Finite Group Databasebeta

← C151×#24129 #50329054

Subgroups — C151×#24129 #50329054

Order 38656.

1204 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555×C151C211yesmaximal, characteristic, complemented
2C2×#474×C151C211yesmaximal, characteristic, complemented
3C2×#442×C151C211yesmaximal, characteristic, complemented
4#2535×C151C211yesmaximal, characteristic, complemented
5#2033×C151C212yesmaximal, complemented
6#1999×C151C212yesmaximal, complemented
7#1993×C151C211yesmaximal, characteristic, complemented
8#1955×C151C211yesmaximal, characteristic, complemented
9#1875×C151C212yesmaximal, complemented
10#1854×C151C212yesmaximal, complemented
11#1804×C151C211yesmaximal, characteristic, complemented
12C22×D8×C151C2211yescharacteristic
13#555×C151—24no—
14#555×C151—24no—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×#124×C151—22no—
17C2×#121×C151—21no—
18C2×#121×C151C2212yes—
19C2×#121×C151C2212yes—
20C2×C4×D4×C151C2211yescharacteristic
21C2×C4×D4×C151C2211yescharacteristic
22C2×C4×D4×C151C2211yescharacteristic
23C2×#117×C151—21no—
24C2×#117×C151C2212yes—
25#474×C151—24no—
26#442×C151—24no—
27C2×#107×C151C2212yes—
28C2×#106×C151C2211yescharacteristic
29C2×C4⋊C8×C151C2211yescharacteristic
30C2×#102×C151C2211yescharacteristic
31C2×#102×C151C2212yes—
32C2×#102×C151C2211yescharacteristic
33C2×#102×C151C2212yes—
34C2×C22⋊C8×C151C2212yes—
35C2×C22⋊C8×C151C2212yes—
36C2×#97×C151C2211yescharacteristic
37#397×C151—21no—
38#392×C151—21no—
39#386×C151—22no—
40#382×C151C2211yescharacteristic
41#380×C151C2212yes—
42#340×C151C2212yes—
43#340×C151C2212yes—
44#340×C151C2212yes—
45C22×Q8×C151—21no—
46C22×D4×C151—22no—
47C22×D4×C151—22no—
48C22×D4×C151D411yescharacteristic
49C22×D4×C151C2311yescharacteristic
50C22×D4×C151C2311yescharacteristic
51C23×C4×C151—21no—
52C23×C4×C151—21no—
53C23×C4×C151C2312yes—
54C2×D8×C151—22no—
55C2×D8×C151—22no—
56C2×D8×C151—22no—
57C22×C8×C151C2311yescharacteristic
58C22×C8×C151C2312yes—
59C22×C8×C151C2311yescharacteristic
60C4⋊Q8×C151—22no—
61C4⋊Q8×C151D414yescomplemented
62#124×C151—216no—
63#121×C151—28no—
64#121×C151—24no—
65#121×C151—24no—
66#121×C151—24no—
67#121×C151—42no—
68C4×D4×C151—22no—
69C4×D4×C151—22no—
70C4×D4×C151—22no—
71C4×D4×C151—22no—
72C4×D4×C151—22no—
73C4×D4×C151D414yescomplemented
74#117×C151—28no—
75#117×C151—24no—
76#117×C151—24no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151D411yescharacteristic
79C2×C4⋊C4×C151C2312yes—
80C2×C4⋊C4×C151C2311yescharacteristic
81C2×C4⋊C4×C151C2311yescharacteristic
82C2×C4⋊C4×C151C2312yes—
83C2×C22⋊C4×C151—22no—
84C2×C22⋊C4×C151—21no—
85C2×C22⋊C4×C151—22no—
86C2×C22⋊C4×C151—21no—
87C2×C22⋊C4×C151—22no—
88C2×C22⋊C4×C151—22no—
89C2×C22⋊C4×C151D412yes—
90C2×C42×C151—22no—
91C2×C42×C151C2311yescharacteristic
92#107×C151—24no—
93#106×C151—22no—
94C4⋊C8×C151—22no—
95#102×C151—22no—
96#102×C151—22no—
97#102×C151—24no—
98#102×C151—42no—
99C22⋊C8×C151—24no—
100C22⋊C8×C151—42no—
101#97×C151—22no—
102#95×C151—22no—
103#95×C151—21no—
