Finite Group Databasebeta

← #25072663

Subgroups — #25072663

Order 11008.

1716 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 195 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#19145045C211yesmaximal, characteristic, complemented
2#19144686C212yesmaximal, complemented
3#19133711C214yesmaximal, complemented
4#19133649C214yesmaximal, complemented
5#19133309C211yesmaximal, characteristic, complemented
6#19133038C212yesmaximal, complemented
7C2×D42×C43C211yesmaximal, characteristic, complemented
8C2×#13063351—22no—
9#14583449C2211yescharacteristic, complemented
10#14583286—22no—
11#14583275C2212yescomplemented
12#14583271C2212yescomplemented
13#14583217—22no—
14#14583189C2214yescomplemented
15#14583094C2214yescomplemented
16#14583033—22no—
17#14583009—24no—
18#14582956C2212yescomplemented
19C4×#10621877C2211yescharacteristic, complemented
20#14582643C2214yescomplemented
21D4⋊(C43⋊C8)C2218yescomplemented
22C23×D4×C43—22no—
23D42×C43—22no—
24D42×C43—24no—
25D42×C43C2214yes—
26C2×#127×C43C2211yescharacteristic
27C2×#121×C43—22no—
28C2×#120×C43—22no—
29C2×C4×D4×C43C2212yes—
30C2×#10621898—21no—
31#13063351—48no—
32C2×#10621883—24no—
33C2×#10621882—24no—
34C2×#10621882—24no—
35C2×#10621877—21no—
36#13063325C2311yescharacteristic, complemented
37#13063322C2311yescharacteristic, complemented
38#13063313—21no—
39#13063299—22no—
40C4×D172—22no—
41#13063256—24no—
42C4⋊(C43⋊C8)C2314yescomplemented
43#13063246—24no—
44#13063245—24no—
45C22⋊(C43⋊C8)—22no—
46#13063236—21no—
47#13063236—22no—
48C4×C43⋊C8C2312yescomplemented
49C25×C43—22no—
50C22×D4×C43—24no—
51C22×D4×C43—28no—
52C22×D4×C43—28no—
53C22×D4×C43—22no—
54C22×D4×C43—416no—
55C22×D4×C43D412yes—
56C22×D4×C43D412yes—
57C23×C4×C43—22no—
58#127×C43—22no—
59#127×C43D412yes—
60#127×C43C2312yes—
61#121×C43—28no—
62#121×C43—28no—
63#120×C43—28no—
64#120×C43—28no—
65C4×D4×C43—22no—
66C4×D4×C43—22no—
67C4×D4×C43D414yes—
68C4×D4×C43C2314yes—
69C2×C4⋊C4×C43D412yes—
70C2×C22⋊C4×C43—22no—
71C2×C22⋊C4×C43—22no—
72C2×C42×C43C2311yescharacteristic
73C24×C43—22no—
74C24×C43—28no—
75C24×C43—22no—
76C24×C43—416no—
77C24×C43—84no—
78C2×D4×C43—28no—
79C2×D4×C43—216no—
80C2×D4×C43—22no—
81C2×D4×C43—24no—
82C2×D4×C43—22no—
83C2×D4×C43—24no—
84C2×D4×C43—416no—
85C2×D4×C43—48no—
86C2×D4×C43—432no—
87C2×D4×C43—416no—
88C2×D4×C43—416no—
89C2×D4×C43—44no—
90C2×D4×C43C2×D414yes—
91C2×D4×C43C2×D414yes—
92C22×C4×C43—24no—
93C22×C4×C43—24no—
94C22×C4×C43—22no—
95C22×C4×C43—48no—
96C22×C4×C43C2×D411yescharacteristic
97C22×C4×C43C2×D412yes—
98C2×C22⋊D43—41no—
99#10621898—44no—
100C2×D172—22no—
101C2×D172—21no—
102C2×C4×D43—41no—
103C2×Dic86—21no—
104C4⋊C4×C43—22no—
105C4⋊C4×C43C2×D414yes—
106C22⋊C4×C43—28no—
107C22⋊C4×C43—24no—
108C22⋊C4×C43—24no—
109C42×C43C2×D411yescharacteristic
110C42×C43C2×D412yes—
111C42×C43C2411yescharacteristic
