Finite Group Databasebeta

← #25132687

Subgroups — #25132687

Order 11008.

1294 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D43C211yesmaximal, characteristic
2C2×#14583185C211yesmaximal, characteristic, complemented
3C2×#14583181C211yesmaximal, characteristic, complemented
4C4×#13063305C211yesmaximal, characteristic
5#19135927C212yesmaximal, complemented
6#19135735C212yesmaximal, complemented
7#19135547C211yesmaximal, characteristic, complemented
8#19134901C212yesmaximal
9#19134789C212yesmaximal
10#19134572C211yesmaximal, characteristic
11C43×#1988C211yesmaximal, characteristic, complemented
12C22×Q8×D43—21no—
13C2×#13063325C2211yescharacteristic
14C2×#13063324—21no—
15C2×#13063324C2212yes—
16C2×#13063323—21no—
17C4⋊Q8×D43—24no—
18C4⋊Q8×D43—24no—
19C2×#13063305C2211yescharacteristic
20C2×C4×Dic86C2211yescharacteristic
21C2×C4⋊C4×D43—21no—
22C2×C4⋊C4×D43C2212yes—
23C2×C42×D43C2211yescharacteristic
24#14583185—24no—
25#14583181—24no—
26C2×#13063257C2212yes—
27C2×#13063256C2212yes—
28C2×C4⋊(C43⋊C8)C2211yescharacteristic
29C2×#13063250C2212yes—
30C2×#13063248C2212yes—
31C2×(C2×D43)⋊C8C2212yes—
32#14582804C2211yescharacteristic
33#14582798C2211yescharacteristic
34#14582784—21no—
35#14582773—21no—
36#14582756C2211yescharacteristic
37C4×C4⋊Dic43—21no—
38C4×(C2×D43)⋊C4—21no—
39#14582661C2212yes—
40#14582661C2212yes—
41#14582653C2212yes—
42C2×C4⋊Q8×C43C2211yescharacteristic
43C2×C4⋊C8×C43C2211yescharacteristic
44C4×C4⋊C4×C43C2211yescharacteristic
45C43×#340C2212yes—
46C43×#340C2212yes—
47C22×Dic86—21no—
48C22×Dic86—22no—
49C22×Dic86C2311yescharacteristic
50C2×Q8×D43—28no—
51C2×Q8×D43—44no—
52C2×Q8×D43—44no—
53C22×C4×D43—22no—
54C22×C4×D43D411yescharacteristic
55C22×C4×D43D411yescharacteristic
56C22×C4×D43C2311yescharacteristic
57C22×C43⋊C8C2312yes—
58#13063325—22no—
59#13063325D414yes—
60#13063324—28no—
61#13063324—24no—
62#13063324—24no—
63#13063323—28no—
64#13063305—22no—
65#13063305Q814yescomplemented
66C4×Dic86—22no—
67C4×Dic86—22no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43C2311yescharacteristic
72C2×C4⋊Dic43C2312yes—
73C2×Dic43⋊C4—22no—
74C2×Dic43⋊C4—22no—
75C2×Dic43⋊C4—21no—
76C4⋊C4×D43—28no—
77C4⋊C4×D43—24no—
78C4⋊C4×D43—24no—
79C2×(C2×D43)⋊C4—21no—
80C2×(C2×D43)⋊C4—21no—
81C2×C4×Dic43—21no—
82C2×C4×Dic43D411yescharacteristic
83C2×C4×Dic43D411yescharacteristic
84C2×C4×Dic43C2311yescharacteristic
85C42×D43—22no—
86C42×D43—22no—
87#13063257—24no—
88#13063256—24no—
89C4⋊(C43⋊C8)—22no—
90#13063250—24no—
91#13063248—24no—
92(C2×D43)⋊C8—24no—
93C43⋊#95—21no—
94C43⋊#95—21no—
95C43⋊#95—21no—
96C43⋊#95—21no—
