Finite Group Databasebeta

← C151×#18814 #50323739

Subgroups — C151×#18814 #50323739

Order 38656.

1262 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
1#3498×C151C211yesmaximal, characteristic, complemented
2#3214×C151C211yesmaximal, characteristic, complemented
3#3205×C151C211yesmaximal, characteristic, complemented
4#3115×C151C211yesmaximal, characteristic, complemented
5#1642×C151C212yesmaximal, complemented
6#1639×C151C212yesmaximal, complemented
7#1621×C151C212yesmaximal, complemented
8#1619×C151C212yesmaximal, complemented
9#1555×C151C211yesmaximal, characteristic, complemented
10#1496×C151C211yesmaximal, characteristic, complemented
11#1488×C151C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)×C151—21no—
13#550×C151—21no—
14#550×C151C2212yes—
15D42×C151—22no—
16D42×C151—22no—
17D42×C151—21no—
18D42×C151C2212yes—
19#535×C151—22no—
20#535×C151—21no—
21#532×C151—21no—
22#532×C151C2211yescharacteristic
23#532×C151C2211yescharacteristic, complemented
24C2×#127×C151—21no—
25C2×#127×C151C2211yescharacteristic
26C4×#41×C151C2211yescharacteristic
27#496×C151—21no—
28#493×C151C2212yescomplemented
29#489×C151—21no—
30#486×C151C2212yescomplemented
31#482×C151—21no—
32#475×C151C2211yescharacteristic, complemented
33#474×C151C2211yescharacteristic, complemented
34#461×C151C2212yescomplemented
35#460×C151C2211yescharacteristic, complemented
36#459×C151—22no—
37#459×C151C2211yescharacteristic, complemented
38#459×C151C2212yescomplemented
39#449×C151—21no—
40#449×C151—22no—
41#447×C151—22no—
42#442×C151—21no—
43#440×C151—21no—
44#432×C151C2211yescharacteristic, complemented
45#432×C151C2211yescharacteristic, complemented
46#431×C151C2211yescharacteristic, complemented
47#331×C151C2212yescomplemented
48#331×C151C2212yescomplemented
49Q8⋊C8×C151C2212yescomplemented
50D4⋊C8×C151C2212yescomplemented
51D4⋊C8×C151C2212yescomplemented
52D4⋊C8×C151C2212yescomplemented
53H(5,𝔽2)×C151—44no—
54H(5,𝔽2)×C151—44no—
55C2×#41×C151—22no—
56C2×#41×C151—21no—
57C2×#41×C151—21no—
58C2×#41×C151—22no—
59C2×#41×C151D411yescharacteristic
60C22×D4×C151—22no—
61C22×D4×C151—24no—
62C22×D4×C151—22no—
63C22×D4×C151—22no—
64C22×D4×C151—22no—
65C22×D4×C151—21no—
66C22×D4×C151—21no—
67C22×D4×C151D412yes—
68C2×SD16×C151—22no—
69C2×SD16×C151—21no—
70C2×D8×C151—21no—
71C2×D8×C151—24no—
72C2×D8×C151—22no—
73C4⋊Q8×C151D411yescharacteristic
74C4⋊Q8×C151C2311yescharacteristic
75#127×C151—22no—
76#127×C151—22no—
77#127×C151—22no—
78#127×C151—21no—
79#127×C151—21no—
80#127×C151D412yes—
81#127×C151D411yescharacteristic
82#127×C151C2311yescharacteristic
83#127×C151C2312yes—
84#124×C151—22no—
85#124×C151—21no—
86#124×C151—21no—
87#121×C151—22no—
88#121×C151—24no—
89#121×C151—24no—
90#121×C151—21no—
91#121×C151—22no—
92#121×C151—22no—
93#121×C151—22no—
94#121×C151—21no—
95#121×C151—21no—
96#121×C151D412yes—
97#120×C151—24no—
98#120×C151—24no—
99#120×C151—24no—
100C4×Q8×C151D411yescharacteristic
101C4×Q8×C151C2311yescharacteristic
102C4×D4×C151—22no—
103C4×D4×C151—21no—
104C4×D4×C151—21no—
105C4×D4×C151—22no—
106C4×D4×C151—21no—
107C4×D4×C151—21no—
108C4×D4×C151—21no—
109C4×D4×C151D411yescharacteristic
110C4×D4×C151C2312yes—
