Finite Group Databasebeta

← C151×#24416 #50329341

Subgroups — C151×#24416 #50329341

Order 38656.

1448 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3487×C151C211yesmaximal, characteristic, complemented
2#3055×C151C212yesmaximal, complemented
3C2×#454×C151C211yesmaximal, characteristic, complemented
4C2×#413×C151C211yesmaximal, characteristic, complemented
5#2431×C151C211yesmaximal, characteristic, complemented
6#2037×C151C212yesmaximal, complemented
7#2002×C151C212yesmaximal, complemented
8#1888×C151C211yesmaximal, characteristic, complemented
9#1871×C151C212yesmaximal, complemented
10#1857×C151C211yesmaximal, characteristic, complemented
11#1793×C151C211yesmaximal, characteristic, complemented
12C2×2−1+4×C151C2211yescharacteristic
13C22×#41×C151C2211yescharacteristic
14C2×AGL(1,ℤ/8)×C151C2212yes—
15C22×M16×C151C2212yes—
16#550×C151—22no—
17#550×C151—22no—
18#548×C151—22no—
19#548×C151—22no—
20#535×C151—22no—
21#535×C151C2212yes—
22C2×#127×C151C2211yescharacteristic
23C2×#124×C151C2211yescharacteristic
24#525×C151—22no—
25#525×C151C2212yes—
26C2×#121×C151C2211yescharacteristic
27#519×C151C2211yescharacteristic
28#469×C151—22no—
29#468×C151—22no—
30#466×C151—22no—
31#465×C151—22no—
32#454×C151—24no—
33#428×C151C2212yes—
34C2×#104×C151C2212yes—
35C2×#104×C151C2212yes—
36#418×C151C2212yes—
37#417×C151C2212yes—
38#413×C151—24no—
39C2×#101×C151C2211yescharacteristic
40C2×#101×C151C2211yescharacteristic
41C2×#100×C151C2212yes—
42#392×C151—22no—
43#391×C151C2211yescharacteristic
44#390×C151C2211yescharacteristic
45#386×C151—22no—
46C4×C22⋊C4×C151C2211yescharacteristic
47#339×C151C2212yes—
48#339×C151C2212yes—
492−1+4×C151—22no—
502−1+4×C151—22no—
512−1+4×C151—42no—
52C2×#41×C151—22no—
53C2×#41×C151—22no—
54C2×#41×C151—22no—
55C2×#41×C151—22no—
56C2×#41×C151—22no—
57C2×#41×C151—22no—
58C2×#41×C151—42no—
59C2×#41×C151C2312yes—
60C22×Q8×C151—22no—
61C22×Q8×C151C2311yescharacteristic
62C22×D4×C151—22no—
63C22×D4×C151—21no—
64C22×D4×C151D412yes—
65C22×D4×C151C2311yescharacteristic
66C22×D4×C151C2311yescharacteristic
67C23×C4×C151D412yes—
68C23×C4×C151C2311yescharacteristic
69AGL(1,ℤ/8)×C151—24no—
70AGL(1,ℤ/8)×C151—42no—
71C2×SD16×C151—22no—
72C2×D8×C151—22no—
73C2×M16×C151—22no—
74C2×M16×C151—24no—
75C2×M16×C151—42no—
76C2×M16×C151C2312yes—
77C2×M16×C151C2312yes—
78C22×C8×C151—22no—
79#127×C151—24no—
80#127×C151—22no—
81#127×C151—42no—
82#124×C151—24no—
83#124×C151—22no—
84#124×C151—22no—
85#124×C151—22no—
86#123×C151—22no—
87C22⋊Q8×C151—22no—
88C22⋊Q8×C151—22no—
89C22⋊Q8×C151D412yescomplemented
90C22⋊Q8×C151D412yescomplemented
91#121×C151—22no—
92#121×C151—22no—
93#121×C151—24no—
94#121×C151—24no—
95#121×C151—22no—
96#121×C151—22no—
97#121×C151—22no—
98#121×C151—22no—
99#121×C151D412yescomplemented
100#121×C151D412yescomplemented
101#120×C151—24no—
102#120×C151—22no—
103C4×Q8×C151—22no—
104C4×Q8×C151—22no—
105C4×D4×C151—24no—
106C4×D4×C151—24no—
107C4×D4×C151—22no—
108C4×D4×C151—22no—
109#117×C151—22no—
