Finite Group Databasebeta

← #25116427

Subgroups — #25116427

Order 11008.

886 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583012C211yesmaximal, characteristic
2#19137566C211yesmaximal, characteristic, complemented
3#19137542C211yesmaximal, characteristic, complemented
4#19135471C211yesmaximal, characteristic
5#19135461C211yesmaximal, characteristic
6#19135223C211yesmaximal, characteristic, complemented
7#19135223C211yesmaximal, characteristic, complemented
8#19135216C211yesmaximal, characteristic, complemented
9#19135216C211yesmaximal, characteristic, complemented
10#19134763C211yesmaximal, characteristic, complemented
11#19134606C211yesmaximal, characteristic, complemented
12#19134596C211yesmaximal, characteristic, complemented
13#19134492C211yesmaximal, characteristic, complemented
14C4×#13063257C211yesmaximal, characteristic, complemented
15C43×#1771C211yesmaximal, characteristic, complemented
16C22×D344C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18C2×C4×D172C2211yescharacteristic
19C2×#13063283—21no—
20#14583012C418yescomplemented
21C2×#13063259C2211yescharacteristic
22C2×#13063259C2211yescharacteristic
23C2×#13063257C2211yescharacteristic
24C2×#13063257C2211yescharacteristic
25C2×#13063257C2211yescharacteristic
26C2×#13063247C2211yescharacteristic
27C2×#13063247C2211yescharacteristic
28C2×#13063247C2211yescharacteristic
29C2×#13063247C2211yescharacteristic
30C2×#13063247C2211yescharacteristic
31C2×#13063247C2211yescharacteristic
32#14582801C2211yescharacteristic
33#14582801C2211yescharacteristic
34#14582801C2211yescharacteristic
35#14582801C2211yescharacteristic
36#14582791—21no—
37#14582791—21no—
38#14582789—21no—
39#14582789—21no—
40#14582769C2211yescharacteristic
41#14582765C2211yescharacteristic
42#14582761—21no—
43#14582756—21no—
44C4×C4⋊Dic43C2211yescharacteristic
45C4×C4⋊Dic43C2211yescharacteristic
46C4×C4⋊Dic43C2211yescharacteristic
47C4×C4⋊Dic43C2211yescharacteristic
48#14582746—21no—
49#14582662C2211yescharacteristic
50#14582662C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582662C2211yescharacteristic
53C2×C4⋊C8×C43C2211yescharacteristic
54C2×#97×C43C2211yescharacteristic
55C2×#97×C43C2211yescharacteristic
56C2×C4×C8×C43C2211yescharacteristic
57C4×C4⋊C4×C43C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C22×D172C2311yescharacteristic
61C22×D172C2311yescharacteristic
62C22×C4×D43—21no—
63C22×C4×D43—21no—
64C2×D344—22no—
65C2×D344—22no—
66C2×D344C2×C414yes—
67C4×D172—22no—
68C4×D172—22no—
69C4×D172C2×C414yes—
70C4×D172C2×C414yes—
71#13063283—44no—
72C2×C4⋊Dic43—21no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×C4⋊Dic43C2311yescharacteristic
78C2×C4⋊Dic43C2311yescharacteristic
79C2×C4⋊Dic43C2311yescharacteristic
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
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×(C2×D43)⋊C4—21no—
88C2×(C2×D43)⋊C4—21no—
89C2×(C2×D43)⋊C4—21no—
90C2×C4×Dic43—21no—
91C2×C4×Dic43—21no—
92C2×C4×Dic43—21no—
93C2×C4×Dic43—21no—
94#13063259—22no—
95#13063259—41no—
96#13063257—22no—
97#13063257—41no—
98#13063257C2×C414yes—
99#13063247—41no—
100#13063247—41no—
101#13063247—41no—
102#13063247—41no—
103#13063247C2×C414yes—
104#13063247C2×C414yes—
105C43⋊#95—21no—
106C43⋊#95—21no—
107C43⋊#95—21no—
108C43⋊#95—21no—
109C43⋊#95—21no—
110C43⋊#95—21no—
111C22×C8×C43C2311yescharacteristic
112C22×C8×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114C22×C8×C43C2311yescharacteristic
115C2×C4⋊C4×C43D411yescharacteristic
116C2×C4⋊C4×C43D411yescharacteristic