112#10621883—48no—
113#10621882—48no—
114#10621882—48no—
115(C2×D43)⋊C4—42no—
116C4⋊Dic43—22no—
117#10621877—24no—
118C2×C43⋊C8—22no—
119C2×C43⋊C8—24no—
120C23×C43—28no—
121C23×C43—24no—
122C23×C43—24no—
123C23×C43—44no—
124C23×C43—48no—
125C23×C43—432no—
126C23×C43—44no—
127C23×C43—44no—
128C23×C43—88no—
129C23×C43—832no—
130C23×C43#12011yescharacteristic
131C22×D43—41no—
132D4×C43—28no—
133D4×C43—24no—
134D4×C43—44no—
135D4×C43—48no—
136D4×C43—432no—
137D4×C43—48no—
138D4×C43—82no—
139D4×C43AGL(1,ℤ/8)18yescomplemented
140C2×C4×C43—22no—
141C2×C4×C43—24no—
142C2×C4×C43—24no—
143C2×C4×C43—48no—
144C2×C4×C43—44no—
145C2×C4×C43—48no—
146C2×C4×C43#12012yes—
147C2×C4×C43#12011yescharacteristic
148C2×C4×C43C22×D411yescharacteristic
149C2×C4×C43C22×D412yes—
150C22⋊D43—82no—
151C2×Dic43—41no—
152D172—42no—
153D172—44no—
154C4×D43—82no—
155Dic86—42no—
156C43⋊C8—24no—
157C43⋊C8—44no—
158#18524—431nomaximal, Sylow
159C22×C43—22no—
160C22×C43—28no—
161C22×C43—416no—
162C22×C43—44no—
163C22×C43—44no—
164C22×C43—48no—
165C22×C43—44no—
166C22×C43—88no—
167C22×C43—832no—
168C22×C43#45312yes—
169C22×C43C2×#12011yescharacteristic
170C2×D43—82no—
171C2×D43—81no—
172C4×C43—22no—
173C4×C43—21no—
174C4×C43—21no—
175C4×C43—44no—
176C4×C43C2×AGL(1,ℤ/8)14yes—
177Dic43—81no—
178C2×D42D4311yescharacteristic, complemented
179#3151—431no—
180#3109—432no—
181#1629—434no—
182#1619—434no—
183#1555—431no—
184#1490—432no—
185C2×C43—22no—
186C2×C43—48no—
187C2×C43—84no—
188C2×C43#300312yes—
189C2×C43C2×#45311yescharacteristic
190D43—161no—
191C23×D4—22no—
192C2×AGL(1,ℤ/8)—862no—
193D42—22no—
194D42—24no—
195D42C2×D4314yes—
196#532—431no—
197C2×#127C2×D4311yescharacteristic
198C2×#121—22no—
199C2×#120—22no—
200C2×C4×D4C2×D4312yes—
201#496—862no—
202#493—432no—
203#492—432no—
204#481—862no—
205#476—434no—
206#459—434no—
207#447—862no—
208#442—864no—
209#432—432no—
210C4×M16—431no—
211#331—434no—
212D4⋊C8—438no—
213C43#1852411yescharacteristic, complemented, Sylow
214C25—22no—
215C2×#41—861no—
216C22×D4—24no—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—416no—
221C22×D4C22⋊D4312yes—
222C22×D4C22⋊D4312yes—
223C23×C4—22no—
224AGL(1,ℤ/8)—1728no—
225C2×SD16—864no—
226C2×D8—864no—
227C2×D8—864no—
228C2×M16—861no—
229C4⋊Q8—431no—
230#127—22no—
231#127—431no—
232#127C22⋊D4312yes—
233#127C22×D4312yes—
234#124—861no—
235#121—28no—
236#121—28no—
237#121—862no—
238#120—28no—
239#120—28no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—862no—
243C4×D4C22⋊D4314yes—
244C4×D4C22×D4314yes—
245C2×C4⋊C4C22⋊D4312yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C42C22×D4311yescharacteristic
249#106—864no—
250C4⋊C8—434no—
251#102—864no—
252#102—864no—
253C22⋊C8—862no—
254#97—861no—
255#97—862no—
256C4×C8—432no—
257C24—22no—
258C24—28no—
259C24—22no—
260C24—416no—
261C24—84no—
