Finite Group Databasebeta

← C31×#24170 #21480929

Subgroups — C31×#24170 #21480929

Order 7936.

1162 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic, complemented
2C31×#3036C211yesmaximal, characteristic, complemented
3C4×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4C31×#2542C211yesmaximal, characteristic, complemented
5C31×#2039C212yesmaximal, complemented
6C31×#2006C212yesmaximal, complemented
7C31×#1997C211yesmaximal, characteristic, complemented
8C31×#1980C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1859C212yesmaximal, complemented
11C31×#1806C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)C2211yescharacteristic
13C2×C31×AGL(1,ℤ/8)C2212yes—
14C31×#556—21no—
15C31×#555—21no—
16C31×#554—21no—
17C31×#553—22no—
18C31×#548—22no—
19C31×#547—21no—
20C31×#542—22no—
21C31×#537C2212yes—
22C31×#533—21no—
23C31×#532—21no—
24C31×#532C2211yescharacteristic
25C2×C31×C4⋊Q8C2211yescharacteristic
26C2×C31×C22⋊Q8—22no—
27C31×#520C2212yes—
28C4×#41×C31C2211yescharacteristic
29C2×C4×D4×C31C2211yescharacteristic
30C31×#463—22no—
31C31×#461—21no—
32C31×#459—21no—
33C31×#442—21no—
34C31×#440—21no—
35C4×SD16×C31—21no—
36C4×D8×C31—21no—
37C31×#428C2211yescharacteristic
38C31×#423C2211yescharacteristic
39C2×C31×#104C2212yes—
40C2×C31×#104C2212yes—
41C2×C31×#104C2212yes—
42C31×#418C2211yescharacteristic
43C31×#417C2212yes—
44C31×#417C2211yescharacteristic
45C4×M16×C31C2211yescharacteristic
46C31×#399—22no—
47C31×#390C2212yes—
48C31×#384C2212yes—
49C4×C4⋊C4×C31C2211yescharacteristic
50C31×#343C2212yes—
51C31×#343C2212yes—
52C31×#339C2212yes—
53C2×#41×C31—22no—
54C2×#41×C31C2311yescharacteristic
55C2×#41×C31C2312yes—
56C22×Q8×C31D412yes—
57C22×D4×C31C2311yescharacteristic
58C23×C4×C31—21no—
59C31×AGL(1,ℤ/8)—22no—
60C31×AGL(1,ℤ/8)—24no—
61C31×AGL(1,ℤ/8)—22no—
62C31×AGL(1,ℤ/8)—42no—
63C2×SD16×C31—21no—
64C2×SD16×C31—22no—
65C2×D8×C31—21no—
66C2×D8×C31—22no—
67C2×M16×C31C2311yescharacteristic
68C2×M16×C31C2312yes—
69C2×M16×C31C2311yescharacteristic
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8—21no—
72C31×C4⋊Q8—21no—
73C31×C4⋊Q8—21no—
74C31×C4⋊Q8D411yescharacteristic
75C31×#127D411yescharacteristic
76C31×#126—24no—
77C31×#125—22no—
78C31×#124—22no—
79C31×#124—24no—
80C31×#124D412yes—
81C31×#123—24no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—24no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—48no—
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—22no—
90C31×#121—21no—
91C31×#121—21no—
92C31×#121—24no—
93C4×Q8×C31—22no—
94C4×Q8×C31—22no—
95C4×Q8×C31D411yescharacteristic
96C4×Q8×C31D411yescharacteristic
97C4×D4×C31—22no—
98C4×D4×C31—22no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—24no—
102C4×D4×C31—21no—
103C4×D4×C31—21no—
104C4×D4×C31—21no—
105C4×D4×C31—21no—
106C4×D4×C31—21no—
107C4×D4×C31—21no—
108C4×D4×C31D411yescharacteristic
109C4×D4×C31D411yescharacteristic
110C31×#117—22no—
