Finite Group Databasebeta

← C31×#15983 #21472742

Subgroups — C31×#15983 #21472742

Order 7936.

1950 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#123C211yesmaximal, characteristic
2C2×C31×#390C211yesmaximal, characteristic
3C2×C31×#387C212yesmaximal, complemented
4C31×#1441C212yesmaximal, complemented
5C31×#1440C212yesmaximal, complemented
6C31×#1439C212yesmaximal, complemented
7C31×#1439C212yesmaximal, complemented
8C31×#1438C211yesmaximal, characteristic
9C31×C22⋊#95C212yesmaximal, complemented
10C23×D4×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic
12C2×C31×#123—24no—
13C2×C31×#123—28no—
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2212yes—
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C22⋊C4×C31C2211yescharacteristic
18C22×C22⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2211yescharacteristic
22C22×C22⋊C4×C31C2212yes—
23C22×C22⋊C4×C31C2211yescharacteristic
24C22×C42×C31C2211yescharacteristic
25C31×#390—24no—
26C31×#387—28no—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2212yes—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C25×C31C2311yescharacteristic
36C25×C31C2311yescharacteristic
37C22×D4×C31—24no—
38C22×D4×C31—24no—
39C22×D4×C31—41no—
40C22×D4×C31—41no—
41C22×D4×C31D414yes—
42C23×C4×C31—22no—
43C23×C4×C31—24no—
44C23×C4×C31—24no—
45C23×C4×C31—41no—
46C23×C4×C31—41no—
47C23×C4×C31C2312yes—
48C23×C4×C31C2312yes—
49C23×C4×C31C2312yes—
50C23×C4×C31C2312yes—
51C23×C4×C31C2312yes—
52C23×C4×C31C2312yes—
53C23×C4×C31C2311yescharacteristic
54C31×#123—232no—
55C2×C4⋊C4×C31—24no—
56C2×C4⋊C4×C31—24no—
57C2×C4⋊C4×C31—24no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—24no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—41no—
62C2×C4⋊C4×C31—41no—
63C2×C4⋊C4×C31D418yes—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—24no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31—41no—
77C2×C22⋊C4×C31—42no—
78C2×C22⋊C4×C31—42no—
79C2×C22⋊C4×C31—42no—
80C2×C22⋊C4×C31—41no—
81C2×C22⋊C4×C31—42no—
82C2×C42×C31—24no—
83C2×C42×C31—22no—
84C31×#95—24no—
85C31×#95—24no—
86C31×#95—24no—
87C31×#95—24no—
88C31×#95—42no—
89C31×#95—42no—
90C31×#95—42no—
91C31×#95—42no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—21no—
97C24×C31—21no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—22no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—22no—
104C24×C31—21no—
105C24×C31—21no—
106C24×C31—22no—
107C24×C31C2×D412yes—
108C24×C31C2×D412yes—
109C24×C31#4114yes—
110C24×C31C2411yescharacteristic
111C2×D4×C31—28no—
112C2×D4×C31—28no—
113C2×D4×C31—44no—
114C2×D4×C31—48no—
115C2×D4×C31—48no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—28no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
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—22no—
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—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—48no—
159C22×C4×C31—48no—
160C22×C4×C31—44no—
161C22×C4×C31—44no—
162C22×C4×C31C2×D414yes—
163C22×C4×C31C2×D414yes—
164C22×C4×C31C2×D414yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31#4114yes—
167C22×C4×C31#4114yes—
168C4⋊C4×C31—216no—
169C4⋊C4×C31—216no—
170C4⋊C4×C31—44no—
171C22⋊C4×C31—28no—
172C22⋊C4×C31—216no—
173C22⋊C4×C31—44no—
174C22⋊C4×C31—48no—
175C22⋊C4×C31—48no—
176C22⋊C4×C31—48no—
177C22⋊C4×C31—44no—
178C42×C31—44no—
179#15983C3111yesmaximal, characteristic, direct product factor, Sylow
180C23×C31—22no—
181C23×C31—24no—
182C23×C31—22no—
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—24no—
