Finite Group Databasebeta

← C31×#25926 #21482685

Subgroups — C31×#25926 #21482685

Order 7936.

1012 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic, complemented
2C31×#3140C211yesmaximal, characteristic, complemented
3C31×#3040C211yesmaximal, characteristic
4C31×#2640C211yesmaximal, characteristic, complemented
5C31×#2028C212yesmaximal, complemented
6C31×#2027C212yesmaximal, complemented
7C31×#1997C211yesmaximal, characteristic, complemented
8C31×#1995C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal
10C31×#1882C212yesmaximal, complemented
11C31×#1759C211yesmaximal, characteristic, complemented
12C2×C31×#137C2211yescharacteristic
13C2×C31×AGL(1,ℤ/8)C2211yescharacteristic
14Q82×C31—21no—
15C31×#557—21no—
16C31×#554—21no—
17C31×#554—22no—
18D4×Q8×C31—22no—
19D4×Q8×C31—21no—
20C31×#544—22no—
21C31×#544C2212yes—
22C31×#533—21no—
23C31×#533C2211yescharacteristic
24C2×C31×C4⋊Q8—21no—
25C2×C31×C4⋊Q8C2211yescharacteristic
26C2×C31×#124C2211yescharacteristic
27C31×#520C2212yes—
28C4×#41×C31C2211yescharacteristic
29C31×#489—21no—
30C31×#489—21no—
31C31×#488—22no—
32C31×#464—21no—
33C31×#464—21no—
34C31×#463—21no—
35C31×#463—21no—
36C31×#423C2211yescharacteristic
37C2×C31×#104C2212yes—
38C31×#419C2212yes—
39C31×#419C2211yescharacteristic
40C31×#418C2212yes—
41C31×#418C2211yescharacteristic
42C4×M16×C31C2211yescharacteristic
43C31×#397—22no—
44C31×#393—22no—
45C31×#391C2212yes—
46C31×#390C2212yes—
47C31×#376C2211yescharacteristic
48C31×#343C2212yes—
49C31×#343C2212yes—
50C31×#339C2212yes—
51C31×#339C2212yes—
52C31×#339C2212yes—
53C2×#41×C31C2311yescharacteristic
54C22×Q8×C31—22no—
55C22×Q8×C31D412yes—
56C22×Q8×C31C2311yescharacteristic
57C22×D4×C31C2311yescharacteristic
58C31×#137—24no—
59C31×AGL(1,ℤ/8)—24no—
60C2×Q16×C31—21no—
61C2×SD16×C31—21no—
62C2×SD16×C31—21no—
63C2×D8×C31—21no—
64C2×M16×C31C2312yes—
65C2×M16×C31C2311yescharacteristic
66C2×M16×C31C2311yescharacteristic
67C31×C4⋊Q8—22no—
68C31×C4⋊Q8—22no—
69C31×C4⋊Q8—24no—
70C31×C4⋊Q8—22no—
71C31×C4⋊Q8—22no—
72C31×C4⋊Q8—21no—
73C31×C4⋊Q8—21no—
74C31×C4⋊Q8—21no—
75C31×C4⋊Q8—21no—
76C31×C4⋊Q8D411yescharacteristic
77C31×C4⋊Q8D411yescharacteristic
78C31×#125—24no—
79C31×#125—22no—
80C31×#125—22no—
81C31×#125D412yes—
82C31×#124—21no—
83C31×#124—21no—
84C31×#124—21no—
85C31×#124—21no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—24no—
89C31×C22⋊Q8—24no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—24no—
93C4×Q8×C31—22no—
94C4×Q8×C31—22no—
95C4×Q8×C31—22no—
96C4×Q8×C31—21no—
97C4×Q8×C31—21no—
98C4×Q8×C31D411yescharacteristic
99C4×Q8×C31D411yescharacteristic
100C4×D4×C31—22no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31—24no—
104C4×D4×C31D411yescharacteristic
105C4×D4×C31D411yescharacteristic
106C31×#117—22no—
107C31×#117—21no—
108C31×#117C2311yescharacteristic
109C31×#117C2312yes—
110C2×C4⋊C4×C31—22no—
111C2×C4⋊C4×C31—22no—
112C2×C4⋊C4×C31D412yes—
113C2×C4⋊C4×C31C2312yes—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—21no—
116C2×C22⋊C4×C31—24no—
117C2×C42×C31—21no—
118C2×C42×C31C2312yes—
119C2×C42×C31C2311yescharacteristic
120C31×C4⋊C8—21no—
121C31×C4⋊C8—21no—
122C31×#104—24no—
