Finite Group Databasebeta

← #25070709

Subgroups — #25070709

Order 11008.

1402 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 167 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×#14583461C211yesmaximal, characteristic
2#19145397C212yesmaximal, complemented
3#19133580C214yesmaximal, complemented
4#19133546C214yesmaximal, complemented
5#19133110C212yesmaximal, complemented
6#19132922C211yesmaximal, characteristic, complemented
7C43×#2561C211yesmaximal, characteristic, complemented
8C23×C22⋊D43—21no—
9#14583461—24no—
10#14583461—22no—
11#14583461C2214yes—
12C2×#13063309—22no—
13C2×#13063299—21no—
14C2×#13063299C2212yes—
15C2×#13063295—22no—
16#14583232—22no—
17#14583212C2214yescomplemented
18#14583147—24no—
19#14583049—22no—
20#14583038C2214yescomplemented
21#14582863C2212yescomplemented
22#14582632C2218yescomplemented
23#14582623C2214yescomplemented
24C2×C22⋊Q8×C43C2211yescharacteristic, complemented
25C22×C22⋊C4×C43C2211yescharacteristic
26C43×#397C2212yescomplemented
27C43×#393C2212yescomplemented
28C43×#388—22no—
29C43×#381—21no—
30C43×#379—21no—
31C4×C22⋊C4×C43C2211yescharacteristic, complemented
32C24×D43—21no—
33C22×D172—21no—
34C22×D172D412yes—
35C22×C22⋊D43—22no—
36C22×C22⋊D43—24no—
37C22×C22⋊D43—22no—
38C22×C22⋊D43—24no—
39C22×C22⋊D43—48no—
40C23×Dic43—21no—
41C2×#10621884—24no—
42C2×#10621877—22no—
43#13063309—28no—
44#13063309—28no—
45#13063299—24no—
46#13063299—24no—
47#13063299—22no—
48#13063299—22no—
49#13063299D414yes—
50#13063299C2314yes—
51#13063295—28no—
52#13063295—28no—
53C2×C4⋊Dic43—21no—
54C2×C4⋊Dic43D412yes—
55C2×(C2×D43)⋊C4—22no—
56C2×(C2×D43)⋊C4—22no—
57#13063258—24no—
58#13063249—24no—
59#13063245—24no—
60C22⋊(C43⋊C8)—22no—
61C22⋊(C43⋊C8)C2314yescomplemented
62C25×C43D411yescharacteristic
63C22×Q8×C43C2311yescharacteristic, complemented
64C23×C4×C43D411yescharacteristic
65C23×C4×C43C2311yescharacteristic
66C22⋊Q8×C43—22no—
67C22⋊Q8×C43—22no—
68C2×C4⋊C4×C43—21no—
69C2×C4⋊C4×C43C2311yescharacteristic, complemented
70C2×C22⋊C4×C43—22no—
71C2×C22⋊C4×C43—22no—
72C2×C22⋊C4×C43—21no—
73C2×C22⋊C4×C43—21no—
74C2×C22⋊C4×C43—22no—
75C2×C22⋊C4×C43—22no—
76C2×C22⋊C4×C43D412yes—
77C2×C22⋊C4×C43C2312yes—
78C2×C42×C43—21no—
79#95×C43—22no—
80#95×C43—22no—
81#95×C43C2312yescomplemented
82C24×C43—21no—
83C24×C43—21no—
84C24×C43—24no—
85C24×C43—42no—
86C24×C43—81no—
87C24×C43C2×D411yescharacteristic
88C24×C43C2×D412yes—
89C23×D43—22no—
90C23×D43—24no—
91C23×D43—44no—
92C23×D43—82no—
93C23×D43—82no—
94C2×Q8×C43—22no—
95C2×Q8×C43—81no—
96C22×C4×C43—21no—
97C22×C4×C43—21no—
98C22×C4×C43—22no—
99C22×C4×C43—21no—
100C22×C4×C43—21no—
101C22×C4×C43—42no—
102C22×C4×C43—81no—
103C22×C4×C43C2×D412yes—
104C22×C4×C43C2×D412yes—
105C22×C4×C43C2×D412yes—
106C22×C4×C43C2×D411yescharacteristic
107C22×C4×C43C2411yescharacteristic
108C2×C22⋊D43—28no—
109C2×C22⋊D43—24no—
110C2×C22⋊D43—24no—
111C2×C22⋊D43—48no—
112C2×C22⋊D43—416no—
113C2×C22⋊D43—48no—
114C2×C22⋊D43—48no—
115C2×C22⋊D43—48no—
116C22×Dic43—24no—
117C22×Dic43—22no—
118C22×Dic43—44no—
119C2×D172—24no—
120C2×D172—22no—
121C2×D172—82no—
122C2×D172—82no—
123C2×D172C2×D414yes—
124C4⋊C4×C43—22no—
125C4⋊C4×C43—42no—
126C22⋊C4×C43—24no—
127C22⋊C4×C43—24no—
128C22⋊C4×C43—42no—
129C22⋊C4×C43—41no—
130C22⋊C4×C43—41no—
131C22⋊C4×C43—44no—
132C22⋊C4×C43—82no—
133C42×C43—81no—
134#10621884—84no—
135(C2×D43)⋊C4—28no—
136(C2×D43)⋊C4—24no—
137(C2×D43)⋊C4—24no—
