Finite Group Databasebeta

← #506355

Subgroups — #506355

Order 512.

521 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 142 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#33637C212yesmaximal, complemented
2#21734C211yesmaximal, characteristic, complemented
3#21710C211yesmaximal, characteristic
4#13511C212yesmaximal, complemented
5#13416C212yesmaximal
6#13140C214yesmaximal, complemented
7#12882C211yesmaximal, characteristic, complemented
8#12825C212yesmaximal, complemented
9#3160C2211yescharacteristic
10#3150C2211yescharacteristic
11#2960—21no—
12C2×#444—21no—
13C2×#443C2212yes—
14#2932—21no—
15#2231—24no—
16#2230—24no—
17#2155C2214yes—
18C2×#364C2212yes—
19#2148C2214yes—
20#2145C2214yes—
21#1994—21no—
22#1975C2212yes—
23#1974—21no—
24#1962—21no—
25#1946—21no—
26#1946C2212yes—
27#1849C2211yescharacteristic
28#1848C2211yescharacteristic
29#1773—21no—
30#1773C2212yes—
31C4×#97C2211yescharacteristic
32#1385C2214yes—
33#1377C2212yes—
34#1375C2212yes—
35C2×#135—21no—
36C2×#135C2312yes—
37C2×#131—22no—
38#533—21no—
39#532—21no—
40C2×M32C2314yes—
41#500—21no—
42#499—21no—
43#499D412yescomplemented
44#498D412yescomplemented
45#497—22no—
46#496—22no—
47#446—42no—
48C8×Q8—42no—
49#444—44no—
50#443—22no—
51#443—22no—
52#443—22no—
53#443—22no—
54#435—44no—
55C2×#108—21no—
56C2×#108—21no—
57C2×#108C2312yes—
58C2×#108C2312yes—
59#427—21no—
60#427C2312yes—
61#424—22no—
62#423—21no—
63#423—21no—
64C2×#104—22no—
65C2×#104—22no—
66#408—22no—
67#405—22no—
68C2×#97D411yescharacteristic
69C2×#97D411yescharacteristic
70C2×#97C2311yescharacteristic
71C2×#97C2311yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C43D411yescharacteristic
75#365—28no—
76#364—22no—
77#364—22no—
78#362—24no—
79#361—24no—
80#360—28no—
81#339—22no—
82#339—22no—
83C2×#41—22no—
84#135—22no—
85#135—22no—
86#135—22no—
87#135—22no—
88#135—42no—
89#135—42no—
90C2×Q16—21no—
91C2×Q16#4112yes—
92C2×SD16—22no—
93C2×SD16—22no—
94C2×D8—21no—
95C2×D8#4112yes—
96#131—42no—
97#131—42no—
98#131—42no—
99#131—42no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—42no—
103C22×C8—42no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2411yescharacteristic
108C4⋊Q8—21no—
109C4⋊Q8—22no—
110#127—21no—
111#125—22no—
112#124—22no—
113C22⋊Q8—42no—
114#121—42no—
115C4×Q8—42no—
116C4×D4—42no—
117#117—22no—
118C2×C42—41no—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122M32—24no—
123M32—24no—
124C2×C16—24no—
125#108—22no—
126#108—24no—
127#108—22no—
128#108—42no—
129#108—42no—
130#107—22no—
131#107—22no—
132#106—22no—
133#106C2×D414yes—
134C4⋊C8—42no—
135C4⋊C8—44no—
136C4⋊C8—42no—
137#104—44no—
138#104—42no—
139#104—42no—
140C22⋊C8—44no—
141#97—22no—
142#97—22no—
143#97—22no—
144#97—42no—
145#97SD1614yes—
146C4×C8—22no—
147C4×C8—42no—
148C4×C8C2×D411yescharacteristic
149C4×C8C2×D411yescharacteristic
150C4×C8#4111yescharacteristic
151C4×C8#4111yescharacteristic
152#41—42no—
153#41—42no—
154#41—42no—
155#41—42no—
156C2×Q8—22no—
157C2×Q8—42no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4C22×D411yescharacteristic
166Q16—22no—
167Q16—22no—
168Q16—42no—
169SD16—44no—
170SD16—42no—
171SD16—42no—
172D8—22no—
173D8—22no—
174D8—42no—
175M16—42no—
176M16—42no—
177M16—42no—
178M16—42no—
179M16—82no—
180M16—82no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8#12111yescharacteristic
190C2×C8#12111yescharacteristic
191C2×C8C2×SD1612yes—
192C2×C8C2×SD1612yes—
193C2×C8AGL(1,ℤ/8)12yes—
194C2×C8AGL(1,ℤ/8)12yes—
195C2×C8C22×D411yescharacteristic
196C2×C8C2×#4112yes—
197C2×C8C2×#4111yescharacteristic
198C2×C82−1+412yes—
199C4⋊C4—24no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C22⋊C4—44no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C42—82no—
211C42#12111yescharacteristic
212C42#12111yescharacteristic
213C42C2×SD1612yes—
214C16—44no—
215C23—42no—
216C23#57511yescharacteristic
217Q8—42no—
218Q8—42no—
219Q8—84no—
220D4—42no—
221D4—42no—
222D4—82no—
223D4—82no—
224D4—162no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—44no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4C2×#12111yescharacteristic
237C2×C4#52511yescharacteristic
238C2×C4#53611yescharacteristic
239C2×C4C22×SD1611yescharacteristic
240C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
242C8—22no—
243C8—22no—
244C8—82no—
245C8—82no—
246C8#46012yes—
247C8#46012yes—
248C8#54812yes—
249C8#55212yes—
250C22—41no—
251C22—84no—
252C22—82no—
253C22#303611yescharacteristic
254C22#310111yescharacteristic
255C22#323511yescharacteristic
256C4—22no—
257C4—41no—
258C4—42no—
259C4—81no—
260C4—84no—
261C4—82no—
262C4#253611yescharacteristic
263C4C2×#46011yescharacteristic
264C4#310911yescharacteristic
265C4#322711yescharacteristic
266C2—21no—
267C2—21no—
268C2—21no—
269C2—162no—
270C2#2820211yescharacteristic
subgroupG/H
centerC4#3227
commutator, Frattini, cosocleC22×C8C24
socleC2#28202
prime radical#3150C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33637C2semidirect
2. #21734C2semidirect
4. #13511C2semidirect
6. #13140C2semidirect
7. #12882C2semidirect
8. #12825C2semidirect
43. #499D4semidirect
44. #498D4semidirect

Maximal subgroups

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

subgroupindex
#336372
#217342
#217102
#135112
#134162
#131402
#128822
#128252