117C2×C42×C43C2311yescharacteristic
118C2×C42×C43C2311yescharacteristic
119C2×C42×C43C2311yescharacteristic
120C4⋊C8×C43—22no—
121#97×C43—22no—
122#97×C43C2×C414yes—
123C4×C8×C43—22no—
124#95×C43D411yescharacteristic
125#95×C43D411yescharacteristic
126C23×D43—21no—
127C23×D43—21no—
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2411yescharacteristic
135C22×Dic43—21no—
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43—21no—
139C22×Dic43—21no—
140C22×Dic43—21no—
141C2×D172—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×D172—21no—
151C2×D172C22×C412yes—
152C2×D172C22×C412yes—
153C2×C4×D43—24no—
154C2×C4×D43—24no—
155C2×C4×D43—42no—
156C2×C4×D43—42no—
157C2×C4×D43—81no—
158C2×C4×D43—81no—
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43—21no—
163C2×C8×C43—21no—
164C2×C8×C43—21no—
165C2×C8×C43—21no—
166C2×C8×C43—21no—
167C2×C8×C43C22×C412yes—
168C2×C8×C43C22×C412yes—
169C2×C8×C43#4112yes—
170C2×C8×C43#4112yes—
171C4⋊C4×C43—22no—
172C4⋊C4×C43—41no—
173C42×C43—21no—
174C42×C43—21no—
175C42×C43—21no—
176C42×C43—21no—
177C42×C43C22×C412yes—
178C42×C43#4112yes—
179(C2×D43)⋊C4—24no—
180(C2×D43)⋊C4—24no—
181(C2×D43)⋊C4—42no—
182(C2×D43)⋊C4—42no—
183(C2×D43)⋊C4—42no—
184(C2×D43)⋊C4—42no—
185(C2×D43)⋊C4—81no—
186(C2×D43)⋊C4—81no—
187(C2×D43)⋊C4—81no—
188(C2×D43)⋊C4—81no—
189C4⋊Dic43—21no—
190C4⋊Dic43—21no—
191C4⋊Dic43—21no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—42no—
200C4⋊Dic43—81no—
201C4⋊Dic43C22×C412yes—
202C4⋊Dic43C22×C412yes—
203C4×Dic43—42no—
204C4×Dic43—42no—
205C4×Dic43—81no—
206C4×Dic43—81no—
207D344—28no—
208C23×C43C22×D411yescharacteristic
209C22×D43—22no—
210C22×D43—22no—
211C22×D43—41no—
212C22×D43—41no—
213C22×D43—41no—
214C22×D43—41no—
215C22×D43—41no—
216C22×D43—41no—
217C22×D43—41no—
218C22×D43—41no—
219C22×D43—41no—
220C22×D43—41no—
221C22×D43—41no—
222C22×D43—41no—
223C2×C4×C43—21no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
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×C43C4×D412yes—
239C2×C4×C43C4×D412yes—
240C2×C4×C43AGL(1,ℤ/8)12yes—
241C2×C4×C43C23×C411yescharacteristic
242C2×C4×C43C2×#4111yescharacteristic
243C2×C4×C43C2×#4111yescharacteristic
244C2×C4×C43C2×#4111yescharacteristic
245C2×C4×C43H(5,𝔽2)11yescharacteristic
246C2×C4×C432−1+411yescharacteristic
247C2×Dic43—22no—
248C2×Dic43—22no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265D172—22no—
266D172—22no—
267D172—22no—
268D172—22no—
269D172—41no—
270D172—41no—
271D172—41no—
272D172—41no—
273C4×D43—48no—
274C4×D43—84no—
275C8×C43—22no—
276C8×C43—41no—
277C8×C43—41no—
278C8×C43#11714yescomplemented
279#22817—431nomaximal, Sylow
280C22×C43C2×C4×D411yescharacteristic
281C22×C43#54711yescharacteristic
282C22×C43#55011yescharacteristic
283C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C43#57511yescharacteristic
285C22×C43#57611yescharacteristic
286C22×C43#57711yescharacteristic
287C2×D43—42no—
288C2×D43—42no—
289C2×D43—42no—
290C2×D43—42no—
291C2×D43—42no—
292C2×D43—42no—
293C2×D43—81no—
294C2×D43—81no—
295C2×D43—81no—
296C2×D43—81no—
297C2×D43—81no—
298C2×D43—81no—
299C2×D43—81no—
300C2×D43—81no—
301C4×C43—22no—
302C4×C43—22no—
303C4×C43—22no—
304C4×C43—41no—
305C4×C43—41no—
306C4×C43—41no—
307C4×C43C2×#11711yescharacteristic
308C4×C43#52011yescharacteristic
309C4×C43#55511yescharacteristic
310C4×C43#55611yescharacteristic
311Dic43—42no—
312Dic43—42no—
313Dic43—81no—
314Dic43—81no—
315Dic43—81no—
316Dic43—81no—
317C2×#442—431no—
318#2357—431no—
319#2354—431no—
320#1972—431no—
321#1970—431no—