111C4×D4×C151C2311yescharacteristic
112#117×C151—21no—
113#117×C151D411yescharacteristic
114C2×C42×C151—21no—
115C2×C42×C151C2311yescharacteristic
116#107×C151—21no—
117#106×C151—21no—
118C4⋊C8×C151—21no—
119C4⋊C8×C151C2312yes—
120C4⋊C8×C151C2311yescharacteristic
121C4⋊C8×C151C2311yescharacteristic
122#103×C151—22no—
123#103×C151—21no—
124#102×C151—21no—
125#102×C151—24no—
126#102×C151—21no—
127#102×C151—21no—
128#102×C151—22no—
129C22⋊C8×C151—21no—
130C22⋊C8×C151—21no—
131C22⋊C8×C151—21no—
132#97×C151—21no—
133#97×C151—21no—
134C4×C8×C151C2312yes—
135C24×C151—24no—
136C24×C151—22no—
137#41×C151—22no—
138#41×C151—22no—
139#41×C151—48no—
140#41×C151—42no—
141#41×C151—42no—
142#41×C151—44no—
143#41×C151—44no—
144#41×C151—82no—
145C2×Q8×C151—21no—
146C2×Q8×C151—21no—
147C2×Q8×C151C2×D411yescharacteristic
148C2×D4×C151—21no—
149C2×D4×C151—24no—
150C2×D4×C151—24no—
151C2×D4×C151—22no—
152C2×D4×C151—24no—
153C2×D4×C151—24no—
154C2×D4×C151—22no—
155C2×D4×C151—21no—
156C2×D4×C151—22no—
157C2×D4×C151—21no—
158C2×D4×C151—22no—
159C2×D4×C151—22no—
160C2×D4×C151—22no—
161C2×D4×C151—21no—
162C2×D4×C151—21no—
163C2×D4×C151—21no—
164C2×D4×C151—48no—
165C2×D4×C151—42no—
166C2×D4×C151—42no—
167C2×D4×C151—416no—
168C2×D4×C151—44no—
169C2×D4×C151—44no—
170C2×D4×C151—41no—
171C2×D4×C151—82no—
172C2×D4×C151—82no—
173C2×D4×C151—82no—
174C2×D4×C151—82no—
175C2×D4×C151—84no—
176C2×D4×C151C2×D412yes—
177C2×D4×C151C2×D411yescharacteristic
178C2×D4×C151C2×D414yes—
179C22×C4×C151—21no—
180C22×C4×C151—24no—
181C22×C4×C151—21no—
182C22×C4×C151—21no—
183C22×C4×C151—22no—
184C22×C4×C151—41no—
185C22×C4×C151C2×D411yescharacteristic
186C22×C4×C151C2×D411yescharacteristic
187C22×C4×C151C2×D411yescharacteristic
188SD16×C151—42no—
189SD16×C151—82no—
190D8×C151—42no—
191D8×C151—82no—
192D8×C151—84no—
193C2×C8×C151—22no—
194C2×C8×C151—21no—
195C2×C8×C151—21no—
196C2×C8×C151—22no—
197C4⋊C4×C151—21no—
198C4⋊C4×C151—21no—
199C4⋊C4×C151—21no—
200C4⋊C4×C151—21no—
201C4⋊C4×C151—21no—
202C4⋊C4×C151C2×D412yes—
203C4⋊C4×C151C2×D411yescharacteristic
204C4⋊C4×C151C2×D411yescharacteristic
205C22⋊C4×C151—24no—
206C22⋊C4×C151—24no—
207C22⋊C4×C151—21no—
208C22⋊C4×C151—21no—
209C22⋊C4×C151—22no—
210C22⋊C4×C151—21no—
211C22⋊C4×C151—22no—
212C22⋊C4×C151—41no—
213C22⋊C4×C151—41no—
214C42×C151—21no—
215C42×C151—21no—
216C42×C151—21no—
217C42×C151C2×D411yescharacteristic
218C42×C151C2×D411yescharacteristic
219C42×C151C2×D411yescharacteristic
220C42×C151C2411yescharacteristic
221C23×C151—21no—
222C23×C151—22no—
223C23×C151—24no—
224C23×C151—44no—
225C23×C151—48no—
226C23×C151—41no—
227C23×C151—82no—
228C23×C151—82no—
229C23×C151—84no—
230C23×C151—84no—
231C23×C151#12011yescharacteristic
232Q8×C151—21no—
233Q8×C151—42no—
234Q8×C151—42no—
235Q8×C151AGL(1,ℤ/8)12yes—
236D4×C151—24no—
237D4×C151—21no—
238D4×C151—22no—
239D4×C151—22no—
240D4×C151—44no—
241D4×C151—44no—
242D4×C151—42no—
243D4×C151—48no—
244D4×C151—44no—
245D4×C151—42no—
246D4×C151—42no—
247D4×C151—44no—
248D4×C151—48no—
249D4×C151—42no—
250D4×C151—42no—
251D4×C151—41no—
252D4×C151—41no—
253D4×C151—42no—
254D4×C151—42no—
255D4×C151—42no—
256D4×C151—82no—
257D4×C151—84no—
