Finite Group Databasebeta

← C31×#16815 #21473574

Subgroups — C31×#16815 #21473574

Order 7936.

1446 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#397C212yesmaximal
2C2×C31×#385C211yesmaximal, characteristic
3C2×C31×#380C211yesmaximal, characteristic
4C31×#1444C211yesmaximal, characteristic, complemented
5C31×#1443C212yesmaximal, complemented
6C31×#1442C211yesmaximal, characteristic, complemented
7C31×#1441C211yesmaximal, characteristic, complemented
8C31×#1441C211yesmaximal, characteristic, complemented
9C31×#1441C212yesmaximal, complemented
10C31×#1440C212yesmaximal
11C31×C22⋊#95C211yesmaximal, characteristic
12C24×C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2211yescharacteristic
21C22×C22⋊C4×C31C2211yescharacteristic
22C31×#397—28no—
23C31×#385—24no—
24C31×#380—24no—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2211yescharacteristic
33C2×C31×#95C2211yescharacteristic
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C2×C31×#95C2212yes—
38C25×C31C2311yescharacteristic
39C23×C4×C31—22no—
40C23×C4×C31—22no—
41C23×C4×C31—42no—
42C23×C4×C31—42no—
43C23×C4×C31D414yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2312yes—
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2312yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2312yes—
53C23×C4×C31C2311yescharacteristic
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—24no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—42no—
63C2×C4⋊C4×C31—42no—
64C2×C4⋊C4×C31—42no—
65C2×C4⋊C4×C31—42no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—42no—
75C2×C22⋊C4×C31—42no—
76C2×C22⋊C4×C31D414yes—
77C31×#95—24no—
78C31×#95—24no—
79C31×#95—24no—
80C31×#95—24no—
81C31×#95—22no—
82C31×#95—22no—
83C31×#95—22no—
84C31×#95—24no—
85C31×#95—22no—
86C31×#95—42no—
87C31×#95—42no—
88C31×#95—42no—
89C31×#95—42no—
90C24×C31—22no—
91C24×C31—21no—
92C24×C31—21no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—21no—
97C24×C31—21no—
98C24×C31C2×D412yes—
99C24×C31C2×D412yes—
100C24×C31#4112yes—
101C24×C31C2411yescharacteristic
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—21no—
109C22×C4×C31—21no—
110C22×C4×C31—22no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—28no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—44no—
157C22×C4×C31—44no—
158C22×C4×C31—48no—
159C22×C4×C31—48no—
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×D412yes—
164C22×C4×C31#4114yes—
165C22×C4×C31#4112yes—
166C22×C4×C31#4112yes—
167C4⋊C4×C31—48no—
168C4⋊C4×C31—44no—
169C4⋊C4×C31—44no—
170C4⋊C4×C31—44no—
171C4⋊C4×C31—44no—
172C22⋊C4×C31—28no—
173C22⋊C4×C31—48no—
174C22⋊C4×C31—44no—
175C22⋊C4×C31—48no—
176#16815C3111yesmaximal, characteristic, direct product factor, Sylow
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—22no—
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—24no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—24no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31#12114yes—
200C23×C31C22×D411yescharacteristic
201C23×C31C22×D411yescharacteristic
202C23×C31C2×#4111yescharacteristic
203C23×C31C2×#4111yescharacteristic
204C23×C31C2×#4111yescharacteristic
205C23×C31C2×#4112yes—
206C23×C31H(5,𝔽2)12yes—
207C23×C31H(5,𝔽2)12yes—
208C23×C312−1+412yes—
209C23×C312−1+411yescharacteristic
210C23×C312−1+411yescharacteristic
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—24no—
214C2×C4×C31—24no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—24no—
226C2×C4×C31—416no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—42no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—42no—
248C2×C4×C31—41no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—41no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31#12314yes—
272C2×#397C2×C3112yes—
273C2×#385C2×C3111yescharacteristic
274C2×#380C2×C3111yescharacteristic
275#1444C2×C3111yescharacteristic, complemented
276#1443C2×C3112yescomplemented
277#1442C2×C3111yescharacteristic, complemented
278#1441C2×C3111yescharacteristic, complemented
279#1441C2×C3111yescharacteristic, complemented
280#1441C2×C3112yescomplemented
281#1440C2×C3112yes—
282C22⋊#95C2×C3111yescharacteristic
283C22×C31—24no—
284C22×C31—24no—
285C22×C31—24no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—44no—
290C22×C31—44no—
291C22×C31—42no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31C2×#12111yescharacteristic
295C22×C31C2×#12311yescharacteristic
296C22×C31#53211yescharacteristic
297C22×C31#53411yescharacteristic
298C22×C31#53411yescharacteristic
299C22×C31#53611yescharacteristic
300C22×C31#53611yescharacteristic
301C22×C31#53712yes—
302C22×C31#53812yes—
303C22×C31#53812yes—
304C22×C31#53912yes—
305C22×C31#53912yes—
306C22×C31#54212yes—
307C22×C31#54612yes—
308C22×C31#54711yescharacteristic
309C22×C31#54711yescharacteristic
310C22×C31#55212yes—
311C22×C31#55211yescharacteristic
312C22×C31#55211yescharacteristic
