Finite Group Databasebeta

← #25082005

Subgroups — #25082005

Order 11008.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145536C211yesmaximal, characteristic, complemented
2#19144385C211yesmaximal, characteristic, complemented
3#19144071C211yesmaximal, characteristic
4#19143704C211yesmaximal, characteristic, complemented
5#19134010C212yesmaximal, complemented
6#19134003C212yesmaximal, complemented
7#19133942C212yesmaximal, complemented
8#19133935C212yesmaximal, complemented
9#19133365C211yesmaximal, characteristic, complemented
10#19133283C211yesmaximal, characteristic, complemented
11C2×C43×#331C211yesmaximal, characteristic, complemented
12C2×#13063351—21no—
13#14583456C2211yescharacteristic
14C4×#10621898C2211yescharacteristic
15#14583271C2212yes—
16#14583245C2212yes—
17#14583207—21no—
18#14583200—21no—
19#14583195C2211yescharacteristic, complemented
20#14583190C2211yescharacteristic, complemented
21#14583185C2211yescharacteristic, complemented
22#14583180C2211yescharacteristic, complemented
23#14583096—21no—
24#14583096—21no—
25#14583096—21no—
26#14583096C2212yes—
27#14583091—21no—
28#14583091C2212yes—
29#14583050—22no—
30#14583023C2212yescomplemented
31#14582964C2212yescomplemented
32C8×C22⋊D43—22no—
33#14582952C2211yescharacteristic
34#14582951—21no—
35#14582937—21no—
36#14582923C2211yescharacteristic
37C4⋊C8×D43—21no—
38#14582904—21no—
39#14582896—21no—
40#14582649C2212yescomplemented
41#14582649C2212yescomplemented
42#14582647C2212yescomplemented
43#14582646C2212yescomplemented
44C2×#127×C43C2211yescharacteristic
45C2×C4⋊C8×C43C2211yescharacteristic
46C2×C4⋊C8×C43C2211yescharacteristic
47C43×#331C2212yes—
48C43×#331C2212yes—
49C2×#10621898D411yescharacteristic
50#13063351—44no—
51C2×#10621883—22no—
52C2×#10621883—22no—
53C2×#10621883—22no—
54C2×#10621882—22no—
55C2×#10621877—21no—
56C2×C8×D43—22no—
57#13063325C2311yescharacteristic
58#13063325C2311yescharacteristic
59#13063317—21no—
60#13063304—21no—
61#13063304—21no—
62C4×Dic86—21no—
63C4×Dic86—21no—
64C4×Dic86D411yescharacteristic
65C4×Dic86C2311yescharacteristic
66C4×D172D411yescharacteristic
67C4×D172C2311yescharacteristic
68C4×C22⋊D43—21no—
69C4×C22⋊D43—21no—
70#13063284—21no—
71#13063284D411yescharacteristic
72#13063282—21no—
73C42×D43—21no—
74#13063259—21no—
75#13063259—21no—
76#13063257—21no—
77#13063257—21no—
78C4⋊(C43⋊C8)—21no—
79C4⋊(C43⋊C8)D411yescharacteristic
80C4⋊(C43⋊C8)D411yescharacteristic
81C4⋊(C43⋊C8)C2312yes—
82Dic43⋊C8—22no—
83#13063250—21no—
84#13063250—21no—
85#13063247—21no—
86#13063247—21no—
87#13063246—22no—
88#13063246—22no—
89#13063246—22no—
90#13063246—22no—
91C22⋊(C43⋊C8)D412yes—
92(C2×D43)⋊C8—22no—
93C4×C43⋊C8C2312yes—
94C8×Dic43—22no—
95C22×D4×C43—21no—
96C22×D4×C43—21no—
97C22×D4×C43—21no—
98C22×C8×C43—21no—
99C22×C8×C43—21no—
100#127×C43—21no—
101#127×C43—21no—
102#127×C43—21no—
103#127×C43C2312yes—