110#117×C151C2312yes—
111C2×C4⋊C4×C151C2311yescharacteristic
112C2×C22⋊C4×C151—21no—
113C2×C22⋊C4×C151—22no—
114C2×C22⋊C4×C151—21no—
115C2×C22⋊C4×C151—22no—
116C2×C42×C151C2311yescharacteristic
117C2×C42×C151C2311yescharacteristic
118#107×C151—22no—
119#106×C151—22no—
120#104×C151—24no—
121#104×C151—42no—
122#103×C151—22no—
123#102×C151—22no—
124#102×C151—22no—
125#102×C151—22no—
126#101×C151—22no—
127#101×C151D414yescomplemented
128#100×C151—24no—
129#95×C151—21no—
130#95×C151—22no—
131C24×C151—21no—
132C24×C151C2×D412yes—
133C24×C151C2×D411yescharacteristic
134#41×C151—22no—
135#41×C151—22no—
136#41×C151—22no—
137#41×C151—22no—
138#41×C151—44no—
139#41×C151—44no—
140#41×C151—44no—
141#41×C151—44no—
142#41×C151—44no—
143#41×C151—44no—
144#41×C151—44no—
145#41×C151—44no—
146C2×Q8×C151—22no—
147C2×Q8×C151—22no—
148C2×Q8×C151—22no—
149C2×Q8×C151—44no—
150C2×Q8×C151—44no—
151C2×Q8×C151C2×D412yes—
152C2×Q8×C151C2×D412yes—
153C2×Q8×C151#4112yes—
154C2×D4×C151—22no—
155C2×D4×C151—24no—
156C2×D4×C151—22no—
157C2×D4×C151—24no—
158C2×D4×C151—24no—
159C2×D4×C151—22no—
160C2×D4×C151—22no—
161C2×D4×C151—24no—
162C2×D4×C151—21no—
163C2×D4×C151—22no—
164C2×D4×C151—22no—
165C2×D4×C151—21no—
166C2×D4×C151—21no—
167C2×D4×C151—21no—
168C2×D4×C151—22no—
169C2×D4×C151—22no—
170C2×D4×C151—42no—
171C2×D4×C151—44no—
172C2×D4×C151—44no—
173C2×D4×C151—42no—
174C2×D4×C151—82no—
175C2×D4×C151—82no—
176C2×D4×C151C2×D412yes—
177C22×C4×C151—21no—
178C22×C4×C151—21no—
179C22×C4×C151—22no—
180C22×C4×C151—24no—
181C22×C4×C151—21no—
182C22×C4×C151—22no—
183C22×C4×C151—22no—
184C22×C4×C151—24no—
185C22×C4×C151—22no—
186C22×C4×C151—21no—
187C22×C4×C151—21no—
188C22×C4×C151—42no—
189C22×C4×C151C2×D411yescharacteristic
190C22×C4×C151C2×D412yes—
191C22×C4×C151C2×D412yes—
192C22×C4×C151C2×D412yes—
193C22×C4×C151C2411yescharacteristic
194SD16×C151—44no—
195D8×C151—44no—
196M16×C151—22no—
197M16×C151—22no—
198M16×C151—22no—
199M16×C151—44no—
200M16×C151C2×D414yes—
201C2×C8×C151—22no—
202C2×C8×C151—22no—
203C2×C8×C151—44no—
204C4⋊C4×C151—22no—
205C4⋊C4×C151—22no—
206C4⋊C4×C151—22no—
207C4⋊C4×C151—22no—
208C4⋊C4×C151—22no—
209C4⋊C4×C151C2×D412yes—
210C4⋊C4×C151C2×D412yes—
211C22⋊C4×C151—24no—
212C22⋊C4×C151—24no—
213C22⋊C4×C151—22no—
214C22⋊C4×C151—22no—
215C22⋊C4×C151—22no—
216C22⋊C4×C151—22no—
217C22⋊C4×C151—22no—
218C22⋊C4×C151—42no—
219C22⋊C4×C151—41no—
220C22⋊C4×C151—41no—
221C22⋊C4×C151—44no—
222C22⋊C4×C151—82no—
223C42×C151—24no—
224C42×C151—22no—
225C42×C151—21no—
226C42×C151—21no—
227C42×C151—22no—
228C23×C151—21no—
229C23×C151—21no—
230C23×C151—22no—
231C23×C151—22no—
232C23×C151—22no—
233C23×C151—22no—
234C23×C151—41no—
235C23×C151—41no—
236C23×C151—42no—
237C23×C151—42no—
238C23×C151—42no—
239C23×C151—42no—
240C23×C151—41no—
241C23×C151—41no—
242C23×C151—41no—
243C23×C151#12012yes—
244C23×C151C22×D411yescharacteristic
245Q8×C151—24no—
246Q8×C151—24no—
247Q8×C151—22no—
248Q8×C151—22no—
249Q8×C151—44no—
250D4×C151—22no—
