Finite Group Databasebeta

← C31×#24659 #21481418

Subgroups — C31×#24659 #21481418

Order 7936.

960 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#461C211yesmaximal, characteristic, complemented
2C2×C31×#432C211yesmaximal, characteristic, complemented
3C31×#2857C211yesmaximal, characteristic, complemented
4C4×C31×#121C211yesmaximal, characteristic
5C31×#2036C212yesmaximal, complemented
6C31×#2000C212yesmaximal
7C31×#1981C211yesmaximal, characteristic, complemented
8C31×#1973C212yesmaximal, complemented
9C31×#1947C211yesmaximal, characteristic, complemented
10C31×#1935C212yesmaximal, complemented
11C31×#1802C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C2×C31×C4⋊Q8C2211yescharacteristic
14C2×C31×C22⋊Q8C2212yes—
15C2×C31×#121C2211yescharacteristic
16C2×C4×D4×C31—21no—
17C2×C4×D4×C31C2211yescharacteristic
18C22×C42×C31C2211yescharacteristic
19C31×#461—24no—
20C31×#432—24no—
21C2×C31×#107C2212yes—
22C2×C31×C4⋊C8C2211yescharacteristic
23C2×C31×C4⋊C8C2211yescharacteristic
24C2×C31×#103C2212yes—
25C2×C31×#103C2212yes—
26C2×C31×#103C2212yes—
27C2×C31×#102C2212yes—
28C2×C31×C22⋊C8C2212yes—
29C2×C31×#97C2212yes—
30C31×#391C2212yes—
31C31×#390C2211yescharacteristic
32C31×#389—21no—
33C31×#389—21no—
34C31×#389C2211yescharacteristic
35C31×#388—21no—
36C31×#386—21no—
37C31×#384C2211yescharacteristic
38C31×#380—21no—
39C31×#380—21no—
40C31×#380C2211yescharacteristic
41C4×C4⋊C4×C31C2211yescharacteristic
42C4×C22⋊C4×C31—21no—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C22×Q8×C31C2312yes—
46C22×D4×C31—21no—
47C22×D4×C31C2311yescharacteristic
48C23×C4×C31—21no—
49C23×C4×C31D411yescharacteristic
50C23×C4×C31D411yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C2×SD16×C31—24no—
53C2×SD16×C31—24no—
54C2×SD16×C31—42no—
55C22×C8×C31C2312yes—
56C22×C8×C31C2312yes—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—22no—
59C31×C22⋊Q8—42no—
60C31×C22⋊Q8—42no—
61C31×#121—22no—
62C31×#121—22no—
63C4×D4×C31—22no—
64C4×D4×C31—22no—
65C4×D4×C31—44no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31C2312yes—
73C2×C4⋊C4×C31C2311yescharacteristic
74C2×C4⋊C4×C31C2311yescharacteristic
75C2×C4⋊C4×C31C2311yescharacteristic
76C2×C22⋊C4×C31—21no—
77C2×C22⋊C4×C31—21no—
78C2×C22⋊C4×C31—22no—
79C2×C22⋊C4×C31—21no—
80C2×C22⋊C4×C31—21no—
81C2×C22⋊C4×C31—21no—
82C2×C42×C31—22no—
83C2×C42×C31—22no—
84C2×C42×C31—21no—
85C2×C42×C31D411yescharacteristic
86C2×C42×C31D411yescharacteristic
87C2×C42×C31C2311yescharacteristic
88C2×C42×C31C2311yescharacteristic
89C31×#107—42no—
90C31×C4⋊C8—22no—
91C31×C4⋊C8D414yescomplemented
92C31×#103—24no—
93C31×#103—24no—
94C31×#103—42no—
95C31×#102—42no—
96C31×C22⋊C8—24no—
97C31×#97—24no—
98C31×#95—21no—
99C31×#95—21no—
100C31×#95—21no—
101C31×#95—22no—
102C31×#95—22no—
103C24×C31—21no—
104C24×C31C2×D411yescharacteristic
105C2×Q8×C31—22no—
106C2×Q8×C31—22no—
107C2×Q8×C31—22no—
108C2×Q8×C31—22no—
109C2×Q8×C31—22no—
110C2×Q8×C31—22no—
111C2×D4×C31—24no—
112C2×D4×C31—21no—
113C2×D4×C31—21no—
114C2×D4×C31—21no—
115C2×D4×C31—21no—
116C2×D4×C31—21no—
117C2×D4×C31—42no—
118C2×D4×C31—42no—
119C2×D4×C31#4112yes—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—41no—
137C22×C4×C31—42no—
138C22×C4×C31—42no—
139C22×C4×C31—42no—
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2411yescharacteristic
146SD16×C31—48no—
147C2×C8×C31—22no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C2×C8×C31C2×D414yes—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—42no—
161C4⋊C4×C31—42no—
162C4⋊C4×C31—44no—
163C4⋊C4×C31—42no—
164C4⋊C4×C31—42no—
165C4⋊C4×C31—42no—
166C4⋊C4×C31#4112yes—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—42no—
169C22⋊C4×C31—42no—
170C22⋊C4×C31—42no—
171C22⋊C4×C31—44no—
172C22⋊C4×C31—42no—
