Finite Group Databasebeta

← C151×#18288 #50323213

Subgroups — C151×#18288 #50323213

Order 38656.

1034 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#539×C151C211yesmaximal, characteristic, complemented
2#3187×C151C211yesmaximal, characteristic, complemented
3#3186×C151C211yesmaximal, characteristic
4#2618×C151C211yesmaximal, characteristic, complemented
5#1609×C151C214yesmaximal, complemented
6#1597×C151C214yesmaximal, complemented
7#1520×C151C211yesmaximal, characteristic, complemented
8#1519×C151C211yesmaximal, characteristic, complemented
9C2×#324×C151C211yesmaximal, characteristic, complemented
10#539×C151—22no—
11#539×C151—22no—
12#539×C151—22no—
13#539×C151C2214yes—
14C2×#126×C151—21no—
15C2×#125×C151—21no—
16C2×#123×C151—21no—
17C2×#123×C151—21no—
18C2×#123×C151—21no—
19C2×C22⋊Q8×C151—21no—
20C2×C22⋊Q8×C151C2211yescharacteristic
21C2×#121×C151C2211yescharacteristic
22C2×#121×C151C2211yescharacteristic
23C2×C4×D4×C151—21no—
24C22×C4⋊C4×C151C2211yescharacteristic
25#483×C151C2212yescomplemented
26#481×C151—21no—
27#480×C151—21no—
28#480×C151C2212yes—
29#466×C151—22no—
30#465×C151—22no—
31#450×C151C2212yescomplemented
32#448×C151—21no—
33#447×C151—21no—
34#447×C151C2212yes—
35C2×C22⋊C8×C151C2211yescharacteristic
36C2×C22⋊C8×C151C2211yescharacteristic
37#398×C151C2212yescomplemented
38#396×C151—22no—
39#394×C151C2212yescomplemented
40#386×C151—21no—
41#380×C151C2211yescharacteristic
42#328×C151C2214yescomplemented
43#327×C151C2214yes—
44#324×C151C2214yescomplemented
45C22×Q8×C151D411yescharacteristic
46C22×D4×C151—21no—
47C22×D4×C151—21no—
48C22×D4×C151D411yescharacteristic
49C22×D4×C151C2311yescharacteristic
50C23×C4×C151—21no—
51C23×C4×C151D411yescharacteristic
52C23×C4×C151D411yescharacteristic
53C23×C4×C151C2311yescharacteristic
54C2×SD16×C151—22no—
55C2×D8×C151—22no—
56C22×C8×C151—21no—
57C22×C8×C151—21no—
58#126×C151—24no—
59#126×C151—24no—
60#125×C151—24no—
61#125×C151—24no—
62#123×C151—24no—
63#123×C151—24no—
64#123×C151—24no—
65#123×C151—24no—
66#123×C151—44no—
67C22⋊Q8×C151—24no—
68C22⋊Q8×C151—21no—
69C22⋊Q8×C151—21no—
70C22⋊Q8×C151—24no—
71C22⋊Q8×C151D412yes—
72C22⋊Q8×C151C2312yes—
73#121×C151—21no—
74#121×C151—21no—
75#121×C151—21no—
76#121×C151—21no—
77#121×C151—21no—
78#121×C151—21no—
79#121×C151D412yes—
80#121×C151C2312yes—
81C4×D4×C151—24no—
82C4×D4×C151—24no—
83C2×C4⋊C4×C151—21no—
84C2×C4⋊C4×C151—21no—
85C2×C4⋊C4×C151—21no—
86C2×C4⋊C4×C151—21no—
87C2×C4⋊C4×C151—21no—
88C2×C4⋊C4×C151—21no—
89C2×C4⋊C4×C151—21no—
90C2×C4⋊C4×C151—21no—
91C2×C4⋊C4×C151—22no—
92C2×C4⋊C4×C151—21no—
93C2×C4⋊C4×C151D412yes—
94C2×C4⋊C4×C151D411yescharacteristic
95C2×C4⋊C4×C151D411yescharacteristic
96C2×C4⋊C4×C151C2311yescharacteristic
97C2×C4⋊C4×C151C2312yes—
98C2×C22⋊C4×C151—21no—
99C2×C22⋊C4×C151—21no—
100C2×C22⋊C4×C151—21no—
101C2×C22⋊C4×C151—21no—
102C2×C22⋊C4×C151—22no—
103C2×C22⋊C4×C151—21no—
104C2×C22⋊C4×C151—22no—
105C2×C22⋊C4×C151—21no—
106C2×C22⋊C4×C151—21no—
107C2×C42×C151—21no—
108#107×C151—22no—
109#106×C151—22no—
110#103×C151—22no—
111#102×C151—22no—
112#102×C151—22no—
113#102×C151—22no—
114C22⋊C8×C151—21no—
115C22⋊C8×C151—21no—
116C22⋊C8×C151C2312yes—
117C22⋊C8×C151C2312yes—
