Finite Group Databasebeta

← #238450

Subgroups — #238450

Order 512.

1800 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1800 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
1C2×#2656C211yesmaximal, characteristic
2#21764C212yesmaximal, complemented
3C2×#1507C212yesmaximal, complemented
4#13965C211yesmaximal, characteristic, complemented
5#10402C218yesmaximal, complemented
6#7909C211yesmaximal, characteristic, complemented
7C22×#121—22no—
8C22×#121C2212yes—
9C2×#407C2212yes—
10#2656—22no—
11#2656—24no—
12#2656C2214yes—
13C2×#399—22no—
14C2×#392—22no—
15C2×#385—22no—
16C2×#380C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18C22×#95C2211yescharacteristic
19#1895—24no—
20#1874C2218yescomplemented
21#1824—22no—
22#1786—22no—
23#1507—28no—
24C2×#327C2218yescomplemented
25#1444C2214yescomplemented
26C4⋊#95—22no—
27#1280C2214yescomplemented
28C23×D4—22no—
29C23×D4—22no—
30C23×D4D412yes—
31C24×C4D412yes—
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34C2×#121—28no—
35C2×#121—22no—
36C2×#121—22no—
37C2×#121—28no—
38C2×#121—24no—
39C2×#121—24no—
40C2×#121—24no—
41C2×#121—24no—
42C2×#121—416no—
43C2×#121D414yes—
44C2×#121C2314yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—24no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C42—21no—
54C2×#102—28no—
55#407—24no—
56#407—24no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8C2314yes—
59#399—28no—
60#399—28no—
61#392—28no—
62#392—28no—
63#385—48no—
64#380—22no—
65#380—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4D414yescomplemented
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—21no—
71C2×#95—24no—
72C2×#95D412yes—
73C2×#95C2312yes—
74C2×#95C2312yes—
75#340—28no—
76#327—216no—
77C25—22no—
78C25C2×D411yescharacteristic
79C22×D4—24no—
80C22×D4—24no—
81C22×D4—28no—
82C22×D4—22no—
83C22×D4—24no—
84C22×D4—24no—
85C22×D4—48no—
86C22×D4—416no—
87C22×D4—44no—
88C22×D4—44no—
89C22×D4—84no—
90C22×D4—84no—
91C22×D4—84no—
92C22×D4C2×D414yes—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—28no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—44no—
100C23×C4—82no—
101C23×C4—82no—
102C23×C4C2×D414yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4#4114yes—
108C23×C4#4112yes—
109C23×C4C2411yescharacteristic
110C2×M16—44no—
111C2×M16—84no—
112C22×C8—24no—
113#121—28no—
114#121—28no—
115#121—24no—
116#121—24no—
117#121—24no—
118#121—24no—
119#121—432no—
120#121—416no—
121#121—44no—
122#121—44no—
123#121—44no—
124#121—44no—
125#121—416no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—48no—
132C2×C4⋊C4—48no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4—84no—
136C2×C4⋊C4C2×D414yes—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—216no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—416no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C42—24no—
150C2×C42—82no—
151#102—88no—
152C22⋊C8—24no—
153C22⋊C8—24no—
154C22⋊C8—44no—
155#95—24no—
156#95—24no—
157#95—22no—
158#95—48no—
159#95—42no—
160#95C2×D414yescomplemented
161C24—24no—
162C24—22no—
163C24—22no—
164C24—24no—
165C24—42no—
166C24—48no—
167C24—48no—
168C24—44no—
169C24—44no—
170C24—44no—
171C24#12011yescharacteristic
172C24#12011yescharacteristic
173C24#12112yes—
174C24#12412yes—
175C24C22×D411yescharacteristic
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—48no—
179C2×D4—416no—
180C2×D4—416no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—416no—
184C2×D4—48no—
185C2×D4—44no—
186C2×D4—84no—
187C2×D4—88no—
188C2×D4—88no—
189C2×D4—88no—
190C2×D4—832no—
191C2×D4—88no—
192C2×D4—816no—
193C2×D4—816no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—28no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—84no—
222C22×C4—816no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4#12012yes—
226C22×C4#12114yes—
227C22×C4#12114yes—
228C22×C4#12112yes—
229C22×C4#12314yes—
230C22×C4#12314yes—
231C22×C4#12412yes—
232C22×C4AGL(1,ℤ/8)18yes—
233C22×C4C22×D412yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4112yes—
236C22×C4C2×#4111yescharacteristic
237M16—88no—
238C2×C8—48no—
239C2×C8—44no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—416no—
243C4⋊C4—48no—
244C4⋊C4—44no—
245C4⋊C4—816no—
246C22⋊C4—216no—
247C22⋊C4—432no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—48no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C42—84no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—24no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—42no—
268C23—42no—
269C23—48no—
270C23—48no—
271C23—44no—
272C23—48no—
273C23—44no—
274C23—44no—
275C23—48no—
276C23—816no—
277C23—88no—
278C23—88no—
279C23—816no—
280C23—88no—
281C23#39412yes—
282C23#39412yes—
283C23#45312yes—
284C23#45612yes—
285C23C2×#12011yescharacteristic
286C23C2×#12111yescharacteristic
287C23C2×#12411yescharacteristic
288C23C2×AGL(1,ℤ/8)14yes—
289D4—48no—
290D4—84no—
291D4—88no—
292D4—832no—
293D4—816no—
294D4—88no—
295D4—88no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—48no—
311C2×C4—48no—
312C2×C4—416no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—48no—
317C2×C4—44no—
318C2×C4—48no—
319C2×C4—84no—
320C2×C4—88no—
321C2×C4—88no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#39414yes—
328C2×C4C2×#12112yes—
329C2×C4C2×#12312yes—
330C8—84no—
331C22—24no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—48no—
345C22—48no—
346C22—42no—
347C22—42no—
348C22—88no—
349C22—88no—
350C22—84no—
351C22—84no—
352C22—84no—
353C22—1616no—
354C22#203914yes—
355C22C2×#39411yescharacteristic
356C22#300312yes—
357C22C2×#45311yescharacteristic
358C22C2×#45611yescharacteristic
359C22#302612yes—
360C22#303812yes—
361C22#313812yes—
362C4—22no—
363C4—22no—
364C4—24no—
365C4—48no—
366C4—48no—
367C4—84no—
368C4—162no—
369C2—22no—
370C2—24no—
371C2—22no—
372C2—42no—
373C2—164no—
374C2#1831612yes—
375C2#1832512yes—
376C2C2×#203911yescharacteristic
377C2#2350511yescharacteristic
378C2#2351511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2656C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21764C2semidirect
3. C2×#1507C2semidirect
4. #13965C2semidirect
5. #10402C2semidirect
6. #7909C2semidirect
20. #1874C22semidirect
24. C2×#327C22semidirect
25. #1444C22semidirect
27. #1280C22semidirect
67. C4×C22⋊C4D4semidirect
160. #95C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26562
#217642
C2×#15072
#139652
#104022
#79092