Finite Group Databasebeta

← #25106334

Subgroups — #25106334

Order 11008.

1234 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583436C211yesmaximal, characteristic
2C2×#14582955C211yesmaximal, characteristic, complemented
3#384×D43C211yesmaximal, characteristic, complemented
4#19135562C212yesmaximal, complemented
5#19135408C212yesmaximal, complemented
6#19134911C212yesmaximal
7#19134899C212yesmaximal, complemented
8#19134600C211yesmaximal, characteristic, complemented
9#19134596C211yesmaximal, characteristic, complemented
10#19134592C211yesmaximal, characteristic, complemented
11C43×#1803C211yesmaximal, characteristic, complemented
12#14583436—24no—
13#14583436C418yescomplemented
14C2×#13063320—21no—
15C2×#13063316C2211yescharacteristic
16C2×#13063314C2211yescharacteristic
17C2×#13063310—21no—
18C2×#13063301C2211yescharacteristic
19C2×#13063300C2211yescharacteristic
20C2×C4×D172—21no—
21C2×C4×D172C2211yescharacteristic
22C2×#13063282—21no—
23C2×C4⋊C4×D43—21no—
24C2×C4⋊C4×D43—21no—
25C2×C4⋊C4×D43C2212yes—
26C2×C42×D43C2211yescharacteristic
27#14582955—24no—
28C2×C4⋊(C43⋊C8)C2211yescharacteristic
29C2×#13063249C2212yes—
30C2×#13063247C2212yes—
31C2×#13063245C2212yes—
32C2×(C2×D43)⋊C8C2212yes—
33C2×#13063235C2212yes—
34#14582769C2211yescharacteristic
35#14582768—21no—
36#14582768—21no—
37#14582768C2212yes—
38#14582662C2212yes—
39#14582661C2212yes—
40#14582661C2212yes—
41C2×C4×Q8×C43C2211yescharacteristic
42C2×C4⋊C8×C43C2211yescharacteristic
43C2×#103×C43C2212yes—
44C43×#384C2211yescharacteristic
45C43×#340C2212yes—
46C22×D172—21no—
47C22×D172C2311yescharacteristic
48C22×C4×D43—21no—
49C22×C4×D43—22no—
50C22×C4×D43D411yescharacteristic
51C22×C4×D43D411yescharacteristic
52C22×C4×D43C2311yescharacteristic
53C22×C43⋊C8C2312yes—
54#13063320—28no—
55#13063316—22no—
56#13063316C2×C414yescomplemented
57#13063314—22no—
58#13063314C2×C414yescomplemented
59#13063310—28no—
60#13063301—22no—
61#13063301C2×C414yes—
62#13063300—22no—
63#13063300C2×C414yes—
64C4×D172—28no—
65C4×D172—22no—
66C4×D172C2×C414yescomplemented
67#13063282—28no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43C2312yes—
72C2×C4⋊Dic43C2311yescharacteristic
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4—22no—
76C2×Dic43⋊C4—22no—
77C2×Dic43⋊C4—22no—
78C4⋊C4×D43—42no—
79C4⋊C4×D43—42no—
80C4⋊C4×D43—44no—
81C4⋊C4×D43—44no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×(C2×D43)⋊C4—21no—
85C2×(C2×D43)⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87C2×C4×Dic43D411yescharacteristic
88C2×C4×Dic43D411yescharacteristic
89C2×C4×Dic43C2311yescharacteristic
90C42×D43C2×C414yes—
91C42×D43D414yes—
92C4⋊(C43⋊C8)—22no—
93#13063249—42no—
94#13063247—42no—
95#13063245—42no—
96(C2×D43)⋊C8—24no—
97#13063235—24no—
