Finite Group Databasebeta

← #511663

Subgroups — #511663

Order 512.

428 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 128 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#33216C212yesmaximal
2#13538C212yesmaximal, complemented
3#13537C212yesmaximal, complemented
4#13494C212yesmaximal
5#13493C211yesmaximal, characteristic
6#13492C211yesmaximal, characteristic
7#13453C212yesmaximal, complemented
8#13453C212yesmaximal, complemented
9#12845C211yesmaximal, characteristic
10C2×#499C2212yes—
11C2×#405C2211yescharacteristic
12#2629—22no—
13C2×#367C2212yes—
14C2×#366C2211yescharacteristic
15C2×#366C2211yescharacteristic
16C2×C22⋊C16C2211yescharacteristic
17C2×C22⋊C16C2211yescharacteristic
18#1992—22no—
19#1943—22no—
20#1942C2212yes—
21#1942C2212yes—
22#1845C2212yes—
23#1809—22no—
24#1808C2212yes—
25#1808C2212yes—
26#1380C2212yes—
27#1380C2212yes—
28#1380C2212yes—
29#1380C2212yes—
30#1380C2211yescharacteristic
31#1380C2211yescharacteristic
32#1380C2212yes—
33#1374C2212yes—
34#1374C2212yes—
35C22×M16D411yescharacteristic
36C23×C8C2311yescharacteristic
37C2×#125—22no—
38C2×#117D411yescharacteristic
39C22×C16C2311yescharacteristic
40C22×C16C2311yescharacteristic
41C22×C16C2312yes—
42#499—42no—
43#499—42no—
44C2×#107—22no—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×#107C2312yes—
48C2×#107C2312yes—
49C2×#106—22no—
50#405—41no—
51#405—41no—
52C2×#97D412yes—
53C2×C4×C8C2312yes—
54#398—42no—
55#398—42no—
56#396—42no—
57#396—42no—
58#389—22no—
59#389—22no—
60#384—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×C4C2×D411yescharacteristic
73C2×M16—24no—
74C2×M16—41no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D412yes—
81C22×C8C2411yescharacteristic
82#125—44no—
83#125—44no—
84#117—41no—
85#117—41no—
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×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4D811yescharacteristic
97C2×C4⋊C4D811yescharacteristic
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4D812yes—
101C2×C42C2×D412yes—
102C2×C16—21no—
103C2×C16—21no—
104C2×C16—22no—
105C2×C16—22no—
106C2×C16—22no—
107C2×C16—22no—
108C2×C16—22no—
109#107—22no—
110#107—22no—
111#107—22no—
112#107—22no—
113#107—22no—
114#107—22no—
115#107—22no—
116#107—22no—
117#107—44no—
118#106—44no—
119#97—42no—
120C4×C8—22no—
121C4×C8—22no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126C24C2×D811yescharacteristic
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4C2×D811yescharacteristic
136C22×C4C2×D812yes—
137C22×C4C2×D812yes—
138C22×C4C22×D411yescharacteristic
139M16—44no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8#13714yes—
148C2×C8H(5,𝔽2)11yescharacteristic
149C2×C8H(5,𝔽2)11yescharacteristic
150C2×C82−1+412yes—
151C2×C82−1+412yes—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C4⋊C4—162no—
165C4⋊C4—162no—
166C4⋊C4—162no—
167C4⋊C4—162no—
168C22⋊C4—42no—
169C22⋊C4—162no—
170C22⋊C4—162no—
171C42—22no—
172C42—22no—
173C16—41no—
174C16—41no—
175C16—42no—
176C23—21no—
177C23—22no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23C22×D811yescharacteristic
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
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#53411yescharacteristic
206C2×C4#53612yes—
207C2×C4C2×#13712yes—
208C2×C4#57511yescharacteristic
209C8—24no—
210C8—41no—
211C8#56311yescharacteristic
212C8#56311yescharacteristic
213C8#56312yes—
214C22—42no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22#318611yescharacteristic
220C22#322612yes—
221C22#341511yescharacteristic
222C22#341612yes—
223C22#341711yescharacteristic
224C4—41no—
225C4—42no—
226C4—42no—
227C4—162no—
228C4—162no—
229C4—162no—
230C4—162no—
231C4#281711yescharacteristic
232C4#319511yescharacteristic
233C4#323712yes—
234C2—81no—
235C2#3278611yescharacteristic
236C2#3376911yescharacteristic
237C2#3377011yescharacteristic
238C2#3380712yes—
239C2#3380812yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13538C2semidirect
3. #13537C2semidirect
7. #13453C2semidirect
8. #13453C2semidirect

Maximal subgroups

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

subgroupindex
#332162
#135382
#135372
#134942
#134932
#134922
#134532
#128452