111C31×#117—22no—
112C31×#117—21no—
113C31×#117C2311yescharacteristic
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31—22no—
116C2×C4⋊C4×C31—24no—
117C2×C4⋊C4×C31D412yes—
118C2×C4⋊C4×C31C2311yescharacteristic
119C2×C4⋊C4×C31C2312yes—
120C2×C22⋊C4×C31—21no—
121C2×C22⋊C4×C31—24no—
122C2×C42×C31—21no—
123C2×C42×C31C2312yes—
124C2×C42×C31C2311yescharacteristic
125C31×#107—21no—
126C31×#106—21no—
127C31×C4⋊C8—21no—
128C31×C4⋊C8—21no—
129C31×#104—24no—
130C31×#104—42no—
131C31×#104—42no—
132C31×#103—21no—
133C31×#103—22no—
134C31×#102—21no—
135C31×#102—22no—
136C31×#102—21no—
137C31×#102—21no—
138C31×#97—21no—
139C4×C8×C31—21no—
140C31×#95—21no—
141C24×C31—21no—
142#41×C31—22no—
143#41×C31—22no—
144#41×C31—21no—
145#41×C31—21no—
146#41×C31—21no—
147#41×C31—22no—
148#41×C31—22no—
149#41×C31—21no—
150#41×C31—84no—
151C2×Q8×C31—22no—
152C2×Q8×C31—22no—
153C2×Q8×C31—44no—
154C2×Q8×C31C2×D412yes—
155C2×Q8×C31C2×D411yescharacteristic
156C2×D4×C31—21no—
157C2×D4×C31—22no—
158C2×D4×C31—24no—
159C2×D4×C31—21no—
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31—21no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31—21no—
166C2×D4×C31C2×D412yes—
167C2×D4×C31C2×D411yescharacteristic
168C2×D4×C31#4112yescomplemented
169C22×C4×C31—22no—
170C22×C4×C31—24no—
171C22×C4×C31—21no—
172C22×C4×C31—21no—
173C22×C4×C31—21no—
174C22×C4×C31—22no—
175C22×C4×C31—21no—
176C22×C4×C31—44no—
177C22×C4×C31—41no—
178C22×C4×C31—41no—
179C22×C4×C31C2×D411yescharacteristic
180C22×C4×C31C2×D414yes—
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2411yescharacteristic
183SD16×C31—24no—
184SD16×C31—44no—
185D8×C31—24no—
186D8×C31—44no—
187M16×C31—21no—
188M16×C31—22no—
189M16×C31—21no—
190M16×C31—22no—
191M16×C31—21no—
192M16×C31#4112yescomplemented
193C2×C8×C31—21no—
194C2×C8×C31—21no—
195C2×C8×C31—22no—
196C4⋊C4×C31—24no—
197C4⋊C4×C31—24no—
198C4⋊C4×C31—24no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—22no—
206C4⋊C4×C31—22no—
207C4⋊C4×C31—21no—
208C4⋊C4×C31—21no—
209C4⋊C4×C31—48no—
210C4⋊C4×C31C2×D411yescharacteristic
211C4⋊C4×C31C2×D411yescharacteristic
212C22⋊C4×C31—24no—
213C22⋊C4×C31—24no—
214C22⋊C4×C31—24no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—48no—
221C22⋊C4×C31—41no—
222C22⋊C4×C31—41no—
223C42×C31—24no—
224C42×C31—21no—
225C42×C31—21no—
226C42×C31—21no—
227C42×C31—22no—
228C42×C31—22no—
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233#24170C3111yesmaximal, characteristic, direct product factor, Sylow
234C23×C31—21no—
235C23×C31—22no—
236C23×C31—22no—
237C23×C31—42no—
238C23×C31—42no—
239C23×C31—41no—
240C23×C31—41no—
241C23×C31C22×D411yescharacteristic
242Q8×C31—21no—
243Q8×C31—21no—
244Q8×C31—22no—
245Q8×C31—22no—
246Q8×C31—44no—
247Q8×C31—82no—
248D4×C31—21no—
249D4×C31—22no—
250D4×C31—21no—
251D4×C31—22no—
252D4×C31—22no—
253D4×C31—22no—
254D4×C31—42no—
255D4×C31—42no—
256D4×C31—41no—
