Finite Group Databasebeta

← #129255

Subgroups — #129255

Order 512.

1116 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2296C211yesmaximal, characteristic
2C2×#1326C214yesmaximal
3#8300C212yesmaximal
4C22×#117C411yescharacteristic
5C22×#117C2211yescharacteristic
6C23×C42C411yescharacteristic
7#2296—28no—
8C2×#389—22no—
9C2×#385C412yes—
10C2×#385C2212yes—
11C2×C4×C4⋊C4—22no—
12#1817—22no—
13C2×#324—22no—
14C2×#323C2214yes—
15#1326—28no—
16C24×C4C2×C411yescharacteristic
17C24×C4C2×C412yes—
18C2×#117—22no—
19C2×#117—24no—
20C2×#117—24no—
21C2×#117—24no—
22C2×#117—42no—
23C2×#117—42no—
24C2×#117C814yescomplemented
25C22×C4⋊C4—21no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4D412yes—
29C22×C22⋊C4C2×C412yes—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2311yescharacteristic
32C22×C42—21no—
33C22×C42—21no—
34C22×C42—24no—
35C22×C42—21no—
36C22×C42—42no—
37C22×C42—42no—
38C22×C42D412yes—
39C2×C22⋊C8—24no—
40#389—216no—
41#385—24no—
42#385—24no—
43#385—24no—
44#385D418yes—
45C4×C4⋊C4—216no—
46C2×#95—22no—
47C2×#95—22no—
48#336—44no—
49#324—44no—
50#323—28no—
51C25C22×C411yescharacteristic
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—48no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C414yes—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C2×C812yes—
69C23×C4M1612yes—
70C23×C4C2×D412yes—
71C22×C8—44no—
72#117—216no—
73#117—216no—
74#117—416no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—28no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—28no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—48no—
84C2×C4⋊C4—48no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4C22⋊C414yes—
94C2×C22⋊C4C2×C814yes—
95C2×C22⋊C4M1614yes—
96C2×C22⋊C4C2×D414yes—
97C2×C42—24no—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—24no—
101C2×C42—24no—
102C2×C42—24no—
103C2×C42—24no—
104C2×C42—42no—
105C2×C42—416no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—42no—
110C22⋊C8—44no—
111C22⋊C8—44no—
112#95—28no—
113#95—28no—
114C24—21no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24—44no—
119C24C2×C22⋊C411yescharacteristic
120C24C2×C22⋊C412yes—
121C24C22×C811yescharacteristic
122C24C2×M1611yescharacteristic
123C24C2×M1612yes—
124C22×C4—28no—
125C22×C4—28no—
126C22×C4—28no—
127C22×C4—28no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—24no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—28no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—48no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—416no—
159C22×C4—48no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—48no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4C22⋊C812yes—
167C22×C4C22⋊C812yes—
168C22×C4#9912yes—
169C22×C4#9912yes—
170C22×C4#10112yes—
171C22×C4#10112yes—
172C22×C4#12014yes—
173C2×C8—84no—
174C2×C8—88no—
175C4⋊C4—216no—
176C4⋊C4—48no—
177C4⋊C4—416no—
178C4⋊C4—416no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—44no—
183C22⋊C4—44no—
184C22⋊C4C22⋊C818yes—
185C42—216no—
186C42—48no—
187C42—48no—
188C42—48no—
189C23—24no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23—44no—
206C23—44no—
207C23#37911yescharacteristic
208C23C2×C22⋊C811yescharacteristic
209C23C2×C22⋊C812yes—
210C23#40711yescharacteristic
211C23#40712yes—
212C23C2×#9911yescharacteristic
213C23C2×#9911yescharacteristic
214C23C2×#10111yescharacteristic
215C23C2×#10111yescharacteristic
216C2×C4—216no—
217C2×C4—216no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—28no—
231C2×C4—28no—
232C2×C4—24no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—48no—
236C2×C4—48no—
237C2×C4—44no—
238C2×C4—48no—
239C2×C4—416no—
240C2×C4—48no—
241C2×C4—44no—
242C2×C4—48no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—48no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4#32414yes—
252C2×C4#32414yes—
253C8—164no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—44no—
267C22—44no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—44no—
273C22C2×#32411yescharacteristic
274C22C2×#32411yescharacteristic
275C22#146311yescharacteristic
276C22#146311yescharacteristic
277C22#177911yescharacteristic
278C22#178111yescharacteristic
279C22#178411yescharacteristic
280C22#212014yes—
281C4—24no—
282C4—24no—
283C4—48no—
284C4—44no—
285C4—48no—
286C4—82no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—42no—
292C2—42no—
293C2#829911yescharacteristic
294C2#1145212yes—
295C2#1151812yes—
296C2#1268912yes—
subgroupG/H
center, socleC23C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
24. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#13262
#83002