98C22×Q8×C43C2311yescharacteristic
99C22×C8×C43C2312yes—
100C4×Q8×C43—22no—
101C4×Q8×C43C2×C414yescomplemented
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—21no—
105C2×C4⋊C4×C43C2312yes—
106C2×C4⋊C4×C43C2311yescharacteristic
107C2×C42×C43—21no—
108C2×C42×C43C2311yescharacteristic
109C4⋊C8×C43—22no—
110#103×C43—42no—
111C23×D43—21no—
112C23×D43C2×D411yescharacteristic
113C2×Q8×C43—21no—
114C2×Q8×C43—21no—
115C2×Q8×C43—21no—
116C2×Q8×C43—21no—
117C2×Q8×C43—21no—
118C2×Q8×C43C22×C412yes—
119C22×C4×C43—21no—
120C22×C4×C43—21no—
121C22×C4×C43—22no—
122C22×C4×C43C2×D411yescharacteristic
123C22×C4×C43C2×D411yescharacteristic
124C22×C4×C43C2411yescharacteristic
125C22×Dic43—21no—
126C22×Dic43—22no—
127C22×Dic43C2×D411yescharacteristic
128C22×Dic43C2×D411yescharacteristic
129C22×Dic43C2×D411yescharacteristic
130C2×D172—21no—
131C2×D172—24no—
132C2×D172—21no—
133C2×D172—21no—
134C2×D172—21no—
135C2×D172—21no—
136C2×D172—42no—
137C2×D172—42no—
138C2×D172C22×C412yes—
139C2×C4×D43—22no—
140C2×C4×D43—22no—
141C2×C4×D43—21no—
142C2×C4×D43—21no—
143C2×C4×D43—21no—
144C2×C4×D43—21no—
145C2×C4×D43—24no—
146C2×C4×D43—22no—
147C2×C4×D43—22no—
148C2×C4×D43—44no—
149C2×C4×D43—44no—
150C2×C4×D43—42no—
151C2×C4×D43—42no—
152C2×C4×D43—44no—
153C2×C4×D43C22⋊C414yes—
154C2×C4×D43C22⋊C414yes—
155C2×C4×D43C22×C412yes—
156C2×C4×D43C2×D412yes—
157C2×C8×C43—22no—
158C2×C8×C43—22no—
159C2×C8×C43—22no—
160C4⋊C4×C43—24no—
161C4⋊C4×C43—21no—
162C4⋊C4×C43—22no—
163C4⋊C4×C43—22no—
164C4⋊C4×C43—42no—
165C4⋊C4×C43—42no—
166C4⋊C4×C43C22×C412yes—
167C42×C43—24no—
168C42×C43C22×C412yes—
169C42×C43C2×D412yes—
170(C2×D43)⋊C4—24no—
171(C2×D43)⋊C4—24no—
172(C2×D43)⋊C4—24no—
173(C2×D43)⋊C4—24no—
174(C2×D43)⋊C4—24no—
175C4⋊Dic43—21no—
176C4⋊Dic43—24no—
177C4⋊Dic43—22no—
178C4⋊Dic43—22no—
179C4⋊Dic43—42no—
180C4⋊Dic43—42no—
181C4⋊Dic43C22×C412yes—
182Dic43⋊C4—24no—
183Dic43⋊C4—24no—
184Dic43⋊C4—44no—
185Dic43⋊C4—44no—
186Dic43⋊C4—44no—
187C4×Dic43C22⋊C414yes—
188C4×Dic43C22⋊C414yes—
189C4×Dic43C22×C412yes—
190C4×Dic43C2×D412yes—
191C2×C43⋊C8—22no—
192C2×C43⋊C8—22no—
193C2×C43⋊C8—22no—
194C23×C43C22×D411yescharacteristic
195C22×D43—21no—
196C22×D43—21no—
197C22×D43—21no—
198C22×D43—21no—
199C22×D43—22no—
200C22×D43—41no—
201C22×D43—41no—
202C22×D43—41no—
203C22×D43—41no—
204C22×D43—41no—
205C22×D43—41no—
206C22×D43C2×C22⋊C412yes—
207C22×D43AGL(1,ℤ/8)12yes—
208C22×D43#13712yes—
209Q8×C43—22no—
210Q8×C43—22no—
211Q8×C43—41no—
212Q8×C43—41no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—22no—
216C2×C4×C43—22no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—42no—
222C2×C4×C43—42no—
223C2×C4×C43—42no—
224C2×C4×C43C2×C22⋊C412yes—
225C2×C4×C43C2×C22⋊C412yes—
226C2×C4×C43AGL(1,ℤ/8)12yes—
227C2×C4×C43#13712yes—
