Finite Group Databasebeta

← #25033365

Subgroups — #25033365

Order 11008.

1710 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582828C214yesmaximal
2C2×#14582768C211yesmaximal, characteristic, complemented
3C2×#14582756C211yesmaximal, characteristic
4C2×#14582756C211yesmaximal, characteristic
5C2×C4×(C2×D43)⋊C4C211yesmaximal, characteristic
6#19132861C214yesmaximal, complemented
7#19132825C211yesmaximal, characteristic, complemented
8#19132824C211yesmaximal, characteristic, complemented
9C43×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D43C2211yescharacteristic
11C22×C4⋊Dic43C2211yescharacteristic
12C22×C4⋊Dic43C2211yescharacteristic
13C22×Dic43⋊C4C2214yes—
14C22×(C2×D43)⋊C4C2212yes—
15C22×(C2×D43)⋊C4C2212yes—
16C22×(C2×D43)⋊C4C2212yes—
17C22×C4×Dic43C2211yescharacteristic
18#14582828—216no—
19#14582768—24no—
20#14582756—24no—
21#14582756—24no—
22C4×(C2×D43)⋊C4—24no—
23C2×C43⋊#95C2214yes—
24C2×C43⋊#95C2214yes—
25C2×C43⋊#95C2212yes—
26C2×C43⋊#95C2212yes—
27C2×C43⋊#95C2212yes—
28C22×C4⋊C4×C43C2211yescharacteristic
29C22×C4⋊C4×C43C2211yescharacteristic
30C22×C42×C43C2211yescharacteristic
31C2×#95×C43C2214yes—
32C24×D43C2311yescharacteristic
33C23×Dic43C2312yes—
34C23×Dic43C2312yes—
35C23×Dic43C2311yescharacteristic
36C23×Dic43C2312yes—
37C22×C4×D43—22no—
38C22×C4×D43—22no—
39C22×C4×D43—22no—
40C22×C4×D43—22no—
41C22×C4×D43—22no—
42C22×C4×D43—22no—
43C22×C4×D43D414yes—
44C2×C4⋊Dic43—22no—
45C2×C4⋊Dic43—22no—
46C2×C4⋊Dic43—22no—
47C2×C4⋊Dic43—22no—
48C2×C4⋊Dic43—22no—
49C2×C4⋊Dic43—22no—
50C2×Dic43⋊C4—28no—
51C2×Dic43⋊C4—28no—
52C2×Dic43⋊C4—28no—
53C2×(C2×D43)⋊C4—24no—
54C2×(C2×D43)⋊C4—24no—
55C2×(C2×D43)⋊C4—28no—
56C2×(C2×D43)⋊C4—28no—
57C2×(C2×D43)⋊C4—24no—
58C2×(C2×D43)⋊C4—24no—
59C2×(C2×D43)⋊C4D418yes—
60C2×C4×Dic43—22no—
61C2×C4×Dic43—22no—
62C2×C4×Dic43D414yes—
63C43⋊#95—28no—
64C43⋊#95—28no—
65C43⋊#95—24no—
66C43⋊#95—24no—
67C43⋊#95—24no—
68C23×C4×C43C2312yes—
69C23×C4×C43C2312yes—
70C23×C4×C43C2312yes—
71C23×C4×C43C2311yescharacteristic
72C2×C4⋊C4×C43—22no—
73C2×C4⋊C4×C43—22no—
74C2×C4⋊C4×C43—22no—
75C2×C4⋊C4×C43—22no—
76C2×C4⋊C4×C43—22no—
77C2×C4⋊C4×C43—22no—
78C2×C42×C43—22no—
79C2×C42×C43—22no—
80C2×C42×C43—22no—
81#95×C43—28no—
82C24×C43C2411yescharacteristic
83C23×D43—21no—
84C23×D43—22no—
85C23×D43—21no—
86C23×D43—21no—
87C23×D43—21no—
88C23×D43—21no—
89C23×D43—21no—
90C23×D43—22no—
91C23×D43—22no—
92C23×D43C2×D414yes—
93C23×D43C2×D412yes—
94C22×C4×C43—22no—
95C22×C4×C43—24no—
96C22×C4×C43—24no—
97C22×C4×C43—22no—
98C22×C4×C43—22no—
99C22×C4×C43—22no—