251D4×C151—22no—
252D4×C151—48no—
253D4×C151—44no—
254D4×C151—42no—
255D4×C151—42no—
256D4×C151—42no—
257D4×C151—42no—
258D4×C151—42no—
259D4×C151—42no—
260D4×C151—82no—
261D4×C151—82no—
262D4×C151—84no—
263D4×C151—84no—
264D4×C151—88no—
265D4×C151—84no—
266C2×C4×C151—22no—
267C2×C4×C151—21no—
268C2×C4×C151—22no—
269C2×C4×C151—21no—
270C2×C4×C151—22no—
271C2×C4×C151—22no—
272C2×C4×C151—22no—
273C2×C4×C151—22no—
274C2×C4×C151—22no—
275C2×C4×C151—22no—
276C2×C4×C151—22no—
277C2×C4×C151—22no—
278C2×C4×C151—44no—
279C2×C4×C151—44no—
280C2×C4×C151—48no—
281C2×C4×C151—41no—
282C2×C4×C151—41no—
283C2×C4×C151—42no—
284C2×C4×C151—42no—
285C2×C4×C151—41no—
286C2×C4×C151—41no—
287C2×C4×C151—42no—
288C2×C4×C151—41no—
289C2×C4×C151—41no—
290C2×C4×C151#12011yescharacteristic
291C2×C4×C151#12011yescharacteristic
292C2×C4×C151#12114yes—
293C2×C4×C151C22×D411yescharacteristic
294C2×C4×C151C22×D412yes—
295C2×C4×C151C2×#4111yescharacteristic
296C2×C4×C151H(5,𝔽2)12yes—
297C8×C151—24no—
298C8×C151—42no—
299C22×C151—21no—
300C22×C151—21no—
301C22×C151—22no—
302C22×C151—22no—
303C22×C151—22no—
304C22×C151—44no—
305C22×C151—42no—
306C22×C151—42no—
307C22×C151—42no—
308C22×C151—42no—
309C22×C151—42no—
310C22×C151—42no—
311C22×C151—81no—
312C22×C151—81no—
313C22×C151—82no—
314C22×C151—82no—
315C22×C151—82no—
316C22×C151—81no—
317C22×C151—81no—
318C22×C151—81no—
319C22×C151C2×#12011yescharacteristic
320C22×C151C2×#12111yescharacteristic
321C22×C151#53411yescharacteristic
322C4×C151—21no—
323C4×C151—21no—
324C4×C151—24no—
325C4×C151—42no—
326C4×C151—42no—
327C4×C151—42no—
328C4×C151—42no—
329C4×C151—81no—
330C4×C151—81no—
331C4×C151D4212yes—
332C4×C151#54612yes—
333C2×C151—21no—
334C2×C151—21no—
335C2×C151—42no—
336C2×C151—82no—
337C2×C151—82no—
338C2×C151—161no—
339C2×C151#241011yescharacteristic
340C2×C151#301811yescharacteristic
341C2×C151#302111yescharacteristic
342#24416C15111yesmaximal, characteristic, direct product factor, Sylow
343C151#2441611yescharacteristic, direct product factor, Sylow
344#3487C2×C15111yescharacteristic, complemented
345#3055C2×C15112yescomplemented
346C2×#454C2×C15111yescharacteristic, complemented
347C2×#413C2×C15111yescharacteristic, complemented
348#2431C2×C15111yescharacteristic, complemented
349#2037C2×C15112yescomplemented
350#2002C2×C15112yescomplemented
351#1888C2×C15111yescharacteristic, complemented
352#1871C2×C15112yescomplemented
353#1857C2×C15111yescharacteristic, complemented
354#1793C2×C15111yescharacteristic, complemented
355C2×2−1+4C22×C15111yescharacteristic
356C22×#41C22×C15111yescharacteristic
357C2×AGL(1,ℤ/8)C22×C15112yes—
358C22×M16C22×C15112yes—
359#550—22no—
360#550—22no—
361#548—22no—
362#548—22no—
363#535—22no—
364#535C22×C15112yes—
365C2×#127C22×C15111yescharacteristic
366C2×#124C22×C15111yescharacteristic
367#525—22no—
368#525C22×C15112yes—
369C2×#121C22×C15111yescharacteristic
370#519C22×C15111yescharacteristic
371#469—22no—
372#468—22no—
373#466—22no—
374#465—22no—
