Finite Group Databasebeta

← #25099630

Subgroups — #25099630

Order 11008.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583012C211yesmaximal, characteristic
2#19137059C211yesmaximal, characteristic, complemented
3C42×D172C211yesmaximal, characteristic, complemented
4#19135223C211yesmaximal, characteristic, complemented
5#19135216C211yesmaximal, characteristic, complemented
6#19135181C211yesmaximal, characteristic, complemented
7#19135174C211yesmaximal, characteristic, complemented
8#19134942C211yesmaximal, characteristic
9#19134942C211yesmaximal, characteristic
10#19134859C211yesmaximal, characteristic, complemented
11#19134845C211yesmaximal, characteristic, complemented
12#19134492C211yesmaximal, characteristic, complemented
13#19134421C211yesmaximal, characteristic, complemented
14#19134421C211yesmaximal, characteristic, complemented
15C43×#1745C211yesmaximal, characteristic, complemented
16C22×D344C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18C2×C4×D172C2211yescharacteristic
19C2×C4×D172C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21C2×#13063282—21no—
22C2×C42×D43—21no—
23#14583012C418yescomplemented
24C2×#13063259C2211yescharacteristic
25C2×#13063257C2211yescharacteristic
26C2×#13063257C2211yescharacteristic
27C2×#13063247C2211yescharacteristic
28C2×#13063247C2211yescharacteristic
29C2×#13063247C2211yescharacteristic
30C2×#13063247C2211yescharacteristic
31C2×#13063247C2211yescharacteristic
32C2×#13063247C2211yescharacteristic
33#14582801C2211yescharacteristic
34#14582801C2211yescharacteristic
35#14582797C2211yescharacteristic
36#14582797C2211yescharacteristic
37#14582790—21no—
38#14582790—21no—
39C4×C4⋊Dic43C2211yescharacteristic
40C4×C4⋊Dic43C2211yescharacteristic
41C4×C4⋊Dic43C2211yescharacteristic
42C4×C4⋊Dic43C2211yescharacteristic
43C4×(C2×D43)⋊C4—21no—
44C4×(C2×D43)⋊C4—21no—
45C4×(C2×D43)⋊C4—21no—
46C4×(C2×D43)⋊C4—21no—
47#14582662C2211yescharacteristic
48#14582662C2211yescharacteristic
49#14582662C2211yescharacteristic
50#14582662C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582662C2211yescharacteristic
53C2×#97×C43C2211yescharacteristic
54C2×#97×C43C2211yescharacteristic
55C43×C43C2211yescharacteristic
56C43×#336C2211yescharacteristic
57C43×#336C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×D172C2311yescharacteristic
61C22×D172C2311yescharacteristic
62C22×C4×D43—21no—
63C22×C4×D43—21no—
64C22×C4×D43—21no—
65C22×C4×D43—21no—
66C2×D344—22no—
67C2×D344—22no—
68C2×D344C2×C414yes—
69C4×D172—22no—
70C4×D172—22no—
71C4×D172—22no—
72C4×D172—22no—
73C4×D172—22no—
74C4×D172—22no—
75C4×D172C2×C414yes—
76C4×D172C2×C414yescomplemented
77#13063282—44no—
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×C4⋊Dic43C2311yescharacteristic
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×(C2×D43)⋊C4—21no—
91C2×(C2×D43)⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×(C2×D43)⋊C4—21no—
