Finite Group Databasebeta

← C151×#24249 #50329174

Subgroups — C151×#24249 #50329174

Order 38656.

1488 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547×C151C211yesmaximal, characteristic, complemented
2C2×#461×C151C211yesmaximal, characteristic, complemented
3C2×#449×C151C212yesmaximal, complemented
4C2×#442×C151C211yesmaximal, characteristic, complemented
5#2535×C151C211yesmaximal, characteristic, complemented
6#2035×C151C212yesmaximal, complemented
7#2000×C151C212yesmaximal, complemented
8#1960×C151C211yesmaximal, characteristic, complemented
9#1955×C151C211yesmaximal, characteristic, complemented
10#1804×C151C211yesmaximal, characteristic, complemented
11#1788×C151C212yesmaximal, complemented
12C22×#41×C151C2212yes—
13C22×SD16×C151C2212yes—
14C22×D8×C151C2211yescharacteristic
15#547×C151—24no—
16#547×C151—24no—
17C2×C4⋊Q8×C151C2211yescharacteristic
18C2×#123×C151—22no—
19C2×C22⋊Q8×C151C2212yes—
20C2×#121×C151—21no—
21C2×#121×C151C2212yes—
22C2×C4×D4×C151C2211yescharacteristic
23C2×C4×D4×C151C2211yescharacteristic
24C2×C4×D4×C151C2211yescharacteristic
25C22×C4⋊C4×C151—21no—
26#461×C151—24no—
27#449×C151—28no—
28#442×C151—24no—
29C2×#106×C151C2211yescharacteristic
30C2×C4⋊C8×C151C2211yescharacteristic
31C2×#103×C151C2212yes—
32C2×#103×C151C2212yes—
33C2×#103×C151C2212yes—
34C2×#102×C151C2211yescharacteristic
35C2×#102×C151C2211yescharacteristic
36C2×#102×C151C2212yes—
37C2×C22⋊C8×C151C2212yes—
38C2×C22⋊C8×C151C2212yes—
39C2×#97×C151C2211yescharacteristic
40#397×C151—21no—
41#392×C151—21no—
42#386×C151—22no—
43#382×C151C2211yescharacteristic
44#380×C151C2212yes—
45#340×C151C2212yes—
46C2×#41×C151—24no—
47C2×#41×C151—24no—
48C2×#41×C151—24no—
49C2×#41×C151—24no—
50C2×#41×C151—42no—
51C2×#41×C151—42no—
52C22×Q8×C151C2312yes—
53C22×D4×C151—22no—
54C22×D4×C151—22no—
55C22×D4×C151D411yescharacteristic
56C22×D4×C151C2311yescharacteristic
57C22×D4×C151C2311yescharacteristic
58C23×C4×C151—21no—
59C23×C4×C151—22no—
60C23×C4×C151—21no—
61C23×C4×C151C2312yes—
62C2×SD16×C151—24no—
63C2×SD16×C151—24no—
64C2×SD16×C151—42no—
65C2×D8×C151—22no—
66C2×D8×C151—22no—
67C2×D8×C151D414yescomplemented
68C22×C8×C151C2312yes—
69C22×C8×C151C2311yescharacteristic
70C22×C8×C151C2311yescharacteristic
71C4⋊Q8×C151—22no—
72C4⋊Q8×C151D414yescomplemented
73#123×C151—216no—
74C22⋊Q8×C151—24no—
75C22⋊Q8×C151—24no—
76#121×C151—28no—
77#121×C151—24no—
78#121×C151—42no—
79C4×D4×C151—22no—
80C4×D4×C151—22no—
81C4×D4×C151—22no—
82C4×D4×C151—22no—
83C4×D4×C151—22no—
84C4×D4×C151D414yescomplemented
85C2×C4⋊C4×C151—28no—
86C2×C4⋊C4×C151—21no—
87C2×C4⋊C4×C151—22no—
88C2×C4⋊C4×C151—21no—
89C2×C4⋊C4×C151D411yescharacteristic
90C2×C4⋊C4×C151C2312yes—
91C2×C4⋊C4×C151C2311yescharacteristic
92C2×C4⋊C4×C151C2311yescharacteristic
93C2×C22⋊C4×C151—21no—
94C2×C22⋊C4×C151—22no—
95C2×C22⋊C4×C151—21no—
96C2×C22⋊C4×C151—22no—
97C2×C22⋊C4×C151—22no—
98C2×C22⋊C4×C151D412yes—
99C2×C42×C151C2311yescharacteristic
100#106×C151—22no—
101C4⋊C8×C151—22no—
102#103×C151—24no—
103#103×C151—24no—
104#103×C151—42no—
105#102×C151—22no—
106#102×C151—24no—
107#102×C151—22no—
108C22⋊C8×C151—24no—
109C22⋊C8×C151—42no—
110#97×C151—22no—
111#95×C151—22no—
112#95×C151—21no—
113#95×C151—21no—
114C24×C151—21no—
115C24×C151—21no—
116C24×C151C2×D412yes—
117#41×C151—48no—
118#41×C151—48no—
119#41×C151—48no—
120#41×C151—48no—
121C2×Q8×C151—22no—
