Finite Group Databasebeta

← C151×#24321 #50329246

Subgroups — C151×#24321 #50329246

Order 38656.

1036 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3503×C151C211yesmaximal, characteristic, complemented
2#3077×C151C211yesmaximal, characteristic, complemented
3C4×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic
4#2563×C151C211yesmaximal, characteristic, complemented
5#2061×C151C212yesmaximal, complemented
6#2027×C151C212yesmaximal, complemented
7#1997×C151C211yesmaximal, characteristic, complemented
8#1986×C151C211yesmaximal, characteristic, complemented
9#1883×C151C212yesmaximal
10#1860×C151C212yesmaximal, complemented
11#1806×C151C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C151C2211yescharacteristic
13Q82×C151—21no—
14#557×C151—21no—
15#554×C151—21no—
16#554×C151—22no—
17D4×Q8×C151—22no—
18D4×Q8×C151—21no—
19#544×C151—22no—
20#544×C151C2212yes—
21#533×C151—21no—
22#533×C151C2211yescharacteristic
23C2×C4⋊Q8×C151—21no—
24C2×C4⋊Q8×C151C2211yescharacteristic
25#520×C151C2212yes—
26C4×#41×C151C2211yescharacteristic
27C2×C4×D4×C151C2211yescharacteristic
28#478×C151—22no—
29#476×C151—21no—
30#474×C151—21no—
31#442×C151—21no—
32#440×C151—21no—
33C4×SD16×C151—21no—
34C4×D8×C151—21no—
35#428×C151C2212yes—
36#428×C151C2211yescharacteristic
37#423×C151C2211yescharacteristic
38C2×#104×C151C2212yes—
39#418×C151C2212yes—
40#418×C151C2211yescharacteristic
41#417×C151C2211yescharacteristic
42C4×M16×C151C2211yescharacteristic
43#393×C151—22no—
44#390×C151C2212yes—
45#382×C151C2212yes—
46#380×C151—22no—
47C4×C4⋊C4×C151C2211yescharacteristic
48#343×C151C2212yes—
49#343×C151C2212yes—
50#339×C151C2212yes—
51#339×C151C2212yes—
52#339×C151C2212yes—
53C2×#41×C151C2311yescharacteristic
54C22×Q8×C151—22no—
55C22×Q8×C151D412yes—
56C22×D4×C151C2311yescharacteristic
57C23×C4×C151—21no—
58AGL(1,ℤ/8)×C151—22no—
59AGL(1,ℤ/8)×C151Q814yescomplemented
60C2×SD16×C151—21no—
61C2×D8×C151—21no—
62C2×M16×C151C2311yescharacteristic
63C2×M16×C151C2312yes—
64C2×M16×C151C2311yescharacteristic
65C4⋊Q8×C151—22no—
66C4⋊Q8×C151—22no—
67C4⋊Q8×C151—24no—
68C4⋊Q8×C151—22no—
69C4⋊Q8×C151—22no—
70C4⋊Q8×C151—21no—
71C4⋊Q8×C151—21no—
72C4⋊Q8×C151—21no—
73C4⋊Q8×C151—21no—
74C4⋊Q8×C151D411yescharacteristic
75C4⋊Q8×C151D411yescharacteristic
76#125×C151—24no—
77#125×C151—22no—
78#125×C151—22no—
79#125×C151D412yes—
80C22⋊Q8×C151—22no—
81C22⋊Q8×C151—22no—
82C22⋊Q8×C151—24no—
83C22⋊Q8×C151—24no—
84C22⋊Q8×C151—21no—
85C22⋊Q8×C151—21no—
86C22⋊Q8×C151—24no—
87C4×Q8×C151—22no—
88C4×Q8×C151—22no—
89C4×Q8×C151—21no—
90C4×Q8×C151—21no—
91C4×Q8×C151—22no—
92C4×Q8×C151D411yescharacteristic
93C4×Q8×C151D411yescharacteristic
94C4×D4×C151—22no—
95C4×D4×C151—21no—
96C4×D4×C151—21no—
97C4×D4×C151—24no—
98C4×D4×C151—21no—
99C4×D4×C151—21no—
100C4×D4×C151—21no—
101C4×D4×C151—21no—
102C4×D4×C151D411yescharacteristic
103C4×D4×C151D411yescharacteristic
104#117×C151—22no—