97C22×Q8×C43—21no—
98C22×C8×C43C2312yes—
99C4⋊Q8×C43—22no—
100C4⋊Q8×C43D414yes—
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43C2311yescharacteristic
103C2×C4⋊C4×C43C2312yes—
104C2×C4⋊C4×C43C2311yescharacteristic
105C2×C42×C43—21no—
106C2×C42×C43C2311yescharacteristic
107C4⋊C8×C43—22no—
108#95×C43—21no—
109C23×D43C2×D411yescharacteristic
110C2×Q8×C43—24no—
111C2×Q8×C43—42no—
112C2×Q8×C43—42no—
113C22×C4×C43—21no—
114C22×C4×C43—22no—
115C22×C4×C43—21no—
116C22×C4×C43C2×D411yescharacteristic
117C22×C4×C43C2×D411yescharacteristic
118C22×C4×C43C2411yescharacteristic
119C22×Dic43—21no—
120C22×Dic43—22no—
121C22×Dic43—21no—
122C22×Dic43C2×D411yescharacteristic
123C22×Dic43C2×D411yescharacteristic
124C22×Dic43C2×D411yescharacteristic
125Q8×D43—432no—
126C2×C4×D43—22no—
127C2×C4×D43—22no—
128C2×C4×D43—22no—
129C2×C4×D43—21no—
130C2×C4×D43—21no—
131C2×C4×D43—21no—
132C2×C4×D43—21no—
133C2×C4×D43—22no—
134C2×C4×D43—28no—
135C2×C4×D43—22no—
136C2×C4×D43—44no—
137C2×C4×D43—44no—
138C2×C4×D43SD1614yes—
139C2×C4×D43C2×Q812yes—
140C2×C4×D43#4112yes—
141C2×Dic86—21no—
142C2×Dic86—21no—
143C2×Dic86—21no—
144C2×Dic86—21no—
145C2×Dic86—28no—
146C2×Dic86—24no—
147C2×Dic86—21no—
148C2×Dic86—44no—
149C2×Dic86—42no—
150C2×Dic86—42no—
151C2×Dic86—44no—
152C2×Dic86C2×Q812yescomplemented
153C2×C8×C43—22no—
154C2×C8×C43—22no—
155C2×C8×C43—22no—
156C4⋊C4×C43—24no—
157C4⋊C4×C43—22no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43—21no—
160C4⋊C4×C43—21no—
161C4⋊C4×C43C2×D414yes—
162C4⋊C4×C43C2×Q812yescomplemented
163C42×C43—42no—
164C42×C43C2×D412yes—
165C42×C43C2×D412yes—
166(C2×D43)⋊C4—24no—
167(C2×D43)⋊C4—42no—
168C4⋊Dic43—24no—
169C4⋊Dic43—22no—
170C4⋊Dic43—24no—
171C4⋊Dic43—21no—
172C4⋊Dic43—21no—
173C4⋊Dic43—42no—
174C4⋊Dic43C2×D414yes—
175Dic43⋊C4—28no—
176Dic43⋊C4—28no—
177Dic43⋊C4—42no—
178C4×Dic43—22no—
179C4×Dic43—22no—
180C4×Dic43—21no—
181C4×Dic43—21no—
182C4×Dic43—42no—
183C2×C43⋊C8—22no—
184C2×C43⋊C8—22no—
185C2×C43⋊C8—22no—
186C23×C43C22×D411yescharacteristic
187C22×D43—21no—
188C22×D43—21no—
189C22×D43—21no—
190C22×D43—21no—
191C22×D43C22⋊Q812yes—
192C22×D43C2×SD1612yes—
193C22×D43#13712yes—
194Q8×C43—48no—
195C2×C4×C43—21no—
196C2×C4×C43—21no—
197C2×C4×C43—24no—
198C2×C4×C43—22no—
199C2×C4×C43—42no—
200C2×C4×C43—41no—
201C2×C4×C43—42no—
202C2×C4×C43—41no—
203C2×C4×C43—41no—
204C2×C4×C43—41no—
205C2×C4×C43—41no—
206C2×C4×C43#12012yes—
207C2×C4×C43#12012yes—
208C2×C4×C43C2×SD1612yes—
209C2×C4×C43AGL(1,ℤ/8)12yes—
210C2×C4×C43C22×D411yescharacteristic
211C2×C4×C43C22×D411yescharacteristic
212C2×C4×C43C22×Q811yescharacteristic
