Finite Group Databasebeta

← #25066586

Subgroups — #25066586

Order 11008.

700 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144651C211yesmaximal, characteristic
2C8×#10621900C211yesmaximal, characteristic, complemented
3#19143304C211yesmaximal, characteristic, complemented
4#19133475C211yesmaximal, characteristic, complemented
5#19133471C211yesmaximal, characteristic, complemented
6#19133459C211yesmaximal, characteristic, complemented
7#19133398C211yesmaximal, characteristic, complemented
8#19133394C211yesmaximal, characteristic, complemented
9D8×C43⋊C8C211yesmaximal, characteristic, complemented
10#19133387C211yesmaximal, characteristic, complemented
11#19133382C211yesmaximal, characteristic, complemented
12#19133031C211yesmaximal, characteristic, complemented
13#19133010C211yesmaximal, characteristic, complemented
14#19132874C211yesmaximal, characteristic, complemented
15C43×#1586C211yesmaximal, characteristic, complemented
16C2×#13063333—21no—
17#14583446C2211yescharacteristic
18C4×#10621900C2211yescharacteristic
19#14583226C411yescharacteristic
20#14583222C411yescharacteristic
21#14583193C411yescharacteristic
22#14583187C411yescharacteristic
23#14583098C411yescharacteristic
24#14583092C411yescharacteristic
25#14583045C411yescharacteristic
26#14583041C411yescharacteristic
27C8×Dic86C2211yescharacteristic
28#14583023C2211yescharacteristic
29C4×#10621873C2211yescharacteristic
30C4×#10621883C2211yescharacteristic
31C4×#10621882C2211yescharacteristic
32#14582978C2211yescharacteristic
33#14582975—21no—
34D4×C43⋊C8C2211yescharacteristic
35D4×C43⋊C8C2211yescharacteristic
36#14582964C2211yescharacteristic
37C8×C22⋊D43—21no—
38(C22⋊D43)⋊C8—21no—
39(C22⋊D43)⋊C8—21no—
40#14582951C2211yescharacteristic
41#14582950C2211yescharacteristic
42#14582949—21no—
43#14582921C2211yescharacteristic
44#14582864—21no—
45C4×#10621872C2211yescharacteristic
46C2×C8×Dic43—21no—
47C4×C8×D43C2211yescharacteristic
48#14582652C2211yescharacteristic
49#14582651C2211yescharacteristic
50#14582649C2211yescharacteristic
51#14582631C2211yescharacteristic
52#14582629C2211yescharacteristic
53#14582628C2211yescharacteristic
54D4⋊(C43⋊C8)C2211yescharacteristic
55D4⋊(C43⋊C8)C2211yescharacteristic
56#14582626C2211yescharacteristic
57#14582621C2211yescharacteristic
58C8×C43⋊C8C2211yescharacteristic
59C8×C43⋊C8C2211yescharacteristic
60C4×D8×C43C2211yescharacteristic
61C43×#436C2211yescharacteristic
62C8×D4×C43C2211yescharacteristic
63C43×#334C2211yescharacteristic
64C43×D4⋊C8C2211yescharacteristic
65C43×D4⋊C8C2211yescharacteristic
66C43×#322C2211yescharacteristic
67C2×#10621898—21no—
68C2×#10621900C2×C411yescharacteristic
69C2×#10621873C2×C411yescharacteristic
70C2×#10621883C2×C411yescharacteristic
71C2×#10621882C2×C411yescharacteristic
72#13063333—44no—
73C2×#10621872—21no—
74C2×#10621872—21no—
75C2×#10621872—21no—
76C22×C43⋊C8—21no—
77C22×C43⋊C8—21no—
78C2×C8×D43—21no—
79C2×C8×D43D411yescharacteristic
80C2×C8×D43D411yescharacteristic
