Finite Group Databasebeta

← #25077850

Subgroups — #25077850

Order 11008.

1066 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 155 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×#14583641C211yesmaximal, characteristic
2#19145594C212yesmaximal, complemented
3#19133900C218yesmaximal, complemented
4#19133219C211yesmaximal, characteristic, complemented
5#19133192C212yesmaximal, complemented
6C43×#3140C211yesmaximal, characteristic, complemented
7#14583641—24no—
8#14583641—22no—
9#14583641C2214yes—
10C2×#13063314—21no—
11C2×#13063313C2212yes—
12C2×#13063311—22no—
13C2×#13063295—22no—
14C2×#13063293—21no—
15#14583291C2214yescomplemented
16#14583261C2214yescomplemented
17#14583121—24no—
18#14583054—22no—
19#14583049—22no—
20#14582956C2212yescomplemented
21#14582639C2214yescomplemented
22#14582638C2218yescomplemented
23C43×#533C2211yescharacteristic, complemented
24C2×#124×C43C2211yescharacteristic
25C43×#489C2214yescomplemented
26C43×#488—22no—
27C43×#419—21no—
28C43×#418—21no—
29C4×M16×C43C2211yescharacteristic, complemented
30C24×D43—21no—
31C22×Dic86D412yes—
32C22×D172—21no—
33C22×D172D412yes—
34C22×C22⋊D43—22no—
35C2×#10621885—24no—
36C2×#10621884—24no—
37#13063314—24no—
38#13063314—24no—
39#13063313—22no—
40#13063313—22no—
41#13063313D414yes—
42#13063313C2314yes—
43#13063311—28no—
44#13063311—28no—
45#13063295—28no—
46#13063295—28no—
47#13063293—24no—
48#13063293—24no—
49C2×C22⋊Dic43—21no—
50C2×(C2×D43)⋊C4—22no—
51C2×(C2×D43)⋊C4—22no—
52C2×(C2×D43)⋊C4—22no—
53C2×C4×Dic43—21no—
54C4⋊(C43⋊C8)—22no—
55#13063248—24no—
56#13063245—24no—
57C22⋊(C43⋊C8)—22no—
58#13063236C2314yescomplemented
59C22×Q8×C43D411yescharacteristic
60C22×D4×C43D411yescharacteristic
61C2×M16×C43—21no—
62C4⋊Q8×C43C2312yescomplemented
63#125×C43—21no—
64#124×C43—22no—
65#124×C43D412yes—
66#124×C43C2312yes—
67C22⋊Q8×C43—22no—
68C4×Q8×C43—22no—
69#117×C43—21no—
70C2×C22⋊C4×C43—22no—
71C2×C42×C43C2311yescharacteristic
72#103×C43—24no—
73#102×C43—24no—
74#97×C43C2312yescomplemented
75C4×C8×C43—21no—
76C24×C43—21no—
77C23×D43—22no—
78C23×D43—24no—
79C23×D43—84no—
80C23×D43—82no—
81C2×Q8×C43—21no—
82C2×Q8×C43—22no—
83C2×Q8×C43—42no—
84C2×Q8×C43C2×D412yes—
85C2×D4×C43—21no—
86C2×D4×C43—42no—
87C2×D4×C43C2×D412yes—
88C22×C4×C43—21no—
89C22×C4×C43C2×D412yes—
90C22×C4×C43C2×D411yescharacteristic
91C2×C22⋊D43—28no—
92C2×C22⋊D43—84no—
93C22×Dic43—22no—
94C2×D172—24no—
95C2×D172—22no—
96C2×D172—82no—
97C2×D172—82no—
98C2×D172C2×D414yes—
99C2×Dic86—22no—
100C2×Dic86—82no—
101C2×Dic86C2×D414yes—
102M16×C43—42no—
103C2×C8×C43—22no—
104C4⋊C4×C43—22no—
105C4⋊C4×C43—22no—
106C4⋊C4×C43—42no—
107C22⋊C4×C43—24no—
108C22⋊C4×C43—24no—
109C22⋊C4×C43—41no—
110C42×C43—41no—
111C42×C43C2×D412yes—
112C42×C43C2×D411yescharacteristic
113C42×C43C2411yescharacteristic