104#95×C151—21no—
105C24×C151—21no—
106C24×C151—21no—
107C24×C151C2×D412yes—
108C2×Q8×C151—24no—
109C2×Q8×C151—82no—
110C2×D4×C151—28no—
111C2×D4×C151—22no—
112C2×D4×C151—21no—
113C2×D4×C151—21no—
114C2×D4×C151—21no—
115C2×D4×C151—21no—
116C2×D4×C151—21no—
117C2×D4×C151—21no—
118C2×D4×C151—21no—
119C2×D4×C151—21no—
120C2×D4×C151—21no—
121C2×D4×C151—21no—
122C2×D4×C151—42no—
123C2×D4×C151—44no—
124C2×D4×C151—44no—
125C2×D4×C151—82no—
126C2×D4×C151—82no—
127C2×D4×C151—82no—
128C2×D4×C151C2×D412yes—
129C2×D4×C151#4112yes—
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—42no—
143C22×C4×C151—82no—
144C22×C4×C151C2×D412yes—
145C22×C4×C151C2×D411yescharacteristic
146C22×C4×C151C2×D411yescharacteristic
147C22×C4×C151C2411yescharacteristic
148D8×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#4112yes—
158C4⋊C4×C151—24no—
159C4⋊C4×C151—22no—
160C4⋊C4×C151—21no—
161C4⋊C4×C151—22no—
162C4⋊C4×C151—21no—
163C4⋊C4×C151—21no—
164C4⋊C4×C151—22no—
165C4⋊C4×C151—22no—
166C4⋊C4×C151C2×D412yes—
167C4⋊C4×C151C2×D414yes—
168C22⋊C4×C151—28no—
169C22⋊C4×C151—28no—
170C22⋊C4×C151—24no—
171C22⋊C4×C151—24no—
172C22⋊C4×C151—44no—
173C22⋊C4×C151—42no—
174C22⋊C4×C151—82no—
175C42×C151—28no—
176C42×C151C2×D412yes—
177C42×C151C2×D412yes—
178C23×C151—22no—
179C23×C151—22no—
180C23×C151—22no—
181C23×C151—22no—
182C23×C151—22no—
183C23×C151—22no—
184C23×C151—22no—
185C23×C151—22no—
186C23×C151—22no—
187C23×C151—41no—
188C23×C151—41no—
189C23×C151—41no—
190C23×C151—41no—
191C23×C151—42no—
192C23×C151—42no—
193C23×C151—42no—
194C23×C151—42no—
195C23×C151C22×D411yescharacteristic
196Q8×C151—84no—
197D4×C151—22no—
198D4×C151—22no—
199D4×C151—22no—
200D4×C151—22no—
201D4×C151—44no—
202D4×C151—41no—
203D4×C151—41no—
204D4×C151—41no—
205D4×C151—41no—
206D4×C151—88no—
207D4×C151—88no—
208C2×C4×C151—22no—
209C2×C4×C151—22no—
210C2×C4×C151—24no—
211C2×C4×C151—22no—
212C2×C4×C151—22no—
213C2×C4×C151—22no—
214C2×C4×C151—22no—
215C2×C4×C151—48no—
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—42no—
223C2×C4×C151—41no—
224C2×C4×C151—42no—
225C2×C4×C151—42no—
226C2×C4×C151—42no—
227C2×C4×C151—42no—
228C2×C4×C151—84no—
229C2×C4×C151#12012yes—
230C2×C4×C151#12012yes—
231C2×C4×C151C22×D411yescharacteristic
232C2×C4×C151C22×D411yescharacteristic
233C2×C4×C151C2×#4111yescharacteristic
234C2×C4×C151C2×#4111yescharacteristic
235C2×C4×C151H(5,𝔽2)11yescharacteristic
236C2×C4×C1512−1+411yescharacteristic
237C8×C151—22no—
238C8×C151—41no—
239C8×C151—42no—
240C22×C151—42no—
241C22×C151—42no—
242C22×C151—42no—
243C22×C151—42no—
244C22×C151—42no—
245C22×C151—42no—
246C22×C151—42no—
247C22×C151—42no—
248C22×C151—42no—
249C22×C151—42no—
250C22×C151—44no—
251C22×C151—42no—
252C22×C151—82no—
253C22×C151—82no—
254C22×C151—81no—
255C22×C151—81no—
256C22×C151—82no—
257C22×C151C2×#12011yescharacteristic
258C22×C151#54611yescharacteristic
259C22×C151#54711yescharacteristic
260C22×C151#57512yes—
261C22×C151#57612yes—
262C4×C151—22no—
263C4×C151—22no—
