Finite Group Databasebeta

← C151×#18801 #50323726

Subgroups — C151×#18801 #50323726

Order 38656.

936 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41×C151C211yesmaximal, characteristic
2#3230×C151C211yesmaximal, characteristic, complemented
3#3079×C151C211yesmaximal, characteristic
4C2×#462×C151C211yesmaximal, characteristic, complemented
5#1649×C151C212yesmaximal, complemented
6#1648×C151C212yesmaximal, complemented
7#1645×C151C212yesmaximal, complemented
8#1644×C151C212yesmaximal, complemented
9#1550×C151C211yesmaximal, characteristic, complemented
10#1487×C151C211yesmaximal, characteristic, complemented
11#1477×C151C211yesmaximal, characteristic
12C22×Q16×C151—21no—
13Q82×C151C2211yescharacteristic
14#557×C151—21no—
15#557×C151C2211yescharacteristic
16#554×C151—22no—
17#554×C151—21no—
18#554×C151—21no—
19#554×C151C2211yescharacteristic
20D4×Q8×C151—21no—
21D4×Q8×C151C2211yescharacteristic
22#533×C151—22no—
23#533×C151—21no—
24#533×C151—21no—
25#533×C151C2211yescharacteristic
26C2×C4⋊Q8×C151C2211yescharacteristic
27C4×#41×C151—21no—
28C4×#41×C151C2211yescharacteristic
29C2×C4×Q8×C151—21no—
30C2×C4×Q8×C151C2211yescharacteristic
31#500×C151C2212yes—
32#487×C151C2212yes—
33#484×C151—22no—
34#477×C151—22no—
35#477×C151—21no—
36#476×C151—21no—
37#475×C151C2212yes—
38#474×C151C2212yes—
39#462×C151—21no—
40#462×C151—21no—
41#462×C151C2212yes—
42#462×C151C2212yes—
43#431×C151C2211yescharacteristic
44#428×C151—21no—
45#423×C151C2211yescharacteristic
46C2×C4⋊C8×C151C2211yescharacteristic
47#417×C151—21no—
48C2×#103×C151—21no—
49#333×C151C2212yes—
50#332×C151C2212yes—
51Q8⋊C8×C151C2212yes—
52Q8⋊C8×C151C2212yes—
53Q8⋊C8×C151C2212yes—
54D4⋊C8×C151C2212yes—
55C2×#41×C151D411yescharacteristic
56C22×Q8×C151—21no—
57C22×Q8×C151—21no—
58C22×Q8×C151D411yescharacteristic
59C2×Q16×C151—22no—
60C2×Q16×C151—22no—
61C2×Q16×C151—44no—
62C2×M16×C151—21no—
63C22×C8×C151—21no—
64C4⋊Q8×C151—22no—
65C4⋊Q8×C151—21no—
66C4⋊Q8×C151—21no—
67C4⋊Q8×C151—21no—
68C4⋊Q8×C151—21no—
69C4⋊Q8×C151—21no—
70C4⋊Q8×C151—22no—
71C4⋊Q8×C151—21no—
72C4⋊Q8×C151—21no—
73C4⋊Q8×C151C2312yes—
74C4⋊Q8×C151C2311yescharacteristic
75C4⋊Q8×C151C2311yescharacteristic
76#125×C151—22no—
77#125×C151—22no—
78#125×C151—22no—
79#125×C151—21no—
80#125×C151—21no—
81#125×C151—21no—
82C22⋊Q8×C151—22no—
83C22⋊Q8×C151—22no—
84C22⋊Q8×C151—22no—
85C22⋊Q8×C151—22no—
86C22⋊Q8×C151—22no—
87C22⋊Q8×C151—21no—
88C22⋊Q8×C151—22no—
89C22⋊Q8×C151—21no—
90C22⋊Q8×C151—21no—
91C22⋊Q8×C151—22no—
92C22⋊Q8×C151D412yes—
93C4×Q8×C151—22no—
94C4×Q8×C151—22no—
95C4×Q8×C151—22no—
96C4×Q8×C151—21no—
97C4×Q8×C151—21no—
98C4×Q8×C151—21no—
99C4×Q8×C151—21no—
100C4×Q8×C151—21no—
101C4×Q8×C151—21no—
102C4×Q8×C151—21no—
103C4×Q8×C151D411yescharacteristic
104C4×Q8×C151D411yescharacteristic
105C4×Q8×C151D411yescharacteristic
106C4×Q8×C151D412yes—
