Finite Group Databasebeta

← #512922

Subgroups — #512922

Order 512.

451 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×#2264C211yesmaximal, characteristic
2C22×Q64C211yesmaximal, characteristic, complemented
3#33721C211yesmaximal, characteristic
4#33458C211yesmaximal, characteristic, complemented
5#13690C214yesmaximal, complemented
6#13688C214yesmaximal, complemented
7#13628C211yesmaximal, characteristic, complemented
8#13618C211yesmaximal, characteristic, complemented
9#13615C211yesmaximal, characteristic
10C2×#510—21no—
11C2×#508C2211yescharacteristic
12C22×Q32C2211yescharacteristic
13C22×Q32C2211yescharacteristic
14#3156—21no—
15#2950—21no—
16#2264—24no—
17C2×Q64—24no—
18C2×Q64C2212yes—
19C2×Q64C2214yes—
20C2×SD64C2212yes—
21C2×M64C2211yescharacteristic
22C22×C32C2211yescharacteristic
23#2251—22no—
24#2249C2212yes—
25#2247C2212yes—
26#2176—22no—
27C4×Q32C2214yes—
28#2158C2211yescharacteristic
29#2141—21no—
30#2108C2211yescharacteristic
31C4⋊C32C2214yes—
32#1416C2212yes—
33#1416C2214yes—
34#1415C2212yes—
35C2×#137—41no—
36C2×#135—21no—
37C22×Q16—21no—
38C22×Q16—21no—
39#536—41no—
40#519—41no—
41#510—44no—
42#508—22no—
43#508—22no—
44C2×Q32—21no—
45C2×Q32—22no—
46C2×Q32—21no—
47C2×Q32—22no—
48C2×Q32—22no—
49C2×Q32—22no—
50C2×Q32D414yes—
51C2×Q32C2312yes—
52C2×Q32C2311yescharacteristic
53C2×Q32C2312yes—
54C2×SD32—21no—
55C2×SD32—22no—
56C2×D16C2311yescharacteristic
57C2×M32D411yescharacteristic
58C22×C16C2311yescharacteristic
59#497—42no—
60#495—22no—
61#494—22no—
62#441—42no—
63C4×Q16—24no—
64#427—21no—
65#419—41no—
66#405D411yescharacteristic
67Q64—28no—
68Q64—24no—
69SD64—24no—
70M64—22no—
71C2×C32—22no—
72C2×C32C2312yes—
73C2×C32C2312yes—
74#367—21no—
75#366C2312yes—
76#358—24no—
77#346D412yes—
78C4×C16C2312yes—
79C2×#41—41no—
80C22×Q8—41no—
81C22×Q8—41no—
82#137—84no—
83#135—42no—
84#135—42no—
85C2×Q16—24no—
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—21no—
89C2×Q16—42no—
90C2×Q16—41no—
91C2×Q16—41no—
92C2×Q16—42no—
93C2×Q16—42no—
94C2×Q16—42no—
95C2×SD16—41no—
96C2×SD16—42no—
97C2×D8—21no—
98C2×M16D811yescharacteristic
99C22×C8C2×D411yescharacteristic
100C4⋊Q8—42no—
101#124—42no—
102#123—82no—
103C22⋊Q8—82no—
104C4×Q8—44no—
105C4×Q8—82no—
106#117—41no—
107#117—82no—
108#117D811yescharacteristic
109Q32—24no—
110Q32—22no—
111Q32—21no—
112Q32—22no—
113Q32—22no—
114Q32—21no—
115Q32—44no—
116Q32—42no—
117Q32C2×D414yes—
118SD32—44no—
119SD32—42no—
120D16—21no—
121D16—21no—
122M32—22no—
123C2×C16C2×D412yes—
124C2×C16C2×D411yescharacteristic
125C2×C16C2×D412yes—
126C2×C16#4112yes—
127C2×C16C2411yescharacteristic
128#107—22no—
129#106—41no—
130#103—44no—
131#97D812yes—
132C4×C8C2×D412yes—
133C32—22no—
134C32—22no—
135#41—82no—
136#41—82no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—41no—
140C2×Q8—44no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8—82no—
144C2×Q8—82no—
145C2×Q8—82no—
146C2×D4—41no—
147C2×D4—81no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4C2×D811yescharacteristic
152Q16—24no—
153Q16—41no—
154Q16—42no—
155Q16—42no—
156Q16—42no—
157Q16—41no—
158Q16—44no—
159Q16—84no—
160Q16—82no—
161SD16—84no—
162SD16—82no—
163D8—41no—
164D8—41no—
165M16—22no—
166C2×C8#12112yes—
167C2×C8C2×D812yes—
168C2×C8C2×D811yescharacteristic
169C2×C8AGL(1,ℤ/8)12yes—
170C2×C8C22×D411yescharacteristic
171C4⋊C4—42no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C4⋊C4D1612yes—
175C22⋊C4—81no—
176C22⋊C4—84no—
177C22⋊C4D1612yes—
178C42—81no—
179C42—84no—
180C42C2×D812yes—
181C16—22no—
182C16#12112yes—
183C16C22×D411yescharacteristic
184C16C2×#4111yescharacteristic
185C23—81no—
186C23C2×D1611yescharacteristic
187Q8—44no—
188Q8—82no—
189Q8—84no—
190Q8—82no—
191Q8—81no—
192Q8—81no—
193Q8—82no—
194Q8—162no—
195D4—81no—
196D4—81no—
197D4—162no—
198C2×C4—82no—
199C2×C4—84no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—162no—
204C2×C4—162no—
205C2×C4#45912yes—
206C2×C4C2×D1614yes—
207C2×C4C2×D1611yescharacteristic
208C2×C4#50912yes—
209C2×C4C22×D811yescharacteristic
210C8—22no—
211C8#45912yes—
212C8C2×#12111yescharacteristic
213C8C2×AGL(1,ℤ/8)11yescharacteristic
214C22—22no—
215C22—161no—
216C22—162no—
217C22#220612yes—
218C22C22×D1611yescharacteristic
219C4—24no—
220C4—84no—
221C4—161no—
222C4—162no—
223C4—162no—
224C4#220612yes—
225C4C2×#45911yescharacteristic
226C4C2×#50911yescharacteristic
227C2—22no—
228C2—321no—
229C2C2×#220611yescharacteristic
230C2#3398412yes—
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radicalC2×#508C22
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×Q64C2semidirect
4. #33458C2semidirect
5. #13690C2semidirect
6. #13688C2semidirect
7. #13628C2semidirect
8. #13618C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22642
C22×Q642
#337212
#334582
#136902
#136882
#136282
#136182
#136152