Finite Group Databasebeta

← #188115

Subgroups — #188115

Order 512.

2632 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 2632 conjugacy classes of subgroups — too many to draw. Only its 318 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×#540C211yesmaximal, characteristic
2#20694C212yesmaximal, complemented
3#17396C214yesmaximal, complemented
4#8553C218yesmaximal, complemented
5C2×#540—216no—
6C2×#540—24no—
7C2×#540—24no—
8C22×#127C411yescharacteristic
9C22×#125C411yescharacteristic
10C22×#123C2214yes—
11C22×#121C414yes—
12C22×#121C2212yes—
13C22×C4×D4C412yes—
14C22×#117C2211yescharacteristic
15C2×#407C2214yes—
16#1822—24no—
17#1821C2218yes—
18#1742—24no—
19#1471—216no—
20#1462C22116yes—
21C23×D4C2×C414yes—
22C23×D4C2×C412yes—
23C23×D4C2×C414yes—
24C24×C4C2×C414yes—
25C24×C4C2311yescharacteristic
26C22×M16—24no—
27#540—232no—
28#540—232no—
29#540—264no—
30C2×#127—28no—
31C2×#127—22no—
32C2×#127—22no—
33C2×#125—28no—
34C2×#125—22no—
35C2×#125—22no—
36C2×#123—216no—
37C2×#123—216no—
38C2×#123—216no—
39C2×#121—216no—
40C2×#121—216no—
41C2×#121—24no—
42C2×#121—28no—
43C2×#121—24no—
44C2×#121—216no—
45C2×#121D4116yes—
46C2×C4×D4—28no—
47C2×C4×D4—28no—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×#117—28no—
51C2×#117D414yes—
52C2×#117D414yes—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4C2×C414yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2314yes—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61#407—28no—
62#407—24no—
63#407C2318yes—
64C2×C22⋊C8—28no—
65#343—216no—
66#323—232no—
67C25C22×C414yes—
68C25C22×C411yescharacteristic
69C22×D4—28no—
70C22×D4—28no—
71C22×D4—28no—
72C22×D4—28no—
73C22×D4—24no—
74C22×D4—22no—
75C22×D4—22no—
76C22×D4—28no—
77C22×D4—48no—
78C22×D4—88no—
79C22×D4—88no—
80C22×D4C22⋊C418yes—
81C22×D4C22⋊C4116yes—
82C23×C4—28no—
83C23×C4—28no—
84C23×C4—28no—
85C23×C4—44no—
86C23×C4—88no—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C412yes—
89C23×C4C22×C414yes—
90C23×C4C22×C412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D418yes—
93C23×C4C2×D412yes—
94C23×C4C2411yescharacteristic
95C2×M16—28no—
96C2×M16—44no—
97C2×M16—416no—
98C22×C8—44no—
99#127—216no—
100#127—216no—
101#125—216no—
102#125—216no—
103#123—2128no—
104#121—2128no—
105#121—216no—
106#121—216no—
107#121—232no—
108C4×D4—232no—
109C4×D4—232no—
110#117—28no—
111#117—28no—
112#117—216no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4C22⋊C418yes—
122C2×C4⋊C4C2×D418yes—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4C22⋊C418yes—
131C2×C22⋊C4C2×D4116yes—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42C22⋊C412yes—
135C2×C42C22⋊C412yes—
136C2×C42C22⋊C412yes—
137C2×C42C22⋊C412yes—
138C22⋊C8—216no—
139C22⋊C8—48no—
140C24—24no—
141C24—24no—
142C24—42no—
143C24—48no—
144C24—44no—
145C24—48no—
146C24—48no—
147C24C2×C22⋊C418yes—
148C24C2×C22⋊C414yes—
149C24C2×C22⋊C412yes—
150C24C23×C411yescharacteristic
151C2×D4—232no—
152C2×D4—232no—
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—48no—
156C2×D4—416no—
157C2×D4—416no—
158C2×D4—48no—
159C2×D4—816no—
160C2×D4—816no—
161C2×D4—816no—
162C2×D4—816no—
163C2×D4—816no—
164C2×D4—816no—
165C22×C4—28no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—232no—
174C22×C4—416no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—48no—
179C22×C4—44no—
180C22×C4—816no—
181C22×C4—816no—
182C22×C4—816no—
183C22×C4C2×C22⋊C414yes—
184C22×C4C2×C22⋊C411yescharacteristic
185C22×C4C2×C22⋊C414yes—
186C22×C4C2×C22⋊C414yes—
187C22×C4C2×C22⋊C418yes—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#12014yes—
190C22×C4#12014yes—
191C22×C4#12018yes—
192C22×C4C22×D412yes—
193C22×C4C22×D414yes—
194M16—416no—
195M16—88no—
196C2×C8—48no—
197C2×C8—88no—
198C4⋊C4—232no—
199C4⋊C4—216no—
200C4⋊C4—28no—
201C4⋊C4—28no—
202C22⋊C4—232no—
203C22⋊C4—216no—
204C22⋊C4—232no—
205C42—24no—
206C42—24no—
207C42—24no—
208C42—24no—
209C23—22no—
210C23—24no—
211C23—24no—
212C23—28no—
213C23—28no—
214C23—216no—
215C23—44no—
216C23—44no—
217C23—44no—
218C23—48no—
219C23—48no—
220C23—48no—
221C23—48no—
222C23—48no—
223C23—48no—
224C23—48no—
225C23—48no—
226C23—88no—
227C23—832no—
228C23#37914yes—
229C23C22×C22⋊C411yescharacteristic
230C23C22×C22⋊C412yes—
231D4—416no—
232D4—416no—
233D4—832no—
234D4—832no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—28no—
238C2×C4—28no—
239C2×C4—28no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—28no—
243C2×C4—48no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—48no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—48no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—48no—
253C2×C4—44no—
254C2×C4—832no—
255C2×C4#37912yes—
256C2×C4#37912yes—
257C2×C4#41014yes—
258C2×C4C2×#12014yes—
259C8—88no—
260C22—28no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—24no—
265C22—28no—
266C22—48no—
267C22—48no—
268C22—48no—
269C22—84no—
270C22—88no—
271C22—816no—
272C22—816no—
273C22—816no—
274C22C2×#37911yescharacteristic
275C22C2×#41012yes—
276C22#288814yes—
277C4—24no—
278C4—22no—
279C4—44no—
280C4—48no—
281C4—44no—
282C4#178214yes—
283C2—22no—
284C2—24no—
285C2—44no—
286C2—88no—
287C2C2×#178211yescharacteristic
288C2#2059212yes—
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#540C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20694C2semidirect
3. #17396C2semidirect
4. #8553C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5402
#206942
#173962
#85532