Finite Group Databasebeta

← C31×#24147 #21480906

Subgroups — C31×#24147 #21480906

Order 7936.

1006 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3509C211yesmaximal, characteristic
2C31×#3078C211yesmaximal, characteristic, complemented
3C4×C31×#137C211yesmaximal, characteristic
4C31×#2546C211yesmaximal, characteristic, complemented
5C31×#2038C212yesmaximal
6C31×#2005C212yesmaximal
7C31×#1986C211yesmaximal, characteristic, complemented
8C31×#1966C211yesmaximal, characteristic, complemented
9C31×#1886C212yesmaximal, complemented
10C31×#1857C212yesmaximal, complemented
11C31×#1806C211yesmaximal, characteristic
12C2×C31×#137C2211yescharacteristic
13C31×#557—21no—
14C31×#557—21no—
15C31×#556—21no—
16C31×#556—21no—
17C31×#552—22no—
18C31×#552—22no—
19C31×#541—22no—
20C31×#536C2212yes—
21C31×#533—21no—
22C31×#533—21no—
23C31×#533C2211yescharacteristic
24C2×C31×C4⋊Q8C2212yes—
25C2×C31×C4⋊Q8C2211yescharacteristic
26C31×#519C2212yes—
27C4×#41×C31C2211yescharacteristic
28C2×C4×Q8×C31C2211yescharacteristic
29C31×#479—22no—
30C31×#477—21no—
31C31×#475—21no—
32C31×#441—21no—
33C31×#440—21no—
34C4×Q16×C31—21no—
35C4×SD16×C31—21no—
36C31×#428C2212yes—
37C31×#428C2211yescharacteristic
38C31×#423C2211yescharacteristic
39C2×C31×#104C2212yes—
40C31×#419C2212yes—
41C31×#419C2211yescharacteristic
42C31×#417C2211yescharacteristic
43C2×C31×#101C2212yes—
44C2×C31×#101C2212yes—
45C4×M16×C31C2211yescharacteristic
46C31×#398—22no—
47C31×#391C2212yes—
48C31×#389—22no—
49C4×C4⋊C4×C31C2211yescharacteristic
50C31×#339C2212yes—
51C31×#339C2212yes—
52C31×#339C2212yes—
53C2×#41×C31C2311yescharacteristic
54C22×Q8×C31C2311yescharacteristic
55C22×Q8×C31C2312yes—
56C31×#137—22no—
57C31×#137—22no—
58C2×Q16×C31—21no—
59C2×SD16×C31—21no—
60C2×M16×C31C2311yescharacteristic
61C2×M16×C31C2312yes—
62C2×M16×C31C2311yescharacteristic
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8—24no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8—42no—
70C31×C4⋊Q8D411yescharacteristic
71C31×C4⋊Q8D411yescharacteristic
72C31×#126—24no—
73C31×#126—24no—
74C31×#125—22no—
75C31×#125—22no—
76C31×#125—24no—
77C31×#125—24no—
78C31×#125D412yes—
79C31×#124—22no—
80C31×#123—24no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C4×Q8×C31—22no—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31—22no—
92C4×Q8×C31—22no—
93C4×Q8×C31—21no—
94C4×Q8×C31—21no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31D411yescharacteristic
100C4×Q8×C31D411yescharacteristic
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31D411yescharacteristic
104C4×D4×C31D411yescharacteristic
105C31×#117—22no—
106C31×#117—22no—
107C31×#117—24no—
108C31×#117—21no—
109C31×#117C2311yescharacteristic
110C31×#117C2312yes—
111C2×C4⋊C4×C31—21no—
112C2×C4⋊C4×C31—21no—
113C2×C4⋊C4×C31—24no—
114C2×C4⋊C4×C31D412yes—
115C2×C4⋊C4×C31D412yes—
116C2×C4⋊C4×C31C2311yescharacteristic
117C2×C42×C31—21no—
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—42no—
128C31×#103—21no—
129C31×#103—21no—
130C31×#103—21no—
131C31×#103—22no—
