Finite Group Databasebeta

← #25081998

Subgroups — #25081998

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#19145533C211yesmaximal, characteristic, complemented
2#19144075C211yesmaximal, characteristic
3#19143706C211yesmaximal, characteristic, complemented
4#19134013C212yesmaximal, complemented
5#19134006C212yesmaximal, complemented
6#19133945C212yesmaximal, complemented
7#19133938C212yesmaximal, complemented
8#19133365C211yesmaximal, characteristic, complemented
9#19133285C211yesmaximal, characteristic, complemented
10C2×#14582642C211yesmaximal, characteristic, complemented
11C43×#3076C211yesmaximal, characteristic, complemented
12C2×#13063350—21no—
13C2×#13063322C2211yescharacteristic
14#14583273C2212yes—
15#14583247C2212yes—
16#14583099—21no—
17#14583099—21no—
18#14583099—21no—
19#14583099C2212yes—
20#14583094—21no—
21#14583094C2212yes—
22#14583049—22no—
23Q8×C43⋊C8C2212yescomplemented
24D4×C43⋊C8—22no—
25D4×C43⋊C8C2212yescomplemented
26#14582952—21no—
27#14582952C2211yescharacteristic
28#14582923C2211yescharacteristic
29C2×C4⋊(C43⋊C8)—21no—
30C2×C4⋊(C43⋊C8)C2211yescharacteristic
31#14582903—21no—
32#14582649C2212yescomplemented
33#14582649C2212yescomplemented
34#14582648C2212yescomplemented
35#14582645C2212yescomplemented
36#14582642C2212yes—
37#14582642C2212yes—
38C43×#533C2211yescharacteristic
39C4×#41×C43C2211yescharacteristic
40C43×#479—21no—
41C43×#478—21no—
42C43×#477C2211yescharacteristic, complemented
43C43×#476C2211yescharacteristic, complemented
44C43×#475C2211yescharacteristic, complemented
45C43×#474C2211yescharacteristic, complemented
46C43×#427—21no—
47C2×C4⋊C8×C43C2211yescharacteristic
48C43×#416—21no—
49C22×D172—21no—
50C22×D172—21no—
51C22×D172—21no—
52#13063350—44no—
53C2×#10621884—22no—
54C2×#10621884—22no—
55C2×#10621884—22no—
56C2×#10621882—22no—
57C2×#10621877—21no—
58C22×C43⋊C8—21no—
59C22×C43⋊C8—22no—
60#13063322—21no—
61#13063322—21no—
62#13063322—21no—
63#13063322C2312yes—
64C4⋊(C43⋊C8)—21no—
65C4⋊(C43⋊C8)—21no—
66C4⋊(C43⋊C8)—22no—
67C4⋊(C43⋊C8)D411yescharacteristic
68C4⋊(C43⋊C8)D411yescharacteristic
69C4⋊(C43⋊C8)C2312yes—
70C4⋊(C43⋊C8)C2312yes—
71#13063245—22no—
72#13063245—22no—
73#13063245—22no—
74#13063245—22no—
75C22⋊(C43⋊C8)—22no—
76C22⋊(C43⋊C8)D412yes—
77C4×C43⋊C8—22no—
78C4×C43⋊C8C2312yes—
79C2×#41×C43D411yescharacteristic
80C22×C8×C43—21no—
81C4⋊Q8×C43C2311yescharacteristic
82C4⋊Q8×C43C2311yescharacteristic
83#125×C43—21no—
84C22⋊Q8×C43—21no—
85C22⋊Q8×C43—21no—
86C4×Q8×C43—21no—
87C4×Q8×C43—21no—
88C4×Q8×C43D411yescharacteristic
89C4×Q8×C43C2311yescharacteristic
90C4×D4×C43—21no—
91C4×D4×C43—21no—
92C4×D4×C43D411yescharacteristic
93C4×D4×C43C2311yescharacteristic
94#117×C43—21no—
95#117×C43—21no—
96#117×C43D411yescharacteristic
97C2×C42×C43—21no—
98C2×C42×C43C2311yescharacteristic
99#107×C43—21no—
100#107×C43—21no—
101#106×C43—21no—