122C2×Q8×C151—22no—
123C2×Q8×C151—22no—
124C2×Q8×C151—22no—
125C2×Q8×C151—22no—
126C2×Q8×C151C2×D414yes—
127C2×D4×C151—28no—
128C2×D4×C151—22no—
129C2×D4×C151—21no—
130C2×D4×C151—21no—
131C2×D4×C151—21no—
132C2×D4×C151—21no—
133C2×D4×C151—21no—
134C2×D4×C151—21no—
135C2×D4×C151—21no—
136C2×D4×C151—21no—
137C2×D4×C151—42no—
138C2×D4×C151—44no—
139C2×D4×C151—44no—
140C2×D4×C151—44no—
141C2×D4×C151—44no—
142C2×D4×C151—82no—
143C2×D4×C151C2×D412yes—
144C2×D4×C151C2×D412yes—
145C2×D4×C151C2×D412yes—
146C2×D4×C151#4112yes—
147C22×C4×C151—28no—
148C22×C4×C151—21no—
149C22×C4×C151—22no—
150C22×C4×C151—24no—
151C22×C4×C151—22no—
152C22×C4×C151—22no—
153C22×C4×C151—22no—
154C22×C4×C151—22no—
155C22×C4×C151—22no—
156C22×C4×C151—22no—
157C22×C4×C151—22no—
158C22×C4×C151—21no—
159C22×C4×C151—44no—
160C22×C4×C151—44no—
161C22×C4×C151—44no—
162C22×C4×C151—42no—
163C22×C4×C151—82no—
164C22×C4×C151C2×D412yes—
165C22×C4×C151C2×D411yescharacteristic
166C22×C4×C151C2×D411yescharacteristic
167C22×C4×C151C2411yescharacteristic
168SD16×C151—48no—
169D8×C151—28no—
170C2×C8×C151—21no—
171C2×C8×C151—21no—
172C2×C8×C151—21no—
173C2×C8×C151—22no—
174C2×C8×C151—22no—
175C2×C8×C151—21no—
176C2×C8×C151—22no—
177C2×C8×C151—21no—
178C2×C8×C151C2×D412yes—
179C4⋊C4×C151—28no—
180C4⋊C4×C151—22no—
181C4⋊C4×C151—24no—
182C4⋊C4×C151—24no—
183C4⋊C4×C151—21no—
184C4⋊C4×C151—22no—
185C4⋊C4×C151—21no—
186C4⋊C4×C151—21no—
187C4⋊C4×C151—22no—
188C4⋊C4×C151C2×D412yes—
189C22⋊C4×C151—28no—
190C22⋊C4×C151—24no—
191C22⋊C4×C151—24no—
192C22⋊C4×C151—44no—
193C22⋊C4×C151—42no—
194C22⋊C4×C151—82no—
195C42×C151C2×D412yes—
196C42×C151C2×D412yes—
197C23×C151—22no—
198C23×C151—22no—
199C23×C151—22no—
200C23×C151—22no—
201C23×C151—22no—
202C23×C151—22no—
203C23×C151—22no—
204C23×C151—22no—
205C23×C151—22no—
206C23×C151—22no—
207C23×C151—22no—
208C23×C151—42no—
209C23×C151—42no—
210C23×C151—41no—
211C23×C151—41no—
212C23×C151—42no—
213C23×C151—42no—
214C23×C151C22×D411yescharacteristic
215Q8×C151—24no—
216Q8×C151—24no—
217Q8×C151—42no—
218Q8×C151—42no—
219D4×C151—22no—
220D4×C151—22no—
221D4×C151—22no—
222D4×C151—22no—
223D4×C151—22no—
224D4×C151—44no—
225D4×C151—416no—
226D4×C151—41no—
227D4×C151—41no—
228D4×C151—88no—
229D4×C151#12114yes—
230C2×C4×C151—22no—
231C2×C4×C151—22no—
232C2×C4×C151—24no—
233C2×C4×C151—22no—
234C2×C4×C151—22no—
235C2×C4×C151—22no—
236C2×C4×C151—22no—
237C2×C4×C151—48no—
238C2×C4×C151—416no—
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—41no—
247C2×C4×C151—42no—
248C2×C4×C151—42no—
249C2×C4×C151—42no—
250C2×C4×C151—42no—
251C2×C4×C151—84no—
252C2×C4×C151#12012yes—
253C2×C4×C151#12012yes—
254C2×C4×C151C22×D411yescharacteristic
255C2×C4×C151C22×D411yescharacteristic
256C2×C4×C151C22×D411yescharacteristic
257C2×C4×C151C2×#4111yescharacteristic
258C2×C4×C151H(5,𝔽2)11yescharacteristic
259C2×C4×C151H(5,𝔽2)11yescharacteristic
260C8×C151—22no—
261C8×C151—41no—
262C8×C151—42no—
263C22×C151—24no—
264C22×C151—42no—
265C22×C151—42no—
266C22×C151—44no—
267C22×C151—44no—
268C22×C151—42no—
269C22×C151—42no—
270C22×C151—42no—
271C22×C151—42no—
272C22×C151—42no—
273C22×C151—44no—
274C22×C151—42no—
275C22×C151—42no—
276C22×C151—42no—
277C22×C151—81no—
278C22×C151—81no—
279C22×C151—82no—
280C22×C151C2×#12011yescharacteristic
281C22×C151D4211yescharacteristic
282C22×C151#54611yescharacteristic
283C22×C151#57512yes—
