Finite Group Databasebeta

← #25079453

Subgroups — #25079453

Order 11008.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145047C211yesmaximal, characteristic
2#19144626C211yesmaximal, characteristic, complemented
3C2×#14583099C211yesmaximal, characteristic, complemented
4#19134079C212yesmaximal, complemented
5#19134067C212yesmaximal, complemented
6#19133955C212yesmaximal, complemented
7#19133943C212yesmaximal, complemented
8#19133347C211yesmaximal, characteristic, complemented
9#19133273C211yesmaximal, characteristic, complemented
10#19133264C211yesmaximal, characteristic, complemented
11Q8×M16×C43C211yesmaximal, characteristic, complemented
12C2×#13063353—21no—
13C2×#13063353—21no—
14C22×#10621884—21no—
15#14583449C2211yescharacteristic
16C2×#13063322C2211yescharacteristic
17#14583288—22no—
18#14583276C2212yes—
19#14583272C2212yes—
20#14583217—21no—
21#14583184C2211yescharacteristic, complemented
22#14583184C2211yescharacteristic, complemented
23#14583099—21no—
24#14583099—21no—
25#14583099C2212yes—
26#14583099C2212yes—
27#14583099C2211yescharacteristic, complemented
28#14583099C2211yescharacteristic, complemented
29#14583038—21no—
30C4×#10621884—21no—
31C4×#10621884—21no—
32#14582952C2211yescharacteristic
33C2×C4⋊(C43⋊C8)C2211yescharacteristic
34C2×#13063245—21no—
35#14582652C2212yescomplemented
36#14582650C2212yescomplemented
37#14582646C2212yescomplemented
38#14582642C2212yes—
39#14582642C2212yes—
40#14582642C2212yescomplemented
41C2×C4×Q8×C43C2211yescharacteristic
42C43×#446—22no—
43C43×#446C2212yescomplemented
44C8×Q8×C43C2212yescomplemented
45C43×#423—21no—
46C43×#423C2211yescharacteristic
47C4×M16×C43—21no—
48C4×M16×C43C2211yescharacteristic
49C2×#10621898—21no—
50C22×D172—21no—
51C22×D172—21no—
52C22×D172—21no—
53#13063353—44no—
54#13063353—44no—
55C2×#10621873—22no—
56C2×#10621873—22no—
57C2×#10621884—22no—
58C2×#10621884—22no—
59C2×#10621884—21no—
60C2×#10621884—21no—
61C2×#10621884—44no—
62C2×D344—22no—
63C2×D344—22no—
64C22×C43⋊C8—21no—
65#13063325C2311yescharacteristic
66#13063322—21no—
67#13063322—21no—
68#13063322—21no—
69#13063322C2312yes—
70#13063322C2311yescharacteristic
71#13063313—21no—
72#13063299—21no—
73#13063299—21no—
74C4×D172—21no—
75C4×D172—21no—
76#13063256—21no—
77#13063256—21no—
78C4⋊(C43⋊C8)—21no—
79C4⋊(C43⋊C8)C2312yes—
80C4⋊(C43⋊C8)C2311yescharacteristic
81C4⋊(C43⋊C8)C2311yescharacteristic
82#13063249—21no—
83#13063249—21no—
84#13063245—22no—
85#13063245—22no—
86#13063245—21no—
87#13063245—21no—
88C22⋊(C43⋊C8)—21no—
89C4×C43⋊C8—21no—
90C22×Q8×C43D411yescharacteristic
91C2×M16×C43—21no—
92C2×M16×C43D411yescharacteristic
93C2×M16×C43D411yescharacteristic
94C4×Q8×C43—21no—
95C4×Q8×C43—21no—
96C4×Q8×C43D411yescharacteristic
97C4×Q8×C43D411yescharacteristic
98C4×Q8×C43C2311yescharacteristic
99C4×Q8×C43C2311yescharacteristic
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43D411yescharacteristic
102C2×C42×C43—21no—
103C2×C42×C43C2311yescharacteristic
104C4⋊C8×C43—22no—
105C4⋊C8×C43—22no—
106C4⋊C8×C43—21no—
107C4⋊C8×C43C2312yes—
108#97×C43—22no—
109#97×C43D412yes—
110C4×C8×C43—22no—
111C4×C8×C43C2312yes—
