Finite Group Databasebeta

← C31×#24177 #21480936

Subgroups — C31×#24177 #21480936

Order 7936.

1070 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×#3507C211yesmaximal, characteristic, complemented
2C31×#3077C211yesmaximal, characteristic, complemented
3C31×#3036C211yesmaximal, characteristic
4C4×C31×#124C211yesmaximal, characteristic, complemented
5C31×#2038C212yesmaximal, complemented
6C31×#2003C212yesmaximal, complemented
7C31×#1994C211yesmaximal, characteristic, complemented
8C31×#1962C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1860C212yesmaximal, complemented
11C31×#1801C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)C2212yes—
13C31×#559—21no—
14C31×#557—21no—
15C31×#556—21no—
16C31×#556—22no—
17C31×#554—21no—
18C31×#552—22no—
19C31×#539—22no—
20C31×#536C2212yes—
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C31×#529—22no—
24C2×C31×#124C2211yescharacteristic
25C31×#520C2212yes—
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C31×#478—22no—
29C31×#476—21no—
30C31×#474—21no—
31C31×#463—22no—
32C31×#461—21no—
33C31×#459—21no—
34C31×#428C2212yes—
35C31×#423C2211yescharacteristic
36C31×#423C2211yescharacteristic
37C2×C31×#104C2212yes—
38C2×C31×#104C2212yes—
39C31×#418C2212yes—
40C31×#417C2212yes—
41C2×C31×#101C2212yes—
42C31×#391C2211yescharacteristic
43C31×#390C2211yescharacteristic
44C31×#388—22no—
45C31×#382C2211yescharacteristic
46C4×C22⋊C4×C31—22no—
47C43×C31C2211yescharacteristic
48C31×#343C2212yes—
49C31×#339C2212yes—
50C31×#339C2212yes—
51C2×#41×C31C2312yes—
52C22×Q8×C31C2311yescharacteristic
53C22×D4×C31C2311yescharacteristic
54C23×C4×C31—21no—
55C31×AGL(1,ℤ/8)—24no—
56C31×AGL(1,ℤ/8)—42no—
57C2×SD16×C31—22no—
58C2×D8×C31—22no—
59C2×M16×C31C2312yes—
60C2×M16×C31C2312yes—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8D411yescharacteristic
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8D411yescharacteristic
65C31×#127D411yescharacteristic
66C31×#126—24no—
67C31×#126—24no—
68C31×#125—22no—
69C31×#125—24no—
70C31×#125D412yes—
71C31×#124—22no—
72C31×#124—22no—
73C31×#124—22no—
74C31×#124—22no—
75C31×#124D412yes—
76C31×#123—24no—
77C31×#123—24no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×#121—21no—
84C31×#121—21no—
85C4×Q8×C31—22no—
86C4×Q8×C31—22no—
87C4×Q8×C31—21no—
88C4×Q8×C31—21no—
89C4×Q8×C31—21no—
90C4×Q8×C31—21no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31—24no—
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C31×#117—24no—
103C31×#117—22no—
104C31×#117C2312yes—
105C2×C4⋊C4×C31—22no—
106C2×C4⋊C4×C31D412yes—
107C2×C4⋊C4×C31C2311yescharacteristic
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C22⋊C4×C31—22no—
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31—22no—
112C2×C42×C31—21no—
113C2×C42×C31—21no—
114C2×C42×C31D412yes—
115C2×C42×C31C2311yescharacteristic
116C2×C42×C31C2311yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×C4⋊C8—21no—
121C31×C4⋊C8—21no—
122C31×C4⋊C8—21no—
123C31×C4⋊C8—21no—
124C31×#104—24no—
125C31×#104—42no—
126C31×#103—22no—
127C31×#102—22no—
128C31×#102—22no—
129C31×#102—22no—
130C31×#101—42no—
131C31×#95—22no—
132C31×#95—22no—
133C24×C31—21no—
134#41×C31—22no—
135#41×C31—22no—
136#41×C31—22no—
137#41×C31—22no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31—21no—
143C2×Q8×C31—21no—
144C2×Q8×C31—21no—
145C2×Q8×C31—21no—
146C2×Q8×C31C2×D412yes—
147C2×D4×C31—22no—
148C2×D4×C31—21no—
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×D412yes—
156C2×D4×C31#4112yes—
157C22×C4×C31—21no—
158C22×C4×C31—21no—
159C22×C4×C31—24no—
160C22×C4×C31—22no—
161C22×C4×C31—21no—
162C22×C4×C31—44no—
163C22×C4×C31—41no—
164C22×C4×C31—41no—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2411yescharacteristic
170SD16×C31—44no—
171D8×C31—44no—
172M16×C31—22no—
173M16×C31—22no—
174M16×C31—22no—
175M16×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31C2×D412yes—
194C4⋊C4×C31C2×D412yes—
195C22⋊C4×C31—28no—
196C22⋊C4×C31—24no—
197C22⋊C4×C31—24no—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—44no—
203C22⋊C4×C31—41no—
204C22⋊C4×C31—41no—