213C2×C4×C43C2×#4111yescharacteristic
214C2×C4×C432−1+411yescharacteristic
215C2×C4×C432−1+411yescharacteristic
216C2×Dic43—21no—
217C2×Dic43—21no—
218C2×Dic43—21no—
219C2×Dic43—21no—
220C2×Dic43—21no—
221C2×Dic43—21no—
222C2×Dic43—24no—
223C2×Dic43—22no—
224C2×Dic43—41no—
225C2×Dic43—41no—
226C2×Dic43—42no—
227C2×Dic43—42no—
228C2×Dic43—41no—
229C2×Dic43—41no—
230C2×Dic43—41no—
231C2×Dic43C22⋊Q812yes—
232C2×Dic43C2×SD1612yes—
233C2×Dic43#13712yes—
234C4×D43—28no—
235C4×D43—22no—
236C4×D43—22no—
237C4×D43—22no—
238C4×D43—22no—
239C4×D43—28no—
240C4×D43—416no—
241Dic86—22no—
242Dic86—22no—
243Dic86—416no—
244Dic86—41no—
245Dic86—41no—
246Dic86—48no—
247C8×C43—42no—
248C43⋊C8—42no—
249#24301—431nomaximal, Sylow
250C22×C43C2×#12011yescharacteristic
251C22×C43C2×C22⋊Q811yescharacteristic
252C22×C43#53611yescharacteristic
253C22×C43C22×SD1611yescharacteristic
254C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C43C2×#13711yescharacteristic
256C22×C43C2×#13711yescharacteristic
257C2×D43—22no—
258C2×D43—22no—
259C2×D43—22no—
260C2×D43—22no—
261C2×D43—22no—
262C2×D43—22no—
263C2×D43#47514yescomplemented
264C4×C43—22no—
265C4×C43—42no—
266C4×C43—44no—
267C4×C43—81no—
268C4×C43#45014yes—
269C4×C43#54811yescharacteristic
270C4×C43#54811yescharacteristic
271C4×C43D4×Q811yescharacteristic
272C4×C43D4×Q811yescharacteristic
273Dic43—22no—
274Dic43—22no—
275Dic43—22no—
276Dic43—42no—
277Dic43—44no—
278Dic43—81no—
279C22×C4⋊Q8—431no—
280C2×#475—431no—
281C2×#475—431no—
282C4×C22⋊Q8—431no—
283#2058—432no—
284#2024—432no—
285#1988—431no—
286#1988D4311yescharacteristic, complemented
287#1872—432no—
288#1853—432no—
289#1797—431no—
290C2×C43#240911yescharacteristic
291C2×C43C2×#45011yescharacteristic
292C2×C43#300411yescharacteristic
293C2×C43C2×#47511yescharacteristic
294C2×C43#307911yescharacteristic
295C2×C43#322911yescharacteristic
296C2×C43#323911yescharacteristic
297D43—24no—
298C23×Q8—861no—
299C2×C4⋊Q8—432no—
300C2×C4⋊Q8—431no—
301C2×C4⋊Q8—861no—
302C2×C4⋊Q8—861no—
303C2×C4⋊Q8—864no—
304C2×C4⋊Q8—864no—
305C2×C4⋊Q8C2×D4311yescharacteristic
306C2×C22⋊Q8—431no—
307C2×C4×Q8—431no—
308C22×C4⋊C4—432no—
309C22×C4⋊C4—861no—
310C22×C42—431no—
311#475—864no—
312#475—864no—
313C2×#106—432no—
314C2×#106—432no—
315C2×C4⋊C8—431no—
316C2×C4⋊C8C2×D4311yescharacteristic
317C2×#103—432no—
318C2×#103—432no—
319C2×C22⋊C8—432no—
320#391—431no—
321#389—431no—
322#387—861no—
323#385—861no—
324#380—431no—
325C4×C4⋊C4—861no—
326C4×C4⋊C4C2×D4311yescharacteristic
327C4×C22⋊C4—861no—
328#340—432no—
329#340—432no—
330#340C2×D4312yes—