228C2×C4×C43C23×C411yescharacteristic
229C2×C4×C43C22×D411yescharacteristic
230C2×C4×C43H(5,𝔽2)11yescharacteristic
231C2×C4×C43H(5,𝔽2)11yescharacteristic
232C2×C4×C432−1+411yescharacteristic
233C2×C4×C432−1+411yescharacteristic
234C2×Dic43—21no—
235C2×Dic43—21no—
236C2×Dic43—21no—
237C2×Dic43—21no—
238C2×Dic43—21no—
239C2×Dic43—21no—
240C2×Dic43—22no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243C2×Dic43—42no—
244C2×Dic43—42no—
245C2×Dic43—42no—
246C2×Dic43C2×C22⋊C412yes—
247C2×Dic43C2×C22⋊C412yes—
248C2×Dic43C2×C22⋊C412yes—
249D172—22no—
250D172—22no—
251D172—48no—
252D172—41no—
253D172—41no—
254C4×D43—28no—
255C4×D43—22no—
256C4×D43—22no—
257C4×D43—22no—
258C4×D43—22no—
259C4×D43—28no—
260C4×D43—48no—
261C4×D43—88no—
262C8×C43—42no—
263C43⋊C8—42no—
264#21867—431nomaximal, Sylow
265C22×C43C22×C22⋊C411yescharacteristic
266C22×C43#53511yescharacteristic
267C22×C43#53611yescharacteristic
268C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C43C2×#13711yescharacteristic
271C22×C43C2×#13711yescharacteristic
272C2×D43—22no—
273C2×D43—22no—
274C2×D43—22no—
275C2×D43—22no—
276C2×D43—22no—
277C2×D43—22no—
278C2×D43—42no—
279C2×D43—42no—
280C2×D43—42no—
281C2×D43—81no—
282C2×D43—81no—
283C2×D43—81no—
284C2×D43—81no—
285C2×D43#41714yes—
286C4×C43—22no—
287C4×C43—42no—
288C4×C43—42no—
289C4×C43—82no—
290C4×C43#41714yes—
291C4×C43#52011yescharacteristic
292C4×C43#52011yescharacteristic
293C4×C43#53711yescharacteristic
294C4×C43#53711yescharacteristic
295Dic43—22no—
296Dic43—22no—
297Dic43—22no—
298Dic43—42no—
299Dic43—82no—
300C2×#529—431no—
301C2×#432—431no—
302C2×#384—431no—
303#1992—432no—
304#1959—432no—
305#1874—432no—
306#1872—432no—
307#1803—431no—
308#1803D4311yescharacteristic, complemented
309#1802—431no—
310#1802—431no—
311C2×C43#231111yescharacteristic
312C2×C43C2×#41711yescharacteristic
313C2×C43C2×#41711yescharacteristic
314C2×C43#321711yescharacteristic
315C2×C43#321811yescharacteristic
316C2×C43#323811yescharacteristic
317C2×C43#323911yescharacteristic
318D43—24no—
319D43—82no—
320#529—438no—
321#529—864no—
322C2×#126—861no—
323C2×#125—431no—
324C2×#124—431no—
325C2×#123—861no—
326C2×C22⋊Q8—431no—
327C2×#121—431no—
328C2×C4×Q8C2×D4311yescharacteristic
329C2×C4×D4—431no—
330C2×C4×D4—861no—
331C2×#117—861no—
332C22×C4⋊C4—432no—
333C22×C4⋊C4—861no—
334C22×C4⋊C4—861no—
335C22×C42—431no—
336#432—864no—
337C2×C4⋊C8—431no—
338C2×C4⋊C8C2×D4311yescharacteristic
339C2×#103—432no—
340C2×#103C2×D4312yes—
341C2×#102—432no—
342C2×#102—432no—
343C2×C22⋊C8—432no—
344C2×#97—432no—
345#384—432no—
346#384—431no—
347#384—861no—
348#384—861no—
349#384C2×D4311yescharacteristic
350#340—432no—
351#340—432no—
352#340—432no—
353#340C2×D4312yes—
354C43#2186711yescharacteristic, complemented, Sylow