105#117×C151—21no—
106#117×C151C2311yescharacteristic
107#117×C151C2312yes—
108C2×C4⋊C4×C151—22no—
109C2×C4⋊C4×C151—22no—
110C2×C4⋊C4×C151—21no—
111C2×C4⋊C4×C151D412yes—
112C2×C4⋊C4×C151C2311yescharacteristic
113C2×C4⋊C4×C151C2312yes—
114C2×C22⋊C4×C151—21no—
115C2×C22⋊C4×C151—24no—
116C2×C42×C151—21no—
117C2×C42×C151C2312yes—
118C2×C42×C151C2311yescharacteristic
119#107×C151—21no—
120#107×C151—22no—
121#106×C151—21no—
122#106×C151—22no—
123C4⋊C8×C151—21no—
124C4⋊C8×C151—21no—
125#104×C151—24no—
126#103×C151—21no—
127#102×C151—21no—
128#102×C151—21no—
129#102×C151—21no—
130#102×C151—22no—
131#102×C151—22no—
132#97×C151—21no—
133C4×C8×C151—21no—
134#95×C151—21no—
135#95×C151—24no—
136C24×C151—21no—
137#41×C151—21no—
138#41×C151—21no—
139#41×C151C2×Q812yes—
140#41×C151#4112yes—
141C2×Q8×C151—24no—
142C2×Q8×C151—21no—
143C2×Q8×C151—22no—
144C2×Q8×C151—21no—
145C2×Q8×C151—22no—
146C2×Q8×C151—82no—
147C2×Q8×C151—84no—
148C2×Q8×C151C2×D411yescharacteristic
149C2×D4×C151—21no—
150C2×D4×C151—21no—
151C2×D4×C151—21no—
152C2×D4×C151—21no—
153C2×D4×C151—21no—
154C2×D4×C151—21no—
155C2×D4×C151C2×D411yescharacteristic
156C2×D4×C151C2×Q812yescomplemented
157C22×C4×C151—21no—
158C22×C4×C151—24no—
159C22×C4×C151—21no—
160C22×C4×C151—21no—
161C22×C4×C151—21no—
162C22×C4×C151—22no—
163C22×C4×C151—44no—
164C22×C4×C151—41no—
165C22×C4×C151—41no—
166C22×C4×C151C2×D411yescharacteristic
167C22×C4×C151C2×D414yes—
168C22×C4×C151C2×D411yescharacteristic
169C22×C4×C151C2411yescharacteristic
170SD16×C151—24no—
171D8×C151—24no—
172M16×C151—21no—
173M16×C151—22no—
174M16×C151—21no—
175M16×C151—21no—
176M16×C151—22no—
177M16×C151C2×Q812yescomplemented
178C2×C8×C151—21no—
179C2×C8×C151—21no—
180C2×C8×C151—22no—
181C4⋊C4×C151—24no—
182C4⋊C4×C151—24no—
183C4⋊C4×C151—24no—
184C4⋊C4×C151—24no—
185C4⋊C4×C151—21no—
186C4⋊C4×C151—22no—
187C4⋊C4×C151—22no—
188C4⋊C4×C151—21no—
189C4⋊C4×C151—21no—
190C4⋊C4×C151—21no—
191C4⋊C4×C151—22no—
192C4⋊C4×C151—22no—
193C4⋊C4×C151—22no—
194C4⋊C4×C151—22no—
195C4⋊C4×C151—21no—
196C4⋊C4×C151—21no—
197C4⋊C4×C151—42no—
198C4⋊C4×C151C2×D412yes—
199C4⋊C4×C151C2×D411yescharacteristic
200C4⋊C4×C151C2×D412yes—
201C4⋊C4×C151C2×D411yescharacteristic
202C22⋊C4×C151—24no—
203C22⋊C4×C151—21no—
204C22⋊C4×C151—22no—
205C22⋊C4×C151—21no—
206C22⋊C4×C151—21no—
207C22⋊C4×C151—41no—
208C22⋊C4×C151—41no—
209C22⋊C4×C151—84no—
210C42×C151—24no—
211C42×C151—21no—
212C42×C151—21no—
213C42×C151—21no—
214C42×C151—22no—
215C42×C151—22no—
216C42×C151—22no—
217C42×C151C2×D411yescharacteristic
218C42×C151C2×D411yescharacteristic
219C42×C151C2×D411yescharacteristic
220C42×C151C2×D411yescharacteristic
221C23×C151—21no—
222C23×C151—22no—
223C23×C151—42no—
224C23×C151—42no—
225C23×C151—41no—
226C23×C151—41no—
227C23×C151C22×D411yescharacteristic
228Q8×C151—21no—
229Q8×C151—44no—
230Q8×C151—42no—
231Q8×C151—42no—
232Q8×C151—82no—
233Q8×C151—84no—
234Q8×C151C22⋊Q812yes—
