Finite Group Databasebeta

← C151×#24151 #50329076

Subgroups — C151×#24151 #50329076

Order 38656.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533×C151C211yesmaximal, characteristic, complemented
2C2×#476×C151C211yesmaximal, characteristic, complemented
3C2×#461×C151C211yesmaximal, characteristic, complemented
4C4×#121×C151C211yesmaximal, characteristic
5#2035×C151C212yesmaximal, complemented
6#2000×C151C212yesmaximal
7#1987×C151C211yesmaximal, characteristic, complemented
8#1981×C151C211yesmaximal, characteristic, complemented
9#1880×C151C212yesmaximal, complemented
10#1865×C151C212yesmaximal, complemented
11#1797×C151C211yesmaximal, characteristic, complemented
12C22×SD16×C151C2212yes—
13#533×C151—24no—
14#533×C151—24no—
15C2×C4⋊Q8×C151C2211yescharacteristic
16C2×C4⋊Q8×C151C2211yescharacteristic
17C2×#125×C151—21no—
18C2×C22⋊Q8×C151—21no—
19C2×C22⋊Q8×C151C2212yes—
20C2×#121×C151C2211yescharacteristic
21C2×C4×Q8×C151—21no—
22C2×C4×Q8×C151C2212yes—
23C2×C4×D4×C151—21no—
24C2×C4×D4×C151C2211yescharacteristic
25C2×#117×C151—21no—
26C22×C42×C151C2211yescharacteristic
27#476×C151—24no—
28#461×C151—24no—
29C2×#106×C151C2212yes—
30C2×C4⋊C8×C151C2211yescharacteristic
31C2×C4⋊C8×C151C2211yescharacteristic
32C2×#103×C151C2212yes—
33C2×#103×C151C2212yes—
34C2×#103×C151C2212yes—
35C2×#102×C151C2212yes—
36C2×C22⋊C8×C151C2212yes—
37#390×C151C2211yescharacteristic
38#389×C151C2211yescharacteristic
39#388×C151—21no—
40#386×C151—21no—
41#380×C151C2211yescharacteristic
42C4×C4⋊C4×C151C2211yescharacteristic
43C4×C22⋊C4×C151—21no—
44#340×C151C2212yes—
45#340×C151C2212yes—
46#336×C151C2212yes—
47C22×Q8×C151—21no—
48C22×Q8×C151C2312yes—
49C22×D4×C151—21no—
50C22×D4×C151C2311yescharacteristic
51C23×C4×C151—21no—
52C23×C4×C151D411yescharacteristic
53C23×C4×C151D411yescharacteristic
54C23×C4×C151C2311yescharacteristic
55C2×SD16×C151—24no—
56C2×SD16×C151—24no—
57C2×SD16×C151—42no—
58C22×C8×C151C2312yes—
59C22×C8×C151C2312yes—
60C4⋊Q8×C151—22no—
61C4⋊Q8×C151—22no—
62C4⋊Q8×C151D414yescomplemented
63C4⋊Q8×C151D414yescomplemented
64#125×C151—28no—
65C22⋊Q8×C151—28no—
66C22⋊Q8×C151—24no—
67C22⋊Q8×C151—24no—
68#121×C151—22no—
69#121×C151—22no—
70C4×Q8×C151—28no—
71C4×Q8×C151—24no—
72C4×Q8×C151—24no—
73C4×D4×C151—22no—
74C4×D4×C151—22no—
75C4×D4×C151—44no—
76#117×C151—28no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—21no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151—22no—
81C2×C4⋊C4×C151—22no—
82C2×C4⋊C4×C151—21no—
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2312yes—
86C2×C4⋊C4×C151C2311yescharacteristic
87C2×C22⋊C4×C151—21no—
88C2×C22⋊C4×C151—21no—
89C2×C22⋊C4×C151—21no—
90C2×C22⋊C4×C151—21no—
91C2×C22⋊C4×C151—22no—
92C2×C22⋊C4×C151—21no—
93C2×C42×C151—22no—
94C2×C42×C151—22no—
95C2×C42×C151—22no—
96C2×C42×C151—21no—
97C2×C42×C151D411yescharacteristic
98C2×C42×C151D411yescharacteristic
99C2×C42×C151C2311yescharacteristic
100C2×C42×C151C2311yescharacteristic
101#106×C151—24no—
102C4⋊C8×C151—22no—
103C4⋊C8×C151—22no—
104#103×C151—24no—
105#103×C151—24no—
106#103×C151—42no—
107#102×C151—24no—
108C22⋊C8×C151—42no—
109#95×C151—21no—
110#95×C151—21no—
111#95×C151—21no—
112C24×C151—21no—
113C24×C151C2×D411yescharacteristic
114C2×Q8×C151—24no—
115C2×Q8×C151—22no—
116C2×Q8×C151—22no—
117C2×Q8×C151—22no—
118C2×Q8×C151—22no—
119C2×Q8×C151—22no—
120C2×Q8×C151—42no—
121C2×Q8×C151—42no—
122C2×Q8×C151C2×D414yes—
