yes—
92C22×C8C22⋊C412yes—
93C22×C8C22⋊C412yes—
94C22×C8C2×D412yes—
95C22×C8C2×D412yes—
96C22×C8C2×D412yes—
97C2×C42—28no—
98C2×C42—28no—
99C2×C42—28no—
100C2×C42—28no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—22no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42C22⋊C412yes—
116C2×C42C22⋊C412yes—
117C2×C42M1612yes—
118C2×C42M1612yes—
119M32—48no—
120C2×C16—44no—
121C4⋊C8—28no—
122C4⋊C8—28no—
123C22⋊C8—28no—
124C22⋊C8—28no—
125#97—28no—
126#97—28no—
127C24—24no—
128C24—21no—
129C24—21no—
130C24—42no—
131C24—42no—
132C24C22⋊C812yes—
133C24C22×C811yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—28no—
140C22×C4—28no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—48no—
154C22×C4—48no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—48no—
164C22×C4—48no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4C22⋊C812yes—
175C22×C4C22⋊C812yes—
176C22×C4C22⋊C812yes—
177C22×C4C22⋊C812yes—
178C22×C4C22⋊C812yes—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C2×C22⋊C411yescharacteristic
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C2×M1611yescharacteristic
183C22×C4C2×M1611yescharacteristic
184C22×C4C2×M1611yescharacteristic
185C2×C8—24no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—24no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8#12014yes—
201C42—28no—
202C42—28no—
203C42—22no—
204C42—22no—
205C42—22no—
206C42—22no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—22no—
211C16—84no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—24no—
216C23—24no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—416no—
224C23—42no—
225C23—42no—
226C23#34914yes—
227C23C2×C22⋊C811yescharacteristic
228C23C2×C22⋊C811yescharacteristic
229C23C2×C22⋊C811yescharacteristic
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—416no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—416no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4#34914yes—
265C2×C4#37911yescharacteristic
266C2×C4#40711yescharacteristic
267C2×C4#40711yescharacteristic
268C2×C4#40711yescharacteristic
269C2×C4#41012yes—
270C2×C4#41312yes—
271C8—24no—
272C8—42no—
273C8—42no—
274C8—42no—
275C22—28no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—24no—
283C22—44no—
284C22—48no—
285C22—44no—
286C22—48no—
287C22—42no—
288C22—42no—
289C22#145812yes—
290C22#177911yescharacteristic
291C22C2×#34912yes—
292C22C2×#34912yes—
293C4—21no—
294C4—21no—
295C4—24no—
296C4—42no—
297C4—42no—
298C4—42no—
299C4—42no—
300C4#146412yes—
301C4#178211yescharacteristic
302C4#178511yescharacteristic
303C2—22no—
304C2—22no—
305C2—22no—
306C2—44no—
307C2#830011yescharacteristic
308C2#1268111yescharacteristic
309C2#1268111yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1779
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12846C2semidirect
3. #11440C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#128462
#114402
#114182