Finite Group Databasebeta

← #149772

Subgroups — #149772

Order 512.

482 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 66 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×#1960C211yesmaximal, characteristic
2#11511C212yesmaximal
3#7353C214yesmaximal, complemented
4C22×C4⋊Q8C411yescharacteristic
5C22×#103—22no—
6C22×#97C411yescharacteristic
7C2×#382C2211yescharacteristic
8#1960—22no—
9#1960C412yes—
10#1960C2212yes—
11#1463—22no—
12#1324—22no—
13#1324C2214yes—
14C23×Q8—21no—
15C23×C8—21no—
16C2×C4⋊Q8—24no—
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8—41no—
19C2×C4⋊Q8—42no—
20C2×C4⋊Q8C2×C412yes—
21C22×C4⋊C4—41no—
22C22×C4⋊C4D412yes—
23C22×C42C2×C411yescharacteristic
24C2×#103—24no—
25C2×#103—24no—
26C2×#103—44no—
27C2×#103—44no—
28#407—42no—
29C2×#97—21no—
30C2×#97—42no—
31C2×#97C2×C412yes—
32#382—41no—
33#382C2×C411yescharacteristic
34#382D412yes—
35#382C2311yescharacteristic
36C2×#95—41no—
37#324—24no—
38#324—42no—
39C22×Q8—24no—
40C22×Q8—22no—
41C22×Q8—42no—
42C22×Q8—44no—
43C22×Q8—42no—
44C22×Q8—42no—
45C22×Q8—41no—
46C23×C4—21no—
47C23×C4—22no—
48C23×C4—41no—
49C23×C4C22⋊C411yescharacteristic
50C2×M16—82no—
51C22×C8—24no—
52C22×C8—22no—
53C22×C8—42no—
54C22×C8—42no—
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57C4⋊Q8—42no—
58C4⋊Q8—48no—
59C4⋊Q8—42no—
60C4⋊Q8M1614yescomplemented
61C2×C4⋊C4—22no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—48no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4C22⋊C412yes—
68C2×C4⋊C4C2×D412yes—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22×C411yescharacteristic
74#103—48no—
75#103—48no—
76C22⋊C8—44no—
77C22⋊C8—82no—
78#97—42no—
79#97—42no—
80#95—41no—
81#95—42no—
82#95—41no—
83C24#9911yescharacteristic
84C2×Q8—24no—
85C2×Q8—44no—
86C2×Q8—44no—
87C2×Q8—48no—
88C2×Q8—48no—
89C2×Q8—44no—
90C2×Q8—42no—
91C2×Q8—84no—
92C2×Q8—84no—
93C2×Q8—81no—
94C2×Q8—81no—
95C2×Q8—82no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—24no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—82no—
111C22×C4—82no—
112C22×C4—84no—
113C22×C4#9911yescharacteristic
114C22×C4C2×C22⋊C411yescharacteristic
115M16—164no—
116C2×C8—24no—
117C2×C8—48no—
118C2×C8—42no—
119C2×C8—42no—
120C2×C8—84no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—44no—
124C4⋊C4—48no—
125C4⋊C4—44no—
126C4⋊C4—82no—
127C42—24no—
128C42—21no—
129C42—21no—
130C42#10012yes—
131C42C2×M1612yes—
132C23—21no—
133C23—24no—
134C23—21no—
135C23#35312yes—
136C23C2×#9911yescharacteristic
137Q8—48no—
138Q8—84no—
139Q8—84no—
140Q8—84no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—48no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—44no—
155C2×C4—88no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—84no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—162no—
162C2×C4#35512yes—
163C2×C4#40711yescharacteristic
164C2×C4C2×#10011yescharacteristic
165C8—44no—
166C8—164no—
167C22—24no—
168C22—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22#140714yes—
173C22#146211yescharacteristic
174C22C2×#35311yescharacteristic
175C22C2×#35511yescharacteristic
176C4—21no—
177C4—22no—
178C4—21no—
179C4—42no—
180C4—42no—
181C4—82no—
182C4—82no—
183C4—84no—
184C2—21no—
185C2—21no—
186C2—24no—
187C2#1150511yescharacteristic
188C2C2×#140712yes—
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7353C2semidirect
60. C4⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19602
#115112
#73532