Finite Group Databasebeta

← #135392

Subgroups — #135392

Order 512.

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

normal subgroups only

This group has 373 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#20830C211yesmaximal, characteristic
2#12846C211yesmaximal, characteristic
3#10821C211yesmaximal, characteristic
4#7264C214yesmaximal
5#2920—21no—
6C22×#97C2211yescharacteristic
7#2112C2212yes—
8#1849C411yescharacteristic, complemented
9#1849C411yescharacteristic, complemented
10#1838—21no—
11#1810—21no—
12#1810C2212yes—
13#1778—21no—
14#1778C412yescomplemented
15C2×#340—21no—
16C2×#340C2212yes—
17C4⋊#95—21no—
18#1377—21no—
19#1377—21no—
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C22×C4⋊C4—22no—
23C22×C42D411yescharacteristic
24C2×M32—22no—
25#432—42no—
26#431—42no—
27C2×#108—21no—
28C2×#108—21no—
29C2×#108C2×C412yes—
30C2×#108C2×C412yes—
31#423—21no—
32#423—21no—
33#407—22no—
34C4×M16—22no—
35C2×#97—24no—
36C2×#97C2×C411yescharacteristic
37C2×#97C2×C411yescharacteristic
38C2×#97D411yescharacteristic
39C2×#97D411yescharacteristic
40C2×#95—42no—
41C22⋊C16—24no—
42#341—22no—
43#341—22no—
44#340—24no—
45#340—24no—
46C23×C4—42no—
47C23×C4SD1612yes—
48C23×C4C2×D411yescharacteristic
49C2×M16—22no—
50C2×M16—22no—
51C22×C8—22no—
52C22×C8—22no—
53C22×C8—41no—
54C22×C8—41no—
55C22×C8—41no—
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22⋊C411yescharacteristic
58C22×C8C22×C411yescharacteristic
59C22×C8C2×D411yescharacteristic
60C22×C8#4114yes—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22⋊C411yescharacteristic
71C2×C42Q1612yes—
72M32—42no—
73M32—42no—
74C2×C16—42no—
75#108—42no—
76#108—42no—
77#108—42no—
78#108—42no—
79C4⋊C8—42no—
80C4⋊C8—42no—
81C22⋊C8—44no—
82#97—22no—
83#97—22no—
84#97—22no—
85#97—22no—
86#97—42no—
87C4×C8—42no—
88#95—84no—
89C24C2×SD1611yescharacteristic
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—44no—
99C22×C4—82no—
100C22×C4—82no—
101C22×C4—82no—
102C22×C4—82no—
103C22×C4—82no—
104C22×C4—82no—
105C22×C4#10312yes—
106C22×C4#10312yes—
107C22×C4C2×C22⋊C411yescharacteristic
108C22×C4#12312yes—
109C22×C4C2×Q1611yescharacteristic
110C22×C4AGL(1,ℤ/8)12yes—
111M16—42no—
112M16—42no—
113M16—42no—
114M16—42no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—44no—
125C2×C8—44no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8#10412yes—
129C2×C8#11712yes—
130C2×C8#12312yes—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4—82no—
134C4⋊C4—82no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—21no—
139C42—24no—
140C16—82no—
141C23—22no—
142C23—22no—
143C23—21no—
144C23—21no—
145C23C2×#10311yescharacteristic
146C23#48112yes—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—82no—
160C2×C4—84no—
161C2×C4—84no—
162C2×C4—84no—
163C2×C4—162no—
164C2×C4—162no—
165C2×C4—162no—
166C2×C4—162no—
167C2×C4#38111yescharacteristic
168C2×C4#41011yescharacteristic
169C2×C4#41011yescharacteristic
170C2×C4#41811yescharacteristic
171C2×C4C2×#10411yescharacteristic
172C2×C4#42111yescharacteristic
173C2×C4#48412yes—
174C8—22no—
175C8—22no—
176C8—22no—
177C8—41no—
178C8—82no—
179C8—82no—
180C22—21no—
181C22—22no—
182C22—21no—
183C22—22no—
184C22—21no—
185C22—21no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22#150311yescharacteristic
190C22#151811yescharacteristic
191C22#182411yescharacteristic
192C22#223812yes—
193C22#223912yes—
194C4—22no—
195C4—42no—
196C4—42no—
197C4—164no—
198C4#150511yescharacteristic
199C4#151411yescharacteristic
200C4#182011yescharacteristic
201C4#182611yescharacteristic
202C2—22no—
203C2—22no—
204C2—21no—
205C2—21no—
206C2#871711yescharacteristic
207C2#1287512yes—
subgroupG/H
center, socleC22#1824
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #1849C4semidirect
9. #1849C4semidirect
14. #1778C4semidirect

Maximal subgroups

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

subgroupindex
#208302
#128462
#108212
#72642