Finite Group Databasebeta

← C151×#24371 #50329296

Subgroups — C151×#24371 #50329296

Order 38656.

1540 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic
2C2×#464×C151C212yesmaximal, complemented
3C2×#452×C151C211yesmaximal, characteristic, complemented
4#2435×C151C211yesmaximal, characteristic, complemented
5#2341×C151C211yesmaximal, characteristic, complemented
6#2054×C151C212yesmaximal, complemented
7#2012×C151C212yesmaximal, complemented
8#1894×C151C211yesmaximal, characteristic, complemented
9#1880×C151C212yesmaximal, complemented
10#1851×C151C211yesmaximal, characteristic, complemented
11#1786×C151C211yesmaximal, characteristic, complemented
12C22×#41×C151C2211yescharacteristic
13C23×D4×C151C2211yescharacteristic
14C2×AGL(1,ℤ/8)×C151—24no—
15C2×AGL(1,ℤ/8)×C151—28no—
16C22×Q16×C151C2211yescharacteristic, complemented
17C22×SD16×C151C2212yes—
18C22×SD16×C151C2211yescharacteristic, complemented
19C22×D8×C151C2212yes—
20C22×M16×C151C2211yescharacteristic
21C2×#124×C151C2212yes—
22C2×C22⋊Q8×C151C2211yescharacteristic
23C2×C4×Q8×C151C2212yescomplemented
24#464×C151—28no—
25#452×C151—24no—
26C2×C4⋊C8×C151C2212yescomplemented
27C2×#103×C151C2211yescharacteristic, complemented
28C2×#103×C151C2212yescomplemented
29C2×#102×C151C2212yes—
30C2×#102×C151C2212yes—
31C2×#102×C151C2211yescharacteristic, complemented
32C2×C22⋊C8×C151C2211yescharacteristic
33#394×C151—21no—
34#394×C151—21no—
35#390×C151—22no—
36#390×C151C2212yes—
37#390×C151C2212yescomplemented
38#380×C151—21no—
39#380×C151C2211yescharacteristic
40#379×C151—22no—
41C4×C22⋊C4×C151—21no—
42C4×C22⋊C4×C151C2211yescharacteristic
43#340×C151C2212yescomplemented
44#336×C151C2212yescomplemented
45C25×C151—21no—
46C2×#41×C151—22no—
47C2×#41×C151—24no—
48C2×#41×C151—42no—
49C2×#41×C151D414yes—
50C22×Q8×C151C2311yescharacteristic
51C22×Q8×C151C2311yescharacteristic
52C22×D4×C151—21no—
53C22×D4×C151—21no—
54C22×D4×C151—24no—
55C22×D4×C151—22no—
56C22×D4×C151—22no—
57C22×D4×C151—42no—
58C22×D4×C151C2312yes—
59C22×D4×C151C2311yescharacteristic
60C23×C4×C151—21no—
61C23×C4×C151C2311yescharacteristic
62AGL(1,ℤ/8)×C151—232no—
63C2×Q16×C151—22no—
64C2×Q16×C151—24no—
65C2×SD16×C151—22no—
66C2×SD16×C151—24no—
67C2×SD16×C151—24no—
68C2×SD16×C151—42no—
69C2×SD16×C151D418yes—
70C2×D8×C151—24no—
71C2×D8×C151—24no—
72C2×D8×C151—24no—
73C2×M16×C151—22no—
74C2×M16×C151—24no—
75C22×C8×C151C2311yescharacteristic
76C22×C8×C151C2311yescharacteristic
77C22×C8×C151C2312yes—
78#124×C151—24no—
79#124×C151—42no—
80C22⋊Q8×C151—22no—
81C22⋊Q8×C151—22no—
82C4×Q8×C151—24no—
83C4×Q8×C151—42no—
84C2×C4⋊C4×C151—22no—
85C2×C4⋊C4×C151—21no—
86C2×C4⋊C4×C151C2311yescharacteristic
87C2×C4⋊C4×C151C2312yes—
88C2×C22⋊C4×C151—22no—
89C2×C22⋊C4×C151—22no—
90C2×C22⋊C4×C151—22no—
91C2×C22⋊C4×C151—22no—
92C2×C22⋊C4×C151—22no—
93C2×C22⋊C4×C151—22no—
94C2×C22⋊C4×C151—21no—
95C2×C22⋊C4×C151—21no—
96C2×C22⋊C4×C151D412yes—
97C2×C42×C151—22no—
98C2×C42×C151C2312yes—
99C4⋊C8×C151—24no—
100#103×C151—24no—
101#103×C151—22no—
102#102×C151—24no—
103#102×C151—22no—
104#102×C151—42no—
105C22⋊C8×C151—22no—
106#95×C151—21no—
107#95×C151D411yescharacteristic
108#95×C151D411yescharacteristic
109C24×C151—21no—
110C24×C151—22no—
111C24×C151—44no—
112C24×C151—44no—
113C24×C151—42no—
114C24×C151—81no—
115C24×C151—81no—
116C24×C151C2×D411yescharacteristic
117#41×C151—28no—
118#41×C151—44no—
