Finite Group Databasebeta

← #122749

Subgroups — #122749

Order 512.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#28676C211yesmaximal, characteristic
2#11494C211yesmaximal, characteristic
3#8621C211yesmaximal, characteristic
4#7081C214yesmaximal
5C2×#500C411yescharacteristic
6C2×C8×Q8C2211yescharacteristic
7#2567—21no—
8#2163—21no—
9#2087C412yes—
10#2087C2212yes—
11#1987—21no—
12#1942—21no—
13#1942—21no—
14C4×#107C411yescharacteristic
15#1568—21no—
16#1467—21no—
17#1322—21no—
18#1322C2212yes—
19#1290C2212yes—
20C2×C4⋊Q8—21no—
21C2×C4×Q8D411yescharacteristic
22#500—41no—
23#500—41no—
24C8×Q8—41no—
25C8×Q8—41no—
26#432—22no—
27C2×#107—21no—
28C2×#107—21no—
29C2×#107C2×C412yes—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8D411yescharacteristic
32C2×C4⋊C8C2311yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36#395—41no—
37#395—41no—
38#391—41no—
39#389—22no—
40#389—22no—
41#384—41no—
42C4×C4⋊C4—21no—
43C4⋊C16—41no—
44C22⋊C16—22no—
45#346—41no—
46#340—22no—
47#340—22no—
48#340C2×C412yes—
49#335—41no—
50#334—41no—
51#324—41no—
52#324—41no—
53C22×Q8—41no—
54C22×Q8SD1611yescharacteristic
55C22×C8C22⋊C411yescharacteristic
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22×C411yescharacteristic
58C22×C8C2×D411yescharacteristic
59C4⋊Q8—42no—
60C4⋊Q8—42no—
61C4×Q8—41no—
62C4×Q8—41no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4D811yescharacteristic
69C2×C4⋊C4D811yescharacteristic
70C2×C4⋊C4SD1611yescharacteristic
71C2×C42—41no—
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22⋊C411yescharacteristic
74C2×C42C2×D411yescharacteristic
75C2×C16—42no—
76#107—24no—
77#107—41no—
78#107—41no—
79#107—42no—
80C4⋊C8—41no—
81C4⋊C8—41no—
82C4⋊C8—41no—
83C4⋊C8—42no—
84C22⋊C8—42no—
85#97—42no—
86C4×C8—21no—
87C4×C8—21no—
88C4×C8—21no—
89C4×C8—21no—
90C4×C8—21no—
91C4×C8—21no—
92#95—41no—
93#95—42no—
94#95—42no—
95C2×Q8—22no—
96C2×Q8—41no—
97C2×Q8—41no—
98C2×Q8—82no—
99C2×Q8—162no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4#10211yescharacteristic
103C22×C4#10211yescharacteristic
104C22×C4#10211yescharacteristic
105C22×C4#10211yescharacteristic
106C22×C4C2×C22⋊C411yescharacteristic
107C22×C4C2×D811yescharacteristic
108C22×C4C2×SD1611yescharacteristic
109C2×C8—22no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—82no—
117C2×C8#9911yescharacteristic
118C2×C8#9911yescharacteristic
119C4⋊C4—24no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—82no—
127C4⋊C4—162no—
128C4⋊C4—162no—
129C42—21no—
130C42—21no—
131C42—21no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—82no—
136C16—82no—
137C23C2×#10211yescharacteristic
138Q8—44no—
139Q8—164no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—44no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4#36212yes—
163C2×C4C2×#9911yescharacteristic
164C2×C4#42111yescharacteristic
165C8—21no—
166C8—41no—
167C8—41no—
168C8—41no—
169C8—162no—
170C8#35612yes—
171C22—21no—
172C22—21no—
173C22#151911yescharacteristic
174C22#214311yescharacteristic
175C22C2×#36211yescharacteristic
176C4—21no—
177C4—22no—
178C4—41no—
179C4—41no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—84no—
184C4—162no—
185C4#212311yescharacteristic
186C4C2×#35611yescharacteristic
187C2—21no—
188C2—21no—
189C2#1186811yescharacteristic
190C2#1188311yescharacteristic
191C2#1189611yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC23C2×#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#286762
#114942
#86212
#70812