112C23×D43—41no—
113C2×Q8×C43—21no—
114C2×Q8×C43—22no—
115C2×Q8×C43—41no—
116C2×Q8×C43C2×D411yescharacteristic
117C2×Q8×C43C2×D411yescharacteristic
118C22×C4×C43—21no—
119C22×C4×C43—21no—
120C22×C4×C43C2×D411yescharacteristic
121C22×C4×C43C2×D411yescharacteristic
122C22×C4×C43#4111yescharacteristic
123C2×C22⋊D43—41no—
124#10621898—42no—
125#10621898—42no—
126C2×D172—21no—
127C2×D172—22no—
128C2×D172—21no—
129C2×D172—22no—
130C2×D172—21no—
131C2×D172—22no—
132C2×D172—41no—
133C2×D172—41no—
134C2×D172—41no—
135C2×D172—42no—
136C2×D172—42no—
137C2×D172—42no—
138C2×D172—42no—
139C2×D172—82no—
140C2×C4×D43—41no—
141C2×Dic86—21no—
142M16×C43—22no—
143M16×C43—42no—
144M16×C43SD1614yescomplemented
145C2×C8×C43—22no—
146C2×C8×C43C2×D412yes—
147C2×C8×C43C2×D412yes—
148C4⋊C4×C43—21no—
149C4⋊C4×C43—21no—
150C4⋊C4×C43—21no—
151C4⋊C4×C43—21no—
152C4⋊C4×C43—21no—
153C4⋊C4×C43C2×D411yescharacteristic
154C4⋊C4×C43C2×D411yescharacteristic
155C42×C43—21no—
156C42×C43—21no—
157C42×C43—21no—
158C42×C43—21no—
159C42×C43C2×D411yescharacteristic
160C42×C43C2×D411yescharacteristic
161C42×C43#4111yescharacteristic
162C42×C43C2411yescharacteristic
163#10621884—42no—
164#10621884—44no—
165#10621884—42no—
166#10621884—44no—
167(C2×D43)⋊C4—41no—
168(C2×D43)⋊C4—41no—
169C4⋊Dic43—21no—
170C4⋊Dic43—21no—
171C2×C43⋊C8—21no—
172C2×C43⋊C8—21no—
173C2×C43⋊C8—22no—
174C2×C43⋊C8—42no—
175D344—44no—
176D344—44no—
177#10621873—44no—
178#10621873—44no—
179C23×C43#12111yescharacteristic
180C22×D43—41no—
181C22×D43—42no—
182C22×D43—82no—
183C22×D43—82no—
184C22×D43—82no—
185Q8×C43—21no—
186Q8×C43—21no—
187Q8×C43—21no—
188Q8×C43—42no—
189Q8×C43AGL(1,ℤ/8)12yescomplemented
190C2×C4×C43—21no—
191C2×C4×C43—21no—
192C2×C4×C43—21no—
193C2×C4×C43—21no—
194C2×C4×C43—21no—
195C2×C4×C43—21no—
196C2×C4×C43—41no—
197C2×C4×C43—42no—
198C2×C4×C43—81no—
199C2×C4×C43#12111yescharacteristic
200C2×C4×C43#12111yescharacteristic
201C2×C4×C43#12111yescharacteristic
202C2×C4×C43C2×SD1612yes—
203C2×C4×C43AGL(1,ℤ/8)12yes—
204C2×C4×C43C22×D411yescharacteristic
205C2×C4×C43C22×D411yescharacteristic
206C2×C4×C43C2×#4111yescharacteristic
207C22⋊D43—82no—
208C2×Dic43—41no—
209D172—42no—
210D172—41no—
211D172—42no—
212D172—41no—
213D172—42no—
214D172—41no—
215D172—41no—
216D172—42no—
217D172—81no—
218D172—82no—
219D172—82no—
220D172—82no—
221D172—82no—
222C4×D43—82no—
223Dic86—41no—
224Dic86—41no—
225C8×C43—22no—
226C8×C43—42no—
227C8×C43C2×SD1614yescomplemented
228C43⋊C8—42no—
229C43⋊C8—41no—
230C43⋊C8—41no—
231#19170—431nomaximal, Sylow
232C22×C43—21no—
233C22×C43—21no—
234C22×C43#46012yes—
235C22×C43C2×#12111yescharacteristic
236C2×D43—82no—
237C2×D43—81no—
238C2×D43—82no—
239C2×D43—81no—
240C2×D43—82no—
241C2×D43—81no—
242C2×D43—164no—
243C4×C43—21no—
244C4×C43—22no—
245C4×C43—21no—
246C4×C43—21no—
247C4×C43—41no—
248C4×C43—41no—
249C4×C43—41no—
250C4×C43#46012yes—
251C4×C43C22×SD1611yescharacteristic
252C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C43#57511yescharacteristic
255Dic43—81no—
256#3151—431no—
257#3103—431no—
258C2×#460—431no—
259Q8×M16D4311yescharacteristic, complemented
260#1688—432no—
261#1686—432no—