100C22×C4×C43—22no—
101C22×C4×C43—22no—
102C22×C4×C43—22no—
103C22×C4×C43—22no—
104C22×C4×C43—22no—
105C22×C4×C43—22no—
106C22×C4×C43—24no—
107C22×C4×C43—22no—
108C22×C4×C43—22no—
109C22×C4×C43—24no—
110C22×C4×C43C2×D414yes—
111C22×C4×C43C2×D412yes—
112C22×C4×C43C2×D412yes—
113C22×C4×C43#4112yes—
114C22×C4×C43#4112yes—
115C22×C4×C43#4112yes—
116C22×C4×C43#4112yes—
117C22×C4×C43#4112yes—
118C22×Dic43—21no—
119C22×Dic43—24no—
120C22×Dic43—24no—
121C22×Dic43—22no—
122C22×Dic43—22no—
123C22×Dic43—22no—
124C22×Dic43—22no—
125C22×Dic43—22no—
126C22×Dic43—21no—
127C22×Dic43—21no—
128C22×Dic43—21no—
129C22×Dic43—21no—
130C22×Dic43—21no—
131C22×Dic43—24no—
132C22×Dic43—22no—
133C22×Dic43—22no—
134C22×Dic43—22no—
135C22×Dic43—22no—
136C22×Dic43—22no—
137C22×Dic43—24no—
138C22×Dic43C2×D412yes—
139C22×Dic43C2×D414yes—
140C22×Dic43C2×D414yes—
141C22×Dic43#4114yes—
142C2×C4×D43—28no—
143C2×C4×D43—28no—
144C2×C4×D43—28no—
145C2×C4×D43—44no—
146C2×C4×D43—44no—
147C2×C4×D43—44no—
148C2×C4×D43—44no—
149C4⋊C4×C43—44no—
150C4⋊C4×C43—44no—
151C42×C43—44no—
152(C2×D43)⋊C4—216no—
153(C2×D43)⋊C4—48no—
154(C2×D43)⋊C4—48no—
155C4⋊Dic43—44no—
156C4⋊Dic43—44no—
157Dic43⋊C4—416no—
158C4×Dic43—28no—
159C23×C43C22×D411yescharacteristic
160C23×C43C22×D411yescharacteristic
161C23×C43C22×D412yes—
162C23×C43C2×#4112yes—
163C23×C43C2×#4111yescharacteristic
164C23×C43C2×#4111yescharacteristic
165C23×C43C2×#4111yescharacteristic
166C23×C43C2×#4111yescharacteristic
167C23×C43C2×#4111yescharacteristic
168C23×C432−1+412yes—
169C23×C432−1+412yes—
170C22×D43—24no—
171C22×D43—22no—
172C22×D43—22no—
173C22×D43—22no—
174C22×D43—24no—
175C22×D43—22no—
176C22×D43—22no—
177C22×D43—22no—
178C22×D43—24no—
179C22×D43—24no—
180C22×D43—24no—
181C22×D43—24no—
182C22×D43—41no—
183C22×D43—41no—
184C22×D43—41no—
185C22×D43—41no—
186C22×D43—42no—
187C22×D43—42no—
188C22×D43—42no—
189C22×D43—42no—
190C22×D43—42no—
191C22×D43—42no—
192C22×D43#12714yes—
193C2×C4×C43—24no—
194C2×C4×C43—24no—
195C2×C4×C43—24no—
196C2×C4×C43—42no—
197C2×C4×C43—42no—
198C2×C4×C43—42no—
199C2×C4×C43—42no—
200C2×C4×C43—44no—
201C2×C4×C43—42no—
202C2×C4×C43—42no—
203C2×C4×C43—42no—
204C2×C4×C43—42no—
205C2×C4×C43—44no—
206C2×C4×C43—42no—
207C2×C4×C43—42no—
208C2×C4×C43—44no—
209C2×C4×C43—44no—
210C2×C4×C43#12114yes—
211C2×C4×C43#12314yes—
212C2×C4×C43#12314yes—
213C2×C4×C43#12314yes—
214C2×C4×C43#12414yes—
215C2×C4×C43#12614yes—
216C2×C4×C43#12614yes—
217C2×Dic43—24no—
218C2×Dic43—24no—
219C2×Dic43—24no—
220C2×Dic43—24no—
221C2×Dic43—24no—
222C2×Dic43—24no—
223C2×Dic43—22no—
