Finite Group Databasebeta

← C31×#24321 #21481080

Subgroups — C31×#24321 #21481080

Order 7936.

1036 conjugacy classes of subgroups.

The subgroup lattice has 628 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×#2563C211yesmaximal, characteristic, complemented
5C31×#2061C212yesmaximal, complemented
6C31×#2027C212yesmaximal, complemented
7C31×#1997C211yesmaximal, characteristic, complemented
8C31×#1986C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal
10C31×#1860C212yesmaximal, 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
25C31×#520C2212yes—
26C4×#41×C31C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C31×#478—22no—
29C31×#476—21no—
30C31×#474—21no—
31C31×#442—21no—
32C31×#440—21no—
33C4×SD16×C31—21no—
34C4×D8×C31—21no—
35C31×#428C2212yes—
36C31×#428C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C2×C31×#104C2212yes—
39C31×#418C2212yes—
40C31×#418C2211yescharacteristic
41C31×#417C2211yescharacteristic
42C4×M16×C31C2211yescharacteristic
43C31×#393—22no—
44C31×#390C2212yes—
45C31×#382C2212yes—
46C31×#380—22no—
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—
87C4×Q8×C31—22no—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×Q8×C31—21no—
91C4×Q8×C31—21no—
92C4×Q8×C31D411yescharacteristic
93C4×Q8×C31D411yescharacteristic
94C4×D4×C31—22no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C4×D4×C31—24no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31D411yescharacteristic
103C4×D4×C31D411yescharacteristic
104C31×#117—22no—
105C31×#117—21no—
106C31×#117C2311yescharacteristic
107C31×#117C2312yes—
108C2×C4⋊C4×C31—22no—
109C2×C4⋊C4×C31—22no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31D412yes—
112C2×C4⋊C4×C31C2311yescharacteristic
113C2×C4⋊C4×C31C2312yes—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31—24no—
116C2×C42×C31—21no—
117C2×C42×C31C2312yes—
118C2×C42×C31C2311yescharacteristic
119C31×#107—21no—
120C31×#107—22no—
121C31×#106—21no—
122C31×#106—22no—
123C31×C4⋊C8—21no—
124C31×C4⋊C8—21no—
125C31×#104—24no—
126C31×#103—21no—
127C31×#102—21no—
128C31×#102—21no—
129C31×#102—21no—
130C31×#102—22no—
131C31×#102—22no—
132C31×#97—21no—
133C4×C8×C31—21no—
134C31×#95—21no—
135C31×#95—24no—
136C24×C31—21no—
137#41×C31—21no—
138#41×C31—21no—
139#41×C31C2×Q812yes—
140#41×C31#4112yes—
141C2×Q8×C31—24no—
142C2×Q8×C31—21no—
143C2×Q8×C31—22no—
144C2×Q8×C31—21no—
145C2×Q8×C31—22no—
146C2×Q8×C31—82no—
147C2×Q8×C31—84no—
148C2×Q8×C31C2×D411yescharacteristic
149C2×D4×C31—21no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31C2×D411yescharacteristic
156C2×D4×C31C2×Q812yescomplemented
157C22×C4×C31—21no—
158C22×C4×C31—24no—
159C22×C4×C31—21no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—22no—
163C22×C4×C31—44no—
164C22×C4×C31—41no—
165C22×C4×C31—41no—
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D414yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2411yescharacteristic
170SD16×C31—24no—
171D8×C31—24no—
172M16×C31—21no—
173M16×C31—22no—
174M16×C31—21no—
175M16×C31—21no—
176M16×C31—22no—
177M16×C31C2×Q812yescomplemented
178C2×C8×C31—21no—
179C2×C8×C31—21no—
180C2×C8×C31—22no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—24no—
183C4⋊C4×C31—24no—
184C4⋊C4×C31—24no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—42no—
198C4⋊C4×C31C2×D412yes—
199C4⋊C4×C31C2×D411yescharacteristic
200C4⋊C4×C31C2×D412yes—
201C4⋊C4×C31C2×D411yescharacteristic
202C22⋊C4×C31—24no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—21no—
205C22⋊C4×C31—21no—
206C22⋊C4×C31—21no—
207C22⋊C4×C31—41no—
208C22⋊C4×C31—41no—
209C22⋊C4×C31—84no—
210C42×C31—24no—
211C42×C31—21no—
212C42×C31—21no—
213C42×C31—21no—
214C42×C31—22no—
215C42×C31—22no—
216C42×C31—22no—
217C42×C31C2×D411yescharacteristic
218C42×C31C2×D411yescharacteristic
219C42×C31C2×D411yescharacteristic
220C42×C31C2×D411yescharacteristic
221#24321C3111yesmaximal, characteristic, direct product factor, Sylow
222C23×C31—21no—
223C23×C31—22no—
224C23×C31—42no—
225C23×C31—42no—
226C23×C31—41no—
227C23×C31—41no—
228C23×C31C22×D411yescharacteristic
229Q8×C31—21no—
230Q8×C31—44no—
231Q8×C31—42no—
232Q8×C31—42no—
