Finite Group Databasebeta

← #139383

Subgroups — #139383

Order 512.

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

normal subgroups only

This group has 334 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#28675C211yesmaximal, characteristic
3#11971C212yesmaximal
4#8617C211yesmaximal, characteristic
5#7296C212yesmaximal
6C2×#500C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2546—21no—
9#2183—24no—
10#2163—22no—
11#2162C2212yes—
12#2086C414yes—
13#1988—21no—
14#1942—22no—
15#1801—21no—
16C4×#107C2211yescharacteristic
17#1567C2212yes—
18#1364—24no—
19#1292—21no—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8—41no—
22C2×C4×Q8D411yescharacteristic
23#500—22no—
24#500—22no—
25C8×Q8—22no—
26C8×Q8—22no—
27C2×#107—21no—
28C2×#107—21no—
29C2×#107C2×C412yes—
30#423—22no—
31C2×C4⋊C8C2×C412yes—
32C2×C4⋊C8D411yescharacteristic
33C2×#101—41no—
34C2×C4×C8D412yes—
35C2×C4×C8C2311yescharacteristic
36#398—42no—
37#391—41no—
38#389—24no—
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—84no—
59C4×Q8—22no—
60C4×Q8—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—44no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4D811yescharacteristic
66C2×C4⋊C4SD1612yes—
67C2×C42—41no—
68C2×C42C22⋊C412yes—
69C2×C42C2×D411yescharacteristic
70C2×C16—24no—
71#107—24no—
72#107—24no—
73#107—41no—
74#107—41no—
75C4⋊C8—24no—
76C4⋊C8—22no—
77C4⋊C8—42no—
78C4⋊C8—42no—
79#101—82no—
80C4×C8—22no—
81C4×C8—22no—
82C4×C8C2×D412yes—
83C4×C8#4112yes—
84#95—44no—
85#95—41no—
86C2×Q8—22no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—21no—
90C2×Q8—82no—
91C2×Q8—82no—
92C2×Q8—81no—
93C2×Q8—81no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4#10212yes—
97C22×C4#10212yes—
98C22×C4C2×C22⋊C411yescharacteristic
99C22×C4C2×D811yescharacteristic
100C22×C4C2×SD1611yescharacteristic
101M16—82no—
102M16—82no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—41no—
106C2×C8—42no—
107C2×C8—82no—
108C2×C8C4×D412yes—
109C2×C8#12011yescharacteristic
110C2×C8#12311yescharacteristic
111C4⋊C4—24no—
112C4⋊C4—22no—
113C4⋊C4—24no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—88no—
118C4⋊C4—82no—
119C42—22no—
120C42—22no—
121C42—82no—
122C42#12112yes—
123C42#13512yes—
124C16—44no—
125C23C2×#10211yescharacteristic
126Q8—42no—
127Q8—42no—
128Q8—162no—
129Q8—162no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—44no—
139C2×C4—82no—
140C2×C4—82no—
141C2×C4—82no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4#38611yescharacteristic
145C2×C4#41611yescharacteristic
146C2×C4#46512yes—
147C2×C4#46612yes—
148C2×C4#50911yescharacteristic
149C2×C4#50912yes—
150C2×C4#51011yescharacteristic
151C8—21no—
152C8—22no—
153C8—42no—
154C8—162no—
155C8#45612yes—
156C22—21no—
157C22—21no—
158C22#189311yescharacteristic
159C22#213911yescharacteristic
160C22#214011yescharacteristic
161C4—22no—
162C4—21no—
163C4—42no—
164C4—42no—
165C4—84no—
166C4—162no—
167C4#188611yescharacteristic
168C4#191511yescharacteristic
169C4#221812yes—
170C4#222012yes—
171C2—21no—
172C2—21no—
173C2#939411yescharacteristic
174C2#1306111yescharacteristic
175C2#1307811yescharacteristic
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
#286752
#119712
#86172
#72962