Finite Group Databasebeta

← #244452

Subgroups — #244452

Order 512.

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

normal subgroups only

This group has 429 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#21677C211yesmaximal, characteristic, complemented
2#21661C211yesmaximal, characteristic
3#10689C212yesmaximal, complemented
4#10630C211yesmaximal, characteristic, complemented
5#10629C211yesmaximal, characteristic
6#10547C212yesmaximal
7#10373C211yesmaximal, characteristic, complemented
8#10372C211yesmaximal, characteristic
9#8256C212yesmaximal, complemented
10#8255C212yesmaximal
11#8137C211yesmaximal, characteristic, complemented
12#3135C2211yescharacteristic
13#2960C2211yescharacteristic
14#2931C2211yescharacteristic
15#2402C2211yescharacteristic
16#1995C2211yescharacteristic
17#1963C2211yescharacteristic
18#1940C2212yes—
19#1938—22no—
20#1917—22no—
21#1838C2211yescharacteristic
22#1832C2211yescharacteristic
23#1771—22no—
24#1770C2212yes—
25#1764—22no—
26#1726C2211yescharacteristic
27#1542C2212yes—
28#1540C2212yes—
29#1503C2212yes—
30#1502C2212yes—
31#1450C2212yes—
32#1303C2212yes—
33#1303C2212yes—
34#1301C2212yes—
35#1301C2212yes—
36#1275C2212yes—
37#1274C2211yescharacteristic
38#1274C2211yescharacteristic
39C2×#131—21no—
40#533C2311yescharacteristic
41#532C2311yescharacteristic
42#491—21no—
43#490—21no—
44#489—22no—
45#446—22no—
46C8×Q8—22no—
47#436—22no—
48C8×D4—22no—
49#431C2312yes—
50#431C2312yes—
51#424—21no—
52#423C2311yescharacteristic
53#423C2311yescharacteristic
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×#104—22no—
57#416—22no—
58#408—21no—
59#405—21no—
60C2×#97—22no—
61C2×#97D412yes—
62C2×#97C2311yescharacteristic
63C2×C4×C8D412yes—
64C2×C4×C8C2311yescharacteristic
65#400—22no—
66#398—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2312yes—
69#376C2311yescharacteristic
70#341—22no—
71#339—22no—
72#336—22no—
73#336—22no—
74#336—22no—
75#333—22no—
76#332—22no—
77#332—22no—
78#331—22no—
79#328—22no—
80#328—22no—
81#327—22no—
82#327—22no—
83#322—22no—
84C82—22no—
85C2×#41—21no—
86#131—44no—
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—41no—
90C22×C8—22no—
91C22×C8—41no—
92C22×C8C2×D412yes—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C4⋊Q8#4111yescharacteristic
96C4⋊Q8#4112yes—
97#127#4111yescharacteristic
98#125—21no—
99#124—21no—
100C22⋊Q8—22no—
101#121—22no—
102C4×Q8—22no—
103C4×D4—22no—
104#117—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C42—22no—
109C2×C42C2×D412yes—
110C2×C42C2411yescharacteristic
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—44no—
118C4⋊C8—42no—
119#104—44no—
120#103—24no—
121#102—24no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—42no—
125#97—21no—
126#97—21no—
127#97—42no—
128#97—41no—
129#97—44no—
130C4×C8—24no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—41no—
134C4×C8#4112yes—
135C4×C8#4112yes—
136#95—22no—
137#95—22no—
138#95—22no—
139#95#4112yes—
140#95#4112yes—
141#41—42no—
142#41—42no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—41no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4C22×D411yescharacteristic
153C22×C4C2×#4112yes—
154M16—42no—
155M16—42no—
156M16—82no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—42no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—82no—
169C2×C8#12412yes—
170C2×C8#12411yescharacteristic
171C2×C8#12411yescharacteristic
172C2×C8#13514yes—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—82no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C22⋊C4—41no—
181C22⋊C4—42no—
182C42—22no—
183C42—22no—
184C42—41no—
185C42—42no—
186C42—42no—
187C42C2×#4111yescharacteristic
188C42C2×#4111yescharacteristic
189C42C2×#4112yes—
190C23—41no—
191C23#52511yescharacteristic
192Q8—42no—
193Q8—82no—
194D4—42no—
195D4—82no—
196D4—82no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4#52511yescharacteristic
212C2×C4C2×#12411yescharacteristic
213C2×C4#52912yes—
214C2×C4#52912yes—
215C2×C4C2×#13512yes—
216C2×C4#57512yes—
217C8—22no—
218C8—42no—
219C8—82no—
220C8—82no—
221C8—82no—
222C8—81no—
223C8—81no—
224C8#48812yes—
225C8#48812yes—
226C22—22no—
227C22—81no—
228C22—82no—
229C22#239811yescharacteristic
230C22#310212yes—
231C4—22no—
232C4—22no—
233C4—82no—
234C4—82no—
235C4—82no—
236C4—81no—
237C4—82no—
238C4#310212yes—
239C4C2×#48811yescharacteristic
240C4#313511yescharacteristic
241C2—22no—
242C2—161no—
243C2#2381811yescharacteristic
244C2#2383412yes—
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22#2398
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21677C2semidirect
3. #10689C2semidirect
4. #10630C2semidirect
7. #10373C2semidirect
9. #8256C2semidirect
11. #8137C2semidirect

Maximal subgroups

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

subgroupindex
#216772
#216612
#106892
#106302
#106292
#105472
#103732
#103722
#82562
#82552
#81372