235D4×C151—22no—
236D4×C151—22no—
237D4×C151—22no—
238D4×C151—21no—
239D4×C151—22no—
240D4×C151—41no—
241D4×C151—41no—
242D4×C151C22⋊Q812yes—
243C2×C4×C151—21no—
244C2×C4×C151—22no—
245C2×C4×C151—22no—
246C2×C4×C151—22no—
247C2×C4×C151—21no—
248C2×C4×C151—24no—
249C2×C4×C151—22no—
250C2×C4×C151—22no—
251C2×C4×C151—21no—
252C2×C4×C151—41no—
253C2×C4×C151—42no—
254C2×C4×C151—42no—
255C2×C4×C151—44no—
256C2×C4×C151—42no—
257C2×C4×C151—42no—
258C2×C4×C151—42no—
259C2×C4×C151—41no—
260C2×C4×C151—42no—
261C2×C4×C151—82no—
262C2×C4×C151—84no—
263C2×C4×C151—82no—
264C2×C4×C151#12011yescharacteristic
265C2×C4×C151#12011yescharacteristic
266C2×C4×C151#12011yescharacteristic
267C2×C4×C151#12011yescharacteristic
268C2×C4×C151C22×D411yescharacteristic
269C2×C4×C151C22×D411yescharacteristic
270C2×C4×C151C22×Q811yescharacteristic
271C2×C4×C151C2×#4111yescharacteristic
272C2×C4×C1512−1+411yescharacteristic
273C2×C4×C1512−1+411yescharacteristic
274C8×C151—22no—
275C8×C151—41no—
276C8×C151—42no—
277C22×C151—21no—
278C22×C151—21no—
279C22×C151—22no—
280C22×C151—44no—
281C22×C151—41no—
282C22×C151—41no—
283C22×C151—42no—
284C22×C151—82no—
285C22×C151—81no—
286C22×C151—81no—
287C22×C151C2×#12011yescharacteristic
288C22×C151#54811yescharacteristic
289C22×C151D4×Q811yescharacteristic
290C4×C151—22no—
291C4×C151—21no—
292C4×C151—21no—
293C4×C151—21no—
294C4×C151—41no—
295C4×C151—42no—
296C4×C151—41no—
297C4×C151—44no—
298C4×C151—42no—
299C4×C151—82no—
300C4×C151—81no—
301C4×C151#45412yes—
302C4×C151C2×C22⋊Q811yescharacteristic
303C4×C151#53611yescharacteristic
304C4×C151#54811yescharacteristic
305C4×C151D4×Q811yescharacteristic
306C2×C151—21no—
307C2×C151—21no—
308C2×C151—42no—
309C2×C151—82no—
310C2×C151#240911yescharacteristic
311C2×C151C2×#45411yescharacteristic
312C2×C151#301911yescharacteristic
313#24321C15111yesmaximal, characteristic, direct product factor, Sylow
314C151#2432111yescharacteristic, direct product factor, Sylow
315#3503C2×C15111yescharacteristic, complemented
316#3077C2×C15111yescharacteristic, complemented
317C4×AGL(1,ℤ/8)C2×C15111yescharacteristic
318#2563C2×C15111yescharacteristic, complemented
319#2061C2×C15112yescomplemented
320#2027C2×C15112yescomplemented
321#1997C2×C15111yescharacteristic, complemented
322#1986C2×C15111yescharacteristic, complemented
323#1883C2×C15112yes—
324#1860C2×C15112yescomplemented
325#1806C2×C15111yescharacteristic, complemented
326C2×AGL(1,ℤ/8)C22×C15111yescharacteristic
327Q82—21no—
328#557—21no—
329#554—21no—
330#554—22no—
331D4×Q8—22no—
332D4×Q8—21no—
333#544—22no—
334#544C22×C15112yes—
335#533—21no—
336#533C22×C15111yescharacteristic
337C2×C4⋊Q8—21no—
338C2×C4⋊Q8C22×C15111yescharacteristic
339#520C22×C15112yes—
340C4×#41C22×C15111yescharacteristic
341C2×C4×D4C22×C15111yescharacteristic
342#478—22no—
343#476—21no—
344#474—21no—
345#442—21no—
346#440—21no—
347C4×SD16—21no—
348C4×D8—21no—