138C4⋊Dic43—24no—
139C4⋊Dic43—22no—
140C4⋊Dic43C2×D414yes—
141#10621877—82no—
142C2×C43⋊C8—24no—
143C23×C43—24no—
144C23×C43—24no—
145C23×C43—22no—
146C23×C43—22no—
147C23×C43—22no—
148C23×C43—22no—
149C23×C43—41no—
150C23×C43—42no—
151C23×C43—42no—
152C23×C43—42no—
153C23×C43—42no—
154C23×C43—81no—
155C23×C43—81no—
156C23×C43—88no—
157C23×C43#12011yescharacteristic
158C23×C43#12011yescharacteristic
159C23×C43AGL(1,ℤ/8)14yes—
160C23×C43C22×D411yescharacteristic
161C22×D43—24no—
162C22×D43—44no—
163C22×D43—44no—
164C22×D43—48no—
165C22×D43—44no—
166C22×D43—84no—
167C22×D43—84no—
168C22×D43—816no—
169Q8×C43—44no—
170C2×C4×C43—22no—
171C2×C4×C43—42no—
172C2×C4×C43—41no—
173C2×C4×C43—42no—
174C2×C4×C43—41no—
175C2×C4×C43—42no—
176C2×C4×C43—42no—
177C2×C4×C43—42no—
178C2×C4×C43—42no—
179C2×C4×C43—42no—
180C2×C4×C43—41no—
181C2×C4×C43—41no—
182C2×C4×C43—42no—
183C2×C4×C43—42no—
184C2×C4×C43—81no—
185C2×C4×C43#12012yes—
186C2×C4×C43C22×D412yes—
187C22⋊D43—416no—
188C22⋊D43—48no—
189C22⋊D43—48no—
190C2×Dic43—24no—
191C2×Dic43—44no—
192C2×Dic43—48no—
193D172—44no—
194D172—82no—
195D172—84no—
196C43⋊C8—44no—
197#18317—431nomaximal, Sylow
198C22×C43—21no—
199C22×C43—21no—
200C22×C43—21no—
201C22×C43—21no—
202C22×C43—24no—
203C22×C43—21no—
204C22×C43—21no—
205C22×C43—21no—
206C22×C43—44no—
207C22×C43—44no—
208C22×C43—48no—
209C22×C43—42no—
210C22×C43—42no—
211C22×C43—82no—
212C22×C43—84no—
213C22×C43#45512yes—
214C22×C43C2×#12011yescharacteristic
215C22×C43C2×AGL(1,ℤ/8)12yes—
216C2×D43—44no—
217C2×D43—44no—
218C2×D43—44no—
219C2×D43—88no—
220C2×D43—816no—
221C4×C43—22no—
222C4×C43—22no—
223C4×C43—42no—
224C4×C43—82no—
225C4×C43—81no—
226C4×C43—81no—
227Dic43—44no—
228C2×#534—431no—
229#3193—432no—
230#2561D4311yescharacteristic, complemented
231#1607—434no—
232#1601—434no—
233#1507—432no—
234#1462—431no—
235C2×C43—22no—
236C2×C43—22no—
237C2×C43—44no—
238C2×C43—41no—
239C2×C43#300311yescharacteristic
240C2×C43C2×#45511yescharacteristic
241C2×C43#302611yescharacteristic
242D43—84no—
243C23×D4—861no—
244#534—434no—
245#534—864no—
246#534—862no—
247C2×#123—862no—
248C2×C22⋊Q8C2×D4311yescharacteristic, complemented
249C2×#121—432no—
250C2×#121—861no—
251C2×#120—862no—
252C22×C22⋊C4C2×D4311yescharacteristic
253#484—862no—
254#480—434no—
255#468—864no—
256#450—862no—
257#448—434no—
258#407—432no—
259#397C2×D4312yescomplemented
260#393C2×D4312yescomplemented
261#388—22no—
262#381—21no—
263#379—21no—
264C4×C22⋊C4C2×D4311yescharacteristic, complemented
265#327—438no—
266#323—434no—
267C43#1831711yescharacteristic, complemented, Sylow
268C25—861no—
269C25C22⋊D4311yescharacteristic
270C22×Q8C22×D4311yescharacteristic, complemented
271C22×D4—432no—
272C22×D4—861no—
273C22×D4—862no—
274C22×D4—864no—
275C22×D4—862no—
276C22×D4—864no—
277C22×D4—1728no—
278C23×C4—861no—
279C23×C4C22⋊D4311yescharacteristic
280C23×C4C22×D4311yescharacteristic
281C2×SD16—864no—
282C2×M16—862no—
283#123—868no—
284#123—868no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287#121—434no—
288#121—434no—
289#121—864no—
290#121—862no—
291#121—862no—
292#121—864no—
293#120—868no—
294#120—868no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—432no—
297C2×C4⋊C4—861no—
298C2×C4⋊C4C22×D4311yescharacteristic, complemented
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—862no—
306C2×C22⋊C4—862no—
307C2×C22⋊C4C22⋊D4312yes—
308C2×C22⋊C4C22×D4312yes—
