Finite Group Databasebeta

← #97037

Subgroups — #97037

Order 512.

1236 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1236 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2619C211yesmaximal, characteristic
2#8300C212yesmaximal, complemented
3#7592C214yesmaximal, complemented
4C22×#127C411yescharacteristic
5C22×#121C412yes—
6C22×#117C2211yescharacteristic
7#2619—28no—
8C2×#394—22no—
9C2×#392—22no—
10C2×#390C2212yes—
11C2×#382C411yescharacteristic
12#1817C2214yes—
13C2×#324—22no—
14C2×#323—22no—
15#1270—28no—
16C23×D4—22no—
17C23×D4—22no—
18C23×D4C2×C412yes—
19C24×C4C2×C411yescharacteristic
20C2×#127—24no—
21C2×#127—44no—
22C2×#121—24no—
23C2×#121—24no—
24C2×#121—44no—
25C2×#121—44no—
26C2×#117—24no—
27C2×#117—44no—
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D412yes—
30C22×C22⋊C4—22no—
31C22×C22⋊C4—24no—
32C22×C22⋊C4D412yes—
33C22×C42C2×C411yescharacteristic
34C22×C42C2311yescharacteristic
35C2×C22⋊C8—24no—
36#394—216no—
37#392—216no—
38#390—24no—
39#390D418yes—
40#382—24no—
41C2×#95—22no—
42#336—28no—
43#324—44no—
44#323—44no—
45C25—22no—
46C25C22⋊C411yescharacteristic
47C22×D4—28no—
48C22×D4—28no—
49C22×D4—22no—
50C22×D4—28no—
51C22×D4—42no—
52C22×D4—42no—
53C22×D4—44no—
54C22×D4—44no—
55C22×D4—44no—
56C22×D4—44no—
57C22×D4—44no—
58C22×D4—88no—
59C22×D4—88no—
60C22×D4C22⋊C414yes—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—24no—
66C23×C4—42no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C412yes—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D412yes—
71C22×C8—44no—
72#127—416no—
73#121—416no—
74#121—416no—
75#117—416no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4C22⋊C414yes—
83C2×C22⋊C4—28no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—216no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—44no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—88no—
91C2×C42—24no—
92C2×C42—22no—
93C2×C42—42no—
94C2×C42C2×D414yes—
95C22⋊C8—44no—
96C22⋊C8—44no—
97#95—28no—
98C24—21no—
99C24—21no—
100C24—22no—
101C24—22no—
102C24—24no—
103C24—44no—
104C24—48no—
105C24—48no—
106C24—44no—
107C24—44no—
108C24—42no—
109C24—42no—
110C24#10012yes—
111C24C2×C22⋊C411yescharacteristic
112C2×D4—28no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—416no—
118C2×D4—416no—
119C2×D4—44no—
120C2×D4—88no—
121C2×D4—88no—
122C2×D4—816no—
123C2×D4—816no—
124C2×D4—816no—
125C2×D4—816no—
126C2×D4—816no—
127C2×D4—816no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—28no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—48no—
142C22×C4—44no—
143C22×C4—48no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—48no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4#9914yes—
151C22×C4#10012yes—
152C22×C4C2×C22⋊C412yes—
153C22×C4#12014yes—
154C22×C4#13114yes—
155C2×C8—84no—
156C2×C8—88no—
157C4⋊C4—48no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C22⋊C4—48no—
161C22⋊C4—416no—
162C22⋊C4—816no—
163C42—28no—
164C42—44no—
165C23—24no—
166C23—22no—
167C23—24no—
168C23—44no—
169C23—44no—
170C23—44no—
171C23—42no—
172C23—41no—
173C23—41no—
174C23—48no—
175C23—48no—
176C23—44no—
177C23—44no—
178C23—44no—
179C23—82no—
180C23—82no—
181C23—84no—
182C23—88no—
183C23—88no—
184C23—84no—
185C23—84no—
186C23—84no—
187C23—84no—
188C23—816no—
189C23—84no—
190C23—84no—
191C23#37911yescharacteristic
192C23#40812yes—
193C23C2×#9912yes—
194C23C2×#10011yescharacteristic
195C23C2×#10011yescharacteristic
196D4—88no—
197D4—88no—
198D4—832no—
199D4—832no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—28no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—416no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—44no—
216C2×C4—48no—
217C2×C4—44no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—84no—
221C2×C4—84no—
222C2×C4#40814yes—
223C8—164no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—42no—
231C22—84no—
232C22—84no—
233C22—82no—
234C22—82no—
235C22—88no—
236C22—84no—
237C22—84no—
238C22—84no—
239C22—88no—
240C22—168no—
241C22—168no—
242C22#147112yes—
243C22#147112yes—
244C22#178011yescharacteristic
245C22#178111yescharacteristic
246C22#178311yescharacteristic
247C22#179514yes—
248C4—24no—
249C4—44no—
250C4—44no—
251C4—42no—
252C4—84no—
253C2—22no—
254C2—22no—
255C2—82no—
256C2—164no—
257C2#830711yescharacteristic
258C2#832612yes—
259C2#851612yes—
260C2#852212yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#2619C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8300C2semidirect
3. #7592C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26192
#83002
#75922