Finite Group Databasebeta

← #391729

Subgroups — #391729

Order 512.

1813 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1813 conjugacy classes of subgroups — too many to draw. Only its 242 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×#2622C211yesmaximal, characteristic
2#14338C212yesmaximal, complemented
3#11203C214yesmaximal, complemented
4#11196C212yesmaximal, complemented
5#11043C214yesmaximal, complemented
6#10965C212yesmaximal
7C22×#120C2211yescharacteristic
8C23×C22⋊C4C2211yescharacteristic
9C22×#99C2214yes—
10#2622—28no—
11C2×#397C414yescomplemented
12C2×#394C414yescomplemented
13C2×#386C2212yes—
14C2×#385C2212yes—
15C2×#380C2214yes—
16C2×#379C2214yes—
17C2×#379C2211yescharacteristic
18C2×#342C2218yes—
19C2×#342C2214yes—
20#1439—28no—
21C22⋊#95C2214yes—
22C26C2311yescharacteristic
23C23×D4C2×C412yes—
24C23×D4C2311yescharacteristic
25C24×C4C2312yes—
26C2×#120—24no—
27C2×#120—44no—
28C22×C4⋊C4—24no—
29C22×C4⋊C4C2×C414yes—
30C22×C22⋊C4—28no—
31C22×C22⋊C4—44no—
32C22×C22⋊C4C2×C412yes—
33C22×C22⋊C4C2×C414yes—
34C22×C22⋊C4C2×C414yes—
35C22×C22⋊C4C2×C414yes—
36C22×C22⋊C4D418yes—
37C22×C22⋊C4C2318yes—
38C22×C22⋊C4C2311yescharacteristic
39C22×C22⋊C4C2312yes—
40C2×#99—48no—
41C2×#99—44no—
42#397—216no—
43#394—216no—
44#386—24no—
45#386—24no—
46#385—24no—
47#385—24no—
48#380—48no—
49#379—24no—
50#379—48no—
51C2×#95—24no—
52C2×#95—28no—
53C2×#95C2×C414yes—
54C2×#95C2×C414yes—
55#342—44no—
56#342—48no—
57C25—24no—
58C25—24no—
59C25—24no—
60C25—84no—
61C25C22×C411yescharacteristic
62C25C22×C412yes—
63C25C2×D414yes—
64C25C2411yescharacteristic
65C22×D4—22no—
66C22×D4—24no—
67C22×D4—24no—
68C22×D4—24no—
69C22×D4—24no—
70C22×D4—44no—
71C22×D4—84no—
72C23×C4—28no—
73C23×C4—28no—
74C23×C4—84no—
75C23×C4C22×C411yescharacteristic
76C23×C4C22×C414yes—
77C23×C4C22×C412yes—
78C23×C4C22×C414yes—
79C23×C4C2×D414yes—
80C23×C4C2×D414yes—
81#120—416no—
82C2×C4⋊C4—28no—
83C2×C4⋊C4—48no—
84C2×C4⋊C4—88no—
85C2×C4⋊C4—84no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—28no—
94C2×C22⋊C4—28no—
95C2×C22⋊C4—28no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—28no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—48no—
102C2×C22⋊C4—48no—
103C2×C22⋊C4—48no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—416no—
106C2×C22⋊C4—88no—
107C2×C22⋊C4—88no—
108C2×C22⋊C4—84no—
109C2×C22⋊C4—84no—
110C2×C22⋊C4C22⋊C4116yes—
111#99—416no—
112#95—28no—
113#95—28no—
114#95—48no—
115#95—88no—
116C24—24no—
117C24—24no—
118C24—24no—
119C24—28no—
120C24—28no—
121C24—28no—
122C24—21no—
123C24—22no—
124C24—24no—
125C24—44no—
126C24—44no—
127C24—48no—
128C24—48no—
129C24—44no—
130C24—44no—
131C24—44no—
132C24—48no—
133C24—44no—
134C24—816no—
135C24—816no—
136C24—816no—
137C24C2×C22⋊C414yes—
138C24C23×C411yescharacteristic
139C24C22×D412yes—
140C24H(5,𝔽2)14yes—
141C24H(5,𝔽2)14yes—
142C242−1+414yes—
143C2×D4—48no—
144C2×D4—44no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—44no—
148C2×D4—44no—
149C2×D4—48no—
150C2×D4—44no—
151C2×D4—88no—
152C2×D4—816no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—28no—
156C22×C4—28no—
157C22×C4—21no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—28no—
164C22×C4—48no—
165C22×C4—48no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—416no—
172C22×C4—416no—
173C22×C4—48no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—816no—
177C22×C4C2×C22⋊C418yes—
178C4⋊C4—88no—
179C4⋊C4—816no—
180C22⋊C4—416no—
181C22⋊C4—48no—
182C22⋊C4—44no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—44no—
186C22⋊C4—48no—
187C22⋊C4—48no—
188C22⋊C4—48no—
189C22⋊C4—48no—
190C22⋊C4—88no—
191C22⋊C4—88no—
192C22⋊C4—832no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—28no—
198C23—28no—
199C23—28no—
200C23—48no—
201C23—48no—
202C23—48no—
203C23—416no—
204C23—48no—
205C23—416no—
206C23—48no—
207C23—48no—
208C23—48no—
209C23—48no—
210C23—41no—
211C23—42no—
212C23—44no—
213C23—44no—
214C23—816no—
215C23—816no—
216C23—88no—
217C23—88no—
218C23—84no—
219C23—88no—
220C23—816no—
221C23—832no—
222C23—816no—
223C23C22×C22⋊C411yescharacteristic
224C23#51812yes—
225C23#52014yes—
226C23#53414yes—
227C23#53512yes—
228C23#53612yes—
229C23#53714yes—
230C23#56014yes—
231C23#56218yes—
232C23#56314yes—
233D4—816no—
234D4—84no—
235D4—84no—
236C2×C4—28no—
237C2×C4—28no—
238C2×C4—28no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—88no—
250C2×C4—88no—
251C2×C4—88no—
252C2×C4—84no—
253C2×C4—816no—
254C2×C4—88no—
255C2×C4—816no—
256C2×C4—88no—
257C2×C4—88no—
258C22—24no—
259C22—28no—
260C22—28no—
261C22—24no—
262C22—24no—
263C22—24no—
264C22—24no—
265C22—44no—
266C22—48no—
267C22—44no—
268C22—44no—
269C22—416no—
270C22—44no—
271C22—44no—
272C22—816no—
273C22—816no—
274C22—816no—
275C22—81no—
276C22—84no—
277C22—82no—
278C22—164no—
279C22#230811yescharacteristic
280C22#231112yes—
281C22#237614yes—
282C22#281314yes—
283C22#281414yes—
284C22#288314yes—
285C4—48no—
286C4—81no—
287C4—84no—
288C4—82no—
289C4—168no—
290C2—24no—
291C2—24no—
292C2—24no—
293C2—84no—
294C2—161no—
295C2#1465811yescharacteristic
296C2#2174912yes—
297C2#2187614yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14338C2semidirect
3. #11203C2semidirect
4. #11196C2semidirect
5. #11043C2semidirect
11. C2×#397C4semidirect
12. C2×#394C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26222
#143382
#112032
#111962
#110432
#109652