Finite Group Databasebeta

← #25107768

Subgroups — #25107768

Order 11008.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic43C211yesmaximal, characteristic, complemented
2C2×#14582802C211yesmaximal, characteristic
3#19135451C212yesmaximal, complemented
4#19135280C212yesmaximal, complemented
5#19135224C212yesmaximal, complemented
6#19134952C212yesmaximal, complemented
7#19134903C211yesmaximal, characteristic, complemented
8#19134790C211yesmaximal, characteristic, complemented
9#19134526C211yesmaximal, characteristic
10C22⋊C8×Dic43C211yesmaximal, characteristic, complemented
11C2×C43×#450C211yesmaximal, characteristic
12C2×Q8×Dic43C2211yescharacteristic, complemented
13C22×C4⋊Dic43C2211yescharacteristic
14C22×C22⋊Dic43—21no—
15C22×C22⋊Dic43—21no—
16C22×C4×Dic43C2211yescharacteristic
17C2×#13063257C2212yescomplemented
18C2×#13063249C2212yescomplemented
19C2×#13063246C2212yes—
20C2×#13063246C2212yes—
21C2×C22⋊(C43⋊C8)C2211yescharacteristic
22C2×C8×Dic43C2212yescomplemented
23#14582806C2211yescharacteristic, complemented
24#14582802—21no—
25#14582802—21no—
26#14582802C2212yes—
27#14582802C2212yes—
28#14582798C2211yescharacteristic, complemented
29#14582785—21no—
30#14582774—21no—
31#14582759C2211yescharacteristic
32C4⋊C4×Dic43—21no—
33C4⋊C4×Dic43C2211yescharacteristic, complemented
34C22⋊C4×Dic43—21no—
35#14582662C2212yescomplemented
36#14582661C2212yescomplemented
37#14582653C2212yescomplemented
38C23×D4×C43C2211yescharacteristic
39C22×SD16×C43C2212yes—
40C2×C22⋊Q8×C43C2211yescharacteristic
41C43×#450C418yescomplemented
42C2×#102×C43C2212yes—
43C2×C22⋊C8×C43C2211yescharacteristic
44C23×Dic43—21no—
45C23×Dic43D411yescharacteristic
46C23×Dic43D411yescharacteristic
47C22×C43⋊C8C2312yes—
48Q8×Dic43—42no—
49C2×C4⋊Dic43—21no—
50C2×C4⋊Dic43—21no—
51C2×C4⋊Dic43—21no—
52C2×C4⋊Dic43—42no—
53C2×C4⋊Dic43C2311yescharacteristic
54C2×C4⋊Dic43C2312yes—
55C2×C4⋊Dic43C2311yescharacteristic
56C2×C22⋊Dic43—21no—
57C2×C22⋊Dic43—22no—
58C2×C22⋊Dic43—22no—
59C2×C22⋊Dic43—22no—
60C2×C22⋊Dic43—22no—
61C2×C22⋊Dic43—44no—
62C2×C22⋊Dic43—44no—
63C2×C4×Dic43—21no—
64C2×C4×Dic43—21no—
65C2×C4×Dic43—42no—
66C2×C4×Dic43D411yescharacteristic
67C2×C4×Dic43D411yescharacteristic
68C2×C4×Dic43C2311yescharacteristic
69C2×C4×Dic43C2311yescharacteristic
70#13063257—42no—
71#13063249—42no—
72#13063246—42no—
73#13063246—42no—
74C22⋊(C43⋊C8)—22no—
75C8×Dic43—42no—
76C43⋊#95—21no—
77C43⋊#95—21no—
78C43⋊#95—21no—
79C43⋊#95—21no—
80C43⋊#95—21no—
81C25×C43—21no—
82C22×Q8×C43C2311yescharacteristic
83C22×D4×C43—21no—
84C22×D4×C43—22no—
85C22×D4×C43—42no—
86C22×D4×C43—42no—
87C22×D4×C43C2×C414yes—
88C22×D4×C43C2312yes—
89C23×C4×C43C2311yescharacteristic
90C2×SD16×C43—42no—
91C2×SD16×C43—42no—
92C2×SD16×C43C2×C418yescomplemented
93C22×C8×C43C2312yes—
94C22⋊Q8×C43—22no—
95C22⋊Q8×C43C2×C414yes—
96C2×C4⋊C4×C43—21no—
97C2×C4⋊C4×C43C2311yescharacteristic
98C2×C22⋊C4×C43—21no—
99#102×C43C2×C418yescomplemented
100C22⋊C8×C43C2×C414yes—
101C24×C43—24no—
102C24×C43—22no—
103C24×C43—42no—
104C24×C43—42no—
105C24×C43—42no—
106C24×C43—42no—
107C24×C43—42no—
108C24×C43—42no—
109C24×C43C2×D411yescharacteristic
110C2×Q8×C43—21no—
111C2×Q8×C43—22no—
112C2×Q8×C43—22no—
