Finite Group Databasebeta

← #25072700

Subgroups — #25072700

Order 11008.

1630 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1630 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
2#19144690C212yesmaximal, complemented
3#19133771C218yesmaximal, complemented
4#19133309C211yesmaximal, characteristic, complemented
5#19133039C212yesmaximal, complemented
6C43×#3138C211yesmaximal, characteristic, complemented
7C22×D4×D43—22no—
8#14583534—24no—
9#14583534—22no—
10#14583534C2214yes—
11C2×#13063296—22no—
12C2×#13063293—22no—
13C2×C4×D172C2212yes—
14#14583232—22no—
15#14583176C2214yescomplemented
16#14583101C2214yescomplemented
17#14583049—22no—
18#14582997—24no—
19#14582956C2212yescomplemented
20#14582643C2214yescomplemented
21#14582628C2218yescomplemented
22C43×#533C2211yescharacteristic, complemented
23C2×#127×C43C2211yescharacteristic
24C43×#490—22no—
25C43×#486C2214yescomplemented
26C43×#418—22no—
27C4×M16×C43C2211yescharacteristic, complemented
28C24×D43—22no—
29C22×D172D412yes—
30C22×C22⋊D43—24no—
31C2×D4×D43—28no—
32C2×D4×D43—28no—
33C2×D4×D43—416no—
34C22×C4×D43—22no—
35C2×#10621884—24no—
36#13063296—28no—
37#13063296—28no—
38#13063293—28no—
39#13063293—28no—
40C4×D172—22no—
41C4×D172—22no—
42C4×D172D414yes—
43C4×D172C2314yes—
44C2×C4⋊Dic43D412yes—
45C2×C22⋊Dic43—22no—
46C2×(C2×D43)⋊C4—22no—
47#13063258—24no—
48C4⋊(C43⋊C8)C2314yescomplemented
49#13063249—24no—
50#13063245—24no—
51C22⋊(C43⋊C8)—22no—
52#13063236—22no—
53C22×D4×C43—22no—
54C22×D4×C43D412yes—
55C2×M16×C43—21no—
56C4⋊Q8×C43C2312yescomplemented
57#127×C43—22no—
58#127×C43D412yes—
59#127×C43C2312yes—
60#125×C43—21no—
61C22⋊Q8×C43—22no—
62C4×Q8×C43—22no—
63#117×C43—21no—
64C2×C42×C43C2311yescharacteristic
65#102×C43—28no—
66#97×C43—21no—
67C4×C8×C43C2312yescomplemented
68C24×C43—22no—
69C23×D43—22no—
70C23×D43—28no—
71C23×D43—416no—
72C23×D43—84no—
73C2×Q8×C43—22no—
74C2×D4×C43—24no—
75C2×D4×C43—22no—
76C2×D4×C43—24no—
77C2×D4×C43—48no—
78C2×D4×C43—44no—
79C2×D4×C43C2×D414yes—
80C22×C4×C43C2×D412yes—
81C22×C4×C43C2×D411yescharacteristic
82C2×C22⋊D43—216no—
83C2×C22⋊D43—416no—
84C22×Dic43—22no—
85D4×D43—432no—
86D4×D43—416no—
87D4×D43—416no—
88C2×D172—22no—
89C2×D172—28no—
90C2×D172C2×D414yes—
91C2×C4×D43—24no—
92C2×C4×D43—24no—
93C2×C4×D43—48no—
94M16×C43—42no—
95C2×C8×C43—22no—
96C4⋊C4×C43—22no—
97C4⋊C4×C43—22no—
98C4⋊C4×C43—42no—
99C22⋊C4×C43—41no—
100C42×C43—41no—
101C42×C43C2×D412yes—
102C42×C43C2×D411yescharacteristic
103C42×C43C2411yescharacteristic
104C22⋊Dic43—28no—
105#10621884—48no—
106(C2×D43)⋊C4—24no—
107(C2×D43)⋊C4—24no—
108C4⋊Dic43—22no—
109C4⋊Dic43C2×D414yes—
110C2×C43⋊C8—24no—
111C23×C43—24no—
112C23×C43—48no—
113C23×C43—44no—
114C23×C43#12011yescharacteristic
115C22×D43—28no—
116C22×D43—24no—
117C22×D43—432no—
118C22×D43—44no—
119C22×D43—44no—
120C22×D43—88no—
121C22×D43—832no—
122Q8×C43—44no—
123D4×C43—44no—
124D4×C43—48no—
125D4×C43—48no—
126D4×C43—82no—
127C2×C4×C43—22no—
128C2×C4×C43—24no—
129C2×C4×C43—42no—
130C2×C4×C43#12012yes—
131C2×C4×C43#12011yescharacteristic
132C2×C4×C43C22×D412yes—
133C2×C4×C43C22×D411yescharacteristic
134C22⋊D43—432no—
135C2×Dic43—24no—
136C2×Dic43—44no—
137D172—28no—
138D172—24no—
139D172AGL(1,ℤ/8)18yes—
140C4×D43—48no—
141C4×D43—48no—
142C8×C43—42no—
143C43⋊C8—44no—
144#18529—431nomaximal, Sylow
145C22×C43—22no—
146C22×C43—48no—
147C22×C43—44no—
148C22×C43—88no—
149C22×C43#45512yes—
150C22×C43C2×#12011yescharacteristic
151C2×D43—28no—
152C2×D43—416no—
153C2×D43—44no—
154C2×D43—44no—
155C2×D43—832no—
156C4×C43—22no—
157C4×C43—21no—
158C4×C43—21no—
159C4×C43—82no—
160C4×C43C2×AGL(1,ℤ/8)14yes—
161Dic43—44no—
162C2×D42—431no—
163#3138D4311yescharacteristic, complemented
164#3110—432no—
165#1639—438no—
166#1555—431no—
167#1490—432no—
168C2×C43—22no—
169C2×C43—84no—
170C2×C43#300312yes—