107C4×Q8×C151C2311yescharacteristic
108C4×Q8×C151C2311yescharacteristic
109C4×Q8×C151C2311yescharacteristic
110C4×D4×C151—22no—
111C4×D4×C151—22no—
112C4×D4×C151—21no—
113C4×D4×C151—21no—
114C4×D4×C151—21no—
115C4×D4×C151—21no—
116C4×D4×C151D411yescharacteristic
117C4×D4×C151C2311yescharacteristic
118#117×C151—22no—
119#117×C151—21no—
120#117×C151—21no—
121#117×C151D411yescharacteristic
122C2×C4⋊C4×C151—22no—
123C2×C4⋊C4×C151—21no—
124C2×C4⋊C4×C151—21no—
125C2×C4⋊C4×C151—21no—
126C2×C4⋊C4×C151—21no—
127C2×C4⋊C4×C151D411yescharacteristic
128C2×C42×C151—22no—
129C2×C42×C151—21no—
130C2×C42×C151—21no—
131C2×C42×C151C2311yescharacteristic
132#107×C151—22no—
133#107×C151—24no—
134#106×C151—22no—
135C4⋊C8×C151—21no—
136C4⋊C8×C151—21no—
137C4⋊C8×C151—21no—
138C4⋊C8×C151C2312yes—
139C4⋊C8×C151C2312yes—
140#103×C151—22no—
141#103×C151—22no—
142#103×C151—22no—
143#103×C151—24no—
144#102×C151—22no—
145C22⋊C8×C151—21no—
146C4×C8×C151C2312yes—
147#41×C151—22no—
148#41×C151Q1614yescomplemented
149C2×Q8×C151—22no—
150C2×Q8×C151—22no—
151C2×Q8×C151—21no—
152C2×Q8×C151—21no—
153C2×Q8×C151—22no—
154C2×Q8×C151—22no—
155C2×Q8×C151—41no—
156C2×Q8×C151—42no—
157C2×Q8×C151—42no—
158C2×Q8×C151—42no—
159C2×Q8×C151—42no—
160C2×Q8×C151—41no—
161C2×Q8×C151C2×D411yescharacteristic
162C2×Q8×C151C2×D411yescharacteristic
163C2×Q8×C151C2×D411yescharacteristic
164C2×Q8×C151C2×D412yes—
165C2×D4×C151—21no—
166C2×D4×C151C2×D411yescharacteristic
167C22×C4×C151—21no—
168C22×C4×C151—22no—
169C22×C4×C151—21no—
170C22×C4×C151—22no—
171C22×C4×C151—21no—
172C22×C4×C151—21no—
173C22×C4×C151—21no—
174C22×C4×C151—41no—
175C22×C4×C151C2×D411yescharacteristic
176C22×C4×C151C2×D411yescharacteristic
177C22×C4×C151C2×D411yescharacteristic
178Q16×C151—44no—
179Q16×C151—44no—
180M16×C151—42no—
181C2×C8×C151—22no—
182C2×C8×C151—22no—
183C2×C8×C151—22no—
184C2×C8×C151—42no—
185C4⋊C4×C151—22no—
186C4⋊C4×C151—22no—
187C4⋊C4×C151—22no—
188C4⋊C4×C151—22no—
189C4⋊C4×C151—22no—
190C4⋊C4×C151—22no—
191C4⋊C4×C151—22no—
192C4⋊C4×C151—22no—
193C4⋊C4×C151—21no—
194C4⋊C4×C151—22no—
195C4⋊C4×C151—21no—
196C4⋊C4×C151—21no—
197C4⋊C4×C151—21no—
198C4⋊C4×C151—22no—
199C4⋊C4×C151—21no—
200C4⋊C4×C151—21no—
201C4⋊C4×C151—21no—
202C4⋊C4×C151—21no—
203C4⋊C4×C151—21no—
204C4⋊C4×C151—21no—
205C4⋊C4×C151—22no—
206C4⋊C4×C151—22no—
207C4⋊C4×C151—41no—
208C4⋊C4×C151—41no—
209C4⋊C4×C151C2×D411yescharacteristic
210C4⋊C4×C151C2×D411yescharacteristic
211C4⋊C4×C151C2×D411yescharacteristic
212C4⋊C4×C151C2×D411yescharacteristic
213C4⋊C4×C151C2×D412yes—
214C22⋊C4×C151—22no—
215C22⋊C4×C151—22no—
216C22⋊C4×C151—22no—
217C22⋊C4×C151—21no—
218C22⋊C4×C151—21no—
219C22⋊C4×C151—21no—
220C42×C151—22no—
221C42×C151—22no—
222C42×C151—21no—
223C42×C151—22no—
224C42×C151—21no—
225C42×C151—21no—
226C42×C151—21no—
227C42×C151—21no—