81#13063324C2×C411yescharacteristic
82#13063321C2×C411yescharacteristic
83#13063311—21no—
84#13063300C2×C411yescharacteristic
85#13063298—21no—
86#13063296C2×C411yescharacteristic
87C4×Dic86C2311yescharacteristic
88#13063290C2×C411yescharacteristic
89C4×D172C2311yescharacteristic
90D4×Dic43C2×C411yescharacteristic
91C4×C22⋊D43—21no—
92C4×C22⋊D43—21no—
93(C22⋊D43)⋊C4—21no—
94#13063283C2×C411yescharacteristic
95#13063281—21no—
96C2×C4×Dic43—21no—
97C42×D43C2311yescharacteristic
98#13063258C2×C411yescharacteristic
99#13063257C2×C411yescharacteristic
100#13063256C2×C411yescharacteristic
101C4⋊(C43⋊C8)C2311yescharacteristic
102C4⋊(C43⋊C8)C2311yescharacteristic
103Dic43⋊C8—21no—
104Dic43⋊C8—21no—
105Dic43⋊C8—21no—
106Dic43⋊C8C2×C411yescharacteristic
107Dic43⋊C8C2×C411yescharacteristic
108#13063250C2×C411yescharacteristic
109#13063248C2×C411yescharacteristic
110#13063247C2×C411yescharacteristic
111#13063246C2×C411yescharacteristic
112#13063246C2×C411yescharacteristic
113#13063245C2×C411yescharacteristic
114C22⋊(C43⋊C8)—21no—
115C22⋊(C43⋊C8)—21no—
116(C2×D43)⋊C8—21no—
117(C2×D43)⋊C8—21no—
118(C2×D43)⋊C8—21no—
119(C2×D43)⋊C8C2×C411yescharacteristic
120(C2×D43)⋊C8C2×C411yescharacteristic
121#13063235—21no—
122C4×C43⋊C8C2311yescharacteristic
123C4×C43⋊C8C2311yescharacteristic
124C4×C43⋊C8C2311yescharacteristic
125C8×Dic43—21no—
126C8×Dic43—21no—
127C8×Dic43—21no—
128C8×Dic43D411yescharacteristic
129C8×Dic43D411yescharacteristic
130C2×D8×C43C2×C411yescharacteristic
131C2×M16×C43—21no—
132C22×C8×C43—21no—
133C4×D4×C43C2311yescharacteristic
134C4×D4×C43C2311yescharacteristic
135#107×C43C2×C411yescharacteristic
136C4⋊C8×C43C2311yescharacteristic
137C4⋊C8×C43C2311yescharacteristic
138#102×C43C2×C411yescharacteristic
139#102×C43C2×C411yescharacteristic
140C22⋊C8×C43—21no—
141C22⋊C8×C43—21no—
142C4×C8×C43C2311yescharacteristic
143C4×C8×C43C2311yescharacteristic
144C4×C8×C43C2311yescharacteristic
145C2×D4×C43C22×C411yescharacteristic
146C2×D4×C43C22×C411yescharacteristic
147C22×C4×C43—21no—
148C22×C4×C43—21no—
149C2×C22⋊D43—21no—
150C2×C22⋊D43—21no—
151C2×C22⋊D43—21no—
152C2×C22⋊D43—21no—
153C22×Dic43—21no—
154C22×Dic43—21no—
155#10621900—22no—
156#10621900—22no—
157#10621898—42no—
158#10621898—42no—
159C2×D172—21no—
160C2×D172C22×C411yescharacteristic
161C2×C4×D43—21no—
162C2×C4×D43C22×C411yescharacteristic
163C2×C4×D43C2×D411yescharacteristic
164C2×C4×D43#4111yescharacteristic
165C2×Dic86—21no—
166C2×Dic86C22×C411yescharacteristic
167D8×C43—22no—
168M16×C43—42no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—42no—
173C2×C8×C43C22×C411yescharacteristic
174C2×C8×C43C22×C411yescharacteristic
175C2×C8×C43C2×D411yescharacteristic
176C2×C8×C43C2×D411yescharacteristic
177C4⋊C4×C43C22×C411yescharacteristic
178C4⋊C4×C43C22×C411yescharacteristic
179C22⋊C4×C43—21no—
180C22⋊C4×C43—21no—
181C42×C43C2411yescharacteristic
182C22⋊Dic43—21no—
183C22⋊Dic43—21no—
184#10621883—22no—
185#10621882—22no—
