Finite Group Databasebeta

← #129285

Subgroups — #129285

Order 512.

628 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 118 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#60126C211yesmaximal, characteristic
2#12847C211yesmaximal, characteristic, complemented
3#12845C211yesmaximal, characteristic, complemented
4#11444C212yesmaximal, complemented
5#11422C212yesmaximal
6#3428C411yescharacteristic
7#2925—24no—
8#2924—24no—
9#2923—22no—
10#2922—22no—
11#2905C411yescharacteristic
12C2×#424C2211yescharacteristic
13#2879C411yescharacteristic
14C2×#408C2211yescharacteristic
15#2873C411yescharacteristic
16C2×#405C2211yescharacteristic
17#2112C2212yes—
18C2×C22⋊C16C2212yes—
19#1377—22no—
20#1374—22no—
21#1323—24no—
22#1315—24no—
23C22×#41C2×C411yescharacteristic
24C22×M16C2×C411yescharacteristic
25C22×M16C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C23×C8C2311yescharacteristic
28#520—22no—
29#520C814yes—
30#519—21no—
31#518—21no—
32#518—22no—
33C4×#41—22no—
34C4×#41—22no—
35C2×#117—22no—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×M32—22no—
39C22×C16—22no—
40#433—28no—
41#433—28no—
42#432—28no—
43#431—28no—
44#424—22no—
45#424—21no—
46#424—21no—
47#424—22no—
48#424D414yes—
49#423—22no—
50C2×C4⋊C8—22no—
51#408—22no—
52#408—22no—
53#408—21no—
54#408—21no—
55#408D414yes—
56#407—22no—
57C2×C22⋊C8—22no—
58#405—22no—
59#405—22no—
60#405—22no—
61#405—22no—
62C4×M16—22no—
63C2×#97D412yes—
64C2×C4×C8D412yes—
65C22⋊C16—22no—
66C22⋊C16—22no—
67C22⋊C16—22no—
68C22⋊C16—22no—
69C2×#41—21no—
70C2×#41—21no—
71C2×#41—42no—
72C2×#41—42no—
73C2×#41C22⋊C412yes—
74C2×#41C2×C812yes—
75C22×Q8M1611yescharacteristic
76C22×D4—21no—
77C22×D4M1611yescharacteristic
78C23×C4—21no—
79C23×C4C22×C411yescharacteristic
80C2×M16—21no—
81C2×M16—21no—
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16C22⋊C412yes—
88C2×M16C22⋊C412yes—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—42no—
92C22×C8C22⋊C412yes—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D412yes—
96C22×C8C2×D412yes—
97C4×Q8—24no—
98C4×Q8—24no—
99C4×D4—28no—
100C4×D4—24no—
101C4×D4—24no—
102C4×D4—28no—
103#117—24no—
104#117—24no—
105#117—21no—
106#117—21no—
107#117—21no—
108#117—21no—
109#117—21no—
110#117—21no—
111#117—24no—
112#117C22⋊C412yes—
113#117C2×C812yes—
114C2×C4⋊C4—24no—
115C2×C4⋊C4C2×C812yes—
116C2×C4⋊C4M1612yes—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4M1612yes—
120C2×C42—24no—
121C2×C42—21no—
122C2×C42C22⋊C412yes—
123M32—44no—
124C2×C16—44no—
125C2×C16—42no—
126C2×C16—42no—
127C4⋊C8—28no—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C22⋊C8—28no—
131C22⋊C8—24no—
132C22⋊C8—24no—
133#97—24no—
134#97—24no—
135C4×C8—24no—
136C4×C8—24no—
137C24—21no—
138C24C2×M1611yescharacteristic
139#41—44no—
140#41—44no—
141#41—42no—
142#41—42no—
143#41—42no—
144#41—42no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8C22⋊C812yes—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4—42no—
152C2×D4—44no—
153C2×D4C22⋊C812yes—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—28no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4C22⋊C814yes—
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C22×C811yescharacteristic
168C22×C4C2×M1611yescharacteristic
169C22×C4C2×M1611yescharacteristic
170M16—42no—
171M16—42no—
172M16—42no—
173M16—42no—
174M16—42no—
175M16—42no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—24no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8#12012yes—
187C2×C8#12012yes—
188C4⋊C4—28no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4C22⋊C814yes—
193C22⋊C4—28no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C42—28no—
199C42—22no—
200C42—22no—
201C42—22no—
202C42—22no—
203C16—82no—
204C16—82no—
205C23—22no—
206C23—22no—
207C23—42no—
208C23—42no—
209C23—82no—
210C23C2×C22⋊C811yescharacteristic
211C23#40711yescharacteristic
212C23#40711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—44no—
217D4—44no—
218D4—42no—
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—22no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—82no—
235C2×C4—88no—
236C2×C4#37911yescharacteristic
237C2×C4C2×C22⋊C811yescharacteristic
238C2×C4C2×C22⋊C811yescharacteristic
239C2×C4#40711yescharacteristic
240C2×C4#41012yes—
241C2×C4#41312yes—
242C8—22no—
243C8—22no—
244C8—42no—
245C8—42no—
246C8—42no—
247C22—22no—
248C22—21no—
249C22—21no—
250C22—22no—
251C22—22no—
252C22—42no—
253C22—44no—
254C22—82no—
255C22—162no—
256C22#145812yes—
257C22#177911yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—44no—
262C4—44no—
263C4#146412yes—
264C4#178211yescharacteristic
265C4#178511yescharacteristic
266C2—22no—
267C2—41no—
268C2—41no—
269C2—82no—
270C2#830011yescharacteristic
271C2#1268512yes—
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1779
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ 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. #12847C2semidirect
3. #12845C2semidirect
4. #11444C2semidirect

Maximal subgroups

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

subgroupindex
#601262
#128472
#128452
#114442
#114222