224C2×Dic43—22no—
225C2×Dic43—22no—
226C2×Dic43—41no—
227C2×Dic43—41no—
228C2×Dic43—41no—
229C2×Dic43—41no—
230C2×Dic43—44no—
231C2×Dic43—42no—
232C2×Dic43—42no—
233C2×Dic43—42no—
234C2×Dic43—42no—
235C2×Dic43—44no—
236C2×Dic43—42no—
237C2×Dic43—42no—
238C2×Dic43—44no—
239C2×Dic43—44no—
240C2×Dic43#12118yes—
241C4×D43—416no—
242#15150—431nomaximal, Sylow
243C22×C43C2×#12112yes—
244C22×C43C2×#12111yescharacteristic
245C22×C43C2×#12311yescharacteristic
246C22×C43C2×#12311yescharacteristic
247C22×C43C2×#12311yescharacteristic
248C22×C43C2×#12411yescharacteristic
249C22×C43C2×#12611yescharacteristic
250C22×C43C2×#12611yescharacteristic
251C22×C43C2×#12711yescharacteristic
252C22×C43#53212yes—
253C22×C43#53211yescharacteristic
254C22×C43#53611yescharacteristic
255C22×C43#53612yes—
256C22×C43#53611yescharacteristic
257C22×C43#54112yes—
258C22×C43#54712yes—
259C22×C43#54712yes—
260C22×C43#54712yes—
261C22×C43#54712yes—
262C22×C43#55212yes—
263C22×C43#55212yes—
264C22×C43#55612yes—
265C22×C43#55612yes—
266C2×D43—24no—
267C2×D43—24no—
268C2×D43—24no—
269C2×D43—42no—
270C2×D43—42no—
271C2×D43—44no—
272C2×D43—44no—
273C2×D43—44no—
274C2×D43—42no—
275C2×D43—42no—
276C2×D43—44no—
277C4×C43—44no—
278C4×C43—82no—
279C4×C43—82no—
280C4×C43#39718yescomplemented
281Dic43—28no—
282Dic43—42no—
283Dic43—82no—
284Dic43—82no—
285C2×#397—434no—
286C2×#384—431no—
287C2×#380—431no—
288C2×#380—431no—
289C2×C4×C22⋊C4—431no—
290#1445—434no—
291C4⋊#95—431no—
292C4⋊#95—431no—
293C4⋊#95D4311yescharacteristic, complemented
294C2×C43C2×#39711yescharacteristic
295C2×C43#248612yes—
296C2×C43#249612yes—
297C2×C43#249612yes—
298C2×C43#249911yescharacteristic
299C2×C43#249911yescharacteristic
300C2×C43#250111yescharacteristic
301C2×C43#250111yescharacteristic
302C2×C43#254712yes—
303C2×C43#262111yescharacteristic
304C2×C43#285611yescharacteristic
305D43—44no—
306C24×C4—431no—
307C22×C4⋊C4—434no—
308C22×C4⋊C4—431no—
309C22×C4⋊C4—431no—
310C22×C4⋊C4C2×D4311yescharacteristic
311C22×C4⋊C4C2×D4311yescharacteristic
312C22×C22⋊C4—432no—
313C22×C22⋊C4—432no—
314C22×C22⋊C4—432no—
315C22×C42—431no—
316C22×C42C2×D4311yescharacteristic
317#397—8616no—
318#384—864no—
319#380—864no—
320#380—864no—
321C4×C22⋊C4—864no—
322C2×#95—434no—
323C2×#95—434no—
324C2×#95—432no—
325C2×#95—432no—
326C2×#95—432no—
327C2×#95C2×D4314yes—
328C43#1515011yescharacteristic, complemented, Sylow
329C25—431no—
330C23×C4—432no—
331C23×C4—431no—
332C23×C4—434no—
333C23×C4—432no—
334C23×C4—432no—
335C23×C4—862no—
336C23×C4—862no—
337C23×C4—862no—
338C23×C4—862no—