123C2×D4×C151—24no—
124C2×D4×C151—21no—
125C2×D4×C151—21no—
126C2×D4×C151—21no—
127C2×D4×C151—21no—
128C2×D4×C151—21no—
129C2×D4×C151—42no—
130C2×D4×C151—42no—
131C2×D4×C151#4112yes—
132C22×C4×C151—22no—
133C22×C4×C151—22no—
134C22×C4×C151—21no—
135C22×C4×C151—22no—
136C22×C4×C151—22no—
137C22×C4×C151—22no—
138C22×C4×C151—21no—
139C22×C4×C151—21no—
140C22×C4×C151—21no—
141C22×C4×C151—21no—
142C22×C4×C151—21no—
143C22×C4×C151—21no—
144C22×C4×C151—21no—
145C22×C4×C151—21no—
146C22×C4×C151—41no—
147C22×C4×C151—41no—
148C22×C4×C151—42no—
149C22×C4×C151—42no—
150C22×C4×C151—42no—
151C22×C4×C151SD1614yescomplemented
152C22×C4×C151C2×D411yescharacteristic
153C22×C4×C151C2×D411yescharacteristic
154C22×C4×C151C2×D411yescharacteristic
155C22×C4×C151C2×D411yescharacteristic
156C22×C4×C151C2×D411yescharacteristic
157C22×C4×C151C2411yescharacteristic
158SD16×C151—48no—
159C2×C8×C151—22no—
160C2×C8×C151—22no—
161C2×C8×C151—22no—
162C2×C8×C151—22no—
163C2×C8×C151—22no—
164C2×C8×C151—22no—
165C4⋊C4×C151—28no—
166C4⋊C4×C151—28no—
167C4⋊C4×C151—24no—
168C4⋊C4×C151—24no—
169C4⋊C4×C151—24no—
170C4⋊C4×C151—22no—
171C4⋊C4×C151—21no—
172C4⋊C4×C151—21no—
173C4⋊C4×C151—21no—
174C4⋊C4×C151—21no—
175C4⋊C4×C151—21no—
176C4⋊C4×C151—42no—
177C4⋊C4×C151C2×D414yes—
178C4⋊C4×C151#4112yes—
179C22⋊C4×C151—28no—
180C22⋊C4×C151—24no—
181C22⋊C4×C151—42no—
182C22⋊C4×C151—42no—
183C22⋊C4×C151—42no—
184C22⋊C4×C151—42no—
185C42×C151—28no—
186C42×C151—22no—
187C42×C151—22no—
188C42×C151—21no—
189C42×C151—21no—
190C42×C151—42no—
191C42×C151C2×D412yes—
192C42×C151C2×D412yes—
193C23×C151—21no—
194C23×C151—21no—
195C23×C151—21no—
196C23×C151—21no—
197C23×C151—21no—
198C23×C151—21no—
199C23×C151—22no—
200C23×C151—41no—
201C23×C151—41no—
202C23×C151—41no—
203C23×C151—41no—
204C23×C151—41no—
205C23×C151—41no—
206C23×C151C2×SD1612yes—
207C23×C151C22×D411yescharacteristic
208Q8×C151—24no—
209Q8×C151—24no—
210Q8×C151—48no—
211Q8×C151—42no—
212Q8×C151—42no—
213D4×C151—22no—
214D4×C151—22no—
215D4×C151—48no—
216D4×C151—41no—
217D4×C151—41no—
218C2×C4×C151—21no—
219C2×C4×C151—21no—
220C2×C4×C151—21no—
221C2×C4×C151—21no—
222C2×C4×C151—28no—
223C2×C4×C151—24no—
224C2×C4×C151—24no—
225C2×C4×C151—21no—
226C2×C4×C151—21no—
227C2×C4×C151—21no—
228C2×C4×C151—22no—
229C2×C4×C151—21no—
230C2×C4×C151—21no—
231C2×C4×C151—21no—
232C2×C4×C151—44no—
233C2×C4×C151—42no—
234C2×C4×C151—42no—
235C2×C4×C151—41no—
236C2×C4×C151—42no—
237C2×C4×C151—42no—
238C2×C4×C151—41no—
239C2×C4×C151—41no—
240C2×C4×C151—41no—
241C2×C4×C151—41no—
242C2×C4×C151—41no—
243C2×C4×C151—41no—
244C2×C4×C151—41no—
245C2×C4×C151—41no—
246C2×C4×C151—41no—
247C2×C4×C151—41no—
248C2×C4×C151—41no—
249C2×C4×C151—84no—
250C2×C4×C151#12012yes—
251C2×C4×C151#12012yes—
252C2×C4×C151C2×SD1612yes—
253C2×C4×C151C2×SD1612yes—
254C2×C4×C151#13712yes—
255C2×C4×C151C22×D411yescharacteristic
256C2×C4×C151C22×D411yescharacteristic
257C2×C4×C151C2×#4111yescharacteristic
258C2×C4×C151C2×#4111yescharacteristic
259C2×C4×C151H(5,𝔽2)11yescharacteristic
260C2×C4×C1512−1+411yescharacteristic
261C8×C151—42no—
262C8×C151—42no—
263C22×C151—22no—
264C22×C151—22no—
265C22×C151—22no—
266C22×C151—41no—
267C22×C151—41no—
268C22×C151—41no—
269C22×C151—41no—
270C22×C151—42no—
271C22×C151—42no—
272C22×C151—42no—
273C22×C151—81no—
274C22×C151—81no—
275C22×C151—81no—
276C22×C151—81no—
277C22×C151C2×#12011yescharacteristic