114C22⋊Dic43—24no—
115#10621885—84no—
116#10621884—84no—
117(C2×D43)⋊C4—28no—
118(C2×D43)⋊C4—24no—
119(C2×D43)⋊C4—24no—
120(C2×D43)⋊C4—24no—
121(C2×D43)⋊C4—24no—
122C4×Dic43—24no—
123C2×C43⋊C8—24no—
124C23×C43—22no—
125C23×C43—44no—
126C23×C43—42no—
127C23×C43#12011yescharacteristic
128C22×D43—24no—
129C22×D43—44no—
130C22×D43—44no—
131C22×D43—44no—
132C22×D43—84no—
133C22×D43—88no—
134C22×D43—816no—
135Q8×C43—42no—
136Q8×C43—44no—
137Q8×C43—81no—
138D4×C43—42no—
139D4×C43—81no—
140C2×C4×C43—22no—
141C2×C4×C43—22no—
142C2×C4×C43—42no—
143C2×C4×C43—42no—
144C2×C4×C43—42no—
145C2×C4×C43#12011yescharacteristic
146C2×C4×C43#12012yes—
147C2×C4×C43C22×D412yes—
148C2×C4×C43C22×D411yescharacteristic
149C22⋊D43—88no—
150C2×Dic43—24no—
151C2×Dic43—44no—
152D172—44no—
153D172—82no—
154D172—84no—
155Dic86—44no—
156Dic86—82no—
157C8×C43—42no—
158C43⋊C8—44no—
159#18997—431nomaximal, Sylow
160C22×C43—22no—
161C22×C43—42no—
162C22×C43—44no—
163C22×C43—84no—
164C22×C43#45512yes—
165C22×C43C2×#12011yescharacteristic
166C2×D43—44no—
167C2×D43—44no—
168C2×D43—44no—
169C2×D43—88no—
170C2×D43—816no—
171C4×C43—21no—
172C4×C43—21no—
173C4×C43—22no—
174C4×C43—22no—
175C4×C43—42no—
176C4×C43—82no—
177Dic43—44no—
178C2×H(3,𝔽4)—431no—
179#3215—432no—
180#3140D4311yescharacteristic, complemented
181#1659—438no—
182#1533—431no—
183#1527—432no—
184C2×C43—22no—
185C2×C43—82no—
186C2×C43C2×#45511yescharacteristic
187C2×C43#301912yes—
188D43—84no—
189H(3,𝔽4)—434no—
190H(3,𝔽4)—864no—
191H(3,𝔽4)—862no—
192#533C2×D4311yescharacteristic, complemented
193C2×#124—432no—
194C2×#124—862no—
195C2×#124—861no—
196C2×#124C2×D4311yescharacteristic
197C2×#120—862no—
198C2×#120—861no—
199#497—434no—
200#489—434no—
201#489C2×D4314yescomplemented
202#488—22no—
203#464—864no—
204#451—862no—
205#450—862no—
206#432—432no—
207#419—21no—
208#418—21no—
209C4×M16C2×D4311yescharacteristic, complemented
210#329—434no—
211#329—438no—
212C43#1899711yescharacteristic, complemented, Sylow
213C25—861no—
214C22×Q8—432no—
215C22×Q8C22⋊D4311yescharacteristic
216C22×D4—432no—
217C22×D4—861no—
218C22×D4—862no—
219C22×D4C22⋊D4311yescharacteristic
220C2×Q16—864no—
221C2×SD16—864no—
222C2×M16—21no—
223C4⋊Q8C22×D4312yescomplemented
224#125—21no—
225#124—22no—
226#124—434no—
227#124—434no—
228#124—868no—
229#124—868no—
230#124—864no—
231#124—864no—
232#124—862no—
233#124—862no—
234#124C22⋊D4312yes—
235#124C22×D4312yes—
236C22⋊Q8—22no—
237#120—868no—
238#120—868no—
239#120—864no—
240#120—864no—
241C4×Q8—22no—
242#117—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—862no—
245C2×C22⋊C4—861no—
246C2×C22⋊C4—862no—
247C2×C22⋊C4—862no—
248C2×C42—861no—
249C2×C42C22×D4311yescharacteristic
250C4⋊C8—862no—
251#103—24no—
252#103—864no—
253#102—24no—
254#102—864no—
255C22⋊C8—862no—
256#97—434no—
