Finite Group Databasebeta

← #25072661

Subgroups — #25072661

Order 11008.

1716 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 171 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×#14583534C211yesmaximal, characteristic, complemented
2#19144682C212yesmaximal, complemented
3#19133709C214yesmaximal, complemented
4#19133647C214yesmaximal, complemented
5#19133309C211yesmaximal, characteristic, complemented
6#19133039C212yesmaximal, complemented
7C43×#3151C211yesmaximal, characteristic, complemented
8C22×D4×D43—22no—
9#14583534—24no—
10#14583534—22no—
11#14583534C2214yes—
12C2×#13063296—22no—
13C2×#13063293—22no—
14C2×C4×D172C2212yes—
15#14583215—22no—
16#14583186C2214yescomplemented
17#14583091C2214yescomplemented
18#14583032—22no—
19#14583009—24no—
20#14582956C2212yescomplemented
21#14582643C2214yescomplemented
22#14582628C2218yescomplemented
23C2×AGL(1,ℤ/8)×C43—22no—
24C43×#532C2211yescharacteristic, complemented
25C2×#127×C43C2211yescharacteristic
26C43×#496—22no—
27C43×#493C2212yescomplemented
28C43×#492C2212yescomplemented
29C4×M16×C43C2211yescharacteristic, complemented
30C24×D43—22no—
31C22×D172D412yes—
32C22×C22⋊D43—24no—
33C2×D4×D43—28no—
34C2×D4×D43—28no—
35C2×D4×D43—416no—
36C22×C4×D43—22no—
37C2×#10621882—24no—
38#13063296—28no—
39#13063296—28no—
40#13063293—28no—
41#13063293—28no—
42C4×D172—22no—
43C4×D172—22no—
44C4×D172D414yes—
45C4×D172C2314yes—
46C2×C4⋊Dic43D412yes—
47C2×C22⋊Dic43—22no—
48C2×(C2×D43)⋊C4—22no—
49#13063256—24no—
50C4⋊(C43⋊C8)C2314yescomplemented
51#13063246—24no—
52#13063245—24no—
53C22⋊(C43⋊C8)—22no—
54#13063236—22no—
55C2×#41×C43—21no—
56C22×D4×C43—22no—
57C22×D4×C43D412yes—
58AGL(1,ℤ/8)×C43—48no—
59C2×SD16×C43—24no—
60C2×D8×C43—24no—
61C2×M16×C43—21no—
62C4⋊Q8×C43C2311yescharacteristic, complemented
63#127×C43—22no—
64#127×C43D412yes—
65#127×C43C2312yes—
66#127×C43C2311yescharacteristic, complemented
67#124×C43—21no—
68#121×C43—22no—
69C4×D4×C43—22no—
70C2×C42×C43C2311yescharacteristic
71#97×C43—21no—
72C4×C8×C43C2312yescomplemented
73C24×C43—22no—
74C23×D43—22no—
75C23×D43—28no—
76C23×D43—416no—
77C23×D43—84no—
78#41×C43—44no—
79C2×Q8×C43—21no—
80C2×D4×C43—24no—
81C2×D4×C43—22no—
82C2×D4×C43—24no—
83C2×D4×C43—22no—
84C2×D4×C43—21no—
85C2×D4×C43—48no—
86C2×D4×C43—41no—
87C2×D4×C43—44no—
88C2×D4×C43C2×D414yes—
89C22×C4×C43—41no—
90C22×C4×C43C2×D412yes—
91C22×C4×C43C2×D411yescharacteristic
92C2×C22⋊D43—216no—
93C2×C22⋊D43—416no—
94C22×Dic43—22no—
95D4×D43—432no—
96D4×D43—416no—
97D4×D43—416no—
98C2×D172—22no—
99C2×D172—28no—
100C2×D172C2×D414yes—
101C2×C4×D43—24no—
102C2×C4×D43—24no—
103C2×C4×D43—48no—
104SD16×C43—48no—
105D8×C43—48no—
106M16×C43—24no—
107C2×C8×C43—22no—
108C4⋊C4×C43—22no—
109C22⋊C4×C43—42no—
110C42×C43C2×D412yes—
111C42×C43C2×D411yescharacteristic
112C42×C43C2411yescharacteristic
113C22⋊Dic43—28no—
114#10621882—48no—
115(C2×D43)⋊C4—24no—