278C22×C151#52511yescharacteristic
279C22×C151#53711yescharacteristic
280C22×C151C22×SD1611yescharacteristic
281C22×C151C2×#13711yescharacteristic
282C22×C151#57511yescharacteristic
283C22×C151#57511yescharacteristic
284C4×C151—22no—
285C4×C151—22no—
286C4×C151—42no—
287C4×C151—41no—
288C4×C151—44no—
289C4×C151—41no—
290C4×C151—44no—
291C4×C151—81no—
292C4×C151—81no—
293C4×C151#44814yes—
294C4×C151#54611yescharacteristic
295C4×C151#54711yescharacteristic
296C4×C151#54811yescharacteristic
297C4×C151#55011yescharacteristic
298C2×C151—42no—
299C2×C151—82no—
300C2×C151#240711yescharacteristic
301C2×C151C2×#44811yescharacteristic
302C2×C151#300511yescharacteristic
303C2×C151#310111yescharacteristic
304C2×C151#311611yescharacteristic
305C2×C151#324611yescharacteristic
306C2×C151#325211yescharacteristic
307#24151C15111yesmaximal, characteristic, direct product factor, Sylow
308C151#2415111yescharacteristic, direct product factor, Sylow
309C2×#533C2×C15111yescharacteristic, complemented
310C2×#476C2×C15111yescharacteristic, complemented
311C2×#461C2×C15111yescharacteristic, complemented
312C4×#121C2×C15111yescharacteristic
313#2035C2×C15112yescomplemented
314#2000C2×C15112yes—
315#1987C2×C15111yescharacteristic, complemented
316#1981C2×C15111yescharacteristic, complemented
317#1880C2×C15112yescomplemented
318#1865C2×C15112yescomplemented
319#1797C2×C15111yescharacteristic, complemented
320C22×SD16C22×C15112yes—
321#533—24no—
322#533—24no—
323C2×C4⋊Q8C22×C15111yescharacteristic
324C2×C4⋊Q8C22×C15111yescharacteristic
325C2×#125—21no—
326C2×C22⋊Q8—21no—
327C2×C22⋊Q8C22×C15112yes—
328C2×#121C22×C15111yescharacteristic
329C2×C4×Q8—21no—
330C2×C4×Q8C22×C15112yes—
331C2×C4×D4—21no—
332C2×C4×D4C22×C15111yescharacteristic
333C2×#117—21no—
334C22×C42C22×C15111yescharacteristic
335#476—24no—
336#461—24no—
337C2×#106C22×C15112yes—
338C2×C4⋊C8C22×C15111yescharacteristic
339C2×C4⋊C8C22×C15111yescharacteristic
340C2×#103C22×C15112yes—
341C2×#103C22×C15112yes—
342C2×#103C22×C15112yes—
343C2×#102C22×C15112yes—
344C2×C22⋊C8C22×C15112yes—
345#390C22×C15111yescharacteristic
346#389C22×C15111yescharacteristic
347#388—21no—
348#386—21no—
349#380C22×C15111yescharacteristic
350C4×C4⋊C4C22×C15111yescharacteristic
351C4×C22⋊C4—21no—
352#340C22×C15112yes—
353#340C22×C15112yes—
354#336C22×C15112yes—
355C22×Q8—21no—
356C22×Q8C23×C15112yes—
357C22×D4—21no—
358C22×D4C23×C15111yescharacteristic
359C23×C4—21no—
360C23×C4D4×C15111yescharacteristic
361C23×C4D4×C15111yescharacteristic
362C23×C4C23×C15111yescharacteristic
363C2×SD16—24no—
364C2×SD16—24no—
365C2×SD16—42no—
366C22×C8C23×C15112yes—
367C22×C8C23×C15112yes—
368C4⋊Q8—22no—
369C4⋊Q8—22no—
370C4⋊Q8D4×C15114yescomplemented
371C4⋊Q8D4×C15114yescomplemented
372#125—28no—
373C22⋊Q8—28no—
374C22⋊Q8—24no—
375C22⋊Q8—24no—
376#121—22no—
377#121—22no—
378C4×Q8—28no—
379C4×Q8—24no—
380C4×Q8—24no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—44no—
384#117—28no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4C23×C15111yescharacteristic
392C2×C4⋊C4C23×C15111yescharacteristic
393C2×C4⋊C4C23×C15111yescharacteristic
394C2×C4⋊C4C23×C15112yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42D4×C15111yescharacteristic
406C2×C42D4×C15111yescharacteristic
407C2×C42C23×C15111yescharacteristic
408C2×C42C23×C15111yescharacteristic
409#106—24no—
410C4⋊C8—22no—
411C4⋊C8—22no—
412#103—24no—
