Finite Group Databasebeta

← C151×#24320 #50329245

Subgroups — C151×#24320 #50329245

Order 38656.

1092 conjugacy classes of subgroups.

The subgroup lattice has 632 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#2542×C151C211yesmaximal, characteristic, complemented
5#2060×C151C212yesmaximal, complemented
6#2027×C151C212yesmaximal, complemented
7#1997×C151C211yesmaximal, characteristic, complemented
8#1986×C151C211yesmaximal, characteristic, complemented
9#1883×C151C212yesmaximal
10#1859×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
25C2×C22⋊Q8×C151—22no—
26#520×C151C2212yes—
27C4×#41×C151C2211yescharacteristic
28C2×C4×D4×C151C2211yescharacteristic
29#478×C151—22no—
30#476×C151—21no—
31#474×C151—21no—
32#442×C151—21no—
33#440×C151—21no—
34C4×SD16×C151—21no—
35C4×D8×C151—21no—
36#428×C151C2212yes—
37#428×C151C2211yescharacteristic
38#423×C151C2211yescharacteristic
39C2×#104×C151C2212yes—
40#418×C151C2212yes—
41#418×C151C2211yescharacteristic
42#417×C151C2211yescharacteristic
43C4×M16×C151C2211yescharacteristic
44#399×C151—22no—
45#390×C151C2212yes—
46#384×C151C2212yes—
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—
87C22⋊Q8×C151—48no—
88C4×Q8×C151—22no—
89C4×Q8×C151—22no—
90C4×Q8×C151—21no—
91C4×Q8×C151—21no—
92C4×Q8×C151—22no—
93C4×Q8×C151D411yescharacteristic
94C4×Q8×C151D411yescharacteristic
95C4×D4×C151—22no—
96C4×D4×C151—21no—
97C4×D4×C151—21no—
98C4×D4×C151—24no—
99C4×D4×C151—21no—
100C4×D4×C151—21no—
101C4×D4×C151—21no—
102C4×D4×C151—21no—
103C4×D4×C151D411yescharacteristic
104C4×D4×C151D411yescharacteristic
105#117×C151—22no—
106#117×C151—21no—
107#117×C151C2311yescharacteristic
108#117×C151C2312yes—
109C2×C4⋊C4×C151—22no—
110C2×C4⋊C4×C151—21no—
111C2×C4⋊C4×C151—22no—
112C2×C4⋊C4×C151—24no—
113C2×C4⋊C4×C151D412yes—
114C2×C4⋊C4×C151C2311yescharacteristic
115C2×C4⋊C4×C151C2312yes—
116C2×C22⋊C4×C151—24no—
117C2×C22⋊C4×C151—21no—
118C2×C42×C151—21no—
119C2×C42×C151C2312yes—
120C2×C42×C151C2311yescharacteristic
121#107×C151—21no—
122#107×C151—22no—
123#106×C151—21no—
124#106×C151—22no—
125C4⋊C8×C151—21no—
126C4⋊C8×C151—21no—
127#104×C151—24no—
128#103×C151—21no—
129#102×C151—21no—
130#102×C151—21no—
131#102×C151—21no—
132#102×C151—22no—
133#102×C151—22no—
134#97×C151—21no—
135C4×C8×C151—21no—
136#95×C151—21no—
137C24×C151—21no—
138#41×C151—21no—
139#41×C151—21no—
140#41×C151C2×Q812yes—
141#41×C151#4112yes—
142C2×Q8×C151—24no—
143C2×Q8×C151—21no—
144C2×Q8×C151—22no—
145C2×Q8×C151—21no—
146C2×Q8×C151—22no—
147C2×Q8×C151—44no—
148C2×Q8×C151—84no—
149C2×Q8×C151C2×D411yescharacteristic
150C2×D4×C151—21no—
151C2×D4×C151—21no—
152C2×D4×C151—21no—
153C2×D4×C151—21no—
154C2×D4×C151—21no—
155C2×D4×C151—21no—
156C2×D4×C151C2×D411yescharacteristic
157C2×D4×C151C2×Q812yescomplemented
158C22×C4×C151—21no—
159C22×C4×C151—22no—
160C22×C4×C151—24no—
161C22×C4×C151—21no—
162C22×C4×C151—21no—
163C22×C4×C151—21no—
164C22×C4×C151—44no—
165C22×C4×C151—41no—
166C22×C4×C151—41no—
167C22×C4×C151C2×D411yescharacteristic
168C22×C4×C151C2×D414yes—
169C22×C4×C151C2×D411yescharacteristic
170C22×C4×C151C2411yescharacteristic
171SD16×C151—24no—
172D8×C151—24no—
173M16×C151—21no—
174M16×C151—22no—
175M16×C151—21no—
176M16×C151—21no—
177M16×C151—22no—
178M16×C151C2×Q812yescomplemented
179C2×C8×C151—21no—