355C22×Q8C22×D4311yescharacteristic
356C22×D4—431no—
357C22×D4—861no—
358C23×C4—431no—
359C23×C4—431no—
360C23×C4—431no—
361C23×C4—862no—
362C23×C4—861no—
363C22×C8—432no—
364C22×C8C22×D4312yes—
365#126—868no—
366#125—434no—
367#125—862no—
368#124—434no—
369#124—862no—
370#123—868no—
371C22⋊Q8—434no—
372C22⋊Q8—862no—
373#121—434no—
374#121—862no—
375C4×Q8—22no—
376C4×Q8C4×D4314yescomplemented
377C4×D4—434no—
378C4×D4—868no—
379C4×D4—862no—
380#117—868no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—431no—
385C2×C4⋊C4—432no—
386C2×C4⋊C4—861no—
387C2×C4⋊C4—861no—
388C2×C4⋊C4—861no—
389C2×C4⋊C4—862no—
390C2×C4⋊C4—862no—
391C2×C4⋊C4—862no—
392C2×C4⋊C4—861no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4—1722no—
395C2×C4⋊C4—1722no—
396C2×C4⋊C4—1724no—
397C2×C4⋊C4—1724no—
398C2×C4⋊C4C22×D4312yes—
399C2×C4⋊C4C22×D4311yescharacteristic
400C2×C22⋊C4—861no—
401C2×C22⋊C4—861no—
402C2×C22⋊C4—861no—
403C2×C22⋊C4—861no—
404C2×C22⋊C4—861no—
405C2×C42—21no—
406C2×C42—431no—
407C2×C42—431no—
408C2×C42—431no—
409C2×C42—434no—
410C2×C42—434no—
411C2×C42C22×D4311yescharacteristic
412C4⋊C8—22no—
413C4⋊C8—862no—
414#103—42no—
415#103—1722no—
416#102—1722no—
417#102—1722no—
418C22⋊C8—864no—
419#97—864no—
420C24—431no—
421C24—861no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C2×C4×D4312yes—
428C2×D4—432no—
429C2×D4—864no—
430C2×D4—861no—
431C2×D4—861no—
432C2×D4—861no—
433C2×D4—861no—
434C2×D4—861no—
435C2×D4—1722no—
436C2×D4—1722no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—431no—
441C22×C4—431no—
442C22×C4—431no—
443C22×C4—434no—
444C22×C4—434no—
445C22×C4—432no—
446C22×C4—432no—
447C22×C4—862no—
448C22×C4—862no—
449C22×C4—862no—
450C22×C4—862no—
451C22×C4—861no—
452C22×C4—861no—
453C22×C4—861no—
454C22×C4—861no—
455C22×C4—861no—
456C22×C4—864no—
457C22×C4—862no—
458C22×C4—1724no—
459C22×C4—1724no—
460C22×C4—1724no—
461C22×C4—1722no—
462C22×C4—1722no—
463C22×C4D4×D4311yescharacteristic
464C22×C4D4×D4311yescharacteristic
465C22×C4C23×D4311yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—862no—
470C2×C8—862no—
471C2×C8—862no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—432no—
479C4⋊C4—864no—
480C4⋊C4—864no—
481C4⋊C4—864no—
482C4⋊C4—861no—
483C4⋊C4—862no—
484C4⋊C4—862no—
485C4⋊C4—1722no—
486C4⋊C4—1724no—
487C4⋊C4—1724no—
488C4⋊C4—1724no—
489C4⋊C4—1722no—
490C4⋊C4C2×C4×D4312yes—
491C22⋊C4—864no—
492C22⋊C4—864no—
493C22⋊C4—864no—
494C22⋊C4—864no—
495C22⋊C4—864no—
496C42—24no—
497C42—434no—
498C42—434no—
499C42—432no—
500C42—432no—
501C42C2×C4×D4312yes—
502C42D4×D4312yes—
503C23—432no—
504C23—432no—
505C23—432no—
506C23—861no—
507C23—861no—
508C23—861no—
509C23—861no—
510C23—862no—
511C23—1721no—
512C23—1721no—
513C23—1721no—
514C23—1721no—