349#428C22×C15112yes—
350#428C22×C15111yescharacteristic
351#423C22×C15111yescharacteristic
352C2×#104C22×C15112yes—
353#418C22×C15112yes—
354#418C22×C15111yescharacteristic
355#417C22×C15111yescharacteristic
356C4×M16C22×C15111yescharacteristic
357#393—22no—
358#390C22×C15112yes—
359#382C22×C15112yes—
360#380—22no—
361C4×C4⋊C4C22×C15111yescharacteristic
362#343C22×C15112yes—
363#343C22×C15112yes—
364#339C22×C15112yes—
365#339C22×C15112yes—
366#339C22×C15112yes—
367C2×#41C23×C15111yescharacteristic
368C22×Q8—22no—
369C22×Q8D4×C15112yes—
370C22×D4C23×C15111yescharacteristic
371C23×C4—21no—
372AGL(1,ℤ/8)—22no—
373AGL(1,ℤ/8)Q8×C15114yescomplemented
374C2×SD16—21no—
375C2×D8—21no—
376C2×M16C23×C15111yescharacteristic
377C2×M16C23×C15112yes—
378C2×M16C23×C15111yescharacteristic
379C4⋊Q8—22no—
380C4⋊Q8—22no—
381C4⋊Q8—24no—
382C4⋊Q8—22no—
383C4⋊Q8—22no—
384C4⋊Q8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8D4×C15111yescharacteristic
389C4⋊Q8D4×C15111yescharacteristic
390#125—24no—
391#125—22no—
392#125—22no—
393#125D4×C15112yes—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—24no—
397C22⋊Q8—24no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—24no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8D4×C15111yescharacteristic
407C4×Q8D4×C15111yescharacteristic
408C4×D4—22no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—24no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4D4×C15111yescharacteristic
417C4×D4D4×C15111yescharacteristic
418#117—22no—
419#117—21no—
420#117C23×C15111yescharacteristic
421#117C23×C15112yes—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4D4×C15112yes—
426C2×C4⋊C4C23×C15111yescharacteristic
427C2×C4⋊C4C23×C15112yes—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—24no—
430C2×C42—21no—
431C2×C42C23×C15112yes—
432C2×C42C23×C15111yescharacteristic
433#107—21no—
434#107—22no—
435#106—21no—
436#106—22no—
437C4⋊C8—21no—
438C4⋊C8—21no—
439#104—24no—
440#103—21no—
441#102—21no—
442#102—21no—
443#102—21no—
444#102—22no—
445#102—22no—
446#97—21no—
447C4×C8—21no—
448#95—21no—
449#95—24no—
450C24—21no—
451#41—21no—
452#41—21no—
453#41C2×Q8×C15112yes—
454#41#41×C15112yes—
455C2×Q8—24no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—82no—
461C2×Q8—84no—
462C2×Q8C2×D4×C15111yescharacteristic
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4C2×D4×C15111yescharacteristic
470C2×D4C2×Q8×C15112yescomplemented
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—44no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4C2×D4×C15111yescharacteristic
481C22×C4C2×D4×C15114yes—
482C22×C4C2×D4×C15111yescharacteristic
483C22×C4C24×C15111yescharacteristic
484SD16—24no—
485D8—24no—
486M16—21no—
487M16—22no—
488M16—21no—
489M16—21no—
490M16—22no—
491M16C2×Q8×C15112yescomplemented
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4C2×D4×C15112yes—
513C4⋊C4C2×D4×C15111yescharacteristic
