Finite Group Databasebeta

← C151×#24120 #50329045

Subgroups — C151×#24120 #50329045

Order 38656.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547×C151C211yesmaximal, characteristic
2C2×#475×C151C211yesmaximal, characteristic, complemented
3C2×C4×SD16×C151C211yesmaximal, characteristic, complemented
4#2538×C151C211yesmaximal, characteristic, complemented
5#2034×C151C212yesmaximal, complemented
6#2000×C151C212yesmaximal, complemented
7#1990×C151C211yesmaximal, characteristic, complemented
8#1954×C151C211yesmaximal, characteristic, complemented
9#1872×C151C212yesmaximal, complemented
10#1853×C151C212yesmaximal, complemented
11#1802×C151C211yesmaximal, characteristic, complemented
12C22×#41×C151—21no—
13C22×SD16×C151C2211yescharacteristic
14#547×C151—24no—
15#547×C151—24no—
16C2×C4⋊Q8×C151C2211yescharacteristic
17C2×#123×C151—22no—
18C2×C22⋊Q8×C151—21no—
19C2×C22⋊Q8×C151C2212yes—
20C2×#121×C151C2212yes—
21C2×C4×Q8×C151C2211yescharacteristic
22C2×C4×D4×C151C2211yescharacteristic
23C2×C4×D4×C151C2211yescharacteristic
24C22×C4⋊C4×C151C2212yes—
25#475×C151—24no—
26C4×SD16×C151—24no—
27C2×#106×C151C2211yescharacteristic
28C2×#106×C151C2212yes—
29C2×C4⋊C8×C151C2211yescharacteristic
30C2×#103×C151C2211yescharacteristic
31C2×#103×C151C2212yes—
32C2×#102×C151C2211yescharacteristic
33C2×#102×C151C2212yes—
34C2×C22⋊C8×C151C2212yes—
35C2×C22⋊C8×C151C2212yes—
36C2×C4×C8×C151C2211yescharacteristic
37#396×C151—22no—
38#390×C151—21no—
39#389×C151—21no—
40#384×C151C2211yescharacteristic
41#380×C151C2212yes—
42#340×C151C2212yes—
43#340×C151C2212yes—
44#340×C151C2212yes—
45C2×#41×C151—28no—
46C2×#41×C151—44no—
47C2×#41×C151—44no—
48C22×Q8×C151—21no—
49C22×Q8×C151C2311yescharacteristic
50C22×D4×C151—22no—
51C22×D4×C151D411yescharacteristic
52C22×D4×C151C2311yescharacteristic
53C23×C4×C151—21no—
54C23×C4×C151—22no—
55C23×C4×C151C2312yes—
56C2×SD16×C151—22no—
57C2×SD16×C151—22no—
58C2×SD16×C151—22no—
59C22×C8×C151C2311yescharacteristic
60C22×C8×C151C2312yes—
61C22×C8×C151C2311yescharacteristic
62C4⋊Q8×C151—22no—
63C4⋊Q8×C151D414yes—
64#123×C151—216no—
65C22⋊Q8×C151—28no—
66C22⋊Q8×C151—24no—
67C22⋊Q8×C151—42no—
68#121×C151—24no—
69#121×C151—24no—
70C4×Q8×C151—22no—
71C4×Q8×C151—22no—
72C4×D4×C151—22no—
73C4×D4×C151—22no—
74C4×D4×C151—22no—
75C4×D4×C151D414yes—
76C2×C4⋊C4×C151—22no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—24no—
79C2×C4⋊C4×C151—24no—
80C2×C4⋊C4×C151—21no—
81C2×C4⋊C4×C151—22no—
82C2×C4⋊C4×C151—21no—
83C2×C4⋊C4×C151—21no—
84C2×C4⋊C4×C151D411yescharacteristic
85C2×C4⋊C4×C151C2312yes—
86C2×C4⋊C4×C151C2311yescharacteristic
87C2×C4⋊C4×C151C2311yescharacteristic
88C2×C4⋊C4×C151C2312yes—
89C2×C22⋊C4×C151—21no—
90C2×C22⋊C4×C151—22no—
91C2×C22⋊C4×C151—22no—
92C2×C22⋊C4×C151—22no—
93C2×C22⋊C4×C151D412yes—
94C2×C42×C151—21no—
95C2×C42×C151C2311yescharacteristic
96#106×C151—22no—
97#106×C151—24no—
98C4⋊C8×C151—22no—
99#103×C151—22no—
100#103×C151—24no—
101#102×C151—22no—
102#102×C151—42no—
103C22⋊C8×C151—24no—
104C22⋊C8×C151—42no—
105C4×C8×C151—22no—
106#95×C151—22no—
107C24×C151—21no—
108C24×C151C2×D412yes—
109#41×C151—432no—
110C2×Q8×C151—24no—
111C2×Q8×C151—21no—
112C2×Q8×C151—21no—
113C2×Q8×C151—21no—
114C2×Q8×C151—21no—
115C2×Q8×C151—21no—
116C2×Q8×C151—42no—
117C2×Q8×C151—42no—
118C2×Q8×C151#4112yescomplemented
