Finite Group Databasebeta

← C31×#24171 #21480930

Subgroups — C31×#24171 #21480930

Order 7936.

1106 conjugacy classes of subgroups.

The subgroup lattice has 670 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×#2563C211yesmaximal, characteristic, complemented
5C31×#2040C212yesmaximal, complemented
6C31×#2006C212yesmaximal, complemented
7C31×#1997C211yesmaximal, characteristic, complemented
8C31×#1980C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1860C212yesmaximal, 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
26C31×#520C2212yes—
27C4×#41×C31C2211yescharacteristic
28C2×C4×D4×C31C2211yescharacteristic
29C31×#463—22no—
30C31×#461—21no—
31C31×#459—21no—
32C31×#442—21no—
33C31×#440—21no—
34C4×SD16×C31—21no—
35C4×D8×C31—21no—
36C31×#428C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C2×C31×#104C2212yes—
39C2×C31×#104C2212yes—
40C2×C31×#104C2212yes—
41C31×#418C2211yescharacteristic
42C31×#417C2212yes—
43C31×#417C2211yescharacteristic
44C4×M16×C31C2211yescharacteristic
45C31×#393—22no—
46C31×#390C2212yes—
47C31×#382C2212yes—
48C31×#380—22no—
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×#121—22no—
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—21no—
90C31×#121—21no—
91C31×#121—24no—
92C4×Q8×C31—22no—
93C4×Q8×C31—22no—
94C4×Q8×C31D411yescharacteristic
95C4×Q8×C31D411yescharacteristic
96C4×D4×C31—22no—
97C4×D4×C31—22no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—24no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31—21no—
104C4×D4×C31—21no—
105C4×D4×C31—21no—
106C4×D4×C31—21no—
107C4×D4×C31D411yescharacteristic
108C4×D4×C31D411yescharacteristic
109C31×#117—22no—
110C31×#117—22no—
111C31×#117—21no—
112C31×#117C2311yescharacteristic
113C2×C4⋊C4×C31—22no—
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31D412yes—
116C2×C4⋊C4×C31C2311yescharacteristic
117C2×C4⋊C4×C31C2312yes—
118C2×C22⋊C4×C31—21no—
119C2×C22⋊C4×C31—24no—
120C2×C42×C31—21no—
121C2×C42×C31C2312yes—
122C2×C42×C31C2311yescharacteristic
123C31×#107—21no—
124C31×#106—21no—
125C31×C4⋊C8—21no—
126C31×C4⋊C8—21no—
127C31×#104—24no—
128C31×#104—42no—
129C31×#104—42no—
130C31×#103—21no—
131C31×#103—22no—
132C31×#102—21no—
133C31×#102—22no—
134C31×#102—21no—
135C31×#102—21no—
136C31×#97—21no—
137C4×C8×C31—21no—
138C31×#95—21no—
139C31×#95—24no—
140C24×C31—21no—
141#41×C31—22no—
142#41×C31—22no—
143#41×C31—21no—
144#41×C31—21no—
145#41×C31—21no—
146#41×C31—22no—
147#41×C31—22no—
148#41×C31—21no—
149#41×C31—84no—
150C2×Q8×C31—22no—
151C2×Q8×C31—22no—
152C2×Q8×C31—82no—
153C2×Q8×C31C2×D412yes—
154C2×Q8×C31C2×D411yescharacteristic
155C2×D4×C31—21no—
156C2×D4×C31—22no—
157C2×D4×C31—24no—
158C2×D4×C31—21no—
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×C31C2×D412yes—
166C2×D4×C31C2×D411yescharacteristic
167C2×D4×C31#4112yescomplemented
168C22×C4×C31—24no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31—22no—
173C22×C4×C31—21no—
174C22×C4×C31—22no—
175C22×C4×C31—44no—
176C22×C4×C31—41no—
177C22×C4×C31—41no—
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D414yes—
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2411yescharacteristic
182SD16×C31—24no—
183SD16×C31—44no—
184D8×C31—24no—
185D8×C31—44no—
186M16×C31—21no—
187M16×C31—22no—
