Finite Group Databasebeta

← #136804

Subgroups — #136804

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#33500C211yesmaximal, characteristic, complemented
2#21663C211yesmaximal, characteristic
3#11633C212yesmaximal
4#11497C211yesmaximal, characteristic, complemented
5#7150C212yesmaximal, complemented
6#3135C411yescharacteristic
7#2960C2211yescharacteristic
8#2931C2211yescharacteristic
9#2183—22no—
10Q8×C16—22no—
11#2164—22no—
12#2163C2212yes—
13#2162C2212yes—
14#1995C411yescharacteristic
15#1963C411yescharacteristic
16#1937—22no—
17#1838C411yescharacteristic
18#1765—22no—
19#1726C2211yescharacteristic
20#1330—22no—
21#1330—22no—
22#1323—22no—
23C2×#131—21no—
24#533C2×C411yescharacteristic
25#532C2×C411yescharacteristic
26#491C812yes—
27#490C812yes—
28#489—22no—
29#446—22no—
30C8×Q8—22no—
31#436—22no—
32C8×D4—22no—
33#424D412yes—
34#423C2×C411yescharacteristic
35#423C2×C411yescharacteristic
36C2×#104—22no—
37#416—22no—
38#408—21no—
39#405D412yes—
40C2×#97—21no—
41C2×#97C2×C411yescharacteristic
42C2×C4×C8—21no—
43C2×C4×C8C2311yescharacteristic
44#376C2×C411yescharacteristic
45C4⋊C16—24no—
46C4⋊C16—22no—
47C4⋊C16D412yes—
48C4⋊C16D412yes—
49C22⋊C16—22no—
50C22⋊C16—22no—
51#346—22no—
52#346—22no—
53C4×C16—22no—
54C4×C16—22no—
55#341—22no—
56#339—22no—
57C2×#41—21no—
58#131—44no—
59C2×M16—21no—
60C2×M16—41no—
61C2×M16C22⋊C412yes—
62C22×C8—21no—
63C22×C8—41no—
64C22×C8C2×D411yescharacteristic
65C22×C8C2×D411yescharacteristic
66C4⋊Q8C2×C811yescharacteristic
67C4⋊Q8M1612yes—
68#127C2×C811yescharacteristic
69#125C2×C812yes—
70#124—21no—
71C22⋊Q8—22no—
72#121—22no—
73C4×Q8—22no—
74C4×D4—22no—
75#117C22⋊C412yes—
76C2×C42—21no—
77C2×C42C22×C411yescharacteristic
78C2×C16—22no—
79C2×C16—22no—
80C2×C16—42no—
81C4⋊C8—24no—
82C4⋊C8—22no—
83C4⋊C8—22no—
84#104—44no—
85#103—24no—
86#102—24no—
87C22⋊C8—42no—
88#97—22no—
89#97—21no—
90#97—42no—
91#97C2×C812yes—
92C4×C8—22no—
93C4×C8—42no—
94C4×C8C2×D412yes—
95C4×C8#4112yes—
96#95—22no—
97#41—42no—
98#41—42no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×D4—21no—
102C2×D4—22no—
103C2×D4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4C2×C22⋊C411yescharacteristic
107C22×C4#13112yes—
108M16—42no—
109M16—42no—
110M16—82no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—82no—
119C2×C8#12011yescharacteristic
120C2×C8#12112yes—
121C2×C8#12311yescharacteristic
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4C22⋊C814yes—
126C22⋊C4—22no—
127C22⋊C4—42no—
128C42—22no—
129C42—22no—
130C42—41no—
131C42C4×D412yes—
132C42C22×C811yescharacteristic
133C42C2×M1611yescharacteristic
134C16—42no—
135C16—42no—
136C16—82no—
137C23—41no—
138C23#40811yescharacteristic
139Q8—44no—
140Q8—42no—
141D4—42no—
142D4—82no—
143D4—82no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—24no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—82no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4#38611yescharacteristic
154C2×C4C2×C22⋊C811yescharacteristic
155C2×C4C8×D412yes—
156C2×C4#43512yes—
157C8—22no—
158C8—21no—
159C8—42no—
160C8—42no—
161C8—81no—
162C8#45412yes—
163C22—21no—
164C22—22no—
165C22—81no—
166C22—82no—
167C22#186911yescharacteristic
168C4—21no—
169C4—22no—
170C4—22no—
171C4—44no—
172C4—82no—
173C4—81no—
174C4#188411yescharacteristic
175C4#188811yescharacteristic
176C2—41no—
177C2—161no—
178C2#896011yescharacteristic
179C2#1298712yes—
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. #33500C2semidirect
4. #11497C2semidirect
5. #7150C2semidirect

Maximal subgroups

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

subgroupindex
#335002
#216632
#116332
#114972
#71502