Finite Group Databasebeta

← #100798

Subgroups — #100798

Order 512.

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

normal subgroups only

This group has 371 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#28518C211yesmaximal, characteristic
2#23363C211yesmaximal, characteristic
3#8594C211yesmaximal, characteristic
4#7720C212yesmaximal
5#7041C212yesmaximal
6C2×#479—22no—
7#2859C2211yescharacteristic
8C4×C4⋊Q8C2211yescharacteristic
9#2096C412yes—
10#2095C412yes—
11#1984—22no—
12#1960—21no—
13#1916—24no—
14#1847—21no—
15#1839C2212yes—
16#1831C2212yes—
17#1767C2211yescharacteristic
18C2×#333—21no—
19#1296—21no—
20#1271—24no—
21C2×C4⋊Q8D411yescharacteristic
22C2×#125—21no—
23C2×C4×Q8D412yes—
24#479—44no—
25#479—44no—
26C2×#107—22no—
27C2×#106—22no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×#103—22no—
32C2×#103—22no—
33C2×#97—21no—
34C2×C4×C8—22no—
35#391C2×C412yes—
36#389—22no—
37#389—22no—
38#389—22no—
39#384C2×C411yescharacteristic
40#384C2×C411yescharacteristic
41#384D411yescharacteristic
42#382—21no—
43C4×C4⋊C4—22no—
44C43C2311yescharacteristic
45#336—24no—
46#336—22no—
47#336—22no—
48#336—22no—
49#336—22no—
50#336C2×C412yes—
51#336D414yes—
52#333—42no—
53C22×Q8C22⋊C412yes—
54C22×C8—22no—
55C22×C8—24no—
56C22×C8—42no—
57C4⋊Q8—22no—
58C4⋊Q8SD1612yes—
59C4⋊Q8Q1612yes—
60#125—42no—
61#125—42no—
62C4×Q8—42no—
63C4×Q8—42no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4C22⋊C412yes—
74C2×C42—22no—
75C2×C42C22×C411yescharacteristic
76C2×C42C2×D411yescharacteristic
77C2×C42C2×D412yes—
78C2×C42C2×D411yescharacteristic
79C2×C42#4112yes—
80#107—44no—
81#106—44no—
82C4⋊C8—24no—
83C4⋊C8—42no—
84C4⋊C8—42no—
85C4⋊C8—82no—
86#103—44no—
87#103—44no—
88#97—42no—
89C4×C8—82no—
90#95—24no—
91#95—42no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8#10314yes—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4C2×C22⋊C411yescharacteristic
101C22×C4C4×D412yes—
102C22×C4#12011yescharacteristic
103C22×C4#12112yes—
104C22×C4#12311yescharacteristic
105C2×C8—24no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—48no—
109C2×C8—44no—
110C2×C8—82no—
111C2×C8—82no—
112C2×C8—82no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—44no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C4⋊C4—82no—
122C4⋊C4—81no—
123C4⋊C4—81no—
124C4⋊C4—82no—
125C42—22no—
126C42—41no—
127C42—42no—
128C42—42no—
129C42—42no—
130C42—42no—
131C42C2×SD1611yescharacteristic
132C42C2×Q1611yescharacteristic
133C42#13512yes—
134C23#38611yescharacteristic
135Q8—44no—
136Q8—82no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—24no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—44no—
151C2×C4—42no—
152C2×C4—82no—
153C2×C4—81no—
154C2×C4—81no—
155C2×C4—82no—
156C2×C4C2×#10311yescharacteristic
157C2×C4#41611yescharacteristic
158C2×C4#45312yes—
159C2×C4#46512yes—
160C2×C4#46712yes—
161C8—44no—
162C8—84no—
163C8—162no—
164C22#188112yes—
165C22#188711yescharacteristic
166C22#189211yescharacteristic
167C22#190512yes—
168C22#191511yescharacteristic
169C4—22no—
170C4—22no—
171C4—21no—
172C4—21no—
173C4—22no—
174C4—41no—
175C4—42no—
176C4—44no—
177C4—82no—
178C4—161no—
179C4—161no—
180C2#900711yescharacteristic
181C2#930412yes—
182C2#933612yes—
183C2#939511yescharacteristic
184C2#942911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#285182
#233632
#85942
#77202
#70412