186C23×C31—24no—
187C23×C31—24no—
188C23×C31—24no—
189C23×C31—24no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—22no—
193C23×C31—24no—
194C23×C31—24no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—41no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31—41no—
221C23×C31#12314yes—
222C23×C31C22×D411yescharacteristic
223C23×C31C22×D412yes—
224C23×C31C2×#4112yes—
225C23×C31C2×#4112yes—
226C23×C31C2×#4112yes—
227C23×C31H(5,𝔽2)11yescharacteristic
228C23×C31H(5,𝔽2)11yescharacteristic
229C23×C31H(5,𝔽2)12yes—
230C23×C312−1+412yes—
231D4×C31—416no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—24no—
235C2×C4×C31—24no—
236C2×C4×C31—24no—
237C2×C4×C31—24no—
238C2×C4×C31—24no—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—24no—
243C2×C4×C31—24no—
244C2×C4×C31—24no—
245C2×C4×C31—24no—
246C2×C4×C31—24no—
247C2×C4×C31—24no—
248C2×C4×C31—416no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
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—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#12014yes—
280C22×#123C2×C3111yescharacteristic
281C2×#390C2×C3111yescharacteristic
282C2×#387C2×C3112yescomplemented
283#1441C2×C3112yescomplemented
284#1440C2×C3112yescomplemented
285#1439C2×C3112yescomplemented
286#1439C2×C3112yescomplemented
287#1438C2×C3111yescharacteristic
288C22⋊#95C2×C3112yescomplemented
289C22×C31—24no—
290C22×C31—28no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—42no—
295C22×C31—44no—
296C22×C31—42no—
297C22×C31—44no—
298C22×C31—42no—
299C22×C31—44no—
300C22×C31—44no—
301C22×C31—42no—
302C22×C31—42no—
303C22×C31—82no—
304C22×C31—82no—
305C22×C31—82no—
306C22×C31—82no—
307C22×C31C2×#12011yescharacteristic
308C22×C31C2×#12311yescharacteristic
309C22×C31#52512yes—
310C22×C31#52912yes—
311C22×C31#53411yescharacteristic
312C22×C31#53712yes—
313C22×C31#53912yes—
314C22×C31#54012yes—
315C22×C31#54212yes—
316C22×C31#54612yes—
317C22×C31#54612yes—
318C22×C31#54612yes—
319C22×C31#54712yes—
320C22×C31#54812yes—
321C22×C31#55012yes—
322C22×C31#55112yes—
323C22×C31#55111yescharacteristic
324C22×C31#55212yes—
325C22×C31#55911yescharacteristic
326C22×C31#56011yescharacteristic
327C22×C31#56311yescharacteristic
328C4×C31—24no—
329C4×C31—44no—
330C4×C31—44no—
331C4×C31—44no—
332C4×C31—44no—
333C4×C31—82no—
334C4×C31—82no—
335C23×D4C22×C3111yescharacteristic
336C24×C4C22×C3111yescharacteristic
337C2×#123—24no—
338C2×#123—28no—
339C22×C4⋊C4C22×C3112yes—
340C22×C4⋊C4C22×C3112yes—
341C22×C4⋊C4C22×C3111yescharacteristic
342C22×C22⋊C4C22×C3111yescharacteristic
343C22×C22⋊C4C22×C3112yes—
344C22×C22⋊C4C22×C3112yes—
345C22×C22⋊C4C22×C3112yes—
346C22×C22⋊C4C22×C3111yescharacteristic
347C22×C22⋊C4C22×C3112yes—
348C22×C22⋊C4C22×C3111yescharacteristic
349C22×C42C22×C3111yescharacteristic
350#390—24no—
351#387—28no—
352C2×#95C22×C3112yes—
353C2×#95C22×C3112yes—
354C2×#95C22×C3112yes—
355C2×#95C22×C3112yes—
356C2×#95C22×C3112yes—
357C2×#95C22×C3112yes—
358C2×#95C22×C3112yes—
359C2×#95C22×C3112yes—
360C2×C31—44no—
361C2×C31—82no—
362C2×C31#240511yescharacteristic
363C2×C31#240712yes—
364C2×C31#241912yes—
365C2×C31#245812yes—
366C2×C31#247112yes—
367C2×C31#270512yes—
368C2×C31#270712yes—
369C2×C31#270911yescharacteristic
370C2×C31#271711yescharacteristic
371C25C23×C3111yescharacteristic
372C25C23×C3111yescharacteristic
373C22×D4—24no—
374C22×D4—24no—
375C22×D4—41no—
376C22×D4—41no—
377C22×D4D4×C3114yes—
378C23×C4—22no—
379C23×C4—24no—
380C23×C4—24no—
381C23×C4—41no—