102#106×C43—21no—
103C4⋊C8×C43—21no—
104C4⋊C8×C43C2311yescharacteristic
105C4⋊C8×C43C2311yescharacteristic
106#103×C43—21no—
107#103×C43—21no—
108#102×C43—21no—
109#102×C43—21no—
110C23×D43—41no—
111#41×C43—22no—
112#41×C43SD1614yescomplemented
113C2×Q8×C43—21no—
114C2×Q8×C43—21no—
115C2×Q8×C43C2×D411yescharacteristic
116C2×D4×C43—21no—
117C2×D4×C43C2×D411yescharacteristic
118C22×C4×C43—21no—
119C22×C4×C43—21no—
120C22×C4×C43—21no—
121C22×C4×C43C2×D411yescharacteristic
122C22×C4×C43C2×D411yescharacteristic
123C22×C4×C43#4111yescharacteristic
124C2×D172—22no—
125C2×D172—22no—
126C2×D172—22no—
127C2×D172—41no—
128C2×D172—41no—
129C2×D172—41no—
130C2×D172—42no—
131C2×D172—42no—
132C2×D172—81no—
133C2×D172—81no—
134C2×D172—82no—
135C2×C8×C43—21no—
136C2×C8×C43—21no—
137C2×C8×C43—42no—
138C4⋊C4×C43—21no—
139C4⋊C4×C43—21no—
140C4⋊C4×C43—21no—
141C4⋊C4×C43—21no—
142C4⋊C4×C43—21no—
143C4⋊C4×C43—21no—
144C4⋊C4×C43—41no—
145C4⋊C4×C43—41no—
146C4⋊C4×C43C2×D411yescharacteristic
147C4⋊C4×C43C2×D411yescharacteristic
148C22⋊C4×C43—21no—
149C22⋊C4×C43—21no—
150C22⋊C4×C43—21no—
151C22⋊C4×C43—41no—
152C42×C43—21no—
153C42×C43—21no—
154C42×C43—21no—
155C42×C43—41no—
156C42×C43C2×D411yescharacteristic
157C42×C43C2×D411yescharacteristic
158C42×C43#4111yescharacteristic
159C42×C43C2411yescharacteristic
160#10621884—44no—
161#10621884—82no—
162#10621884—82no—
163#10621882—44no—
164#10621877—42no—
165C2×C43⋊C8—22no—
166C2×C43⋊C8—22no—
167C2×C43⋊C8—82no—
168C2×C43⋊C8—81no—
169C2×C43⋊C8C2×D412yes—
170C2×C43⋊C8C2×D412yes—
171C23×C43—21no—
172C23×C43#12111yescharacteristic
173C22×D43—42no—
174C22×D43—82no—
175C22×D43—82no—
176C22×D43—82no—
177Q8×C43—21no—
178Q8×C43—41no—
179Q8×C43—41no—
180Q8×C43—81no—
181Q8×C43C2×SD1612yes—
182D4×C43—24no—
183D4×C43—21no—
184D4×C43C2×SD1612yes—
185C2×C4×C43—21no—
186C2×C4×C43—21no—
187C2×C4×C43—21no—
188C2×C4×C43—24no—
189C2×C4×C43—21no—
190C2×C4×C43—21no—
191C2×C4×C43—41no—
192C2×C4×C43—41no—
193C2×C4×C43—41no—
194C2×C4×C43—41no—
195C2×C4×C43—41no—
196C2×C4×C43—81no—
197C2×C4×C43#12111yescharacteristic
198C2×C4×C43#12111yescharacteristic
199C2×C4×C43#12111yescharacteristic
200C2×C4×C43C2×SD1612yes—
201C2×C4×C43#13512yes—
202C2×C4×C43C22×D411yescharacteristic
203C2×C4×C43C22×D411yescharacteristic
204C2×C4×C43C2×#4111yescharacteristic
205D172—42no—
206D172—42no—
207D172—42no—
208D172—42no—
209D172—82no—
210D172—82no—
211D172—82no—
212D172—82no—
213C8×C43—41no—
214C8×C43—41no—
215C43⋊C8—22no—
216C43⋊C8—22no—
217C43⋊C8—42no—
218C43⋊C8—42no—
219#19409—431nomaximal, Sylow
220C22×C43—21no—
221C22×C43—21no—
222C22×C43—22no—
223C22×C43—41no—
224C22×C43—41no—
225C22×C43#46512yes—
226C22×C43C2×#12111yescharacteristic
227C2×D43—82no—
228C2×D43—82no—
229C2×D43—82no—
230C2×D43—164no—
231C4×C43—21no—
232C4×C43—22no—
233C4×C43—21no—
234C4×C43—21no—
235C4×C43—41no—
236C4×C43—41no—
237C4×C43—81no—