284C22×C151#57712yes—
285C4×C151—22no—
286C4×C151—22no—
287C4×C151—42no—
288C4×C151—44no—
289C4×C151—42no—
290C4×C151—81no—
291C4×C151—82no—
292C4×C151C2×#12111yescharacteristic
293C4×C151#53511yescharacteristic
294C4×C151D4211yescharacteristic
295C4×C151#54611yescharacteristic
296C2×C151—44no—
297C2×C151—82no—
298C2×C151—82no—
299C2×C151#240811yescharacteristic
300C2×C151#300511yescharacteristic
301C2×C151#301011yescharacteristic
302C2×C151#326712yes—
303C2×C151#330712yes—
304#24249C15111yesmaximal, characteristic, direct product factor, Sylow
305C151#2424911yescharacteristic, direct product factor, Sylow
306C2×#547C2×C15111yescharacteristic, complemented
307C2×#461C2×C15111yescharacteristic, complemented
308C2×#449C2×C15112yescomplemented
309C2×#442C2×C15111yescharacteristic, complemented
310#2535C2×C15111yescharacteristic, complemented
311#2035C2×C15112yescomplemented
312#2000C2×C15112yescomplemented
313#1960C2×C15111yescharacteristic, complemented
314#1955C2×C15111yescharacteristic, complemented
315#1804C2×C15111yescharacteristic, complemented
316#1788C2×C15112yescomplemented
317C22×#41C22×C15112yes—
318C22×SD16C22×C15112yes—
319C22×D8C22×C15111yescharacteristic
320#547—24no—
321#547—24no—
322C2×C4⋊Q8C22×C15111yescharacteristic
323C2×#123—22no—
324C2×C22⋊Q8C22×C15112yes—
325C2×#121—21no—
326C2×#121C22×C15112yes—
327C2×C4×D4C22×C15111yescharacteristic
328C2×C4×D4C22×C15111yescharacteristic
329C2×C4×D4C22×C15111yescharacteristic
330C22×C4⋊C4—21no—
331#461—24no—
332#449—28no—
333#442—24no—
334C2×#106C22×C15111yescharacteristic
335C2×C4⋊C8C22×C15111yescharacteristic
336C2×#103C22×C15112yes—
337C2×#103C22×C15112yes—
338C2×#103C22×C15112yes—
339C2×#102C22×C15112yes—
340C2×#102C22×C15111yescharacteristic
341C2×#102C22×C15111yescharacteristic
342C2×C22⋊C8C22×C15112yes—
343C2×C22⋊C8C22×C15112yes—
344C2×#97C22×C15111yescharacteristic
345#397—21no—
346#392—21no—
347#386—22no—
348#382C22×C15111yescharacteristic
349#380C22×C15112yes—
350#340C22×C15112yes—
351C2×#41—24no—
352C2×#41—24no—
353C2×#41—24no—
354C2×#41—24no—
355C2×#41—42no—
356C2×#41—42no—
357C22×Q8C23×C15112yes—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4D4×C15111yescharacteristic
361C22×D4C23×C15111yescharacteristic
362C22×D4C23×C15111yescharacteristic
363C23×C4—21no—
364C23×C4—22no—
365C23×C4—21no—
366C23×C4C23×C15112yes—
367C2×SD16—24no—
368C2×SD16—24no—
369C2×SD16—42no—
370C2×D8—22no—
371C2×D8—22no—
372C2×D8D4×C15114yescomplemented
373C22×C8C23×C15112yes—
374C22×C8C23×C15111yescharacteristic
375C22×C8C23×C15111yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8D4×C15114yescomplemented
378#123—216no—
379C22⋊Q8—24no—
380C22⋊Q8—24no—
381#121—28no—
382#121—24no—
383#121—42no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4D4×C15114yescomplemented
390C2×C4⋊C4—28no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4D4×C15111yescharacteristic
395C2×C4⋊C4C23×C15111yescharacteristic
396C2×C4⋊C4C23×C15112yes—
397C2×C4⋊C4C23×C15111yescharacteristic
398C2×C22⋊C4—22no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4D4×C15112yes—
404C2×C42C23×C15111yescharacteristic
405#106—22no—
406C4⋊C8—22no—
407#103—24no—
408#103—24no—
409#103—42no—
410#102—22no—
411#102—24no—
412#102—22no—
413C22⋊C8—24no—
414C22⋊C8—42no—
415#97—22no—
416#95—22no—
417#95—21no—
418#95—21no—
419C24—21no—
420C24—21no—
421C24C2×D4×C15112yes—
