Finite Group Databasebeta

← #303967

Subgroups — #303967

Order 512.

953 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 953 conjugacy classes of subgroups — too many to draw. Only its 176 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#59827C211yesmaximal, characteristic
2#29141C212yesmaximal, complemented
3#28509C211yesmaximal, characteristic
4#10437C214yesmaximal, complemented
5#9673C212yesmaximal, complemented
6#8445C214yesmaximal
7C4×#331C211yesmaximal, characteristic
8C2×#559C2212yes—
9C2×#476—22no—
10#2868C2211yescharacteristic
11#2724—22no—
12#2582C2212yes—
13#2547—22no—
14#2486—22no—
15#2447—24no—
16D4×C4⋊C4—22no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2094—22no—
20#2024—24no—
21#1957C2212yes—
22#1956—22no—
23#1844C2212yes—
24#1834—22no—
25#1803C2218yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#331C2212yes—
29C2×Q8⋊C8C2218yes—
30#1296C2212yes—
31#559—28no—
32#559—44no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8D412yes—
36C2×#127C2311yescharacteristic
37C2×#124—24no—
38C2×#123—24no—
39C2×C4×Q8C2314yes—
40C2×C4×D4—24no—
41C2×C4×D4—22no—
42C2×C4×D4D412yes—
43C22×C4⋊C4—22no—
44#476—48no—
45C2×#106—24no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8C2314yes—
48C2×#103—28no—
49C2×#102—24no—
50C2×C4×C8—22no—
51C2×C4×C8C2312yes—
52#398—24no—
53#397—24no—
54#397—28no—
55#391—24no—
56#391D414yes—
57#390—22no—
58#390D412yes—
59#389—24no—
60#389—22no—
61#389—22no—
62#385—28no—
63#384—22no—
64#384—21no—
65#384D412yes—
66#384C2312yes—
67#384C2311yescharacteristic
68#381—24no—
69#380—24no—
70#380—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C43C2311yescharacteristic
74#340—28no—
75#336—21no—
76#336—22no—
77#331—24no—
78Q8⋊C8—48no—
79C22×Q8—41no—
80C22×Q8C2×D414yes—
81C22×D4—22no—
82C22×D4C2×D412yes—
83C23×C4—24no—
84C23×C4—22no—
85C22×C8—22no—
86C22×C8—24no—
87C4⋊Q8—24no—
88C4⋊Q8—24no—
89C4⋊Q8—44no—
90C4⋊Q8—84no—
91#127—22no—
92#127#4114yes—
93#124—416no—
94#123—48no—
95#123—48no—
96C4×Q8—28no—
97C4×Q8—28no—
98C4×D4—44no—
99C4×D4—416no—
100C4×D4—44no—
101C4×D4—44no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—84no—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D412yes—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—22no—
120C2×C42—24no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2411yescharacteristic
126#106—48no—
127C4⋊C8—24no—
128C4⋊C8—24no—
129C4⋊C8—24no—
130#103—88no—
131#102—48no—
132C4×C8—24no—
133C4×C8—44no—
134#95—28no—
135#95—24no—
136#95—24no—
137#95—24no—
138C24—22no—
139C2×Q8—28no—
140C2×Q8—24no—
141C2×Q8—24no—
142C2×Q8—28no—
143C2×Q8—82no—
144C2×Q8—84no—
145C2×D4—22no—
146C2×D4—28no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—44no—
152C2×D4#12414yes—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—84no—
161C22×C4—84no—
162C22×C4#12011yescharacteristic
163C22×C4#12012yes—
164C22×C4#12011yescharacteristic
165C22×C4C22×D411yescharacteristic
166C22×C4C22×D412yes—
167C2×C8—24no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—44no—
171C2×C8—44no—
172C2×C8—44no—
173C4⋊C4—28no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—48no—
184C4⋊C4—82no—
185C4⋊C4—84no—
186C4⋊C4—84no—
187C4⋊C4—82no—
188C22⋊C4—416no—
189C22⋊C4—48no—
190C22⋊C4—48no—
191C22⋊C4—82no—
192C42—24no—
193C42—22no—
194C42—41no—
195C42—48no—
196C42#13514yes—
197C42#13714yes—
198C42C2×#4112yes—
199C42H(5,𝔽2)12yes—
200C23—24no—
201C23—44no—
202C23—44no—
203C23—42no—
204C23—42no—
205C23C2×#12011yescharacteristic
206Q8—48no—
207Q8—48no—
208Q8—164no—
209D4—44no—
210D4—48no—
211D4—48no—
212D4—82no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—88no—
228C2×C4—84no—
229C2×C4—82no—
230C2×C4—168no—
231C2×C4#45214yes—
232C2×C4#45214yes—
233C2×C4#45312yes—
234C2×C4C2×#12411yescharacteristic
235C2×C4#53511yescharacteristic
236C2×C4#54614yes—
237C2×C4C2×#13512yes—
238C2×C4C2×#13712yes—
239C2×C4#57714yes—
240C8—44no—
241C8—84no—
242C22—44no—
243C22—48no—
244C22—82no—
245C22—82no—
246C22—84no—
247C22#243511yescharacteristic
248C22C2×#45212yes—
249C22#301012yes—
250C22C2×#45311yescharacteristic
251C22#302211yescharacteristic
252C4—21no—
253C4—21no—
254C4—24no—
255C4—41no—
256C4—84no—
257C4—84no—
258C4—82no—
259C4—162no—
260C4#163014yes—
261C4#330314yes—
262C4#333114yes—
263C2—84no—
264C2C2×#163011yescharacteristic
265C2#1871811yescharacteristic
266C2#1890712yes—
267C2#2587011yescharacteristic
268C2#2588612yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29141C2semidirect
4. #10437C2semidirect
5. #9673C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#291412
#285092
#104372
#96732
#84452
C4×#3312