Finite Group Databasebeta

← C151×#24357 #50329282

Subgroups — C151×#24357 #50329282

Order 38656.

1526 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×SD16×C151C211yesmaximal, characteristic
2C2×#464×C151C212yesmaximal, complemented
3C2×#451×C151C211yesmaximal, characteristic, complemented
4#2435×C151C211yesmaximal, characteristic
5Q8×C22⋊C4×C151C211yesmaximal, characteristic, complemented
6#2055×C151C212yesmaximal, complemented
7#2015×C151C212yesmaximal, complemented
8#1892×C151C211yesmaximal, characteristic, complemented
9#1880×C151C212yesmaximal, complemented
10#1850×C151C211yesmaximal, characteristic, complemented
11#1786×C151C211yesmaximal, characteristic
12C23×Q8×C151C2211yescharacteristic
13C23×D4×C151C2211yescharacteristic
14C22×Q16×C151C2211yescharacteristic, complemented
15C22×Q16×C151C2211yescharacteristic, complemented
16C22×SD16×C151—24no—
17C22×SD16×C151—28no—
18C22×SD16×C151C2212yes—
19C22×SD16×C151C2212yes—
20C23×C8×C151C2211yescharacteristic
21C2×#124×C151C2212yes—
22C2×C22⋊Q8×C151C2211yescharacteristic
23C2×C4×Q8×C151C2212yescomplemented
24#464×C151—28no—
25#451×C151—24no—
26C2×C4⋊C8×C151C2212yescomplemented
27C2×#103×C151C2211yescharacteristic, complemented
28C2×#103×C151C2212yescomplemented
29C2×#103×C151C2211yescharacteristic, complemented
30C2×#102×C151C2212yes—
31C2×#102×C151C2212yes—
32C2×C22⋊C8×C151C2211yescharacteristic
33#394×C151—22no—
34#391×C151—22no—
35#391×C151C2212yescomplemented
36#390×C151C2212yes—
37#380×C151—21no—
38#380×C151C2211yescharacteristic
39#379×C151—21no—
40#379×C151—21no—
41C4×C22⋊C4×C151—21no—
42C4×C22⋊C4×C151C2211yescharacteristic
43#340×C151C2212yescomplemented
44#336×C151C2212yescomplemented
45C25×C151—21no—
46C22×Q8×C151—24no—
47C22×Q8×C151—21no—
48C22×Q8×C151—22no—
49C22×Q8×C151—42no—
50C22×Q8×C151D414yes—
51C22×Q8×C151C2311yescharacteristic
52C22×Q8×C151C2311yescharacteristic
53C22×Q8×C151C2311yescharacteristic
54C22×D4×C151—21no—
55C22×D4×C151—24no—
56C22×D4×C151—22no—
57C22×D4×C151—22no—
58C22×D4×C151—42no—
59C22×D4×C151C2312yes—
60C23×C4×C151—21no—
61C23×C4×C151C2311yescharacteristic
62C2×Q16×C151—22no—
63C2×Q16×C151—22no—
64C2×Q16×C151—24no—
65C2×Q16×C151—42no—
66C2×SD16×C151—232no—
67C2×SD16×C151—24no—
68C2×SD16×C151—24no—
69C2×SD16×C151—24no—
70C2×SD16×C151—24no—
71C2×SD16×C151—24no—
72C2×SD16×C151D418yes—
73C22×C8×C151—22no—
74C22×C8×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—21no—
93C2×C22⋊C4×C151—21no—
94C2×C22⋊C4×C151D411yescharacteristic
95C2×C22⋊C4×C151D411yescharacteristic
96C2×C42×C151—22no—
97C2×C42×C151C2312yes—
98C4⋊C8×C151—24no—
99#103×C151—24no—
100#103×C151—22no—
101#103×C151—22no—
102#102×C151—24no—
103#102×C151—42no—
104C22⋊C8×C151—22no—
105#95×C151—22no—
106#95×C151—22no—
107#95×C151—21no—
108#95×C151D412yes—
109C24×C151—22no—
110C24×C151—44no—
111C24×C151—42no—
112C24×C151—42no—
113C24×C151—42no—
114C24×C151—82no—
115C24×C151C2×D411yescharacteristic
116C2×Q8×C151—24no—
117C2×Q8×C151—21no—
118C2×Q8×C151—21no—
119C2×Q8×C151—22no—
120C2×Q8×C151—28no—
121C2×Q8×C151—22no—
122C2×Q8×C151—22no—
123C2×Q8×C151—21no—
124C2×Q8×C151—21no—
125C2×Q8×C151—22no—
126C2×Q8×C151—22no—
127C2×Q8×C151—44no—