123C31×#103—21no—
124C31×#103—21no—
125C31×#103—22no—
126C31×#103—22no—
127C31×#102—21no—
128C31×#102—21no—
129C31×#102—22no—
130C31×#102—22no—
131C31×#97—21no—
132C4×C8×C31—21no—
133C31×#95—21no—
134C31×#95—24no—
135C31×#95—21no—
136C24×C31—21no—
137#41×C31—22no—
138#41×C31#4114yes—
139C2×Q8×C31—24no—
140C2×Q8×C31—21no—
141C2×Q8×C31—22no—
142C2×Q8×C31—21no—
143C2×Q8×C31—22no—
144C2×Q8×C31—21no—
145C2×Q8×C31—21no—
146C2×Q8×C31—22no—
147C2×Q8×C31—22no—
148C2×Q8×C31—82no—
149C2×Q8×C31—84no—
150C2×Q8×C31C2×D411yescharacteristic
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—22no—
155C2×D4×C31—22no—
156C2×D4×C31C2×D411yescharacteristic
157C22×C4×C31—21no—
158C22×C4×C31—24no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—22no—
162C22×C4×C31—21no—
163C22×C4×C31C2×D414yes—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2411yescharacteristic
167Q16×C31—42no—
168SD16×C31—42no—
169SD16×C31—42no—
170D8×C31—42no—
171M16×C31—22no—
172M16×C31—22no—
173M16×C31—22no—
174M16×C31—22no—
175C2×C8×C31—22no—
176C2×C8×C31—21no—
177C2×C8×C31—21no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31C2×D412yes—
193C4⋊C4×C31C2×D412yes—
194C4⋊C4×C31C2×D411yescharacteristic
195C4⋊C4×C31C2×D411yescharacteristic
196C22⋊C4×C31—24no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—21no—
199C22⋊C4×C31—21no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—41no—
202C22⋊C4×C31—41no—
203C22⋊C4×C31—41no—
204C22⋊C4×C31—41no—
205C22⋊C4×C31—84no—
206C42×C31—24no—
207C42×C31—21no—
208C42×C31—21no—
209C42×C31—21no—
210C42×C31—22no—
211C42×C31—22no—
212C42×C31—22no—
213C42×C31C2×D411yescharacteristic
214C42×C31C2×D411yescharacteristic
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217#25926C3111yesmaximal, characteristic, direct product factor, Sylow
218C23×C31—21no—
219C23×C31—42no—
220C23×C31—42no—
221C23×C31—41no—
222C23×C31—42no—
223C23×C31C22×D411yescharacteristic
224Q8×C31—21no—
225Q8×C31—42no—
226Q8×C31—44no—
227Q8×C31—42no—
228Q8×C31—41no—
229Q8×C31—41no—
230Q8×C31—42no—
231Q8×C31—82no—
232Q8×C31—84no—
233Q8×C31#12412yes—
234D4×C31—24no—
235D4×C31—21no—
236D4×C31—41no—
237D4×C31—41no—
238D4×C31—42no—
239D4×C31#12412yes—
240C2×C4×C31—21no—
241C2×C4×C31—24no—
242C2×C4×C31—22no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—24no—
246C2×C4×C31—22no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—44no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—84no—
258C2×C4×C31#12011yescharacteristic
259C2×C4×C31#12011yescharacteristic
260C2×C4×C31#12011yescharacteristic
261C2×C4×C31#12011yescharacteristic
262C2×C4×C31C22×D411yescharacteristic
263C2×C4×C31C22×D411yescharacteristic
264C2×C4×C31C2×#4112yes—
265C2×C4×C312−1+412yes—
266C8×C31—41no—
267C8×C31—41no—
268C8×C31—42no—
269#3503C2×C3111yescharacteristic, complemented
270#3140C2×C3111yescharacteristic, complemented
271#3040C2×C3111yescharacteristic
272#2640C2×C3111yescharacteristic, complemented
273#2028C2×C3112yescomplemented
274#2027C2×C3112yescomplemented
275#1997C2×C3111yescharacteristic, complemented
276#1995C2×C3111yescharacteristic, complemented
277#1883C2×C3112yes—
278#1882C2×C3112yescomplemented