339C23×C4—862no—
340C23×C4—862no—
341C23×C4C22×D4312yes—
342C23×C4C22×D4312yes—
343C23×C4C22×D4312yes—
344C23×C4C22×D4311yescharacteristic
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—868no—
352C2×C4⋊C4—868no—
353C2×C4⋊C4—868no—
354C2×C4⋊C4—862no—
355C2×C4⋊C4—862no—
356C2×C4⋊C4—862no—
357C2×C4⋊C4—862no—
358C2×C4⋊C4—862no—
359C2×C4⋊C4—862no—
360C2×C22⋊C4—438no—
361C2×C22⋊C4—864no—
362C2×C22⋊C4—864no—
363C2×C22⋊C4—864no—
364C2×C22⋊C4—864no—
365C2×C22⋊C4—868no—
366C2×C22⋊C4—868no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—434no—
371C2×C42—862no—
372C2×C42—862no—
373#95—28no—
374#95—864no—
375#95—864no—
376#95—864no—
377#95—868no—
378#95—868no—
379C24—434no—
380C24—432no—
381C24—861no—
382C24—862no—
383C24—861no—
384C24—861no—
385C24—861no—
386C24—861no—
387C24—861no—
388C24—862no—
389C24—862no—
390C24C23×D4311yescharacteristic
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—434no—
408C22×C4—434no—
409C22×C4—434no—
410C22×C4—432no—
411C22×C4—862no—
412C22×C4—862no—
413C22×C4—862no—
414C22×C4—862no—
415C22×C4—861no—
416C22×C4—861no—
417C22×C4—861no—
418C22×C4—861no—
419C22×C4—861no—
420C22×C4—861no—
421C22×C4—868no—
422C22×C4—868no—
423C22×C4—868no—
424C22×C4—864no—
425C22×C4—864no—
426C22×C4—862no—
427C22×C4—862no—
428C22×C4—862no—
429C22×C4—864no—
430C22×C4—862no—
431C22×C4—862no—
432C22×C4—862no—
433C22×C4—864no—
434C22×C4—1724no—
435C22×C4—1724no—
436C22×C4—1724no—
437C22×C4—1724no—
438C22×C4#1062189812yes—
439C22×C4#1062189812yes—
440C22×C4#1062189812yes—
441C22×C4D4×D4314yes—
442C22×C4D4×D4312yes—
443C22×C4D4×D4312yes—
444C22×C4#1062190012yes—
445C22×C4#1062190212yes—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—17216no—
449C4⋊C4—1724no—
450C4⋊C4—1724no—
451C22⋊C4—8616no—
452C22⋊C4—1728no—
453C22⋊C4—1728no—
454C42—44no—
455C42—868no—
456C23—434no—
457C23—864no—
458C23—864no—
459C23—864no—
460C23—862no—
461C23—862no—
462C23—862no—
463C23—864no—
464C23—862no—
465C23—862no—
466C23—862no—
467C23—864no—
468C23—864no—
469C23—1722no—
470C23—1722no—
471C23—1722no—
472C23—1722no—
473C23—1721no—
474C23—1721no—
475C23—1721no—
476C23—1721no—
477C23—1722no—
478C23—1722no—
479C23C2×D4×D4312yes—
480C23C2×D4×D4311yescharacteristic
481C23C2×D4×D4311yescharacteristic
482C23#41×D4312yes—
483C23C2×#1062190011yescharacteristic
484C23C2×#1062189811yescharacteristic
485C23C2×#1062189811yescharacteristic
486C23C2×#1062189811yescharacteristic
487C23C2×#1062190211yescharacteristic
488C23#1306337212yes—
489C23#1306337312yes—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—438no—