128C2×Q8×C151—44no—
129C2×Q8×C151—48no—
130C2×Q8×C151C2×D412yes—
131C2×Q8×C151C2×D412yes—
132C2×Q8×C151C2×D414yes—
133C2×D4×C151—28no—
134C2×D4×C151—22no—
135C2×D4×C151—22no—
136C2×D4×C151—22no—
137C2×D4×C151—22no—
138C2×D4×C151—22no—
139C2×D4×C151—48no—
140C2×D4×C151—44no—
141C2×D4×C151—44no—
142C2×D4×C151—42no—
143C2×D4×C151C2×D414yes—
144C22×C4×C151—21no—
145C22×C4×C151—22no—
146C22×C4×C151—22no—
147C22×C4×C151—24no—
148C22×C4×C151—21no—
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—
160Q16×C151—44no—
161SD16×C151—216no—
162SD16×C151—216no—
163C2×C8×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—22no—
178C22⋊C4×C151—22no—
179C22⋊C4×C151—44no—
180C22⋊C4×C151—44no—
181C22⋊C4×C151—42no—
182C22⋊C4×C151—42no—
183C22⋊C4×C151—82no—
184C22⋊C4×C151—82no—
185C42×C151—22no—
186C42×C151—22no—
187C42×C151—44no—
188C23×C151—22no—
189C23×C151—22no—
190C23×C151—24no—
191C23×C151—48no—
192C23×C151—42no—
193C23×C151—42no—
194C23×C151—42no—
195C23×C151—42no—
196C23×C151—42no—
197C23×C151—42no—
198C23×C151—88no—
199C23×C151—88no—
200C23×C151—84no—
201C23×C151—84no—
202C23×C151#12112yes—
203C23×C151#13512yes—
204C23×C151#13712yes—
205C23×C151C22×D411yescharacteristic
206Q8×C151—22no—
207Q8×C151—24no—
208Q8×C151—22no—
209Q8×C151—22no—
210Q8×C151—48no—
211Q8×C151—42no—
212Q8×C151—41no—
213Q8×C151—41no—
214Q8×C151—42no—
215Q8×C151#12014yes—
216D4×C151—24no—
217D4×C151—24no—
218D4×C151—48no—
219D4×C151—42no—
220D4×C151—42no—
221C2×C4×C151—22no—
222C2×C4×C151—24no—
223C2×C4×C151—22no—
224C2×C4×C151—22no—
225C2×C4×C151—22no—
226C2×C4×C151—24no—
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—22no—
234C2×C4×C151—48no—
235C2×C4×C151—42no—
236C2×C4×C151—41no—
237C2×C4×C151—41no—
238C2×C4×C151—42no—
239C2×C4×C151—42no—
240C2×C4×C151—42no—
241C2×C4×C151—42no—
242C2×C4×C151—42no—
243C2×C4×C151#12112yes—
244C2×C4×C151C22×D411yescharacteristic
245C2×C4×C151C22×D412yes—
246C2×C4×C151C2×#4111yescharacteristic
247C2×C4×C151H(5,𝔽2)12yes—
248C8×C151—24no—
249C8×C151—41no—
250C8×C151—41no—
251C22×C151—22no—
252C22×C151—24no—
253C22×C151—24no—
254C22×C151—22no—
255C22×C151—44no—
256C22×C151—44no—
257C22×C151—44no—
258C22×C151—816no—
259C22×C151—82no—
260C22×C151—82no—
261C22×C151—82no—
262C22×C151—82no—
263C22×C151#46414yescomplemented
264C22×C151C2×#12111yescharacteristic
265C22×C151D4212yes—
266C22×C151C2×#13511yescharacteristic
267C22×C151C2×#13711yescharacteristic
268C22×C151#57712yes—
269C4×C151—22no—
270C4×C151—24no—
271C4×C151—42no—
272C4×C151—42no—
273C4×C151—42no—
274C4×C151—81no—
275C4×C151—82no—
276C4×C151—81no—
277C4×C151C2×#12011yescharacteristic
278C4×C151#53411yescharacteristic
279C4×C151#54612yes—
280C2×C151—24no—
281C2×C151—84no—
282C2×C151#241011yescharacteristic
283C2×C151C2×#46411yescharacteristic
284C2×C151#304311yescharacteristic
285C2×C151#326612yes—
286C2×C151#328212yes—
287#24357C15111yesmaximal, characteristic, direct product factor, Sylow
288C151#2435711yescharacteristic, direct product factor, Sylow
289C23×SD16C2×C15111yescharacteristic
290C2×#464C2×C15112yescomplemented
291C2×#451C2×C15111yescharacteristic, complemented
292#2435C2×C15111yescharacteristic