375#454—24no—
376#428C22×C15112yes—
377C2×#104C22×C15112yes—
378C2×#104C22×C15112yes—
379#418C22×C15112yes—
380#417C22×C15112yes—
381#413—24no—
382C2×#101C22×C15111yescharacteristic
383C2×#101C22×C15111yescharacteristic
384C2×#100C22×C15112yes—
385#392—22no—
386#391C22×C15111yescharacteristic
387#390C22×C15111yescharacteristic
388#386—22no—
389C4×C22⋊C4C22×C15111yescharacteristic
390#339C22×C15112yes—
391#339C22×C15112yes—
3922−1+4—22no—
3932−1+4—22no—
3942−1+4—42no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—42no—
402C2×#41C23×C15112yes—
403C22×Q8—22no—
404C22×Q8C23×C15111yescharacteristic
405C22×D4—22no—
406C22×D4—21no—
407C22×D4D4×C15112yes—
408C22×D4C23×C15111yescharacteristic
409C22×D4C23×C15111yescharacteristic
410C23×C4D4×C15112yes—
411C23×C4C23×C15111yescharacteristic
412AGL(1,ℤ/8)—24no—
413AGL(1,ℤ/8)—42no—
414C2×SD16—22no—
415C2×D8—22no—
416C2×M16—22no—
417C2×M16—24no—
418C2×M16—42no—
419C2×M16C23×C15112yes—
420C2×M16C23×C15112yes—
421C22×C8—22no—
422#127—24no—
423#127—22no—
424#127—42no—
425#124—24no—
426#124—22no—
427#124—22no—
428#124—22no—
429#123—22no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8D4×C15112yescomplemented
433C22⋊Q8D4×C15112yescomplemented
434#121—22no—
435#121—22no—
436#121—24no—
437#121—24no—
438#121—22no—
439#121—22no—
440#121—22no—
441#121—22no—
442#121D4×C15112yescomplemented
443#121D4×C15112yescomplemented
444#120—24no—
445#120—22no—
446C4×Q8—22no—
447C4×Q8—22no—
448C4×D4—24no—
449C4×D4—24no—
450C4×D4—22no—
451C4×D4—22no—
452#117—22no—
453#117C23×C15112yes—
454C2×C4⋊C4C23×C15111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—22no—
458C2×C22⋊C4—22no—
459C2×C42C23×C15111yescharacteristic
460C2×C42C23×C15111yescharacteristic
461#107—22no—
462#106—22no—
463#104—24no—
464#104—42no—
465#103—22no—
466#102—22no—
467#102—22no—
468#102—22no—
469#101—22no—
470#101D4×C15114yescomplemented
471#100—24no—
472#95—21no—
473#95—22no—
474C24—21no—
475C24C2×D4×C15112yes—
476C24C2×D4×C15111yescharacteristic
477#41—22no—
478#41—22no—
479#41—22no—
480#41—22no—
481#41—44no—
482#41—44no—
483#41—44no—
484#41—44no—
485#41—44no—
486#41—44no—
487#41—44no—
488#41—44no—
489C2×Q8—22no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8C2×D4×C15112yes—
495C2×Q8C2×D4×C15112yes—
496C2×Q8#41×C15112yes—
497C2×D4—22no—
498C2×D4—24no—
499C2×D4—22no—
500C2×D4—24no—
501C2×D4—24no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—24no—
505C2×D4—21no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—21no—
511C2×D4—22no—
512C2×D4—22no—
513C2×D4—42no—
514C2×D4—44no—
515C2×D4—44no—
516C2×D4—42no—
517C2×D4—82no—
518C2×D4—82no—
519C2×D4C2×D4×C15112yes—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—24no—
524C22×C4—21no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—24no—
528C22×C4—22no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—42no—
532C22×C4C2×D4×C15111yescharacteristic
533C22×C4C2×D4×C15112yes—
534C22×C4C2×D4×C15112yes—
535C22×C4C2×D4×C15112yes—
