Finite Group Databasebeta

← #511659

Subgroups — #511659

Order 512.

450 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 130 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#33208C212yesmaximal
2#13539C212yesmaximal, complemented
3#13539C212yesmaximal, complemented
4#13494C212yesmaximal
5#13493C211yesmaximal, characteristic
6#13492C211yesmaximal, characteristic
7#13453C212yesmaximal, complemented
8#13453C212yesmaximal, complemented
9#12844C211yesmaximal, characteristic
10C2×#499C2212yes—
11C22×C4×C8C2211yescharacteristic
12#2851—22no—
13C2×#367C2212yes—
14C2×#366C2211yescharacteristic
15C2×#366C2211yescharacteristic
16C2×C22⋊C16C2211yescharacteristic
17C2×C22⋊C16C2211yescharacteristic
18#1991—22no—
19#1942C2212yes—
20#1942C2212yes—
21#1941—22no—
22#1846C2212yes—
23#1808C2212yes—
24#1808C2212yes—
25#1807—22no—
26#1380C2212yes—
27#1380C2212yes—
28#1380C2212yes—
29#1380C2212yes—
30#1380C2211yescharacteristic
31#1380C2211yescharacteristic
32#1380C2212yes—
33#1374C2212yes—
34#1374C2212yes—
35C23×C8D411yescharacteristic
36C23×C8C2311yescharacteristic
37C2×#125—22no—
38C22×C42D411yescharacteristic
39C22×C16C2311yescharacteristic
40C22×C16C2311yescharacteristic
41C22×C16C2312yes—
42#499—42no—
43#499—42no—
44C2×#107C2312yes—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#107C2312yes—
48C2×#106—22no—
49C2×#106—22no—
50C2×C4×C8—41no—
51C2×C4×C8—41no—
52C2×C4×C8D412yes—
53C2×C4×C8C2312yes—
54#389—22no—
55#389—22no—
56#387—42no—
57#387—42no—
58#385—42no—
59#385—42no—
60#382—22no—
61#380—22no—
62#380—22no—
63#367—42no—
64#366—41no—
65#366—41no—
66C22⋊C16—41no—
67C22⋊C16—41no—
68#340—22no—
69#340—22no—
70#340—22no—
71#340—22no—
72C23×C4D811yescharacteristic
73C23×C4D811yescharacteristic
74C23×C4C2×D411yescharacteristic
75C22×C8—24no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D412yes—
83C22×C8C2411yescharacteristic
84#125—44no—
85#125—44no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42D812yes—
99C2×C42C2×D412yes—
100C2×C16—21no—
101C2×C16—21no—
102C2×C16—22no—
103C2×C16—22no—
104C2×C16—22no—
105C2×C16—22no—
106C2×C16—22no—
107#107—22no—
108#107—22no—
109#107—22no—
110#107—22no—
111#107—22no—
112#107—22no—
113#107—22no—
114#107—22no—
115#106—44no—
116#106—44no—
117C4×C8—22no—
118C4×C8—22no—
119C4×C8—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126C24C2×D811yescharacteristic
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4C2×D811yescharacteristic
141C22×C4C2×D812yes—
142C22×C4C2×D812yes—
143C22×C4C22×D411yescharacteristic
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—22no—
148C2×C8—44no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—42no—
152C2×C8#13714yes—
153C2×C8H(5,𝔽2)11yescharacteristic
154C2×C8H(5,𝔽2)11yescharacteristic
155C2×C82−1+412yes—
156C2×C82−1+412yes—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C4⋊C4—84no—
169C22⋊C4—84no—
170C22⋊C4—84no—
171C42—22no—
172C42—22no—
173C42—42no—
174C16—41no—
175C16—41no—
176C16—42no—
177C23—22no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23#50812yes—
182C23C22×D811yescharacteristic
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4#53411yescharacteristic
209C2×C4#53612yes—
210C2×C4C2×#13712yes—
211C2×C4#57511yescharacteristic
212C8—24no—
213C8—41no—
214C8#56311yescharacteristic
215C8#56311yescharacteristic
216C8#56312yes—
217C22—24no—
218C22—22no—
219C22—42no—
220C22—41no—
221C22—41no—
222C22#318611yescharacteristic
223C22#322612yes—
224C22C2×#50811yescharacteristic
225C22#341412yes—
226C22#341411yescharacteristic
227C4—41no—
228C4—42no—
229C4—42no—
230C4—162no—
231C4—162no—
232C4—162no—
233C4—162no—
234C4#281711yescharacteristic
235C4#319511yescharacteristic
236C4#323712yes—
237C2—42no—
238C2#3278611yescharacteristic
239C2#3375711yescharacteristic
240C2#3376311yescharacteristic
241C2#3379812yes—
242C2#3380612yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13539C2semidirect
3. #13539C2semidirect
7. #13453C2semidirect
8. #13453C2semidirect

Maximal subgroups

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

subgroupindex
#332082
#135392
#134942
#134932
#134922
#134532
#128442