Finite Group Databasebeta

← C31×#24320 #21481079

Subgroups — C31×#24320 #21481079

Order 7936.

1092 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic, complemented
2C31×#3077C211yesmaximal, characteristic, complemented
3C4×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic
4C31×#2542C211yesmaximal, characteristic, complemented
5C31×#2060C212yesmaximal, complemented
6C31×#2027C212yesmaximal, complemented
7C31×#1997C211yesmaximal, characteristic, complemented
8C31×#1986C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal
10C31×#1859C212yesmaximal, complemented
11C31×#1806C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)C2211yescharacteristic
13Q82×C31—21no—
14C31×#557—21no—
15C31×#554—21no—
16C31×#554—22no—
17D4×Q8×C31—22no—
18D4×Q8×C31—21no—
19C31×#544—22no—
20C31×#544C2212yes—
21C31×#533—21no—
22C31×#533C2211yescharacteristic
23C2×C31×C4⋊Q8—21no—
24C2×C31×C4⋊Q8C2211yescharacteristic
25C2×C31×C22⋊Q8—22no—
26C31×#520C2212yes—
27C4×#41×C31C2211yescharacteristic
28C2×C4×D4×C31C2211yescharacteristic
29C31×#478—22no—
30C31×#476—21no—
31C31×#474—21no—
32C31×#442—21no—
33C31×#440—21no—
34C4×SD16×C31—21no—
35C4×D8×C31—21no—
36C31×#428C2212yes—
37C31×#428C2211yescharacteristic
38C31×#423C2211yescharacteristic
39C2×C31×#104C2212yes—
40C31×#418C2212yes—
41C31×#418C2211yescharacteristic
42C31×#417C2211yescharacteristic
43C4×M16×C31C2211yescharacteristic
44C31×#399—22no—
45C31×#390C2212yes—
46C31×#384C2212yes—
47C4×C4⋊C4×C31C2211yescharacteristic
48C31×#343C2212yes—
49C31×#343C2212yes—
50C31×#339C2212yes—
51C31×#339C2212yes—
52C31×#339C2212yes—
53C2×#41×C31C2311yescharacteristic
54C22×Q8×C31—22no—
55C22×Q8×C31D412yes—
56C22×D4×C31C2311yescharacteristic
57C23×C4×C31—21no—
58C31×AGL(1,ℤ/8)—22no—
59C31×AGL(1,ℤ/8)Q814yescomplemented
60C2×SD16×C31—21no—
61C2×D8×C31—21no—
62C2×M16×C31C2311yescharacteristic
63C2×M16×C31C2312yes—
64C2×M16×C31C2311yescharacteristic
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8—22no—
67C31×C4⋊Q8—24no—
68C31×C4⋊Q8—22no—
69C31×C4⋊Q8—22no—
70C31×C4⋊Q8—21no—
71C31×C4⋊Q8—21no—
72C31×C4⋊Q8—21no—
73C31×C4⋊Q8—21no—
74C31×C4⋊Q8D411yescharacteristic
75C31×C4⋊Q8D411yescharacteristic
76C31×#125—24no—
77C31×#125—22no—
78C31×#125—22no—
79C31×#125D412yes—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—24no—
83C31×C22⋊Q8—24no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8—24no—
87C31×C22⋊Q8—48no—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×Q8×C31D411yescharacteristic
94C4×Q8×C31D411yescharacteristic
95C4×D4×C31—22no—
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—24no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31D411yescharacteristic
104C4×D4×C31D411yescharacteristic
105C31×#117—22no—
106C31×#117—21no—
107C31×#117C2311yescharacteristic
108C31×#117C2312yes—
109C2×C4⋊C4×C31—22no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31—22no—
112C2×C4⋊C4×C31—24no—
113C2×C4⋊C4×C31D412yes—
114C2×C4⋊C4×C31C2311yescharacteristic
115C2×C4⋊C4×C31C2312yes—
116C2×C22⋊C4×C31—21no—
117C2×C22⋊C4×C31—24no—
118C2×C42×C31—21no—
119C2×C42×C31C2312yes—
120C2×C42×C31C2311yescharacteristic
121C31×#107—21no—
122C31×#107—22no—
123C31×#106—21no—
124C31×#106—22no—
125C31×C4⋊C8—21no—
126C31×C4⋊C8—21no—
127C31×#104—24no—
128C31×#103—21no—
129C31×#102—21no—
130C31×#102—21no—
131C31×#102—21no—
132C31×#102—22no—
133C31×#102—22no—
134C31×#97—21no—
135C4×C8×C31—21no—
136C31×#95—21no—
