Finite Group Databasebeta

← #134135

Subgroups — #134135

Order 512.

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

normal subgroups only

This group has 734 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#35576C211yesmaximal, characteristic
2#11514C214yesmaximal
3#8551C212yesmaximal
4C2×#544C411yescharacteristic
5C22×C4⋊Q8C2211yescharacteristic
6#2583—22no—
7#2551—24no—
8#2550—24no—
9#2434—22no—
10#2317C2212yes—
11#2296C412yes—
12#2274C411yescharacteristic
13#1784—22no—
14#1463—22no—
15#1463C2214yes—
16#1325—24no—
17#1324—24no—
18C23×Q8D412yes—
19#544—44no—
20C2×C4⋊Q8—24no—
21C2×C4⋊Q8—22no—
22C2×C4⋊Q8—24no—
23C2×C4⋊Q8—44no—
24C2×#125—22no—
25C2×C22⋊Q8—22no—
26C2×#117C2×C412yes—
27C2×#117C2×C412yes—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D412yes—
30C22×C4⋊C4C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32C2×#101—44no—
33#407—24no—
34#398—24no—
35#395—24no—
36#395—28no—
37#391—24no—
38#389—28no—
39#389—22no—
40#389—22no—
41#389—24no—
42#385—22no—
43#385—22no—
44#385—24no—
45#384D414yes—
46#382D414yes—
47C4×C4⋊C4—24no—
48C4×C4⋊C4—24no—
49C4×C22⋊C4—22no—
50#324—28no—
51#324—44no—
52C22×Q8—22no—
53C22×Q8—24no—
54C22×Q8—24no—
55C22×Q8—42no—
56C22×Q8—84no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C412yes—
60C23×C4C22×C411yescharacteristic
61C23×C4C2×D412yes—
62C2×M16—44no—
63C4⋊Q8—416no—
64C4⋊Q8—48no—
65C4⋊Q8—88no—
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—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—28no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4C22⋊C412yes—
84C2×C4⋊C4C22⋊C412yes—
85C2×C4⋊C4C2×D414yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—24no—
91C2×C42—42no—
92C2×C42C22⋊C414yes—
93#101—88no—
94C22⋊C8—44no—
95C22⋊C8—44no—
96#95—24no—
97#95—24no—
98#95—24no—
99#95—28no—
100C24C2×C22⋊C411yescharacteristic
101C2×Q8—48no—
102C2×Q8—48no—
103C2×Q8—48no—
104C2×Q8—44no—
105C2×Q8—84no—
106C2×Q8—84no—
107C2×Q8—84no—
108C2×Q8—84no—
109C2×Q8—84no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—82no—
126C22×C4—82no—
127C22×C4#9914yes—
128C22×C4C2×C22⋊C412yes—
129C22×C4#12014yes—
130M16—88no—
131C2×C8—84no—
132C4⋊C4—28no—
133C4⋊C4—28no—
134C4⋊C4—44no—
135C4⋊C4—48no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—48no—
139C4⋊C4—44no—
140C4⋊C4—48no—
141C4⋊C4—48no—
142C4⋊C4—88no—
143C22⋊C4—28no—
144C22⋊C4—44no—
145C42—24no—
146C42—44no—
147C42—44no—
148C42—84no—
149C42#10118yescomplemented
150C23—22no—
151C23—41no—
152C23—41no—
153C23#37911yescharacteristic
154C23C2×#9912yes—
155Q8—816no—
156Q8—164no—
157C2×C4—28no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—28no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4C2×#10114yes—
186C8—164no—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—44no—
193C22#178111yescharacteristic
194C22#178412yes—
195C4—24no—
196C4—48no—
197C4—44no—
198C4—42no—
199C4—44no—
200C4—42no—
201C4—82no—
202C4—82no—
203C2—22no—
204C2—42no—
205C2#855111yescharacteristic
206C2#1270711yescharacteristic
207C2#1271011yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
149. C42#101semidirect

Maximal subgroups

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

subgroupindex
#355762
#115142
#85512