279#1759C2×C3111yescharacteristic, complemented
280C22×C31—21no—
281C22×C31—21no—
282C22×C31—22no—
283C22×C31—41no—
284C22×C31—41no—
285C22×C31—84no—
286C22×C31—81no—
287C22×C31—81no—
288C22×C31—81no—
289C22×C31C2×#12011yescharacteristic
290C22×C31#54812yes—
291C4×C31—22no—
292C4×C31—21no—
293C4×C31—21no—
294C4×C31—21no—
295C4×C31—21no—
296C4×C31—41no—
297C4×C31—44no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—82no—
302C4×C31—81no—
303C4×C31C2×#12411yescharacteristic
304C4×C31#53611yescharacteristic
305C4×C31#54812yes—
306C2×#137C22×C3111yescharacteristic
307C2×AGL(1,ℤ/8)C22×C3111yescharacteristic
308Q82—21no—
309#557—21no—
310#554—21no—
311#554—22no—
312D4×Q8—22no—
313D4×Q8—21no—
314#544—22no—
315#544C22×C3112yes—
316#533—21no—
317#533C22×C3111yescharacteristic
318C2×C4⋊Q8—21no—
319C2×C4⋊Q8C22×C3111yescharacteristic
320C2×#124C22×C3111yescharacteristic
321#520C22×C3112yes—
322C4×#41C22×C3111yescharacteristic
323#489—21no—
324#489—21no—
325#488—22no—
326#464—21no—
327#464—21no—
328#463—21no—
329#463—21no—
330#423C22×C3111yescharacteristic
331C2×#104C22×C3112yes—
332#419C22×C3112yes—
333#419C22×C3111yescharacteristic
334#418C22×C3112yes—
335#418C22×C3111yescharacteristic
336C4×M16C22×C3111yescharacteristic
337#397—22no—
338#393—22no—
339#391C22×C3112yes—
340#390C22×C3112yes—
341#376C22×C3111yescharacteristic
342#343C22×C3112yes—
343#343C22×C3112yes—
344#339C22×C3112yes—
345#339C22×C3112yes—
346#339C22×C3112yes—
347C2×C31—22no—
348C2×C31—42no—
349C2×C31—161no—
350C2×C31#243611yescharacteristic
351C2×C31#301911yescharacteristic
352C2×C31#302011yescharacteristic
353C2×#41C23×C3111yescharacteristic
354C22×Q8—22no—
355C22×Q8D4×C3112yes—
356C22×Q8C23×C3111yescharacteristic
357C22×D4C23×C3111yescharacteristic
358#137—24no—
359AGL(1,ℤ/8)—24no—
360C2×Q16—21no—
361C2×SD16—21no—
362C2×SD16—21no—
363C2×D8—21no—
364C2×M16C23×C3111yescharacteristic
365C2×M16C23×C3111yescharacteristic
366C2×M16C23×C3112yes—
367C4⋊Q8—22no—
368C4⋊Q8—22no—
369C4⋊Q8—24no—
370C4⋊Q8—22no—
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8D4×C3111yescharacteristic
377C4⋊Q8D4×C3111yescharacteristic
378#125—24no—
379#125—22no—
380#125—22no—
381#125D4×C3112yes—
382#124—21no—
383#124—21no—
384#124—21no—
385#124—21no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—24no—
389C22⋊Q8—24no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—24no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—22no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8D4×C3111yescharacteristic
399C4×Q8D4×C3111yescharacteristic
400C4×D4—22no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—24no—
404C4×D4D4×C3111yescharacteristic
405C4×D4D4×C3111yescharacteristic
406#117—22no—
407#117—21no—
408#117C23×C3111yescharacteristic
409#117C23×C3112yes—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4D4×C3112yes—
413C2×C4⋊C4C23×C3112yes—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—24no—
417C2×C42—21no—
418C2×C42C23×C3112yes—
419C2×C42C23×C3111yescharacteristic
420C4⋊C8—21no—
421C4⋊C8—21no—
422#104—24no—
423#103—21no—
424#103—21no—
425#103—22no—
426#103—22no—
427#102—21no—
428#102—21no—
429#102—22no—
430#102—22no—