331#340C2×D4312yes—
332#336—432no—
333C43#2430111yescharacteristic, complemented, Sylow
334C22×Q8—21no—
335C22×Q8—431no—
336C22×Q8—862no—
337C22×Q8—868no—
338C22×Q8—861no—
339C22×Q8—1724no—
340C22×Q8—1724no—
341C23×C4—431no—
342C23×C4—431no—
343C23×C4—431no—
344C23×C4—862no—
345C22×C8—432no—
346C22×C8C22×D4312yes—
347C4⋊Q8—22no—
348C4⋊Q8—434no—
349C4⋊Q8—864no—
350C4⋊Q8—868no—
351C4⋊Q8—868no—
352C4⋊Q8—864no—
353C4⋊Q8—862no—
354C4⋊Q8D17214yes—
355C22⋊Q8—434no—
356C22⋊Q8—862no—
357C4×Q8—862no—
358C4×Q8—862no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—432no—
361C2×C4⋊C4—431no—
362C2×C4⋊C4—864no—
363C2×C4⋊C4—868no—
364C2×C4⋊C4—862no—
365C2×C4⋊C4—862no—
366C2×C4⋊C4—861no—
367C2×C4⋊C4—861no—
368C2×C4⋊C4—864no—
369C2×C4⋊C4—861no—
370C2×C4⋊C4—861no—
371C2×C4⋊C4C22×D4311yescharacteristic
372C2×C4⋊C4C22×D4312yes—
373C2×C4⋊C4C22×D4311yescharacteristic
374C2×C22⋊C4—861no—
375C2×C22⋊C4—861no—
376C2×C42—21no—
377C2×C42—431no—
378C2×C42—431no—
379C2×C42—431no—
380C2×C42—862no—
381C2×C42—862no—
382C2×C42—861no—
383C2×C42C22×D4311yescharacteristic
384#106—864no—
385#106—864no—
386C4⋊C8—22no—
387C4⋊C8—862no—
388#103—864no—
389#103—864no—
390C22⋊C8—864no—
391#95—21no—
392#95—861no—
393#95—861no—
394#95—861no—
395#95—861no—
396C24—431no—
397C2×Q8—24no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—432no—
401C2×Q8—868no—
402C2×Q8—864no—
403C2×Q8—861no—
404C2×Q8—861no—
405C2×Q8—861no—
406C2×Q8—861no—
407C2×Q8—861no—
408C2×Q8—1724no—
409C2×Q8—1724no—
410C2×Q8—17232no—
411C2×Q8—1722no—
412C2×Q8—1722no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—431no—
417C22×C4—431no—
418C22×C4—431no—
419C22×C4—434no—
420C22×C4—432no—
421C22×C4—432no—
422C22×C4—862no—
423C22×C4—862no—
424C22×C4—868no—
425C22×C4—862no—
426C22×C4—862no—
427C22×C4—862no—
428C22×C4—861no—
429C22×C4—861no—
430C22×C4—861no—
431C22×C4—861no—
432C22×C4—861no—
433C22×C4—862no—
434C22×C4—861no—
435C22×C4—1724no—
436C22×C4—1724no—
437C22×C4D4×D4311yescharacteristic
438C22×C4D4×D4311yescharacteristic
439C22×C4C23×D4311yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—862no—
444C2×C8—862no—
445C2×C8—862no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—434no—
452C4⋊C4—864no—
453C4⋊C4—868no—
454C4⋊C4—868no—
455C4⋊C4—864no—
456C4⋊C4—862no—
457C4⋊C4—861no—
458C4⋊C4—861no—
459C4⋊C4—1722no—
460C4⋊C4—1722no—
461C4⋊C4C2×D17214yes—
462C4⋊C4Q8×D4312yescomplemented
463C22⋊C4—864no—
464C22⋊C4—1722no—
465C42—42no—
466C42—862no—
467C42—862no—
468C42—861no—
469C42—861no—
470C42—1722no—
471C42C2×D17212yes—
472C42D4×D4312yes—
473C23—432no—
474C23—432no—
475C23—432no—
476C23—861no—