322#1926—431no—
323#1926—431no—
324#1925—431no—
325#1925—431no—
326#1847—431no—
327#1804—431no—
328#1802—431no—
329#1776—431no—
330C4×#106—431no—
331#1771D4311yescharacteristic, complemented
332C2×C43#235411yescharacteristic
333C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
334C2×C43#294611yescharacteristic
335C2×C43#334511yescharacteristic
336C2×C43#334711yescharacteristic
337C2×C43#339911yescharacteristic
338C2×C43#340111yescharacteristic
339D43—82no—
340D43—82no—
341C22×D8—431no—
342C2×C4×D4—431no—
343C2×C4×D4—431no—
344C2×#117—861no—
345#442—438no—
346C2×#107—431no—
347C2×#107—431no—
348C2×#106—431no—
349C2×#106—431no—
350C2×#106—431no—
351C2×C4⋊C8C2×D4311yescharacteristic
352C2×#102—431no—
353C2×#102—431no—
354C2×#102—431no—
355C2×#102—431no—
356C2×#102—431no—
357C2×#102—431no—
358C2×#97C2×D4311yescharacteristic
359C2×#97C2×D4311yescharacteristic
360C2×C4×C8C2×D4311yescharacteristic
361#390—431no—
362#390—431no—
363#390—431no—
364#390—431no—
365#388—861no—
366#388—861no—
367#388—861no—
368#388—861no—
369#384—431no—
370#382—431no—
371#381—861no—
372#380—861no—
373C4×C4⋊C4—431no—
374C4×C4⋊C4—431no—
375C4×C4⋊C4—431no—
376C4×C4⋊C4—431no—
377C4×C4⋊C4C2×D4311yescharacteristic
378C4×C22⋊C4—861no—
379#340—431no—
380#340—431no—
381#340—431no—
382#340—431no—
383#336C2×D4311yescharacteristic
384#336C2×D4311yescharacteristic
385C43#2281711yescharacteristic, complemented, Sylow
386C22×D4—431no—
387C22×D4—431no—
388C23×C4—861no—
389C23×C4—861no—
390C2×D8—434no—
391C2×D8—862no—
392C2×D8—862no—
393C22×C8C22×D4311yescharacteristic
394C22×C8C22×D4311yescharacteristic
395C22×C8C22×D4311yescharacteristic
396C22×C8C22×D4311yescharacteristic
397C4×D4—434no—
398C4×D4—434no—
399C4×D4—862no—
400C4×D4—862no—
401#117—1724no—
402C2×C4⋊C4—431no—
403C2×C4⋊C4—431no—
404C2×C4⋊C4—431no—
405C2×C4⋊C4—431no—
406C2×C4⋊C4—431no—
407C2×C4⋊C4—431no—
408C2×C4⋊C4—861no—
409C2×C4⋊C4—861no—
410C2×C4⋊C4D17211yescharacteristic
411C2×C4⋊C4D17211yescharacteristic
412C2×C22⋊C4—861no—
413C2×C22⋊C4—861no—
414C2×C22⋊C4—861no—
415C2×C22⋊C4—861no—
416C2×C22⋊C4—861no—
417C2×C22⋊C4—861no—
418C2×C22⋊C4—861no—
419C2×C22⋊C4—861no—
420C2×C22⋊C4—861no—
421C2×C22⋊C4—861no—
422C2×C42—861no—
423C2×C42—861no—
424C2×C42—861no—
425C2×C42—861no—
426C2×C42C22×D4311yescharacteristic
427C2×C42C22×D4311yescharacteristic
428C2×C42C22×D4311yescharacteristic
429#107—862no—
430#107—1721no—
431#106—434no—
432#106—862no—
433#106—1721no—
434C4⋊C8—22no—
435#102—434no—
436#102—434no—
437#102—1721no—
438#102—1721no—
439#102—1721no—
440#102—1721no—
441#97—22no—
442#97C4×D4314yes—
443C4×C8—22no—
444#95—861no—
445#95—861no—
446#95—861no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95D17211yescharacteristic
451#95D17211yescharacteristic
452C24—861no—
453C24—861no—
454C2×D4—432no—
455C2×D4—432no—
456C2×D4—861no—
457C2×D4—861no—
458C2×D4—861no—
459C2×D4—861no—
460C2×D4—861no—
461C2×D4—861no—
462C2×D4—861no—
463C2×D4—861no—
464C2×D4—861no—
465C2×D4—861no—
466C22×C4—861no—
467C22×C4—861no—
468C22×C4—864no—
469C22×C4—864no—
470C22×C4—861no—
471C22×C4—861no—
472C22×C4—861no—
473C22×C4—861no—
474C22×C4—3441no—
475C22×C4—3441no—
476C22×C4—1722no—
477C22×C4—1722no—
478C22×C4C2×D17211yescharacteristic
479C22×C4C2×D17211yescharacteristic
480C22×C4C2×D17211yescharacteristic
481C22×C4C2×D17211yescharacteristic
482C22×C4C2×D17211yescharacteristic
483C22×C4C2×D17211yescharacteristic
484C22×C4C23×D4311yescharacteristic
485D8—868no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8C2×C4×D4312yes—
495C2×C8C2×C4×D4312yes—