94C2×C4×Dic43—21no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43—21no—
97C2×C4×Dic43—21no—
98C42×D43—44no—
99#13063259—22no—
100#13063257—22no—
101#13063257C2×C414yescomplemented
102#13063247—22no—
103#13063247—22no—
104#13063247—22no—
105#13063247—22no—
106#13063247C2×C414yes—
107#13063247C2×C414yescomplemented
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C22×C8×C43C2311yescharacteristic
115C22×C8×C43C2311yescharacteristic
116C22×C8×C43C2311yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C2×C42×C43D411yescharacteristic
119C2×C42×C43D411yescharacteristic
120C2×C42×C43D411yescharacteristic
121C2×C42×C43D411yescharacteristic
122C2×C42×C43C2311yescharacteristic
123C2×C42×C43C2311yescharacteristic
124C2×C42×C43C2311yescharacteristic
125#97×C43—22no—
126#97×C43C2×C414yescomplemented
127C23×D43—21no—
128C23×D43—21no—
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2411yescharacteristic
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43—21no—
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43—21no—
142C2×D172—21no—
143C2×D172—21no—
144C2×D172—21no—
145C2×D172—21no—
146C2×D172—21no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172—21no—
150C2×D172C22×C412yes—
151C2×D172C22×C412yes—
152C2×D172#4112yes—
153C2×D172#4112yes—
154C2×C4×D43—24no—
155C2×C4×D43—24no—
156C2×C4×D43—24no—
157C2×C4×D43—24no—
158C2×C4×D43—42no—
159C2×C4×D43—42no—
160C2×C4×D43—42no—
161C2×C4×D43—42no—
162C2×C4×D43—42no—
163C2×C4×D43—42no—
164C2×C4×D43—42no—
165C2×C4×D43—42no—
166C2×C8×C43—21no—
167C2×C8×C43—21no—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43C22×C412yes—
177C2×C8×C43C22×C412yes—
178C42×C43—22no—
179C42×C43—22no—
180C42×C43—21no—
181C42×C43—21no—
182C42×C43—21no—
183C42×C43—21no—
184C42×C43—41no—
185C42×C43—41no—
186C42×C43C22×C412yes—
187C42×C43#4112yes—
188(C2×D43)⋊C4—24no—
189(C2×D43)⋊C4—24no—
190(C2×D43)⋊C4—24no—
191(C2×D43)⋊C4—24no—
192(C2×D43)⋊C4—42no—
193(C2×D43)⋊C4—42no—
194(C2×D43)⋊C4—42no—
195(C2×D43)⋊C4—42no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43—21no—
203C4⋊Dic43—21no—
204C4⋊Dic43C22×C412yes—
205C4⋊Dic43C22×C412yes—
206C4⋊Dic43#4112yes—
207C4⋊Dic43#4112yes—
208Dic43⋊C4—42no—
209Dic43⋊C4—42no—
210C4×Dic43—42no—
211C4×Dic43—42no—
212C4×Dic43—42no—
213C4×Dic43—42no—
214D344—28no—
215C23×C43C22×D411yescharacteristic
216C22×D43—22no—
217C22×D43—22no—
218C22×D43—22no—
219C22×D43—22no—
220C22×D43—41no—
221C22×D43—41no—
222C22×D43—41no—
223C22×D43—41no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—41no—
227C22×D43—41no—
228C22×D43—41no—
229C22×D43—41no—
230C2×C4×C43—21no—
231C2×C4×C43—21no—
232C2×C4×C43—21no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—21no—
244C2×C4×C43C4×D412yes—
245C2×C4×C43C4×D412yes—
246C2×C4×C43AGL(1,ℤ/8)12yes—
247C2×C4×C43AGL(1,ℤ/8)12yes—
248C2×C4×C43C23×C411yescharacteristic
249C2×C4×C43C2×#4111yescharacteristic
250C2×C4×C43C2×#4111yescharacteristic
251C2×C4×C43C2×#4111yescharacteristic