264C4×C151—42no—
265C4×C151—42no—
266C4×C151—44no—
267C4×C151—81no—
268C4×C151—82no—
269C4×C151#52511yescharacteristic
270C4×C151#53711yescharacteristic
271C4×C151#54811yescharacteristic
272C4×C151#55011yescharacteristic
273C2×C151—82no—
274C2×C151—82no—
275C2×C151—82no—
276C2×C151#240711yescharacteristic
277C2×C151#300511yescharacteristic
278C2×C151#300811yescharacteristic
279C2×C151#331912yes—
280C2×C151#334712yes—
281#24129C15111yesmaximal, characteristic, direct product factor, Sylow
282C151#2412911yescharacteristic, direct product factor, Sylow
283C2×#555C2×C15111yescharacteristic, complemented
284C2×#474C2×C15111yescharacteristic, complemented
285C2×#442C2×C15111yescharacteristic, complemented
286#2535C2×C15111yescharacteristic, complemented
287#2033C2×C15112yescomplemented
288#1999C2×C15112yescomplemented
289#1993C2×C15111yescharacteristic, complemented
290#1955C2×C15111yescharacteristic, complemented
291#1875C2×C15112yescomplemented
292#1854C2×C15112yescomplemented
293#1804C2×C15111yescharacteristic, complemented
294C22×D8C22×C15111yescharacteristic
295#555—24no—
296#555—24no—
297C2×C4⋊Q8C22×C15111yescharacteristic
298C2×#124—22no—
299C2×#121—21no—
300C2×#121C22×C15112yes—
301C2×#121C22×C15112yes—
302C2×C4×D4C22×C15111yescharacteristic
303C2×C4×D4C22×C15111yescharacteristic
304C2×C4×D4C22×C15111yescharacteristic
305C2×#117—21no—
306C2×#117C22×C15112yes—
307#474—24no—
308#442—24no—
309C2×#107C22×C15112yes—
310C2×#106C22×C15111yescharacteristic
311C2×C4⋊C8C22×C15111yescharacteristic
312C2×#102C22×C15111yescharacteristic
313C2×#102C22×C15112yes—
314C2×#102C22×C15111yescharacteristic
315C2×#102C22×C15112yes—
316C2×C22⋊C8C22×C15112yes—
317C2×C22⋊C8C22×C15112yes—
318C2×#97C22×C15111yescharacteristic
319#397—21no—
320#392—21no—
321#386—22no—
322#382C22×C15111yescharacteristic
323#380C22×C15112yes—
324#340C22×C15112yes—
325#340C22×C15112yes—
326#340C22×C15112yes—
327C22×Q8—21no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4D4×C15111yescharacteristic
331C22×D4C23×C15111yescharacteristic
332C22×D4C23×C15111yescharacteristic
333C23×C4—21no—
334C23×C4—21no—
335C23×C4C23×C15112yes—
336C2×D8—22no—
337C2×D8—22no—
338C2×D8—22no—
339C22×C8C23×C15111yescharacteristic
340C22×C8C23×C15112yes—
341C22×C8C23×C15111yescharacteristic
342C4⋊Q8—22no—
343C4⋊Q8D4×C15114yescomplemented
344#124—216no—
345#121—28no—
346#121—24no—
347#121—24no—
348#121—24no—
349#121—42no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4D4×C15114yescomplemented
356#117—28no—
357#117—24no—
358#117—24no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4D4×C15111yescharacteristic
361C2×C4⋊C4C23×C15111yescharacteristic
362C2×C4⋊C4C23×C15112yes—
363C2×C4⋊C4C23×C15111yescharacteristic
364C2×C4⋊C4C23×C15112yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4D4×C15112yes—
372C2×C42—22no—
373C2×C42C23×C15111yescharacteristic
374#107—24no—
375#106—22no—
376C4⋊C8—22no—
377#102—22no—
378#102—22no—
379#102—24no—
380#102—42no—
381C22⋊C8—24no—