113C2×Q8×C43C22×C412yes—
114C2×D4×C43—24no—
115C2×D4×C43—22no—
116C2×D4×C43—22no—
117C2×D4×C43—22no—
118C2×D4×C43—22no—
119C2×D4×C43—22no—
120C2×D4×C43—48no—
121C2×D4×C43—48no—
122C2×D4×C43—42no—
123C2×D4×C43—42no—
124C2×D4×C43C22×C414yes—
125C22×C4×C43—22no—
126C22×C4×C43—22no—
127C22×C4×C43—21no—
128C22×C4×C43—21no—
129C22×C4×C43C22×C412yes—
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43#4112yes—
132C22×C4×C43C2411yescharacteristic
133C22×Dic43—22no—
134C22×Dic43—22no—
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—22no—
138C22×Dic43—41no—
139C22×Dic43—41no—
140C22×Dic43—41no—
141C22×Dic43—42no—
142C22×Dic43—42no—
143C22×Dic43—42no—
144C22×Dic43SD1614yescomplemented
145C22×Dic43C2×D411yescharacteristic
146C22×Dic43C2×D411yescharacteristic
147C22×Dic43C2×D411yescharacteristic
148C22×Dic43C2×D411yescharacteristic
149SD16×C43—48no—
150C2×C8×C43—22no—
151C2×C8×C43—22no—
152C2×C8×C43C22×C414yes—
153C4⋊C4×C43—24no—
154C4⋊C4×C43—21no—
155C4⋊C4×C43C22×C412yes—
156C22⋊C4×C43—24no—
157C22⋊Dic43—44no—
158C22⋊Dic43—44no—
159C22⋊Dic43—44no—
160C22⋊Dic43—44no—
161C22⋊Dic43—42no—
162C4⋊Dic43—22no—
163C4⋊Dic43—22no—
164C4⋊Dic43—22no—
165C4⋊Dic43—22no—
166C4⋊Dic43—42no—
167C4⋊Dic43—42no—
168C4⋊Dic43—42no—
169C4×Dic43—22no—
170C4×Dic43—22no—
171C4×Dic43—41no—
172C4×Dic43—41no—
173C4×Dic43—42no—
174C4×Dic43—42no—
175C2×C43⋊C8—22no—
176C2×C43⋊C8—22no—
177C2×C43⋊C8—22no—
178C23×C43—21no—
179C23×C43—21no—
180C23×C43—21no—
181C23×C43—21no—
182C23×C43—24no—
183C23×C43—48no—
184C23×C43—48no—
185C23×C43—42no—
186C23×C43—42no—
187C23×C43—42no—
188C23×C43—42no—
189C23×C43—42no—
190C23×C43—42no—
191C23×C43—48no—
192C23×C43—84no—
193C23×C43—84no—
194C23×C43—84no—
195C23×C43—84no—
196C23×C43C4×D412yes—
197C23×C43C2×SD1612yes—
198C23×C43#13512yes—
199C23×C43C22×D411yescharacteristic
200Q8×C43—24no—
201Q8×C43—42no—
202D4×C43—24no—
203D4×C43—24no—
204D4×C43—48no—
205D4×C43—42no—
206D4×C43—42no—
207C2×C4×C43—21no—
208C2×C4×C43—21no—
209C2×C4×C43—24no—
210C2×C4×C43—24no—
211C2×C4×C43—22no—
212C2×C4×C43—22no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43C4×D412yes—
218C2×C4×C43C23×C411yescharacteristic
219C2×C4×C43C2×#4111yescharacteristic
220C2×C4×C43H(5,𝔽2)12yes—
221C2×C4×C43H(5,𝔽2)12yes—
222C2×Dic43—21no—
223C2×Dic43—21no—
224C2×Dic43—21no—
225C2×Dic43—21no—
226C2×Dic43—21no—
227C2×Dic43—21no—
228C2×Dic43—21no—
229C2×Dic43—21no—
230C2×Dic43—21no—
231C2×Dic43—44no—
232C2×Dic43—44no—
233C2×Dic43—41no—
234C2×Dic43—42no—
235C2×Dic43—41no—
236C2×Dic43—42no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—42no—
241C2×Dic43—41no—
242C2×Dic43—84no—
243C2×Dic43C2×SD1612yes—
244C2×Dic43C2×SD1612yes—
245C2×Dic43AGL(1,ℤ/8)12yes—
246C8×C43—24no—
247C43⋊C8—42no—
248#22043—431nomaximal, Sylow
249C22×C43—22no—
250C22×C43—22no—
251C22×C43—22no—
252C22×C43—22no—
253C22×C43—22no—
254C22×C43—22no—
255C22×C43—44no—
256C22×C43—44no—
257C22×C43—44no—
258C22×C43—816no—
259C22×C43—82no—
260C22×C43—82no—
261C22×C43—82no—
262C22×C43—82no—
263C22×C43C4×SD1614yes—
264C22×C43C2×C4×D411yescharacteristic
265C22×C43#53411yescharacteristic
266C22×C43#53511yescharacteristic
267C22×C43C22×SD1611yescharacteristic
268C22×C43C2×#13511yescharacteristic