257#97C22×D4312yescomplemented
258C4×C8—21no—
259C24—21no—
260C24—864no—
261C24—862no—
262C24—3444no—
263C24—3442no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—42no—
267C2×Q8—434no—
268C2×Q8—862no—
269C2×Q8—3442no—
270C2×Q8C2×C22⋊D4312yes—
271C2×D4—21no—
272C2×D4—42no—
273C2×D4—434no—
274C2×D4—864no—
275C2×D4—868no—
276C2×D4—862no—
277C2×D4—3442no—
278C2×D4—3442no—
279C2×D4—3444no—
280C2×D4C2×C22⋊D4312yes—
281C22×C4—21no—
282C22×C4—862no—
283C22×C4D4×D4312yes—
284C22×C4C2×C22⋊D4311yescharacteristic
285Q16—3444no—
286SD16—3444no—
287M16—42no—
288C2×C8—22no—
289C2×C8—864no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—41no—
296C22⋊C4—868no—
297C22⋊C4—864no—
298C22⋊C4—864no—
299C22⋊C4—864no—
300C22⋊C4—864no—
301C22⋊C4—864no—
302C42—41no—
303C42—864no—
304C42D4×D4312yes—
305C42C2×C22⋊D4311yescharacteristic
306C42C23×D4311yescharacteristic
307C23—22no—
308C23—44no—
309C23—42no—
310C23—864no—
311C23—3448no—
312C23—3444no—
313C23—34416no—
314C23—1724no—
315C23—1724no—
316C23—1724no—
317C23#1306329311yescharacteristic
318Q8—42no—
319Q8—44no—
320Q8—81no—
321Q8—3442no—
322Q8—1724no—
323D4—42no—
324D4—81no—
325D4—3442no—
326D4—3444no—
327D4—3448no—
328D4—1724no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—864no—
335C2×C4—1724no—
336C2×C4#1306329311yescharacteristic
337C2×C4#1306329312yes—
338C2×C4C2×D4×D4312yes—
339C2×C4C22×C22⋊D4311yescharacteristic
340C8—42no—
341C8—1724no—
342C22—22no—
343C22—44no—
344C22—42no—
345C22—84no—
346C22—3448no—
347C22—34416no—
348C22—1724no—
349C22—1724no—
350C22—1724no—
351C22#1458307612yes—
352C22C2×#1306329311yescharacteristic
353C4—22no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—82no—
359C4—1724no—
360C2—22no—
361C2—82no—
362C2—3444no—
363C2C2×#1458307611yescharacteristic
364C2#1914395012yes—
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3140C2
socleC22×C43C2×#120
prime radicalC2×#14583641C2
hypercenter#3140D43
hypocenterC43#18997
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3140

Complements

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

NH ≅ G/Ntype
2. #19145594C2semidirect
3. #19133900C2semidirect
4. #19133219C2semidirect
5. #19133192C2semidirect
6. C43×#3140C2semidirect
15. #14583291C22semidirect
16. #14583261C22semidirect
20. #14582956C22semidirect
21. #14582639C22semidirect
22. #14582638C22semidirect
23. C43×#533C22semidirect
25. C43×#489C22semidirect
29. C4×M16×C43C22semidirect
58. #13063236C23semidirect
62. C4⋊Q8×C43C23semidirect
74. #97×C43C23semidirect
180. #3140D43semidirect
192. #533C2×D43semidirect
201. #489C2×D43semidirect
209. C4×M16C2×D43semidirect
212. C43#18997semidirect
223. C4⋊Q8C22×D43semidirect
257. #97C22×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145836412
#191455942
#191339002
#191332192
#191331922
C43×#31402
#1899743