Finite Group Databasebeta

← #136805

Subgroups — #136805

Order 512.

312 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 88 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#33501C211yesmaximal, characteristic, complemented
2#21660C211yesmaximal, characteristic
3#11635C212yesmaximal
4#11497C211yesmaximal, characteristic, complemented
5#7149C212yesmaximal, complemented
6#3131C411yescharacteristic
7#2960C2211yescharacteristic
8#2931C2211yescharacteristic
9#2183—24no—
10#2164—22no—
11#2163C2212yes—
12#2163C2212yes—
13#1994C411yescharacteristic
14#1962C411yescharacteristic
15#1938—22no—
16#1837C411yescharacteristic
17#1765—22no—
18C4×#97C2211yescharacteristic
19#1330—24no—
20#1323—22no—
21C2×#131—21no—
22#533C2×C411yescharacteristic
23#532C2×C411yescharacteristic
24#488—22no—
25#487C812yes—
26#486C812yes—
27#446—22no—
28C8×Q8—22no—
29#436—22no—
30C8×D4—22no—
31#424D412yes—
32#423C2×C411yescharacteristic
33#423C2×C411yescharacteristic
34C2×#104—22no—
35#416—22no—
36#408—21no—
37#405D412yes—
38C2×#97—21no—
39C2×#97—21no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8C2311yescharacteristic
42C43C2×C411yescharacteristic
43C4⋊C16—24no—
44C4⋊C16—22no—
45C4⋊C16D414yes—
46C22⋊C16—22no—
47C22⋊C16—22no—
48#346—22no—
49#346—22no—
50#346—22no—
51C4×C16—22no—
52#341—22no—
53#339—22no—
54C2×#41—21no—
55#131—44no—
56C2×M16—21no—
57C2×M16—41no—
58C2×M16C22⋊C412yes—
59C22×C8—21no—
60C22×C8—41no—
61C22×C8C2×D411yescharacteristic
62C22×C8C2×D411yescharacteristic
63C4⋊Q8C2×C811yescharacteristic
64C4⋊Q8C2×C812yes—
65#127C2×C811yescharacteristic
66#125M1612yes—
67#124—21no—
68C22⋊Q8—22no—
69#121—22no—
70C4×Q8—22no—
71C4×D4—22no—
72#117C22⋊C412yes—
73C2×C42—21no—
74C2×C42—22no—
75C2×C42C22×C411yescharacteristic
76C2×C16—24no—
77C2×C16—42no—
78C4⋊C8—24no—
79C4⋊C8—22no—
80C4⋊C8—22no—
81#104—44no—
82#103—24no—
83#102—24no—
84C22⋊C8—42no—
85#97—22no—
86#97—42no—
87#97—42no—
88C4×C8—22no—
89C4×C8—21no—
90C4×C8C2×C812yes—
91C4×C8C2×D412yes—
92C4×C8#4112yes—
93#41—42no—
94#41—42no—
95C2×Q8—22no—
96C2×Q8—21no—
97C2×D4—21no—
98C2×D4—22no—
99C2×D4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4C2×C22⋊C411yescharacteristic
103C22×C4#13112yes—
104M16—42no—
105M16—42no—
106M16—82no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—82no—
115C2×C8#12011yescharacteristic
116C2×C8#12112yes—
117C2×C8#12311yescharacteristic
118C4⋊C4—22no—
119C4⋊C4—24no—
120C4⋊C4—22no—
121C4⋊C4C22⋊C814yes—
122C22⋊C4—22no—
123C22⋊C4—42no—
124C42—22no—
125C42—22no—
126C42—41no—
127C42—82no—
128C42C4×D412yes—
129C42C22×C811yescharacteristic
130C42C2×M1611yescharacteristic
131C16—44no—
132C16—82no—
133C23—41no—
134C23#40811yescharacteristic
135Q8—42no—
136Q8—82no—
137D4—42no—
138D4—82no—
139D4—82no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—24no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—82no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4#38611yescharacteristic
150C2×C4C2×C22⋊C811yescharacteristic
151C2×C4C8×D412yes—
152C2×C4#43512yes—
153C8—22no—
154C8—21no—
155C8—42no—
156C8—42no—
157C8—81no—
158C8#45412yes—
159C22—21no—
160C22—22no—
161C22—81no—
162C22—82no—
163C22#186911yescharacteristic
164C4—21no—
165C4—22no—
166C4—22no—
167C4—44no—
168C4—82no—
169C4—81no—
170C4#188411yescharacteristic
171C4#188811yescharacteristic
172C2—41no—
173C2—161no—
174C2#896011yescharacteristic
175C2#1298612yes—
subgroupG/H
centerC2×C4#386
commutatorC42C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radical#532C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33501C2semidirect
4. #11497C2semidirect
5. #7149C2semidirect

Maximal subgroups

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

subgroupindex
#335012
#216602
#116352
#114972
#71492