Finite Group Databasebeta

← #135404

Subgroups — #135404

Order 512.

304 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 74 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#20821C211yesmaximal, characteristic
2#12608C211yesmaximal, characteristic
3#12606C211yesmaximal, characteristic
4#7266C214yesmaximal, complemented
5C2×#405C2211yescharacteristic
6#2314C412yes—
7C2×#349—21no—
8#2112—21no—
9C2×#347C2212yes—
10C2×#347C2212yes—
11#1846C411yescharacteristic
12#1845C411yescharacteristic
13#1836C2212yes—
14#1809—21no—
15#1808—21no—
16#1776—21no—
17C4×#107—21no—
18#1376—21no—
19#1375—21no—
20C22×M16D411yescharacteristic
21C23×C8D411yescharacteristic
22C2×#117C2×C411yescharacteristic
23C2×M32—22no—
24C2×M32—22no—
25C2×#107—21no—
26C2×#107—21no—
27C2×#107—21no—
28C2×#106—21no—
29#405—24no—
30C2×#97D411yescharacteristic
31C2×#97C2311yescharacteristic
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8D411yescharacteristic
34#388—42no—
35#387—42no—
36#384C2×C412yes—
37#382C2×C412yes—
38#380—22no—
39C4×C4⋊C4—22no—
40#349—24no—
41C22⋊C16—24no—
42#347—24no—
43#347—24no—
44#340—22no—
45#340—22no—
46C23×C4C22⋊C411yescharacteristic
47C2×M16—22no—
48C2×M16—22no—
49C2×M16—41no—
50C22×C8—22no—
51C22×C8—22no—
52C22×C8—41no—
53C22×C8C22⋊C411yescharacteristic
54C22×C8C22⋊C411yescharacteristic
55C22×C8C2×D411yescharacteristic
56C22×C8C2×D411yescharacteristic
57#117—22no—
58#117—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—42no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—42no—
65C2×C42—42no—
66C2×C42C22⋊C411yescharacteristic
67C2×C42C22×C411yescharacteristic
68M32—24no—
69M32—42no—
70M32—42no—
71M32—42no—
72C2×C16—24no—
73C2×C16—42no—
74#107—42no—
75#107—42no—
76#107—42no—
77#106—42no—
78#97—22no—
79#97#4114yes—
80C4×C8—22no—
81C4×C8—22no—
82#95—42no—
83#95—42no—
84#95—42no—
85C24#10011yescharacteristic
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4#10011yescharacteristic
97C22×C4C2×C22⋊C411yescharacteristic
98M16—44no—
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—44no—
108C2×C8#10412yes—
109C2×C8#10412yes—
110C2×C8#12312yes—
111C2×C8#12312yes—
112C4⋊C4—24no—
113C4⋊C4—42no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—162no—
118C22⋊C4—24no—
119C22⋊C4—162no—
120C42—21no—
121C42—21no—
122C42—162no—
123C42#10112yes—
124C42#11712yes—
125C16—44no—
126C16—44no—
127C23—21no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23C2×#10011yescharacteristic
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4#38111yescharacteristic
153C2×C4C2×#10111yescharacteristic
154C2×C4C2×#10411yescharacteristic
155C2×C4C2×#10411yescharacteristic
156C2×C4#42111yescharacteristic
157C2×C4#42111yescharacteristic
158C8—22no—
159C8—22no—
160C8—22no—
161C8—22no—
162C22—21no—
163C22—21no—
164C22—42no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—81no—
169C22#153011yescharacteristic
170C22#153011yescharacteristic
171C22#182111yescharacteristic
172C4—42no—
173C4—42no—
174C4—41no—
175C4—162no—
176C4—162no—
177C4#152611yescharacteristic
178C4#153711yescharacteristic
179C4#182611yescharacteristic
180C4#182611yescharacteristic
181C2—21no—
182C2—21no—
183C2—81no—
184C2#872211yescharacteristic
185C2#1191311yescharacteristic
186C2#1191411yescharacteristic
subgroupG/H
center, socleC22#1530
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7266C2semidirect

Maximal subgroups

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

subgroupindex
#208212
#126082
#126062
#72662