Finite Group Databasebeta

← #328052

Subgroups — #328052

Order 512.

680 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 148 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#37643C211yesmaximal, characteristic
2#29939C212yesmaximal
3#27975C211yesmaximal, characteristic
4#10581C214yesmaximal, complemented
5#10355C214yesmaximal, complemented
6#8729C211yesmaximal, characteristic
7#8709C212yesmaximal
8#2857C2211yescharacteristic
9#2845—22no—
10#2629—24no—
11#2629—22no—
12#2552—22no—
13#2552—22no—
14#2552C2214yes—
15#2285C2212yes—
16#2285C2211yescharacteristic
17#2100—22no—
18#2083—22no—
19#2076—24no—
20#2074—22no—
21#1987C2214yes—
22#1956C2212yes—
23#1947C2214yes—
24#1873—22no—
25#1834C2212yes—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28#1296C2214yes—
29#1279C2218yes—
30C2×C4⋊Q8C2311yescharacteristic
31C2×#125—22no—
32C2×#125—24no—
33C2×C22⋊Q8—21no—
34C2×#117—22no—
35C2×#117D412yes—
36C22×C42C2311yescharacteristic
37C2×#107—24no—
38C2×C4⋊C8C2314yes—
39C2×#103—24no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—22no—
42C2×C4×C8C2312yes—
43#398—28no—
44#398—28no—
45#398—28no—
46#396—28no—
47#391—21no—
48#389—24no—
49#389—24no—
50#389—24no—
51#389—22no—
52#384—22no—
53#384—24no—
54#384D412yes—
55#384D412yes—
56#384C2311yescharacteristic
57#384C2312yes—
58#382—22no—
59#382D412yes—
60#380—22no—
61#380—22no—
62#380—22no—
63#380D414yes—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4C2314yes—
67#340—24no—
68#340—24no—
69#340—24no—
70#340—24no—
71#336—22no—
72#336—21no—
73C22×Q8—21no—
74C23×C4C2×D412yes—
75C23×C4C2×D411yescharacteristic
76C22×C8—24no—
77C22×C8—22no—
78C4⋊Q8—24no—
79#125—84no—
80#125—84no—
81#125—84no—
82C22⋊Q8—44no—
83#117—82no—
84#117—82no—
85#117—82no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4C2×D414yes—
100C2×C4⋊C4C2×D414yes—
101C2×C4⋊C4C2×D414yes—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—41no—
105C2×C42—24no—
106C2×C42—22no—
107C2×C42—24no—
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42C2411yescharacteristic
111#107—84no—
112C4⋊C8—28no—
113#103—84no—
114C22⋊C8—42no—
115C22⋊C8—82no—
116C4×C8—24no—
117#95—28no—
118#95—24no—
119#95—24no—
120#95—24no—
121#95—42no—
122C24#12011yescharacteristic
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4#12012yes—
139C22×C4#12011yescharacteristic
140C22×C4C22×D411yescharacteristic
141C22×C4C22×D412yes—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—44no—
146C2×C8—44no—
147C2×C8—44no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—28no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—88no—
164C4⋊C4—88no—
165C4⋊C4—88no—
166C4⋊C4—82no—
167C22⋊C4—82no—
168C22⋊C4—84no—
169C22⋊C4—82no—
170C42—24no—
171C42—22no—
172C42—44no—
173C42—44no—
174C42—44no—
175C422−1+414yes—
176C23—21no—
177C23—22no—
178C23—22no—
179C23—21no—
180C23—21no—
181C23C2×#12011yescharacteristic
182Q8—84no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4#45412yes—
208C2×C4#53612yes—
209C2×C4#53614yes—
210C2×C4#57612yes—
211C2×C4#57712yes—
212C2×C4#57814yes—
213C8—82no—
214C8—84no—
215C22—41no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22—42no—
220C22#261411yescharacteristic
221C22#301112yes—
222C22#301212yes—
223C22C2×#45411yescharacteristic
224C22#301911yescharacteristic
225C4—22no—
226C4—22no—
227C4—42no—
228C4—84no—
229C4—84no—
230C4—84no—
231C4—161no—
232C4—161no—
233C4#324214yes—
234C4#324314yes—
235C2—81no—
236C2#1869611yescharacteristic
237C2#1873211yescharacteristic
238C2#1894112yes—
239C2#2964912yes—
240C2#2965111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10581C2semidirect
5. #10355C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#299392
#279752
#105812
#103552
#87292
#87092