Finite Group Databasebeta

← #105809

Subgroups — #105809

Order 512.

1490 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1490 conjugacy classes of subgroups — too many to draw. Only its 194 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×#2312C211yesmaximal, characteristic
2C2×#1819C212yesmaximal
3C2×#1304C214yesmaximal
4C23×C22⋊C4C2211yescharacteristic
5#2312—28no—
6C2×#388—24no—
7C2×#380C412yescomplemented
8C2×#379C2214yes—
9C2×#379C2212yes—
10C2×C4×C22⋊C4C412yescomplemented
11#1819—24no—
12#1819C418yescomplemented
13C2×#342—24no—
14#1304—28no—
15C26C2311yescharacteristic
16C24×C4C2×C411yescharacteristic
17C24×C4C2×C411yescharacteristic
18C22×C4⋊C4—22no—
19C22×C22⋊C4—24no—
20C22×C22⋊C4—24no—
21C22×C22⋊C4—22no—
22C22×C22⋊C4—22no—
23C22×C22⋊C4—24no—
24C22×C22⋊C4—24no—
25C22×C22⋊C4—24no—
26C22×C22⋊C4C2×C414yes—
27C22×C22⋊C4C2×C414yes—
28C22×C22⋊C4D412yes—
29C22×C22⋊C4D414yes—
30C22×C22⋊C4Q812yes—
31C22×C42—22no—
32#388—232no—
33#380—28no—
34#379—24no—
35#379—24no—
36#379—24no—
37#379—24no—
38#379C2×C418yes—
39#379D418yes—
40C4×C22⋊C4—28no—
41C2×#95—22no—
42C2×#95—22no—
43#342—216no—
44C25—44no—
45C25—82no—
46C25—82no—
47C25C22×C411yescharacteristic
48C25C22×C412yes—
49C25C2×D412yes—
50C25C2×D411yescharacteristic
51C25C2×D414yes—
52C25C2×Q811yescharacteristic
53C25#4114yes—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—24no—
59C23×C4—44no—
60C23×C4—82no—
61C23×C4—82no—
62C23×C4C4212yes—
63C23×C4C4212yes—
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C412yes—
66C23×C4C4⋊C412yes—
67C23×C4C4⋊C412yes—
68C2×C4⋊C4—28no—
69C2×C4⋊C4—84no—
70C2×C22⋊C4—28no—
71C2×C22⋊C4—28no—
72C2×C22⋊C4—28no—
73C2×C22⋊C4—28no—
74C2×C22⋊C4—28no—
75C2×C22⋊C4—28no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—28no—
78C2×C22⋊C4—28no—
79C2×C22⋊C4—28no—
80C2×C22⋊C4—28no—
81C2×C22⋊C4—28no—
82C2×C22⋊C4—48no—
83C2×C22⋊C4—416no—
84C2×C22⋊C4—44no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—44no—
87C2×C22⋊C4—84no—
88C2×C22⋊C4—84no—
89C2×C22⋊C4—88no—
90C2×C22⋊C4—88no—
91C2×C22⋊C4C22⋊C418yes—
92C2×C22⋊C4C22⋊C418yes—
93C2×C22⋊C4C4⋊C418yes—
94C2×C42—28no—
95C2×C42—84no—
96#95—28no—
97#95—28no—
98C24—24no—
99C24—28no—
100C24—42no—
101C24—42no—
102C24—44no—
103C24—44no—
104C24—48no—
105C24—48no—
106C24—44no—
107C24—42no—
108C24—48no—
109C24—42no—
110C24—48no—
111C24—82no—
112C24—82no—
113C24—84no—
114C24—84no—
115C24—81no—
116C24—81no—
117C24—81no—
118C24—81no—
119C24—832no—
120C24C2×C4211yescharacteristic
121C24C2×C22⋊C411yescharacteristic
122C24C2×C22⋊C412yes—
123C24C2×C22⋊C412yes—
124C24C2×C4⋊C411yescharacteristic
125C24C2×C4⋊C412yes—
126C24#11712yes—
127C24C4×D414yes—
128C24C4×D414yes—
129C24#12014yes—
130C24#12114yes—
131C24C22⋊Q814yes—
132C24#12314yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—28no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—48no—
145C22×C4—48no—
146C22×C4—48no—
147C22×C4—48no—
148C22×C4—84no—
149C22×C4—84no—
150C22×C4—84no—
151C22×C4—84no—
152C22×C4—82no—
153C22×C4—82no—
154C22×C4—82no—
155C22×C4—82no—
156C22×C4—88no—
157C22×C4—88no—
158C22×C4—88no—
159C22×C4#9514yes—
160C22×C4#9514yes—
161C4⋊C4—88no—
162C22⋊C4—216no—
163C22⋊C4—216no—
164C22⋊C4—48no—
165C22⋊C4—48no—
166C22⋊C4—416no—
167C22⋊C4—88no—
168C22⋊C4—88no—
169C22⋊C4—88no—
170C22⋊C4—88no—
171C22⋊C4—88no—
172C22⋊C4—88no—
173C22⋊C4—88no—
174C42—88no—
175C23—22no—
176C23—22no—
177C23—28no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—28no—
182C23—28no—
183C23—22no—
184C23—22no—
185C23—24no—
186C23—24no—
187C23—24no—
188C23—28no—
189C23—41no—
190C23—41no—
191C23—42no—
192C23—42no—
193C23—41no—
194C23—41no—
195C23—44no—
196C23—44no—
197C23—44no—
198C23—44no—
199C23—44no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—416no—
214C23—816no—
215C23—832no—
216C23—816no—
217C23—832no—
218C23—88no—
219C23—88no—
220C23C2×#9511yescharacteristic
221C23C4×C22⋊C412yes—
222C23#37911yescharacteristic
223C23#38012yes—
224C23#38111yescharacteristic
225C23#38514yes—
226C23#38614yes—
227C23#41014yes—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—48no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—84no—
242C2×C4—84no—
243C2×C4—816no—
244C2×C4—88no—
245C2×C4—88no—
246C2×C4—88no—
247C2×C4—1616no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—28no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—24no—
258C22—24no—
259C22—24no—
260C22—24no—
261C22—24no—
262C22—22no—
263C22—22no—
264C22—24no—
265C22—48no—
266C22—44no—
267C22—44no—
268C22—416no—
269C22—48no—
270C22—88no—
271C22—84no—
272C22—84no—
273C22—88no—
274C22—88no—
275C22—88no—
276C22—88no—
277C22C22⋊#9511yescharacteristic
278C22#174012yes—
279C22#178212yes—
280C22#182012yes—
281C22#182812yes—
282C22#182812yes—
283C4—44no—
284C4—44no—
285C4—88no—
286C4—168no—
287C2—22no—
288C2—22no—
289C2—44no—
290C2—42no—
291C2—42no—
292C2—44no—
293C2#875014yes—
294C2#1072611yescharacteristic
295C2#1076811yescharacteristic
296C2#1076811yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC24C2×C42
Frattini, prime radical, cosocleC26C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C26 ◁ G
LECSG ▷ C26 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#380C4semidirect
10. C2×C4×C22⋊C4C4semidirect
12. #1819C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23122
C2×#18192
C2×#13042