Finite Group Databasebeta

← #112264

Subgroups — #112264

Order 512.

940 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2629C211yesmaximal, characteristic
2C2×#1534C212yesmaximal, complemented
3C2×#1464C212yesmaximal
4#8040C212yesmaximal, complemented
5C22×#125C2211yescharacteristic
6C22×#117C2211yescharacteristic
7C2×#432C2214yes—
8#2629—24no—
9#2629—24no—
10C2×#398—22no—
11C2×#396—22no—
12C2×#389C412yes—
13C2×#384C2211yescharacteristic
14C2×#380C412yes—
15#1534—28no—
16C2×#330—24no—
17#1464—28no—
18C2×#324—22no—
19C2×#323—22no—
20#1296—24no—
21C24×C4C2×C411yescharacteristic
22C2×#125—22no—
23C2×#125—24no—
24C2×#125—24no—
25C2×#125D414yes—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—24no—
29C2×#117—24no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4D412yes—
36C22×C22⋊C4—22no—
37C22×C22⋊C4D412yes—
38C22×C42C2×C411yescharacteristic
39C22×C42C2311yescharacteristic
40#432—24no—
41#432—24no—
42#432—24no—
43#432D418yes—
44C2×C4⋊C8—24no—
45C2×C22⋊C8—24no—
46C2×#97—24no—
47#398—216no—
48#396—216no—
49#389—24no—
50#389—24no—
51#384—22no—
52#384D414yes—
53#380—24no—
54#380—24no—
55C2×#95—22no—
56C2×#95—22no—
57#330—216no—
58#324—28no—
59#323—28no—
60C25C22⋊C411yescharacteristic
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—42no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C412yes—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D412yes—
73C22×C8—44no—
74#125—24no—
75#125—28no—
76#125—24no—
77#125—44no—
78#125—42no—
79#125—42no—
80#117—42no—
81#117—44no—
82#117—44no—
83#117—42no—
84#117—42no—
85#117—42no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—84no—
102C2×C4⋊C4—84no—
103C2×C4⋊C4C22⋊C414yes—
104C2×C4⋊C4C22⋊C414yes—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—84no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—24no—
114C2×C42C2×D412yes—
115C2×C42C2×D414yes—
116C2×C42C2×D412yes—
117C2×C42#4114yes—
118C4⋊C8—28no—
119C4⋊C8—44no—
120C22⋊C8—28no—
121C22⋊C8—44no—
122#97—28no—
123#97—44no—
124#95—28no—
125#95—28no—
126C24—21no—
127C24—21no—
128C24—22no—
129C24—22no—
130C24—42no—
131C24—42no—
132C24#10112yes—
133C24C2×C22⋊C411yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—48no—
151C22×C4—48no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4#10112yes—
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C4×D414yes—
169C22×C4#12014yes—
170C22×C4#12012yes—
171C22×C4#12114yes—
172C22×C4#12312yes—
173C2×C8—48no—
174C2×C8—84no—
175C2×C8—84no—
176C4⋊C4—28no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—48no—
182C4⋊C4—44no—
183C4⋊C4—48no—
184C4⋊C4—88no—
185C4⋊C4—88no—
186C22⋊C4—44no—
187C22⋊C4—44no—
188C22⋊C4—88no—
189C42—42no—
190C42—42no—
191C42#12012yes—
192C42#12114yes—
193C42#12114yes—
194C42#12312yes—
195C42#12412yes—
196C42#12712yes—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—24no—
201C23—44no—
202C23—44no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—41no—
207C23—41no—
208C23—82no—
209C23—82no—
210C23—82no—
211C23—82no—
212C23#37911yescharacteristic
213C23#38612yes—
214C23C2×#10111yescharacteristic
215C23C2×#10111yescharacteristic
216C23#41312yes—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—48no—
231C2×C4—48no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—88no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4#38612yes—
250C2×C4#39012yes—
251C2×C4#39214yes—
252C2×C4#39414yes—
253C8—88no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—44no—
261C22—42no—
262C22—84no—
263C22—84no—
264C22—81no—
265C22—81no—
266C22—82no—
267C22#143811yescharacteristic
268C22#178411yescharacteristic
269C22#178511yescharacteristic
270C22#179512yes—
271C22#179612yes—
272C22#190112yes—
273C22#190112yes—
274C4—22no—
275C4—22no—
276C4—24no—
277C4—24no—
278C4—42no—
279C4—84no—
280C4—82no—
281C4—84no—
282C2—22no—
283C2—22no—
284C2—82no—
285C2#852512yes—
286C2#852611yescharacteristic
287C2#852711yescharacteristic
288C2#1096811yescharacteristic
289C2#1098612yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1534C2semidirect
4. #8040C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#15342
C2×#14642
#80402