258D4×C151—84no—
259D4×C151—82no—
260D4×C151AGL(1,ℤ/8)12yes—
261C2×C4×C151—24no—
262C2×C4×C151—21no—
263C2×C4×C151—21no—
264C2×C4×C151—21no—
265C2×C4×C151—22no—
266C2×C4×C151—22no—
267C2×C4×C151—21no—
268C2×C4×C151—21no—
269C2×C4×C151—41no—
270C2×C4×C151—48no—
271C2×C4×C151—42no—
272C2×C4×C151—42no—
273C2×C4×C151—42no—
274C2×C4×C151—42no—
275C2×C4×C151—41no—
276C2×C4×C151—82no—
277C2×C4×C151#12011yescharacteristic
278C2×C4×C151#12011yescharacteristic
279C2×C4×C151#12011yescharacteristic
280C2×C4×C151C22×D411yescharacteristic
281C2×C4×C151C22×D411yescharacteristic
282C2×C4×C151C22×D411yescharacteristic
283C8×C151—42no—
284C8×C151—41no—
285C8×C151—41no—
286C8×C151—42no—
287C22×C151—21no—
288C22×C151—22no—
289C22×C151—21no—
290C22×C151—21no—
291C22×C151—41no—
292C22×C151—41no—
293C22×C151—44no—
294C22×C151—42no—
295C22×C151—42no—
296C22×C151—44no—
297C22×C151—44no—
298C22×C151—42no—
299C22×C151—81no—
300C22×C151—81no—
301C22×C151—81no—
302C22×C151—164no—
303C22×C151—168no—
304C22×C151C2×#12011yescharacteristic
305C4×C151—21no—
306C4×C151—22no—
307C4×C151—21no—
308C4×C151—21no—
309C4×C151—21no—
310C4×C151—42no—
311C4×C151—41no—
312C4×C151—41no—
313C4×C151—81no—
314C4×C151C2×AGL(1,ℤ/8)11yescharacteristic
315C4×C151C2×AGL(1,ℤ/8)11yescharacteristic
316C4×C151#57612yes—
317C2×C151—41no—
318C2×C151—42no—
319C2×C151—82no—
320C2×C151—84no—
321C2×C151—161no—
322C2×C151#300311yescharacteristic
323C2×C151#300811yescharacteristic
324C2×C151#301911yescharacteristic
325#18814C15111yesmaximal, characteristic, direct product factor, Sylow
326C151#1881411yescharacteristic, direct product factor, Sylow
327#3498C2×C15111yescharacteristic, complemented
328#3214C2×C15111yescharacteristic, complemented
329#3205C2×C15111yescharacteristic, complemented
330#3115C2×C15111yescharacteristic, complemented
331#1642C2×C15112yescomplemented
332#1639C2×C15112yescomplemented
333#1621C2×C15112yescomplemented
334#1619C2×C15112yescomplemented
335#1555C2×C15111yescharacteristic, complemented
336#1496C2×C15111yescharacteristic, complemented
337#1488C2×C15111yescharacteristic, complemented
338C2×H(5,𝔽2)—21no—
339#550—21no—
340#550C22×C15112yes—
341D42—22no—
342D42—22no—
343D42—21no—
344D42C22×C15112yes—
345#535—22no—
346#535—21no—
347#532—21no—
348#532C22×C15111yescharacteristic
349#532C22×C15111yescharacteristic, complemented
350C2×#127—21no—
351C2×#127C22×C15111yescharacteristic
352C4×#41C22×C15111yescharacteristic
353#496—21no—
354#493C22×C15112yescomplemented
355#489—21no—
356#486C22×C15112yescomplemented
357#482—21no—
358#475C22×C15111yescharacteristic, complemented
359#474C22×C15111yescharacteristic, complemented
360#461C22×C15112yescomplemented
361#460C22×C15111yescharacteristic, complemented
362#459—22no—
363#459C22×C15111yescharacteristic, complemented
364#459C22×C15112yescomplemented
365#449—21no—
366#449—22no—
367#447—22no—
368#442—21no—
369#440—21no—
370#432C22×C15111yescharacteristic, complemented
371#432C22×C15111yescharacteristic, complemented
372#431C22×C15111yescharacteristic, complemented
373#331C22×C15112yescomplemented
374#331C22×C15112yescomplemented
375Q8⋊C8C22×C15112yescomplemented
