Finite Group Databasebeta

← #97024

Subgroups — #97024

Order 512.

1240 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2328C211yesmaximal, characteristic
2#8699C212yesmaximal, complemented
3#7592C214yesmaximal, complemented
4C22×#127C411yescharacteristic
5C22×C4×D4C412yes—
6#2328—28no—
7C2×#390C2212yes—
8C2×#386—24no—
9C2×#382C411yescharacteristic
10C2×#376C2211yescharacteristic
11#1817C2214yes—
12#1280—24no—
13#1270—28no—
14C23×D4—22no—
15C23×D4C2×C412yes—
16C24×C4—22no—
17C2×#127—24no—
18C2×#127—24no—
19C2×#127—22no—
20C2×#127C814yescomplemented
21C2×C4×D4—24no—
22C2×C4×D4—24no—
23C2×C4×D4—44no—
24C2×C4×D4—44no—
25C22×C4⋊C4C2×C412yes—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—24no—
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2311yescharacteristic
31C2×C22⋊C8—24no—
32#390—24no—
33#390D418yes—
34#386—232no—
35#382—24no—
36#376—24no—
37C2×#95—22no—
38C2×#95D414yes—
39#336—28no—
40C25—22no—
41C22×D4—28no—
42C22×D4—28no—
43C22×D4—28no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—42no—
48C22×D4—42no—
49C22×D4—42no—
50C22×D4—42no—
51C22×D4—88no—
52C22×D4C22⋊C414yes—
53C22×D4C2×C814yes—
54C22×D4M1614yes—
55C23×C4—22no—
56C23×C4—28no—
57C23×C4—44no—
58C23×C4—44no—
59C23×C4—88no—
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C412yes—
62C23×C4C22×C411yescharacteristic
63C23×C4C2×D412yes—
64C22×C8—44no—
65#127—216no—
66#127—216no—
67C4×D4—416no—
68C4×D4—416no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4C22⋊C414yes—
73C2×C22⋊C4—28no—
74C2×C22⋊C4—216no—
75C2×C22⋊C4—44no—
76C2×C22⋊C4—44no—
77C2×C22⋊C4—88no—
78C2×C42—24no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—42no—
83C2×C42C2×C812yes—
84C2×C42M1612yes—
85C2×C42C2×D414yes—
86C22⋊C8—48no—
87#95—28no—
88#95—28no—
89C24—24no—
90C24—24no—
91C24—24no—
92C24—48no—
93C24—48no—
94C24—42no—
95C24—42no—
96C24—42no—
97C24—42no—
98C24C2×C22⋊C411yescharacteristic
99C2×D4—232no—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4—44no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—44no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—816no—
116C2×D4—816no—
117C2×D4—816no—
118C2×D4C22⋊C818yes—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—44no—
128C22×C4—416no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—48no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—816no—
135C22×C4—816no—
136C22×C4—816no—
137C22×C4C22⋊C814yes—
138C22×C4#10014yes—
139C22×C4C2×C22⋊C412yes—
140C22×C4#12014yes—
141C22×C4C22×C811yescharacteristic
142C22×C4C2×M1612yes—
143C22×C4C2×M1611yescharacteristic
144C2×C8—84no—
145C2×C8—88no—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C22⋊C4—416no—
149C22⋊C4—816no—
150C42—28no—
151C42—24no—
152C42—24no—
153C42—44no—
154C23—24no—
155C23—28no—
156C23—416no—
157C23—416no—
158C23—48no—
159C23—48no—
160C23—44no—
161C23—44no—
162C23—44no—
163C23—44no—
164C23—44no—
165C23—44no—
166C23—84no—
167C23—84no—
168C23—88no—
169C23—88no—
170C23—84no—
171C23—84no—
172C23#37911yescharacteristic
173C23C2×C22⋊C811yescharacteristic
174C23#40711yescharacteristic
175C23#41012yes—
176C23C2×#10012yes—
177D4—48no—
178D4—48no—
179D4—48no—
180D4—48no—
181D4—832no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—28no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—48no—
199C2×C4—44no—
200C2×C4—84no—
201C2×C4—832no—
202C2×C4—84no—
203C2×C4—84no—
204C2×C4C2×C22⋊C812yes—
205C2×C4#40712yes—
206C8—164no—
207C22—24no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—48no—
213C22—416no—
214C22—88no—
215C22—88no—
216C22—88no—
217C22—88no—
218C22—88no—
219C22—84no—
220C22—84no—
221C22#146012yes—
222C22#146612yes—
223C22#177911yescharacteristic
224C22#178211yescharacteristic
225C22#178311yescharacteristic
226C22#179514yes—
227C4—22no—
228C4—22no—
229C4—44no—
230C4—44no—
231C4—44no—
232C4—84no—
233C2—22no—
234C2—22no—
235C2—88no—
236C2#830211yescharacteristic
237C2#832212yes—
238C2#851812yes—
239C2#852212yes—
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC22×#127C4
LCSG ▷ C22×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. #8699C2semidirect
3. #7592C2semidirect
20. C2×#127C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23282
#86992
#75922