Finite Group Databasebeta

← #139683

Subgroups — #139683

Order 512.

340 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 96 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#33496C211yesmaximal, characteristic
2#28187C211yesmaximal, characteristic
3#11658C212yesmaximal
4#8617C211yesmaximal, characteristic
5#7217C212yesmaximal
6C2×C8×Q8C2211yescharacteristic
7C2×C4×Q16C2211yescharacteristic
8#2582—21no—
9Q8×C16—24no—
10#2162—22no—
11#2162C2212yes—
12#2087C412yes—
13#2056C412yes—
14#1933—22no—
15#1834C2211yescharacteristic
16#1803—21no—
17#1801—21no—
18#1567C2212yes—
19#1332—24no—
20#1292—21no—
21C22×Q16C2×C411yescharacteristic
22C2×C4×Q8—21no—
23C2×C4×Q8D411yescharacteristic
24C8×Q8—22no—
25C8×Q8—22no—
26C4×Q16—22no—
27C4×Q16—22no—
28C2×#107C2×C411yescharacteristic
29#423—22no—
30C2×C4⋊C8C2×C412yes—
31C2×C4⋊C8D411yescharacteristic
32C2×#103—22no—
33C2×#103—21no—
34C2×#101—41no—
35C2×C4×C8D412yes—
36C2×C4×C8C2311yescharacteristic
37#398—42no—
38#391—22no—
39#391—41no—
40#389—22no—
41#389—41no—
42#384—21no—
43C4⋊C16—24no—
44C4⋊C16—22no—
45C22⋊C16—22no—
46C4×C16—24no—
47C4×C16D414yes—
48#343—41no—
49#340—22no—
50#340—21no—
51#340C2×C412yes—
52#335—22no—
53#335—22no—
54C22×Q8—21no—
55C22×Q8D811yescharacteristic
56C2×Q16—22no—
57C2×Q16—42no—
58C2×M16—42no—
59C22×C8C22⋊C412yes—
60C22×C8C22×C411yescharacteristic
61C22×C8C2×D411yescharacteristic
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—42no—
65C4×Q8—42no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4D811yescharacteristic
73C2×C4⋊C4SD1612yes—
74C2×C42—41no—
75C2×C42C22⋊C412yes—
76C2×C42C2×D411yescharacteristic
77C2×C16—24no—
78#107—22no—
79C4⋊C8—24no—
80C4⋊C8—22no—
81C4⋊C8—42no—
82C4⋊C8—42no—
83#103—44no—
84#103—41no—
85#103—41no—
86#101—82no—
87C4×C8—22no—
88C4×C8—22no—
89C4×C8C2×D412yes—
90C4×C8#4112yes—
91#95—42no—
92#95—42no—
93#95—42no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—41no—
98C2×Q8—41no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8SD3212yes—
103C2×Q8Q3212yes—
104C22×C4—41no—
105C22×C4—42no—
106C22×C4—41no—
107C22×C4#10212yes—
108C22×C4#10212yes—
109C22×C4C2×C22⋊C411yescharacteristic
110C22×C4C2×D811yescharacteristic
111C22×C4C2×SD1611yescharacteristic
112Q16—24no—
113Q16—42no—
114M16—82no—
115M16—82no—
116C2×C8—22no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—42no—
120C2×C8—82no—
121C2×C8C4×D412yes—
122C2×C8#12011yescharacteristic
123C2×C8#12311yescharacteristic
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—82no—
131C4⋊C4—82no—
132C4⋊C4—161no—
133C4⋊C4—162no—
134C42—22no—
135C42—22no—
136C42—82no—
137C42#12112yes—
138C42#13512yes—
139C16—44no—
140C23C2×#10211yescharacteristic
141Q8—24no—
142Q8—24no—
143Q8—42no—
144Q8—42no—
145Q8—81no—
146Q8—81no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—82no—
162C2×C4—81no—
163C2×C4#35814yes—
164C2×C4#38611yescharacteristic
165C2×C4#41611yescharacteristic
166C2×C4#46512yes—
167C2×C4#46612yes—
168C2×C4C2×SD3211yescharacteristic
169C2×C4C2×Q3211yescharacteristic
170C2×C4#50812yes—
171C8—22no—
172C8—21no—
173C8—21no—
174C8—42no—
175C8—162no—
176C22—21no—
177C22—21no—
178C22#189311yescharacteristic
179C22C2×#35811yescharacteristic
180C22#213811yescharacteristic
181C4—24no—
182C4—21no—
183C4—22no—
184C4—42no—
185C4—82no—
186C4—162no—
187C4—161no—
188C4#189111yescharacteristic
189C4#191211yescharacteristic
190C4#221612yes—
191C4#221712yes—
192C2—21no—
193C2—21no—
194C2#948611yescharacteristic
195C2#1306011yescharacteristic
196C2#1307611yescharacteristic
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
#334962
#281872
#116582
#86172
#72172