376D4⋊C8C22×C15112yescomplemented
377D4⋊C8C22×C15112yescomplemented
378D4⋊C8C22×C15112yescomplemented
379H(5,𝔽2)—44no—
380H(5,𝔽2)—44no—
381C2×#41—22no—
382C2×#41—22no—
383C2×#41—21no—
384C2×#41—21no—
385C2×#41D4×C15111yescharacteristic
386C22×D4—22no—
387C22×D4—24no—
388C22×D4—22no—
389C22×D4—22no—
390C22×D4—22no—
391C22×D4—21no—
392C22×D4—21no—
393C22×D4D4×C15112yes—
394C2×SD16—21no—
395C2×SD16—22no—
396C2×D8—21no—
397C2×D8—24no—
398C2×D8—22no—
399C4⋊Q8D4×C15111yescharacteristic
400C4⋊Q8C23×C15111yescharacteristic
401#127—22no—
402#127—22no—
403#127—22no—
404#127—21no—
405#127—21no—
406#127D4×C15111yescharacteristic
407#127D4×C15112yes—
408#127C23×C15111yescharacteristic
409#127C23×C15112yes—
410#124—22no—
411#124—21no—
412#124—21no—
413#121—22no—
414#121—24no—
415#121—24no—
416#121—21no—
417#121—22no—
418#121—22no—
419#121—22no—
420#121—21no—
421#121—21no—
422#121D4×C15112yes—
423#120—24no—
424#120—24no—
425#120—24no—
426C4×Q8D4×C15111yescharacteristic
427C4×Q8C23×C15111yescharacteristic
428C4×D4—22no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—22no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4D4×C15111yescharacteristic
436C4×D4C23×C15111yescharacteristic
437C4×D4C23×C15112yes—
438#117—21no—
439#117D4×C15111yescharacteristic
440C2×C42—21no—
441C2×C42C23×C15111yescharacteristic
442#107—21no—
443#106—21no—
444C4⋊C8—21no—
445C4⋊C8C23×C15111yescharacteristic
446C4⋊C8C23×C15111yescharacteristic
447C4⋊C8C23×C15112yes—
448#103—21no—
449#103—22no—
450#102—21no—
451#102—24no—
452#102—21no—
453#102—21no—
454#102—22no—
455C22⋊C8—21no—
456C22⋊C8—21no—
457C22⋊C8—21no—
458#97—21no—
459#97—21no—
460C4×C8C23×C15112yes—
461C24—24no—
462C24—22no—
463#41—22no—
464#41—22no—
465#41—48no—
466#41—42no—
467#41—42no—
468#41—44no—
469#41—44no—
470#41—82no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8C2×D4×C15111yescharacteristic
474C2×D4—21no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—22no—
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—48no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—416no—
494C2×D4—44no—
495C2×D4—44no—
496C2×D4—41no—
497C2×D4—82no—
498C2×D4—82no—
499C2×D4—82no—
500C2×D4—82no—
501C2×D4—84no—
502C2×D4C2×D4×C15112yes—
503C2×D4C2×D4×C15111yescharacteristic
504C2×D4C2×D4×C15114yes—
505C22×C4—21no—
506C22×C4—24no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—41no—
511C22×C4C2×D4×C15111yescharacteristic
512C22×C4C2×D4×C15111yescharacteristic
513C22×C4C2×D4×C15111yescharacteristic
514SD16—42no—
515SD16—82no—
516D8—42no—
517D8—82no—
518D8—84no—
519C2×C8—22no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4C2×D4×C15112yes—
529C4⋊C4C2×D4×C15111yescharacteristic
530C4⋊C4C2×D4×C15111yescharacteristic
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—22no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42C2×D4×C15111yescharacteristic
544C42C2×D4×C15111yescharacteristic
545C42C2×D4×C15111yescharacteristic
546C42C24×C15111yescharacteristic
547C23—21no—
548C23—22no—
549C23—24no—
550C23—44no—
551C23—48no—
552C23—41no—
553C23—82no—
554C23—82no—
555C23—84no—
556C23—84no—
557C23#120×C15111yescharacteristic
558Q8—21no—
559Q8—42no—
560Q8—42no—