180C2×C8×C151—21no—
181C2×C8×C151—22no—
182C4⋊C4×C151—24no—
183C4⋊C4×C151—24no—
184C4⋊C4×C151—24no—
185C4⋊C4×C151—24no—
186C4⋊C4×C151—24no—
187C4⋊C4×C151—21no—
188C4⋊C4×C151—22no—
189C4⋊C4×C151—22no—
190C4⋊C4×C151—21no—
191C4⋊C4×C151—21no—
192C4⋊C4×C151—21no—
193C4⋊C4×C151—22no—
194C4⋊C4×C151—22no—
195C4⋊C4×C151—22no—
196C4⋊C4×C151—22no—
197C4⋊C4×C151—21no—
198C4⋊C4×C151—21no—
199C4⋊C4×C151—48no—
200C4⋊C4×C151C2×D412yes—
201C4⋊C4×C151C2×D411yescharacteristic
202C4⋊C4×C151C2×D412yes—
203C4⋊C4×C151C2×D411yescharacteristic
204C22⋊C4×C151—24no—
205C22⋊C4×C151—21no—
206C22⋊C4×C151—22no—
207C22⋊C4×C151—21no—
208C22⋊C4×C151—21no—
209C22⋊C4×C151—48no—
210C22⋊C4×C151—41no—
211C22⋊C4×C151—41no—
212C42×C151—24no—
213C42×C151—21no—
214C42×C151—21no—
215C42×C151—21no—
216C42×C151—22no—
217C42×C151—22no—
218C42×C151—22no—
219C42×C151C2×D411yescharacteristic
220C42×C151C2×D411yescharacteristic
221C42×C151C2×D411yescharacteristic
222C42×C151C2×D411yescharacteristic
223C23×C151—21no—
224C23×C151—22no—
225C23×C151—42no—
226C23×C151—42no—
227C23×C151—41no—
228C23×C151—41no—
229C23×C151C22×D411yescharacteristic
230Q8×C151—21no—
231Q8×C151—42no—
232Q8×C151—44no—
233Q8×C151—42no—
234Q8×C151—82no—
235Q8×C151—84no—
236Q8×C151C22⋊Q812yescomplemented
237D4×C151—22no—
238D4×C151—22no—
239D4×C151—22no—
240D4×C151—21no—
241D4×C151—22no—
242D4×C151—41no—
243D4×C151—41no—
244D4×C151C22⋊Q812yescomplemented
245C2×C4×C151—21no—
246C2×C4×C151—22no—
247C2×C4×C151—22no—
248C2×C4×C151—22no—
249C2×C4×C151—24no—
250C2×C4×C151—21no—
251C2×C4×C151—24no—
252C2×C4×C151—22no—
253C2×C4×C151—22no—
254C2×C4×C151—21no—
255C2×C4×C151—41no—
256C2×C4×C151—42no—
257C2×C4×C151—42no—
258C2×C4×C151—42no—
259C2×C4×C151—42no—
260C2×C4×C151—44no—
261C2×C4×C151—42no—
262C2×C4×C151—41no—
263C2×C4×C151—82no—
264C2×C4×C151—84no—
265C2×C4×C151—82no—
266C2×C4×C151#12011yescharacteristic
267C2×C4×C151#12011yescharacteristic
268C2×C4×C151#12011yescharacteristic
269C2×C4×C151#12011yescharacteristic
270C2×C4×C151C22×D411yescharacteristic
271C2×C4×C151C22×D411yescharacteristic
272C2×C4×C151C22×Q811yescharacteristic
273C2×C4×C151C2×#4111yescharacteristic
274C2×C4×C1512−1+411yescharacteristic
275C2×C4×C1512−1+411yescharacteristic
276C8×C151—22no—
277C8×C151—41no—
278C8×C151—42no—
279C22×C151—21no—
280C22×C151—21no—
281C22×C151—22no—
282C22×C151—44no—
283C22×C151—41no—
284C22×C151—41no—
285C22×C151—42no—
286C22×C151—82no—
287C22×C151—81no—
288C22×C151—81no—
289C22×C151C2×#12011yescharacteristic
290C22×C151#54811yescharacteristic
291C22×C151D4×Q811yescharacteristic
292C4×C151—22no—
293C4×C151—21no—
294C4×C151—21no—
295C4×C151—21no—
296C4×C151—44no—
297C4×C151—41no—
298C4×C151—44no—
299C4×C151—42no—
300C4×C151—41no—
301C4×C151—42no—
302C4×C151—81no—
303C4×C151#45312yes—
304C4×C151C2×C22⋊Q811yescharacteristic
305C4×C151#53611yescharacteristic
306C4×C151#54811yescharacteristic
307C4×C151D4×Q811yescharacteristic
308C2×C151—21no—
309C2×C151—21no—
310C2×C151—42no—
311C2×C151—82no—
312C2×C151#240911yescharacteristic
313C2×C151C2×#45311yescharacteristic
314C2×C151#302011yescharacteristic
315#24320C15111yesmaximal, characteristic, direct product factor, Sylow
316C151#2432011yescharacteristic, direct product factor, Sylow
317#3503C2×C15111yescharacteristic, complemented