233Q8×C31—82no—
234Q8×C31—84no—
235Q8×C31C22⋊Q812yes—
236D4×C31—22no—
237D4×C31—22no—
238D4×C31—22no—
239D4×C31—21no—
240D4×C31—22no—
241D4×C31—41no—
242D4×C31—41no—
243D4×C31C22⋊Q812yes—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—21no—
251C2×C4×C31—24no—
252C2×C4×C31—22no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—44no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—41no—
261C2×C4×C31—42no—
262C2×C4×C31—82no—
263C2×C4×C31—84no—
264C2×C4×C31—82no—
265C2×C4×C31#12011yescharacteristic
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31#12011yescharacteristic
268C2×C4×C31#12011yescharacteristic
269C2×C4×C31C22×D411yescharacteristic
270C2×C4×C31C22×D411yescharacteristic
271C2×C4×C31C22×Q811yescharacteristic
272C2×C4×C31C2×#4111yescharacteristic
273C2×C4×C312−1+411yescharacteristic
274C2×C4×C312−1+411yescharacteristic
275C8×C31—22no—
276C8×C31—41no—
277C8×C31—42no—
278#3503C2×C3111yescharacteristic, complemented
279#3077C2×C3111yescharacteristic, complemented
280C4×AGL(1,ℤ/8)C2×C3111yescharacteristic
281#2563C2×C3111yescharacteristic, complemented
282#2061C2×C3112yescomplemented
283#2027C2×C3112yescomplemented
284#1997C2×C3111yescharacteristic, complemented
285#1986C2×C3111yescharacteristic, complemented
286#1883C2×C3112yes—
287#1860C2×C3112yescomplemented
288#1806C2×C3111yescharacteristic, complemented
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—22no—
292C22×C31—44no—
293C22×C31—41no—
294C22×C31—41no—
295C22×C31—42no—
296C22×C31—82no—
297C22×C31—81no—
298C22×C31—81no—
299C22×C31C2×#12011yescharacteristic
300C22×C31#54811yescharacteristic
301C22×C31D4×Q811yescharacteristic
302C4×C31—22no—
303C4×C31—21no—
304C4×C31—21no—
305C4×C31—21no—
306C4×C31—41no—
307C4×C31—42no—
308C4×C31—41no—
309C4×C31—44no—
310C4×C31—42no—
311C4×C31—82no—
312C4×C31—81no—
313C4×C31#45412yes—
314C4×C31C2×C22⋊Q811yescharacteristic
315C4×C31#53611yescharacteristic
316C4×C31#54811yescharacteristic
317C4×C31D4×Q811yescharacteristic
318C2×AGL(1,ℤ/8)C22×C3111yescharacteristic
319Q82—21no—
320#557—21no—
321#554—21no—
322#554—22no—
323D4×Q8—22no—
324D4×Q8—21no—
325#544—22no—
326#544C22×C3112yes—
327#533—21no—
328#533C22×C3111yescharacteristic
329C2×C4⋊Q8—21no—
330C2×C4⋊Q8C22×C3111yescharacteristic
331#520C22×C3112yes—
332C4×#41C22×C3111yescharacteristic
333C2×C4×D4C22×C3111yescharacteristic
334#478—22no—
335#476—21no—
336#474—21no—
337#442—21no—
338#440—21no—
339C4×SD16—21no—
340C4×D8—21no—
341#428C22×C3112yes—
342#428C22×C3111yescharacteristic
343#423C22×C3111yescharacteristic
344C2×#104C22×C3112yes—
345#418C22×C3112yes—
346#418C22×C3111yescharacteristic
347#417C22×C3111yescharacteristic
348C4×M16C22×C3111yescharacteristic
349#393—22no—
350#390C22×C3112yes—
351#382C22×C3112yes—
352#380—22no—
353C4×C4⋊C4C22×C3111yescharacteristic
354#343C22×C3112yes—
355#343C22×C3112yes—
356#339C22×C3112yes—
357#339C22×C3112yes—
358#339C22×C3112yes—
359C2×C31—21no—
360C2×C31—21no—
361C2×C31—42no—
362C2×C31—82no—
363C2×C31#240911yescharacteristic
364C2×C31C2×#45411yescharacteristic
365C2×C31#301911yescharacteristic
366C2×#41C23×C3111yescharacteristic
367C22×Q8—22no—
368C22×Q8D4×C3112yes—
369C22×D4C23×C3111yescharacteristic
370C23×C4—21no—
371AGL(1,ℤ/8)—22no—
372AGL(1,ℤ/8)Q8×C3114yescomplemented
373C2×SD16—21no—
374C2×D8—21no—
375C2×M16C23×C3111yescharacteristic
376C2×M16C23×C3112yes—
377C2×M16C23×C3111yescharacteristic
378C4⋊Q8—22no—
379C4⋊Q8—22no—
380C4⋊Q8—24no—
381C4⋊Q8—22no—
382C4⋊Q8—22no—
383C4⋊Q8—21no—
384C4⋊Q8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8D4×C3111yescharacteristic
388C4⋊Q8D4×C3111yescharacteristic
389#125—24no—
390#125—22no—
391#125—22no—
392#125D4×C3112yes—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—24no—
396C22⋊Q8—24no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—24no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8D4×C3111yescharacteristic
406C4×Q8D4×C3111yescharacteristic
407C4×D4—22no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—24no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4D4×C3111yescharacteristic
416C4×D4D4×C3111yescharacteristic
417#117—22no—