186(C2×D43)⋊C4—21no—
187(C2×D43)⋊C4—21no—
188(C2×D43)⋊C4—21no—
189C4⋊Dic43C22×C411yescharacteristic
190C4⋊Dic43C22×C411yescharacteristic
191Dic43⋊C4—21no—
192Dic43⋊C4—21no—
193Dic43⋊C4—21no—
194C4×Dic43—21no—
195C4×Dic43—21no—
196C4×Dic43—21no—
197C4×Dic43C22×C411yescharacteristic
198C4×Dic43C2×D411yescharacteristic
199C4×Dic43#4111yescharacteristic
200C2×C43⋊C8—21no—
201C2×C43⋊C8—21no—
202C2×C43⋊C8—21no—
203C2×C43⋊C8—42no—
204C2×C43⋊C8—42no—
205C2×C43⋊C8C22×C411yescharacteristic
206C2×C43⋊C8C22×C411yescharacteristic
207C2×C43⋊C8C2×D411yescharacteristic
208C2×C43⋊C8C2×D411yescharacteristic
209#10621873—22no—
210#10621872—24no—
211#10621872—42no—
212#10621872—42no—
213C8×D43—22no—
214C8×D43—41no—
215C8×D43—42no—
216C23×C43—21no—
217C23×C43—21no—
218C22×D43—21no—
219C22×D43C4×D411yescharacteristic
220D4×C43—21no—
221D4×C43—21no—
222D4×C43—21no—
223D4×C43#13112yescomplemented
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43C23×C411yescharacteristic
231C2×C4×C43C22×D411yescharacteristic
232C2×C4×C43C2×#4111yescharacteristic
233C22⋊D43—24no—
234C22⋊D43—42no—
235C22⋊D43—42no—
236C22⋊D43—42no—
237C2×Dic43—24no—
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—21no—
241C2×Dic43—42no—
242C2×Dic43C4×D411yescharacteristic
243C2×Dic43C4×D411yescharacteristic
244C2×Dic43C4×D411yescharacteristic
245D172—21no—
246D172—21no—
247D172—42no—
248C4×D43—21no—
249C4×D43—21no—
250C4×D43—21no—
251C4×D43—21no—
252C4×D43—22no—
253C4×D43—41no—
254C4×D43—41no—
255Dic86—21no—
256Dic86—42no—
257Dic86#13112yescomplemented
258C8×C43—22no—
259C8×C43—21no—
260C8×C43—21no—
261C8×C43—21no—
262C8×C43—21no—
263C8×C43—41no—
264C8×C43—41no—
265C43⋊C8—22no—
266C43⋊C8—21no—
267C43⋊C8—21no—
268C43⋊C8—21no—
269C43⋊C8—41no—
270C43⋊C8—41no—
271C43⋊C8AGL(1,ℤ/8)12yes—
272#17912—431nomaximal, Sylow
273C22×C43—22no—
274C22×C43—41no—
275C22×C43—41no—
276C22×C43—41no—
277C22×C43—41no—
278C22×C43—41no—
279C22×C43C2×C4×D411yescharacteristic
280C2×D43—21no—
281C2×D43—21no—
282C2×D43—21no—
283C2×D43—41no—
284C2×D43—41no—
285C2×D43—41no—
286C4×C43—21no—
287C4×C43—41no—
288C4×C43—41no—
289C4×C43C2×#13111yescharacteristic
290C4×C43C2×#13111yescharacteristic
291C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C43#57511yescharacteristic
293Dic43—22no—
294Dic43—21no—
295Dic43—21no—
296Dic43—41no—
297Dic43—41no—
298Dic43#44312yes—
299#3106—431no—
300C8×#41—431no—
301#2931—431no—
302#1588—431no—
303#1588—431no—
304#1586—431no—
305#1586D4311yescharacteristic, complemented
306C8×SD16—431no—
307C8×SD16—431no—
308C8×D8—431no—
309C8×D8—431no—
310#1574—431no—
311#1489—431no—
312#1484—431no—
313C8×M16—431no—
314C2×C43—42no—
315C2×C43—81no—
316C2×C43#292911yescharacteristic
317C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C43C2×#44311yescharacteristic
319D43—41no—
320D43—81no—
321C2×#131—861no—
322#532—431no—
323C4×#41—431no—
324#482—431no—
325#482—431no—
326#477—431no—
327#476—431no—
328#460—431no—
329#459—431no—
330#449—431no—
331#449—431no—
