Finite Group Databasebeta

← #97484

Subgroups — #97484

Order 512.

660 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 98 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#35640C211yesmaximal, characteristic
2#8306C211yesmaximal, characteristic
3#8306C211yesmaximal, characteristic
4#7646C214yesmaximal
5C2×#544C411yescharacteristic
6C22×#125C2211yescharacteristic
7#2583—22no—
8#2552—24no—
9#2550—24no—
10#2502—22no—
11#2317C412yes—
12#2296C2212yes—
13#2274C411yescharacteristic
14#1817C2212yes—
15#1817C2212yes—
16C2×#324—22no—
17C2×#324—22no—
18#1271—24no—
19#1271—24no—
20#544—44no—
21C2×C4⋊Q8—22no—
22C2×#125—22no—
23C2×#125—24no—
24C2×#125—44no—
25C2×C22⋊Q8—22no—
26C2×#117C2×C412yes—
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#398—24no—
36#398—28no—
37#395—24no—
38#391—24no—
39#389—28no—
40#389—24no—
41#389D414yes—
42#389D414yes—
43#385—24no—
44#385—22no—
45#385—22no—
46#384—24no—
47#384—22no—
48#382—22no—
49C4×C4⋊C4—24no—
50C4×C4⋊C4—24no—
51C4×C22⋊C4—22no—
52#336—24no—
53#336—24no—
54#324—44no—
55#324—44no—
56C22×Q8—22no—
57C23×C4C22⋊C412yes—
58C23×C4C22⋊C412yes—
59C23×C4C22×C411yescharacteristic
60C23×C4C2×D411yescharacteristic
61C23×C4C2×D411yescharacteristic
62C22×C8—42no—
63C22×C8—42no—
64C4⋊Q8—48no—
65#125—416no—
66#125—48no—
67C22⋊Q8—44no—
68C22⋊Q8—44no—
69#117—28no—
70#117—44no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—28no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4C22⋊C414yes—
87C2×C4⋊C4C22⋊C412yes—
88C2×C4⋊C4C22⋊C412yes—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—24no—
94C2×C42—42no—
95C2×C42C2×D414yes—
96C22⋊C8—44no—
97C22⋊C8—44no—
98#95—24no—
99#95—28no—
100#95—24no—
101#95—24no—
102C24C2×C22⋊C411yescharacteristic
103C2×Q8—44no—
104C2×Q8—84no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4#9914yes—
122C22×C4C2×C22⋊C412yes—
123C22×C4#12012yes—
124C22×C4#12012yes—
125C2×C8—82no—
126C2×C8—82no—
127C2×C8—84no—
128C2×C8—84no—
129C4⋊C4—28no—
130C4⋊C4—24no—
131C4⋊C4—48no—
132C4⋊C4—48no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—44no—
136C4⋊C4—44no—
137C4⋊C4—48no—
138C4⋊C4—84no—
139C4⋊C4—84no—
140C4⋊C4—88no—
141C4⋊C4#10118yes—
142C22⋊C4—28no—
143C22⋊C4—44no—
144C42—28no—
145C42—44no—
146C42—44no—
147C42—84no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23#37911yescharacteristic
152C23C2×#9912yes—
153Q8—164no—
154C2×C4—28no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—28no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4C2×#10114yes—
179C8—162no—
180C8—162no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—44no—
185C22#178111yescharacteristic
186C22#178412yes—
187C22#179412yes—
188C22#179412yes—
189C4—24no—
190C4—42no—
191C4—44no—
192C4—42no—
193C4—48no—
194C4—82no—
195C4—82no—
196C4—82no—
197C2—42no—
198C2#846211yescharacteristic
199C2#846411yescharacteristic
200C2#847814yes—
201C2#855111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#356402
#83062
#76462