418#117—21no—
419#117C23×C3111yescharacteristic
420#117C23×C3112yes—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4D4×C3112yes—
425C2×C4⋊C4C23×C3111yescharacteristic
426C2×C4⋊C4C23×C3112yes—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—24no—
429C2×C42—21no—
430C2×C42C23×C3112yes—
431C2×C42C23×C3111yescharacteristic
432#107—21no—
433#107—22no—
434#106—21no—
435#106—22no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438#104—24no—
439#103—21no—
440#102—21no—
441#102—21no—
442#102—21no—
443#102—22no—
444#102—22no—
445#97—21no—
446C4×C8—21no—
447#95—21no—
448#95—24no—
449C31#2432111yescharacteristic, direct product factor, Sylow
450C24—21no—
451#41—21no—
452#41—21no—
453#41C2×Q8×C3112yes—
454#41#41×C3112yes—
455C2×Q8—24no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—82no—
461C2×Q8—84no—
462C2×Q8C2×D4×C3111yescharacteristic
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4C2×D4×C3111yescharacteristic
470C2×D4C2×Q8×C3112yescomplemented
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—44no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4C2×D4×C3111yescharacteristic
481C22×C4C2×D4×C3114yes—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C24×C3111yescharacteristic
484SD16—24no—
485D8—24no—
486M16—21no—
487M16—22no—
488M16—21no—
489M16—21no—
490M16—22no—
491M16C2×Q8×C3112yescomplemented
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4C2×D4×C3112yes—
513C4⋊C4C2×D4×C3111yescharacteristic
514C4⋊C4C2×D4×C3112yes—
515C4⋊C4C2×D4×C3111yescharacteristic
516C22⋊C4—24no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—84no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42C2×D4×C3111yescharacteristic
532C42C2×D4×C3111yescharacteristic
533C42C2×D4×C3111yescharacteristic
534C42C2×D4×C3111yescharacteristic
535C23—21no—
536C23—22no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23C22×D4×C3111yescharacteristic
542Q8—21no—
543Q8—44no—
544Q8—42no—
545Q8—42no—
546Q8—82no—
547Q8—84no—
548Q8C31×C22⋊Q812yes—
549D4—22no—
550D4—22no—
551D4—22no—
552D4—21no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4C31×C22⋊Q812yes—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—82no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4C31×#12011yescharacteristic
579C2×C4C31×#12011yescharacteristic
580C2×C4C31×#12011yescharacteristic
581C2×C4C31×#12011yescharacteristic
582C2×C4C22×D4×C3111yescharacteristic
583C2×C4C22×D4×C3111yescharacteristic
584C2×C4C22×Q8×C3111yescharacteristic
585C2×C4C2×#41×C3111yescharacteristic
586C2×C4C31×2−1+411yescharacteristic
587C2×C4C31×2−1+411yescharacteristic
588C8—22no—
589C8—41no—
590C8—42no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—44no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—82no—
599C22—81no—
600C22—81no—
601C22C2×C31×#12011yescharacteristic
602C22C31×#54811yescharacteristic
603C22D4×Q8×C3111yescharacteristic
604C4—22no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—81no—
615C4C31×#45412yes—
616C4C2×C31×C22⋊Q811yescharacteristic
617C4C31×#53611yescharacteristic
618C4C31×#54811yescharacteristic
619C4D4×Q8×C3111yescharacteristic
620C2—21no—
621C2—21no—
622C2—42no—
623C2—82no—
624C2C31×#240911yescharacteristic
625C2C2×C31×#45411yescharacteristic
626C2C31×#301911yescharacteristic
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×#2563C2semidirect
5. C31×#2061C2semidirect
6. C31×#2027C2semidirect
7. C31×#1997C2semidirect
8. C31×#1986C2semidirect
10. C31×#1860C2semidirect
11. C31×#1806C2semidirect
59. C31×AGL(1,ℤ/8)Q8semidirect
156. C2×D4×C31C2×Q8semidirect
177. M16×C31C2×Q8semidirect
221. #24321C31direct (N × H)
278. #3503C2×C31semidirect
279. #3077C2×C31semidirect
281. #2563C2×C31semidirect
282. #2061C2×C31semidirect
283. #2027C2×C31semidirect
284. #1997C2×C31semidirect
285. #1986C2×C31semidirect
287. #1860C2×C31semidirect
288. #1806C2×C31semidirect
372. AGL(1,ℤ/8)Q8×C31semidirect
449. C31#24321direct (N × H)
470. C2×D4C2×Q8×C31semidirect
491. M16C2×Q8×C31semidirect

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×#25632
C31×#20612
C31×#20272
C31×#19972
C31×#19862
C31×#18832
C31×#18602
C31×#18062
#2432131