119C2×D4×C151—28no—
120C2×D4×C151—22no—
121C2×D4×C151—22no—
122C2×D4×C151—21no—
123C2×D4×C151—21no—
124C2×D4×C151—21no—
125C2×D4×C151—21no—
126C2×D4×C151—21no—
127C2×D4×C151—44no—
128C2×D4×C151—44no—
129C2×D4×C151SD1614yescomplemented
130C2×D4×C151C2×D412yes—
131C22×C4×C151—28no—
132C22×C4×C151—21no—
133C22×C4×C151—21no—
134C22×C4×C151—22no—
135C22×C4×C151—24no—
136C22×C4×C151—22no—
137C22×C4×C151—22no—
138C22×C4×C151—22no—
139C22×C4×C151—22no—
140C22×C4×C151—22no—
141C22×C4×C151—22no—
142C22×C4×C151—21no—
143C22×C4×C151—22no—
144C22×C4×C151—44no—
145C22×C4×C151—44no—
146C22×C4×C151—42no—
147C22×C4×C151—42no—
148C22×C4×C151C2×D412yes—
149C22×C4×C151C2×D411yescharacteristic
150C22×C4×C151C2×D411yescharacteristic
151C22×C4×C151C2411yescharacteristic
152SD16×C151—28no—
153C2×C8×C151—21no—
154C2×C8×C151—21no—
155C2×C8×C151—21no—
156C2×C8×C151—22no—
157C2×C8×C151—22no—
158C2×C8×C151—21no—
159C2×C8×C151—22no—
160C2×C8×C151—21no—
161C2×C8×C151#4112yescomplemented
162C4⋊C4×C151—28no—
163C4⋊C4×C151—24no—
164C4⋊C4×C151—22no—
165C4⋊C4×C151—21no—
166C4⋊C4×C151—22no—
167C4⋊C4×C151—21no—
168C4⋊C4×C151—21no—
169C4⋊C4×C151—22no—
170C4⋊C4×C151—22no—
171C4⋊C4×C151—44no—
172C4⋊C4×C151—42no—
173C4⋊C4×C151—42no—
174C4⋊C4×C151—81no—
175C4⋊C4×C151C2×D412yes—
176C4⋊C4×C151C2×D414yes—
177C22⋊C4×C151—28no—
178C22⋊C4×C151—24no—
179C22⋊C4×C151—24no—
180C22⋊C4×C151—44no—
181C22⋊C4×C151—82no—
182C42×C151—42no—
183C42×C151C2×D412yes—
184C42×C151C2×D412yes—
185C23×C151—22no—
186C23×C151—22no—
187C23×C151—22no—
188C23×C151—22no—
189C23×C151—22no—
190C23×C151—22no—
191C23×C151—22no—
192C23×C151—41no—
193C23×C151—41no—
194C23×C151—41no—
195C23×C151—41no—
196C23×C151—41no—
197C23×C151—41no—
198C23×C151C2×SD1614yes—
199C23×C151C22×D411yescharacteristic
200Q8×C151—22no—
201Q8×C151—22no—
202Q8×C151—41no—
203Q8×C151—41no—
204Q8×C151—48no—
205D4×C151—22no—
206D4×C151—22no—
207D4×C151—28no—
208D4×C151—416no—
209D4×C151—41no—
210D4×C151—41no—
211C2×C4×C151—21no—
212C2×C4×C151—21no—
213C2×C4×C151—24no—
214C2×C4×C151—22no—
215C2×C4×C151—22no—
216C2×C4×C151—22no—
217C2×C4×C151—22no—
218C2×C4×C151—22no—
219C2×C4×C151—416no—
220C2×C4×C151—41no—
221C2×C4×C151—41no—
222C2×C4×C151—42no—
223C2×C4×C151—41no—
224C2×C4×C151—42no—
225C2×C4×C151—41no—
226C2×C4×C151—42no—
227C2×C4×C151—42no—
228C2×C4×C151—42no—
229C2×C4×C151—42no—
230C2×C4×C151—48no—
231C2×C4×C151—41no—
232C2×C4×C151—42no—
233C2×C4×C151—42no—
234C2×C4×C151—41no—
235C2×C4×C151—42no—
236C2×C4×C151—41no—
237C2×C4×C151#12012yes—
238C2×C4×C151#12012yes—
239C2×C4×C151C2×SD1612yes—
240C2×C4×C151AGL(1,ℤ/8)12yes—
241C2×C4×C151C22×D411yescharacteristic
242C2×C4×C151C22×D411yescharacteristic
243C2×C4×C151C2×#4111yescharacteristic
244C2×C4×C151C2×#4111yescharacteristic
245C2×C4×C151H(5,𝔽2)11yescharacteristic
246C2×C4×C1512−1+411yescharacteristic
247C8×C151—22no—
248C8×C151—41no—
249C8×C151—42no—
250C22×C151—24no—
251C22×C151—24no—
252C22×C151—24no—
253C22×C151—42no—
254C22×C151—42no—
255C22×C151—42no—
256C22×C151—42no—
257C22×C151—42no—
258C22×C151—42no—
259C22×C151—42no—
260C22×C151—81no—
261C22×C151—81no—
262C22×C151—81no—
263C22×C151—81no—
264C22×C151C2×#12011yescharacteristic
265C22×C151#54611yescharacteristic
266C22×C151#54711yescharacteristic
267C22×C151C22×SD1611yescharacteristic
