Finite Group Databasebeta

← C151×#24132 #50329057

Subgroups — C151×#24132 #50329057

Order 38656.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547×C151C211yesmaximal, characteristic
2C2×#477×C151C211yesmaximal, characteristic, complemented
3C2×#440×C151C211yesmaximal, characteristic, complemented
4#2536×C151C211yesmaximal, characteristic, complemented
5#2033×C151C212yesmaximal, complemented
6#2000×C151C212yesmaximal, complemented
7#1993×C151C211yesmaximal, characteristic, complemented
8#1955×C151C211yesmaximal, characteristic, complemented
9#1872×C151C212yesmaximal, complemented
10#1852×C151C212yesmaximal, complemented
11#1804×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#477×C151—24no—
26#440×C151—24no—
27C2×#107×C151C2211yescharacteristic
28C2×#107×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×#97×C151C2211yescharacteristic
37#398×C151—21no—
38#394×C151—21no—
39#388×C151—22no—
40#382×C151C2211yescharacteristic
41#380×C151C2212yes—
42#340×C151C2212yes—
43#340×C151C2212yes—
44#340×C151C2212yes—
45C2×#41×C151—28no—
46C2×#41×C151—48no—
47C22×Q8×C151—21no—
48C22×Q8×C151C2311yescharacteristic
49C22×D4×C151—22no—
50C22×D4×C151D411yescharacteristic
51C22×D4×C151C2311yescharacteristic
52C23×C4×C151—21no—
53C23×C4×C151—22no—
54C23×C4×C151C2312yes—
55C2×SD16×C151—22no—
56C2×SD16×C151—22no—
57C2×SD16×C151—22no—
58C22×C8×C151C2311yescharacteristic
59C22×C8×C151C2312yes—
60C22×C8×C151C2311yescharacteristic
61C4⋊Q8×C151—22no—
62C4⋊Q8×C151D414yes—
63#123×C151—216no—
64C22⋊Q8×C151—28no—
65C22⋊Q8×C151—24no—
66C22⋊Q8×C151—42no—
67#121×C151—24no—
68#121×C151—24no—
69C4×Q8×C151—22no—
70C4×Q8×C151—22no—
71C4×D4×C151—22no—
72C4×D4×C151—22no—
73C4×D4×C151—22no—
74C4×D4×C151D414yes—
75C2×C4⋊C4×C151—22no—
76C2×C4⋊C4×C151—21no—
77C2×C4⋊C4×C151—24no—
78C2×C4⋊C4×C151—24no—
79C2×C4⋊C4×C151—22no—
80C2×C4⋊C4×C151—21no—
81C2×C4⋊C4×C151D411yescharacteristic
82C2×C4⋊C4×C151C2312yes—
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2312yes—
86C2×C22⋊C4×C151—21no—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151—22no—
89C2×C22⋊C4×C151—22no—
90C2×C22⋊C4×C151D412yes—
91C2×C42×C151—21no—
92C2×C42×C151C2311yescharacteristic
93#107×C151—22no—
94#107×C151—24no—
95C4⋊C8×C151—22no—
96#103×C151—22no—
97#103×C151—24no—
98#102×C151—22no—
99#102×C151—42no—
100C22⋊C8×C151—24no—
101C22⋊C8×C151—42no—
102#97×C151—22no—
103#95×C151—22no—
104#95×C151—21no—
105#95×C151—21no—
106C24×C151—21no—
107C24×C151C2×D412yes—
108#41×C151—432no—
109C2×Q8×C151—24no—
110C2×Q8×C151—21no—
111C2×Q8×C151—21no—
112C2×Q8×C151—21no—
113C2×Q8×C151—21no—
114C2×Q8×C151—21no—
115C2×Q8×C151—44no—
116C2×Q8×C151#4112yescomplemented
117C2×D4×C151—24no—
118C2×D4×C151—28no—
119C2×D4×C151—22no—
120C2×D4×C151—21no—
121C2×D4×C151—21no—
122C2×D4×C151—21no—
123C2×D4×C151—21no—
124C2×D4×C151—21no—
125C2×D4×C151—44no—
126C2×D4×C151—44no—
127C2×D4×C151C2×D412yes—
128C22×C4×C151—28no—
