Finite Group Databasebeta

← #129276

Subgroups — #129276

Order 512.

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

normal subgroups only

This group has 710 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
1C22×#433C211yesmaximal, characteristic
2#12844C212yesmaximal, complemented
3C2×#1322C212yesmaximal, complemented
4C2×#1314C212yesmaximal
5C22×#117C411yescharacteristic
6C2×#433—24no—
7C2×#433—24no—
8C2×#433—24no—
9C22×C4⋊C8C411yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C22×C22⋊C8C411yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×#97C411yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×C22⋊C16C2214yes—
16#1374—24no—
17#1322—24no—
18#1314—24no—
19C24×C4C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C23×C8C2×C411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×#117—22no—
25C2×#117—22no—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—22no—
30C22×C4⋊C4C811yescharacteristic
31C22×C4⋊C4C811yescharacteristic
32C22×C22⋊C4C811yescharacteristic
33C22×C22⋊C4C811yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C16—24no—
37#433—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×C4×C8—22no—
54C2×C4×C8—22no—
55C2×C4×C8D414yes—
56C22⋊C16—24no—
57C22⋊C16—24no—
58C25C2×C811yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4C22⋊C412yes—
66C23×C4C2×C811yescharacteristic
67C23×C4C2×C811yescharacteristic
68C23×C4C2×C811yescharacteristic
69C23×C4C2×C811yescharacteristic
70C23×C4C2×C811yescharacteristic
71C23×C4M1612yes—
72C23×C4C22×C411yescharacteristic
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8C22⋊C412yes—
84C22×C8C22⋊C412yes—
85C22×C8C22⋊C412yes—
86C22×C8C2×D412yes—
87C22×C8C2×D412yes—
88C22×C8C2×D412yes—
89#117—28no—
90#117—28no—
91#117—28no—
92#117—28no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42C22⋊C412yes—
114C2×C42C22⋊C412yes—
115C2×C42M1612yes—
116C2×C42M1612yes—
117C2×C16—48no—
118C2×C16—44no—
119C4⋊C8—28no—
120C4⋊C8—28no—
121C22⋊C8—28no—
122C22⋊C8—28no—
123#97—28no—
124C4×C8—28no—
125C24—21no—
126C24—21no—
127C24—24no—
128C24—42no—
129C24—42no—
130C24C22⋊C812yes—
131C24C22×C811yescharacteristic
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4C22⋊C812yes—
163C22×C4C22⋊C812yes—
164C22×C4C22⋊C812yes—
165C22×C4C22⋊C812yes—
166C22×C4C22⋊C812yes—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C2×M1611yescharacteristic
171C22×C4C2×M1611yescharacteristic
172C22×C4C2×M1611yescharacteristic
173C2×C8—24no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—24no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8#12014yes—
189C4⋊C4—28no—
190C4⋊C4—28no—
191C22⋊C4—28no—
192C22⋊C4—28no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—22no—
201C16—84no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—24no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—88no—
215C23C2×C22⋊C811yescharacteristic
216C23C2×C22⋊C811yescharacteristic
217C23C2×C22⋊C811yescharacteristic
218C23#50414yes—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4#37911yescharacteristic
252C2×C4#40711yescharacteristic
253C2×C4#40711yescharacteristic
254C2×C4#40711yescharacteristic
255C2×C4#41012yes—
256C2×C4#41312yes—
257C8—24no—
258C8—42no—
259C8—42no—
260C8—42no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—28no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—84no—
270C22—84no—
271C22—84no—
272C22#145812yes—
273C22#177911yescharacteristic
274C22#211312yes—
275C22#211312yes—
276C22#211312yes—
277C22#211512yes—
278C4—21no—
279C4—21no—
280C4—42no—
281C4—42no—
282C4—42no—
283C4—42no—
284C4—42no—
285C4#146412yes—
286C4#178211yescharacteristic
287C4#178511yescharacteristic
288C2—22no—
289C2—22no—
290C2—82no—
291C2#830011yescharacteristic
292C2#1146114yes—
293C2#1268011yescharacteristic
294C2#1268211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12844C2semidirect
3. C2×#1322C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
#128442
C2×#13222
C2×#13142