188M16×C31—21no—
189M16×C31—22no—
190M16×C31—21no—
191M16×C31#4112yescomplemented
192C2×C8×C31—21no—
193C2×C8×C31—21no—
194C2×C8×C31—22no—
195C4⋊C4×C31—24no—
196C4⋊C4×C31—24no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—22no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31—21no—
207C4⋊C4×C31—42no—
208C4⋊C4×C31C2×D411yescharacteristic
209C4⋊C4×C31C2×D411yescharacteristic
210C22⋊C4×C31—24no—
211C22⋊C4×C31—24no—
212C22⋊C4×C31—24no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—41no—
219C22⋊C4×C31—41no—
220C22⋊C4×C31—84no—
221C42×C31—24no—
222C42×C31—21no—
223C42×C31—21no—
224C42×C31—21no—
225C42×C31—22no—
226C42×C31—22no—
227C42×C31C2×D411yescharacteristic
228C42×C31C2×D411yescharacteristic
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231#24171C3111yesmaximal, characteristic, direct product factor, Sylow
232C23×C31—21no—
233C23×C31—22no—
234C23×C31—22no—
235C23×C31—42no—
236C23×C31—42no—
237C23×C31—41no—
238C23×C31—41no—
239C23×C31C22×D411yescharacteristic
240Q8×C31—21no—
241Q8×C31—21no—
242Q8×C31—22no—
243Q8×C31—22no—
244Q8×C31—44no—
245Q8×C31—82no—
246D4×C31—21no—
247D4×C31—22no—
248D4×C31—21no—
249D4×C31—22no—
250D4×C31—22no—
251D4×C31—22no—
252D4×C31—42no—
253D4×C31—42no—
254D4×C31—41no—
255D4×C31—42no—
256D4×C31—41no—
257D4×C31—41no—
258D4×C31—41no—
259D4×C31—42no—
260D4×C31—84no—
261C2×C4×C31—22no—
262C2×C4×C31—21no—
263C2×C4×C31—22no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—24no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—41no—
272C2×C4×C31—41no—
273C2×C4×C31—42no—
274C2×C4×C31—44no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—41no—
279C2×C4×C31—42no—
280C2×C4×C31—82no—
281C2×C4×C31—84no—
282C2×C4×C31—82no—
283C2×C4×C31#12011yescharacteristic
284C2×C4×C31#12011yescharacteristic
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31#12011yescharacteristic
287C2×C4×C31C22×D411yescharacteristic
288C2×C4×C31C22×D411yescharacteristic
289C2×C4×C31C2×#4111yescharacteristic
290C2×C4×C31C2×#4111yescharacteristic
291C2×C4×C31H(5,𝔽2)11yescharacteristic
292C2×C4×C312−1+411yescharacteristic
293C8×C31—22no—
294C8×C31—41no—
295C8×C31—42no—
296#3511C2×C3111yescharacteristic, complemented
297#3036C2×C3111yescharacteristic, complemented
298C4×AGL(1,ℤ/8)C2×C3111yescharacteristic, complemented
299#2563C2×C3111yescharacteristic, complemented
300#2040C2×C3112yescomplemented
301#2006C2×C3112yescomplemented
302#1997C2×C3111yescharacteristic, complemented
303#1980C2×C3111yescharacteristic, complemented
304#1883C2×C3112yescomplemented
305#1860C2×C3112yescomplemented
306#1806C2×C3111yescharacteristic, complemented
307C22×C31—21no—
308C22×C31—21no—
309C22×C31—44no—
310C22×C31—42no—
311C22×C31—41no—
312C22×C31—42no—
313C22×C31—41no—
314C22×C31—42no—
315C22×C31—41no—
316C22×C31—42no—
317C22×C31—82no—
318C22×C31—81no—
319C22×C31—81no—
320C22×C31C2×#12011yescharacteristic
321C22×C31#54811yescharacteristic
322C22×C31#55011yescharacteristic
323C4×C31—21no—
324C4×C31—21no—
325C4×C31—21no—
326C4×C31—42no—
327C4×C31—41no—
328C4×C31—41no—
329C4×C31—41no—
330C4×C31—44no—
331C4×C31—82no—
332C4×C31—81no—
333C4×C31#45412yes—
334C4×C31#52511yescharacteristic
335C4×C31#53711yescharacteristic
336C4×C31#54611yescharacteristic
337C4×C31#54711yescharacteristic