129C22×C4×C151—21no—
130C22×C4×C151—21no—
131C22×C4×C151—22no—
132C22×C4×C151—24no—
133C22×C4×C151—22no—
134C22×C4×C151—22no—
135C22×C4×C151—22no—
136C22×C4×C151—22no—
137C22×C4×C151—22no—
138C22×C4×C151—22no—
139C22×C4×C151—22no—
140C22×C4×C151—21no—
141C22×C4×C151—44no—
142C22×C4×C151—44no—
143C22×C4×C151—44no—
144C22×C4×C151C2×D412yes—
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2×D411yescharacteristic
147C22×C4×C151C2411yescharacteristic
148SD16×C151—28no—
149C2×C8×C151—21no—
150C2×C8×C151—21no—
151C2×C8×C151—21no—
152C2×C8×C151—22no—
153C2×C8×C151—22no—
154C2×C8×C151—21no—
155C2×C8×C151—22no—
156C2×C8×C151—21no—
157C2×C8×C151#4112yescomplemented
158C4⋊C4×C151—28no—
159C4⋊C4×C151—24no—
160C4⋊C4×C151—22no—
161C4⋊C4×C151—21no—
162C4⋊C4×C151—22no—
163C4⋊C4×C151—21no—
164C4⋊C4×C151—21no—
165C4⋊C4×C151—22no—
166C4⋊C4×C151—22no—
167C4⋊C4×C151—44no—
168C4⋊C4×C151—42no—
169C4⋊C4×C151C2×D412yes—
170C4⋊C4×C151C2×D414yes—
171C22⋊C4×C151—28no—
172C22⋊C4×C151—24no—
173C22⋊C4×C151—24no—
174C22⋊C4×C151—44no—
175C22⋊C4×C151—44no—
176C42×C151—42no—
177C42×C151C2×D412yes—
178C42×C151C2×D412yes—
179C23×C151—22no—
180C23×C151—22no—
181C23×C151—22no—
182C23×C151—22no—
183C23×C151—22no—
184C23×C151—22no—
185C23×C151—22no—
186C23×C151—41no—
187C23×C151—41no—
188C23×C151—42no—
189C23×C151—42no—
190C23×C151AGL(1,ℤ/8)14yes—
191C23×C151C22×D411yescharacteristic
192Q8×C151—22no—
193Q8×C151—22no—
194Q8×C151—48no—
195Q8×C151—41no—
196Q8×C151—41no—
197D4×C151—28no—
198D4×C151—22no—
199D4×C151—22no—
200D4×C151—416no—
201D4×C151—41no—
202D4×C151—41no—
203C2×C4×C151—22no—
204C2×C4×C151—22no—
205C2×C4×C151—24no—
206C2×C4×C151—22no—
207C2×C4×C151—22no—
208C2×C4×C151—22no—
209C2×C4×C151—22no—
210C2×C4×C151—22no—
211C2×C4×C151—48no—
212C2×C4×C151—416no—
213C2×C4×C151—41no—
214C2×C4×C151—41no—
215C2×C4×C151—42no—
216C2×C4×C151—42no—
217C2×C4×C151—42no—
218C2×C4×C151—42no—
219C2×C4×C151—42no—
220C2×C4×C151—42no—
221C2×C4×C151—42no—
222C2×C4×C151—41no—
223C2×C4×C151—42no—
224C2×C4×C151—42no—
225C2×C4×C151—42no—
226C2×C4×C151—42no—
227C2×C4×C151#12012yes—
228C2×C4×C151#12012yes—
229C2×C4×C151C22×D411yescharacteristic
230C2×C4×C151C22×D411yescharacteristic
231C2×C4×C151C2×#4111yescharacteristic
232C2×C4×C151C2×#4111yescharacteristic
233C2×C4×C151H(5,𝔽2)11yescharacteristic
234C2×C4×C1512−1+411yescharacteristic
235C8×C151—22no—
236C8×C151—41no—
237C8×C151—42no—
238C22×C151—24no—
239C22×C151—24no—
240C22×C151—24no—
241C22×C151—42no—
242C22×C151—42no—
243C22×C151—42no—
244C22×C151—42no—
245C22×C151—42no—
246C22×C151—42no—
247C22×C151—42no—
248C22×C151—82no—
249C22×C151—82no—
250C22×C151C2×#12011yescharacteristic
251C22×C151#54611yescharacteristic
252C22×C151#54711yescharacteristic
253C22×C151C2×AGL(1,ℤ/8)12yes—
254C22×C151#57812yes—
255C4×C151—22no—
256C4×C151—22no—
257C4×C151—42no—
258C4×C151—42no—
259C4×C151—42no—
260C4×C151—44no—
261C4×C151—81no—
262C4×C151—82no—