262#41—1724no—
263C2×Q8—861no—
264C2×D4—28no—
265C2×D4—216no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—48no—
271C2×D4—44no—
272C2×D4—432no—
273C2×D4—416no—
274C2×D4—416no—
275C2×D4—416no—
276C2×D4—862no—
277C2×D4—861no—
278C2×D4—1721no—
279C2×D4C2×C22⋊D4314yes—
280C2×D4C2×C22⋊D4314yes—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—48no—
285C22×C4—1721no—
286C22×C4C2×C22⋊D4312yes—
287C22×C4C2×C22⋊D4311yescharacteristic
288SD16—1728no—
289D8—1728no—
290D8—1728no—
291M16—864no—
292C2×C8—862no—
293C2×C8—864no—
294C4⋊C4—22no—
295C4⋊C4—862no—
296C4⋊C4C2×C22⋊D4314yes—
297C22⋊C4—28no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—1722no—
301C42C2×C22⋊D4311yescharacteristic
302C42C2×C22⋊D4312yes—
303C42C23×D4311yescharacteristic
304C23—28no—
305C23—24no—
306C23—24no—
307C23—44no—
308C23—48no—
309C23—432no—
310C23—44no—
311C23—44no—
312C23—88no—
313C23—832no—
314C23—1721no—
315C23#1306329411yescharacteristic
316Q8—1722no—
317D4—28no—
318D4—24no—
319D4—44no—
320D4—48no—
321D4—432no—
322D4—48no—
323D4—82no—
324D4—3442no—
325D4—1722no—
326D4—1724no—
327D4#1306335118yescomplemented
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—48no—
334C2×C4—3442no—
335C2×C4—1721no—
336C2×C4#1306329411yescharacteristic
337C2×C4#1306329412yes—
338C2×C4C22×C22⋊D4311yescharacteristic
339C2×C4C22×C22⋊D4312yes—
340C8—864no—
341C8—1724no—
342C22—22no—
343C22—28no—
344C22—416no—
345C22—44no—
346C22—44no—
347C22—48no—
348C22—44no—
349C22—88no—
350C22—832no—
351C22—3442no—
352C22—3441no—
353C22#1458306712yes—
354C22C2×#1306329411yescharacteristic
355C4—22no—
356C4—21no—
357C4—21no—
358C4—44no—
359C4—3441no—
360C4C2×#1306335114yes—
361C2—22no—
362C2—48no—
363C2—84no—
364C2—6881no—
365C2#1914382212yes—
366C2C2×#1458306711yescharacteristic
subgroupG/H
centerC22C2×#13063294
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC2×D42×C43C2
socleC22×C43C2×#120
hypercenterC2×D42D43
hypocenterC43#18524
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. #19145045C2semidirect
2. #19144686C2semidirect
3. #19133711C2semidirect
4. #19133649C2semidirect
5. #19133309C2semidirect
6. #19133038C2semidirect
7. C2×D42×C43C2semidirect
9. #14583449C22semidirect
11. #14583275C22semidirect
12. #14583271C22semidirect
14. #14583189C22semidirect
15. #14583094C22semidirect
18. #14582956C22semidirect
19. C4×#10621877C22semidirect
20. #14582643C22semidirect
21. D4⋊(C43⋊C8)C22semidirect
36. #13063325C23semidirect
37. #13063322C23semidirect
42. C4⋊(C43⋊C8)C23semidirect
48. C4×C43⋊C8C23semidirect
139. D4×C43AGL(1,ℤ/8)semidirect
178. C2×D42D43semidirect
213. C43#18524semidirect
327. D4#13063351semidirect

Maximal subgroups

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

subgroupindex
#191450452
#191446862
#191337112
#191336492
#191333092
#191330382
C2×D42×C432
#1852443