477C23—861no—
478C23—861no—
479C23—861no—
480C23C2×D4×D4311yescharacteristic
481Q8—48no—
482Q8—862no—
483Q8—862no—
484Q8—17216no—
485Q8—1721no—
486Q8—1721no—
487Q8—1728no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—432no—
500C2×C4—432no—
501C2×C4—432no—
502C2×C4—868no—
503C2×C4—861no—
504C2×C4—861no—
505C2×C4—861no—
506C2×C4—861no—
507C2×C4—861no—
508C2×C4—861no—
509C2×C4—862no—
510C2×C4—862no—
511C2×C4—868no—
512C2×C4—862no—
513C2×C4—862no—
514C2×C4—864no—
515C2×C4—862no—
516C2×C4—17216no—
517C2×C4—1721no—
518C2×C4—1721no—
519C2×C4—1721no—
520C2×C4—1721no—
521C2×C4—1722no—
522C2×C4—1722no—
523C2×C4—1721no—
524C2×C4#1306329512yes—
525C2×C4#1306329512yes—
526C2×C4SD16×D4312yes—
527C2×C4#1306334912yes—
528C2×C4C2×D4×D4311yescharacteristic
529C2×C4C22×D17211yescharacteristic
530C2×C4C2×Q8×D4311yescharacteristic
531C2×C4#41×D4311yescharacteristic
532C2×C4#1306337211yescharacteristic
533C2×C4#1306337311yescharacteristic
534C8—42no—
535C8—1722no—
536C22—434no—
537C22—862no—
538C22—862no—
539C22—862no—
540C22—862no—
541C22—862no—
542C22—862no—
543C22C2×#1306329511yescharacteristic
544C22C22⋊Q8×D4311yescharacteristic
545C22#1458347211yescharacteristic
546C22C2×SD16×D4311yescharacteristic
547C22C2×#1306334911yescharacteristic
548C22#137×D4311yescharacteristic
549C22#137×D4311yescharacteristic
550C4—22no—
551C4—44no—
552C4—42no—
553C4—81no—
554C4—862no—
555C4—862no—
556C4—862no—
557C4—3441no—
558C4—1724no—
559C4—1722no—
560C4#1458305114yes—
561C4#1458355711yescharacteristic
562C4#1458356111yescharacteristic
563C4#1458356411yescharacteristic
564C4Q8×D17211yescharacteristic
565C2—864no—
566C2#1913795811yescharacteristic
567C2C2×#1458305111yescharacteristic
568C2#1914383111yescharacteristic
569C2#475×D4311yescharacteristic
570C2#1914440311yescharacteristic
571C2#1914573811yescharacteristic
572C2#1914582411yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1988C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
hypercenter#1988D43
hypocenterC43#24301
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C2×#14583185C2semidirect
3. C2×#14583181C2semidirect
5. #19135927C2semidirect
6. #19135735C2semidirect
7. #19135547C2semidirect
11. C43×#1988C2semidirect
65. #13063305Q8semidirect
152. C2×Dic86C2×Q8semidirect
162. C4⋊C4×C43C2×Q8semidirect
263. C2×D43#475semidirect
286. #1988D43semidirect
333. C43#24301semidirect
462. C4⋊C4Q8×D43semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D432
C2×#145831852
C2×#145831812
C4×#130633052
#191359272
#191357352
#191355472
#191349012
#191347892
#191345722
C43×#19882
#2430143