205C42×C31—24no—
206C42×C31—22no—
207C42×C31—22no—
208C42×C31—22no—
209C42×C31—21no—
210C42×C31—21no—
211C42×C31—21no—
212C42×C31—42no—
213C42×C31—42no—
214C42×C31C2×D411yescharacteristic
215C42×C31C2×D411yescharacteristic
216C42×C31C2×D411yescharacteristic
217C42×C31C2×D411yescharacteristic
218C42×C31#4112yes—
219#24177C3111yesmaximal, characteristic, direct product factor, Sylow
220C23×C31—22no—
221C23×C31—22no—
222C23×C31—42no—
223C23×C31—42no—
224C23×C31—41no—
225C23×C31—41no—
226C23×C31C22×D411yescharacteristic
227Q8×C31—22no—
228Q8×C31—22no—
229Q8×C31—41no—
230Q8×C31—41no—
231Q8×C31—41no—
232Q8×C31—41no—
233Q8×C31—42no—
234Q8×C31—81no—
235D4×C31—22no—
236D4×C31—22no—
237D4×C31—22no—
238D4×C31—22no—
239D4×C31—42no—
240D4×C31—42no—
241D4×C31—41no—
242D4×C31—41no—
243D4×C31—42no—
244D4×C31—81no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—22no—
254C2×C4×C31—22no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—41no—
265C2×C4×C31—82no—
266C2×C4×C31—84no—
267C2×C4×C31—82no—
268C2×C4×C31#12011yescharacteristic
269C2×C4×C31#12011yescharacteristic
270C2×C4×C31#12011yescharacteristic
271C2×C4×C31#12011yescharacteristic
272C2×C4×C31C22×D411yescharacteristic
273C2×C4×C31C22×D411yescharacteristic
274C2×C4×C31C2×#4111yescharacteristic
275C2×C4×C31C2×#4111yescharacteristic
276C2×C4×C31H(5,𝔽2)11yescharacteristic
277C2×C4×C312−1+411yescharacteristic
278C8×C31—42no—
279C8×C31—42no—
280#3507C2×C3111yescharacteristic, complemented
281#3077C2×C3111yescharacteristic, complemented
282#3036C2×C3111yescharacteristic
283C4×#124C2×C3111yescharacteristic, complemented
284#2038C2×C3112yescomplemented
285#2003C2×C3112yescomplemented
286#1994C2×C3111yescharacteristic, complemented
287#1962C2×C3111yescharacteristic, complemented
288#1883C2×C3112yescomplemented
289#1860C2×C3112yescomplemented
290#1801C2×C3111yescharacteristic, complemented
291C22×C31—21no—
292C22×C31—21no—
293C22×C31—44no—
294C22×C31—42no—
295C22×C31—42no—
296C22×C31—42no—
297C22×C31—42no—
298C22×C31—82no—
299C22×C31—81no—
300C22×C31—81no—
301C22×C31C2×#12011yescharacteristic
302C22×C31#52511yescharacteristic
303C22×C31#53711yescharacteristic
304C4×C31—21no—
305C4×C31—21no—
306C4×C31—21no—
307C4×C31—42no—
308C4×C31—42no—
309C4×C31—42no—
310C4×C31—42no—
311C4×C31—42no—
312C4×C31—42no—
313C4×C31—42no—
314C4×C31—81no—
315C4×C31#45412yes—
316C4×C31#54611yescharacteristic
317C4×C31#54711yescharacteristic
318C4×C31#54811yescharacteristic
319C4×C31#55011yescharacteristic
320C2×AGL(1,ℤ/8)C22×C3112yes—
321#559—21no—
322#557—21no—
323#556—21no—
324#556—22no—
325#554—21no—
326#552—22no—
327#539—22no—
328#536C22×C3112yes—
329#533C22×C3111yescharacteristic
330#532C22×C3111yescharacteristic
331#529—22no—
332C2×#124C22×C3111yescharacteristic
333#520C22×C3112yes—
334C2×C4×Q8C22×C3111yescharacteristic
335C2×C4×D4C22×C3111yescharacteristic
336#478—22no—
337#476—21no—
338#474—21no—
339#463—22no—
340#461—21no—
341#459—21no—
342#428C22×C3112yes—
343#423C22×C3111yescharacteristic
344#423C22×C3111yescharacteristic
345C2×#104C22×C3112yes—
346C2×#104C22×C3112yes—
347#418C22×C3112yes—
348#417C22×C3112yes—
349C2×#101C22×C3112yes—
350#391C22×C3111yescharacteristic
351#390C22×C3111yescharacteristic
352#388—22no—
353#382C22×C3111yescharacteristic
354C4×C22⋊C4—22no—
355C43C22×C3111yescharacteristic
356#343C22×C3112yes—
357#339C22×C3112yes—
358#339C22×C3112yes—
359C2×C31—21no—
360C2×C31—21no—
361C2×C31—82no—
362C2×C31—82no—
363C2×C31#240711yescharacteristic
364C2×C31C2×#45411yescharacteristic
365C2×C31#301811yescharacteristic
366C2×#41C23×C3112yes—
367C22×Q8C23×C3111yescharacteristic
368C22×D4C23×C3111yescharacteristic
369C23×C4—21no—
370AGL(1,ℤ/8)—24no—
371AGL(1,ℤ/8)—42no—
372C2×SD16—22no—
373C2×D8—22no—
374C2×M16C23×C3112yes—
375C2×M16C23×C3112yes—
376C4⋊Q8—22no—
377C4⋊Q8D4×C3111yescharacteristic
378C4⋊Q8D4×C3111yescharacteristic
379C4⋊Q8D4×C3111yescharacteristic
380#127D4×C3111yescharacteristic
381#126—24no—
382#126—24no—
383#125—22no—
384#125—24no—
385#125D4×C3112yes—
386#124—22no—
387#124—22no—
388#124—22no—
389#124—22no—
390#124D4×C3112yes—
