Finite Group Databasebeta

← #136770

Subgroups — #136770

Order 512.

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

normal subgroups only

This group has 323 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#33494C211yesmaximal, characteristic
2#21661C211yesmaximal, characteristic
3#11640C212yesmaximal, complemented
4#11490C211yesmaximal, characteristic
5#7150C212yesmaximal, complemented
6#3135C411yescharacteristic
7#2960C2211yescharacteristic
8#2931C2211yescharacteristic
9#2168—24no—
10#2163C2212yes—
11#2162C2212yes—
12#2113—22no—
13#1995C411yescharacteristic
14#1963C411yescharacteristic
15#1938—22no—
16#1838C411yescharacteristic
17#1764—22no—
18#1726C2211yescharacteristic
19Q8⋊C16—24no—
20#1315—22no—
21C2×#131D412yes—
22#533C2×C411yescharacteristic
23#532C2×C411yescharacteristic
24C2×M32—22no—
25C22×C16—22no—
26#491—21no—
27#490—21no—
28#489C814yes—
29#446—22no—
30C8×Q8—22no—
31#436—22no—
32C8×D4—22no—
33#424—21no—
34#423C2×C411yescharacteristic
35#423C2×C411yescharacteristic
36C2×#104—22no—
37#416—22no—
38#408D412yes—
39#405—21no—
40C2×#97—21no—
41C2×#97C2×C411yescharacteristic
42C2×C4×C8—21no—
43C2×C4×C8C2311yescharacteristic
44#376C2×C411yescharacteristic
45C4⋊C16—22no—
46C4⋊C16D414yes—
47C22⋊C16—24no—
48C22⋊C16—22no—
49C22⋊C16—22no—
50#346—22no—
51C4×C16—22no—
52#341—22no—
53#339—22no—
54C2×#41C22⋊C412yes—
55#131—42no—
56#131—42no—
57C2×M16—22no—
58C2×M16—21no—
59C2×M16C22⋊C412yes—
60C22×C8—21no—
61C22×C8—22no—
62C22×C8C2×D411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C4⋊Q8C2×C812yes—
65C4⋊Q8M1611yescharacteristic
66#127M1611yescharacteristic
67#125—21no—
68#124C2×C812yes—
69C22⋊Q8—22no—
70#121—22no—
71C4×Q8—22no—
72C4×D4—22no—
73#117—21no—
74C2×C42—21no—
75C2×C42C22×C411yescharacteristic
76M32—82no—
77C2×C16—24no—
78C2×C16—42no—
79C2×C16—82no—
80C4⋊C8—22no—
81C4⋊C8—22no—
82C4⋊C8—42no—
83#104—44no—
84#103—24no—
85#102—24no—
86C22⋊C8—24no—
87#97—21no—
88#97—42no—
89#97—41no—
90#97C2×C812yes—
91C4×C8—42no—
92C4×C8—41no—
93C4×C8C2×D412yes—
94C4×C8#4112yes—
95#95—22no—
96#41—42no—
97#41—42no—
98C2×Q8—22no—
99C2×Q8C22⋊C812yes—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4C22⋊C812yes—
103C22×C4—22no—
104C22×C4—41no—
105C22×C4C2×C22⋊C411yescharacteristic
106C22×C4#13112yes—
107M16—42no—
108M16—42no—
109M16—82no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—42no—
117C2×C8—42no—
118C2×C8—41no—
119C2×C8#12011yescharacteristic
120C2×C8#12112yes—
121C2×C8#12311yescharacteristic
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—42no—
126C22⋊C4—24no—
127C22⋊C4—41no—
128C42—22no—
129C42—41no—
130C42—41no—
131C42C4×D412yes—
132C42C22×C811yescharacteristic
133C42C2×M1611yescharacteristic
134C16—44no—
135C16—82no—
136C23—22no—
137C23#40811yescharacteristic
138Q8—42no—
139Q8—44no—
140D4—42no—
141D4—82no—
142D4—82no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—82no—
152C2×C4#38611yescharacteristic
153C2×C4C2×C22⋊C811yescharacteristic
154C2×C4C8×D412yes—
155C2×C4#43512yes—
156C8—22no—
157C8—21no—
158C8—42no—
159C8—42no—
160C8—81no—
161C8#45512yes—
162C22—22no—
163C22—21no—
164C22—44no—
165C22—42no—
166C22#186911yescharacteristic
167C4—22no—
168C4—22no—
169C4—21no—
170C4—42no—
171C4—82no—
172C4—82no—
173C4#188411yescharacteristic
174C4#188511yescharacteristic
175C2—41no—
176C2—82no—
177C2#895711yescharacteristic
178C2#1297912yes—
subgroupG/H
centerC2×C8#120
commutatorC42C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radical#532C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ 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
3. #11640C2semidirect
5. #7150C2semidirect

Maximal subgroups

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

subgroupindex
#334942
#216612
#116402
#114902
#71502