Finite Group Databasebeta

← #238790

Subgroups — #238790

Order 512.

1160 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 202 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#37639C211yesmaximal, characteristic
2#21893C212yesmaximal, complemented
3#20760C211yesmaximal, characteristic, complemented
4C2×#1492C212yesmaximal, complemented
5#10431C218yesmaximal, complemented
6#8041C211yesmaximal, characteristic, complemented
7C2×#532C2212yes—
8C2×#423C2212yes—
9C22×C4×C8C2211yescharacteristic
10#2819—22no—
11#2619—22no—
12#2615—24no—
13#2563—22no—
14#2541—22no—
15#2537C2214yes—
16#2285C2211yescharacteristic
17#2285C2211yescharacteristic
18#1987—22no—
19#1982—22no—
20#1895—24no—
21#1835—22no—
22#1834C2214yes—
23#1802C2218yes—
24#1735—22no—
25#1492—28no—
26C2×D4⋊C8C2218yes—
27#1296C2214yes—
28C22×#41—22no—
29C22×M16—22no—
30C23×C8—21no—
31#532—24no—
32#532—24no—
33#532—24no—
34#532—24no—
35C2×C4⋊Q8D412yes—
36C2×#127—22no—
37C2×#127D412yes—
38C2×#124—22no—
39C2×#121—22no—
40C2×#121—28no—
41C2×#121D414yes—
42C2×C4×D4—22no—
43C2×C4×D4C2314yes—
44C2×#117—21no—
45C2×#117D412yes—
46C22×C42C2311yescharacteristic
47#423—24no—
48#423—24no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2314yes—
51C2×#102—28no—
52C2×C4×C8—21no—
53C2×C4×C8—22no—
54C2×C4×C8D414yescomplemented
55C2×C4×C8C2312yes—
56#394—28no—
57#393—28no—
58#392—28no—
59#392—28no—
60#390—24no—
61#384C2312yes—
62#384C2312yes—
63#382—22no—
64#382—21no—
65#382D412yes—
66#380—22no—
67#380—28no—
68#380—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71#340—28no—
72#340—28no—
73D4⋊C8—216no—
74C2×#41—44no—
75C2×#41—44no—
76C2×#41—84no—
77C2×#41—84no—
78C22×Q8—22no—
79C22×D4—24no—
80C22×D4—28no—
81C22×D4—22no—
82C22×D4—22no—
83C22×D4C2×D414yes—
84C23×C4—24no—
85C23×C4C2×D412yes—
86C23×C4C2×D411yescharacteristic
87C2×M16—44no—
88C2×M16—84no—
89C22×C8—24no—
90C22×C8—22no—
91C22×C8—24no—
92C22×C8—42no—
93C22×C8—42no—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96#127—24no—
97#127—24no—
98#127—84no—
99#124—44no—
100#124—44no—
101#121—28no—
102#121—28no—
103#121—44no—
104#121—44no—
105#121—88no—
106#121—88no—
107C4×D4—24no—
108C4×D4—24no—
109C4×D4—24no—
110C4×D4—24no—
111C4×D4—44no—
112C4×D4—44no—
113#117—44no—
114#117—44no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—216no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—41no—
126C2×C42—22no—
127C2×C42—41no—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42#4112yes—
132C2×C42#4114yes—
133C2×C42C2411yescharacteristic
134C4⋊C8—24no—
135C4⋊C8—24no—
136C4⋊C8—44no—
137#102—88no—
138C4×C8—24no—
139C4×C8—22no—
140C4×C8C2×D414yes—
141#95—28no—
142#95—24no—
143#95—42no—
144C24—24no—
145C24#12011yescharacteristic
146#41—816no—
147#41—88no—
148#41—88no—
149C2×Q8—44no—
150C2×Q8—84no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—48no—
156C2×D4—44no—
157C2×D4—44no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—84no—
161C2×D4—88no—
162C2×D4—816no—
163C2×D4—88no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—42no—
175C22×C4#12012yes—
176C22×C4#12011yescharacteristic
177C22×C4#12114yes—
178C22×C4#12112yes—
179C22×C4#12314yes—
180C22×C4#12412yes—
181C22×C4C22×D411yescharacteristic
182C22×C4C22×D412yes—
183M16—88no—
184C2×C8—24no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—48no—
188C2×C8—44no—
189C2×C8—48no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C22⋊C4—48no—
202C22⋊C4—82no—
203C22⋊C4—816no—
204C22⋊C4—82no—
205C42—44no—
206C42—82no—
207C42#12112yes—
208C42#12114yes—
209C42#12314yes—
210C42#12412yes—
211C42C22×D411yescharacteristic
212C42C2×#4111yescharacteristic
213C42C2×#4112yes—
214C23—24no—
215C23—21no—
216C23—21no—
217C23—44no—
218C23—48no—
219C23—48no—
220C23—44no—
221C23—44no—
222C23#39412yes—
223C23C2×#12011yescharacteristic
224Q8—88no—
225D4—48no—
226D4—44no—
227D4—44no—
228D4—88no—
229D4—816no—
230D4—88no—
231D4—1616no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—28no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—44no—
241C2×C4—48no—
242C2×C4—48no—
243C2×C4—44no—
244C2×C4—88no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4#39414yes—
250C2×C4#39412yes—
251C2×C4#45412yes—
252C2×C4#45512yes—
253C2×C4C2×#12111yescharacteristic
254C2×C4C2×#12112yes—
255C2×C4C2×#12312yes—
256C2×C4C2×#12411yescharacteristic
257C2×C4#57614yes—
258C2×C4#57714yes—
259C8—44no—
260C8—84no—
261C22—24no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—88no—
266C22—88no—
267C22—84no—
268C22—84no—
269C22—84no—
270C22C2×#39411yescharacteristic
271C22#301214yes—
272C22C2×#45511yescharacteristic
273C22C2×#45411yescharacteristic
274C4—24no—
275C4—21no—
276C4—21no—
277C4—84no—
278C4—84no—
279C4—162no—
280C4#204014yes—
281C4#305712yes—
282C4#305812yes—
283C4#309614yes—
284C2—42no—
285C2—164no—
286C2#1869112yes—
287C2#1869612yes—
288C2#2348612yes—
289C2C2×#204011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#37639C2
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
2. #21893C2semidirect
3. #20760C2semidirect
4. C2×#1492C2semidirect
5. #10431C2semidirect
6. #8041C2semidirect
54. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#376392
#218932
#207602
C2×#14922
#104312
#80412