Finite Group Databasebeta

← #144033

Subgroups — #144033

Order 512.

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

normal subgroups only

This group has 353 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#19150C211yesmaximal, characteristic
3C2×#1694C211yesmaximal, characteristic
4#13268C211yesmaximal, characteristic
5#11940C211yesmaximal, characteristic
6#11936C211yesmaximal, characteristic
7#7158C211yesmaximal, characteristic
8#3160C2211yescharacteristic
9C2×#500C2211yescharacteristic
10#3104—21no—
11C2×#462—21no—
12C2×#446C2211yescharacteristic
13#2255—22no—
14#2254—21no—
15#2252—21no—
16#2159—22no—
17#2150C2211yescharacteristic
18C4×M32C2211yescharacteristic
19#1696—21no—
20#1694—22no—
21#1694—21no—
22#1694—22no—
23#1692—22no—
24#1564C2211yescharacteristic
25C2×#334C2211yescharacteristic
26#1542—21no—
27C2×#332—21no—
28#1377—22no—
29#1364—22no—
30#1363—22no—
31C22×Q16—41no—
32#533—21no—
33C2×C4⋊Q8—21no—
34C2×C4×Q8D411yescharacteristic
35C2×M32—21no—
36C2×M32D412yes—
37#500—21no—
38#500—21no—
39#500—21no—
40#500D412yes—
41#500D411yescharacteristic
42#500D411yescharacteristic
43#499—21no—
44#499—21no—
45#498—21no—
46#484—41no—
47#477—22no—
48#462—22no—
49#462—24no—
50#462—42no—
51#451—41no—
52#446—21no—
53#446—21no—
54#446—21no—
55#446—21no—
56C4×Q16—42no—
57#431—21no—
58#427—21no—
59#427—21no—
60C2×#107—21no—
61C2×#107—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8D411yescharacteristic
65C2×#103—41no—
66C2×#97—21no—
67C2×C4×C8C2311yescharacteristic
68#367—24no—
69#366—24no—
70C4⋊C16—21no—
71C4⋊C16—21no—
72#346—21no—
73C4×C16—21no—
74#335D412yes—
75#334—21no—
76#334—21no—
77#334D412yes—
78#333—22no—
79#332—24no—
80#332—22no—
81C22×Q8—41no—
82C22×Q8D811yescharacteristic
83C2×Q16—44no—
84C2×Q16—42no—
85C2×Q16—84no—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8C2×D411yescharacteristic
89C22×C8#4111yescharacteristic
90C4⋊Q8—22no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96#125—41no—
97C22⋊Q8—41no—
98C22⋊Q8—41no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—41no—
104C4×Q8—41no—
105#117—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4D811yescharacteristic
110C2×C42—21no—
111C2×C42C2×D411yescharacteristic
112M32—22no—
113M32—42no—
114M32Q1614yescomplemented
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—41no—
127#106—41no—
128C4⋊C8—21no—
129C4⋊C8—21no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135#103—44no—
136#103—42no—
137#103—42no—
138C22⋊C8—41no—
139#97—41no—
140#97—41no—
141C4×C8—41no—
142C4×C8C2×D411yescharacteristic
143C4×C8C2×D411yescharacteristic
144C4×C8C2×D411yescharacteristic
145C4×C8C2×D411yescharacteristic
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—42no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—41no—
152C2×Q8—82no—
153C2×Q8—82no—
154C2×Q8—81no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—81no—
158C22×C4#12111yescharacteristic
159C22×C4C2×D811yescharacteristic
160C22×C4#13511yescharacteristic
161Q16—84no—
162Q16—88no—
163C2×C8—22no—
164C2×C8—41no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—82no—
169C2×C8#12011yescharacteristic
170C2×C8#12111yescharacteristic
171C2×C8#12111yescharacteristic
172C2×C8#12711yescharacteristic
173C2×C8C2×Q1612yes—
174C2×C8#13712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C22⋊C4—81no—
190C42—41no—
191C42—41no—
192C42—81no—
193C42#12011yescharacteristic
194C42#12011yescharacteristic
195C42AGL(1,ℤ/8)11yescharacteristic
196C42AGL(1,ℤ/8)11yescharacteristic
197C16—24no—
198C16—81no—
199C23#46611yescharacteristic
200Q8—42no—
201Q8—42no—
202Q8—82no—
203Q8—82no—
204Q8—82no—
205Q8—161no—
206Q8—162no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—162no—
217C2×C4#39211yescharacteristic
218C2×C4#44711yescharacteristic
219C2×C4#44711yescharacteristic
220C2×C4#44911yescharacteristic
221C2×C4#44911yescharacteristic
222C2×C4#46212yes—
223C2×C4#46911yescharacteristic
224C2×C4#47312yes—
225C8—21no—
226C8—21no—
227C8—82no—
228C8—82no—
229C8—81no—
230C8#45112yes—
231C8#49411yescharacteristic
232C8#49711yescharacteristic
233C22—21no—
234C22—21no—
235C22#167412yes—
236C22#199911yescharacteristic
237C4—21no—
238C4—21no—
239C4—21no—
240C4—81no—
241C4—81no—
242C4—162no—
243C4—161no—
244C4—161no—
245C4#169412yes—
246C4#201611yescharacteristic
247C4#202011yescharacteristic
248C4#219012yes—
249C2—22no—
250C2#1026511yescharacteristic
251C2#1331611yescharacteristic
252C2#1332411yescharacteristic
subgroupG/H
center, socleC22#1999
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#466
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
114. M32Q16semidirect

Maximal subgroups

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

subgroupindex
#337312
#191502
C2×#16942
#132682
#119402
#119362
#71582