293Q8×C22⋊C4C2×C15111yescharacteristic, complemented
294#2055C2×C15112yescomplemented
295#2015C2×C15112yescomplemented
296#1892C2×C15111yescharacteristic, complemented
297#1880C2×C15112yescomplemented
298#1850C2×C15111yescharacteristic, complemented
299#1786C2×C15111yescharacteristic
300C23×Q8C22×C15111yescharacteristic
301C23×D4C22×C15111yescharacteristic
302C22×Q16C22×C15111yescharacteristic, complemented
303C22×Q16C22×C15111yescharacteristic, complemented
304C22×SD16—24no—
305C22×SD16—28no—
306C22×SD16C22×C15112yes—
307C22×SD16C22×C15112yes—
308C23×C8C22×C15111yescharacteristic
309C2×#124C22×C15112yes—
310C2×C22⋊Q8C22×C15111yescharacteristic
311C2×C4×Q8C22×C15112yescomplemented
312#464—28no—
313#451—24no—
314C2×C4⋊C8C22×C15112yescomplemented
315C2×#103C22×C15111yescharacteristic, complemented
316C2×#103C22×C15111yescharacteristic, complemented
317C2×#103C22×C15112yescomplemented
318C2×#102C22×C15112yes—
319C2×#102C22×C15112yes—
320C2×C22⋊C8C22×C15111yescharacteristic
321#394—22no—
322#391—22no—
323#391C22×C15112yescomplemented
324#390C22×C15112yes—
325#380—21no—
326#380C22×C15111yescharacteristic
327#379—21no—
328#379—21no—
329C4×C22⋊C4—21no—
330C4×C22⋊C4C22×C15111yescharacteristic
331#340C22×C15112yescomplemented
332#336C22×C15112yescomplemented
333C25—21no—
334C22×Q8—22no—
335C22×Q8—21no—
336C22×Q8—24no—
337C22×Q8—42no—
338C22×Q8D4×C15114yes—
339C22×Q8C23×C15111yescharacteristic
340C22×Q8C23×C15111yescharacteristic
341C22×Q8C23×C15111yescharacteristic
342C22×D4—24no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—21no—
346C22×D4—42no—
347C22×D4C23×C15112yes—
348C23×C4—21no—
349C23×C4C23×C15111yescharacteristic
350C2×Q16—22no—
351C2×Q16—22no—
352C2×Q16—24no—
353C2×Q16—42no—
354C2×SD16—232no—
355C2×SD16—24no—
356C2×SD16—24no—
357C2×SD16—24no—
358C2×SD16—24no—
359C2×SD16—24no—
360C2×SD16D4×C15118yes—
361C22×C8—22no—
362C22×C8—24no—
363C22×C8C23×C15111yescharacteristic
364C22×C8C23×C15111yescharacteristic
365C22×C8C23×C15112yes—
366#124—24no—
367#124—42no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C4×Q8—24no—
371C4×Q8—42no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C23×C15112yes—
375C2×C4⋊C4C23×C15111yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4D4×C15111yescharacteristic
383C2×C22⋊C4D4×C15111yescharacteristic
384C2×C42—22no—
385C2×C42C23×C15112yes—
386C4⋊C8—24no—
387#103—22no—
388#103—22no—
389#103—24no—
390#102—24no—
391#102—42no—
392C22⋊C8—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95D4×C15112yes—
397C24—22no—
398C24—44no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—82no—
403C24C2×D4×C15111yescharacteristic
404C2×Q8—24no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—28no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—48no—
418C2×Q8C2×D4×C15112yes—
419C2×Q8C2×D4×C15112yes—
420C2×Q8C2×D4×C15114yes—
421C2×D4—28no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—48no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4C2×D4×C15114yes—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—44no—
441C22×C4C2×D4×C15112yes—
442C22×C4C2×D4×C15112yes—
443C22×C4C2×D4×C15111yescharacteristic
444C22×C4C2×D4×C15112yes—
445C22×C4#41×C15112yes—
