Finite Group Databasebeta

← #511281

Subgroups — #511281

Order 512.

434 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 132 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#27839C212yesmaximal
2#13516C212yesmaximal, complemented
3#13505C212yesmaximal, complemented
4#13493C211yesmaximal, characteristic
5#13493C211yesmaximal, characteristic
6#13176C212yesmaximal, complemented
7#13111C212yesmaximal, complemented
8#12913C211yesmaximal, characteristic
9#12836C212yesmaximal
10C22×M32C2211yescharacteristic
11#2496—22no—
12C2×#367C2212yes—
13C2×#366C2212yes—
14C2×#366C2211yescharacteristic
15C2×#366C2211yescharacteristic
16C2×C4⋊C16C2212yes—
17C2×C22⋊C16C2211yescharacteristic
18#2099—22no—
19#2075—22no—
20#1942C2212yes—
21#1942C2212yes—
22#1918—22no—
23#1845C2212yes—
24#1808C2212yes—
25#1808C2212yes—
26C4×#107C2212yes—
27C8×C22⋊C4C2211yescharacteristic
28#1380C2212yes—
29#1380C2212yes—
30#1380C2211yescharacteristic
31#1380C2212yes—
32#1380C2212yes—
33#1380C2211yescharacteristic
34#1374C2212yes—
35C23×C8C2311yescharacteristic
36C2×#126—42no—
37C2×M32—24no—
38C2×M32—41no—
39C22×C16C2311yescharacteristic
40C22×C16C2311yescharacteristic
41C22×C16C2312yes—
42C2×#107—22no—
43C2×#107C2312yes—
44C2×#107C2312yes—
45C2×#107C2312yes—
46C2×#107C2312yes—
47C2×C22⋊C8D411yescharacteristic
48C2×C4×C8C2312yes—
49#400—42no—
50#397—42no—
51#389—22no—
52#389—22no—
53#387—42no—
54#384—22no—
55#380—22no—
56#380—22no—
57C4×C4⋊C4—22no—
58C4×C22⋊C4D411yescharacteristic
59#367—24no—
60#366—42no—
61#366—41no—
62#366—41no—
63C4⋊C16—24no—
64C22⋊C16—41no—
65#340—22no—
66#340—22no—
67#340—22no—
68#340—22no—
69#340—22no—
70#336D412yes—
71C23×C4C2×D411yescharacteristic
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—22no—
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D412yes—
78C22×C8C2411yescharacteristic
79#126—88no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4D812yes—
91C2×C42—42no—
92C2×C42C2×D412yes—
93M32—44no—
94C2×C16—22no—
95C2×C16—22no—
96C2×C16—22no—
97C2×C16—21no—
98C2×C16—21no—
99C2×C16—22no—
100C2×C16#4114yes—
101#107—22no—
102#107—22no—
103#107—22no—
104#107—22no—
105#107—22no—
106#107—22no—
107#107—22no—
108#107—22no—
109#107—44no—
110C22⋊C8—41no—
111C4×C8—22no—
112C4×C8—22no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95D811yescharacteristic
119#95D811yescharacteristic
120C24C2×D811yescharacteristic
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4C2×D811yescharacteristic
130C22×C4C2×D812yes—
131C22×C4C2×D812yes—
132C22×C4C22×D411yescharacteristic
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—42no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8C2×#4112yes—
142C2×C8H(5,𝔽2)11yescharacteristic
143C2×C82−1+411yescharacteristic
144C2×C82−1+412yes—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—84no—
154C4⋊C4—84no—
155C4⋊C4—84no—
156C4⋊C4—84no—
157C22⋊C4—24no—
158C22⋊C4—84no—
159C22⋊C4—84no—
160C42—22no—
161C42—22no—
162C42—84no—
163C16—24no—
164C16—41no—
165C16—41no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23C22×D811yescharacteristic
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4#50914yes—
195C2×C4#53711yescharacteristic
196C2×C4#54712yes—
197C2×C4#57511yescharacteristic
198C2×C4#57812yes—
199C8—42no—
200C8—41no—
201C8#54112yes—
202C8#55111yescharacteristic
203C8#56311yescharacteristic
204C22—41no—
205C22—41no—
206C22—42no—
207C22—42no—
208C22—41no—
209C22#324511yescharacteristic
210C22#333412yes—
211C22C2×#50912yes—
212C22#341711yescharacteristic
213C22#341711yescharacteristic
214C4—24no—
215C4—42no—
216C4—41no—
217C4—162no—
218C4—162no—
219C4—162no—
220C4—162no—
221C4#272211yescharacteristic
222C4#326111yescharacteristic
223C4#334912yes—
224C2—81no—
225C2#3240611yescharacteristic
226C2#3381911yescharacteristic
227C2#3382711yescharacteristic
228C2#3391012yes—
229C2#3392012yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13516C2semidirect
3. #13505C2semidirect
6. #13176C2semidirect
7. #13111C2semidirect

Maximal subgroups

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

subgroupindex
#278392
#135162
#135052
#134932
#131762
#131112
#129132
#128362