Finite Group Databasebeta

← #145277

Subgroups — #145277

Order 512.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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#19259C211yesmaximal, characteristic
2#12021C211yesmaximal, characteristic, complemented
3#11987C211yesmaximal, characteristic, complemented
4#11676C211yesmaximal, characteristic, complemented
5#11583C211yesmaximal, characteristic
6#8622C211yesmaximal, characteristic
7#7246C211yesmaximal, characteristic, complemented
8#3170C2211yescharacteristic
9#3110—21no—
10C2×C8×D4C2211yescharacteristic
11#2150C2211yescharacteristic
12C4×M32C2211yescharacteristic
13#1679—22no—
14#1673—21no—
15#1669—21no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1568C2211yescharacteristic
19#1552—21no—
20#1466—21no—
21#1368—22no—
22#1365—21no—
23#1362—21no—
24#1332—21no—
25#1332—21no—
26D4⋊C16—21no—
27D4⋊C16—21no—
28#1290—21no—
29C23×C8—21no—
30#533—21no—
31C2×C4×D4D411yescharacteristic
32C2×M32—21no—
33C2×M32—21no—
34#500D411yescharacteristic
35#494D411yescharacteristic
36#487—21no—
37#484—41no—
38#474—22no—
39#467—21no—
40#467—21no—
41#461—22no—
42#450—41no—
43#440—42no—
44C4×Q16—21no—
45C4×Q16—21no—
46C8×D4—21no—
47C8×D4—21no—
48C8×D4—21no—
49C8×D4—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53C2×C4⋊C8D411yescharacteristic
54C2×C22⋊C8—21no—
55C2×C4×C8C2311yescharacteristic
56C4⋊C16—21no—
57C4⋊C16—21no—
58#346—21no—
59C4×C16D411yescharacteristic
60C4×C16D411yescharacteristic
61#335—21no—
62#335—21no—
63#335—21no—
64#334—21no—
65#334—21no—
66#334D412yes—
67#332—22no—
68#332—22no—
69#323—42no—
70C22×D4SD1611yescharacteristic
71C23×C4—21no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×SD16—42no—
75C22×C8—21no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—81no—
79C22×C8—81no—
80C22×C8#4111yescharacteristic
81C22×C8#4111yescharacteristic
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84#125—41no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93#117—41no—
94C2×C4⋊C4SD1611yescharacteristic
95C2×C22⋊C4—21no—
96C2×C42C2×D411yescharacteristic
97M32—42no—
98M32—81no—
99C2×C16—21no—
100C2×C16—21no—
101C2×C16—41no—
102#107—21no—
103#107—21no—
104#107—42no—
105C4⋊C8—21no—
106C4⋊C8—22no—
107C4⋊C8—21no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#103—42no—
111#103—41no—
112#103—41no—
113#102—42no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119#97—41no—
120#97—41no—
121#97—41no—
122C4×C8C2×D411yescharacteristic
123C4×C8C2×D411yescharacteristic
124C4×C8C2×D411yescharacteristic
125C4×C8#4111yescharacteristic
126C24—21no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—42no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—81no—
136C22×C4—81no—
137C22×C4#12311yescharacteristic
138C22×C4C2×SD1611yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140Q16—41no—
141Q16—41no—
142Q16—81no—
143SD16—84no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—42no—
149C2×C8—82no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8—81no—
154C2×C8—81no—
155C2×C8#12111yescharacteristic
156C2×C8#12111yescharacteristic
157C2×C8#12311yescharacteristic
158C2×C8#12411yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—81no—
170C42—81no—
171C42#12011yescharacteristic
172C42#12111yescharacteristic
173C42#13511yescharacteristic
174C42AGL(1,ℤ/8)11yescharacteristic
175C16—22no—
176C16—41no—
177C16—81no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—81no—
182C23—81no—
183C23#48111yescharacteristic
184Q8—82no—
185Q8—81no—
186Q8—81no—
187D4—22no—
188D4—22no—
189D4—41no—
190D4—41no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4#39411yescharacteristic
203C2×C4#44911yescharacteristic
204C2×C4#45011yescharacteristic
205C2×C4#46511yescharacteristic
206C2×C4#46911yescharacteristic
207C2×C4#48211yescharacteristic
208C8—21no—
209C8—21no—
210C8—21no—
211C8—82no—
212C8—81no—
213C8—161no—
214C8—161no—
215C8#47111yescharacteristic
216C8#47211yescharacteristic
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—82no—
222C22—82no—
223C22—81no—
224C22—81no—
225C22#203411yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—81no—
232C4—161no—
233C4—161no—
234C4#205011yescharacteristic
235C4#205111yescharacteristic
236C2—41no—
237C2—82no—
238C2#1045711yescharacteristic
239C2#1339511yescharacteristic
240C2#1340511yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12021C2semidirect
3. #11987C2semidirect
4. #11676C2semidirect
7. #7246C2semidirect

Maximal subgroups

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

subgroupindex
#192592
#120212
#119872
#116762
#115832
#86222
#72462