238C4×C43—81no—
239C4×C43#46512yes—
240C4×C43C22×SD1611yescharacteristic
241C4×C43C2×#13511yescharacteristic
242C4×C43#57611yescharacteristic
243C4×C43#57711yescharacteristic
244#3208—431no—
245#3076D4311yescharacteristic, complemented
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×#331—431no—
255C2×C43—22no—
256C2×C43—42no—
257C2×C43#304511yescharacteristic
258C2×C43C2×#46511yescharacteristic
259C2×C43#306811yescharacteristic
260D43—162no—
261C2×AGL(1,ℤ/8)—861no—
262#533C2×D4311yescharacteristic
263C2×#127—431no—
264C4×#41C2×D4311yescharacteristic
265#492—432no—
266#486—432no—
267#479—21no—
268#478—21no—
269#477C2×D4311yescharacteristic, complemented
270#476C2×D4311yescharacteristic, complemented
271#475C2×D4311yescharacteristic, complemented
272#474C2×D4311yescharacteristic, complemented
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—21no—
286#423—431no—
287C2×C4⋊C8—431no—
288C2×C4⋊C8—861no—
289C2×C4⋊C8C2×D4311yescharacteristic
290#418—861no—
291#416—21no—
292#334—432no—
293#334—432no—
294#333—432no—
295#332—432no—
296#331—432no—
297#331—432no—
298C43#1940911yescharacteristic, complemented, Sylow
299C2×#41C22⋊D4311yescharacteristic
300C22×D4—861no—
301C22×D4—861no—
302C22×D4—861no—
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—862no—
311C22×C8—861no—
312C4⋊Q8C22×D4311yescharacteristic
313C4⋊Q8C22×D4311yescharacteristic
314#127—432no—
315#127—861no—
316#127—861no—
317#127—861no—
318#125—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8C22⋊D4311yescharacteristic
324C4×Q8C22×D4311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D4311yescharacteristic
328C4×D4C22×D4311yescharacteristic
329#117—21no—
330#117—21no—
331#117C22⋊D4311yescharacteristic
332C2×C42—21no—
333C2×C42C22×D4311yescharacteristic
334#107—21no—
335#107—21no—
336#106—21no—
337#106—21no—
338C4⋊C8—21no—
339C4⋊C8—431no—
340C4⋊C8—431no—
341C4⋊C8—432no—
342C4⋊C8—432no—
343C4⋊C8—862no—
344C4⋊C8—861no—
345C4⋊C8—861no—
346C4⋊C8C22×D4311yescharacteristic
347C4⋊C8C22×D4311yescharacteristic
348#103—21no—
349#103—21no—
350#102—21no—
351#102—21no—
352#102—862no—
353#102—862no—
354#102—862no—
355#102—862no—
356C22⋊C8—432no—
357C22⋊C8—862no—
358C4×C8—432no—
359C4×C8—862no—
360C24—1721no—
361#41—22no—
362#41#1062188414yescomplemented
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×C22⋊D4311yescharacteristic
366C2×D4—21no—
367C2×D4—862no—
368C2×D4—862no—
369C2×D4—862no—
370C2×D4—3441no—
371C2×D4—3441no—
372C2×D4—3442no—
373C2×D4—1721no—
374C2×D4—1721no—
375C2×D4—1721no—
376C2×D4—1722no—
377C2×D4—1722no—
378C2×D4C2×C22⋊D4311yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D4311yescharacteristic
383C22×C4#1062190011yescharacteristic
384C22×C4C2×C22⋊D4311yescharacteristic
385SD16—3442no—
386SD16—3442no—
387SD16—1724no—
388D8—1724no—
389M16—1722no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—42no—