431#97—21no—
432C4×C8—21no—
433#95—21no—
434#95—21no—
435#95—24no—
436C31#2592611yescharacteristic, direct product factor, Sylow
437C24—21no—
438#41—22no—
439#41#41×C3114yes—
440C2×Q8—24no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—82no—
450C2×Q8—84no—
451C2×Q8C2×D4×C3111yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4C2×D4×C3111yescharacteristic
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4C2×D4×C3114yes—
465C22×C4C2×D4×C3111yescharacteristic
466C22×C4C2×D4×C3111yescharacteristic
467C22×C4C24×C3111yescharacteristic
468Q16—42no—
469SD16—42no—
470SD16—42no—
471D8—42no—
472M16—22no—
473M16—22no—
474M16—22no—
475M16—22no—
476C2×C8—22no—
477C2×C8—21no—
478C2×C8—21no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4C2×D4×C3112yes—
494C4⋊C4C2×D4×C3112yes—
495C4⋊C4C2×D4×C3111yescharacteristic
496C4⋊C4C2×D4×C3111yescharacteristic
497C22⋊C4—24no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—84no—
507C42—24no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42C2×D4×C3111yescharacteristic
515C42C2×D4×C3111yescharacteristic
516C42C2×D4×C3111yescharacteristic
517C42C2×D4×C3111yescharacteristic
518C23—21no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23C22×D4×C3111yescharacteristic
524Q8—21no—
525Q8—42no—
526Q8—44no—
527Q8—42no—
528Q8—41no—
529Q8—41no—
530Q8—42no—
531Q8—82no—
532Q8—84no—
533Q8C31×#12412yes—
534D4—24no—
535D4—21no—
536D4—41no—
537D4—41no—
538D4—42no—
539D4C31×#12412yes—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4C31×#12011yescharacteristic
559C2×C4C31×#12011yescharacteristic
560C2×C4C31×#12011yescharacteristic
561C2×C4C31×#12011yescharacteristic
562C2×C4C22×D4×C3111yescharacteristic
563C2×C4C22×D4×C3111yescharacteristic
564C2×C4C2×#41×C3112yes—
565C2×C4C31×2−1+412yes—
566C8—41no—
567C8—41no—
568C8—42no—
569C22—21no—
570C22—21no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—84no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22C2×C31×#12011yescharacteristic
579C22C31×#54812yes—
580C4—22no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—44no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—82no—
591C4—81no—
592C4C2×C31×#12411yescharacteristic
593C4C31×#53611yescharacteristic
594C4C31×#54812yes—
595C2—22no—
596C2—42no—
597C2—161no—
598C2C31×#243611yescharacteristic
599C2C31×#301911yescharacteristic
600C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×AGL(1,ℤ/8)C22
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×#3503C2semidirect
2. C31×#3140C2semidirect
4. C31×#2640C2semidirect
5. C31×#2028C2semidirect
6. C31×#2027C2semidirect
7. C31×#1997C2semidirect
8. C31×#1995C2semidirect
10. C31×#1882C2semidirect
11. C31×#1759C2semidirect
217. #25926C31direct (N × H)
269. #3503C2×C31semidirect
270. #3140C2×C31semidirect
272. #2640C2×C31semidirect
273. #2028C2×C31semidirect
274. #2027C2×C31semidirect
275. #1997C2×C31semidirect
276. #1995C2×C31semidirect
278. #1882C2×C31semidirect
279. #1759C2×C31semidirect
436. C31#25926direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#31402
C31×#30402
C31×#26402
C31×#20282
C31×#20272
C31×#19972
C31×#19952
C31×#18832
C31×#18822
C31×#17592
#2592631