118#95×C151—22no—
119#95×C151C2312yes—
120C24×C151—21no—
121C24×C151—21no—
122C24×C151C2×D411yescharacteristic
123C2×Q8×C151—21no—
124C2×Q8×C151—81no—
125C2×Q8×C151C2×D412yes—
126C2×D4×C151—22no—
127C2×D4×C151—22no—
128C2×D4×C151—21no—
129C2×D4×C151—21no—
130C2×D4×C151—21no—
131C2×D4×C151—42no—
132C2×D4×C151—42no—
133C2×D4×C151—41no—
134C2×D4×C151—41no—
135C2×D4×C151—42no—
136C2×D4×C151—42no—
137C2×D4×C151—81no—
138C2×D4×C151—81no—
139C2×D4×C151C2×D412yes—
140C22×C4×C151—21no—
141C22×C4×C151—21no—
142C22×C4×C151—21no—
143C22×C4×C151—21no—
144C22×C4×C151—21no—
145C22×C4×C151—21no—
146C22×C4×C151—22no—
147C22×C4×C151—22no—
148C22×C4×C151—24no—
149C22×C4×C151—21no—
150C22×C4×C151—21no—
151C22×C4×C151—22no—
152C22×C4×C151—81no—
153C22×C4×C151—81no—
154C22×C4×C151—82no—
155C22×C4×C151C2×D412yes—
156C22×C4×C151C2×D412yes—
157C22×C4×C151C2×D411yescharacteristic
158C22×C4×C151C2×D411yescharacteristic
159C22×C4×C151C2×D411yescharacteristic
160C22×C4×C151C2×D411yescharacteristic
161C22×C4×C151C2×D411yescharacteristic
162C22×C4×C151C2411yescharacteristic
163SD16×C151—82no—
164D8×C151—82no—
165C2×C8×C151—22no—
166C2×C8×C151—22no—
167C2×C8×C151—81no—
168C2×C8×C151—81no—
169C4⋊C4×C151—24no—
170C4⋊C4×C151—24no—
171C4⋊C4×C151—24no—
172C4⋊C4×C151—22no—
173C4⋊C4×C151—22no—
174C4⋊C4×C151—22no—
175C4⋊C4×C151—21no—
176C4⋊C4×C151—21no—
177C4⋊C4×C151—22no—
178C4⋊C4×C151—22no—
179C4⋊C4×C151—22no—
180C4⋊C4×C151—21no—
181C4⋊C4×C151—21no—
182C4⋊C4×C151—44no—
183C4⋊C4×C151—81no—
184C4⋊C4×C151C2×D412yes—
185C4⋊C4×C151C2×D412yes—
186C22⋊C4×C151—24no—
187C22⋊C4×C151—24no—
188C22⋊C4×C151—24no—
189C22⋊C4×C151—22no—
190C22⋊C4×C151—22no—
191C22⋊C4×C151—22no—
192C22⋊C4×C151—22no—
193C22⋊C4×C151—44no—
194C22⋊C4×C151—42no—
195C22⋊C4×C151—41no—
196C22⋊C4×C151—41no—
197C22⋊C4×C151—82no—
198C42×C151—24no—
199C23×C151—22no—
200C23×C151—41no—
201C23×C151—41no—
202C23×C151—42no—
203C23×C151—42no—
204C23×C151—42no—
205C23×C151—42no—
206C23×C151—42no—
207C23×C151—41no—
208C23×C151—41no—
209C23×C151—41no—
210C23×C151#12011yescharacteristic
211C23×C151#12011yescharacteristic
212C23×C151#13512yes—
213C23×C151AGL(1,ℤ/8)12yes—
214C23×C151C22×D411yescharacteristic
215Q8×C151—42no—
216Q8×C151—81no—
217D4×C151—42no—
218D4×C151—42no—
219D4×C151—42no—
220D4×C151—42no—
221D4×C151—42no—
222D4×C151—44no—
223D4×C151—81no—
224D4×C151—82no—
225D4×C151—82no—
226C2×C4×C151—22no—
227C2×C4×C151—22no—
228C2×C4×C151—22no—
229C2×C4×C151—22no—
230C2×C4×C151—22no—
231C2×C4×C151—22no—
232C2×C4×C151—22no—
233C2×C4×C151—41no—
234C2×C4×C151—41no—
235C2×C4×C151—41no—
236C2×C4×C151—41no—
237C2×C4×C151—41no—
238C2×C4×C151—41no—
239C2×C4×C151—42no—
240C2×C4×C151—42no—
241C2×C4×C151—42no—
242C2×C4×C151—42no—
243C2×C4×C151—42no—
244C2×C4×C151—42no—
245C2×C4×C151—42no—
246C2×C4×C151—42no—
247C2×C4×C151—42no—
248C2×C4×C151—41no—
249C2×C4×C151—41no—
250C2×C4×C151—42no—
251C2×C4×C151—41no—
252C2×C4×C151—81no—
253C2×C4×C151—84no—
254C2×C4×C151#12011yescharacteristic
255C2×C4×C151#12011yescharacteristic
256C2×C4×C151C22×D411yescharacteristic
257C2×C4×C151C22×D411yescharacteristic
258C8×C151—42no—
259C8×C151—42no—
260C22×C151—21no—
261C22×C151—21no—
262C22×C151—21no—
263C22×C151—21no—