171C2×C43C2×#45511yescharacteristic
172D43—48no—
173C23×D4—862no—
174D42—434no—
175D42—864no—
176D42—862no—
177#533C2×D4311yescharacteristic, complemented
178C2×#127C2×D4311yescharacteristic
179C2×#121—862no—
180C2×#120—862no—
181C2×C4×D4—432no—
182#490—22no—
183#486C2×D4314yescomplemented
184#484—862no—
185#474—434no—
186#461—434no—
187#450—862no—
188#440—864no—
189#432—432no—
190#418—22no—
191C4×M16C2×D4311yescharacteristic, complemented
192#331—434no—
193D4⋊C8—438no—
194C43#1852911yescharacteristic, complemented, Sylow
195C25—862no—
196C22×D4—22no—
197C22×D4—432no—
198C22×D4—868no—
199C22×D4—868no—
200C22×D4—864no—
201C22×D4—17216no—
202C22×D4C22⋊D4312yes—
203C23×C4—862no—
204C2×SD16—864no—
205C2×M16—21no—
206C4⋊Q8C22×D4312yescomplemented
207#127—22no—
208#127C22⋊D4312yes—
209#127C22×D4312yes—
210#125—21no—
211C22⋊Q8—22no—
212#121—868no—
213#121—868no—
214#120—868no—
215#120—868no—
216C4×Q8—22no—
217C4×D4—434no—
218C4×D4—434no—
219C4×D4—862no—
220C4×D4—862no—
221#117—21no—
222C2×C4⋊C4—432no—
223C2×C22⋊C4—862no—
224C2×C22⋊C4—862no—
225C2×C42C22×D4311yescharacteristic
226#107—864no—
227C4⋊C8—434no—
228#103—864no—
229#102—28no—
230#102—864no—
231C22⋊C8—862no—
232#97—21no—
233#97—862no—
234C4×C8C22×D4312yescomplemented
235C24—22no—
236C24—868no—
237C24—862no—
238C24—3444no—
239C24—17216no—
240C2×Q8—22no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—48no—
245C2×D4—44no—
246C2×D4—434no—
247C2×D4—868no—
248C2×D4—8616no—
249C2×D4—862no—
250C2×D4—17216no—
251C2×D4—17232no—
252C2×D4—17216no—
253C2×D4—17216no—
254C2×D4C2×C22⋊D4314yes—
255C22×C4—862no—
256C22×C4—864no—
257C22×C4—864no—
258C22×C4—1728no—
259C22×C4D4×D4312yes—
260C22×C4C2×C22⋊D4311yescharacteristic
261SD16—1728no—
262M16—42no—
263C2×C8—22no—
264C2×C8—864no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—434no—
269C4⋊C4—862no—
270C22⋊C4—41no—
271C22⋊C4—868no—
272C22⋊C4—864no—
273C22⋊C4—864no—
274C42—41no—
275C42D4×D4312yes—
276C42C2×C22⋊D4311yescharacteristic
277C42C23×D4311yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—868no—
282C23—864no—
283C23—3448no—
284C23—34432no—
285C23—1724no—
286C23—17232no—
287C23—1724no—
288C23#1306329311yescharacteristic
289Q8—44no—
290D4—44no—
291D4—48no—
292D4—48no—
293D4—82no—
294D4—438no—
295D4—868no—
296D4—864no—
297D4—17232no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—864no—
302C2×C4—1728no—
303C2×C4—1724no—
304C2×C4—1728no—
305C2×C4#1306329311yescharacteristic
306C2×C4#1306329312yes—
307C2×C4C2×D4×D4312yes—
308C2×C4C22×C22⋊D4311yescharacteristic
309C8—42no—
310C8—1724no—
311C22—22no—
312C22—48no—
313C22—44no—
314C22—88no—
315C22—868no—
316C22—34432no—
317C22—17216no—
318C22—1724no—
319C22—1724no—
320C22#1458307612yes—
321C22C2×#1306329311yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—82no—
326C4—1724no—
327C4AGL(1,ℤ/8)×D4314yes—
328C2—22no—
329C2—84no—
330C2—1728no—
331C2#1914382012yes—
332C2C2×#1458307611yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3138C2
socleC22×C43C2×#120
prime radicalC2×#14583534C2
hypercenter#3138D43
hypocenterC43#18529
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
2. #19144690C2semidirect
3. #19133771C2semidirect
4. #19133309C2semidirect
5. #19133039C2semidirect
6. C43×#3138C2semidirect
15. #14583176C22semidirect
16. #14583101C22semidirect
19. #14582956C22semidirect
20. #14582643C22semidirect
21. #14582628C22semidirect
22. C43×#533C22semidirect
25. C43×#486C22semidirect
27. C4×M16×C43C22semidirect
48. C4⋊(C43⋊C8)C23semidirect
56. C4⋊Q8×C43C23semidirect
67. C4×C8×C43C23semidirect
163. #3138D43semidirect
177. #533C2×D43semidirect
183. #486C2×D43semidirect
191. C4×M16C2×D43semidirect
194. C43#18529semidirect
206. C4⋊Q8C22×D43semidirect
234. 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
#191446902
#191337712
#191333092
#191330392
C43×#31382
#1852943