Finite Group Databasebeta

← #134563

Subgroups — #134563

Order 512.

311 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C4×#491C211yesmaximal, characteristic
2#12658C212yesmaximal
3C2×#1356C212yesmaximal
4#7266C212yesmaximal
5C2×#491C2211yescharacteristic
6C4×C4⋊Q8C411yescharacteristic
7C4×#125C411yescharacteristic
8#2150—22no—
9C2×#347C2214yes—
10#1992C411yescharacteristic
11#1960C411yescharacteristic
12#1935—22no—
13#1836C2211yescharacteristic
14C4×#103—22no—
15C4×#97C2211yescharacteristic
16#1375—22no—
17#1356—24no—
18C2×C4⋊Q8C2×C411yescharacteristic
19C2×#125C2×C411yescharacteristic
20C2×C4×Q8—21no—
21C2×M32—24no—
22#491—22no—
23#491D412yes—
24#491Q812yes—
25C2×#103—22no—
26C2×#103—22no—
27C2×#97C2×C411yescharacteristic
28C2×#97D412yes—
29C2×#97C2311yescharacteristic
30C2×C4×C8D412yes—
31#391—21no—
32#389—21no—
33#389—41no—
34#387—41no—
35#384C2×C411yescharacteristic
36#382C2×C411yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—41no—
40C43C2×C411yescharacteristic
41C4⋊C16—24no—
42C4⋊C16—24no—
43#347—24no—
44#347—24no—
45#340—22no—
46#340—22no—
47C22×Q8—21no—
48C22×C8C22⋊C412yes—
49C22×C8C2×D412yes—
50C4⋊Q8—21no—
51C4⋊Q8—41no—
52C4⋊Q8C4⋊C412yes—
53#125—21no—
54#125—41no—
55#125C4⋊C412yes—
56C4×Q8—42no—
57C4×Q8—42no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42—41no—
69C2×C42—41no—
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22⋊C411yescharacteristic
72C2×C42C22×C411yescharacteristic
73M32—44no—
74M32—44no—
75C2×C16—28no—
76#103—24no—
77#103—24no—
78#103—44no—
79#97—41no—
80#97—42no—
81#97C2×D411yescharacteristic
82#97C2×Q811yescharacteristic
83#97#4112yes—
84C4×C8—22no—
85C4×C8—22no—
86#95—41no—
87#95—41no—
88#95—42no—
89#95—41no—
90C2×Q8—22no—
91C2×Q8—41no—
92C2×Q8—42no—
93C2×Q8—42no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4#10011yescharacteristic
101C22×C4#10011yescharacteristic
102C22×C4C2×C22⋊C411yescharacteristic
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8#12112yes—
109C2×C8C22⋊Q812yes—
110C4⋊C4—22no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4—42no—
115C4⋊C4—41no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—82no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C42—21no—
122C42—22no—
123C42—42no—
124C42—82no—
125C42—81no—
126C42—81no—
127C42—81no—
128C42#10412yes—
129C42#10412yes—
130C42C2×C4⋊C411yescharacteristic
131C42#11711yescharacteristic
132C16—48no—
133C23C2×#10011yescharacteristic
134Q8—44no—
135Q8—82no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—82no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4#32911yescharacteristic
157C2×C4#32912yes—
158C2×C4#32911yescharacteristic
159C2×C4#38011yescharacteristic
160C2×C4#41311yescharacteristic
161C2×C4C2×#10412yes—
162C2×C4#42112yes—
163C8—22no—
164C8—22no—
165C8—42no—
166C22—21no—
167C22—21no—
168C22C2×#32911yescharacteristic
169C22#153611yescharacteristic
170C22#180011yescharacteristic
171C4—21no—
172C4—22no—
173C4—41no—
174C4—41no—
175C4—81no—
176C4—82no—
177C4—82no—
178C4—161no—
179C4—161no—
180C4#135612yescomplemented
181C4#152912yes—
182C4#180612yes—
183C2—21no—
184C2—21no—
185C2#857911yescharacteristic
186C2C2×#135611yescharacteristic
187C2#1191111yescharacteristic
subgroupG/H
center, socleC22C2×#329
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
180. C4#1356semidirect

Maximal subgroups

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

subgroupindex
C4×#4912
#126582
C2×#13562
#72662