Finite Group Databasebeta

← #508451

Subgroups — #508451

Order 512.

508 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×#2227C211yesmaximal, characteristic
2C2×#2225C211yesmaximal, characteristic
3#33313C211yesmaximal, characteristic
4#21276C211yesmaximal, characteristic
5#13507C214yesmaximal
6#13483C214yesmaximal
7#13276C211yesmaximal, characteristic
8#13275C211yesmaximal, characteristic
9#12879C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×#498—21no—
13C2×C4×Q16C2211yescharacteristic
14#2862—21no—
15#2295—21no—
16#2227—24no—
17#2225—24no—
18C2×#367C2212yes—
19C2×#366C2212yes—
20C2×C4⋊C16C2211yescharacteristic
21C2×C4⋊C16C2211yescharacteristic
22C2×#358C2212yes—
23C2×#358C2212yes—
24#1969C2212yes—
25#1942C2212yes—
26#1933—22no—
27#1846—22no—
28#1846C2214yes—
29C4×#107C2211yescharacteristic
30C4×#103—21no—
31C42×C8C2211yescharacteristic
32#1380C2214yes—
33#1380C2214yes—
34#1374C2214yes—
35C22×Q16C2311yescharacteristic
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4×Q8—21no—
39C22×C16C2312yes—
40C22×C16C2312yes—
41#500—22no—
42#500—22no—
43#500—22no—
44#500—22no—
45#498—44no—
46C4×Q16—22no—
47C4×Q16D414yes—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107C2312yes—
51C2×#107C2312yes—
52C2×#107C2312yes—
53C2×#107C2311yescharacteristic
54C2×#106—24no—
55C2×#103—21no—
56C2×#103—22no—
57C2×C4×C8D412yes—
58C2×C4×C8D411yescharacteristic
59C2×C4×C8C2312yes—
60C2×C4×C8C2311yescharacteristic
61#391—22no—
62#391—41no—
63#391—41no—
64#391—42no—
65#389—22no—
66#387—42no—
67#382—24no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—41no—
70#376—41no—
71C43D411yescharacteristic
72#367—24no—
73#366—24no—
74C4⋊C16—22no—
75C4⋊C16—22no—
76#358—24no—
77#358—24no—
78#340—21no—
79#340—22no—
80C22×Q8—21no—
81C22×Q8—41no—
82C22×Q8—41no—
83C2×Q16—21no—
84C2×Q16—22no—
85C2×Q16—22no—
86C2×Q16C2×D412yes—
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D412yes—
89C22×C8C2411yescharacteristic
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4×Q8—24no—
95C4×Q8—42no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42D811yescharacteristic
107C2×C42D811yescharacteristic
108C2×C42D812yes—
109C2×C42C2×D412yes—
110C2×C42C2×D411yescharacteristic
111C2×C16—22no—
112C2×C16—22no—
113C2×C16—24no—
114C2×C16—24no—
115#107—21no—
116#107—22no—
117#107—22no—
118#107—22no—
119#107—22no—
120#107—22no—
121#107—22no—
122#107—42no—
123#107—42no—
124#107C2×D412yes—
125#106—48no—
126#103—24no—
127#103—44no—
128C4×C8—24no—
129C4×C8—22no—
130C4×C8—22no—
131C4×C8—22no—
132C4×C8C2×D412yes—
133C4×C8#4112yes—
134#95—42no—
135#95—42no—
136#95—41no—
137#95—42no—
138#95—44no—
139#95—44no—
140C2×Q8—22no—
141C2×Q8—41no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—84no—
145C2×Q8—82no—
146C2×Q8—82no—
147C2×Q8—82no—
148C2×Q8—82no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C2×D812yes—
155C22×C4C2×D812yes—
156C22×C4C2×D811yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4C22×D411yescharacteristic
159Q16—22no—
160Q16—22no—
161Q16—42no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8#12112yes—
166C2×C8#13714yes—
167C2×C8C22×D411yescharacteristic
168C2×C8C2×#4111yescharacteristic
169C2×C82−1+412yes—
170C2×C82−1+412yes—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—44no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—84no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—42no—
188C42—44no—
189C42—82no—
190C42#12112yes—
191C42AGL(1,ℤ/8)12yes—
192C16—42no—
193C16—42no—
194C23C22×D811yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—82no—
198Q8—164no—
199Q8—164no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4#45912yes—
220C2×C4#45912yes—
221C2×C4#50812yes—
222C2×C4#51012yes—
223C2×C4C2×#12111yescharacteristic
224C2×C4#53612yes—
225C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
226C2×C4C2×#13712yes—
227C8—24no—
228C8—22no—
229C8#53611yescharacteristic
230C8#53611yescharacteristic
231C8#54112yes—
232C22C2×#45911yescharacteristic
233C22#322612yes—
234C22C2×#50811yescharacteristic
235C22C2×#51011yescharacteristic
236C22#341412yes—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4—82no—
242C4—162no—
243C4—162no—
244C4—162no—
245C4#221114yes—
246C4#262111yescharacteristic
247C4#303911yescharacteristic
248C4#323912yes—
249C2#2986411yescharacteristic
250C2C2×#221111yescharacteristic
251C2#3358411yescharacteristic
252C2#3379912yes—
253C2#3380312yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#22272
C2×#22252
#333132
#212762
#135072
#134832
#132762
#132752
#128792