173C42×C31—22no—
174C42×C31—22no—
175C42×C31—21no—
176C42×C31—21no—
177C42×C31—42no—
178C42×C31C2×D412yes—
179C42×C31#4112yes—
180#24659C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—21no—
182C23×C31—21no—
183C23×C31—21no—
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—22no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31#13712yes—
195C23×C31C22×D411yescharacteristic
196Q8×C31—42no—
197Q8×C31—42no—
198Q8×C31—42no—
199Q8×C31—42no—
200D4×C31—22no—
201D4×C31—22no—
202D4×C31—48no—
203D4×C31—41no—
204D4×C31—41no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—22no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—44no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—44no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—42no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—42no—
237C2×C4×C31—84no—
238C2×C4×C31#12112yes—
239C2×C4×C31#12112yes—
240C2×C4×C31AGL(1,ℤ/8)12yes—
241C2×C4×C31#13712yes—
242C2×C4×C31C22×D411yescharacteristic
243C2×C4×C31C2×#4111yescharacteristic
244C2×C4×C31C2×#4111yescharacteristic
245C2×C4×C31C2×#4111yescharacteristic
246C2×C4×C31H(5,𝔽2)11yescharacteristic
247C2×C4×C31H(5,𝔽2)11yescharacteristic
248C8×C31—24no—
249C8×C31—42no—
250C2×#461C2×C3111yescharacteristic, complemented
251C2×#432C2×C3111yescharacteristic, complemented
252#2857C2×C3111yescharacteristic, complemented
253C4×#121C2×C3111yescharacteristic
254#2036C2×C3112yescomplemented
255#2000C2×C3112yes—
256#1981C2×C3111yescharacteristic, complemented
257#1973C2×C3112yescomplemented
258#1947C2×C3111yescharacteristic, complemented
259#1935C2×C3112yescomplemented
260#1802C2×C3111yescharacteristic, complemented
261C22×C31—22no—
262C22×C31—22no—
263C22×C31—22no—
264C22×C31—41no—
265C22×C31—41no—
266C22×C31—41no—
267C22×C31—41no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—42no—
271C22×C31—81no—
272C22×C31—81no—
273C22×C31—81no—
274C22×C31—81no—
275C22×C31C2×#12111yescharacteristic
276C22×C31#52511yescharacteristic
277C22×C31#53511yescharacteristic
278C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C31C2×#13711yescharacteristic
280C22×C31C2×#13711yescharacteristic
281C22×C31#57511yescharacteristic
282C4×C31—22no—
283C4×C31—42no—
284C4×C31—41no—
285C4×C31—41no—
286C4×C31—41no—
287C4×C31—82no—
288C4×C31—82no—
289C4×C31—81no—
290C4×C31—81no—
291C4×C31#46814yes—
292C4×C31#54611yescharacteristic
293C4×C31#55011yescharacteristic
294C4×C31#55311yescharacteristic
295C4×C31#55511yescharacteristic
296C22×SD16C22×C3112yes—
297C2×C4⋊Q8C22×C3111yescharacteristic
298C2×C22⋊Q8C22×C3112yes—
299C2×#121C22×C3111yescharacteristic
300C2×C4×D4—21no—
301C2×C4×D4C22×C3111yescharacteristic
302C22×C42C22×C3111yescharacteristic
303#461—24no—
304#432—24no—
305C2×#107C22×C3112yes—
306C2×C4⋊C8C22×C3111yescharacteristic
307C2×C4⋊C8C22×C3111yescharacteristic
308C2×#103C22×C3112yes—
309C2×#103C22×C3112yes—
310C2×#103C22×C3112yes—
311C2×#102C22×C3112yes—
312C2×C22⋊C8C22×C3112yes—
313C2×#97C22×C3112yes—
314#391C22×C3112yes—
315#390C22×C3111yescharacteristic
316#389—21no—
317#389—21no—
318#389C22×C3111yescharacteristic
319#388—21no—
320#386—21no—
321#384C22×C3111yescharacteristic
322#380—21no—
323#380—21no—
324#380C22×C3111yescharacteristic
325C4×C4⋊C4C22×C3111yescharacteristic
326C4×C22⋊C4—21no—
327#340C22×C3112yes—
328#340C22×C3112yes—
329C2×C31—42no—
330C2×C31—82no—
331C2×C31#241411yescharacteristic
332C2×C31C2×#46811yescharacteristic
333C2×C31#305611yescharacteristic
334C2×C31#311011yescharacteristic
335C2×C31#311611yescharacteristic
336C2×C31#321411yescharacteristic
337C2×C31#321811yescharacteristic
338C22×Q8C23×C3112yes—
339C22×D4—21no—
340C22×D4C23×C3111yescharacteristic
341C23×C4—21no—
342C23×C4D4×C3111yescharacteristic