496C2×C8#1062190012yes—
497C2×C8#1062190212yes—
498C4⋊C4—22no—
499C4⋊C4—41no—
500C4⋊C4—432no—
501C4⋊C4—432no—
502C4⋊C4—861no—
503C4⋊C4—861no—
504C4⋊C4—861no—
505C4⋊C4—861no—
506C4⋊C4—861no—
507C4⋊C4—861no—
508C4⋊C4—861no—
509C4⋊C4—861no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—3441no—
513C4⋊C4—1722no—
514C22⋊C4—864no—
515C22⋊C4—864no—
516C22⋊C4—3441no—
517C22⋊C4—3441no—
518C22⋊C4—3441no—
519C22⋊C4—3441no—
520C22⋊C4—1722no—
521C22⋊C4—1722no—
522C22⋊C4—1722no—
523C22⋊C4—1722no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—3441no—
529C42—3441no—
530C42—1722no—
531C42—1722no—
532C42C2×C4×D4312yes—
533C42#1062189812yes—
534C23—862no—
535C23—862no—
536C23—1721no—
537C23—1721no—
538C23—1721no—
539C23—1721no—
540C23—1721no—
541C23—1721no—
542C23—1721no—
543C23—1721no—
544C23—1721no—
545C23—1721no—
546C23—1721no—
547C23—1721no—
548C23C22×D17211yescharacteristic
549D4—862no—
550D4—862no—
551D4—862no—
552D4—862no—
553D4—1721no—
554D4—1721no—
555D4—1721no—
556D4—1721no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—862no—
573C2×C4—862no—
574C2×C4—3444no—
575C2×C4—1728no—
576C2×C4—1721no—
577C2×C4—1721no—
578C2×C4—1721no—
579C2×C4—1721no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4C4×D17212yes—
593C2×C4C4×D17212yes—
594C2×C4#1306335312yes—
595C2×C4C22×C4×D4311yescharacteristic
596C2×C4C2×#1062190011yescharacteristic
597C2×C4C2×#1062189811yescharacteristic
598C2×C4C2×#1062190211yescharacteristic
599C2×C4#1306337111yescharacteristic
600C2×C4#1306337211yescharacteristic
601C8—22no—
602C8—41no—
603C8—41no—
604C8#1306328314yescomplemented
605C22—3441no—
606C22—3441no—
607C22—3441no—
608C22—3441no—
609C22—3441no—
610C22—3441no—
611C22—3441no—
612C22—3441no—
613C22—1722no—
614C22—1722no—
615C22—1722no—
616C22—1722no—
617C22—1722no—
618C22—1722no—
619C22C2×C4×D17211yescharacteristic
620C22#1458355311yescharacteristic
621C22#1458357111yescharacteristic
622C22C2×#1306335311yescharacteristic
623C22#1458369911yescharacteristic
624C22#1458370411yescharacteristic
625C22#1458371111yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—3441no—
633C4—3441no—
634C4—3441no—
635C4—3441no—
636C4—1722no—
637C4—1722no—
638C4C2×#1306328311yescharacteristic
639C4#1458337911yescharacteristic
640C4#1458361411yescharacteristic
641C4#1458362211yescharacteristic
642C2—3442no—
643C2—3442no—
644C2#1913754211yescharacteristic
645C2C4×#1306335311yescharacteristic
646C2#1914342911yescharacteristic
647C2#1914706711yescharacteristic
648C2#1914709711yescharacteristic
649C2#1914763911yescharacteristic
650C2#1914766311yescharacteristic
subgroupG/H
centerC23C22×D172
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#1771C2
socleC23×C43C22×D4
prime radicalC22×D344C22
cosocleC22×C4×C43C24
hypercenter#1771D43
hypocenterC43#22817
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #19137566C2semidirect
3. #19137542C2semidirect
6. #19135223C2semidirect
7. #19135223C2semidirect
8. #19135216C2semidirect
9. #19135216C2semidirect
10. #19134763C2semidirect
11. #19134606C2semidirect
12. #19134596C2semidirect
13. #19134492C2semidirect
14. C4×#13063257C2semidirect
15. C43×#1771C2semidirect
20. #14583012C4semidirect
278. C8×C43#117semidirect
331. #1771D43semidirect
385. C43#22817semidirect
604. C8#13063283semidirect

Maximal subgroups

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

subgroupindex
C2×#145830122
#191375662
#191375422
#191354712
#191354612
#191352232
#191352162
#191347632
#191346062
#191345962
#191344922
C4×#130632572
C43×#17712
#2281743