Finite Group Databasebeta

← #507404

Subgroups — #507404

Order 512.

485 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 130 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#33260C211yesmaximal, characteristic, complemented
2#21708C211yesmaximal, characteristic
3#13541C212yesmaximal, complemented
4#13410C212yesmaximal
5#13153C214yesmaximal, complemented
6#12823C212yesmaximal, complemented
7#12820C212yesmaximal, complemented
8#12660C211yesmaximal, characteristic, complemented
9#3150C2211yescharacteristic
10C2×#443—21no—
11C2×#443C2212yes—
12#2931—21no—
13#2647—21no—
14C2×#364C2212yes—
15C2×#360C2214yes—
16#2139C2214yes—
17C2×#347C2212yes—
18#1975—21no—
19#1975C2212yes—
20#1963—21no—
21#1944C2212yes—
22#1943C2212yes—
23#1941—21no—
24#1941—21no—
25#1849C2211yescharacteristic
26#1846C2211yescharacteristic
27#1844C2211yescharacteristic
28#1836—21no—
29#1726C2211yescharacteristic
30#1387C2214yes—
31#1383C2214yes—
32#1377C2212yes—
33C2×#135—21no—
34C2×#135C2312yes—
35C2×#131—22no—
36#532—21no—
37C2×M32C2314yes—
38#497—22no—
39#496—22no—
40#443—24no—
41#443—24no—
42#443—44no—
43#436—42no—
44C8×D4—42no—
45C2×#108—21no—
46C2×#108C2312yes—
47C2×#107C2312yes—
48C2×#106—21no—
49C2×#106—21no—
50C2×#106C2312yes—
51#423—21no—
52C2×#104—22no—
53C2×#104—22no—
54#408—22no—
55C2×#97D411yescharacteristic
56C2×#97C2311yescharacteristic
57C2×#97C2311yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#400—42no—
62#398—42no—
63#389—22no—
64#389—22no—
65#384—21no—
66#382—21no—
67#376D411yescharacteristic
68#364—24no—
69#360—44no—
70#358—24no—
71#357—24no—
72#347—42no—
73#340—22no—
74#340—22no—
75C2×#41—22no—
76#135—22no—
77#135—22no—
78#135—22no—
79#135—22no—
80#135—42no—
81#135—42no—
82C2×Q16—21no—
83C2×Q16#4112yes—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×D8—21no—
87C2×D8#4112yes—
88#131—44no—
89#131—44no—
90C2×M16—22no—
91C2×M16—42no—
92C22×C8—42no—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2411yescharacteristic
97C4⋊Q8—21no—
98#127—21no—
99#124—22no—
100#121—42no—
101C4×D4—42no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110M32—24no—
111M32—24no—
112C2×C16—24no—
113#108—22no—
114#108—22no—
115#108—42no—
116#107—24no—
117#106—22no—
118#106—22no—
119#106—41no—
120#106—41no—
121#106—42no—
122C4⋊C8—42no—
123#104—44no—
124#104—44no—
125C22⋊C8—44no—
126#97—22no—
127#97—22no—
128#97—21no—
129#97—21no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—22no—
133C4×C8#4114yes—
134#95—22no—
135#95—42no—
136#95—42no—
137#95—42no—
138#41—42no—
139#41—42no—
140#41—42no—
141#41—42no—
142C2×Q8—22no—
143C2×D4—22no—
144C2×D4—42no—
145C2×D4—42no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4AGL(1,ℤ/8)12yes—
152C22×C4C22×D411yescharacteristic
153Q16—22no—
154Q16—22no—
155Q16—42no—
156SD16—44no—
157SD16—42no—
158SD16—42no—
159D8—22no—
160D8—22no—
161D8—42no—
162M16—42no—
163M16—42no—
164M16—84no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—42no—
171C2×C8—84no—
172C2×C8#12412yes—
173C2×C8AGL(1,ℤ/8)11yescharacteristic
174C2×C8AGL(1,ℤ/8)11yescharacteristic
175C2×C8AGL(1,ℤ/8)12yes—
176C2×C8#13712yes—
177C2×C8C2×#4112yes—
178C2×C8C2×#4112yes—
179C2×C8H(5,𝔽2)11yescharacteristic
180C2×C82−1+411yescharacteristic
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—162no—
187C4⋊C4—162no—
188C4⋊C4—162no—
189C22⋊C4—44no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42#12412yes—
196C16—44no—
197C23—42no—
198C23C2×AGL(1,ℤ/8)11yescharacteristic
199Q8—42no—
200Q8—42no—
201D4—42no—
202D4—42no—
203D4—84no—
204D4—82no—
205D4—82no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4#52511yescharacteristic
222C2×C4C2×#12411yescharacteristic
223C2×C4#53711yescharacteristic
224C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
225C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
226C2×C4C2×#13711yescharacteristic
227C8—22no—
228C8—41no—
229C8—82no—
230C8#49012yes—
231C8#49012yes—
232C8#55212yes—
233C8#55312yes—
234C22—22no—
235C22—84no—
236C22—82no—
237C22#310911yescharacteristic
238C22#314111yescharacteristic
239C22#325311yescharacteristic
240C4—42no—
241C4—42no—
242C4—81no—
243C4—82no—
244C4—162no—
245C4—162no—
246C4#256211yescharacteristic
247C4#311011yescharacteristic
248C4C2×#49011yescharacteristic
249C4#325411yescharacteristic
250C2—21no—
251C2—21no—
252C2—162no—
253C2#2913011yescharacteristic
254C2#3369112yes—
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radical#3150C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33260C2semidirect
3. #13541C2semidirect
5. #13153C2semidirect
6. #12823C2semidirect
7. #12820C2semidirect
8. #12660C2semidirect

Maximal subgroups

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

subgroupindex
#332602
#217082
#135412
#134102
#131532
#128232
#128202
#126602