561Q8AGL(1,ℤ/8)×C15112yes—
562D4—24no—
563D4—21no—
564D4—22no—
565D4—22no—
566D4—44no—
567D4—44no—
568D4—42no—
569D4—48no—
570D4—44no—
571D4—42no—
572D4—42no—
573D4—44no—
574D4—48no—
575D4—42no—
576D4—42no—
577D4—41no—
578D4—41no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—82no—
583D4—84no—
584D4—84no—
585D4—82no—
586D4AGL(1,ℤ/8)×C15112yes—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—82no—
603C2×C4#120×C15111yescharacteristic
604C2×C4#120×C15111yescharacteristic
605C2×C4#120×C15111yescharacteristic
606C2×C4C22×D4×C15111yescharacteristic
607C2×C4C22×D4×C15111yescharacteristic
608C2×C4C22×D4×C15111yescharacteristic
609C8—42no—
610C8—41no—
611C8—41no—
612C8—42no—
613C22—21no—
614C22—22no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—41no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—164no—
629C22—168no—
630C22C2×#120×C15111yescharacteristic
631C4—21no—
632C4—22no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4C2×AGL(1,ℤ/8)×C15111yescharacteristic
641C4C2×AGL(1,ℤ/8)×C15111yescharacteristic
642C4#576×C15112yes—
643C2—41no—
644C2—42no—
645C2—82no—
646C2—84no—
647C2—161no—
648C2#3003×C15111yescharacteristic
649C2#3008×C15111yescharacteristic
650C2#3019×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3498×C151C2semidirect
2. #3214×C151C2semidirect
3. #3205×C151C2semidirect
4. #3115×C151C2semidirect
5. #1642×C151C2semidirect
6. #1639×C151C2semidirect
7. #1621×C151C2semidirect
8. #1619×C151C2semidirect
9. #1555×C151C2semidirect
10. #1496×C151C2semidirect
11. #1488×C151C2semidirect
23. #532×C151C22semidirect
28. #493×C151C22semidirect
30. #486×C151C22semidirect
32. #475×C151C22semidirect
33. #474×C151C22semidirect
34. #461×C151C22semidirect
35. #460×C151C22semidirect
37. #459×C151C22semidirect
38. #459×C151C22semidirect
44. #432×C151C22semidirect
45. #432×C151C22semidirect
46. #431×C151C22semidirect
47. #331×C151C22semidirect
48. #331×C151C22semidirect
49. Q8⋊C8×C151C22semidirect
50. D4⋊C8×C151C22semidirect
51. D4⋊C8×C151C22semidirect
52. D4⋊C8×C151C22semidirect
325. #18814C151direct (N × H)
326. C151#18814direct (N × H)
327. #3498C2×C151semidirect
328. #3214C2×C151semidirect
329. #3205C2×C151semidirect
330. #3115C2×C151semidirect
331. #1642C2×C151semidirect
332. #1639C2×C151semidirect
333. #1621C2×C151semidirect
334. #1619C2×C151semidirect
335. #1555C2×C151semidirect
336. #1496C2×C151semidirect
337. #1488C2×C151semidirect
349. #532C22×C151semidirect
354. #493C22×C151semidirect
356. #486C22×C151semidirect
358. #475C22×C151semidirect
359. #474C22×C151semidirect
360. #461C22×C151semidirect
361. #460C22×C151semidirect
363. #459C22×C151semidirect
364. #459C22×C151semidirect
370. #432C22×C151semidirect
371. #432C22×C151semidirect
372. #431C22×C151semidirect
373. #331C22×C151semidirect
374. #331C22×C151semidirect
375. Q8⋊C8C22×C151semidirect
376. D4⋊C8C22×C151semidirect
377. D4⋊C8C22×C151semidirect
378. D4⋊C8C22×C151semidirect

Maximal subgroups

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

subgroupindex
#3498×C1512
#3214×C1512
#3205×C1512
#3115×C1512
#1642×C1512
#1639×C1512
#1621×C1512
#1619×C1512
#1555×C1512
#1496×C1512
#1488×C1512
#18814151