446C22×C4C24×C15111yescharacteristic
447Q16—44no—
448Q16—44no—
449SD16—216no—
450SD16—216no—
451C2×C8—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8C2×D4×C15114yes—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C42—22no—
474C42—22no—
475C42—44no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—48no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—88no—
487C23—88no—
488C23—84no—
489C23—84no—
490C23#121×C15112yes—
491C23#135×C15112yes—
492C23#137×C15112yes—
493C23C22×D4×C15111yescharacteristic
494Q8—22no—
495Q8—24no—
496Q8—22no—
497Q8—22no—
498Q8—48no—
499Q8—42no—
500Q8—41no—
501Q8—41no—
502Q8—42no—
503Q8#120×C15114yes—
504D4—24no—
505D4—24no—
506D4—48no—
507D4—42no—
508D4—42no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4#121×C15112yes—
532C2×C4C22×D4×C15111yescharacteristic
533C2×C4C22×D4×C15112yes—
534C2×C4C2×#41×C15111yescharacteristic
535C2×C4H(5,𝔽2)×C15112yes—
536C8—24no—
537C8—41no—
538C8—41no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—22no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—816no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22#464×C15114yescomplemented
552C22C2×#121×C15111yescharacteristic
553C22D42×C15112yes—
554C22C2×#135×C15111yescharacteristic
555C22C2×#137×C15111yescharacteristic
556C22#577×C15112yes—
557C4—22no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—82no—
564C4—81no—
565C4C2×#120×C15111yescharacteristic
566C4#534×C15111yescharacteristic
567C4#546×C15112yes—
568C2—24no—
569C2—84no—
570C2#2410×C15111yescharacteristic
571C2C2×#464×C15111yescharacteristic
572C2#3043×C15111yescharacteristic
573C2#3266×C15112yes—
574C2#3282×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC23×D4×C151C22
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×#451×C151C2semidirect
5. Q8×C22⋊C4×C151C2semidirect
6. #2055×C151C2semidirect
7. #2015×C151C2semidirect
8. #1892×C151C2semidirect
9. #1880×C151C2semidirect
10. #1850×C151C2semidirect
14. C22×Q16×C151C22semidirect
15. C22×Q16×C151C22semidirect
23. C2×C4×Q8×C151C22semidirect
26. C2×C4⋊C8×C151C22semidirect
27. C2×#103×C151C22semidirect
28. C2×#103×C151C22semidirect
29. C2×#103×C151C22semidirect
35. #391×C151C22semidirect
43. #340×C151C22semidirect
44. #336×C151C22semidirect
263. C22×C151#464semidirect
287. #24357C151direct (N × H)
288. C151#24357direct (N × H)
290. C2×#464C2×C151semidirect
291. C2×#451C2×C151semidirect
293. Q8×C22⋊C4C2×C151semidirect
294. #2055C2×C151semidirect
295. #2015C2×C151semidirect
296. #1892C2×C151semidirect
297. #1880C2×C151semidirect
298. #1850C2×C151semidirect
302. C22×Q16C22×C151semidirect
303. C22×Q16C22×C151semidirect
311. C2×C4×Q8C22×C151semidirect
314. C2×C4⋊C8C22×C151semidirect
315. C2×#103C22×C151semidirect
316. C2×#103C22×C151semidirect
317. C2×#103C22×C151semidirect
323. #391C22×C151semidirect
331. #340C22×C151semidirect
332. #336C22×C151semidirect
551. C22#464×C151semidirect

Maximal subgroups

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

subgroupindex
C23×SD16×C1512
C2×#464×C1512
C2×#451×C1512
#2435×C1512
Q8×C22⋊C4×C1512
#2055×C1512
#2015×C1512
#1892×C1512
#1880×C1512
#1850×C1512
#1786×C1512
#24357151