104C2×C42×C43C2311yescharacteristic
105C4⋊C8×C43—21no—
106C4⋊C8×C43—21no—
107C4⋊C8×C43C2312yes—
108C4⋊C8×C43C2311yescharacteristic
109C4⋊C8×C43C2311yescharacteristic
110C24×C43—41no—
111C2×D4×C43—22no—
112C2×D4×C43—22no—
113C2×D4×C43—22no—
114C2×D4×C43—41no—
115C2×D4×C43—41no—
116C2×D4×C43—41no—
117C2×D4×C43—42no—
118C2×D4×C43—42no—
119C2×D4×C43—81no—
120C2×D4×C43—81no—
121C2×D4×C43—82no—
122C22×C4×C43C2×D411yescharacteristic
123C22×C4×C43C2×D411yescharacteristic
124C22×C4×C43#4111yescharacteristic
125C2×C22⋊D43—21no—
126#10621898—22no—
127#10621898SD1614yescomplemented
128C2×D172C2×D411yescharacteristic
129C2×C4×D43—21no—
130C2×C4×D43—21no—
131C2×C4×D43—21no—
132C2×Dic86—21no—
133C2×Dic86—21no—
134C2×Dic86C2×D411yescharacteristic
135C2×C8×C43—22no—
136C2×C8×C43—21no—
137C2×C8×C43—21no—
138C2×C8×C43—42no—
139C2×C8×C43—81no—
140C42×C43C2×D411yescharacteristic
141C42×C43C2×D411yescharacteristic
142C42×C43#4111yescharacteristic
143C42×C43C2411yescharacteristic
144C22⋊Dic43—21no—
145C22⋊Dic43—41no—
146#10621883—44no—
147#10621883—82no—
148#10621883—82no—
149#10621882—44no—
150(C2×D43)⋊C4—21no—
151(C2×D43)⋊C4—21no—
152C4⋊Dic43—21no—
153C4⋊Dic43—21no—
154C4⋊Dic43—21no—
155C4⋊Dic43—21no—
156C4⋊Dic43C2×D411yescharacteristic
157C4⋊Dic43C2×D411yescharacteristic
158Dic43⋊C4—21no—
159Dic43⋊C4—21no—
160Dic43⋊C4—41no—
161Dic43⋊C4—41no—
162C4×Dic43—21no—
163C4×Dic43—21no—
164C4×Dic43—21no—
165C4×Dic43—41no—
166#10621877—42no—
167C2×C43⋊C8—22no—
168C2×C43⋊C8C2×D412yes—
169C2×C43⋊C8C2×D412yes—
170C8×D43—82no—
171C23×C43—42no—
172C23×C43—82no—
173C23×C43—82no—
174C23×C43—82no—
175C23×C43#12111yescharacteristic
176C22×D43—21no—
177D4×C43—42no—
178D4×C43—42no—
179D4×C43—42no—
180D4×C43—42no—
181D4×C43—82no—
182D4×C43—82no—
183D4×C43—82no—
184D4×C43—82no—
185C2×C4×C43—21no—
186C2×C4×C43—21no—
187C2×C4×C43—41no—
188C2×C4×C43#12111yescharacteristic
189C2×C4×C43#12111yescharacteristic
190C2×C4×C43#12111yescharacteristic
191C2×C4×C43C2×SD1612yes—
192C2×C4×C43#13512yes—
193C2×C4×C43C22×D411yescharacteristic
194C2×C4×C43C22×D411yescharacteristic
195C2×C4×C43C2×#4111yescharacteristic
196C22⋊D43—24no—
197C2×Dic43—21no—
198C2×Dic43—21no—
199C2×Dic43—21no—
200C2×Dic43—41no—
201C2×Dic43—41no—
202D172—21no—
203D172C2×SD1612yes—
204C4×D43—24no—
205C4×D43—41no—
206C4×D43—41no—
207C4×D43—81no—
208Dic86—21no—
209Dic86—41no—
210Dic86—41no—
211Dic86—81no—
212Dic86C2×SD1612yes—
213C8×C43—41no—
214C8×C43—41no—
215C8×C43—42no—
216C43⋊C8—22no—
217C43⋊C8—22no—
218C43⋊C8—42no—
219#19409—431nomaximal, Sylow
220C22×C43—21no—
221C22×C43—21no—
222C22×C43—82no—
223C22×C43—82no—
224C22×C43—82no—
225C22×C43—164no—
226C22×C43#46512yes—
227C22×C43C2×#12111yescharacteristic
228C2×D43—22no—
229C2×D43—41no—
230C2×D43—41no—
231C4×C43—21no—
232C4×C43—21no—
233C4×C43—21no—
234C4×C43#46512yes—
235C4×C43C22×SD1611yescharacteristic
236C4×C43C2×#13511yescharacteristic