332C8×Q8—431no—
333C8×Q8—431no—
334C4×SD16—431no—
335C4×SD16—431no—
336C4×D8—431no—
337C4×D8C2×D4311yescharacteristic
338#436—431no—
339#436—861no—
340#436C2×D4311yescharacteristic
341C8×D4—431no—
342C8×D4—431no—
343C8×D4—431no—
344C8×D4—861no—
345C8×D4—861no—
346C8×D4—861no—
347C8×D4C2×D4311yescharacteristic
348#431—431no—
349#431—431no—
350#431—861no—
351#423—431no—
352#408—861no—
353C4×M16—431no—
354C2×C4×C8—431no—
355C2×C4×C8—861no—
356#335—431no—
357#335—431no—
358#334—431no—
359#334C2×D4311yescharacteristic
360Q8⋊C8—431no—
361Q8⋊C8—431no—
362D4⋊C8—431no—
363D4⋊C8—431no—
364D4⋊C8—431no—
365D4⋊C8—431no—
366D4⋊C8C2×D4311yescharacteristic
367D4⋊C8C2×D4311yescharacteristic
368#322—431no—
369#322C2×D4311yescharacteristic
370C82—431no—
371C82—431no—
372C43#1791211yescharacteristic, complemented, Sylow
373C2×#41—431no—
374C2×#41—861no—
375C2×SD16—431no—
376C2×SD16—431no—
377C2×D8—431no—
378C2×D8C4×D4311yescharacteristic
379#131—1724no—
380C2×M16—21no—
381C2×M16—861no—
382C2×M16—861no—
383C2×M16—861no—
384C22×C8—21no—
385C22×C8—431no—
386C22×C8—431no—
387C22×C8—861no—
388C22×C8—861no—
389C22×C8—861no—
390C4⋊Q8—431no—
391#127—431no—
392#124—861no—
393#121—431no—
394#121—431no—
395#121—861no—
396C4×Q8—431no—
397C4×Q8—431no—
398C4×D4—431no—
399C4×D4—431no—
400C4×D4—861no—
401C4×D4—861no—
402C4×D4—861no—
403C4×D4C22×D4311yescharacteristic
404C4×D4C22×D4311yescharacteristic
405#117—431no—
406#117—861no—
407C2×C42—431no—
408C2×C42—861no—
409#107—431no—
410#107C4×D4311yescharacteristic
411#106—431no—
412#106—431no—
413C4⋊C8—431no—
414C4⋊C8—431no—
415C4⋊C8—431no—
416C4⋊C8—431no—
417C4⋊C8—861no—
418C4⋊C8—861no—
419C4⋊C8—861no—
420C4⋊C8C22×D4311yescharacteristic
421C4⋊C8C22×D4311yescharacteristic
422#103—431no—
423#103—431no—
424#102—431no—
425#102—431no—
426#102—431no—
427#102—431no—
428#102C4×D4311yescharacteristic
429#102C4×D4311yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—431no—
433C22⋊C8—431no—
434C22⋊C8—861no—
435C22⋊C8—861no—
436C22⋊C8—861no—
437C22⋊C8—861no—
438C22⋊C8—861no—
439#97—861no—
440C4×C8—431no—
441C4×C8—431no—
442C4×C8—431no—
443C4×C8—431no—
444C4×C8—431no—
445C4×C8—861no—
446C4×C8—861no—
447C4×C8—861no—
448C4×C8C22×D4311yescharacteristic
449C4×C8C22×D4311yescharacteristic
450C4×C8C22×D4311yescharacteristic
451#41—862no—
452#41—862no—
453#41—1722no—
454#41—1722no—
455C2×Q8—431no—
456C2×Q8—861no—
457C2×D4—431no—
458C2×D4—861no—
459C2×D4—861no—
460C2×D4—861no—
461C2×D4—861no—
462C2×D4—861no—
463C2×D4C2×C4×D4311yescharacteristic
464C2×D4C2×C4×D4311yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—431no—
468C22×C4—431no—
469C22×C4—431no—
470C22×C4—861no—
471C22×C4—861no—
472C22×C4—861no—
473SD16—862no—
474SD16—862no—
475D8—22no—
476D8—862no—
477M16—42no—
478M16—864no—
479M16—1722no—
480M16—1722no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—431no—
486C2×C8—431no—
487C2×C8—431no—
488C2×C8—431no—
489C2×C8—862no—
490C2×C8—861no—
491C2×C8—861no—
492C2×C8—861no—
493C2×C8—1721no—