119#41×C151—48no—
120C2×Q8×C151—21no—
121C2×Q8×C151—22no—
122C2×Q8×C151—22no—
123C2×Q8×C151—21no—
124C2×Q8×C151—21no—
125C2×Q8×C151—22no—
126C2×Q8×C151C2×D412yes—
127C2×Q8×C151C2×D414yes—
128C2×D4×C151—28no—
129C2×D4×C151—24no—
130C2×D4×C151—22no—
131C2×D4×C151—22no—
132C2×D4×C151—22no—
133C2×D4×C151—22no—
134C2×D4×C151—21no—
135C2×D4×C151—22no—
136C2×D4×C151—22no—
137C2×D4×C151—22no—
138C2×D4×C151—48no—
139C2×D4×C151—44no—
140C2×D4×C151—44no—
141C2×D4×C151—44no—
142C2×D4×C151—42no—
143C2×D4×C151C2×D414yes—
144C2×D4×C151C2×D412yes—
145C22×C4×C151—21no—
146C22×C4×C151—22no—
147C22×C4×C151—22no—
148C22×C4×C151—24no—
149C22×C4×C151—22no—
150C22×C4×C151—21no—
151C22×C4×C151—21no—
152C22×C4×C151—44no—
153C22×C4×C151C2×D412yes—
154C22×C4×C151C2×D412yes—
155C22×C4×C151C2×D411yescharacteristic
156C22×C4×C151C2×D412yes—
157C22×C4×C151#4112yes—
158C22×C4×C151C2411yescharacteristic
159Q16×C151—44no—
160SD16×C151—216no—
161SD16×C151—44no—
162D8×C151—216no—
163M16×C151—28no—
164C2×C8×C151—22no—
165C2×C8×C151—22no—
166C2×C8×C151—21no—
167C2×C8×C151—21no—
168C2×C8×C151—22no—
169C2×C8×C151—22no—
170C2×C8×C151C2×D414yes—
171C4⋊C4×C151—22no—
172C4⋊C4×C151—21no—
173C4⋊C4×C151—21no—
174C4⋊C4×C151—22no—
175C4⋊C4×C151—44no—
176C4⋊C4×C151—42no—
177C22⋊C4×C151—42no—
178C22⋊C4×C151—44no—
179C22⋊C4×C151—44no—
180C22⋊C4×C151—44no—
181C22⋊C4×C151—42no—
182C22⋊C4×C151—82no—
183C22⋊C4×C151—82no—
184C22⋊C4×C151—82no—
185C22⋊C4×C151—81no—
186C42×C151—22no—
187C42×C151—22no—
188C42×C151—44no—
189C23×C151—22no—
190C23×C151—22no—
191C23×C151—21no—
192C23×C151—21no—
193C23×C151—22no—
194C23×C151—24no—
195C23×C151—48no—
196C23×C151—42no—
197C23×C151—42no—
198C23×C151—42no—
199C23×C151—42no—
200C23×C151—42no—
201C23×C151—42no—
202C23×C151—42no—
203C23×C151—41no—
204C23×C151—41no—
205C23×C151—42no—
206C23×C151—84no—
207C23×C151—84no—
208C23×C151—88no—
209C23×C151—88no—
210C23×C151#12112yes—
211C23×C151C22×D411yescharacteristic
212Q8×C151—22no—
213Q8×C151—24no—
214Q8×C151—41no—
215Q8×C151—41no—
216Q8×C151—42no—
217Q8×C151#12014yes—
218D4×C151—22no—
219D4×C151—24no—
220D4×C151—22no—
221D4×C151—24no—
222D4×C151—48no—
223D4×C151—48no—
224D4×C151—42no—
225D4×C151—42no—
226D4×C151—42no—
227C2×C4×C151—22no—
228C2×C4×C151—24no—
229C2×C4×C151—22no—
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—22no—
238C2×C4×C151—22no—
239C2×C4×C151—48no—
240C2×C4×C151—41no—
241C2×C4×C151—41no—
242C2×C4×C151—42no—
243C2×C4×C151—42no—
244C2×C4×C151—42no—
245C2×C4×C151—42no—
246C2×C4×C151—41no—
247C2×C4×C151—41no—
248C2×C4×C151#12112yes—
249C2×C4×C151C22×D411yescharacteristic
250C2×C4×C151C22×D412yes—
251C2×C4×C151C2×#4111yescharacteristic
252C2×C4×C151H(5,𝔽2)12yes—
253C8×C151—24no—
254C8×C151—41no—
255C8×C151—41no—
256C22×C151—22no—
257C22×C151—22no—
258C22×C151—22no—
259C22×C151—42no—
260C22×C151—42no—
261C22×C151—42no—
262C22×C151—44no—
263C22×C151—44no—
264C22×C151—44no—
265C22×C151—42no—
266C22×C151—42no—
267C22×C151—816no—
268C22×C151—82no—
269C22×C151—82no—
270C22×C151—82no—
271C22×C151—82no—
272C22×C151—82no—
273C22×C151—82no—
274C22×C151C2×#12111yescharacteristic
275C22×C151D4212yes—
276C22×C151#57512yes—
277C22×C151#57611yescharacteristic
278C22×C151#57711yescharacteristic
279C4×C151—22no—
280C4×C151—24no—
281C4×C151—42no—
282C4×C151—42no—
283C4×C151—81no—
284C4×C151—82no—
285C4×C151—81no—
286C4×C151C2×#12011yescharacteristic