132C31×#103—22no—
133C31×#102—21no—
134C31×#101—24no—
135C31×#101—42no—
136C31×#97—21no—
137C4×C8×C31—21no—
138C31×#95—21no—
139C31×#95—24no—
140#41×C31—21no—
141#41×C31—21no—
142#41×C31—21no—
143#41×C31—21no—
144C2×Q8×C31—21no—
145C2×Q8×C31—21no—
146C2×Q8×C31—22no—
147C2×Q8×C31—22no—
148C2×Q8×C31—22no—
149C2×Q8×C31—22no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—22no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—21no—
157C2×Q8×C31C2×D411yescharacteristic
158C2×Q8×C31#4112yes—
159C2×D4×C31—21no—
160C2×D4×C31C2×D411yescharacteristic
161C22×C4×C31—22no—
162C22×C4×C31—21no—
163C22×C4×C31—21no—
164C22×C4×C31—21no—
165C22×C4×C31—21no—
166C22×C4×C31—21no—
167C22×C4×C31C2×D414yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171Q16×C31—24no—
172SD16×C31—24no—
173M16×C31—21no—
174M16×C31—22no—
175M16×C31—21no—
176M16×C31—22no—
177M16×C31—21no—
178M16×C31#4112yes—
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—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—48no—
200C4⋊C4×C31—41no—
201C4⋊C4×C31—41no—
202C4⋊C4×C31C2×D411yescharacteristic
203C4⋊C4×C31C2×D412yes—
204C4⋊C4×C31C2×D411yescharacteristic
205C4⋊C4×C31C2×D412yes—
206C22⋊C4×C31—24no—
207C22⋊C4×C31—24no—
208C22⋊C4×C31—21no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—21no—
211C22⋊C4×C31—21no—
212C42×C31—24no—
213C42×C31—24no—
214C42×C31—21no—
215C42×C31—21no—
216C42×C31—21no—
217C42×C31—22no—
218C42×C31—22no—
219C42×C31—22no—
220C42×C31—41no—
221C42×C31—41no—
222C42×C31C2×D411yescharacteristic
223C42×C31C2×D411yescharacteristic
224C42×C31C2×D411yescharacteristic
225C42×C31C2×D411yescharacteristic
226#24147C3111yesmaximal, characteristic, direct product factor, Sylow
227C23×C31—21no—
228C23×C31C22×D411yescharacteristic
229Q8×C31—22no—
230Q8×C31—21no—
231Q8×C31—21no—
232Q8×C31—22no—
233Q8×C31—42no—
234Q8×C31—42no—
235Q8×C31—42no—
236Q8×C31—41no—
237Q8×C31—42no—
238Q8×C31—42no—
239Q8×C31—41no—
240Q8×C31—42no—
241D4×C31—21no—
242D4×C31—21no—
243D4×C31—41no—
244D4×C31—41no—
245C2×C4×C31—22no—
246C2×C4×C31—24no—
247C2×C4×C31—21no—
248C2×C4×C31—21no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—24no—
252C2×C4×C31—21no—
253C2×C4×C31—22no—
254C2×C4×C31—24no—
255C2×C4×C31—24no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—41no—
260C2×C4×C31—42no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—41no—
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×C31C2×#4111yescharacteristic
274C2×C4×C31C2×#4111yescharacteristic
275C2×C4×C31H(5,𝔽2)11yescharacteristic
276C2×C4×C312−1+411yescharacteristic
277C8×C31—22no—
278C8×C31—41no—
279C8×C31—42no—
280#3509C2×C3111yescharacteristic
281#3078C2×C3111yescharacteristic, complemented
282C4×#137C2×C3111yescharacteristic
283#2546C2×C3111yescharacteristic, complemented
284#2038C2×C3112yes—
285#2005C2×C3112yes—
286#1986C2×C3111yescharacteristic, complemented
287#1966C2×C3111yescharacteristic, complemented
288#1886C2×C3112yescomplemented
289#1857C2×C3112yescomplemented
290#1806C2×C3111yescharacteristic
291C22×C31—21no—
