Finite Group Databasebeta

← #143389

Subgroups — #143389

Order 512.

374 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 84 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#33731C211yesmaximal, characteristic
2#28683C211yesmaximal, characteristic
3#19150C211yesmaximal, characteristic
4#13278C211yesmaximal, characteristic
5#11940C211yesmaximal, characteristic
6#11611C211yesmaximal, characteristic
7#7175C211yesmaximal, characteristic
8#3160C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#3104—21no—
11C2×#446C2211yescharacteristic
12#2546—21no—
13#2255—22no—
14#2254—21no—
15#2252—21no—
16#2159—22no—
17#2150C2211yescharacteristic
18C4×M32C2211yescharacteristic
19#2087—22no—
20#1988—21no—
21#1944—22no—
22C4×#106C2211yescharacteristic
23#1696—21no—
24#1694—21no—
25#1692—22no—
26#1564C2211yescharacteristic
27#1542—21no—
28#1386—22no—
29#1364—22no—
30#1338—22no—
31#533—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—41no—
34C2×C4×Q8D411yescharacteristic
35C2×M32—21no—
36C2×M32—21no—
37#500—21no—
38#500—21no—
39#500—21no—
40#500—21no—
41#500D411yescharacteristic
42#500D411yescharacteristic
43#499—21no—
44#499—21no—
45#498—21no—
46#484—41no—
47#477—22no—
48#462—22no—
49#451—41no—
50#446—21no—
51#446—21no—
52#446—21no—
53#446—21no—
54C4×Q16—42no—
55#431—21no—
56#427—21no—
57#427—21no—
58C2×#107—21no—
59C2×#107D412yes—
60C2×#106D412yes—
61C2×C4⋊C8—21no—
62C2×C4⋊C8D411yescharacteristic
63C2×#97—21no—
64C2×C4×C8C2311yescharacteristic
65#398—42no—
66#391—41no—
67#389—24no—
68C4×C4⋊C4—21no—
69#367—24no—
70#366—24no—
71C4⋊C16—21no—
72C4⋊C16—21no—
73#346D412yes—
74C4×C16—21no—
75#340—24no—
76#340—21no—
77#335D412yes—
78#334—21no—
79#333—22no—
80#332—22no—
81C22×Q8—41no—
82C22×Q8D811yescharacteristic
83C2×Q16—42no—
84C22×C8—21no—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8—41no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—84no—
94#125—41no—
95C22⋊Q8—41no—
96C22⋊Q8—41no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—41no—
102C4×Q8—41no—
103#117—41no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4D811yescharacteristic
110C2×C42—21no—
111C2×C42—41no—
112C2×C42C2×D411yescharacteristic
113M32—24no—
114M32—42no—
115C2×C16—22no—
116C2×C16—41no—
117#107—21no—
118#107—22no—
119#107—21no—
120#107—22no—
121#107—21no—
122#107—21no—
123#107—42no—
124#107—41no—
125#107—41no—
126#106—22no—
127#106—41no—
128#106—41no—
129#106Q1614yescomplemented
130C4⋊C8—21no—
131C4⋊C8—22no—
132C4⋊C8—21no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135#103—42no—
136#103—42no—
137C22⋊C8—41no—
138#97—41no—
139#97—41no—
140C4×C8—41no—
141C4×C8C2×D411yescharacteristic
142C4×C8C2×D411yescharacteristic
143C4×C8C2×D411yescharacteristic
144C4×C8#4111yescharacteristic
145#95—44no—
146#95—41no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—42no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—82no—
153C2×Q8—82no—
154C2×Q8—81no—
155C2×Q8—81no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4#12011yescharacteristic
161C22×C4C2×D811yescharacteristic
162C22×C4AGL(1,ℤ/8)11yescharacteristic
163Q16—84no—
164C2×C8—22no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8#12011yescharacteristic
169C2×C8#12111yescharacteristic
170C2×C8#12111yescharacteristic
171C2×C8#12711yescharacteristic
172C2×C8C2×Q1612yes—
173C2×C8#13712yes—
174C4⋊C4—21no—
175C4⋊C4—24no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—41no—
186C4⋊C4—88no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C22⋊C4—81no—
192C42—41no—
193C42—41no—
194C42—82no—
195C42—81no—
196C42#12011yescharacteristic
197C42#12111yescharacteristic
198C42#13511yescharacteristic
199C42AGL(1,ℤ/8)11yescharacteristic
200C16—24no—
201C16—81no—
202C23#44711yescharacteristic
203Q8—42no—
204Q8—81no—
205Q8—82no—
206Q8—161no—
207Q8—162no—
208Q8—162no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4#39211yescharacteristic
224C2×C4#44711yescharacteristic
225C2×C4#44911yescharacteristic
226C2×C4#44911yescharacteristic
227C2×C4#45112yes—
228C2×C4#45612yes—
229C2×C4#46611yescharacteristic
230C2×C4#46911yescharacteristic
231C8—21no—
232C8—21no—
233C8—82no—
234C8—81no—
235C8#46212yes—
236C8#49411yescharacteristic
237C8#49711yescharacteristic
238C22—21no—
239C22—21no—
240C22#164912yes—
241C22#199911yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—84no—
246C4—81no—
247C4—81no—
248C4—162no—
249C4—161no—
250C4—161no—
251C4#164812yes—
252C4#200511yescharacteristic
253C4#201611yescharacteristic
254C4#219012yes—
255C2—22no—
256C2#1006711yescharacteristic
257C2#1331111yescharacteristic
258C2#1331611yescharacteristic
subgroupG/H
center, socleC22#1999
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#447
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
129. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#337312
#286832
#191502
#132782
#119402
#116112
#71752