263C4×C151#52511yescharacteristic
264C4×C151#53711yescharacteristic
265C4×C151#54811yescharacteristic
266C4×C151#55011yescharacteristic
267C2×C151—44no—
268C2×C151—82no—
269C2×C151#240711yescharacteristic
270C2×C151#300311yescharacteristic
271C2×C151#301111yescharacteristic
272C2×C151#330812yes—
273C2×C151#334312yes—
274#24132C15111yesmaximal, characteristic, direct product factor, Sylow
275C151#2413211yescharacteristic, direct product factor, Sylow
276C2×#547C2×C15111yescharacteristic
277C2×#477C2×C15111yescharacteristic, complemented
278C2×#440C2×C15111yescharacteristic, complemented
279#2536C2×C15111yescharacteristic, complemented
280#2033C2×C15112yescomplemented
281#2000C2×C15112yescomplemented
282#1993C2×C15111yescharacteristic, complemented
283#1955C2×C15111yescharacteristic, complemented
284#1872C2×C15112yescomplemented
285#1852C2×C15112yescomplemented
286#1804C2×C15111yescharacteristic, complemented
287C22×#41—21no—
288C22×SD16C22×C15111yescharacteristic
289#547—24no—
290#547—24no—
291C2×C4⋊Q8C22×C15111yescharacteristic
292C2×#123—22no—
293C2×C22⋊Q8—21no—
294C2×C22⋊Q8C22×C15112yes—
295C2×#121C22×C15112yes—
296C2×C4×Q8C22×C15111yescharacteristic
297C2×C4×D4C22×C15111yescharacteristic
298C2×C4×D4C22×C15111yescharacteristic
299C22×C4⋊C4C22×C15112yes—
300#477—24no—
301#440—24no—
302C2×#107C22×C15112yes—
303C2×#107C22×C15111yescharacteristic
304C2×C4⋊C8C22×C15111yescharacteristic
305C2×#103C22×C15112yes—
306C2×#103C22×C15111yescharacteristic
307C2×#102C22×C15111yescharacteristic
308C2×#102C22×C15112yes—
309C2×C22⋊C8C22×C15112yes—
310C2×C22⋊C8C22×C15112yes—
311C2×#97C22×C15111yescharacteristic
312#398—21no—
313#394—21no—
314#388—22no—
315#382C22×C15111yescharacteristic
316#380C22×C15112yes—
317#340C22×C15112yes—
318#340C22×C15112yes—
319#340C22×C15112yes—
320C2×#41—28no—
321C2×#41—48no—
322C22×Q8—21no—
323C22×Q8C23×C15111yescharacteristic
324C22×D4—22no—
325C22×D4D4×C15111yescharacteristic
326C22×D4C23×C15111yescharacteristic
327C23×C4—22no—
328C23×C4—21no—
329C23×C4C23×C15112yes—
330C2×SD16—22no—
331C2×SD16—22no—
332C2×SD16—22no—
333C22×C8C23×C15111yescharacteristic
334C22×C8C23×C15112yes—
335C22×C8C23×C15111yescharacteristic
336C4⋊Q8—22no—
337C4⋊Q8D4×C15114yes—
338#123—216no—
339C22⋊Q8—28no—
340C22⋊Q8—24no—
341C22⋊Q8—42no—
342#121—24no—
343#121—24no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4D4×C15114yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4D4×C15111yescharacteristic
357C2×C4⋊C4C23×C15111yescharacteristic
358C2×C4⋊C4C23×C15112yes—
359C2×C4⋊C4C23×C15111yescharacteristic
360C2×C4⋊C4C23×C15112yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4D4×C15112yes—
366C2×C42—21no—
367C2×C42C23×C15111yescharacteristic
368#107—22no—
369#107—24no—
370C4⋊C8—22no—
371#103—22no—
372#103—24no—
373#102—22no—
374#102—42no—
375C22⋊C8—24no—
376C22⋊C8—42no—
377#97—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381C24—21no—
382C24C2×D4×C15112yes—