292C22×C31—21no—
293C22×C31—41no—
294C22×C31—41no—
295C22×C31—41no—
296C22×C31C2×#12011yescharacteristic
297C22×C31#54611yescharacteristic
298C22×C31#54711yescharacteristic
299C4×C31—21no—
300C4×C31—21no—
301C4×C31—21no—
302C4×C31—44no—
303C4×C31—44no—
304C4×C31—41no—
305C4×C31—42no—
306C4×C31—41no—
307C4×C31—42no—
308C4×C31—41no—
309C4×C31—42no—
310C4×C31—81no—
311C4×C31#45612yes—
312C4×C31#52511yescharacteristic
313C4×C31#53711yescharacteristic
314C4×C31#54811yescharacteristic
315C4×C31#55011yescharacteristic
316C2×#137C22×C3111yescharacteristic
317#557—21no—
318#557—21no—
319#556—21no—
320#556—21no—
321#552—22no—
322#552—22no—
323#541—22no—
324#536C22×C3112yes—
325#533—21no—
326#533—21no—
327#533C22×C3111yescharacteristic
328C2×C4⋊Q8C22×C3112yes—
329C2×C4⋊Q8C22×C3111yescharacteristic
330#519C22×C3112yes—
331C4×#41C22×C3111yescharacteristic
332C2×C4×Q8C22×C3111yescharacteristic
333#479—22no—
334#477—21no—
335#475—21no—
336#441—21no—
337#440—21no—
338C4×Q16—21no—
339C4×SD16—21no—
340#428C22×C3112yes—
341#428C22×C3111yescharacteristic
342#423C22×C3111yescharacteristic
343C2×#104C22×C3112yes—
344#419C22×C3112yes—
345#419C22×C3111yescharacteristic
346#417C22×C3111yescharacteristic
347C2×#101C22×C3112yes—
348C2×#101C22×C3112yes—
349C4×M16C22×C3111yescharacteristic
350#398—22no—
351#391C22×C3112yes—
352#389—22no—
353C4×C4⋊C4C22×C3111yescharacteristic
354#339C22×C3112yes—
355#339C22×C3112yes—
356#339C22×C3112yes—
357C2×C31—21no—
358C2×C31—21no—
359C2×C31—81no—
360C2×C31#240711yescharacteristic
361C2×C31C2×#45611yescharacteristic
362C2×C31#302011yescharacteristic
363C2×#41C23×C3111yescharacteristic
364C22×Q8C23×C3111yescharacteristic
365C22×Q8C23×C3112yes—
366#137—22no—
367#137—22no—
368C2×Q16—21no—
369C2×SD16—21no—
370C2×M16C23×C3111yescharacteristic
371C2×M16C23×C3112yes—
372C2×M16C23×C3111yescharacteristic
373C4⋊Q8—22no—
374C4⋊Q8—24no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8—42no—
380C4⋊Q8D4×C3111yescharacteristic
381C4⋊Q8D4×C3111yescharacteristic
382#126—24no—
383#126—24no—
384#125—22no—
385#125—22no—
386#125—24no—
387#125—24no—
388#125D4×C3112yes—
389#124—22no—
390#123—24no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8D4×C3111yescharacteristic
410C4×Q8D4×C3111yescharacteristic
411C4×D4—21no—
412C4×D4—21no—
413C4×D4D4×C3111yescharacteristic
414C4×D4D4×C3111yescharacteristic
415#117—22no—
416#117—22no—
417#117—24no—
418#117—21no—
419#117C23×C3111yescharacteristic
420#117C23×C3112yes—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—24no—
424C2×C4⋊C4D4×C3112yes—
425C2×C4⋊C4D4×C3112yes—
426C2×C4⋊C4C23×C3111yescharacteristic
427C2×C42—21no—
428C2×C42—21no—
429C2×C42C23×C3112yes—
430C2×C42C23×C3111yescharacteristic
431#107—21no—
432#107—22no—
433#106—21no—
434#106—22no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437#104—42no—
438#103—21no—
439#103—21no—
440#103—21no—
441#103—22no—
442#103—22no—
443#102—21no—
444#101—24no—
445#101—42no—
446#97—21no—
447C4×C8—21no—
448#95—21no—
449#95—24no—