494C2×C8—1722no—
495C2×C8—1722no—
496C2×C8—1722no—
497C2×C8C2×C4×D4311yescharacteristic
498C2×C8C2×C4×D4311yescharacteristic
499C2×C8D4×D4311yescharacteristic
500C2×C8D4×D4311yescharacteristic
501C4⋊C4—431no—
502C4⋊C4—431no—
503C4⋊C4—861no—
504C4⋊C4—861no—
505C4⋊C4—861no—
506C4⋊C4C2×C4×D4311yescharacteristic
507C4⋊C4C2×C4×D4311yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—861no—
511C22⋊C4—861no—
512C22⋊C4—861no—
513C22⋊C4—861no—
514C22⋊C4—861no—
515C42—431no—
516C42—431no—
517C42—431no—
518C42—861no—
519C42—861no—
520C42—861no—
521C42C23×D4311yescharacteristic
522C23—21no—
523C23—21no—
524C23—431no—
525C23—861no—
526Q8—432no—
527Q8—861no—
528Q8—1722no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—864no—
533D4—861no—
534D4—861no—
535D4—1722no—
536D4—1722no—
537D4—1722no—
538D4—1722no—
539D4#1306333312yescomplemented
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—431no—
547C2×C4—431no—
548C2×C4—431no—
549C2×C4—862no—
550C2×C4—864no—
551C2×C4—861no—
552C2×C4—861no—
553C2×C4—861no—
554C2×C4—861no—
555C2×C4—861no—
556C2×C4—861no—
557C2×C4—861no—
558C2×C4—1722no—
559C2×C4—1721no—
560C2×C4—1721no—
561C2×C4C22×C4×D4311yescharacteristic
562C2×C4C2×D4×D4311yescharacteristic
563C2×C4#41×D4311yescharacteristic
564C8—22no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—432no—
572C8—862no—
573C8—861no—
574C8—861no—
575C8—861no—
576C8—1721no—
577C8—1721no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—861no—
585C22—861no—
586C22—861no—
587C22—1721no—
588C22—1721no—
589C22—1721no—
590C22C4×D4×D4311yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—432no—
595C4—861no—
596C4—862no—
597C4—861no—
598C4—1721no—
599C4—1721no—
600C4#131×D4311yescharacteristic
601C4C2×#1306333311yescharacteristic
602C4AGL(1,ℤ/8)×D4311yescharacteristic
603C4#1458369511yescharacteristic
604C2—42no—
605C2—81no—
606C2—3441no—
607C2—1721no—
608C2#1914328611yescharacteristic
609C2#1914340611yescharacteristic
610C2#443×D4311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC42C23×D43
FittingC43×#1586C2
socleC22×C43C2×C4×D4
prime radical#19144651C2
cosocleC42×C43C24
hypercenter#1586D43
hypocenterC43#17912
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. C8×#10621900C2semidirect
3. #19143304C2semidirect
4. #19133475C2semidirect
5. #19133471C2semidirect
6. #19133459C2semidirect
7. #19133398C2semidirect
8. #19133394C2semidirect
9. D8×C43⋊C8C2semidirect
10. #19133387C2semidirect
11. #19133382C2semidirect
12. #19133031C2semidirect
13. #19133010C2semidirect
14. #19132874C2semidirect
15. C43×#1586C2semidirect
223. D4×C43#131semidirect
257. Dic86#131semidirect
305. #1586D43semidirect
372. C43#17912semidirect
539. D4#13063333semidirect

Maximal subgroups

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

subgroupindex
#191446512
C8×#106219002
#191433042
#191334752
#191334712
#191334592
#191333982
#191333942
D8×C43⋊C82
#191333872
#191333822
#191330312
#191330102
#191328742
C43×#15862
#1791243