269C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C43#57511yescharacteristic
271C4×C43—22no—
272C4×C43—42no—
273C4×C43—42no—
274C4×C43#51812yes—
275C4×C43#53812yes—
276Dic43—22no—
277Dic43—22no—
278Dic43—22no—
279Dic43—41no—
280Dic43—81no—
281Dic43—82no—
282Dic43—81no—
283C2×#450D4311yescharacteristic
284C4×C22⋊Q8—431no—
285C2×#390—431no—
286#1968—432no—
287#1934—432no—
288#1926—432no—
289#1880—432no—
290#1872—431no—
291#1853—431no—
292#1786—431no—
293C4×C22⋊C8—431no—
294C2×C43—24no—
295C2×C43—84no—
296C2×C43#232111yescharacteristic
297C2×C43C2×C4×SD1611yescharacteristic
298C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
299C2×C43#318711yescharacteristic
300C2×C43#318811yescharacteristic
301C2×C43#320811yescharacteristic
302C2×C43#321111yescharacteristic
303C23×D4C2×D4311yescharacteristic
304C22×SD16C2×D4312yes—
305C2×C22⋊Q8C2×D4311yescharacteristic
306C2×C4×Q8—431no—
307C22×C4⋊C4—431no—
308C22×C22⋊C4—861no—
309C22×C22⋊C4—861no—
310C22×C42—431no—
311#450Dic4318yescomplemented
312C2×#106—432no—
313C2×#103—432no—
314C2×#102—432no—
315C2×#102—432no—
316C2×#102C2×D4312yes—
317C2×C22⋊C8—431no—
318C2×C22⋊C8C2×D4311yescharacteristic
319C2×C4×C8—432no—
320#391—431no—
321#390—432no—
322#390—432no—
323#390—861no—
324#390—861no—
325#389—431no—
326#387—861no—
327#385—861no—
328#380—431no—
329C4×C4⋊C4—431no—
330C4×C4⋊C4—861no—
331C4×C22⋊C4—861no—
332#340—432no—
333#340—432no—
334#336—432no—
335C43#2204311yescharacteristic, complemented, Sylow
336C25—21no—
337C22×Q8C22×D4311yescharacteristic
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C2×Dic4314yes—
343C22×D4C22×D4312yes—
344C23×C4—431no—
345C23×C4—431no—
346C23×C4—861no—
347C23×C4C22×D4311yescharacteristic
348C2×SD16—42no—
349C2×SD16—42no—
350C2×SD16C2×Dic4318yescomplemented
351C22×C8—432no—
352C22×C8C22×D4312yes—
353C22⋊Q8—22no—
354C22⋊Q8C2×Dic4314yes—
355C4×Q8—1722no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—432no—
358C2×C4⋊C4—431no—
359C2×C4⋊C4—431no—
360C2×C4⋊C4—861no—
361C2×C4⋊C4—861no—
362C2×C4⋊C4—861no—
363C2×C4⋊C4—1722no—
364C2×C4⋊C4C22×D4311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—862no—
367C2×C22⋊C4—862no—
368C2×C22⋊C4—862no—
369C2×C22⋊C4—862no—
370C2×C22⋊C4—861no—
371C2×C22⋊C4—1724no—
372C2×C22⋊C4—1724no—
373C2×C42—431no—
374C2×C42—431no—
375C2×C42—431no—
376C2×C42—431no—
377C2×C42—861no—
378C2×C42—861no—
379C2×C42—1722no—
380#106—1722no—
381#103—1722no—
382#102—1722no—
383#102—1722no—
384#102C2×Dic4318yescomplemented
385C22⋊C8—862no—
386C22⋊C8C2×Dic4314yes—
387C4×C8—1722no—
388#95—861no—
389#95—861no—
390#95—861no—
391#95—861no—
392#95—861no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24D4×D4311yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C22×Dic4312yes—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C22×Dic4314yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—431no—
422C22×C4—431no—
423C22×C4—431no—
424C22×C4—434no—
425C22×C4—431no—
426C22×C4—862no—
427C22×C4—862no—
428C22×C4—861no—
429C22×C4—861no—
430C22×C4—862no—
431C22×C4—1721no—
432C22×C4—1721no—
433C22×C4—1721no—
434C22×C4—1722no—
435C22×C4—1722no—
436C22×C4—1722no—
437C22×C4D4×D4311yescharacteristic
438C22×C4#1062190012yes—