318#3077C2×C15111yescharacteristic, complemented
319C4×AGL(1,ℤ/8)C2×C15111yescharacteristic
320#2542C2×C15111yescharacteristic, complemented
321#2060C2×C15112yescomplemented
322#2027C2×C15112yescomplemented
323#1997C2×C15111yescharacteristic, complemented
324#1986C2×C15111yescharacteristic, complemented
325#1883C2×C15112yes—
326#1859C2×C15112yescomplemented
327#1806C2×C15111yescharacteristic, complemented
328C2×AGL(1,ℤ/8)C22×C15111yescharacteristic
329Q82—21no—
330#557—21no—
331#554—21no—
332#554—22no—
333D4×Q8—22no—
334D4×Q8—21no—
335#544—22no—
336#544C22×C15112yes—
337#533—21no—
338#533C22×C15111yescharacteristic
339C2×C4⋊Q8—21no—
340C2×C4⋊Q8C22×C15111yescharacteristic
341C2×C22⋊Q8—22no—
342#520C22×C15112yes—
343C4×#41C22×C15111yescharacteristic
344C2×C4×D4C22×C15111yescharacteristic
345#478—22no—
346#476—21no—
347#474—21no—
348#442—21no—
349#440—21no—
350C4×SD16—21no—
351C4×D8—21no—
352#428C22×C15112yes—
353#428C22×C15111yescharacteristic
354#423C22×C15111yescharacteristic
355C2×#104C22×C15112yes—
356#418C22×C15112yes—
357#418C22×C15111yescharacteristic
358#417C22×C15111yescharacteristic
359C4×M16C22×C15111yescharacteristic
360#399—22no—
361#390C22×C15112yes—
362#384C22×C15112yes—
363C4×C4⋊C4C22×C15111yescharacteristic
364#343C22×C15112yes—
365#343C22×C15112yes—
366#339C22×C15112yes—
367#339C22×C15112yes—
368#339C22×C15112yes—
369C2×#41C23×C15111yescharacteristic
370C22×Q8—22no—
371C22×Q8D4×C15112yes—
372C22×D4C23×C15111yescharacteristic
373C23×C4—21no—
374AGL(1,ℤ/8)—22no—
375AGL(1,ℤ/8)Q8×C15114yescomplemented
376C2×SD16—21no—
377C2×D8—21no—
378C2×M16C23×C15111yescharacteristic
379C2×M16C23×C15112yes—
380C2×M16C23×C15111yescharacteristic
381C4⋊Q8—22no—
382C4⋊Q8—22no—
383C4⋊Q8—24no—
384C4⋊Q8—22no—
385C4⋊Q8—22no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8D4×C15111yescharacteristic
391C4⋊Q8D4×C15111yescharacteristic
392#125—24no—
393#125—22no—
394#125—22no—
395#125D4×C15112yes—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—24no—
399C22⋊Q8—24no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—24no—
403C22⋊Q8—48no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8D4×C15111yescharacteristic
410C4×Q8D4×C15111yescharacteristic
411C4×D4—22no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—24no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4D4×C15111yescharacteristic
420C4×D4D4×C15111yescharacteristic
421#117—22no—
422#117—21no—
423#117C23×C15111yescharacteristic
424#117C23×C15112yes—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—24no—
429C2×C4⋊C4D4×C15112yes—
430C2×C4⋊C4C23×C15111yescharacteristic
431C2×C4⋊C4C23×C15112yes—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—24no—
434C2×C42—21no—
435C2×C42C23×C15112yes—
436C2×C42C23×C15111yescharacteristic
437#107—21no—
438#107—22no—
439#106—21no—
440#106—22no—
441C4⋊C8—21no—
442C4⋊C8—21no—
443#104—24no—
444#103—21no—
445#102—21no—
446#102—21no—
447#102—21no—
448#102—22no—
449#102—22no—
450#97—21no—
451C4×C8—21no—
452#95—21no—
453C24—21no—
454#41—21no—
455#41—21no—
456#41C2×Q8×C15112yes—
457#41#41×C15112yes—
458C2×Q8—24no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—44no—
464C2×Q8—84no—
465C2×Q8C2×D4×C15111yescharacteristic
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×D4×C15111yescharacteristic
473C2×D4C2×Q8×C15112yescomplemented