116(C2×D43)⋊C4—24no—
117C4⋊Dic43—22no—
118C4⋊Dic43C2×D414yes—
119C2×C43⋊C8—24no—
120C23×C43—24no—
121C23×C43—48no—
122C23×C43—44no—
123C23×C43—41no—
124C23×C43#12011yescharacteristic
125C22×D43—28no—
126C22×D43—24no—
127C22×D43—432no—
128C22×D43—44no—
129C22×D43—44no—
130C22×D43—88no—
131C22×D43—832no—
132Q8×C43—42no—
133D4×C43—44no—
134D4×C43—48no—
135D4×C43—42no—
136D4×C43—48no—
137D4×C43—44no—
138D4×C43—82no—
139D4×C43—82no—
140C2×C4×C43—22no—
141C2×C4×C43—24no—
142C2×C4×C43—41no—
143C2×C4×C43—82no—
144C2×C4×C43#12012yes—
145C2×C4×C43#12011yescharacteristic
146C2×C4×C43C22×D412yes—
147C2×C4×C43C22×D411yescharacteristic
148C22⋊D43—432no—
149C2×Dic43—24no—
150C2×Dic43—44no—
151D172—28no—
152D172—24no—
153D172AGL(1,ℤ/8)18yescomplemented
154C4×D43—48no—
155C4×D43—48no—
156C8×C43—24no—
157C43⋊C8—44no—
158#18524—431nomaximal, Sylow
159C22×C43—22no—
160C22×C43—48no—
161C22×C43—44no—
162C22×C43—88no—
163C22×C43—82no—
164C22×C43—81no—
165C22×C43#45312yes—
166C22×C43C2×#12011yescharacteristic
167C2×D43—28no—
168C2×D43—416no—
169C2×D43—44no—
170C2×D43—44no—
171C2×D43—832no—
172C4×C43—22no—
173C4×C43—21no—
174C4×C43—21no—
175C4×C43—81no—
176C4×C43C2×AGL(1,ℤ/8)14yes—
177Dic43—44no—
178C2×D42—431no—
179#3151D4311yescharacteristic, complemented
180#3109—432no—
181#1629—434no—
182#1619—434no—
183#1555—431no—
184#1490—432no—
185C2×C43—22no—
186C2×C43—84no—
187C2×C43—161no—
188C2×C43#300312yes—
189C2×C43C2×#45311yescharacteristic
190D43—48no—
191C23×D4—862no—
192C2×AGL(1,ℤ/8)—22no—
193D42—434no—
194D42—864no—
195D42—862no—
196#532C2×D4311yescharacteristic, complemented
197C2×#127C2×D4311yescharacteristic
198C2×#121—862no—
199C2×#120—862no—
200C2×C4×D4—432no—
201#496—22no—
202#493C2×D4312yescomplemented
203#492C2×D4312yescomplemented
204#481—862no—
205#476—434no—
206#459—434no—
207#447—862no—
208#442—864no—
209#432—432no—
210C4×M16C2×D4311yescharacteristic, complemented
211#331—434no—
212D4⋊C8—438no—
213C43#1852411yescharacteristic, complemented, Sylow
214C25—862no—
215C2×#41—21no—
216C22×D4—22no—
217C22×D4—432no—
218C22×D4—868no—
219C22×D4—868no—
220C22×D4—864no—
221C22×D4—17216no—
222C22×D4C22⋊D4312yes—
223C23×C4—862no—
224AGL(1,ℤ/8)—48no—
225C2×SD16—24no—
226C2×D8—24no—
227C2×D8—864no—
228C2×M16—21no—
229C4⋊Q8C22×D4311yescharacteristic, complemented
230#127—22no—
231#127C22⋊D4312yes—
232#127C22×D4312yes—
233#127C22×D4311yescharacteristic, complemented
234#124—21no—
235#121—22no—
236#121—868no—
237#121—868no—
238#120—868no—
239#120—868no—
240C4×D4—22no—
241C4×D4—434no—
242C4×D4—434no—
243C4×D4—862no—
244C4×D4—862no—
245C2×C4⋊C4—432no—
246C2×C22⋊C4—862no—
247C2×C22⋊C4—862no—
248C2×C42C22×D4311yescharacteristic
249#106—864no—
250C4⋊C8—434no—
251#102—864no—
252#102—864no—
253C22⋊C8—862no—
254#97—21no—
255#97—862no—
256C4×C8C22×D4312yescomplemented
257C24—22no—
258C24—868no—
259C24—862no—
260C24—3444no—
