Finite Group Databasebeta

← #303995

Subgroups — #303995

Order 512.

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

normal subgroups only

This group has 973 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#28717C212yesmaximal, complemented
3#28508C211yesmaximal, characteristic
4#10439C214yesmaximal, complemented
5#9673C212yesmaximal, complemented
6#8445C214yesmaximal
7C4×#331C211yesmaximal, characteristic
8C2×#559C2212yes—
9C2×#487—22no—
10C2×#474—22no—
11#2868C2211yescharacteristic
12#2724—22no—
13#2582C2212yes—
14#2547—22no—
15#2486—22no—
16#2447—24no—
17D4×C4⋊C4—22no—
18C4×C4⋊Q8C2212yes—
19C4×#127C2211yescharacteristic
20#2093—22no—
21#2072—24no—
22#1990—22no—
23#1957C2212yes—
24#1845C2212yes—
25#1803C2218yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#331C2212yes—
29C2×Q8⋊C8C2218yes—
30#1296C2212yes—
31#559—28no—
32#559—44no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—21no—
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#487—48no—
45#474—48no—
46C2×#107—24no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8C2314yes—
49C2×#103—28no—
50C2×#102—24no—
51C2×C4×C8—22no—
52C2×C4×C8C2312yes—
53#398—24no—
54#397—24no—
55#397—28no—
56#391—24no—
57#391D414yes—
58#390—22no—
59#390D412yes—
60#389—24no—
61#389—22no—
62#389—22no—
63#385—28no—
64#384—22no—
65#384—21no—
66#384D412yes—
67#384C2312yes—
68#384C2311yescharacteristic
69#381—24no—
70#380—24no—
71#380—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C43C2311yescharacteristic
75#340—28no—
76#336—22no—
77#336—21no—
78#331—24no—
79Q8⋊C8—48no—
80C22×Q8—41no—
81C22×Q8C2×D414yes—
82C22×D4—22no—
83C22×D4C2×D412yes—
84C23×C4—24no—
85C23×C4—22no—
86C22×C8—22no—
87C22×C8—24no—
88C4⋊Q8—24no—
89C4⋊Q8—24no—
90C4⋊Q8—44no—
91C4⋊Q8—44no—
92C4⋊Q8—44no—
93#127—22no—
94#127#4114yes—
95#124—416no—
96#123—48no—
97#123—48no—
98C4×Q8—28no—
99C4×Q8—28no—
100C4×D4—44no—
101C4×D4—416no—
102C4×D4—44no—
103C4×D4—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—84no—
116C2×C4⋊C4C2×D414yes—
117C2×C4⋊C4C2×D412yes—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—22no—
122C2×C42—24no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D412yes—
127C2×C42C2411yescharacteristic
128#107—48no—
129C4⋊C8—24no—
130C4⋊C8—24no—
131C4⋊C8—24no—
132#103—416no—
133#102—48no—
134C4×C8—24no—
135C4×C8—44no—
136#95—28no—
137#95—24no—
138#95—24no—
139#95—24no—
140C24—22no—
141C2×Q8—28no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—28no—
145C2×Q8—84no—
146C2×Q8—82no—
147C2×D4—22no—
148C2×D4—28no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—44no—
154C2×D4#12414yes—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—48no—
159C22×C4—48no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—84no—
163C22×C4—84no—
164C22×C4#12011yescharacteristic
165C22×C4#12012yes—
166C22×C4#12011yescharacteristic
167C22×C4C22×D411yescharacteristic
168C22×C4C22×D412yes—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—44no—
173C2×C8—44no—
174C2×C8—44no—
175C4⋊C4—28no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C22⋊C4—416no—
192C22⋊C4—48no—
193C22⋊C4—48no—
194C22⋊C4—82no—
195C42—24no—
196C42—22no—
197C42—41no—
198C42—48no—
199C42#13514yes—
200C42AGL(1,ℤ/8)14yes—
201C42C2×#4112yes—
202C42H(5,𝔽2)12yes—
203C23—24no—
204C23—44no—
205C23—44no—
206C23—42no—
207C23—42no—
208C23C2×#12011yescharacteristic
209Q8—48no—
210Q8—48no—
211Q8—164no—
212D4—44no—
213D4—48no—
214D4—48no—
215D4—82no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—88no—
231C2×C4—84no—
232C2×C4—82no—
233C2×C4—168no—
234C2×C4#44914yes—
235C2×C4#44914yes—
236C2×C4#45312yes—
237C2×C4C2×#12411yescharacteristic
238C2×C4#53511yescharacteristic
239C2×C4#54614yes—
240C2×C4C2×#13512yes—
241C2×C4C2×AGL(1,ℤ/8)12yes—
242C2×C4#57814yes—
243C8—44no—
244C8—84no—
245C22—44no—
246C22—48no—
247C22—82no—
248C22—82no—
249C22—84no—
250C22#243511yescharacteristic
251C22C2×#44912yes—
252C22#301112yes—
253C22C2×#45311yescharacteristic
254C22#302211yescharacteristic
255C4—21no—
256C4—21no—
257C4—24no—
258C4—41no—
259C4—84no—
260C4—84no—
261C4—82no—
262C4—162no—
263C4#162014yes—
264C4#330214yes—
265C4#333014yes—
266C2—84no—
267C2C2×#162011yescharacteristic
268C2#1871911yescharacteristic
269C2#1890612yes—
270C2#2587011yescharacteristic
271C2#2588212yes—
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. #28717C2semidirect
4. #10439C2semidirect
5. #9673C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#287172
#285082
#104392
#96732
#84452
C4×#3312