Finite Group Databasebeta

← C31×#16812 #21473571

Subgroups — C31×#16812 #21473571

Order 7936.

1486 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#398C211yesmaximal, characteristic, complemented
2C2×C31×#397C211yesmaximal, characteristic
3C2×C31×#395C211yesmaximal, characteristic, complemented
4C2×C31×#393C211yesmaximal, characteristic
5C2×C31×#391C211yesmaximal, characteristic, complemented
6C2×C31×#388C211yesmaximal, characteristic
7C31×#1445C212yesmaximal, complemented
8C31×#1445C212yesmaximal, complemented
9C31×#1439C212yesmaximal
10C31×#1439C212yesmaximal
11C31×#1434C211yesmaximal, characteristic, complemented
12C23×Q8×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2212yes—
16C22×C4⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2211yescharacteristic
22C22×C42×C31C2211yescharacteristic
23C31×#398—24no—
24C31×#397—24no—
25C31×#395—24no—
26C31×#393—24no—
27C31×#391—24no—
28C31×#388—24no—
29C2×C31×#95C2212yes—
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2211yescharacteristic
36C2×C31×#95C2211yescharacteristic
37C2×C31×#95C2212yes—
38C2×C31×#95C2212yes—
39C2×C31×#95C2211yescharacteristic
40C2×C31×#95C2211yescharacteristic
41C25×C31C2311yescharacteristic
42C22×Q8×C31—22no—
43C22×Q8×C31—22no—
44C22×Q8×C31—22no—
45C22×Q8×C31—24no—
46C22×Q8×C31—42no—
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2312yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2312yes—
53C23×C4×C31C2312yes—
54C23×C4×C31C2311yescharacteristic
55C23×C4×C31C2312yes—
56C23×C4×C31C2311yescharacteristic
57C2×C4⋊C4×C31—24no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—24no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—24no—
66C2×C4⋊C4×C31—24no—
67C2×C4⋊C4×C31—42no—
68C2×C4⋊C4×C31—42no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—22no—
75C2×C22⋊C4×C31—22no—
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—24no—
78C2×C22⋊C4×C31—22no—
79C2×C22⋊C4×C31—24no—
80C2×C22⋊C4×C31—42no—
81C2×C22⋊C4×C31—42no—
82C2×C22⋊C4×C31—42no—
83C2×C42×C31—42no—
84C2×C42×C31D414yes—
85C31×#95—24no—
86C31×#95—24no—
87C31×#95—24no—
88C31×#95—22no—
89C31×#95—22no—
90C31×#95—24no—
91C31×#95—24no—
92C31×#95—24no—
93C31×#95—42no—
94C31×#95—42no—
95C31×#95D414yes—
96C31×#95D414yes—
97C24×C31—21no—
98C24×C31—21no—
99C24×C31—21no—
100C24×C31—21no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—21no—
104C24×C31—22no—
105C24×C31—21no—
106C24×C31—22no—
107C24×C31#4112yes—
108C24×C31C2411yescharacteristic
109C2×Q8×C31—28no—
110C2×Q8×C31—48no—
111C2×Q8×C31—48no—
112C2×Q8×C31—44no—
113C2×Q8×C31—44no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
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—22no—
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—21no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—22no—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D414yes—
168C22×C4×C31C2×D414yes—
169C22×C4×C31C2×D412yes—
170C22×C4×C31C2×D412yes—
171C22×C4×C31C2×D412yes—
172C22×C4×C31#4112yes—
173C22×C4×C31#4114yes—
174C4⋊C4×C31—48no—
175C4⋊C4×C31—48no—
176C4⋊C4×C31—44no—
177C4⋊C4×C31—44no—
178C22⋊C4×C31—28no—
179C22⋊C4×C31—48no—
180C22⋊C4×C31—48no—
181C22⋊C4×C31—44no—
182C22⋊C4×C31—44no—
183C42×C31—44no—
184#16812C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—22no—
186C23×C31—24no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—24no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—41no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31C22×D411yescharacteristic
209C23×C31C22×D411yescharacteristic
210C23×C31C22×D411yescharacteristic
211C23×C31C2×#4111yescharacteristic