338C2×AGL(1,ℤ/8)C22×C3111yescharacteristic
339C2×AGL(1,ℤ/8)C22×C3112yes—
340#556—21no—
341#555—21no—
342#554—21no—
343#553—22no—
344#548—22no—
345#547—21no—
346#542—22no—
347#537C22×C3112yes—
348#533—21no—
349#532—21no—
350#532C22×C3111yescharacteristic
351C2×C4⋊Q8C22×C3111yescharacteristic
352#520C22×C3112yes—
353C4×#41C22×C3111yescharacteristic
354C2×C4×D4C22×C3111yescharacteristic
355#463—22no—
356#461—21no—
357#459—21no—
358#442—21no—
359#440—21no—
360C4×SD16—21no—
361C4×D8—21no—
362#428C22×C3111yescharacteristic
363#423C22×C3111yescharacteristic
364C2×#104C22×C3112yes—
365C2×#104C22×C3112yes—
366C2×#104C22×C3112yes—
367#418C22×C3111yescharacteristic
368#417C22×C3112yes—
369#417C22×C3111yescharacteristic
370C4×M16C22×C3111yescharacteristic
371#393—22no—
372#390C22×C3112yes—
373#382C22×C3112yes—
374#380—22no—
375C4×C4⋊C4C22×C3111yescharacteristic
376#343C22×C3112yes—
377#343C22×C3112yes—
378#339C22×C3112yes—
379C2×C31—21no—
380C2×C31—21no—
381C2×C31—81no—
382C2×C31—82no—
383C2×C31—82no—
384C2×C31#240711yescharacteristic
385C2×C31C2×#45411yescharacteristic
386C2×C31#301911yescharacteristic
387C2×#41—22no—
388C2×#41C23×C3111yescharacteristic
389C2×#41C23×C3112yes—
390C22×Q8D4×C3112yes—
391C22×D4C23×C3111yescharacteristic
392C23×C4—21no—
393AGL(1,ℤ/8)—22no—
394AGL(1,ℤ/8)—24no—
395AGL(1,ℤ/8)—22no—
396AGL(1,ℤ/8)—42no—
397C2×SD16—21no—
398C2×SD16—22no—
399C2×D8—21no—
400C2×D8—22no—
401C2×M16C23×C3112yes—
402C2×M16C23×C3111yescharacteristic
403C2×M16C23×C3111yescharacteristic
404C4⋊Q8—21no—
405C4⋊Q8—21no—
406C4⋊Q8—21no—
407C4⋊Q8—21no—
408C4⋊Q8D4×C3111yescharacteristic
409#127D4×C3111yescharacteristic
410#126—24no—
411#125—22no—
412#124—22no—
413#124—24no—
414#124D4×C3112yes—
415#123—24no—
416C22⋊Q8—22no—
417C22⋊Q8—24no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420#121—22no—
421#121—22no—
422#121—22no—
423#121—21no—
424#121—21no—
425#121—24no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8D4×C3111yescharacteristic
429C4×Q8D4×C3111yescharacteristic
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—24no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4D4×C3111yescharacteristic
442C4×D4D4×C3111yescharacteristic
443#117—22no—
444#117—22no—
445#117—21no—
446#117C23×C3111yescharacteristic
447C2×C4⋊C4—22no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4D4×C3112yes—
450C2×C4⋊C4C23×C3111yescharacteristic
451C2×C4⋊C4C23×C3112yes—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—24no—
454C2×C42—21no—
455C2×C42C23×C3112yes—
456C2×C42C23×C3111yescharacteristic
457#107—21no—
458#106—21no—
459C4⋊C8—21no—
460C4⋊C8—21no—
461#104—24no—
462#104—42no—
463#104—42no—
464#103—21no—
465#103—22no—
466#102—21no—
467#102—22no—
468#102—21no—
469#102—21no—
470#97—21no—
471C4×C8—21no—
472#95—21no—
473#95—24no—
474C31#2417111yescharacteristic, direct product factor, Sylow
475C24—21no—
476#41—22no—
477#41—22no—
478#41—21no—
479#41—21no—
480#41—21no—
481#41—22no—
482#41—22no—
483#41—21no—
484#41—84no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—82no—
488C2×Q8C2×D4×C3112yes—
489C2×Q8C2×D4×C3111yescharacteristic
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—24no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4C2×D4×C3112yes—
