Finite Group Databasebeta

← #238768

Subgroups — #238768

Order 512.

1164 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1164 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#21859C212yesmaximal, complemented
3#20764C211yesmaximal, characteristic, complemented
4C2×#1484C212yesmaximal, complemented
5#10431C218yesmaximal, complemented
6#8038C211yesmaximal, characteristic, complemented
7C2×#532C2212yes—
8C2×#431C2212yes—
9C2×C4×M16C2211yescharacteristic
10#2819—22no—
11#2619—22no—
12#2615—24no—
13#2563—22no—
14#2541—22no—
15#2537C2214yes—
16#2285C2211yescharacteristic
17C2×#384C2211yescharacteristic
18#1990—22no—
19#1957—22no—
20#1874—24no—
21#1836—22no—
22#1834C2214yes—
23#1802C2218yes—
24#1736—22no—
25#1484—28no—
26C2×D4⋊C8C2218yes—
27#1296C2214yes—
28C22×#41—22no—
29C22×M16—21no—
30#532—24no—
31#532—24no—
32#532—24no—
33#532—24no—
34C2×C4⋊Q8D412yes—
35C2×#127—22no—
36C2×#127D412yes—
37C2×#124—22no—
38C2×#121—22no—
39C2×#121—24no—
40C2×#121—24no—
41C2×#121D414yes—
42C2×C4×D4—22no—
43C2×C4×D4C2314yes—
44C2×#117D412yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C42C2311yescharacteristic
48#431—24no—
49#431—24no—
50C2×C4⋊C8C2314yes—
51C2×#102—28no—
52C2×C22⋊C8—22no—
53C4×M16—22no—
54C4×M16D414yescomplemented
55C2×#97—21no—
56C2×C4×C8—22no—
57C2×C4×C8C2312yes—
58#394—28no—
59#393—28no—
60#392—28no—
61#392—28no—
62#390—24no—
63#384—21no—
64#384—22no—
65#384C2312yes—
66#384C2312yes—
67#382—22no—
68#382D412yes—
69#380—22no—
70#380—24no—
71#380—24no—
72C4×C4⋊C4—22no—
73#340—28no—
74#340—28no—
75D4⋊C8—216no—
76C2×#41—44no—
77C2×#41—44no—
78C2×#41—84no—
79C2×#41—84no—
80C22×Q8—22no—
81C22×D4—24no—
82C22×D4—28no—
83C22×D4—22no—
84C22×D4—22no—
85C22×D4C2×D414yes—
86C23×C4—24no—
87C23×C4—41no—
88C23×C4C2×D412yes—
89C23×C4C2×D411yescharacteristic
90C2×M16—24no—
91C2×M16—42no—
92C2×M16—42no—
93C22×C8—22no—
94C22×C8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97#127—24no—
98#127—24no—
99#127—84no—
100#124—44no—
101#124—44no—
102#121—28no—
103#121—28no—
104#121—44no—
105#121—44no—
106#121—88no—
107#121—88no—
108C4×D4—24no—
109C4×D4—24no—
110C4×D4—24no—
111C4×D4—24no—
112C4×D4—44no—
113C4×D4—44no—
114#117—44no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—84no—
124C2×C4⋊C4C2×D414yes—
125C2×C4⋊C4C2×D414yes—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—22no—
131C2×C42—22no—
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42#4112yes—
136C2×C42#4114yes—
137C2×C42C2411yescharacteristic
138C4⋊C8—24no—
139C4⋊C8—24no—
140#102—88no—
141C22⋊C8—44no—
142#97—24no—
143C4×C8—22no—
144C4×C8—44no—
145C4×C8C2×D414yes—
146#95—28no—
147#95—24no—
148C24—24no—
149C24#12011yescharacteristic
150#41—816no—
151#41—816no—
152C2×Q8—44no—
153C2×Q8—82no—
154C2×Q8—82no—
155C2×D4—24no—
156C2×D4—24no—
157C2×D4—28no—
158C2×D4—28no—
159C2×D4—48no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—82no—
165C2×D4—82no—
166C2×D4—84no—
167C2×D4—84no—
168C2×D4—88no—
169C2×D4—88no—
170C2×D4—88no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—42no—
182C22×C4—82no—
183C22×C4—84no—
184C22×C4#12012yes—
185C22×C4#12011yescharacteristic
186C22×C4#12114yes—
187C22×C4#12112yes—
188C22×C4#12314yes—
189C22×C4#12412yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D412yes—
192M16—48no—
193C2×C8—24no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—44no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—84no—
208C4⋊C4—82no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C22⋊C4—48no—
212C22⋊C4—82no—
213C22⋊C4—88no—
214C22⋊C4—88no—
215C42—44no—
216C42#12114yes—
217C42#12112yes—
218C42#12314yes—
219C42#12412yes—
220C42C22×D411yescharacteristic
221C42C2×#4111yescharacteristic
222C42C2×#4112yes—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—44no—
228C23—44no—
229C23—44no—
230C23—44no—
231C23—44no—
232C23—44no—
233C23—44no—
234C23#39412yes—
235C23C2×#12011yescharacteristic
236Q8—88no—
237D4—48no—
238D4—48no—
239D4—88no—
240D4—816no—
241D4—88no—
242D4—1616no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—28no—
248C2×C4—28no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—88no—
258C2×C4—88no—
259C2×C4—84no—
260C2×C4—82no—
261C2×C4—164no—
262C2×C4#39414yes—
263C2×C4#39412yes—
264C2×C4#45412yes—
265C2×C4#45512yes—
266C2×C4C2×#12111yescharacteristic
267C2×C4C2×#12112yes—
268C2×C4C2×#12312yes—
269C2×C4C2×#12411yescharacteristic
270C2×C4#57614yes—
271C2×C4#57714yes—
272C8—44no—
273C8—84no—
274C22—24no—
275C22—22no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—84no—
280C22—84no—
281C22—84no—
282C22—84no—
283C22—84no—
284C22—84no—
285C22—84no—
286C22C2×#39411yescharacteristic
287C22#300812yes—
288C22#300912yes—
289C22C2×#45511yescharacteristic
290C22C2×#45411yescharacteristic
291C4—24no—
292C4—21no—
293C4—21no—
294C4—84no—
295C4—84no—
296C4—162no—
297C4#204014yes—
298C4#306214yes—
299C4#309614yes—
300C2—42no—
301C2—164no—
302C2#1872212yes—
303C2#1872711yescharacteristic
304C2#1872811yescharacteristic
305C2#2348312yes—
306C2C2×#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. #21859C2semidirect
3. #20764C2semidirect
4. C2×#1484C2semidirect
5. #10431C2semidirect
6. #8038C2semidirect
54. C4×M16D4semidirect

Maximal subgroups

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

subgroupindex
#376392
#218592
#207642
C2×#14842
#104312
#80382