264C22×C151—21no—
265C22×C151—21no—
266C22×C151—42no—
267C22×C151—42no—
268C22×C151—42no—
269C22×C151—42no—
270C22×C151—42no—
271C22×C151—84no—
272C22×C151—82no—
273C22×C151—82no—
274C22×C151—81no—
275C22×C151—81no—
276C22×C151—81no—
277C22×C151#44912yes—
278C22×C151#44912yes—
279C22×C151C2×#12011yescharacteristic
280C22×C151C2×#13511yescharacteristic
281C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
282C4×C151—21no—
283C4×C151—21no—
284C4×C151—21no—
285C4×C151—21no—
286C4×C151—42no—
287C4×C151—42no—
288C4×C151—42no—
289C4×C151—42no—
290C4×C151—42no—
291C4×C151—82no—
292C4×C151—81no—
293C2×C151—21no—
294C2×C151—21no—
295C2×C151—22no—
296C2×C151—22no—
297C2×C151—82no—
298C2×C151—161no—
299C2×C151C2×#44911yescharacteristic
300C2×C151#302011yescharacteristic
301C2×C151#302611yescharacteristic
302#18288C15111yesmaximal, characteristic, direct product factor, Sylow
303C151#1828811yescharacteristic, direct product factor, Sylow
304C2×#539C2×C15111yescharacteristic, complemented
305#3187C2×C15111yescharacteristic, complemented
306#3186C2×C15111yescharacteristic
307#2618C2×C15111yescharacteristic, complemented
308#1609C2×C15114yescomplemented
309#1597C2×C15114yescomplemented
310#1520C2×C15111yescharacteristic, complemented
311#1519C2×C15111yescharacteristic, complemented
312C2×#324C2×C15111yescharacteristic, complemented
313#539—22no—
314#539—22no—
315#539—22no—
316#539C22×C15114yes—
317C2×#126—21no—
318C2×#125—21no—
319C2×#123—21no—
320C2×#123—21no—
321C2×#123—21no—
322C2×C22⋊Q8—21no—
323C2×C22⋊Q8C22×C15111yescharacteristic
324C2×#121C22×C15111yescharacteristic
325C2×#121C22×C15111yescharacteristic
326C2×C4×D4—21no—
327C22×C4⋊C4C22×C15111yescharacteristic
328#483C22×C15112yescomplemented
329#481—21no—
330#480—21no—
331#480C22×C15112yes—
332#466—22no—
333#465—22no—
334#450C22×C15112yescomplemented
335#448—21no—
336#447—21no—
337#447C22×C15112yes—
338C2×C22⋊C8C22×C15111yescharacteristic
339C2×C22⋊C8C22×C15111yescharacteristic
340#398C22×C15112yescomplemented
341#396—22no—
342#394C22×C15112yescomplemented
343#386—21no—
344#380C22×C15111yescharacteristic
345#328C22×C15114yescomplemented
346#327C22×C15114yes—
347#324C22×C15114yescomplemented
348C22×Q8D4×C15111yescharacteristic
349C22×D4—21no—
350C22×D4—21no—
351C22×D4D4×C15111yescharacteristic
352C22×D4C23×C15111yescharacteristic
353C23×C4—21no—
354C23×C4D4×C15111yescharacteristic
355C23×C4D4×C15111yescharacteristic
356C23×C4C23×C15111yescharacteristic
357C2×SD16—22no—
358C2×D8—22no—
359C22×C8—21no—
360C22×C8—21no—
361#126—24no—
362#126—24no—
363#125—24no—
364#125—24no—
365#123—24no—
366#123—24no—
367#123—24no—
368#123—24no—
369#123—44no—
370C22⋊Q8—24no—
371C22⋊Q8—24no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8D4×C15112yes—
375C22⋊Q8C23×C15112yes—
376#121—21no—
377#121—21no—
378#121—21no—
379#121—21no—
380#121—21no—
381#121—21no—
382#121D4×C15112yes—
383#121C23×C15112yes—
384C4×D4—24no—
385C4×D4—24no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4D4×C15112yes—
397C2×C4⋊C4D4×C15111yescharacteristic
398C2×C4⋊C4D4×C15111yescharacteristic
399C2×C4⋊C4C23×C15111yescharacteristic
400C2×C4⋊C4C23×C15112yes—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C42—21no—
