Finite Group Databasebeta

← #139391

Subgroups — #139391

Order 512.

315 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 84 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#11974C212yesmaximal
4#8623C211yesmaximal, characteristic
5#7301C212yesmaximal
6C2×#500C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2567—21no—
9#2183—24no—
10#2163—21no—
11#2163C2212yes—
12#2162—21no—
13#2087C412yes—
14#2087C412yes—
15#1987—21no—
16#1942—21no—
17#1942—21no—
18C4×#107C2211yescharacteristic
19#1568C2212yes—
20#1467—21no—
21#1364—24no—
22#1290—21no—
23C2×C4⋊Q8—21no—
24C2×C4×Q8D411yescharacteristic
25#500—22no—
26#500—22no—
27C8×Q8—22no—
28C8×Q8—22no—
29#432—22no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107C2×C412yes—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8C2311yescharacteristic
39#395—41no—
40#395—41no—
41#391—41no—
42#389—22no—
43#389—22no—
44#384—41no—
45C4×C4⋊C4—21no—
46C4⋊C16—22no—
47C4⋊C16—22no—
48C4⋊C16D414yes—
49C22⋊C16—22no—
50#346—22no—
51#346—22no—
52C4×C16—22no—
53#340—22no—
54#340—22no—
55#340C2×C412yes—
56#335—22no—
57#334—22no—
58#324—41no—
59#324—41no—
60C22×Q8—41no—
61C22×Q8SD1611yescharacteristic
62C22×C8C22⋊C411yescharacteristic
63C22×C8C22⋊C411yescharacteristic
64C22×C8C22×C411yescharacteristic
65C22×C8C2×D411yescharacteristic
66C4⋊Q8—42no—
67C4⋊Q8—42no—
68C4×Q8—22no—
69C4×Q8—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4D811yescharacteristic
76C2×C4⋊C4D811yescharacteristic
77C2×C4⋊C4SD1611yescharacteristic
78C2×C42—41no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C2×D411yescharacteristic
82C2×C16—24no—
83#107—24no—
84#107—24no—
85#107—41no—
86#107—41no—
87C4⋊C8—22no—
88C4⋊C8—22no—
89C4⋊C8—22no—
90C4⋊C8—42no—
91C22⋊C8—42no—
92#97—42no—
93C4×C8—21no—
94C4×C8—21no—
95C4×C8—21no—
96C4×C8—21no—
97C4×C8C2×D412yes—
98C4×C8#4112yes—
99#95—42no—
100#95—42no—
101#95—41no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—41no—
105C2×Q8—82no—
106C2×Q8—162no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4#10211yescharacteristic
110C22×C4#10211yescharacteristic
111C22×C4#10211yescharacteristic
112C22×C4#10211yescharacteristic
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4C2×D811yescharacteristic
115C22×C4C2×SD1611yescharacteristic
116C2×C8—22no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—82no—
123C2×C8C4×D412yes—
124C2×C8#12011yescharacteristic
125C2×C8#12311yescharacteristic
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—24no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—82no—
134C4⋊C4—162no—
135C4⋊C4—162no—
136C42—21no—
137C42—21no—
138C42—21no—
139C42—21no—
140C42—82no—
141C42#12112yes—
142C42#13512yes—
143C16—44no—
144C23C2×#10211yescharacteristic
145Q8—44no—
146Q8—164no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—44no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4#36212yes—
168C2×C4#38611yescharacteristic
169C2×C4#41611yescharacteristic
170C2×C4#46512yes—
171C2×C4#46612yes—
172C8—21no—
173C8—22no—
174C8—41no—
175C8—41no—
176C8—162no—
177C8#45612yes—
178C22—21no—
179C22—21no—
180C22#189311yescharacteristic
181C22C2×#36211yescharacteristic
182C22#214811yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—21no—
187C4—41no—
188C4—41no—
189C4—41no—
190C4—84no—
191C4—162no—
192C4#188611yescharacteristic
193C4#191511yescharacteristic
194C2—21no—
195C2—21no—
196C2#939411yescharacteristic
197C2#1306611yescharacteristic
198C2#1307411yescharacteristic
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
#119742
#86232
#73012