257D4×C31—42no—
258D4×C31—41no—
259D4×C31—41no—
260D4×C31—41no—
261D4×C31—42no—
262D4×C31—84no—
263C2×C4×C31—22no—
264C2×C4×C31—24no—
265C2×C4×C31—21no—
266C2×C4×C31—22no—
267C2×C4×C31—21no—
268C2×C4×C31—21no—
269C2×C4×C31—24no—
270C2×C4×C31—41no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—41no—
277C2×C4×C31—41no—
278C2×C4×C31—42no—
279C2×C4×C31—44no—
280C2×C4×C31—42no—
281C2×C4×C31—41no—
282C2×C4×C31—82no—
283C2×C4×C31—84no—
284C2×C4×C31—82no—
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31#12011yescharacteristic
287C2×C4×C31#12011yescharacteristic
288C2×C4×C31#12011yescharacteristic
289C2×C4×C31C22×D411yescharacteristic
290C2×C4×C31C22×D411yescharacteristic
291C2×C4×C31C2×#4111yescharacteristic
292C2×C4×C31C2×#4111yescharacteristic
293C2×C4×C31H(5,𝔽2)11yescharacteristic
294C2×C4×C312−1+411yescharacteristic
295C8×C31—22no—
296C8×C31—41no—
297C8×C31—42no—
298#3511C2×C3111yescharacteristic, complemented
299#3036C2×C3111yescharacteristic, complemented
300C4×AGL(1,ℤ/8)C2×C3111yescharacteristic, complemented
301#2542C2×C3111yescharacteristic, complemented
302#2039C2×C3112yescomplemented
303#2006C2×C3112yescomplemented
304#1997C2×C3111yescharacteristic, complemented
305#1980C2×C3111yescharacteristic, complemented
306#1883C2×C3112yescomplemented
307#1859C2×C3112yescomplemented
308#1806C2×C3111yescharacteristic, complemented
309C22×C31—21no—
310C22×C31—21no—
311C22×C31—44no—
312C22×C31—42no—
313C22×C31—41no—
314C22×C31—42no—
315C22×C31—41no—
316C22×C31—42no—
317C22×C31—41no—
318C22×C31—42no—
319C22×C31—82no—
320C22×C31—81no—
321C22×C31—81no—
322C22×C31C2×#12011yescharacteristic
323C22×C31#54811yescharacteristic
324C22×C31#55011yescharacteristic
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—21no—
328C4×C31—44no—
329C4×C31—42no—
330C4×C31—41no—
331C4×C31—41no—
332C4×C31—44no—
333C4×C31—41no—
334C4×C31—81no—
335C4×C31#45312yes—
336C4×C31#52511yescharacteristic
337C4×C31#53711yescharacteristic
338C4×C31#54611yescharacteristic
339C4×C31#54711yescharacteristic
340C2×AGL(1,ℤ/8)C22×C3111yescharacteristic
341C2×AGL(1,ℤ/8)C22×C3112yes—
342#556—21no—
343#555—21no—
344#554—21no—
345#553—22no—
346#548—22no—
347#547—21no—
348#542—22no—
349#537C22×C3112yes—
350#533—21no—
351#532—21no—
352#532C22×C3111yescharacteristic
353C2×C4⋊Q8C22×C3111yescharacteristic
354C2×C22⋊Q8—22no—
355#520C22×C3112yes—
356C4×#41C22×C3111yescharacteristic
357C2×C4×D4C22×C3111yescharacteristic
358#463—22no—
359#461—21no—
360#459—21no—
361#442—21no—
362#440—21no—
363C4×SD16—21no—
364C4×D8—21no—
365#428C22×C3111yescharacteristic
366#423C22×C3111yescharacteristic
367C2×#104C22×C3112yes—
368C2×#104C22×C3112yes—
369C2×#104C22×C3112yes—
370#418C22×C3111yescharacteristic
371#417C22×C3112yes—
372#417C22×C3111yescharacteristic
373C4×M16C22×C3111yescharacteristic
374#399—22no—
375#390C22×C3112yes—
376#384C22×C3112yes—
377C4×C4⋊C4C22×C3111yescharacteristic