413#103—24no—
414#103—42no—
415#102—24no—
416C22⋊C8—42no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—21no—
421C24C2×D4×C15111yescharacteristic
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8C2×D4×C15114yes—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4#41×C15112yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4SD16×C15114yescomplemented
460C22×C4C2×D4×C15111yescharacteristic
461C22×C4C2×D4×C15111yescharacteristic
462C22×C4C2×D4×C15111yescharacteristic
463C22×C4C2×D4×C15111yescharacteristic
464C22×C4C2×D4×C15111yescharacteristic
465C22×C4C24×C15111yescharacteristic
466SD16—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4C2×D4×C15114yes—
486C4⋊C4#41×C15112yes—
487C22⋊C4—28no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C42—28no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—42no—
499C42C2×D4×C15112yes—
500C42C2×D4×C15112yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×SD16×C15112yes—
515C23C22×D4×C15111yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—42no—
520Q8—42no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4#120×C15112yes—
559C2×C4#120×C15112yes—
560C2×C4C2×SD16×C15112yes—
561C2×C4C2×SD16×C15112yes—
562C2×C4#137×C15112yes—
563C2×C4C22×D4×C15111yescharacteristic
564C2×C4C22×D4×C15111yescharacteristic
565C2×C4C2×#41×C15111yescharacteristic
566C2×C4C2×#41×C15111yescharacteristic
567C2×C4H(5,𝔽2)×C15111yescharacteristic
568C2×C42−1+4×C15111yescharacteristic
569C8—42no—
570C8—42no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#120×C15111yescharacteristic
586C22#525×C15111yescharacteristic
587C22#537×C15111yescharacteristic
588C22C22×SD16×C15111yescharacteristic
589C22C2×#137×C15111yescharacteristic
590C22#575×C15111yescharacteristic
591C22#575×C15111yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—41no—
596C4—44no—
597C4—41no—
598C4—44no—
599C4—81no—
600C4—81no—
601C4#448×C15114yes—
602C4#546×C15111yescharacteristic
603C4#547×C15111yescharacteristic
604C4#548×C15111yescharacteristic
605C4#550×C15111yescharacteristic
606C2—42no—
607C2—82no—
608C2#2407×C15111yescharacteristic
609C2C2×#448×C15111yescharacteristic
610C2#3005×C15111yescharacteristic
611C2#3101×C15111yescharacteristic
612C2#3116×C15111yescharacteristic
613C2#3246×C15111yescharacteristic
614C2#3252×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC2×#121×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C22×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#533×C151C2semidirect
2. C2×#476×C151C2semidirect
3. C2×#461×C151C2semidirect
5. #2035×C151C2semidirect
7. #1987×C151C2semidirect
8. #1981×C151C2semidirect
9. #1880×C151C2semidirect
10. #1865×C151C2semidirect
11. #1797×C151C2semidirect
62. C4⋊Q8×C151D4semidirect
63. C4⋊Q8×C151D4semidirect
151. C22×C4×C151SD16semidirect
307. #24151C151direct (N × H)
308. C151#24151direct (N × H)
309. C2×#533C2×C151semidirect
310. C2×#476C2×C151semidirect
311. C2×#461C2×C151semidirect
313. #2035C2×C151semidirect
315. #1987C2×C151semidirect
316. #1981C2×C151semidirect
317. #1880C2×C151semidirect
318. #1865C2×C151semidirect
319. #1797C2×C151semidirect
370. C4⋊Q8D4×C151semidirect
371. C4⋊Q8D4×C151semidirect
459. C22×C4SD16×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#533×C1512
C2×#476×C1512
C2×#461×C1512
C4×#121×C1512
#2035×C1512
#2000×C1512
#1987×C1512
#1981×C1512
#1880×C1512
#1865×C1512
#1797×C1512
#24151151