450C31#2414711yescharacteristic, direct product factor, Sylow
451#41—21no—
452#41—21no—
453#41—21no—
454#41—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—21no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C2×D4×C3111yescharacteristic
469C2×Q8#41×C3112yes—
470C2×D4—21no—
471C2×D4C2×D4×C3111yescharacteristic
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4C2×D4×C3114yes—
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3111yescharacteristic
481C22×C4C24×C3111yescharacteristic
482Q16—24no—
483SD16—24no—
484M16—21no—
485M16—22no—
486M16—21no—
487M16—22no—
488M16—21no—
489M16#41×C3112yes—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—48no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C2×D4×C3111yescharacteristic
514C4⋊C4C2×D4×C3112yes—
515C4⋊C4C2×D4×C3111yescharacteristic
516C4⋊C4C2×D4×C3112yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—21no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C42—24no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—41no—
532C42—41no—
533C42C2×D4×C3111yescharacteristic
534C42C2×D4×C3111yescharacteristic
535C42C2×D4×C3111yescharacteristic
536C42C2×D4×C3111yescharacteristic
537C23—21no—
538C23C22×D4×C3111yescharacteristic
539Q8—22no—
540Q8—21no—
541Q8—21no—
542Q8—22no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—41no—
548Q8—42no—
549Q8—42no—
550Q8—41no—
551D4—21no—
552D4—21no—
553D4—41no—
554D4—41no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4C31×#12011yescharacteristic
578C2×C4C31×#12011yescharacteristic
579C2×C4C31×#12011yescharacteristic
580C2×C4C31×#12011yescharacteristic
581C2×C4C22×D4×C3111yescharacteristic
582C2×C4C22×D4×C3111yescharacteristic
583C2×C4C2×#41×C3111yescharacteristic
584C2×C4C2×#41×C3111yescharacteristic
585C2×C4C31×H(5,𝔽2)11yescharacteristic
586C2×C4C31×2−1+411yescharacteristic
587C8—22no—
588C8—41no—
589C8—42no—
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×C31×#12011yescharacteristic
596C22C31×#54611yescharacteristic
597C22C31×#54711yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—44no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4C31×#45612yes—
611C4C31×#52511yescharacteristic
612C4C31×#53711yescharacteristic
613C4C31×#54811yescharacteristic
614C4C31×#55011yescharacteristic
615C2—21no—
616C2—21no—
617C2—81no—
618C2C31×#240711yescharacteristic
619C2C2×C31×#45611yescharacteristic
620C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×#41×C31C23
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
2. C31×#3078C2semidirect
4. C31×#2546C2semidirect
7. C31×#1986C2semidirect
8. C31×#1966C2semidirect
9. C31×#1886C2semidirect
10. C31×#1857C2semidirect
226. #24147C31direct (N × H)
281. #3078C2×C31semidirect
283. #2546C2×C31semidirect
286. #1986C2×C31semidirect
287. #1966C2×C31semidirect
288. #1886C2×C31semidirect
289. #1857C2×C31semidirect
450. C31#24147direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35092
C31×#30782
C4×C31×#1372
C31×#25462
C31×#20382
C31×#20052
C31×#19862
C31×#19662
C31×#18862
C31×#18572
C31×#18062
#2414731