237C4×C43#57611yescharacteristic
238C4×C43#57711yescharacteristic
239Dic43—22no—
240Dic43—41no—
241Dic43—41no—
242Dic43—81no—
243Dic43—81no—
244#3208—431no—
245#3076—431no—
246#3038—431no—
247#2988—431no—
248#1678—432no—
249#1677—432no—
250#1666—432no—
251#1665—432no—
252#1570—431no—
253#1550—431no—
254C2×#331D4311yescharacteristic, complemented
255C2×C43—22no—
256C2×C43—162no—
257C2×C43#304511yescharacteristic
258C2×C43C2×#46511yescharacteristic
259C2×C43#306811yescharacteristic
260D43—42no—
261C2×AGL(1,ℤ/8)—861no—
262#533—431no—
263C2×#127C2×D4311yescharacteristic
264C4×#41—431no—
265#492—432no—
266#486—432no—
267#479—861no—
268#478—861no—
269#477—431no—
270#476—431no—
271#475—431no—
272#474—431no—
273#460—432no—
274#460—861no—
275#460—861no—
276#460—861no—
277#459—432no—
278#459—861no—
279#450—862no—
280C8×Q8—432no—
281C8×D4—432no—
282C8×D4—862no—
283#431—431no—
284#431—861no—
285#427—861no—
286#423—431no—
287C2×C4⋊C8—861no—
288C2×C4⋊C8C2×D4311yescharacteristic
289C2×C4⋊C8C2×D4311yescharacteristic
290#418—861no—
291#416—861no—
292#334—432no—
293#334—432no—
294#333—432no—
295#332—432no—
296#331C2×D4312yes—
297#331C2×D4312yes—
298C43#1940911yescharacteristic, complemented, Sylow
299C2×#41—431no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303AGL(1,ℤ/8)—1724no—
304C2×SD16—862no—
305C2×SD16—862no—
306C2×SD16—862no—
307C2×D8—862no—
308C2×M16—861no—
309C22×C8—21no—
310C22×C8—21no—
311C22×C8—862no—
312C4⋊Q8—431no—
313C4⋊Q8—431no—
314#127—21no—
315#127—21no—
316#127—21no—
317#127C22×D4312yes—
318#125—861no—
319C22⋊Q8—861no—
320C22⋊Q8—861no—
321C4×Q8—431no—
322C4×Q8—431no—
323C4×Q8—861no—
324C4×Q8—861no—
325C4×D4—431no—
326C4×D4—431no—
327C4×D4—861no—
328C4×D4—861no—
329#117—431no—
330#117—861no—
331#117—861no—
332C2×C42—861no—
333C2×C42C22×D4311yescharacteristic
334#107—861no—
335#107—861no—
336#106—861no—
337#106—861no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—431no—
341C4⋊C8—431no—
342C4⋊C8—432no—
343C4⋊C8—862no—
344C4⋊C8—861no—
345C4⋊C8C22×D4312yes—
346C4⋊C8C22×D4311yescharacteristic
347C4⋊C8C22×D4311yescharacteristic
348#103—861no—
349#103—861no—
350#102—862no—
351#102—862no—
352#102—861no—
353#102—861no—
354#102—862no—
355#102—862no—
356C22⋊C8—432no—
357C22⋊C8—862no—
358C4×C8—432no—
359C4×C8—862no—
360C24—41no—
361#41—434no—
362#41—862no—
363C2×Q8—431no—
364C2×Q8—861no—
365C2×Q8—861no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4—82no—
377C2×D4—431no—
378C2×D4—861no—
379C22×C4—861no—
380C22×C4—861no—
381C22×C4—861no—
382C22×C4#1062189811yescharacteristic
383C22×C4D4×D4311yescharacteristic
384C22×C4D4×D4311yescharacteristic
385SD16—3442no—
386SD16—3442no—
387SD16—1724no—
388D8—1724no—
389M16—1722no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—42no—
394C2×C8—81no—
395C2×C8—432no—
396C2×C8—432no—
397C2×C8—862no—
398C2×C8—3442no—
399C4⋊C4—431no—
400C4⋊C4—431no—