393C2×C8—432no—
394C2×C8—432no—
395C2×C8—862no—
396C2×C8—862no—
397C2×C8—3442no—
398C2×C8—3441no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×C22⋊D4311yescharacteristic
408C4⋊C4C2×C22⋊D4311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—41no—
417C42D4×D4311yescharacteristic
418C42#1062190011yescharacteristic
419C42C2×C22⋊D4311yescharacteristic
420C42C23×D4311yescharacteristic
421C23—21no—
422C23—3442no—
423C23—3442no—
424C23—3442no—
425C23—1722no—
426C23#1306329711yescharacteristic
427Q8—21no—
428Q8—41no—
429Q8—41no—
430Q8—81no—
431Q8C2×#1062188412yes—
432D4—24no—
433D4—21no—
434D4—3442no—
435D4—3442no—
436D4—3442no—
437D4—3442no—
438D4—1722no—
439D4—1722no—
440D4—1722no—
441D4—1722no—
442D4C2×#1062188412yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4#1306329711yescharacteristic
456C2×C4#1306329711yescharacteristic
457C2×C4#1306329711yescharacteristic
458C2×C4C2×#1062188412yes—
459C2×C4#1306334512yes—
460C2×C4C2×D4×D4311yescharacteristic
461C2×C4C22×C22⋊D4311yescharacteristic
462C2×C4C2×#1062190011yescharacteristic
463C8—41no—
464C8—41no—
465C8—862no—
466C8—862no—
467C8—1722no—
468C8—1722no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—3442no—
475C22—3442no—
476C22—3442no—
477C22—6884no—
478C22#1458313012yes—
479C22C2×#1306329711yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—81no—
488C4#1458313012yes—
489C4C22×#1062188411yescharacteristic
490C4C2×#1306334511yescharacteristic
491C4#1458370111yescharacteristic
492C4#1458370611yescharacteristic
493C2—22no—
494C2—42no—
495C2—6882no—
496C2#1914413211yescharacteristic
497C2C2×#1458313011yescharacteristic
498C2#1914432511yescharacteristic
subgroupG/H
centerC22C2×#13063297
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3076C2
socleC22×C43C2×#121
prime radical#19144075C2
hypercenter#3076D43
hypocenterC43#19409
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #19145533C2semidirect
3. #19143706C2semidirect
4. #19134013C2semidirect
5. #19134006C2semidirect
6. #19133945C2semidirect
7. #19133938C2semidirect
8. #19133365C2semidirect
9. #19133285C2semidirect
10. C2×#14582642C2semidirect
11. C43×#3076C2semidirect
23. Q8×C43⋊C8C22semidirect
25. D4×C43⋊C8C22semidirect
32. #14582649C22semidirect
33. #14582649C22semidirect
34. #14582648C22semidirect
35. #14582645C22semidirect
42. C43×#477C22semidirect
43. C43×#476C22semidirect
44. C43×#475C22semidirect
45. C43×#474C22semidirect
112. #41×C43SD16semidirect
245. #3076D43semidirect
269. #477C2×D43semidirect
270. #476C2×D43semidirect
271. #475C2×D43semidirect
272. #474C2×D43semidirect
298. C43#19409semidirect
362. #41#10621884semidirect

Maximal subgroups

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

subgroupindex
#191455332
#191440752
#191437062
#191340132
#191340062
#191339452
#191339382
#191333652
#191332852
C2×#145826422
C43×#30762
#1940943