501C2×D4C2×D4×C3111yescharacteristic
502C2×D4#41×C3112yescomplemented
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—44no—
511C22×C4—41no—
512C22×C4—41no—
513C22×C4C2×D4×C3111yescharacteristic
514C22×C4C2×D4×C3114yes—
515C22×C4C2×D4×C3111yescharacteristic
516C22×C4C24×C3111yescharacteristic
517SD16—24no—
518SD16—44no—
519D8—24no—
520D8—44no—
521M16—21no—
522M16—22no—
523M16—21no—
524M16—22no—
525M16—21no—
526M16#41×C3112yescomplemented
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—42no—
543C4⋊C4C2×D4×C3111yescharacteristic
544C4⋊C4C2×D4×C3111yescharacteristic
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—84no—
556C42—24no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—22no—
561C42—22no—
562C42C2×D4×C3111yescharacteristic
563C42C2×D4×C3111yescharacteristic
564C42C2×D4×C3111yescharacteristic
565C42C2×D4×C3111yescharacteristic
566C23—21no—
567C23—22no—
568C23—22no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23C22×D4×C3111yescharacteristic
574Q8—21no—
575Q8—21no—
576Q8—22no—
577Q8—22no—
578Q8—44no—
579Q8—82no—
580D4—21no—
581D4—22no—
582D4—21no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—42no—
587D4—42no—
588D4—41no—
589D4—42no—
590D4—41no—
591D4—42no—
592D4—41no—
593D4—41no—
594D4—84no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—82no—
615C2×C4—84no—
616C2×C4—82no—
617C2×C4C31×#12011yescharacteristic
618C2×C4C31×#12011yescharacteristic
619C2×C4C31×#12011yescharacteristic
620C2×C4C31×#12011yescharacteristic
621C2×C4C22×D4×C3111yescharacteristic
622C2×C4C22×D4×C3111yescharacteristic
623C2×C4C2×#41×C3111yescharacteristic
624C2×C4C2×#41×C3111yescharacteristic
625C2×C4C31×H(5,𝔽2)11yescharacteristic
626C2×C4C31×2−1+411yescharacteristic
627C8—22no—
628C8—41no—
629C8—42no—
630C22—21no—
631C22—21no—
632C22—44no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22C2×C31×#12011yescharacteristic
644C22C31×#54811yescharacteristic
645C22C31×#55011yescharacteristic
646C4—21no—
647C4—21no—
648C4—21no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—44no—
654C4—82no—
655C4—81no—
656C4C31×#45412yes—
657C4C31×#52511yescharacteristic
658C4C31×#53711yescharacteristic
659C4C31×#54611yescharacteristic
660C4C31×#54711yescharacteristic
661C2—21no—
662C2—21no—
663C2—82no—
664C2—81no—
665C2—82no—
666C2C31×#240711yescharacteristic
667C2C2×C31×#45411yescharacteristic
668C2C31×#301911yescharacteristic
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×#2563C2semidirect
5. C31×#2040C2semidirect
6. C31×#2006C2semidirect
7. C31×#1997C2semidirect
8. C31×#1980C2semidirect
9. C31×#1883C2semidirect
10. C31×#1860C2semidirect
11. C31×#1806C2semidirect
167. C2×D4×C31#41semidirect
191. M16×C31#41semidirect
231. #24171C31direct (N × H)
296. #3511C2×C31semidirect
297. #3036C2×C31semidirect
298. C4×AGL(1,ℤ/8)C2×C31semidirect
299. #2563C2×C31semidirect
300. #2040C2×C31semidirect
301. #2006C2×C31semidirect
302. #1997C2×C31semidirect
303. #1980C2×C31semidirect
304. #1883C2×C31semidirect
305. #1860C2×C31semidirect
306. #1806C2×C31semidirect
474. C31#24171direct (N × H)
502. C2×D4#41×C31semidirect
526. 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×#25632
C31×#20402
C31×#20062
C31×#19972
C31×#19802
C31×#18832
C31×#18602
C31×#18062
#2417131