401C4⋊C4—861no—
402C4⋊C4—861no—
403C4⋊C4—861no—
404C4⋊C4—861no—
405C4⋊C4—861no—
406C4⋊C4—861no—
407C4⋊C4—1721no—
408C4⋊C4—1721no—
409C22⋊C4—861no—
410C22⋊C4—861no—
411C22⋊C4—861no—
412C22⋊C4—1721no—
413C42—861no—
414C42—861no—
415C42—861no—
416C42—1721no—
417C42#1062189811yescharacteristic
418C42D4×D4311yescharacteristic
419C42D4×D4311yescharacteristic
420C42C23×D4311yescharacteristic
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—861no—
426C23#1306329811yescharacteristic
427Q8—432no—
428Q8—861no—
429Q8—3441no—
430Q8—1721no—
431Q8—1721no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—432no—
441D4—864no—
442D4—861no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—41no—
446C2×C4—861no—
447C2×C4—864no—
448C2×C4—861no—
449C2×C4—861no—
450C2×C4—3441no—
451C2×C4—1721no—
452C2×C4—1721no—
453C2×C4—1721no—
454C2×C4—1721no—
455C2×C4#1306329811yescharacteristic
456C2×C4#1306329811yescharacteristic
457C2×C4#1306329811yescharacteristic
458C2×C4SD16×D4312yes—
459C2×C4#1306334412yes—
460C2×C4C2×D4×D4311yescharacteristic
461C2×C4C2×D4×D4311yescharacteristic
462C2×C4C2×#1062189811yescharacteristic
463C8—41no—
464C8—41no—
465C8—42no—
466C8—862no—
467C8—862no—
468C8—1722no—
469C22—21no—
470C22—21no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—164no—
475C22—862no—
476C22—1721no—
477C22—1721no—
478C22#1458312712yes—
479C22C2×#1306329811yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—862no—
484C4—3441no—
485C4—3441no—
486C4—1721no—
487C4—1721no—
488C4#1458312712yes—
489C4C2×SD16×D4311yescharacteristic
490C4C2×#1306334411yescharacteristic
491C4#1458370011yescharacteristic
492C4#1458370511yescharacteristic
493C2—22no—
494C2—162no—
495C2—1722no—
496C2#1914413911yescharacteristic
497C2C2×#1458312711yescharacteristic
498C2#1914432211yescharacteristic
subgroupG/H
centerC22C2×#13063298
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC2×C43×#331C2
socleC22×C43C2×#121
prime radical#19144071C2
hypercenterC2×#331D43
hypocenterC43#19409
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
1. #19145536C2semidirect
2. #19144385C2semidirect
4. #19143704C2semidirect
5. #19134010C2semidirect
6. #19134003C2semidirect
7. #19133942C2semidirect
8. #19133935C2semidirect
9. #19133365C2semidirect
10. #19133283C2semidirect
11. C2×C43×#331C2semidirect
19. #14583195C22semidirect
20. #14583190C22semidirect
21. #14583185C22semidirect
22. #14583180C22semidirect
30. #14583023C22semidirect
31. #14582964C22semidirect
40. #14582649C22semidirect
41. #14582649C22semidirect
42. #14582647C22semidirect
43. #14582646C22semidirect
127. #10621898SD16semidirect
254. C2×#331D43semidirect
298. C43#19409semidirect

Maximal subgroups

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

subgroupindex
#191455362
#191443852
#191440712
#191437042
#191340102
#191340032
#191339422
#191339352
#191333652
#191332832
C2×C43×#3312
#1940943