Finite Group Databasebeta

← #139384

Subgroups — #139384

Order 512.

324 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 90 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#33497C211yesmaximal, characteristic
2#28676C211yesmaximal, characteristic
3#11972C212yesmaximal
4#8617C211yesmaximal, characteristic
5#7296C212yesmaximal
6C2×#500C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2567—21no—
9#2183—24no—
10#2163—22no—
11#2162C2212yes—
12#2087C414yes—
13#1987—21no—
14#1942—22no—
15#1801—21no—
16C4×#107C2211yescharacteristic
17#1567C2212yes—
18#1365—24no—
19#1292—21no—
20C2×C4⋊Q8—21no—
21C2×C4×Q8D411yescharacteristic
22#500—22no—
23#500—22no—
24C8×Q8—22no—
25C8×Q8—22no—
26C2×#107—21no—
27C2×#107—21no—
28C2×#107C2×C412yes—
29#423—22no—
30C2×C4⋊C8C2×C412yes—
31C2×C4⋊C8D411yescharacteristic
32C2×#101—41no—
33C2×C4×C8D412yes—
34C2×C4×C8C2311yescharacteristic
35#395—42no—
36#391—41no—
37#389—24no—
38#384—41no—
39C4×C4⋊C4—21no—
40C4⋊C16—24no—
41C4⋊C16D414yes—
42C22⋊C16—22no—
43#346—24no—
44C4×C16—22no—
45#343—41no—
46#340—24no—
47#340C2×C412yes—
48#335—22no—
49#335—22no—
50C22×Q8—41no—
51C22×Q8D811yescharacteristic
52C2×M16—42no—
53C22×C8C22⋊C412yes—
54C22×C8C22×C411yescharacteristic
55C22×C8C2×D411yescharacteristic
56C4⋊Q8—42no—
57C4⋊Q8—42no—
58C4×Q8—22no—
59C4×Q8—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—44no—
64C2×C4⋊C4D811yescharacteristic
65C2×C4⋊C4SD1612yes—
66C2×C42—41no—
67C2×C42C22⋊C412yes—
68C2×C42C2×D411yescharacteristic
69C2×C16—24no—
70#107—24no—
71#107—24no—
72#107—41no—
73#107—41no—
74C4⋊C8—24no—
75C4⋊C8—22no—
76C4⋊C8—42no—
77C4⋊C8—42no—
78#101—82no—
79C4×C8—22no—
80C4×C8—22no—
81C4×C8C2×D412yes—
82C4×C8#4112yes—
83#95—44no—
84#95—41no—
85C2×Q8—22no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—82no—
90C2×Q8—162no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4#10212yes—
94C22×C4#10212yes—
95C22×C4C2×C22⋊C411yescharacteristic
96C22×C4C2×D811yescharacteristic
97C22×C4C2×SD1611yescharacteristic
98M16—82no—
99M16—82no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—41no—
103C2×C8—42no—
104C2×C8—82no—
105C2×C8C4×D412yes—
106C2×C8#12011yescharacteristic
107C2×C8#12311yescharacteristic
108C4⋊C4—24no—
109C4⋊C4—22no—
110C4⋊C4—24no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4—82no—
115C4⋊C4—164no—
116C42—22no—
117C42—22no—
118C42—81no—
119C42—81no—
120C42#12112yes—
121C42#13512yes—
122C16—44no—
123C23C2×#10211yescharacteristic
124Q8—42no—
125Q8—42no—
126Q8—164no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—44no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4—84no—
139C2×C4—82no—
140C2×C4#38611yescharacteristic
141C2×C4#41611yescharacteristic
142C2×C4#46512yes—
143C2×C4#46612yes—
144C2×C4#50911yescharacteristic
145C2×C4#51011yescharacteristic
146C2×C4#51012yes—
147C8—21no—
148C8—22no—
149C8—42no—
150C8—162no—
151C8#45612yes—
152C22—21no—
153C22—21no—
154C22#189311yescharacteristic
155C22#213911yescharacteristic
156C22#214111yescharacteristic
157C4—22no—
158C4—21no—
159C4—42no—
160C4—42no—
161C4—84no—
162C4—162no—
163C4#188611yescharacteristic
164C4#191511yescharacteristic
165C4#221912yes—
166C4#222012yes—
167C2—21no—
168C2—21no—
169C2#939411yescharacteristic
170C2#1306211yescharacteristic
171C2#1307911yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23C2×#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#334972
#286762
#119722
#86172
#72962