383#41—432no—
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—44no—
391C2×Q8#41×C15112yescomplemented
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4C2×D4×C15112yes—
403C22×C4—28no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4C2×D4×C15112yes—
420C22×C4C2×D4×C15111yescharacteristic
421C22×C4C2×D4×C15111yescharacteristic
422C22×C4C24×C15111yescharacteristic
423SD16—28no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8#41×C15112yescomplemented
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4C2×D4×C15112yes—
445C4⋊C4C2×D4×C15114yes—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C42—42no—
452C42C2×D4×C15112yes—
453C42C2×D4×C15112yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23AGL(1,ℤ/8)×C15114yes—
466C23C22×D4×C15111yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—48no—
470Q8—41no—
471Q8—41no—
472D4—28no—
473D4—22no—
474D4—22no—
475D4—416no—
476D4—41no—
477D4—41no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—416no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4#120×C15112yes—
503C2×C4#120×C15112yes—
504C2×C4C22×D4×C15111yescharacteristic
505C2×C4C22×D4×C15111yescharacteristic
506C2×C4C2×#41×C15111yescharacteristic
507C2×C4C2×#41×C15111yescharacteristic
508C2×C4H(5,𝔽2)×C15111yescharacteristic
509C2×C42−1+4×C15111yescharacteristic
510C8—22no—
511C8—41no—
512C8—42no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22C2×#120×C15111yescharacteristic
526C22#546×C15111yescharacteristic
527C22#547×C15111yescharacteristic
528C22C2×AGL(1,ℤ/8)×C15112yes—
529C22#578×C15112yes—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—44no—
536C4—81no—
537C4—82no—
538C4#525×C15111yescharacteristic
539C4#537×C15111yescharacteristic
540C4#548×C15111yescharacteristic
541C4#550×C15111yescharacteristic
542C2—44no—
543C2—82no—
544C2#2407×C15111yescharacteristic
545C2#3003×C15111yescharacteristic
546C2#3011×C15111yescharacteristic
547C2#3308×C15112yes—
548C2#3343×C15112yes—
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×#477×C151C2semidirect
3. C2×#440×C151C2semidirect
4. #2536×C151C2semidirect
5. #2033×C151C2semidirect
6. #2000×C151C2semidirect
7. #1993×C151C2semidirect
8. #1955×C151C2semidirect
9. #1872×C151C2semidirect
10. #1852×C151C2semidirect
11. #1804×C151C2semidirect
116. C2×Q8×C151#41semidirect
157. C2×C8×C151#41semidirect
274. #24132C151direct (N × H)
275. C151#24132direct (N × H)
277. C2×#477C2×C151semidirect
278. C2×#440C2×C151semidirect
279. #2536C2×C151semidirect
280. #2033C2×C151semidirect
281. #2000C2×C151semidirect
282. #1993C2×C151semidirect
283. #1955C2×C151semidirect
284. #1872C2×C151semidirect
285. #1852C2×C151semidirect
286. #1804C2×C151semidirect
391. C2×Q8#41×C151semidirect
432. 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×#477×C1512
C2×#440×C1512
#2536×C1512
#2033×C1512
#2000×C1512
#1993×C1512
#1955×C1512
#1872×C1512
#1852×C1512
#1804×C1512
#24132151