309C2×C42—21no—
310#107—864no—
311#103—864no—
312#102—864no—
313C22⋊C8—434no—
314C22⋊C8—862no—
315#95—22no—
316#95—22no—
317#95C22×D4312yescomplemented
318C24—21no—
319C24—21no—
320C24—24no—
321C24—42no—
322C24—81no—
323C24—864no—
324C24—862no—
325C24—3442no—
326C24—3442no—
327C24—1724no—
328C24C2×C22⋊D4311yescharacteristic
329C24C2×C22⋊D4312yes—
330C2×Q8—22no—
331C2×Q8—81no—
332C2×D4—434no—
333C2×D4—864no—
334C2×D4—868no—
335C2×D4—864no—
336C2×D4—864no—
337C2×D4—862no—
338C2×D4—3442no—
339C2×D4—3442no—
340C2×D4—1728no—
341C2×D4—17216no—
342C2×D4—1728no—
343C2×D4—1728no—
344C2×D4—1728no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—42no—
351C22×C4—81no—
352C22×C4—864no—
353C22×C4—862no—
354C22×C4—1724no—
355C22×C4D4×D4312yes—
356C22×C4D4×D4312yes—
357C22×C4C2×C22⋊D4312yes—
358C22×C4C2×C22⋊D4311yescharacteristic
359C22×C4C23×D4311yescharacteristic
360SD16—3444no—
361M16—3442no—
362C2×C8—864no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—434no—
366C4⋊C4—864no—
367C4⋊C4—862no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—44no—
374C22⋊C4—82no—
375C22⋊C4—868no—
376C22⋊C4—864no—
377C22⋊C4—864no—
378C42—81no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—81no—
391C23—81no—
392C23—88no—
393C23—864no—
394C23—34416no—
395C23—3444no—
396C23—3444no—
397C23—1724no—
398C23—1724no—
399C23—1728no—
400C23—1724no—
401C23#1306329311yescharacteristic
402C23#1306329311yescharacteristic
403C23#1306335014yes—
404C23C22×C22⋊D4311yescharacteristic
405Q8—44no—
406D4—3442no—
407D4—3444no—
408D4—1724no—
409D4—17216no—
410D4—1728no—
411D4—1728no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—81no—
427C2×C4—864no—
428C2×C4—1724no—
429C2×C4—1728no—
430C2×C4#1306329312yes—
431C2×C4C2×D4×D4312yes—
432C8—1724no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—21no—
440C22—21no—
441C22—44no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—82no—
447C22—84no—
448C22—3448no—
449C22—34416no—
450C22—1724no—
451C22—1724no—
452C22—1724no—
453C22#1458307612yes—
454C22C2×#1306329311yescharacteristic
455C22C2×#1306335012yes—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4—1724no—
463C2—22no—
464C2—22no—
465C2—41no—
466C2—44no—
467C2—3444no—
468C2#1914382111yescharacteristic
469C2C2×#1458307611yescharacteristic
470C2#1914399111yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2561C2
socleC22×C43C2×#120
prime radicalC2×#14583461C2
hypercenter#2561D43
hypocenterC43#18317
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #19145397C2semidirect
3. #19133580C2semidirect
4. #19133546C2semidirect
5. #19133110C2semidirect
6. #19132922C2semidirect
7. C43×#2561C2semidirect
17. #14583212C22semidirect
20. #14583038C22semidirect
21. #14582863C22semidirect
22. #14582632C22semidirect
23. #14582623C22semidirect
24. C2×C22⋊Q8×C43C22semidirect
26. C43×#397C22semidirect
27. C43×#393C22semidirect
31. C4×C22⋊C4×C43C22semidirect
61. C22⋊(C43⋊C8)C23semidirect
63. C22×Q8×C43C23semidirect
69. C2×C4⋊C4×C43C23semidirect
81. #95×C43C23semidirect
230. #2561D43semidirect
248. C2×C22⋊Q8C2×D43semidirect
259. #397C2×D43semidirect
260. #393C2×D43semidirect
264. C4×C22⋊C4C2×D43semidirect
267. C43#18317semidirect
270. C22×Q8C22×D43semidirect
298. C2×C4⋊C4C22×D43semidirect
317. #95C22×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145834612
#191453972
#191335802
#191335462
#191331102
#191329222
C43×#25612
#1831743