508C2×C4—864no—
509C2×C4—864no—
510C2×C4—864no—
511C2×C4—864no—
512C2×C4—864no—
513C2×C4—864no—
514C2×C4—862no—
515C2×C4—862no—
516C2×C4—862no—
517C2×C4—1724no—
518C2×C4—1724no—
519C2×C4—1722no—
520C2×C4—1722no—
521C2×C4—1722no—
522C2×C4—1722no—
523C2×C4—1724no—
524C2×C4—1722no—
525C2×C4—1722no—
526C2×C4—1724no—
527C2×C4—1721no—
528C2×C4—1721no—
529C2×C4—1721no—
530C2×C4—1721no—
531C2×C4—17216no—
532C2×C4#1306329814yes—
533C2×C4#1306330714yes—
534C2×C4#1306331014yes—
535C2×C4#1306331014yes—
536C2×C4#1306331114yes—
537C2×C4#1306331914yes—
538C2×C4#1306332014yes—
539C22—864no—
540C22—864no—
541C22—864no—
542C22—1724no—
543C22—1722no—
544C22—1722no—
545C22—1724no—
546C22—1724no—
547C22—1724no—
548C22—1722no—
549C22—1722no—
550C22#121×D4312yes—
551C22C2×#1306329811yescharacteristic
552C22C2×#1306330711yescharacteristic
553C22C2×#1306331011yescharacteristic
554C22C2×#1306331011yescharacteristic
555C22C2×#1306331111yescharacteristic
556C22C2×#1306331911yescharacteristic
557C22C2×#1306332011yescharacteristic
558C22#127×D4311yescharacteristic
559C22#1458344612yes—
560C22#1458344811yescharacteristic
561C22#1458346911yescharacteristic
562C22#1458347212yes—
563C22#1458347411yescharacteristic
564C22#1458350712yes—
565C22#1458354912yes—
566C22#1458355012yes—
567C22#1458355212yes—
568C22#1458355412yes—
569C22#1458358712yes—
570C22#1458359012yes—
571C22#1458361712yes—
572C22#1458362012yes—
573C4—44no—
574C4—82no—
575C4—82no—
576C4—868no—
577C4—3442no—
578C4—3442no—
579C4—1722no—
580C4#1458282818yescomplemented
581C2—1724no—
582C2C2×#1458282811yescharacteristic
583C2#1913890112yes—
584C2#1913901112yes—
585C2#1913901812yes—
586C2#1913904711yescharacteristic
587C2#1913905111yescharacteristic
588C2#1913906311yescharacteristic
589C2#1913906511yescharacteristic
590C2#1913958912yes—
591C2#1914036611yescharacteristic
592C2#1914293911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D43
commutator, socle, cosocleC24×C43C24
FittingC43×C4⋊#95C2
prime radicalC24×D43C23
hypercenterC4⋊#95D43
hypocenterC43#15150
LCSG ▷ C24×C43 ▷ C43
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#14582768C2semidirect
6. #19132861C2semidirect
7. #19132825C2semidirect
8. #19132824C2semidirect
9. C43×C4⋊#95C2semidirect
280. C4×C43#397semidirect
293. C4⋊#95D43semidirect
328. C43#15150semidirect
580. C4#14582828semidirect

Maximal subgroups

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

subgroupindex
C2×#145828282
C2×#145827682
C2×#145827562
C2×C4×(C2×D43)⋊C42
#191328612
#191328252
#191328242
C43×C4⋊#952
#1515043