Finite Group Databasebeta

← #97485

Subgroups — #97485

Order 512.

730 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#35649C211yesmaximal, characteristic
2#8306C211yesmaximal, characteristic, complemented
3#8306C211yesmaximal, characteristic, complemented
4#7645C214yesmaximal, complemented
5C2×#537C411yescharacteristic
6C22×#125C2211yescharacteristic
7#2591—22no—
8#2504—22no—
9#2490—24no—
10#2489—24no—
11#2311C412yes—
12#2297C2212yes—
13#2274C411yescharacteristic
14#1817C2212yes—
15#1817C2212yes—
16C2×#324—22no—
17C2×#324—22no—
18#1272—24no—
19#1270—24no—
20C22×#41C2×C412yes—
21#537—44no—
22C2×#125—24no—
23C2×#125—44no—
24C2×C22⋊Q8—22no—
25C2×#121—22no—
26C2×#121—22no—
27C2×#117C2×C412yes—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D412yes—
31C22×C4⋊C4D412yes—
32C22×C42C2311yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35#400—24no—
36#397—28no—
37#394—24no—
38#391—22no—
39#391D414yes—
40#390—24no—
41#390—22no—
42#390D414yes—
43#388—28no—
44#386—24no—
45#386—22no—
46#386—22no—
47#382—24no—
48#380—24no—
49C4×C4⋊C4—24no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—24no—
52#336—24no—
53#336—24no—
54#324—44no—
55#324—44no—
56C2×#41—44no—
57C2×#41—44no—
58C2×#41—44no—
59C22×Q8C22⋊C412yes—
60C22×D4—22no—
61C22×D4—22no—
62C22×D4C22⋊C412yes—
63C23×C4—22no—
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C412yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C22×C8—42no—
70C22×C8—42no—
71#125—416no—
72C22⋊Q8—48no—
73#121—48no—
74#121—44no—
75#121—44no—
76#117—44no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4C22⋊C414yes—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—28no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C42—22no—
95C2×C42—24no—
96C2×C42—42no—
97C2×C42C2×D414yes—
98C22⋊C8—44no—
99C22⋊C8—44no—
100#95—24no—
101#95—28no—
102#95—24no—
103#95—24no—
104C24—22no—
105C24C2×C22⋊C411yescharacteristic
106#41—816no—
107C2×Q8—24no—
108C2×Q8—42no—
109C2×Q8—42no—
110C2×D4—24no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—84no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4#9914yes—
137C22×C4C2×C22⋊C412yes—
138C22×C4#12012yes—
139C22×C4#12012yes—
140C2×C8—82no—
141C2×C8—82no—
142C2×C8—84no—
143C2×C8—84no—
144C4⋊C4—28no—
145C4⋊C4—48no—
146C4⋊C4—48no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C22⋊C4—48no—
151C22⋊C4—44no—
152C22⋊C4—84no—
153C22⋊C4—84no—
154C22⋊C4—88no—
155C42—44no—
156C42—44no—
157C42—84no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—44no—
162C23—44no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23#37911yescharacteristic
167C23C2×#9912yes—
168Q8—84no—
169D4—84no—
170D4—88no—
171D4—164no—
172C2×C4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—88no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4#41314yes—
197C8—162no—
198C8—162no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—44no—
203C22—82no—
204C22—82no—
205C22—84no—
206C22—84no—
207C22—82no—
208C22#178111yescharacteristic
209C22#178512yes—
210C22#179412yes—
211C22#179412yes—
212C4—24no—
213C4—42no—
214C4—44no—
215C4—42no—
216C4—82no—
217C4—84no—
218C4—82no—
219C2—42no—
220C2—162no—
221C2#846211yescharacteristic
222C2#846411yescharacteristic
223C2#847614yes—
224C2#855211yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#537C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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. #8306C2semidirect
3. #8306C2semidirect
4. #7645C2semidirect

Maximal subgroups

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

subgroupindex
#356492
#83062
#76452