313C22×C31#55312yes—
314C22×C31#55612yes—
315C22×C31#56312yes—
316C22×C31#56312yes—
317C4×C31—24no—
318C4×C31—42no—
319C4×C31—44no—
320C4×C31—42no—
321C4×C31—42no—
322C4×C31—82no—
323C4×C31—81no—
324C4×C31—81no—
325C4×C31—82no—
326C4×C31—82no—
327C24×C4C22×C3111yescharacteristic
328C22×C4⋊C4C22×C3111yescharacteristic
329C22×C4⋊C4C22×C3112yes—
330C22×C4⋊C4C22×C3111yescharacteristic
331C22×C4⋊C4C22×C3111yescharacteristic
332C22×C4⋊C4C22×C3111yescharacteristic
333C22×C22⋊C4C22×C3112yes—
334C22×C22⋊C4C22×C3112yes—
335C22×C22⋊C4C22×C3111yescharacteristic
336C22×C22⋊C4C22×C3111yescharacteristic
337#397—28no—
338#385—24no—
339#380—24no—
340C2×#95C22×C3112yes—
341C2×#95C22×C3112yes—
342C2×#95C22×C3112yes—
343C2×#95C22×C3111yescharacteristic
344C2×#95C22×C3111yescharacteristic
345C2×#95C22×C3112yes—
346C2×#95C22×C3112yes—
347C2×#95C22×C3111yescharacteristic
348C2×#95C22×C3111yescharacteristic
349C2×#95C22×C3112yes—
350C2×#95C22×C3112yes—
351C2×#95C22×C3112yes—
352C2×#95C22×C3112yes—
353C2×C31—44no—
354C2×C31#248611yescharacteristic
355C2×C31#249911yescharacteristic
356C2×C31#254311yescharacteristic
357C2×C31#261811yescharacteristic
358C2×C31#262211yescharacteristic
359C2×C31#266311yescharacteristic
360C2×C31#269512yes—
361C2×C31#271512yes—
362C2×C31#277012yes—
363C2×C31#278712yes—
364C2×C31#281711yescharacteristic
365C25C23×C3111yescharacteristic
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—42no—
369C23×C4—42no—
370C23×C4D4×C3114yes—
371C23×C4C23×C3112yes—
372C23×C4C23×C3111yescharacteristic
373C23×C4C23×C3112yes—
374C23×C4C23×C3111yescharacteristic
375C23×C4C23×C3112yes—
376C23×C4C23×C3111yescharacteristic
377C23×C4C23×C3111yescharacteristic
378C23×C4C23×C3111yescharacteristic
379C23×C4C23×C3112yes—
380C23×C4C23×C3111yescharacteristic
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—42no—
390C2×C4⋊C4—42no—
391C2×C4⋊C4—42no—
392C2×C4⋊C4—42no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—42no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4D4×C3114yes—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—22no—
411#95—24no—
412#95—22no—
413#95—42no—
414#95—42no—
415#95—42no—
416#95—42no—
417C31#1681511yescharacteristic, direct product factor, Sylow
418C24—22no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—21no—
425C24—21no—
426C24C2×D4×C3112yes—
427C24C2×D4×C3112yes—
428C24#41×C3112yes—
429C24C24×C3111yescharacteristic
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4C2×D4×C3112yes—
489C22×C4C2×D4×C3112yes—
490C22×C4C2×D4×C3112yes—
491C22×C4C2×D4×C3112yes—
492C22×C4#41×C3114yes—
493C22×C4#41×C3112yes—
494C22×C4#41×C3112yes—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C22⋊C4—28no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C31×#12114yes—
527C23C22×D4×C3111yescharacteristic
528C23C22×D4×C3111yescharacteristic
529C23C2×#41×C3111yescharacteristic
530C23C2×#41×C3111yescharacteristic
531C23C2×#41×C3111yescharacteristic
532C23C2×#41×C3112yes—
533C23C31×H(5,𝔽2)12yes—
534C23C31×H(5,𝔽2)12yes—
535C23C31×2−1+412yes—
536C23C31×2−1+411yescharacteristic
537C23C31×2−1+411yescharacteristic
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—416no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4C31×#12314yes—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22C2×C31×#12111yescharacteristic
611C22C2×C31×#12311yescharacteristic
612C22C31×#53211yescharacteristic
613C22C31×#53411yescharacteristic
614C22C31×#53411yescharacteristic
615C22C31×#53611yescharacteristic
616C22C31×#53611yescharacteristic
617C22C31×#53712yes—
618C22C31×#53812yes—
619C22C31×#53812yes—
620C22C31×#53912yes—
621C22C31×#53912yes—
622C22C31×#54212yes—
623C22C31×#54612yes—
624C22C31×#54711yescharacteristic
625C22C31×#54711yescharacteristic
626C22C31×#55212yes—
627C22C31×#55211yescharacteristic
628C22C31×#55211yescharacteristic
629C22C31×#55312yes—
630C22C31×#55612yes—
631C22C31×#56312yes—
632C22C31×#56312yes—
633C4—24no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—82no—
643C2—44no—
644C2C31×#248611yescharacteristic
645C2C31×#249911yescharacteristic
646C2C31×#254311yescharacteristic
647C2C31×#261811yescharacteristic
648C2C31×#262211yescharacteristic
649C2C31×#266311yescharacteristic
650C2C31×#269512yes—
651C2C31×#271512yes—
652C2C31×#277012yes—
653C2C31×#278712yes—
654C2C31×#281711yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C31×#1444C2semidirect
5. C31×#1443C2semidirect
6. C31×#1442C2semidirect
7. C31×#1441C2semidirect
8. C31×#1441C2semidirect
9. C31×#1441C2semidirect
176. #16815C31direct (N × H)
275. #1444C2×C31semidirect
276. #1443C2×C31semidirect
277. #1442C2×C31semidirect
278. #1441C2×C31semidirect
279. #1441C2×C31semidirect
280. #1441C2×C31semidirect
417. C31#16815direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3972
C2×C31×#3852
C2×C31×#3802
C31×#14442
C31×#14432
C31×#14422
C31×#14412
C31×#14402
C31×C22⋊#952
#1681531