422#41—48no—
423#41—48no—
424#41—48no—
425#41—48no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8C2×D4×C15114yes—
432C2×D4—28no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—82no—
448C2×D4C2×D4×C15112yes—
449C2×D4C2×D4×C15112yes—
450C2×D4C2×D4×C15112yes—
451C2×D4#41×C15112yes—
452C22×C4—28no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—82no—
469C22×C4C2×D4×C15112yes—
470C22×C4C2×D4×C15111yescharacteristic
471C22×C4C2×D4×C15111yescharacteristic
472C22×C4C24×C15111yescharacteristic
473SD16—48no—
474D8—28no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—21no—
481C2×C8—22no—
482C2×C8—21no—
483C2×C8C2×D4×C15112yes—
484C4⋊C4—28no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4C2×D4×C15112yes—
494C22⋊C4—28no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—82no—
500C42C2×D4×C15112yes—
501C42C2×D4×C15112yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23C22×D4×C15111yescharacteristic
520Q8—24no—
521Q8—24no—
522Q8—42no—
523Q8—42no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4—44no—
530D4—416no—
531D4—41no—
532D4—41no—
533D4—88no—
534D4#121×C15114yes—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—416no—
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—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4#120×C15112yes—
558C2×C4#120×C15112yes—
559C2×C4C22×D4×C15111yescharacteristic
560C2×C4C22×D4×C15111yescharacteristic
561C2×C4C22×D4×C15111yescharacteristic
562C2×C4C2×#41×C15111yescharacteristic
563C2×C4H(5,𝔽2)×C15111yescharacteristic
564C2×C4H(5,𝔽2)×C15111yescharacteristic
565C8—22no—
566C8—41no—
567C8—42no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—82no—
585C22C2×#120×C15111yescharacteristic
586C22D42×C15111yescharacteristic
587C22#546×C15111yescharacteristic
588C22#575×C15112yes—
589C22#577×C15112yes—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—44no—
594C4—42no—
595C4—81no—
596C4—82no—
597C4C2×#121×C15111yescharacteristic
598C4#535×C15111yescharacteristic
599C4D42×C15111yescharacteristic
600C4#546×C15111yescharacteristic
601C2—44no—
602C2—82no—
603C2—82no—
604C2#2408×C15111yescharacteristic
605C2#3005×C15111yescharacteristic
606C2#3010×C15111yescharacteristic
607C2#3267×C15112yes—
608C2#3307×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×D4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#547×C151C2semidirect
2. C2×#461×C151C2semidirect
3. C2×#449×C151C2semidirect
4. C2×#442×C151C2semidirect
5. #2535×C151C2semidirect
6. #2035×C151C2semidirect
7. #2000×C151C2semidirect
8. #1960×C151C2semidirect
9. #1955×C151C2semidirect
10. #1804×C151C2semidirect
11. #1788×C151C2semidirect
67. C2×D8×C151D4semidirect
72. C4⋊Q8×C151D4semidirect
84. C4×D4×C151D4semidirect
304. #24249C151direct (N × H)
305. C151#24249direct (N × H)
306. C2×#547C2×C151semidirect
307. C2×#461C2×C151semidirect
308. C2×#449C2×C151semidirect
309. C2×#442C2×C151semidirect
310. #2535C2×C151semidirect
311. #2035C2×C151semidirect
312. #2000C2×C151semidirect
313. #1960C2×C151semidirect
314. #1955C2×C151semidirect
315. #1804C2×C151semidirect
316. #1788C2×C151semidirect
372. C2×D8D4×C151semidirect
377. C4⋊Q8D4×C151semidirect
389. C4×D4D4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#547×C1512
C2×#461×C1512
C2×#449×C1512
C2×#442×C1512
#2535×C1512
#2035×C1512
#2000×C1512
#1960×C1512
#1955×C1512
#1804×C1512
#1788×C1512
#24249151