515C23—1721no—
516C23—1721no—
517C23C2×D4×D4311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—862no—
523D4—862no—
524D4—1728no—
525D4—1721no—
526D4—1721no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—432no—
539C2×C4—432no—
540C2×C4—432no—
541C2×C4—868no—
542C2×C4—861no—
543C2×C4—861no—
544C2×C4—861no—
545C2×C4—861no—
546C2×C4—861no—
547C2×C4—861no—
548C2×C4—862no—
549C2×C4—862no—
550C2×C4—868no—
551C2×C4—862no—
552C2×C4—862no—
553C2×C4—862no—
554C2×C4—3448no—
555C2×C4—1722no—
556C2×C4—1722no—
557C2×C4—1722no—
558C2×C4—1721no—
559C2×C4—1721no—
560C2×C4—1728no—
561C2×C4C22⋊C4×D4312yes—
562C2×C4C22⋊C4×D4312yes—
563C2×C4#1306335212yes—
564C2×C4#1306335712yes—
565C2×C4C22×C4×D4311yescharacteristic
566C2×C4C2×D4×D4311yescharacteristic
567C2×C4#1306337011yescharacteristic
568C2×C4#1306337111yescharacteristic
569C2×C4#1306337211yescharacteristic
570C2×C4#1306337311yescharacteristic
571C8—42no—
572C8—1722no—
573C22—434no—
574C22—862no—
575C22—862no—
576C22—862no—
577C22—862no—
578C22—862no—
579C22—862no—
580C22—3441no—
581C22—3441no—
582C22—3441no—
583C22—3441no—
584C22—1722no—
585C22—1722no—
586C22—1722no—
587C22C2×C22⋊C4×D4311yescharacteristic
588C22#1458346511yescharacteristic
589C22#1458347211yescharacteristic
590C22AGL(1,ℤ/8)×D4311yescharacteristic
591C22C2×#1306335211yescharacteristic
592C22#137×D4311yescharacteristic
593C22C2×#1306335711yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—862no—
599C4—862no—
600C4—862no—
601C4—3442no—
602C4—1722no—
603C4#1458289914yes—
604C4#1458337911yescharacteristic
605C4#1458338011yescharacteristic
606C4#1458347811yescharacteristic
607C4#1458347911yescharacteristic
608C2—864no—
609C2—3442no—
610C2#1913718411yescharacteristic
611C2#417×D4311yescharacteristic
612C2C2×#1458289911yescharacteristic
613C2#1914561711yescharacteristic
614C2#1914562611yescharacteristic
615C2#1914582011yescharacteristic
616C2#1914582911yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1803C2
socleC23×C43C22×D4
prime radicalC2×#13063300C22
cosocleC22×C4×C43C24
hypercenter#1803D43
hypocenterC43#21867
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#14582955C2semidirect
3. #384×D43C2semidirect
4. #19135562C2semidirect
5. #19135408C2semidirect
7. #19134899C2semidirect
8. #19134600C2semidirect
9. #19134596C2semidirect
10. #19134592C2semidirect
11. C43×#1803C2semidirect
13. #14583436C4semidirect
56. #13063316C2×C4semidirect
58. #13063314C2×C4semidirect
66. C4×D172C2×C4semidirect
101. C4×Q8×C43C2×C4semidirect
308. #1803D43semidirect
354. C43#21867semidirect
376. C4×Q8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145834362
C2×#145829552
#384×D432
#191355622
#191354082
#191349112
#191348992
#191346002
#191345962
#191345922
C43×#18032
#2186743