287C4×C151#53411yescharacteristic
288C4×C151#54612yes—
289C2×C151—42no—
290C2×C151—82no—
291C2×C151—84no—
292C2×C151#241011yescharacteristic
293C2×C151#304011yescharacteristic
294C2×C151#304511yescharacteristic
295C2×C151#326812yes—
296C2×C151#327512yes—
297#24371C15111yesmaximal, characteristic, direct product factor, Sylow
298C151#2437111yescharacteristic, direct product factor, Sylow
299C22×AGL(1,ℤ/8)C2×C15111yescharacteristic
300C2×#464C2×C15112yescomplemented
301C2×#452C2×C15111yescharacteristic, complemented
302#2435C2×C15111yescharacteristic, complemented
303#2341C2×C15111yescharacteristic, complemented
304#2054C2×C15112yescomplemented
305#2012C2×C15112yescomplemented
306#1894C2×C15111yescharacteristic, complemented
307#1880C2×C15112yescomplemented
308#1851C2×C15111yescharacteristic, complemented
309#1786C2×C15111yescharacteristic, complemented
310C22×#41C22×C15111yescharacteristic
311C23×D4C22×C15111yescharacteristic
312C2×AGL(1,ℤ/8)—24no—
313C2×AGL(1,ℤ/8)—28no—
314C22×Q16C22×C15111yescharacteristic, complemented
315C22×SD16C22×C15112yes—
316C22×SD16C22×C15111yescharacteristic, complemented
317C22×D8C22×C15112yes—
318C22×M16C22×C15111yescharacteristic
319C2×#124C22×C15112yes—
320C2×C22⋊Q8C22×C15111yescharacteristic
321C2×C4×Q8C22×C15112yescomplemented
322#464—28no—
323#452—24no—
324C2×C4⋊C8C22×C15112yescomplemented
325C2×#103C22×C15111yescharacteristic, complemented
326C2×#103C22×C15112yescomplemented
327C2×#102C22×C15112yes—
328C2×#102C22×C15112yes—
329C2×#102C22×C15111yescharacteristic, complemented
330C2×C22⋊C8C22×C15111yescharacteristic
331#394—21no—
332#394—21no—
333#390—22no—
334#390C22×C15112yes—
335#390C22×C15112yescomplemented
336#380—21no—
337#380C22×C15111yescharacteristic
338#379—22no—
339C4×C22⋊C4—21no—
340C4×C22⋊C4C22×C15111yescharacteristic
341#340C22×C15112yescomplemented
342#336C22×C15112yescomplemented
343C25—21no—
344C2×#41—22no—
345C2×#41—24no—
346C2×#41—42no—
347C2×#41D4×C15114yes—
348C22×Q8C23×C15111yescharacteristic
349C22×Q8C23×C15111yescharacteristic
350C22×D4—24no—
351C22×D4—21no—
352C22×D4—22no—
353C22×D4—22no—
354C22×D4—21no—
355C22×D4—42no—
356C22×D4C23×C15112yes—
357C22×D4C23×C15111yescharacteristic
358C23×C4—21no—
359C23×C4C23×C15111yescharacteristic
360AGL(1,ℤ/8)—232no—
361C2×Q16—22no—
362C2×Q16—24no—
363C2×SD16—24no—
364C2×SD16—24no—
365C2×SD16—22no—
366C2×SD16—42no—
367C2×SD16D4×C15118yes—
368C2×D8—24no—
369C2×D8—24no—
370C2×D8—24no—
371C2×M16—22no—
372C2×M16—24no—
373C22×C8C23×C15111yescharacteristic
374C22×C8C23×C15111yescharacteristic
375C22×C8C23×C15112yes—
376#124—24no—
377#124—42no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C4×Q8—24no—
381C4×Q8—42no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4C23×C15112yes—
385C2×C4⋊C4C23×C15111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4D4×C15112yes—
395C2×C42—22no—
396C2×C42C23×C15112yes—
397C4⋊C8—24no—
398#103—22no—
399#103—24no—
400#102—24no—
401#102—22no—
402#102—42no—
403C22⋊C8—22no—
404#95—21no—
405#95D4×C15111yescharacteristic
406#95D4×C15111yescharacteristic
407C24—21no—
408C24—22no—
409C24—44no—
410C24—44no—
411C24—42no—
412C24—81no—
413C24—81no—
414C24C2×D4×C15111yescharacteristic
415#41—28no—
416#41—44no—
417#41—48no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8C2×D4×C15112yes—
425C2×Q8C2×D4×C15114yes—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4C2×D4×C15114yes—