268C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C151#57711yescharacteristic
270C22×C151#57811yescharacteristic
271C4×C151—22no—
272C4×C151—42no—
273C4×C151—42no—
274C4×C151—42no—
275C4×C151—44no—
276C4×C151—82no—
277C4×C151—81no—
278C4×C151#45014yes—
279C4×C151#52511yescharacteristic
280C4×C151#53711yescharacteristic
281C4×C151#54811yescharacteristic
282C4×C151#55011yescharacteristic
283C2×C151—44no—
284C2×C151—82no—
285C2×C151#240711yescharacteristic
286C2×C151C2×#45011yescharacteristic
287C2×C151#301311yescharacteristic
288C2×C151#329811yescharacteristic
289C2×C151#333211yescharacteristic
290C2×C151#333511yescharacteristic
291C2×C151#335311yescharacteristic
292#24120C15111yesmaximal, characteristic, direct product factor, Sylow
293C151#2412011yescharacteristic, direct product factor, Sylow
294C2×#547C2×C15111yescharacteristic
295C2×#475C2×C15111yescharacteristic, complemented
296C2×C4×SD16C2×C15111yescharacteristic, complemented
297#2538C2×C15111yescharacteristic, complemented
298#2034C2×C15112yescomplemented
299#2000C2×C15112yescomplemented
300#1990C2×C15111yescharacteristic, complemented
301#1954C2×C15111yescharacteristic, complemented
302#1872C2×C15112yescomplemented
303#1853C2×C15112yescomplemented
304#1802C2×C15111yescharacteristic, complemented
305C22×#41—21no—
306C22×SD16C22×C15111yescharacteristic
307#547—24no—
308#547—24no—
309C2×C4⋊Q8C22×C15111yescharacteristic
310C2×#123—22no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8C22×C15112yes—
313C2×#121C22×C15112yes—
314C2×C4×Q8C22×C15111yescharacteristic
315C2×C4×D4C22×C15111yescharacteristic
316C2×C4×D4C22×C15111yescharacteristic
317C22×C4⋊C4C22×C15112yes—
318#475—24no—
319C4×SD16—24no—
320C2×#106C22×C15112yes—
321C2×#106C22×C15111yescharacteristic
322C2×C4⋊C8C22×C15111yescharacteristic
323C2×#103C22×C15112yes—
324C2×#103C22×C15111yescharacteristic
325C2×#102C22×C15111yescharacteristic
326C2×#102C22×C15112yes—
327C2×C22⋊C8C22×C15112yes—
328C2×C22⋊C8C22×C15112yes—
329C2×C4×C8C22×C15111yescharacteristic
330#396—22no—
331#390—21no—
332#389—21no—
333#384C22×C15111yescharacteristic
334#380C22×C15112yes—
335#340C22×C15112yes—
336#340C22×C15112yes—
337#340C22×C15112yes—
338C2×#41—28no—
339C2×#41—44no—
340C2×#41—44no—
341C22×Q8—21no—
342C22×Q8C23×C15111yescharacteristic
343C22×D4—22no—
344C22×D4D4×C15111yescharacteristic
345C22×D4C23×C15111yescharacteristic
346C23×C4—22no—
347C23×C4—21no—
348C23×C4C23×C15112yes—
349C2×SD16—22no—
350C2×SD16—22no—
351C2×SD16—22no—
352C22×C8C23×C15111yescharacteristic
353C22×C8C23×C15112yes—
354C22×C8C23×C15111yescharacteristic
355C4⋊Q8—22no—
356C4⋊Q8D4×C15114yes—
357#123—216no—
358C22⋊Q8—28no—
359C22⋊Q8—24no—
360C22⋊Q8—42no—
361#121—24no—
362#121—24no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4D4×C15114yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4D4×C15111yescharacteristic
378C2×C4⋊C4C23×C15111yescharacteristic
379C2×C4⋊C4C23×C15112yes—
380C2×C4⋊C4C23×C15111yescharacteristic
381C2×C4⋊C4C23×C15112yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4D4×C15112yes—
387C2×C42—21no—
388C2×C42C23×C15111yescharacteristic
389#106—22no—
390#106—24no—
391C4⋊C8—22no—
392#103—22no—
393#103—24no—
394#102—22no—
395#102—42no—
396C22⋊C8—24no—
397C22⋊C8—42no—
398C4×C8—22no—
399#95—22no—
400C24—21no—
401C24C2×D4×C15112yes—