212C23×C31C2×#4112yes—
213C23×C31C2×#4111yescharacteristic
214C23×C31H(5,𝔽2)12yes—
215C23×C31H(5,𝔽2)11yescharacteristic
216C23×C312−1+412yes—
217C23×C312−1+412yes—
218C23×C312−1+411yescharacteristic
219Q8×C31—416no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—24no—
223C2×C4×C31—24no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—24no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—24no—
235C2×C4×C31—24no—
236C2×C4×C31—24no—
237C2×C4×C31—22no—
238C2×C4×C31—24no—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—22no—
242C2×C4×C31—41no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—41no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—41no—
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—41no—
263C2×C4×C31—42no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—42no—
267C2×C4×C31—41no—
268C2×C4×C31—41no—
269C2×C4×C31—41no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—42no—
279C2×C4×C31#12714yes—
280C2×#398C2×C3111yescharacteristic, complemented
281C2×#397C2×C3111yescharacteristic
282C2×#395C2×C3111yescharacteristic, complemented
283C2×#393C2×C3111yescharacteristic
284C2×#391C2×C3111yescharacteristic, complemented
285C2×#388C2×C3111yescharacteristic
286#1445C2×C3112yescomplemented
287#1445C2×C3112yescomplemented
288#1439C2×C3112yes—
289#1439C2×C3112yes—
290#1434C2×C3111yescharacteristic, complemented
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—44no—
294C22×C31—44no—
295C22×C31—42no—
296C22×C31—81no—
297C22×C31—81no—
298C22×C31—82no—
299C22×C31—82no—
300C22×C31—81no—
301C22×C31—81no—
302C22×C31#52511yescharacteristic
303C22×C31#52511yescharacteristic
304C22×C31C2×#12711yescharacteristic
305C22×C31#53411yescharacteristic
306C22×C31#53611yescharacteristic
307C22×C31#53611yescharacteristic
308C22×C31#53612yes—
309C22×C31#53612yes—
310C22×C31#53711yescharacteristic
311C22×C31#53912yes—
312C22×C31#54112yes—
313C22×C31#54212yes—
314C22×C31#54212yes—
315C22×C31#54611yescharacteristic
316C22×C31#54612yes—
317C22×C31#54612yes—
318C22×C31#54611yescharacteristic
319C22×C31#54811yescharacteristic
320C22×C31#54811yescharacteristic
321C22×C31#55212yes—
322C22×C31#55612yes—
323C22×C31#56212yes—
324C22×C31#56312yes—
325C4×C31—24no—
326C4×C31—42no—
327C4×C31—44no—
328C4×C31—44no—
329C4×C31—44no—
330C4×C31—42no—
331C4×C31—42no—
332C4×C31—81no—
333C4×C31—81no—
334C4×C31—82no—
335C23×Q8C22×C3111yescharacteristic
336C22×C4⋊C4C22×C3111yescharacteristic
337C22×C4⋊C4C22×C3111yescharacteristic
338C22×C4⋊C4C22×C3112yes—
339C22×C4⋊C4C22×C3112yes—
340C22×C22⋊C4C22×C3111yescharacteristic
341C22×C22⋊C4C22×C3111yescharacteristic
342C22×C22⋊C4C22×C3112yes—
343C22×C22⋊C4C22×C3112yes—
344C22×C22⋊C4C22×C3111yescharacteristic
345C22×C42C22×C3111yescharacteristic
346#398—24no—
347#397—24no—
348#395—24no—
349#393—24no—
350#391—24no—
351#388—24no—
352C2×#95C22×C3112yes—
353C2×#95C22×C3112yes—
354C2×#95C22×C3112yes—
355C2×#95C22×C3112yes—
356C2×#95C22×C3112yes—
357C2×#95C22×C3112yes—
358C2×#95C22×C3111yescharacteristic
359C2×#95C22×C3111yescharacteristic
360C2×#95C22×C3112yes—
361C2×#95C22×C3112yes—
362C2×#95C22×C3111yescharacteristic
363C2×#95C22×C3111yescharacteristic
364C2×C31—82no—
365C2×C31#253911yescharacteristic
366C2×C31#254011yescharacteristic
367C2×C31#259911yescharacteristic
368C2×C31#263711yescharacteristic
369C2×C31#263811yescharacteristic
370C2×C31#265811yescharacteristic
371C2×C31#268712yes—
372C2×C31#269012yes—
373C2×C31#270012yes—
374C2×C31#271612yes—
375C2×C31#281411yescharacteristic
376C25C23×C3111yescharacteristic
377C22×Q8—22no—
378C22×Q8—22no—