137C24×C31—21no—
138#41×C31—21no—
139#41×C31—21no—
140#41×C31C2×Q812yes—
141#41×C31#4112yes—
142C2×Q8×C31—24no—
143C2×Q8×C31—21no—
144C2×Q8×C31—22no—
145C2×Q8×C31—21no—
146C2×Q8×C31—22no—
147C2×Q8×C31—44no—
148C2×Q8×C31—84no—
149C2×Q8×C31C2×D411yescharacteristic
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31C2×D411yescharacteristic
157C2×D4×C31C2×Q812yescomplemented
158C22×C4×C31—21no—
159C22×C4×C31—22no—
160C22×C4×C31—24no—
161C22×C4×C31—21no—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31—44no—
165C22×C4×C31—41no—
166C22×C4×C31—41no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D414yes—
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171SD16×C31—24no—
172D8×C31—24no—
173M16×C31—21no—
174M16×C31—22no—
175M16×C31—21no—
176M16×C31—21no—
177M16×C31—22no—
178M16×C31C2×Q812yescomplemented
179C2×C8×C31—21no—
180C2×C8×C31—21no—
181C2×C8×C31—22no—
182C4⋊C4×C31—24no—
183C4⋊C4×C31—24no—
184C4⋊C4×C31—24no—
185C4⋊C4×C31—24no—
186C4⋊C4×C31—24no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—48no—
200C4⋊C4×C31C2×D412yes—
201C4⋊C4×C31C2×D411yescharacteristic
202C4⋊C4×C31C2×D412yes—
203C4⋊C4×C31C2×D411yescharacteristic
204C22⋊C4×C31—24no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—21no—
207C22⋊C4×C31—21no—
208C22⋊C4×C31—21no—
209C22⋊C4×C31—48no—
210C22⋊C4×C31—41no—
211C22⋊C4×C31—41no—
212C42×C31—24no—
213C42×C31—21no—
214C42×C31—21no—
215C42×C31—21no—
216C42×C31—22no—
217C42×C31—22no—
218C42×C31—22no—
219C42×C31C2×D411yescharacteristic
220C42×C31C2×D411yescharacteristic
221C42×C31C2×D411yescharacteristic
222C42×C31C2×D411yescharacteristic
223#24320C3111yesmaximal, characteristic, direct product factor, Sylow
224C23×C31—21no—
225C23×C31—22no—
226C23×C31—42no—
227C23×C31—42no—
228C23×C31—41no—
229C23×C31—41no—
230C23×C31C22×D411yescharacteristic
231Q8×C31—21no—
232Q8×C31—42no—
233Q8×C31—44no—
234Q8×C31—42no—
235Q8×C31—82no—
236Q8×C31—84no—
237Q8×C31C22⋊Q812yescomplemented
238D4×C31—22no—
239D4×C31—22no—
240D4×C31—22no—
241D4×C31—21no—
242D4×C31—22no—
243D4×C31—41no—
244D4×C31—41no—
245D4×C31C22⋊Q812yescomplemented
246C2×C4×C31—21no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—24no—
251C2×C4×C31—22no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—24no—
255C2×C4×C31—22no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—44no—
262C2×C4×C31—42no—
263C2×C4×C31—41no—
264C2×C4×C31—82no—
265C2×C4×C31—84no—
266C2×C4×C31—82no—
267C2×C4×C31#12011yescharacteristic
268C2×C4×C31#12011yescharacteristic
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31C22×D411yescharacteristic
272C2×C4×C31C22×D411yescharacteristic
273C2×C4×C31C22×Q811yescharacteristic
274C2×C4×C31C2×#4111yescharacteristic
275C2×C4×C312−1+411yescharacteristic
276C2×C4×C312−1+411yescharacteristic
277C8×C31—22no—
278C8×C31—41no—
279C8×C31—42no—
280#3503C2×C3111yescharacteristic, complemented
281#3077C2×C3111yescharacteristic, complemented
282C4×AGL(1,ℤ/8)C2×C3111yescharacteristic
283#2542C2×C3111yescharacteristic, complemented
284#2060C2×C3112yescomplemented
285#2027C2×C3112yescomplemented
286#1997C2×C3111yescharacteristic, complemented
287#1986C2×C3111yescharacteristic, complemented
288#1883C2×C3112yes—
289#1859C2×C3112yescomplemented
290#1806C2×C3111yescharacteristic, complemented
291C22×C31—21no—
292C22×C31—21no—
293C22×C31—22no—
294C22×C31—44no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—42no—
298C22×C31—82no—