411#107—22no—
412#106—22no—
413#103—22no—
414#102—22no—
415#102—22no—
416#102—22no—
417C22⋊C8—21no—
418C22⋊C8—21no—
419C22⋊C8C23×C15112yes—
420C22⋊C8C23×C15112yes—
421#95—22no—
422#95C23×C15112yes—
423C24—21no—
424C24—21no—
425C24C2×D4×C15111yescharacteristic
426C2×Q8—21no—
427C2×Q8—81no—
428C2×Q8C2×D4×C15112yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4C2×D4×C15112yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—82no—
458C22×C4C2×D4×C15112yes—
459C22×C4C2×D4×C15112yes—
460C22×C4C2×D4×C15111yescharacteristic
461C22×C4C2×D4×C15111yescharacteristic
462C22×C4C2×D4×C15111yescharacteristic
463C22×C4C2×D4×C15111yescharacteristic
464C22×C4C2×D4×C15111yescharacteristic
465C22×C4C24×C15111yescharacteristic
466SD16—82no—
467D8—82no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—81no—
471C2×C8—81no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—44no—
486C4⋊C4—81no—
487C4⋊C4C2×D4×C15112yes—
488C4⋊C4C2×D4×C15112yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—82no—
501C42—24no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23#120×C15111yescharacteristic
514C23#120×C15111yescharacteristic
515C23#135×C15112yes—
516C23AGL(1,ℤ/8)×C15112yes—
517C23C22×D4×C15111yescharacteristic
518Q8—42no—
519Q8—81no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—42no—
525D4—42no—
526D4—81no—
527D4—82no—
528D4—82no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—81no—
556C2×C4—84no—
557C2×C4#120×C15111yescharacteristic
558C2×C4#120×C15111yescharacteristic
559C2×C4C22×D4×C15111yescharacteristic
560C2×C4C22×D4×C15111yescharacteristic
561C8—42no—
562C8—42no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22#449×C15112yes—
581C22#449×C15112yes—
582C22C2×#120×C15111yescharacteristic
583C22C2×#135×C15111yescharacteristic
584C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—82no—
595C4—81no—
596C2—21no—
597C2—22no—
598C2—21no—
599C2—22no—
600C2—82no—
601C2—161no—
602C2C2×#449×C15111yescharacteristic
603C2#3020×C15111yescharacteristic
604C2#3026×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC22×C4C24×C151
prime radical#3186×C151C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#539×C151C2semidirect
2. #3187×C151C2semidirect
4. #2618×C151C2semidirect
5. #1609×C151C2semidirect
6. #1597×C151C2semidirect
7. #1520×C151C2semidirect
8. #1519×C151C2semidirect
9. C2×#324×C151C2semidirect
25. #483×C151C22semidirect
31. #450×C151C22semidirect
37. #398×C151C22semidirect
39. #394×C151C22semidirect
42. #328×C151C22semidirect
44. #324×C151C22semidirect
302. #18288C151direct (N × H)
303. C151#18288direct (N × H)
304. C2×#539C2×C151semidirect
305. #3187C2×C151semidirect
307. #2618C2×C151semidirect
308. #1609C2×C151semidirect
309. #1597C2×C151semidirect
310. #1520C2×C151semidirect
311. #1519C2×C151semidirect
312. C2×#324C2×C151semidirect
328. #483C22×C151semidirect
334. #450C22×C151semidirect
340. #398C22×C151semidirect
342. #394C22×C151semidirect
345. #328C22×C151semidirect
347. #324C22×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#539×C1512
#3187×C1512
#3186×C1512
#2618×C1512
#1609×C1512
#1597×C1512
#1520×C1512
#1519×C1512
C2×#324×C1512
#18288151