Finite Group Databasebeta

← #135545

Subgroups — #135545

Order 512.

467 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 60 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#21798C211yesmaximal, characteristic
2#12699C212yesmaximal
3#7317C212yesmaximal, complemented
4#7315C212yesmaximal, complemented
5C22×#124C2211yescharacteristic
6#2920C412yes—
7C2×#355—24no—
8#1967—24no—
9#1784C2214yes—
10#1784C2211yescharacteristic
11#1784C2211yescharacteristic
12#1783C412yes—
13C23×Q8C2311yescharacteristic
14C23×D4C2×C411yescharacteristic
15C2×#124—22no—
16C2×#124—42no—
17C2×#124—42no—
18C2×#124D414yes—
19C22×C22⋊C4—22no—
20C22×C42C2×C411yescharacteristic
21#432—42no—
22#431—42no—
23#423—22no—
24C2×#101—24no—
25C2×#101—22no—
26C2×#101—22no—
27C2×#101—24no—
28C2×#100—24no—
29#407—24no—
30#407C2×C412yes—
31#407C2×C412yes—
32C4×M16—22no—
33#355—44no—
34#355—44no—
35C25—21no—
36C22×Q8—42no—
37C22×Q8—82no—
38C22×Q8C2×D412yes—
39C22×Q8#4112yes—
40C22×D4—21no—
41C22×D4—42no—
42C22×D4—82no—
43C22×D4C22⋊C412yes—
44C23×C4—21no—
45C23×C4C22×C411yescharacteristic
46C23×C4#4112yes—
47C2×M16—22no—
48C2×M16—22no—
49C2×M16—44no—
50#124—24no—
51#124—24no—
52#124—44no—
53#124—42no—
54#124—44no—
55#124—42no—
56C2×C22⋊C4—28no—
57C2×C22⋊C4—44no—
58C2×C22⋊C4—42no—
59C2×C22⋊C4—42no—
60C2×C42—21no—
61C2×C42—22no—
62C2×C42—22no—
63C2×C42C22⋊C412yes—
64C4⋊C8—44no—
65#101—48no—
66#101—82no—
67#101—82no—
68#101—84no—
69#100—84no—
70C22⋊C8—42no—
71C22⋊C8—42no—
72C22⋊C8—84no—
73#97—42no—
74C4×C8—42no—
75C24—22no—
76C24—42no—
77C24—82no—
78C24—84no—
79C24#11711yescharacteristic
80C2×Q8—24no—
81C2×Q8—42no—
82C2×Q8—42no—
83C2×Q8—44no—
84C2×Q8—42no—
85C2×Q8—88no—
86C2×D4—24no—
87C2×D4—44no—
88C2×D4—42no—
89C2×D4—82no—
90C2×D4—88no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—42no—
97C22×C4—44no—
98C22×C4—42no—
99C22×C4—82no—
100C22×C4C2×C22⋊C411yescharacteristic
101C22×C4#11711yescharacteristic
102C22×C4#12314yes—
103M16—44no—
104M16—44no—
105M16—84no—
106M16—84no—
107C2×C8—42no—
108C2×C8—42no—
109C2×C8—84no—
110C22⋊C4—48no—
111C22⋊C4—48no—
112C42—21no—
113C42—22no—
114C42—21no—
115C42—22no—
116C42#9912yes—
117C42#9912yes—
118C23—21no—
119C23—22no—
120C23—42no—
121C23—42no—
122C23—44no—
123C23—84no—
124C23—84no—
125C23—84no—
126C23—84no—
127C23—164no—
128C23—164no—
129C23#38111yescharacteristic
130Q8—84no—
131Q8—84no—
132D4—84no—
133D4—84no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—44no—
143C2×C4—44no—
144C2×C4—82no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4—84no—
148C2×C4C2×#9912yes—
149C8—82no—
150C8—82no—
151C8—164no—
152C22—22no—
153C22—24no—
154C22—41no—
155C22—41no—
156C22—42no—
157C22—41no—
158C22—84no—
159C22—82no—
160C22—164no—
161C22—168no—
162C22#181911yescharacteristic
163C22#182812yes—
164C4—22no—
165C4—42no—
166C4—42no—
167C4—44no—
168C4—41no—
169C2—21no—
170C2—22no—
171C2—42no—
172C2—162no—
173C2#874811yescharacteristic
subgroupG/H
center, socleC2#8748
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7317C2semidirect
4. #7315C2semidirect

Maximal subgroups

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

subgroupindex
#217982
#126992
#73172
#73152