Finite Group Databasebeta

← #135270

Subgroups — #135270

Order 512.

370 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 114 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#20444C211yesmaximal, characteristic
2#12789C211yesmaximal, characteristic
3#8156C211yesmaximal, characteristic
4#7181C212yesmaximal, complemented
5#7180C212yesmaximal, complemented
6C2×#430C411yescharacteristic
7#2911C411yescharacteristic
8C2×#405—21no—
9C2×#405C2211yescharacteristic
10C2×#364C2211yescharacteristic
11C2×#364C2211yescharacteristic
12#2112—21no—
13#1773—21no—
14#1773C412yes—
15#1772—21no—
16#1772C2212yes—
17#1733—21no—
18#1458—21no—
19#1450C2212yes—
20#1376—21no—
21#1281—21no—
22C22×M16—21no—
23C22×M16C2×C411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C2×#117D411yescharacteristic
27C2×M32—21no—
28C2×M32—21no—
29#431—22no—
30#430—22no—
31#430C814yescomplemented
32C2×#108—21no—
33C2×#108C2×C412yes—
34#428—24no—
35#427—21no—
36#427C2×C412yes—
37#405—22no—
38#405—22no—
39#405—22no—
40#405—22no—
41#405—41no—
42#405—41no—
43C2×#97—41no—
44C2×#97C2×C411yescharacteristic
45C2×#97D411yescharacteristic
46C2×C4×C8—41no—
47C2×C4×C8D411yescharacteristic
48C2×C4×C8C2311yescharacteristic
49#364—22no—
50#364—22no—
51C22⋊C16—42no—
52#339—22no—
53#339—22no—
54#324—22no—
55#323—22no—
56#322—22no—
57C82—22no—
58C23×C4C22⋊C411yescharacteristic
59C2×M16—22no—
60C2×M16—22no—
61C2×M16—24no—
62C2×M16—22no—
63C2×M16—42no—
64C2×M16—41no—
65C2×M16C2×C812yes—
66C2×M16M1612yes—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8SD1612yes—
75C22×C8Q1612yes—
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78#117—22no—
79#117—22no—
80#117—22no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—41no—
88C2×C42—41no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C2×D411yescharacteristic
91M32—41no—
92M32—41no—
93M32—41no—
94M32—41no—
95C2×C16—41no—
96C2×C16—41no—
97#108—24no—
98#108—22no—
99#108C2×C814yes—
100#107—22no—
101#107—22no—
102#106—22no—
103#106M1614yes—
104C4⋊C8—42no—
105C22⋊C8—24no—
106#97—22no—
107#97—22no—
108#97—42no—
109#97—42no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—42no—
113C4×C8—42no—
114C4×C8—42no—
115C4×C8#4112yes—
116C4×C8#4112yes—
117C24#10411yescharacteristic
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4C22⋊C812yes—
126C22×C4#10312yes—
127C22×C4#10411yescharacteristic
128C22×C4C2×C22⋊C411yescharacteristic
129M16—24no—
130M16—24no—
131M16—24no—
132M16—44no—
133M16—42no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—44no—
143C2×C8—42no—
144C2×C8—44no—
145C2×C8C22⋊C812yes—
146C2×C8#10312yes—
147C2×C8#11712yes—
148C2×C8#12311yescharacteristic
149C2×C8#12311yescharacteristic
150C2×C8C22×C811yescharacteristic
151C2×C8C2×M1612yes—
152C2×C8C2×M1611yescharacteristic
153C2×C8C2×SD1611yescharacteristic
154C2×C8C2×Q1611yescharacteristic
155C2×C8AGL(1,ℤ/8)11yescharacteristic
156C2×C8AGL(1,ℤ/8)11yescharacteristic
157C4⋊C4—24no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—42no—
167C42—42no—
168C42#12312yes—
169C16—81no—
170C16—81no—
171C23—21no—
172C23—21no—
173C23—81no—
174C23Q8⋊C812yes—
175C23C2×#10411yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—44no—
185C2×C4—81no—
186C2×C4—84no—
187C2×C4—81no—
188C2×C4Q8⋊C812yes—
189C2×C4#38111yescharacteristic
190C2×C4C2×C22⋊C811yescharacteristic
191C2×C4#40711yescharacteristic
192C2×C4C2×#10311yescharacteristic
193C2×C4#41811yescharacteristic
194C2×C4#42111yescharacteristic
195C8—22no—
196C8—22no—
197C8—44no—
198C8—84no—
199C8#43112yes—
200C8#43212yes—
201C8#48111yescharacteristic
202C8#48111yescharacteristic
203C8#48411yescharacteristic
204C8#48411yescharacteristic
205C22—22no—
206C22—41no—
207C22—42no—
208C22—42no—
209C22—81no—
210C22C2×Q8⋊C811yescharacteristic
211C22#149011yescharacteristic
212C22#182611yescharacteristic
213C4—22no—
214C4—42no—
215C4—81no—
216C4—84no—
217C4#147811yescharacteristic
218C4#148711yescharacteristic
219C4#181711yescharacteristic
220C4#182411yescharacteristic
221C2—21no—
222C2—21no—
223C2—21no—
224C2—42no—
225C2#870711yescharacteristic
subgroupG/H
centerC8#481
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
socleC2#8707
prime radicalC24#104
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7181C2semidirect
5. #7180C2semidirect
31. #430C8semidirect

Maximal subgroups

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

subgroupindex
#204442
#127892
#81562
#71812
#71802