402#41—432no—
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8#41×C15112yescomplemented
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4SD16×C15114yescomplemented
423C2×D4C2×D4×C15112yes—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4C2×D4×C15112yes—
442C22×C4C2×D4×C15111yescharacteristic
443C22×C4C2×D4×C15111yescharacteristic
444C22×C4C24×C15111yescharacteristic
445SD16—28no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—21no—
454C2×C8#41×C15112yescomplemented
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4C2×D4×C15112yes—
469C4⋊C4C2×D4×C15114yes—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C42—42no—
476C42C2×D4×C15112yes—
477C42C2×D4×C15112yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×SD16×C15114yes—
492C23C22×D4×C15111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—48no—
496Q8—41no—
497Q8—41no—
498D4—28no—
499D4—22no—
500D4—22no—
501D4—416no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—416no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4#120×C15112yes—
531C2×C4#120×C15112yes—
532C2×C4C2×SD16×C15112yes—
533C2×C4AGL(1,ℤ/8)×C15112yes—
534C2×C4C22×D4×C15111yescharacteristic
535C2×C4C22×D4×C15111yescharacteristic
536C2×C4C2×#41×C15111yescharacteristic
537C2×C4C2×#41×C15111yescharacteristic
538C2×C4H(5,𝔽2)×C15111yescharacteristic
539C2×C42−1+4×C15111yescharacteristic
540C8—22no—
541C8—41no—
542C8—42no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#120×C15111yescharacteristic
558C22#546×C15111yescharacteristic
559C22#547×C15111yescharacteristic
560C22C22×SD16×C15111yescharacteristic
561C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
562C22#577×C15111yescharacteristic
563C22#578×C15111yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—44no—
569C4—81no—
570C4—82no—
571C4#450×C15114yes—
572C4#525×C15111yescharacteristic
573C4#537×C15111yescharacteristic
574C4#548×C15111yescharacteristic
575C4#550×C15111yescharacteristic
576C2—44no—
577C2—82no—
578C2#2407×C15111yescharacteristic
579C2C2×#450×C15111yescharacteristic
580C2#3013×C15111yescharacteristic
581C2#3298×C15111yescharacteristic
582C2#3332×C15111yescharacteristic
583C2#3335×C15111yescharacteristic
584C2#3353×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×#547×C151C2
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
2. C2×#475×C151C2semidirect
3. C2×C4×SD16×C151C2semidirect
4. #2538×C151C2semidirect
5. #2034×C151C2semidirect
6. #2000×C151C2semidirect
7. #1990×C151C2semidirect
8. #1954×C151C2semidirect
9. #1872×C151C2semidirect
10. #1853×C151C2semidirect
11. #1802×C151C2semidirect
118. C2×Q8×C151#41semidirect
129. C2×D4×C151SD16semidirect
161. C2×C8×C151#41semidirect
292. #24120C151direct (N × H)
293. C151#24120direct (N × H)
295. C2×#475C2×C151semidirect
296. C2×C4×SD16C2×C151semidirect
297. #2538C2×C151semidirect
298. #2034C2×C151semidirect
299. #2000C2×C151semidirect
300. #1990C2×C151semidirect
301. #1954C2×C151semidirect
302. #1872C2×C151semidirect
303. #1853C2×C151semidirect
304. #1802C2×C151semidirect
411. C2×Q8#41×C151semidirect
422. C2×D4SD16×C151semidirect
454. C2×C8#41×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×#475×C1512
C2×C4×SD16×C1512
#2538×C1512
#2034×C1512
#2000×C1512
#1990×C1512
#1954×C1512
#1872×C1512
#1853×C1512
#1802×C1512
#24120151