299C22×C31—81no—
300C22×C31—81no—
301C22×C31C2×#12011yescharacteristic
302C22×C31#54811yescharacteristic
303C22×C31D4×Q811yescharacteristic
304C4×C31—22no—
305C4×C31—21no—
306C4×C31—21no—
307C4×C31—21no—
308C4×C31—44no—
309C4×C31—41no—
310C4×C31—44no—
311C4×C31—42no—
312C4×C31—41no—
313C4×C31—42no—
314C4×C31—81no—
315C4×C31#45312yes—
316C4×C31C2×C22⋊Q811yescharacteristic
317C4×C31#53611yescharacteristic
318C4×C31#54811yescharacteristic
319C4×C31D4×Q811yescharacteristic
320C2×AGL(1,ℤ/8)C22×C3111yescharacteristic
321Q82—21no—
322#557—21no—
323#554—21no—
324#554—22no—
325D4×Q8—22no—
326D4×Q8—21no—
327#544—22no—
328#544C22×C3112yes—
329#533—21no—
330#533C22×C3111yescharacteristic
331C2×C4⋊Q8—21no—
332C2×C4⋊Q8C22×C3111yescharacteristic
333C2×C22⋊Q8—22no—
334#520C22×C3112yes—
335C4×#41C22×C3111yescharacteristic
336C2×C4×D4C22×C3111yescharacteristic
337#478—22no—
338#476—21no—
339#474—21no—
340#442—21no—
341#440—21no—
342C4×SD16—21no—
343C4×D8—21no—
344#428C22×C3112yes—
345#428C22×C3111yescharacteristic
346#423C22×C3111yescharacteristic
347C2×#104C22×C3112yes—
348#418C22×C3112yes—
349#418C22×C3111yescharacteristic
350#417C22×C3111yescharacteristic
351C4×M16C22×C3111yescharacteristic
352#399—22no—
353#390C22×C3112yes—
354#384C22×C3112yes—
355C4×C4⋊C4C22×C3111yescharacteristic
356#343C22×C3112yes—
357#343C22×C3112yes—
358#339C22×C3112yes—
359#339C22×C3112yes—
360#339C22×C3112yes—
361C2×C31—21no—
362C2×C31—21no—
363C2×C31—42no—
364C2×C31—82no—
365C2×C31#240911yescharacteristic
366C2×C31C2×#45311yescharacteristic
367C2×C31#302011yescharacteristic
368C2×#41C23×C3111yescharacteristic
369C22×Q8—22no—
370C22×Q8D4×C3112yes—
371C22×D4C23×C3111yescharacteristic
372C23×C4—21no—
373AGL(1,ℤ/8)—22no—
374AGL(1,ℤ/8)Q8×C3114yescomplemented
375C2×SD16—21no—
376C2×D8—21no—
377C2×M16C23×C3111yescharacteristic
378C2×M16C23×C3112yes—
379C2×M16C23×C3111yescharacteristic
380C4⋊Q8—22no—
381C4⋊Q8—22no—
382C4⋊Q8—24no—
383C4⋊Q8—22no—
384C4⋊Q8—22no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8D4×C3111yescharacteristic
390C4⋊Q8D4×C3111yescharacteristic
391#125—24no—
392#125—22no—
393#125—22no—
394#125D4×C3112yes—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—24no—
398C22⋊Q8—24no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—24no—
402C22⋊Q8—48no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8D4×C3111yescharacteristic
409C4×Q8D4×C3111yescharacteristic
410C4×D4—22no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—24no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4D4×C3111yescharacteristic
419C4×D4D4×C3111yescharacteristic
420#117—22no—
421#117—21no—
422#117C23×C3111yescharacteristic
423#117C23×C3112yes—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—24no—
428C2×C4⋊C4D4×C3112yes—
429C2×C4⋊C4C23×C3111yescharacteristic
430C2×C4⋊C4C23×C3112yes—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—24no—
433C2×C42—21no—
434C2×C42C23×C3112yes—
435C2×C42C23×C3111yescharacteristic
436#107—21no—
437#107—22no—
438#106—21no—
439#106—22no—
440C4⋊C8—21no—
441C4⋊C8—21no—
442#104—24no—
443#103—21no—
444#102—21no—
445#102—21no—
446#102—21no—
447#102—22no—
448#102—22no—
449#97—21no—
450C4×C8—21no—
451#95—21no—
452C31#2432011yescharacteristic, direct product factor, Sylow
453C24—21no—
454#41—21no—
455#41—21no—
456#41C2×Q8×C3112yes—
457#41#41×C3112yes—
458C2×Q8—24no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—44no—
464C2×Q8—84no—