262#1668—432no—
263#1666—432no—
264#1565—431no—
265#1547—431no—
266#1545—431no—
267C2×C43—22no—
268C2×C43C2×#46011yescharacteristic
269C2×C43#305511yescharacteristic
270C2×C43#306811yescharacteristic
271D43—161no—
272D43—162no—
273C2×AGL(1,ℤ/8)—861no—
274C2×AGL(1,ℤ/8)—861no—
275C22×SD16—861no—
276#532—431no—
277C2×#127—431no—
278C2×C4×Q8C2×D4311yescharacteristic
279#496—862no—
280#493—432no—
281#492—432no—
282#481—861no—
283#475—431no—
284#475—431no—
285#460—431no—
286#460—431no—
287#460—432no—
288#460—432no—
289#460—861no—
290#460—861no—
291#448—861no—
292#446—22no—
293#446C2×D4312yescomplemented
294C8×Q8C2×D4312yescomplemented
295C4×SD16—861no—
296C4×SD16—861no—
297#431—431no—
298#423—21no—
299#423C2×D4311yescharacteristic
300C2×C4⋊C8—431no—
301C2×#102—861no—
302C4×M16—21no—
303C4×M16C2×D4311yescharacteristic
304#335—432no—
305#334—432no—
306#332—432no—
307#331—432no—
308#331—432no—
309#331—432no—
310C43#1917011yescharacteristic, complemented, Sylow
311C2×#41—861no—
312C22×Q8D17211yescharacteristic
313C22×D4—861no—
314C22×D4—861no—
315C22×D4—861no—
316AGL(1,ℤ/8)—1724no—
317AGL(1,ℤ/8)—1724no—
318C2×SD16—862no—
319C2×SD16—862no—
320C2×SD16—861no—
321C2×SD16—861no—
322C2×SD16—862no—
323C2×SD16—862no—
324C2×SD16—1724no—
325C2×D8—862no—
326C2×D8—862no—
327C2×M16—21no—
328C2×M16C22⋊D4311yescharacteristic
329C2×M16C22⋊D4311yescharacteristic
330C22×C8—861no—
331C4⋊Q8—431no—
332#127—432no—
333#127—431no—
334#127—861no—
335#127—861no—
336#127—861no—
337#124—861no—
338#121—861no—
339#121—861no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8D17211yescharacteristic
343C4×Q8D17211yescharacteristic
344C4×Q8C22×D4311yescharacteristic
345C4×Q8C22×D4311yescharacteristic
346C4×D4—861no—
347C4×D4—861no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4D17211yescharacteristic
350C2×C42—21no—
351C2×C42C22×D4311yescharacteristic
352#106—861no—
353#106—861no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—21no—
357C4⋊C8—431no—
358C4⋊C8—431no—
359C4⋊C8—432no—
360C4⋊C8—861no—
361C4⋊C8C22×D4312yes—
362#103—861no—
363#103—861no—
364#102—862no—
365#102—862no—
366#102—861no—
367#102—861no—
368C22⋊C8—861no—
369#97—22no—
370#97C22⋊D4312yes—
371C4×C8—22no—
372C4×C8—861no—
373C4×C8C22×D4312yes—
374C24—1721no—
375#41—1722no—
376#41—1722no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—861no—
381C2×Q8C2×D17211yescharacteristic
382C2×Q8C2×D17211yescharacteristic
383C2×D4—861no—
384C2×D4—862no—
385C2×D4—861no—
386C2×D4—862no—
387C2×D4—861no—
388C2×D4—862no—
389C2×D4—3442no—
390C2×D4—1722no—
391C2×D4—1722no—
392C2×D4—1721no—
393C2×D4—1721no—
394C2×D4—1721no—
395C2×D4—1721no—
396C2×D4—1722no—
397C2×D4—1722no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—1721no—
401C22×C4C2×D17211yescharacteristic
402C22×C4#1062189811yescharacteristic
403C22×C4C2×C22⋊D4311yescharacteristic
404SD16—1724no—
405SD16—1724no—
406SD16—1722no—
407SD16—1724no—
408SD16—1722no—
409SD16—1724no—
410D8—1724no—
411D8—1724no—
412M16—22no—
413M16—42no—
414M16#1062188414yescomplemented
415C2×C8—22no—
416C2×C8—861no—
417C2×C8—861no—
418C2×C8—862no—
419C2×C8—1722no—
420C2×C8C2×C22⋊D4312yes—