379C22×Q8—22no—
380C22×Q8—24no—
381C22×Q8—42no—
382C23×C4C23×C3111yescharacteristic
383C23×C4C23×C3112yes—
384C23×C4C23×C3111yescharacteristic
385C23×C4C23×C3112yes—
386C23×C4C23×C3112yes—
387C23×C4C23×C3111yescharacteristic
388C23×C4C23×C3111yescharacteristic
389C23×C4C23×C3111yescharacteristic
390C23×C4C23×C3112yes—
391C23×C4C23×C3111yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—42no—
403C2×C4⋊C4—42no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4—42no—
418C2×C42—42no—
419C2×C42D4×C3114yes—
420#95—22no—
421#95—22no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—42no—
429#95—42no—
430#95D4×C3114yes—
431#95D4×C3114yes—
432C31#1681211yescharacteristic, direct product factor, Sylow
433C24—21no—
434C24—21no—
435C24—21no—
436C24—21no—
437C24—22no—
438C24—22no—
439C24—21no—
440C24—22no—
441C24—21no—
442C24—22no—
443C24#41×C3112yes—
444C24C24×C3111yescharacteristic
445C2×Q8—28no—
446C2×Q8—48no—
447C2×Q8—48no—
448C2×Q8—44no—
449C2×Q8—44no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4C2×D4×C3112yes—
502C22×C4C2×D4×C3112yes—
503C22×C4C2×D4×C3114yes—
504C22×C4C2×D4×C3114yes—
505C22×C4C2×D4×C3112yes—
506C22×C4C2×D4×C3112yes—
507C22×C4C2×D4×C3112yes—
508C22×C4#41×C3112yes—
509C22×C4#41×C3114yes—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C22⋊C4—28no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C42—44no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C22×D4×C3111yescharacteristic
544C23C22×D4×C3111yescharacteristic
545C23C22×D4×C3111yescharacteristic
546C23C2×#41×C3111yescharacteristic
547C23C2×#41×C3112yes—
548C23C2×#41×C3111yescharacteristic
549C23C31×H(5,𝔽2)12yes—
550C23C31×H(5,𝔽2)11yescharacteristic
551C23C31×2−1+412yes—
552C23C31×2−1+412yes—
553C23C31×2−1+411yescharacteristic
554Q8—416no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4C31×#12714yes—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—82no—
623C22—82no—
624C22—81no—
625C22—81no—
626C22C31×#52511yescharacteristic
627C22C31×#52511yescharacteristic
628C22C2×C31×#12711yescharacteristic
629C22C31×#53411yescharacteristic
630C22C31×#53611yescharacteristic
631C22C31×#53611yescharacteristic
632C22C31×#53612yes—
633C22C31×#53612yes—
634C22C31×#53711yescharacteristic
635C22C31×#53912yes—
636C22C31×#54112yes—
637C22C31×#54212yes—
638C22C31×#54212yes—
639C22C31×#54611yescharacteristic
640C22C31×#54612yes—
641C22C31×#54612yes—
642C22C31×#54611yescharacteristic
643C22C31×#54811yescharacteristic
644C22C31×#54811yescharacteristic
645C22C31×#55212yes—
646C22C31×#55612yes—
647C22C31×#56212yes—
648C22C31×#56312yes—
649C4—24no—
650C4—42no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—82no—
659C2—82no—
660C2C31×#253911yescharacteristic
661C2C31×#254011yescharacteristic
662C2C31×#259911yescharacteristic
663C2C31×#263711yescharacteristic
664C2C31×#263811yescharacteristic
665C2C31×#265811yescharacteristic
666C2C31×#268712yes—
667C2C31×#269012yes—
668C2C31×#270012yes—
669C2C31×#271612yes—
670C2C31×#281411yescharacteristic
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
1. C2×C31×#398C2semidirect
3. C2×C31×#395C2semidirect
5. C2×C31×#391C2semidirect
7. C31×#1445C2semidirect
8. C31×#1445C2semidirect
11. C31×#1434C2semidirect
184. #16812C31direct (N × H)
280. C2×#398C2×C31semidirect
282. C2×#395C2×C31semidirect
284. C2×#391C2×C31semidirect
286. #1445C2×C31semidirect
287. #1445C2×C31semidirect
290. #1434C2×C31semidirect
432. C31#16812direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3982
C2×C31×#3972
C2×C31×#3952
C2×C31×#3932
C2×C31×#3912
C2×C31×#3882
C31×#14452
C31×#14392
C31×#14342
#1681231