228C42×C151—21no—
229C42×C151—21no—
230C42×C151C2×D411yescharacteristic
231C42×C151C2×D411yescharacteristic
232C42×C151C2×D411yescharacteristic
233C42×C151C2411yescharacteristic
234C23×C151—21no—
235C23×C151#12011yescharacteristic
236Q8×C151—21no—
237Q8×C151—21no—
238Q8×C151—21no—
239Q8×C151—21no—
240Q8×C151—42no—
241Q8×C151—42no—
242Q8×C151—44no—
243Q8×C151—41no—
244Q8×C151—41no—
245Q8×C151—42no—
246Q8×C151—42no—
247Q8×C151—82no—
248Q8×C151—82no—
249Q8×C151C2×Q1612yescomplemented
250Q8×C151AGL(1,ℤ/8)12yes—
251D4×C151—24no—
252D4×C151—21no—
253D4×C151C2×Q1612yescomplemented
254C2×C4×C151—21no—
255C2×C4×C151—21no—
256C2×C4×C151—21no—
257C2×C4×C151—21no—
258C2×C4×C151—24no—
259C2×C4×C151—21no—
260C2×C4×C151—22no—
261C2×C4×C151—21no—
262C2×C4×C151—21no—
263C2×C4×C151—21no—
264C2×C4×C151—22no—
265C2×C4×C151—21no—
266C2×C4×C151—41no—
267C2×C4×C151—42no—
268C2×C4×C151—44no—
269C2×C4×C151—41no—
270C2×C4×C151—41no—
271C2×C4×C151—42no—
272C2×C4×C151—81no—
273C2×C4×C151—81no—
274C2×C4×C151—82no—
275C2×C4×C151—82no—
276C2×C4×C151#12011yescharacteristic
277C2×C4×C151#12011yescharacteristic
278C2×C4×C151#12011yescharacteristic
279C2×C4×C151C2×Q1612yes—
280C2×C4×C151#13712yes—
281C2×C4×C151C22×D411yescharacteristic
282C2×C4×C151C22×D411yescharacteristic
283C2×C4×C151C22×D411yescharacteristic
284C8×C151—42no—
285C8×C151—42no—
286C8×C151—42no—
287C22×C151—21no—
288C22×C151—21no—
289C22×C151—22no—
290C22×C151—41no—
291C22×C151—41no—
292C22×C151#45112yes—
293C22×C151C2×#12011yescharacteristic
294C4×C151—22no—
295C4×C151—21no—
296C4×C151—22no—
297C4×C151—21no—
298C4×C151—21no—
299C4×C151—41no—
300C4×C151—42no—
301C4×C151—41no—
302C4×C151—41no—
303C4×C151—42no—
304C4×C151—41no—
305C4×C151—41no—
306C4×C151—82no—
307C4×C151#45112yes—
308C4×C151C22×Q1611yescharacteristic
309C4×C151C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C151C2×#13711yescharacteristic
311C4×C151#57511yescharacteristic
312C2×C151—22no—
313C2×C151—42no—
314C2×C151C2×#45111yescharacteristic
315C2×C151#300611yescharacteristic
316C2×C151#302211yescharacteristic
317#18801C15111yesmaximal, characteristic, direct product factor, Sylow
318C151#1880111yescharacteristic, direct product factor, Sylow
319Q8×#41C2×C15111yescharacteristic
320#3230C2×C15111yescharacteristic, complemented
321#3079C2×C15111yescharacteristic
322C2×#462C2×C15111yescharacteristic, complemented
323#1649C2×C15112yescomplemented
324#1648C2×C15112yescomplemented
325#1645C2×C15112yescomplemented
326#1644C2×C15112yescomplemented
327#1550C2×C15111yescharacteristic, complemented
328#1487C2×C15111yescharacteristic, complemented
329#1477C2×C15111yescharacteristic
330C22×Q16—21no—
331Q82C22×C15111yescharacteristic
332#557—21no—
333#557C22×C15111yescharacteristic
334#554—22no—
335#554—21no—
336#554—21no—
337#554C22×C15111yescharacteristic
338D4×Q8—21no—