378#343C22×C3112yes—
379#343C22×C3112yes—
380#339C22×C3112yes—
381C2×C31—21no—
382C2×C31—21no—
383C2×C31—81no—
384C2×C31—82no—
385C2×C31—82no—
386C2×C31#240711yescharacteristic
387C2×C31C2×#45311yescharacteristic
388C2×C31#302011yescharacteristic
389C2×#41—22no—
390C2×#41C23×C3111yescharacteristic
391C2×#41C23×C3112yes—
392C22×Q8D4×C3112yes—
393C22×D4C23×C3111yescharacteristic
394C23×C4—21no—
395AGL(1,ℤ/8)—22no—
396AGL(1,ℤ/8)—24no—
397AGL(1,ℤ/8)—22no—
398AGL(1,ℤ/8)—42no—
399C2×SD16—21no—
400C2×SD16—22no—
401C2×D8—21no—
402C2×D8—22no—
403C2×M16C23×C3112yes—
404C2×M16C23×C3111yescharacteristic
405C2×M16C23×C3111yescharacteristic
406C4⋊Q8—21no—
407C4⋊Q8—21no—
408C4⋊Q8—21no—
409C4⋊Q8—21no—
410C4⋊Q8D4×C3111yescharacteristic
411#127D4×C3111yescharacteristic
412#126—24no—
413#125—22no—
414#124—22no—
415#124—24no—
416#124D4×C3112yes—
417#123—24no—
418C22⋊Q8—22no—
419C22⋊Q8—24no—
420C22⋊Q8—22no—
421C22⋊Q8—22no—
422C22⋊Q8—48no—
423#121—22no—
424#121—22no—
425#121—22no—
426#121—21no—
427#121—21no—
428#121—24no—
429C4×Q8—22no—
430C4×Q8—22no—
431C4×Q8D4×C3111yescharacteristic
432C4×Q8D4×C3111yescharacteristic
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—24no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4D4×C3111yescharacteristic
445C4×D4D4×C3111yescharacteristic
446#117—22no—
447#117—22no—
448#117—21no—
449#117C23×C3111yescharacteristic
450C2×C4⋊C4—21no—
451C2×C4⋊C4—22no—
452C2×C4⋊C4—24no—
453C2×C4⋊C4D4×C3112yes—
454C2×C4⋊C4C23×C3111yescharacteristic
455C2×C4⋊C4C23×C3112yes—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—24no—
458C2×C42—21no—
459C2×C42C23×C3112yes—
460C2×C42C23×C3111yescharacteristic
461#107—21no—
462#106—21no—
463C4⋊C8—21no—
464C4⋊C8—21no—
465#104—24no—
466#104—42no—
467#104—42no—
468#103—21no—
469#103—22no—
470#102—21no—
471#102—22no—
472#102—21no—
473#102—21no—
474#97—21no—
475C4×C8—21no—
476#95—21no—
477C31#2417011yescharacteristic, direct product factor, Sylow
478C24—21no—
479#41—22no—
480#41—22no—
481#41—21no—
482#41—21no—
483#41—21no—
484#41—22no—
485#41—22no—
486#41—21no—
487#41—84no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—44no—
491C2×Q8C2×D4×C3112yes—
492C2×Q8C2×D4×C3111yescharacteristic
493C2×D4—21no—
494C2×D4—22no—
495C2×D4—24no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4C2×D4×C3112yes—
504C2×D4C2×D4×C3111yescharacteristic
505C2×D4#41×C3112yescomplemented
506C22×C4—22no—
507C22×C4—24no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4—44no—
514C22×C4—41no—
515C22×C4—41no—
516C22×C4C2×D4×C3111yescharacteristic
517C22×C4C2×D4×C3114yes—
518C22×C4C2×D4×C3111yescharacteristic
519C22×C4C24×C3111yescharacteristic
520SD16—24no—
521SD16—44no—
522D8—24no—
523D8—44no—
524M16—21no—
525M16—22no—
526M16—21no—
527M16—22no—
528M16—21no—
529M16#41×C3112yescomplemented
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—48no—
547C4⋊C4C2×D4×C3111yescharacteristic