252C2×C4×C43C2×#4111yescharacteristic
253C2×C4×C43C2×#4111yescharacteristic
254C2×Dic43—22no—
255C2×Dic43—22no—
256C2×Dic43—22no—
257C2×Dic43—22no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268C2×Dic43—41no—
269C2×Dic43—41no—
270C2×Dic43—41no—
271C2×Dic43—41no—
272D172—22no—
273D172—22no—
274D172—22no—
275D172—22no—
276D172—22no—
277D172—41no—
278D172—41no—
279D172#11714yescomplemented
280C4×D43—48no—
281C4×D43—48no—
282C4×D43—48no—
283C4×D43—48no—
284C8×C43—22no—
285C8×C43—22no—
286C8×C43—41no—
287C8×C43—41no—
288#21122—431nomaximal, Sylow
289C22×C43C2×C4×D411yescharacteristic
290C22×C43#52511yescharacteristic
291C22×C43#52511yescharacteristic
292C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C43#57511yescharacteristic
295C22×C43#57511yescharacteristic
296C2×D43—24no—
297C2×D43—42no—
298C2×D43—42no—
299C2×D43—42no—
300C2×D43—42no—
301C2×D43—42no—
302C2×D43—42no—
303C2×D43—42no—
304C2×D43—42no—
305C2×D43—42no—
306C2×D43—81no—
307C2×D43—81no—
308C2×D43—81no—
309C2×D43—81no—
310C4×C43—22no—
311C4×C43—22no—
312C4×C43—22no—
313C4×C43—22no—
314C4×C43—41no—
315C4×C43—41no—
316C4×C43C2×#11711yescharacteristic
317C4×C43C4×#4111yescharacteristic
318C4×C43#52911yescharacteristic
319C4×C43#52911yescharacteristic
320Dic43—42no—
321Dic43—42no—
322Dic43—42no—
323Dic43—42no—
324Dic43—81no—
325Dic43—81no—
326C2×#442—431no—
327#2293—431no—
328C42×D4—431no—
329#1926—431no—
330#1925—431no—
331#1920—431no—
332#1919—431no—
333#1879—431no—
334#1879—431no—
335#1864—431no—
336#1862—431no—
337#1776—431no—
338#1762—431no—
339#1762—431no—
340#1745D4311yescharacteristic, complemented
341C2×C43#229011yescharacteristic
342C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
343C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
344C2×C43#310911yescharacteristic
345C2×C43#311111yescharacteristic
346C2×C43#311411yescharacteristic
347C2×C43#311511yescharacteristic
348D43—44no—
349D43—82no—
350C22×D8—431no—
351C2×C4×D4—431no—
352C2×C4×D4—431no—
353C2×C4×D4—431no—
354C2×C4×D4—431no—
355C2×#117—861no—
356C22×C42—861no—
357#442—438no—
358C2×#107—431no—
359C2×#106—431no—
360C2×#106—431no—
361C2×#102—431no—
362C2×#102—431no—
363C2×#102—431no—
364C2×#102—431no—
365C2×#102—431no—
366C2×#102—431no—
367C2×#97C2×D4311yescharacteristic
368C2×#97C2×D4311yescharacteristic
369#390—431no—
370#390—431no—
371#389—431no—
372#389—431no—
373#388—861no—
374#388—861no—
375C4×C4⋊C4—431no—
376C4×C4⋊C4—431no—
377C4×C4⋊C4—431no—
378C4×C4⋊C4—431no—
379C4×C22⋊C4—861no—
380C4×C22⋊C4—861no—
381C4×C22⋊C4—861no—
382C4×C22⋊C4—861no—
383C43C2×D4311yescharacteristic
384#340—431no—
385#340—431no—
386#340—431no—
387#340—431no—
388#340—431no—
389#340—431no—
390#336C2×D4311yescharacteristic
391#336C2×D4311yescharacteristic
392#336C2×D4311yescharacteristic
393#336C2×D4311yescharacteristic
394C43#2112211yescharacteristic, complemented, Sylow
395C22×D4—431no—
396C22×D4—431no—
397C23×C4—861no—
398C23×C4—861no—