339D4×Q8C22×C15111yescharacteristic
340#533—22no—
341#533—21no—
342#533—21no—
343#533C22×C15111yescharacteristic
344C2×C4⋊Q8C22×C15111yescharacteristic
345C4×#41—21no—
346C4×#41C22×C15111yescharacteristic
347C2×C4×Q8—21no—
348C2×C4×Q8C22×C15111yescharacteristic
349#500C22×C15112yes—
350#487C22×C15112yes—
351#484—22no—
352#477—21no—
353#477—22no—
354#476—21no—
355#475C22×C15112yes—
356#474C22×C15112yes—
357#462—21no—
358#462—21no—
359#462C22×C15112yes—
360#462C22×C15112yes—
361#431C22×C15111yescharacteristic
362#428—21no—
363#423C22×C15111yescharacteristic
364C2×C4⋊C8C22×C15111yescharacteristic
365#417—21no—
366C2×#103—21no—
367#333C22×C15112yes—
368#332C22×C15112yes—
369Q8⋊C8C22×C15112yes—
370Q8⋊C8C22×C15112yes—
371Q8⋊C8C22×C15112yes—
372D4⋊C8C22×C15112yes—
373C2×#41D4×C15111yescharacteristic
374C22×Q8—21no—
375C22×Q8—21no—
376C22×Q8D4×C15111yescharacteristic
377C2×Q16—22no—
378C2×Q16—22no—
379C2×Q16—44no—
380C2×M16—21no—
381C22×C8—21no—
382C4⋊Q8—22no—
383C4⋊Q8—21no—
384C4⋊Q8—22no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8—21no—
391C4⋊Q8C23×C15112yes—
392C4⋊Q8C23×C15111yescharacteristic
393C4⋊Q8C23×C15111yescharacteristic
394#125—22no—
395#125—22no—
396#125—22no—
397#125—21no—
398#125—21no—
399#125—21no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405C22⋊Q8—21no—
406C22⋊Q8—22no—
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8D4×C15112yes—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8D4×C15111yescharacteristic
422C4×Q8D4×C15111yescharacteristic
423C4×Q8D4×C15111yescharacteristic
424C4×Q8D4×C15112yes—
425C4×Q8C23×C15111yescharacteristic
426C4×Q8C23×C15111yescharacteristic
427C4×Q8C23×C15111yescharacteristic
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4D4×C15111yescharacteristic
435C4×D4C23×C15111yescharacteristic
436#117—22no—
437#117—21no—
438#117—21no—
439#117D4×C15111yescharacteristic
440C2×C4⋊C4—22no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4D4×C15111yescharacteristic
446C2×C42—22no—
447C2×C42—21no—
448C2×C42—21no—
449C2×C42C23×C15111yescharacteristic
450#107—22no—
451#107—24no—
452#106—22no—
453C4⋊C8—21no—
454C4⋊C8—21no—
455C4⋊C8—21no—
456C4⋊C8C23×C15112yes—
457C4⋊C8C23×C15112yes—
458#103—22no—
459#103—24no—
460#103—22no—
461#103—22no—
462#102—22no—
463C22⋊C8—21no—
464C4×C8C23×C15112yes—
465#41—22no—
466#41Q16×C15114yescomplemented
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—41no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—42no—
477C2×Q8—42no—
478C2×Q8—41no—
479C2×Q8C2×D4×C15111yescharacteristic
480C2×Q8C2×D4×C15111yescharacteristic
481C2×Q8C2×D4×C15111yescharacteristic
482C2×Q8C2×D4×C15112yes—
483C2×D4—21no—
484C2×D4C2×D4×C15111yescharacteristic
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—41no—
493C22×C4C2×D4×C15111yescharacteristic
494C22×C4C2×D4×C15111yescharacteristic