514C4⋊C4C2×D4×C15112yes—
515C4⋊C4C2×D4×C15111yescharacteristic
516C22⋊C4—24no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—84no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42C2×D4×C15111yescharacteristic
532C42C2×D4×C15111yescharacteristic
533C42C2×D4×C15111yescharacteristic
534C42C2×D4×C15111yescharacteristic
535C23—21no—
536C23—22no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23C22×D4×C15111yescharacteristic
542Q8—21no—
543Q8—44no—
544Q8—42no—
545Q8—42no—
546Q8—82no—
547Q8—84no—
548Q8C22⋊Q8×C15112yes—
549D4—22no—
550D4—22no—
551D4—22no—
552D4—21no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4C22⋊Q8×C15112yes—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—82no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4#120×C15111yescharacteristic
579C2×C4#120×C15111yescharacteristic
580C2×C4#120×C15111yescharacteristic
581C2×C4#120×C15111yescharacteristic
582C2×C4C22×D4×C15111yescharacteristic
583C2×C4C22×D4×C15111yescharacteristic
584C2×C4C22×Q8×C15111yescharacteristic
585C2×C4C2×#41×C15111yescharacteristic
586C2×C42−1+4×C15111yescharacteristic
587C2×C42−1+4×C15111yescharacteristic
588C8—22no—
589C8—41no—
590C8—42no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—44no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—82no—
599C22—81no—
600C22—81no—
601C22C2×#120×C15111yescharacteristic
602C22#548×C15111yescharacteristic
603C22D4×Q8×C15111yescharacteristic
604C4—22no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—81no—
615C4#454×C15112yes—
616C4C2×C22⋊Q8×C15111yescharacteristic
617C4#536×C15111yescharacteristic
618C4#548×C15111yescharacteristic
619C4D4×Q8×C15111yescharacteristic
620C2—21no—
621C2—21no—
622C2—42no—
623C2—82no—
624C2#2409×C15111yescharacteristic
625C2C2×#454×C15111yescharacteristic
626C2#3019×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×AGL(1,ℤ/8)×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3503×C151C2semidirect
2. #3077×C151C2semidirect
4. #2563×C151C2semidirect
5. #2061×C151C2semidirect
6. #2027×C151C2semidirect
7. #1997×C151C2semidirect
8. #1986×C151C2semidirect
10. #1860×C151C2semidirect
11. #1806×C151C2semidirect
59. AGL(1,ℤ/8)×C151Q8semidirect
156. C2×D4×C151C2×Q8semidirect
177. M16×C151C2×Q8semidirect
313. #24321C151direct (N × H)
314. C151#24321direct (N × H)
315. #3503C2×C151semidirect
316. #3077C2×C151semidirect
318. #2563C2×C151semidirect
319. #2061C2×C151semidirect
320. #2027C2×C151semidirect
321. #1997C2×C151semidirect
322. #1986C2×C151semidirect
324. #1860C2×C151semidirect
325. #1806C2×C151semidirect
373. AGL(1,ℤ/8)Q8×C151semidirect
470. C2×D4C2×Q8×C151semidirect
491. M16C2×Q8×C151semidirect

Maximal subgroups

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

subgroupindex
#3503×C1512
#3077×C1512
C4×AGL(1,ℤ/8)×C1512
#2563×C1512
#2061×C1512
#2027×C1512
#1997×C1512
#1986×C1512
#1883×C1512
#1860×C1512
#1806×C1512
#24321151