399C23×C4—861no—
400C23×C4—861no—
401C2×D8—434no—
402C2×D8—862no—
403C2×D8—862no—
404C22×C8C22×D4311yescharacteristic
405C22×C8C22×D4311yescharacteristic
406C22×C8C22×D4311yescharacteristic
407C22×C8C22×D4311yescharacteristic
408C4×D4—434no—
409C4×D4—434no—
410C4×D4—862no—
411C4×D4—862no—
412C4×D4—862no—
413C4×D4—862no—
414C4×D4—862no—
415C4×D4—862no—
416#117—1724no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—431no—
419C2×C4⋊C4—431no—
420C2×C4⋊C4—431no—
421C2×C4⋊C4—431no—
422C2×C4⋊C4—431no—
423C2×C4⋊C4—861no—
424C2×C4⋊C4—861no—
425C2×C22⋊C4—861no—
426C2×C22⋊C4—861no—
427C2×C22⋊C4—861no—
428C2×C22⋊C4—861no—
429C2×C22⋊C4—861no—
430C2×C22⋊C4—861no—
431C2×C22⋊C4—861no—
432C2×C22⋊C4—861no—
433C2×C42—861no—
434C2×C42—861no—
435C2×C42—861no—
436C2×C42—861no—
437C2×C42—1724no—
438C2×C42D17211yescharacteristic
439C2×C42D17211yescharacteristic
440C2×C42D17211yescharacteristic
441C2×C42D17211yescharacteristic
442C2×C42C22×D4311yescharacteristic
443C2×C42C22×D4311yescharacteristic
444C2×C42C22×D4311yescharacteristic
445#107—862no—
446#106—434no—
447#106—862no—
448#102—434no—
449#102—434no—
450#102—862no—
451#102—862no—
452#102—862no—
453#102—862no—
454#97—22no—
455#97C4×D4314yescomplemented
456#95—861no—
457#95—861no—
458#95—861no—
459#95—861no—
460#95—861no—
461#95—861no—
462C24—861no—
463C24—861no—
464C2×D4—432no—
465C2×D4—432no—
466C2×D4—432no—
467C2×D4—432no—
468C2×D4—861no—
469C2×D4—861no—
470C2×D4—861no—
471C2×D4—861no—
472C2×D4—861no—
473C2×D4—861no—
474C2×D4—861no—
475C2×D4—861no—
476C22×C4—864no—
477C22×C4—864no—
478C22×C4—861no—
479C22×C4—861no—
480C22×C4—861no—
481C22×C4—861no—
482C22×C4—864no—
483C22×C4—864no—
484C22×C4—861no—
485C22×C4—861no—
486C22×C4—1722no—
487C22×C4—1722no—
488C22×C4—1722no—
489C22×C4—1722no—
490C22×C4—1722no—
491C22×C4—1722no—
492C22×C4—1722no—
493C22×C4—1722no—
494C22×C4C2×D17211yescharacteristic
495C22×C4C2×D17211yescharacteristic
496C22×C4C2×D17211yescharacteristic
497C22×C4C2×D17211yescharacteristic
498C22×C4C2×D17211yescharacteristic
499C22×C4C2×D17211yescharacteristic
500C22×C4C23×D4311yescharacteristic
501D8—868no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8C2×C4×D4312yes—
513C2×C8C2×C4×D4312yes—
514C4⋊C4—432no—
515C4⋊C4—432no—
516C4⋊C4—432no—
517C4⋊C4—432no—
518C4⋊C4—861no—
519C4⋊C4—861no—
520C4⋊C4—861no—
521C4⋊C4—861no—
522C4⋊C4—861no—
523C4⋊C4—861no—
524C4⋊C4—861no—
525C4⋊C4—861no—
526C4⋊C4—1722no—
527C4⋊C4—1722no—
528C22⋊C4—864no—
529C22⋊C4—864no—
530C22⋊C4—864no—
531C22⋊C4—864no—
532C22⋊C4—1722no—
533C22⋊C4—1722no—
534C22⋊C4—1722no—
535C22⋊C4—1722no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—41no—
543C42—41no—
544C42—1722no—
545C42—1722no—
546C42—1722no—
547C42—1722no—
548C42C2×C4×D4312yes—
549C42#1062189812yes—
550C23—862no—
551C23—862no—
552C23—862no—
553C23—862no—
554C23—1721no—
555C23—1721no—
556C23—1721no—
557C23—1721no—
558C23—1721no—
559C23—1721no—
560C23—1721no—
561C23—1721no—