442C2×D4C2×D4×C15112yes—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—44no—
451C22×C4C2×D4×C15112yes—
452C22×C4C2×D4×C15112yes—
453C22×C4C2×D4×C15111yescharacteristic
454C22×C4C2×D4×C15112yes—
455C22×C4#41×C15112yes—
456C22×C4C24×C15111yescharacteristic
457Q16—44no—
458SD16—216no—
459SD16—44no—
460D8—216no—
461M16—28no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8C2×D4×C15114yes—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—81no—
484C42—22no—
485C42—22no—
486C42—44no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—24no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—84no—
505C23—84no—
506C23—88no—
507C23—88no—
508C23#121×C15112yes—
509C23C22×D4×C15111yescharacteristic
510Q8—22no—
511Q8—24no—
512Q8—41no—
513Q8—41no—
514Q8—42no—
515Q8#120×C15114yes—
516D4—22no—
517D4—24no—
518D4—22no—
519D4—24no—
520D4—48no—
521D4—48no—
522D4—42no—
523D4—42no—
524D4—42no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4#121×C15112yes—
547C2×C4C22×D4×C15111yescharacteristic
548C2×C4C22×D4×C15112yes—
549C2×C4C2×#41×C15111yescharacteristic
550C2×C4H(5,𝔽2)×C15112yes—
551C8—24no—
552C8—41no—
553C8—41no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—816no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22C2×#121×C15111yescharacteristic
573C22D42×C15112yes—
574C22#575×C15112yes—
575C22#576×C15111yescharacteristic
576C22#577×C15111yescharacteristic
577C4—22no—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—81no—
582C4—82no—
583C4—81no—
584C4C2×#120×C15111yescharacteristic
585C4#534×C15111yescharacteristic
586C4#546×C15112yes—
587C2—42no—
588C2—84no—
589C2—82no—
590C2#2410×C15111yescharacteristic
591C2#3040×C15111yescharacteristic
592C2#3045×C15111yescharacteristic
593C2#3268×C15112yes—
594C2#3275×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC22×AGL(1,ℤ/8)×C151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C4×C22⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#464×C151C2semidirect
3. C2×#452×C151C2semidirect
4. #2435×C151C2semidirect
5. #2341×C151C2semidirect
6. #2054×C151C2semidirect
7. #2012×C151C2semidirect
8. #1894×C151C2semidirect
9. #1880×C151C2semidirect
10. #1851×C151C2semidirect
11. #1786×C151C2semidirect
16. C22×Q16×C151C22semidirect
18. C22×SD16×C151C22semidirect
23. C2×C4×Q8×C151C22semidirect
26. C2×C4⋊C8×C151C22semidirect
27. C2×#103×C151C22semidirect
28. C2×#103×C151C22semidirect
31. C2×#102×C151C22semidirect
37. #390×C151C22semidirect
43. #340×C151C22semidirect
44. #336×C151C22semidirect
297. #24371C151direct (N × H)
298. C151#24371direct (N × H)
300. C2×#464C2×C151semidirect
301. C2×#452C2×C151semidirect
302. #2435C2×C151semidirect
303. #2341C2×C151semidirect
304. #2054C2×C151semidirect
305. #2012C2×C151semidirect
306. #1894C2×C151semidirect
307. #1880C2×C151semidirect
308. #1851C2×C151semidirect
309. #1786C2×C151semidirect
314. C22×Q16C22×C151semidirect
316. C22×SD16C22×C151semidirect
321. C2×C4×Q8C22×C151semidirect
324. C2×C4⋊C8C22×C151semidirect
325. C2×#103C22×C151semidirect
326. C2×#103C22×C151semidirect
329. C2×#102C22×C151semidirect
335. #390C22×C151semidirect
341. #340C22×C151semidirect
342. #336C22×C151semidirect

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)×C1512
C2×#464×C1512
C2×#452×C1512
#2435×C1512
#2341×C1512
#2054×C1512
#2012×C1512
#1894×C1512
#1880×C1512
#1851×C1512
#1786×C1512
#24371151