343C23×C4D4×C3111yescharacteristic
344C23×C4C23×C3111yescharacteristic
345C2×SD16—24no—
346C2×SD16—24no—
347C2×SD16—42no—
348C22×C8C23×C3112yes—
349C22×C8C23×C3112yes—
350C4⋊Q8—22no—
351C4⋊Q8—22no—
352C22⋊Q8—42no—
353C22⋊Q8—42no—
354#121—22no—
355#121—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—44no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4C23×C3111yescharacteristic
366C2×C4⋊C4C23×C3112yes—
367C2×C4⋊C4C23×C3111yescharacteristic
368C2×C4⋊C4C23×C3111yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42D4×C3111yescharacteristic
379C2×C42D4×C3111yescharacteristic
380C2×C42C23×C3111yescharacteristic
381C2×C42C23×C3111yescharacteristic
382#107—42no—
383C4⋊C8—22no—
384C4⋊C8D4×C3114yescomplemented
385#103—24no—
386#103—24no—
387#103—42no—
388#102—42no—
389C22⋊C8—24no—
390#97—24no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—22no—
395#95—22no—
396C31#2465911yescharacteristic, direct product factor, Sylow
397C24—21no—
398C24C2×D4×C3111yescharacteristic
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4#41×C3112yes—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C2×D4×C3111yescharacteristic
439C22×C4C24×C3111yescharacteristic
440SD16—48no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8C2×D4×C3114yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4#41×C3112yes—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—42no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42C2×D4×C3112yes—
473C42#41×C3112yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C31×#13712yes—
488C23C22×D4×C3111yescharacteristic
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493D4—22no—
494D4—22no—
495D4—48no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—84no—
531C2×C4C31×#12112yes—
532C2×C4C31×#12112yes—
533C2×C4C31×AGL(1,ℤ/8)12yes—
534C2×C4C31×#13712yes—
535C2×C4C22×D4×C3111yescharacteristic
536C2×C4C2×#41×C3111yescharacteristic
537C2×C4C2×#41×C3111yescharacteristic
538C2×C4C2×#41×C3111yescharacteristic
539C2×C4C31×H(5,𝔽2)11yescharacteristic
540C2×C4C31×H(5,𝔽2)11yescharacteristic
541C8—24no—
542C8—42no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×C31×#12111yescharacteristic
558C22C31×#52511yescharacteristic
559C22C31×#53511yescharacteristic
560C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
561C22C2×C31×#13711yescharacteristic
562C22C2×C31×#13711yescharacteristic
563C22C31×#57511yescharacteristic
564C4—22no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—82no—
570C4—82no—
571C4—81no—
572C4—81no—
573C4C31×#46814yes—
574C4C31×#54611yescharacteristic
575C4C31×#55011yescharacteristic
576C4C31×#55311yescharacteristic
577C4C31×#55511yescharacteristic
578C2—42no—
579C2—82no—
580C2C31×#241411yescharacteristic
581C2C2×C31×#46811yescharacteristic
582C2C31×#305611yescharacteristic
583C2C31×#311011yescharacteristic
584C2C31×#311611yescharacteristic
585C2C31×#321411yescharacteristic
586C2C31×#321811yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#461C2semidirect
2. C2×C31×#432C2semidirect
3. C31×#2857C2semidirect
5. C31×#2036C2semidirect
7. C31×#1981C2semidirect
8. C31×#1973C2semidirect
9. C31×#1947C2semidirect
10. C31×#1935C2semidirect
11. C31×#1802C2semidirect
91. C31×C4⋊C8D4semidirect
180. #24659C31direct (N × H)
250. C2×#461C2×C31semidirect
251. C2×#432C2×C31semidirect
252. #2857C2×C31semidirect
254. #2036C2×C31semidirect
256. #1981C2×C31semidirect
257. #1973C2×C31semidirect
258. #1947C2×C31semidirect
259. #1935C2×C31semidirect
260. #1802C2×C31semidirect
384. C4⋊C8D4×C31semidirect
396. C31#24659direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4612
C2×C31×#4322
C31×#28572
C4×C31×#1212
C31×#20362
C31×#20002
C31×#19812
C31×#19732
C31×#19472
C31×#19352
C31×#18022
#2465931