465C2×Q8C2×D4×C3111yescharacteristic
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×D4×C3111yescharacteristic
473C2×D4C2×Q8×C3112yescomplemented
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—44no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4C2×D4×C3111yescharacteristic
484C22×C4C2×D4×C3114yes—
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C24×C3111yescharacteristic
487SD16—24no—
488D8—24no—
489M16—21no—
490M16—22no—
491M16—21no—
492M16—21no—
493M16—22no—
494M16C2×Q8×C3112yescomplemented
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—48no—
516C4⋊C4C2×D4×C3112yes—
517C4⋊C4C2×D4×C3111yescharacteristic
518C4⋊C4C2×D4×C3112yes—
519C4⋊C4C2×D4×C3111yescharacteristic
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—48no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42C2×D4×C3111yescharacteristic
536C42C2×D4×C3111yescharacteristic
537C42C2×D4×C3111yescharacteristic
538C42C2×D4×C3111yescharacteristic
539C23—21no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23C22×D4×C3111yescharacteristic
546Q8—21no—
547Q8—42no—
548Q8—44no—
549Q8—42no—
550Q8—82no—
551Q8—84no—
552Q8C31×C22⋊Q812yescomplemented
553D4—22no—
554D4—22no—
555D4—22no—
556D4—21no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4C31×C22⋊Q812yescomplemented
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4—82no—
582C2×C4C31×#12011yescharacteristic
583C2×C4C31×#12011yescharacteristic
584C2×C4C31×#12011yescharacteristic
585C2×C4C31×#12011yescharacteristic
586C2×C4C22×D4×C3111yescharacteristic
587C2×C4C22×D4×C3111yescharacteristic
588C2×C4C22×Q8×C3111yescharacteristic
589C2×C4C2×#41×C3111yescharacteristic
590C2×C4C31×2−1+411yescharacteristic
591C2×C4C31×2−1+411yescharacteristic
592C8—22no—
593C8—41no—
594C8—42no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—44no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22C2×C31×#12011yescharacteristic
606C22C31×#54811yescharacteristic
607C22D4×Q8×C3111yescharacteristic
608C4—22no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—44no—
613C4—41no—
614C4—44no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4C31×#45312yes—
620C4C2×C31×C22⋊Q811yescharacteristic
621C4C31×#53611yescharacteristic
622C4C31×#54811yescharacteristic
623C4D4×Q8×C3111yescharacteristic
624C2—21no—
625C2—21no—
626C2—42no—
627C2—82no—
628C2C31×#240911yescharacteristic
629C2C2×C31×#45311yescharacteristic
630C2C31×#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×#3077C2semidirect
4. C31×#2542C2semidirect
5. C31×#2060C2semidirect
6. C31×#2027C2semidirect
7. C31×#1997C2semidirect
8. C31×#1986C2semidirect
10. C31×#1859C2semidirect
11. C31×#1806C2semidirect
59. C31×AGL(1,ℤ/8)Q8semidirect
157. C2×D4×C31C2×Q8semidirect
178. M16×C31C2×Q8semidirect
223. #24320C31direct (N × H)
237. Q8×C31C22⋊Q8semidirect
245. D4×C31C22⋊Q8semidirect
280. #3503C2×C31semidirect
281. #3077C2×C31semidirect
283. #2542C2×C31semidirect
284. #2060C2×C31semidirect
285. #2027C2×C31semidirect
286. #1997C2×C31semidirect
287. #1986C2×C31semidirect
289. #1859C2×C31semidirect
290. #1806C2×C31semidirect
374. AGL(1,ℤ/8)Q8×C31semidirect
452. C31#24320direct (N × H)
473. C2×D4C2×Q8×C31semidirect
494. M16C2×Q8×C31semidirect
552. Q8C31×C22⋊Q8semidirect
560. D4C31×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#30772
C4×C31×AGL(1,ℤ/8)2
C31×#25422
C31×#20602
C31×#20272
C31×#19972
C31×#19862
C31×#18832
C31×#18592
C31×#18062
#2432031