Finite Group Databasebeta

← #130431

Subgroups — #130431

Order 512.

730 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 98 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#62245C211yesmaximal, characteristic
2#13190C211yesmaximal, characteristic, complemented
3#11982C212yesmaximal, complemented
4#11678C212yesmaximal, complemented
5#8622C211yesmaximal, characteristic, complemented
6#3496—21no—
7#3334—22no—
8#3294—24no—
9D4×D8—22no—
10#3245—21no—
11#3186C414yes—
12#3169C2212yes—
13C2×#466—22no—
14C2×C8×D4C2211yescharacteristic
15C2×C4⋊C16C2211yescharacteristic, complemented
16C2×C22⋊C16—22no—
17C2×#346C2211yescharacteristic, complemented
18#1568C2212yescomplemented
19#1466—21no—
20#1365—22no—
21#1361—22no—
22#1333—24no—
23#1290—21no—
24C2×H(5,𝔽2)—41no—
25C22×D8—22no—
26C23×C8D412yes—
27#547—24no—
28#546—44no—
29D42—24no—
30#537—41no—
31#534—42no—
32#532—22no—
33C2×#121—24no—
34C2×C4×D4D411yescharacteristic
35C22×C16—21no—
36C22×C16—21no—
37#500D412yes—
38#493D412yes—
39#486—22no—
40#480—28no—
41#474—24no—
42#466—24no—
43#466—24no—
44#466—24no—
45#466—22no—
46#466C2×C414yes—
47#459—24no—
48#447—28no—
49C4×D8—22no—
50C4×D8D414yes—
51C8×D4—22no—
52C8×D4—22no—
53#432—22no—
54C2×#107—22no—
55C2×C4⋊C8D411yescharacteristic
56C2×#102—24no—
57C2×C22⋊C8C2×C412yes—
58C2×C4×C8C2311yescharacteristic
59C4⋊C16—22no—
60C22⋊C16—44no—
61#346—22no—
62#335—22no—
63#334—22no—
64#323—42no—
65H(5,𝔽2)—88no—
66C2×#41—44no—
67C2×#41—42no—
68C22×D4—24no—
69C22×D4—44no—
70C22×D4—44no—
71C22×D4SD1611yescharacteristic
72C23×C4C22⋊C412yes—
73C2×D8—24no—
74C2×D8—22no—
75C2×D8—82no—
76C2×D8—82no—
77C2×D8C22⋊C414yes—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8C22⋊C412yes—
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C4⋊Q8—22no—
86#127—22no—
87#124—42no—
88#123—48no—
89C22⋊Q8—44no—
90#121—28no—
91#121—24no—
92#121—44no—
93#121—44no—
94#121—44no—
95#121—42no—
96#120—48no—
97C4×D4—22no—
98C4×D4—24no—
99C4×D4—22no—
100C4×D4—42no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4SD1611yescharacteristic
103C2×C22⋊C4—44no—
104C2×C22⋊C4D812yes—
105C2×C42C2×D411yescharacteristic
106C2×C16—22no—
107C2×C16—41no—
108C2×C16—41no—
109C2×C16—42no—
110C2×C16—41no—
111#107—24no—
112#107—22no—
113#107C22⋊C414yes—
114C4⋊C8—22no—
115C4⋊C8—42no—
116#102—28no—
117#102—24no—
118#102—24no—
119C22⋊C8—24no—
120C22⋊C8—42no—
121#97—42no—
122C4×C8C2×D412yes—
123C4×C8C2×D412yes—
124C24—44no—
125C24#10212yes—
126#41—88no—
127#41—816no—
128C2×Q8—42no—
129C2×D4—24no—
130C2×D4—24no—
131C2×D4—22no—
132C2×D4—44no—
133C2×D4—48no—
134C2×D4—44no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—84no—
138C2×D4—84no—
139C2×D4—816no—
140C2×D4—88no—
141C2×D4—88no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4#10212yes—
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C2×D811yescharacteristic
151C22×C4C2×SD1611yescharacteristic
152D8—44no—
153D8—42no—
154D8—42no—
155D8—84no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—22no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8C2×C22⋊C412yes—
166C2×C8#12011yescharacteristic
167C2×C8#12011yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—24no—
170C4⋊C4—44no—
171C4⋊C4—42no—
172C22⋊C4—24no—
173C22⋊C4—48no—
174C22⋊C4—44no—
175C22⋊C4—44no—
176C42#12012yes—
177C42AGL(1,ℤ/8)12yes—
178C16—42no—
179C16—81no—
180C23—22no—
181C23—22no—
182C23—42no—
183C23—42no—
184C23—44no—
185C23—88no—
186C23—164no—
187C23—164no—
188C23#36114yes—
189C23C2×#10211yescharacteristic
190Q8—84no—
191D4—28no—
192D4—44no—
193D4—44no—
194D4—84no—
195D4—816no—
196D4—88no—
197D4—162no—
198D4—164no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—82no—
209C2×C4—88no—
210C2×C4—88no—
211C2×C4#37911yescharacteristic
212C2×C4#41811yescharacteristic
213C2×C4#44712yes—
214C2×C4#45012yes—
215C8—21no—
216C8—22no—
217C8—42no—
218C8—162no—
219C8#45311yescharacteristic
220C8#45511yescharacteristic
221C22—21no—
222C22—21no—
223C22—24no—
224C22—44no—
225C22—42no—
226C22—42no—
227C22—82no—
228C22—84no—
229C22—84no—
230C22—84no—
231C22—328no—
232C22#178611yescharacteristic
233C22C2×#36112yes—
234C4—21no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—84no—
239C4#179111yescharacteristic
240C4#179211yescharacteristic
241C2—22no—
242C2—44no—
243C2—164no—
244C2#835311yescharacteristic
245C2#1273611yescharacteristic
246C2#1274511yescharacteristic
subgroupG/H
center, socleC22#1786
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#62245C2
LCSG ▷ C22×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. #13190C2semidirect
3. #11982C2semidirect
4. #11678C2semidirect
5. #8622C2semidirect
15. C2×C4⋊C16C22semidirect
17. C2×#346C22semidirect
18. #1568C22semidirect

Maximal subgroups

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

subgroupindex
#622452
#131902
#119822
#116782
#86222