536C22×C4C24×C15111yescharacteristic
537SD16—44no—
538D8—44no—
539M16—22no—
540M16—22no—
541M16—22no—
542M16—44no—
543M16C2×D4×C15114yes—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—44no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4C2×D4×C15112yes—
553C4⋊C4C2×D4×C15112yes—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—42no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—44no—
565C22⋊C4—82no—
566C42—24no—
567C42—22no—
568C42—21no—
569C42—21no—
570C42—22no—
571C23—21no—
572C23—21no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—41no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23#120×C15112yes—
587C23C22×D4×C15111yescharacteristic
588Q8—24no—
589Q8—24no—
590Q8—22no—
591Q8—22no—
592Q8—44no—
593D4—22no—
594D4—22no—
595D4—48no—
596D4—44no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—82no—
604D4—82no—
605D4—84no—
606D4—84no—
607D4—88no—
608D4—84no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4#120×C15111yescharacteristic
634C2×C4#120×C15111yescharacteristic
635C2×C4#121×C15114yes—
636C2×C4C22×D4×C15111yescharacteristic
637C2×C4C22×D4×C15112yes—
638C2×C4C2×#41×C15111yescharacteristic
639C2×C4H(5,𝔽2)×C15112yes—
640C8—24no—
641C8—42no—
642C22—21no—
643C22—21no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—81no—
655C22—81no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22C2×#120×C15111yescharacteristic
663C22C2×#121×C15111yescharacteristic
664C22#534×C15111yescharacteristic
665C4—21no—
666C4—21no—
667C4—24no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4D42×C15112yes—
675C4#546×C15112yes—
676C2—21no—
677C2—21no—
678C2—42no—
679C2—82no—
680C2—82no—
681C2—161no—
682C2#2410×C15111yescharacteristic
683C2#3018×C15111yescharacteristic
684C2#3021×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3487×C151C2semidirect
2. #3055×C151C2semidirect
3. C2×#454×C151C2semidirect
4. C2×#413×C151C2semidirect
5. #2431×C151C2semidirect
6. #2037×C151C2semidirect
7. #2002×C151C2semidirect
8. #1888×C151C2semidirect
9. #1871×C151C2semidirect
10. #1857×C151C2semidirect
11. #1793×C151C2semidirect
89. C22⋊Q8×C151D4semidirect
90. C22⋊Q8×C151D4semidirect
99. #121×C151D4semidirect
100. #121×C151D4semidirect
127. #101×C151D4semidirect
342. #24416C151direct (N × H)
343. C151#24416direct (N × H)
344. #3487C2×C151semidirect
345. #3055C2×C151semidirect
346. C2×#454C2×C151semidirect
347. C2×#413C2×C151semidirect
348. #2431C2×C151semidirect
349. #2037C2×C151semidirect
350. #2002C2×C151semidirect
351. #1888C2×C151semidirect
352. #1871C2×C151semidirect
353. #1857C2×C151semidirect
354. #1793C2×C151semidirect
432. C22⋊Q8D4×C151semidirect
433. C22⋊Q8D4×C151semidirect
442. #121D4×C151semidirect
443. #121D4×C151semidirect
470. #101D4×C151semidirect

Maximal subgroups

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

subgroupindex
#3487×C1512
#3055×C1512
C2×#454×C1512
C2×#413×C1512
#2431×C1512
#2037×C1512
#2002×C1512
#1888×C1512
#1871×C1512
#1857×C1512
#1793×C1512
#24416151