562C23—1721no—
563C23—1721no—
564C23C22×D17211yescharacteristic
565D4—434no—
566D4—862no—
567D4—862no—
568D4—862no—
569D4—862no—
570D4—862no—
571D4—1721no—
572D4—1721no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—862no—
588C2×C4—862no—
589C2×C4—862no—
590C2×C4—862no—
591C2×C4—1728no—
592C2×C4—1728no—
593C2×C4—1728no—
594C2×C4—1728no—
595C2×C4—1721no—
596C2×C4—1721no—
597C2×C4—1721no—
598C2×C4—1721no—
599C2×C4—1721no—
600C2×C4—1721no—
601C2×C4—1721no—
602C2×C4—1721no—
603C2×C4—1721no—
604C2×C4—1721no—
605C2×C4—1721no—
606C2×C4—1721no—
607C2×C4—1721no—
608C2×C4—1721no—
609C2×C4C4×D17212yes—
610C2×C4C4×D17212yes—
611C2×C4#1306335312yes—
612C2×C4#1306335312yes—
613C2×C4C22×C4×D4311yescharacteristic
614C2×C4#41×D4311yescharacteristic
615C2×C4#41×D4311yescharacteristic
616C2×C4#41×D4311yescharacteristic
617C2×C4#41×D4311yescharacteristic
618C2×C4C2×#1062189811yescharacteristic
619C8—22no—
620C8—22no—
621C8—41no—
622C8—41no—
623C22—864no—
624C22—3441no—
625C22—3441no—
626C22—3441no—
627C22—3441no—
628C22—1722no—
629C22—1722no—
630C22—1722no—
631C22—1722no—
632C22—1722no—
633C22—1722no—
634C22—1722no—
635C22—1722no—
636C22—1722no—
637C22C2×C4×D17211yescharacteristic
638C22#1458341111yescharacteristic
639C22#1458341111yescharacteristic
640C22C2×#1306335311yescharacteristic
641C22C2×#1306335311yescharacteristic
642C22#1458369911yescharacteristic
643C22#1458369911yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—3441no—
651C4—3441no—
652C4—1722no—
653C4—1722no—
654C4—1722no—
655C4—1722no—
656C4#117×D4311yescharacteristic
657C4#1458336711yescharacteristic
658C4#1458342711yescharacteristic
659C4#1458342711yescharacteristic
660C2—3442no—
661C2—1724no—
662C2#1913703911yescharacteristic
663C2C4×#1306335311yescharacteristic
664C2C4×#1306335311yescharacteristic
665C2#1914468711yescharacteristic
666C2#1914470611yescharacteristic
667C2#1914473511yescharacteristic
668C2#1914474711yescharacteristic
subgroupG/H
centerC23C22×D172
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1745C2
socleC23×C43C22×D4
prime radicalC22×D344C22
cosocleC22×C4×C43C24
hypercenter#1745D43
hypocenterC43#21122
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
2. #19137059C2semidirect
3. C42×D172C2semidirect
4. #19135223C2semidirect
5. #19135216C2semidirect
6. #19135181C2semidirect
7. #19135174C2semidirect
10. #19134859C2semidirect
11. #19134845C2semidirect
12. #19134492C2semidirect
13. #19134421C2semidirect
14. #19134421C2semidirect
15. C43×#1745C2semidirect
23. #14583012C4semidirect
76. C4×D172C2×C4semidirect
101. #13063257C2×C4semidirect
107. #13063247C2×C4semidirect
126. #97×C43C2×C4semidirect
279. D172#117semidirect
340. #1745D43semidirect
394. C43#21122semidirect
455. #97C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145830122
#191370592
C42×D1722
#191352232
#191352162
#191351812
#191351742
#191349422
#191348592
#191348452
#191344922
#191344212
C43×#17452
#2112243