421C2×C8C2×C22⋊D4312yes—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—861no—
428C4⋊C4—861no—
429C4⋊C4C2×D17211yescharacteristic
430C4⋊C4C2×D17211yescharacteristic
431C22⋊C4—1721no—
432C22⋊C4—1721no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42C2×D17211yescharacteristic
438C42#1062189811yescharacteristic
439C42C2×C22⋊D4311yescharacteristic
440C42C23×D4311yescharacteristic
441C23—3442no—
442C23—3442no—
443C23—3442no—
444C23—1721no—
445C23—1722no—
446C23#1306329911yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—1721no—
452Q8—1721no—
453Q8#1306335312yescomplemented
454D4—3442no—
455D4—3441no—
456D4—3442no—
457D4—3442no—
458D4—3442no—
459D4—3442no—
460D4—1721no—
461D4—1722no—
462D4—1721no—
463D4—1722no—
464D4—1722no—
465D4—1721no—
466D4—1721no—
467D4—1722no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—81no—
477C2×C4—3442no—
478C2×C4—1721no—
479C2×C4#1306329911yescharacteristic
480C2×C4#1306329911yescharacteristic
481C2×C4#1306329911yescharacteristic
482C2×C4C2×#1062188412yes—
483C2×C4#1306335012yes—
484C2×C4C22×C22⋊D4311yescharacteristic
485C2×C4C22×D17211yescharacteristic
486C2×C4C2×#1062189811yescharacteristic
487C8—22no—
488C8—42no—
489C8—1722no—
490C8—1721no—
491C8—1721no—
492C8C2×#1062188414yescomplemented
493C22—21no—
494C22—21no—
495C22—3442no—
496C22—3441no—
497C22—3441no—
498C22—3442no—
499C22—3442no—
500C22—3441no—
501C22—6884no—
502C22#1458309912yes—
503C22C2×#1306329911yescharacteristic
504C4—21no—
505C4—22no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—3441no—
512C4#1458309912yes—
513C4C22×#1062188411yescharacteristic
514C4C2×#1306335011yescharacteristic
515C4C2×#1306335311yescharacteristic
516C4#1458369911yescharacteristic
517C2—22no—
518C2—6882no—
519C2—6881no—
520C2C2×#1458309911yescharacteristic
521C2#1914421411yescharacteristic
522C2#1914432311yescharacteristic
subgroupG/H
centerC22C2×#13063299
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingQ8×M16×C43C2
socleC22×C43C2×#121
prime radical#19145047C2
hypercenterQ8×M16D43
hypocenterC43#19170
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #19144626C2semidirect
3. C2×#14583099C2semidirect
4. #19134079C2semidirect
5. #19134067C2semidirect
6. #19133955C2semidirect
7. #19133943C2semidirect
8. #19133347C2semidirect
9. #19133273C2semidirect
10. #19133264C2semidirect
11. Q8×M16×C43C2semidirect
21. #14583184C22semidirect
22. #14583184C22semidirect
27. #14583099C22semidirect
28. #14583099C22semidirect
35. #14582652C22semidirect
36. #14582650C22semidirect
37. #14582646C22semidirect
40. #14582642C22semidirect
43. C43×#446C22semidirect
44. C8×Q8×C43C22semidirect
144. M16×C43SD16semidirect
189. Q8×C43AGL(1,ℤ/8)semidirect
227. C8×C43C2×SD16semidirect
259. Q8×M16D43semidirect
293. #446C2×D43semidirect
294. C8×Q8C2×D43semidirect
310. C43#19170semidirect
414. M16#10621884semidirect
453. Q8#13063353semidirect
492. C8C2×#10621884semidirect

Maximal subgroups

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

subgroupindex
#191450472
#191446262
C2×#145830992
#191340792
#191340672
#191339552
#191339432
#191333472
#191332732
#191332642
Q8×M16×C432
#1917043