382C23×C4—41no—
383C23×C4C23×C3112yes—
384C23×C4C23×C3112yes—
385C23×C4C23×C3112yes—
386C23×C4C23×C3112yes—
387C23×C4C23×C3112yes—
388C23×C4C23×C3112yes—
389C23×C4C23×C3111yescharacteristic
390#123—232no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—41no—
399C2×C4⋊C4D4×C3118yes—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—41no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—41no—
417C2×C22⋊C4—42no—
418C2×C42—24no—
419C2×C42—22no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—42no—
425#95—42no—
426#95—42no—
427#95—42no—
428C31#1598311yescharacteristic, direct product factor, Sylow
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—21no—
434C24—21no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—21no—
442C24—21no—
443C24—22no—
444C24C2×D4×C3112yes—
445C24C2×D4×C3112yes—
446C24#41×C3114yes—
447C24C24×C3111yescharacteristic
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—44no—
451C2×D4—48no—
452C2×D4—48no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
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—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4C2×D4×C3114yes—
500C22×C4C2×D4×C3114yes—
501C22×C4C2×D4×C3114yes—
502C22×C4C2×D4×C3112yes—
503C22×C4#41×C3114yes—
504C22×C4#41×C3114yes—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—44no—
508C22⋊C4—28no—
509C22⋊C4—216no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C42—44no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C31×#12314yes—
558C23C22×D4×C3111yescharacteristic
559C23C22×D4×C3112yes—
560C23C2×#41×C3112yes—
561C23C2×#41×C3112yes—
562C23C2×#41×C3112yes—
563C23C31×H(5,𝔽2)11yescharacteristic
564C23C31×H(5,𝔽2)11yescharacteristic
565C23C31×H(5,𝔽2)12yes—
566C23C31×2−1+412yes—
567D4—416no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—416no—
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×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
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×C4—42no—
615C2×C4C31×#12014yes—
616C22—24no—
617C22—28no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×C31×#12011yescharacteristic
635C22C2×C31×#12311yescharacteristic
636C22C31×#52512yes—
637C22C31×#52912yes—
638C22C31×#53411yescharacteristic
639C22C31×#53712yes—
640C22C31×#53912yes—
641C22C31×#54012yes—
642C22C31×#54212yes—
643C22C31×#54612yes—
644C22C31×#54612yes—
645C22C31×#54612yes—
646C22C31×#54712yes—
647C22C31×#54812yes—
648C22C31×#55012yes—
649C22C31×#55112yes—
650C22C31×#55111yescharacteristic
651C22C31×#55212yes—
652C22C31×#55911yescharacteristic
653C22C31×#56011yescharacteristic
654C22C31×#56311yescharacteristic
655C4—24no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—82no—
661C4—82no—
662C2—44no—
663C2—82no—
664C2C31×#240511yescharacteristic
665C2C31×#240712yes—
666C2C31×#241912yes—
667C2C31×#245812yes—
668C2C31×#247112yes—
669C2C31×#270512yes—
670C2C31×#270712yes—
671C2C31×#270911yescharacteristic
672C2C31×#271711yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#387C2semidirect
4. C31×#1441C2semidirect
5. C31×#1440C2semidirect
6. C31×#1439C2semidirect
7. C31×#1439C2semidirect
9. C31×C22⋊#95C2semidirect
179. #15983C31direct (N × H)
282. C2×#387C2×C31semidirect
283. #1441C2×C31semidirect
284. #1440C2×C31semidirect
285. #1439C2×C31semidirect
286. #1439C2×C31semidirect
288. C22⋊#95C2×C31semidirect
428. C31#15983direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1232
C2×C31×#3902
C2×C31×#3872
C31×#14412
C31×#14402
C31×#14392
C31×#14382
C31×C22⋊#952
#1598331