391#123—24no—
392#123—24no—
393C22⋊Q8—24no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398#121—21no—
399#121—21no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—24no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417#117—24no—
418#117—22no—
419#117C23×C3112yes—
420C2×C4⋊C4—22no—
421C2×C4⋊C4D4×C3112yes—
422C2×C4⋊C4C23×C3111yescharacteristic
423C2×C4⋊C4C23×C3111yescharacteristic
424C2×C22⋊C4—22no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—22no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42D4×C3112yes—
430C2×C42C23×C3111yescharacteristic
431C2×C42C23×C3111yescharacteristic
432C2×C42C23×C3111yescharacteristic
433#107—22no—
434#106—22no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—21no—
439#104—24no—
440#104—42no—
441#103—22no—
442#102—22no—
443#102—22no—
444#102—22no—
445#101—42no—
446#95—22no—
447#95—22no—
448C31#2417711yescharacteristic, direct product factor, Sylow
449C24—21no—
450#41—22no—
451#41—22no—
452#41—22no—
453#41—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8C2×D4×C3112yes—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4C2×D4×C3112yes—
472C2×D4#41×C3112yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—44no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3112yes—
484C22×C4C2×D4×C3111yescharacteristic
485C22×C4C24×C3111yescharacteristic
486SD16—44no—
487D8—44no—
488M16—22no—
489M16—22no—
490M16—22no—
491M16—22no—
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4C2×D4×C3112yes—
510C4⋊C4C2×D4×C3112yes—
511C22⋊C4—28no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—44no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—42no—
529C42—42no—
530C42C2×D4×C3111yescharacteristic
531C42C2×D4×C3111yescharacteristic
532C42C2×D4×C3111yescharacteristic
533C42C2×D4×C3111yescharacteristic
534C42#41×C3112yes—
535C23—22no—
536C23—22no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23C22×D4×C3111yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—42no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—81no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—22no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—41no—
558D4—41no—
559D4—81no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—82no—
581C2×C4—84no—
582C2×C4—82no—
583C2×C4C31×#12011yescharacteristic
584C2×C4C31×#12011yescharacteristic
585C2×C4C31×#12011yescharacteristic
586C2×C4C31×#12011yescharacteristic
587C2×C4C22×D4×C3111yescharacteristic
588C2×C4C22×D4×C3111yescharacteristic
589C2×C4C2×#41×C3111yescharacteristic
590C2×C4C2×#41×C3111yescharacteristic
591C2×C4C31×H(5,𝔽2)11yescharacteristic
592C2×C4C31×2−1+411yescharacteristic
593C8—42no—
594C8—42no—
595C22—21no—
596C22—21no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22C2×C31×#12011yescharacteristic
606C22C31×#52511yescharacteristic
607C22C31×#53711yescharacteristic
608C4—21no—
609C4—21no—
610C4—21no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—81no—
619C4C31×#45412yes—
620C4C31×#54611yescharacteristic
621C4C31×#54711yescharacteristic
622C4C31×#54811yescharacteristic
623C4C31×#55011yescharacteristic
624C2—21no—
625C2—21no—
626C2—82no—
627C2—82no—
628C2C31×#240711yescharacteristic
629C2C2×C31×#45411yescharacteristic
630C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3036C2
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×#3507C2semidirect
2. C31×#3077C2semidirect
4. C4×C31×#124C2semidirect
5. C31×#2038C2semidirect
6. C31×#2003C2semidirect
7. C31×#1994C2semidirect
8. C31×#1962C2semidirect
9. C31×#1883C2semidirect
10. C31×#1860C2semidirect
11. C31×#1801C2semidirect
219. #24177C31direct (N × H)
280. #3507C2×C31semidirect
281. #3077C2×C31semidirect
283. C4×#124C2×C31semidirect
284. #2038C2×C31semidirect
285. #2003C2×C31semidirect
286. #1994C2×C31semidirect
287. #1962C2×C31semidirect
288. #1883C2×C31semidirect
289. #1860C2×C31semidirect
290. #1801C2×C31semidirect
448. C31#24177direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35072
C31×#30772
C31×#30362
C4×C31×#1242
C31×#20382
C31×#20032
C31×#19942
C31×#19622
C31×#18832
C31×#18602
C31×#18012
#2417731