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—44no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4C2×D4×C15111yescharacteristic
484C22×C4C2×D4×C15114yes—
485C22×C4C2×D4×C15111yescharacteristic
486C22×C4C24×C15111yescharacteristic
487SD16—24no—
488D8—24no—
489M16—21no—
490M16—22no—
491M16—21no—
492M16—21no—
493M16—22no—
494M16C2×Q8×C15112yescomplemented
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—48no—
516C4⋊C4C2×D4×C15112yes—
517C4⋊C4C2×D4×C15111yescharacteristic
518C4⋊C4C2×D4×C15112yes—
519C4⋊C4C2×D4×C15111yescharacteristic
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—48no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42C2×D4×C15111yescharacteristic
536C42C2×D4×C15111yescharacteristic
537C42C2×D4×C15111yescharacteristic
538C42C2×D4×C15111yescharacteristic
539C23—21no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23C22×D4×C15111yescharacteristic
546Q8—21no—
547Q8—42no—
548Q8—44no—
549Q8—42no—
550Q8—82no—
551Q8—84no—
552Q8C22⋊Q8×C15112yescomplemented
553D4—22no—
554D4—22no—
555D4—22no—
556D4—21no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4C22⋊Q8×C15112yescomplemented
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4—82no—
582C2×C4#120×C15111yescharacteristic
583C2×C4#120×C15111yescharacteristic
584C2×C4#120×C15111yescharacteristic
585C2×C4#120×C15111yescharacteristic
586C2×C4C22×D4×C15111yescharacteristic
587C2×C4C22×D4×C15111yescharacteristic
588C2×C4C22×Q8×C15111yescharacteristic
589C2×C4C2×#41×C15111yescharacteristic
590C2×C42−1+4×C15111yescharacteristic
591C2×C42−1+4×C15111yescharacteristic
592C8—22no—
593C8—41no—
594C8—42no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—44no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22C2×#120×C15111yescharacteristic
606C22#548×C15111yescharacteristic
607C22D4×Q8×C15111yescharacteristic
608C4—22no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—44no—
613C4—41no—
614C4—44no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4#453×C15112yes—
620C4C2×C22⋊Q8×C15111yescharacteristic
621C4#536×C15111yescharacteristic
622C4#548×C15111yescharacteristic
623C4D4×Q8×C15111yescharacteristic
624C2—21no—
625C2—21no—
626C2—42no—
627C2—82no—
628C2#2409×C15111yescharacteristic
629C2C2×#453×C15111yescharacteristic
630C2#3020×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. #2542×C151C2semidirect
5. #2060×C151C2semidirect
6. #2027×C151C2semidirect
7. #1997×C151C2semidirect
8. #1986×C151C2semidirect
10. #1859×C151C2semidirect
11. #1806×C151C2semidirect
59. AGL(1,ℤ/8)×C151Q8semidirect
157. C2×D4×C151C2×Q8semidirect
178. M16×C151C2×Q8semidirect
236. Q8×C151C22⋊Q8semidirect
244. D4×C151C22⋊Q8semidirect
315. #24320C151direct (N × H)
316. C151#24320direct (N × H)
317. #3503C2×C151semidirect
318. #3077C2×C151semidirect
320. #2542C2×C151semidirect
321. #2060C2×C151semidirect
322. #2027C2×C151semidirect
323. #1997C2×C151semidirect
324. #1986C2×C151semidirect
326. #1859C2×C151semidirect
327. #1806C2×C151semidirect
375. AGL(1,ℤ/8)Q8×C151semidirect
473. C2×D4C2×Q8×C151semidirect
494. M16C2×Q8×C151semidirect
552. Q8C22⋊Q8×C151semidirect
560. D4C22⋊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
#2542×C1512
#2060×C1512
#2027×C1512
#1997×C1512
#1986×C1512
#1883×C1512
#1859×C1512
#1806×C1512
#24320151