495C22×C4C2×D4×C15111yescharacteristic
496Q16—44no—
497Q16—44no—
498M16—42no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—42no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4C2×D4×C15111yescharacteristic
528C4⋊C4C2×D4×C15111yescharacteristic
529C4⋊C4C2×D4×C15111yescharacteristic
530C4⋊C4C2×D4×C15111yescharacteristic
531C4⋊C4C2×D4×C15112yes—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42C2×D4×C15111yescharacteristic
549C42C2×D4×C15111yescharacteristic
550C42C2×D4×C15111yescharacteristic
551C42C24×C15111yescharacteristic
552C23—21no—
553C23#120×C15111yescharacteristic
554Q8—21no—
555Q8—21no—
556Q8—21no—
557Q8—21no—
558Q8—42no—
559Q8—42no—
560Q8—44no—
561Q8—41no—
562Q8—41no—
563Q8—42no—
564Q8—42no—
565Q8—82no—
566Q8—82no—
567Q8C2×Q16×C15112yescomplemented
568Q8AGL(1,ℤ/8)×C15112yes—
569D4—24no—
570D4—21no—
571D4C2×Q16×C15112yescomplemented
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4#120×C15111yescharacteristic
595C2×C4#120×C15111yescharacteristic
596C2×C4#120×C15111yescharacteristic
597C2×C4C2×Q16×C15112yes—
598C2×C4#137×C15112yes—
599C2×C4C22×D4×C15111yescharacteristic
600C2×C4C22×D4×C15111yescharacteristic
601C2×C4C22×D4×C15111yescharacteristic
602C8—42no—
603C8—42no—
604C8—42no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22#451×C15112yes—
611C22C2×#120×C15111yescharacteristic
612C4—22no—
613C4—21no—
614C4—22no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—82no—
625C4#451×C15112yes—
626C4C22×Q16×C15111yescharacteristic
627C4C2×AGL(1,ℤ/8)×C15111yescharacteristic
628C4C2×#137×C15111yescharacteristic
629C4#575×C15111yescharacteristic
630C2—22no—
631C2—42no—
632C2C2×#451×C15111yescharacteristic
633C2#3006×C15111yescharacteristic
634C2#3022×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#120
commutator, Frattini, cosocleC42C24×C151
prime radicalC2×#41×C151D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3230×C151C2semidirect
4. C2×#462×C151C2semidirect
5. #1649×C151C2semidirect
6. #1648×C151C2semidirect
7. #1645×C151C2semidirect
8. #1644×C151C2semidirect
9. #1550×C151C2semidirect
10. #1487×C151C2semidirect
148. #41×C151Q16semidirect
249. Q8×C151C2×Q16semidirect
253. D4×C151C2×Q16semidirect
317. #18801C151direct (N × H)
318. C151#18801direct (N × H)
320. #3230C2×C151semidirect
322. C2×#462C2×C151semidirect
323. #1649C2×C151semidirect
324. #1648C2×C151semidirect
325. #1645C2×C151semidirect
326. #1644C2×C151semidirect
327. #1550C2×C151semidirect
328. #1487C2×C151semidirect
466. #41Q16×C151semidirect
567. Q8C2×Q16×C151semidirect
571. D4C2×Q16×C151semidirect

Maximal subgroups

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

subgroupindex
Q8×#41×C1512
#3230×C1512
#3079×C1512
C2×#462×C1512
#1649×C1512
#1648×C1512
#1645×C1512
#1644×C1512
#1550×C1512
#1487×C1512
#1477×C1512
#18801151