548C4⋊C4C2×D4×C3111yescharacteristic
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—48no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C42—24no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—22no—
565C42—22no—
566C42C2×D4×C3111yescharacteristic
567C42C2×D4×C3111yescharacteristic
568C42C2×D4×C3111yescharacteristic
569C42C2×D4×C3111yescharacteristic
570C23—21no—
571C23—22no—
572C23—22no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23C22×D4×C3111yescharacteristic
578Q8—21no—
579Q8—21no—
580Q8—22no—
581Q8—22no—
582Q8—44no—
583Q8—82no—
584D4—21no—
585D4—22no—
586D4—21no—
587D4—22no—
588D4—22no—
589D4—22no—
590D4—42no—
591D4—42no—
592D4—41no—
593D4—42no—
594D4—41no—
595D4—42no—
596D4—41no—
597D4—41no—
598D4—84no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—82no—
619C2×C4—84no—
620C2×C4—82no—
621C2×C4C31×#12011yescharacteristic
622C2×C4C31×#12011yescharacteristic
623C2×C4C31×#12011yescharacteristic
624C2×C4C31×#12011yescharacteristic
625C2×C4C22×D4×C3111yescharacteristic
626C2×C4C22×D4×C3111yescharacteristic
627C2×C4C2×#41×C3111yescharacteristic
628C2×C4C2×#41×C3111yescharacteristic
629C2×C4C31×H(5,𝔽2)11yescharacteristic
630C2×C4C31×2−1+411yescharacteristic
631C8—22no—
632C8—41no—
633C8—42no—
634C22—21no—
635C22—21no—
636C22—44no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—41no—
641C22—42no—
642C22—41no—
643C22—42no—
644C22—82no—
645C22—81no—
646C22—81no—
647C22C2×C31×#12011yescharacteristic
648C22C31×#54811yescharacteristic
649C22C31×#55011yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—44no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—44no—
658C4—41no—
659C4—81no—
660C4C31×#45312yes—
661C4C31×#52511yescharacteristic
662C4C31×#53711yescharacteristic
663C4C31×#54611yescharacteristic
664C4C31×#54711yescharacteristic
665C2—21no—
666C2—21no—
667C2—82no—
668C2—81no—
669C2—82no—
670C2C31×#240711yescharacteristic
671C2C2×C31×#45311yescharacteristic
672C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3511C2semidirect
2. C31×#3036C2semidirect
3. C4×C31×AGL(1,ℤ/8)C2semidirect
4. C31×#2542C2semidirect
5. C31×#2039C2semidirect
6. C31×#2006C2semidirect
7. C31×#1997C2semidirect
8. C31×#1980C2semidirect
9. C31×#1883C2semidirect
10. C31×#1859C2semidirect
11. C31×#1806C2semidirect
168. C2×D4×C31#41semidirect
192. M16×C31#41semidirect
233. #24170C31direct (N × H)
298. #3511C2×C31semidirect
299. #3036C2×C31semidirect
300. C4×AGL(1,ℤ/8)C2×C31semidirect
301. #2542C2×C31semidirect
302. #2039C2×C31semidirect
303. #2006C2×C31semidirect
304. #1997C2×C31semidirect
305. #1980C2×C31semidirect
306. #1883C2×C31semidirect
307. #1859C2×C31semidirect
308. #1806C2×C31semidirect
477. C31#24170direct (N × H)
505. C2×D4#41×C31semidirect
529. M16#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#30362
C4×C31×AGL(1,ℤ/8)2
C31×#25422
C31×#20392
C31×#20062
C31×#19972
C31×#19802
C31×#18832
C31×#18592
C31×#18062
#2417031