439C22×C4C22×Dic4312yes—
440C22×C4C23×D4311yescharacteristic
441SD16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—862no—
445C2×C8—862no—
446C2×C8—862no—
447C2×C8C22×Dic4314yes—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—862no—
451C4⋊C4—862no—
452C4⋊C4—862no—
453C4⋊C4—862no—
454C4⋊C4—1722no—
455C4⋊C4—1722no—
456C4⋊C4—1722no—
457C4⋊C4C22×Dic4312yes—
458C22⋊C4—24no—
459C22⋊C4—1724no—
460C22⋊C4—1724no—
461C22⋊C4—1724no—
462C22⋊C4—1724no—
463C22⋊C4—1722no—
464C42—862no—
465C42—862no—
466C42—1721no—
467C42—1721no—
468C42—1722no—
469C42—1722no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23D4×Dic4312yes—
489C23SD16×D4312yes—
490C23#1306334612yes—
491C23C2×D4×D4311yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—432no—
510C2×C4—432no—
511C2×C4—432no—
512C2×C4—861no—
513C2×C4—861no—
514C2×C4—861no—
515C2×C4—861no—
516C2×C4—861no—
517C2×C4—861no—
518C2×C4—861no—
519C2×C4—861no—
520C2×C4—861no—
521C2×C4—3444no—
522C2×C4—1724no—
523C2×C4—1724no—
524C2×C4—1722no—
525C2×C4—1721no—
526C2×C4—1721no—
527C2×C4—1722no—
528C2×C4—1721no—
529C2×C4—1721no—
530C2×C4—1721no—
531C2×C4—1722no—
532C2×C4—1721no—
533C2×C4D4×Dic4312yes—
534C2×C4C23×Dic4311yescharacteristic
535C2×C4C2×#1062190011yescharacteristic
536C2×C4#1306337012yes—
537C2×C4#1306337012yes—
538C8—24no—
539C8—1722no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22SD16×Dic4314yes—
555C22C2×D4×Dic4311yescharacteristic
556C22#1458345911yescharacteristic
557C22#1458346411yescharacteristic
558C22C2×SD16×D4311yescharacteristic
559C22C2×#1306334611yescharacteristic
560C22AGL(1,ℤ/8)×D4311yescharacteristic
561C22#1458369511yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—862no—
566C4—862no—
567C4—862no—
568C4—3442no—
569C4—3441no—
570C4—3441no—
571C4—1721no—
572C4#1458337412yes—
573C4#1458348412yes—
574C2—24no—
575C2—84no—
576C2#1913725611yescharacteristic
577C2C2×SD16×Dic4311yescharacteristic
578C2AGL(1,ℤ/8)×Dic4311yescharacteristic
579C2#1914534911yescharacteristic
580C2#1914536311yescharacteristic
581C2#1914553411yescharacteristic
582C2#1914556311yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#450C2
socleC23×C43C22×D4
prime radicalC23×D4×C43C22
cosocleC22×C4×C43C24
hypercenterC2×#450D43
hypocenterC43#22043
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×Dic43C2semidirect
3. #19135451C2semidirect
4. #19135280C2semidirect
5. #19135224C2semidirect
6. #19134952C2semidirect
7. #19134903C2semidirect
8. #19134790C2semidirect
10. C22⋊C8×Dic43C2semidirect
12. C2×Q8×Dic43C22semidirect
17. C2×#13063257C22semidirect
18. C2×#13063249C22semidirect
22. C2×C8×Dic43C22semidirect
23. #14582806C22semidirect
28. #14582798C22semidirect
33. C4⋊C4×Dic43C22semidirect
35. #14582662C22semidirect
36. #14582661C22semidirect
37. #14582653C22semidirect
41. C43×#450C4semidirect
92. C2×SD16×C43C2×C4semidirect
99. #102×C43C2×C4semidirect
144. C22×Dic43SD16semidirect
311. #450Dic43semidirect
335. C43#22043semidirect
350. C2×SD16C2×Dic43semidirect
384. #102C2×Dic43semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic432
C2×#145828022
#191354512
#191352802
#191352242
#191349522
#191349032
#191347902
#191345262
C22⋊C8×Dic432
C2×C43×#4502
#2204343