382C22⋊C8—42no—
383#97—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387C24—21no—
388C24—21no—
389C24C2×D4×C15112yes—
390C2×Q8—24no—
391C2×Q8—82no—
392C2×D4—28no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4C2×D4×C15112yes—
411C2×D4#41×C15112yes—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—82no—
426C22×C4C2×D4×C15112yes—
427C22×C4C2×D4×C15111yescharacteristic
428C22×C4C2×D4×C15111yescharacteristic
429C22×C4C24×C15111yescharacteristic
430D8—28no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—21no—
439C2×C8#41×C15112yes—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4C2×D4×C15112yes—
449C4⋊C4C2×D4×C15114yes—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—82no—
457C42—28no—
458C42C2×D4×C15112yes—
459C42C2×D4×C15112yes—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23C22×D4×C15111yescharacteristic
478Q8—84no—
479D4—22no—
480D4—22no—
481D4—22no—
482D4—22no—
483D4—44no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—41no—
488D4—88no—
489D4—88no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4#120×C15112yes—
512C2×C4#120×C15112yes—
513C2×C4C22×D4×C15111yescharacteristic
514C2×C4C22×D4×C15111yescharacteristic
515C2×C4C2×#41×C15111yescharacteristic
516C2×C4C2×#41×C15111yescharacteristic
517C2×C4H(5,𝔽2)×C15111yescharacteristic
518C2×C42−1+4×C15111yescharacteristic
519C8—22no—
520C8—41no—
521C8—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22C2×#120×C15111yescharacteristic
540C22#546×C15111yescharacteristic
541C22#547×C15111yescharacteristic
542C22#575×C15112yes—
543C22#576×C15112yes—
544C4—22no—
545C4—22no—
546C4—42no—
547C4—42no—
548C4—44no—
549C4—81no—
550C4—82no—
551C4#525×C15111yescharacteristic
552C4#537×C15111yescharacteristic
553C4#548×C15111yescharacteristic
554C4#550×C15111yescharacteristic
555C2—82no—
556C2—82no—
557C2—82no—
558C2#2407×C15111yescharacteristic
559C2#3005×C15111yescharacteristic
560C2#3008×C15111yescharacteristic
561C2#3319×C15112yes—
562C2#3347×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×#555×C151C2semidirect
2. C2×#474×C151C2semidirect
3. C2×#442×C151C2semidirect
4. #2535×C151C2semidirect
5. #2033×C151C2semidirect
6. #1999×C151C2semidirect
7. #1993×C151C2semidirect
8. #1955×C151C2semidirect
9. #1875×C151C2semidirect
10. #1854×C151C2semidirect
11. #1804×C151C2semidirect
61. C4⋊Q8×C151D4semidirect
73. C4×D4×C151D4semidirect
281. #24129C151direct (N × H)
282. C151#24129direct (N × H)
283. C2×#555C2×C151semidirect
284. C2×#474C2×C151semidirect
285. C2×#442C2×C151semidirect
286. #2535C2×C151semidirect
287. #2033C2×C151semidirect
288. #1999C2×C151semidirect
289. #1993C2×C151semidirect
290. #1955C2×C151semidirect
291. #1875C2×C151semidirect
292. #1854C2×C151semidirect
293. #1804C2×C151semidirect
343. C4⋊Q8D4×C151semidirect
355. C4×D4D4×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#555×C1512
C2×#474×C1512
C2×#442×C1512
#2535×C1512
#2033×C1512
#1999×C1512
#1993×C1512
#1955×C1512
#1875×C1512
#1854×C1512
#1804×C1512
#24129151