261C24—17216no—
262#41—44no—
263C2×Q8—21no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—21no—
269C2×D4—48no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—434no—
273C2×D4—868no—
274C2×D4—8616no—
275C2×D4—862no—
276C2×D4—17216no—
277C2×D4—17232no—
278C2×D4—17216no—
279C2×D4—17216no—
280C2×D4C2×C22⋊D4314yes—
281C22×C4—41no—
282C22×C4—862no—
283C22×C4—864no—
284C22×C4—864no—
285C22×C4—1728no—
286C22×C4D4×D4312yes—
287C22×C4C2×C22⋊D4311yescharacteristic
288SD16—48no—
289D8—48no—
290D8—1728no—
291M16—24no—
292C2×C8—22no—
293C2×C8—864no—
294C4⋊C4—22no—
295C4⋊C4—434no—
296C4⋊C4—862no—
297C22⋊C4—42no—
298C22⋊C4—868no—
299C22⋊C4—864no—
300C22⋊C4—864no—
301C42D4×D4312yes—
302C42C2×C22⋊D4311yescharacteristic
303C42C23×D4311yescharacteristic
304C23—24no—
305C23—48no—
306C23—44no—
307C23—41no—
308C23—868no—
309C23—864no—
310C23—3448no—
311C23—34432no—
312C23—1724no—
313C23—17232no—
314C23—1724no—
315C23#1306329311yescharacteristic
316Q8—42no—
317D4—44no—
318D4—48no—
319D4—42no—
320D4—48no—
321D4—44no—
322D4—82no—
323D4—82no—
324D4—438no—
325D4—868no—
326D4—864no—
327D4—17232no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—864no—
333C2×C4—1728no—
334C2×C4—1724no—
335C2×C4—1728no—
336C2×C4#1306329311yescharacteristic
337C2×C4#1306329312yes—
338C2×C4C2×D4×D4312yes—
339C2×C4C22×C22⋊D4311yescharacteristic
340C8—24no—
341C8—1724no—
342C22—22no—
343C22—48no—
344C22—44no—
345C22—88no—
346C22—82no—
347C22—81no—
348C22—868no—
349C22—34432no—
350C22—17216no—
351C22—1724no—
352C22—1724no—
353C22#1458306612yes—
354C22C2×#1306329311yescharacteristic
355C4—22no—
356C4—21no—
357C4—21no—
358C4—81no—
359C4—1724no—
360C4AGL(1,ℤ/8)×D4314yes—
361C2—22no—
362C2—84no—
363C2—161no—
364C2—1728no—
365C2#1914382012yes—
366C2C2×#1458306611yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3151C2
socleC22×C43C2×#120
hypercenter#3151D43
hypocenterC43#18524
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. C2×#14583534C2semidirect
2. #19144682C2semidirect
3. #19133709C2semidirect
4. #19133647C2semidirect
5. #19133309C2semidirect
6. #19133039C2semidirect
7. C43×#3151C2semidirect
16. #14583186C22semidirect
17. #14583091C22semidirect
20. #14582956C22semidirect
21. #14582643C22semidirect
22. #14582628C22semidirect
24. C43×#532C22semidirect
27. C43×#493C22semidirect
28. C43×#492C22semidirect
29. C4×M16×C43C22semidirect
50. C4⋊(C43⋊C8)C23semidirect
62. C4⋊Q8×C43C23semidirect
66. #127×C43C23semidirect
72. C4×C8×C43C23semidirect
153. D172AGL(1,ℤ/8)semidirect
179. #3151D43semidirect
196. #532C2×D43semidirect
202. #493C2×D43semidirect
203. #492C2×D43semidirect
210. C4×M16C2×D43semidirect
213. C43#18524semidirect
229. C4⋊Q8C22×D43semidirect
233. #127C22×D43semidirect
256. C4×C8C22×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145835342
#191446822
#191337092
#191336472
#191333092
#191330392
C43×#31512
#1852443