Finite Group Databasebeta

← C31×#23973 #21480732

Subgroups — C31×#23973 #21480732

Order 7936.

1488 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#553C211yesmaximal, characteristic, complemented
2C2×C31×#447C211yesmaximal, characteristic, complemented
3C2×C31×#418C211yesmaximal, characteristic, complemented
4C31×#2408C211yesmaximal, characteristic, complemented
5C31×#2025C212yesmaximal, complemented
6C31×#1999C212yesmaximal, complemented
7C31×#1895C211yesmaximal, characteristic, complemented
8C31×#1879C212yesmaximal, complemented
9C31×#1874C212yesmaximal, complemented
10C31×#1870C211yesmaximal, characteristic, complemented
11C31×#1825C211yesmaximal, characteristic, complemented
12C23×D4×C31C2211yescharacteristic
13C22×D8×C31C2212yes—
14C22×M16×C31C2211yescharacteristic
15C31×#553—24no—
16C31×#553—24no—
17C2×C31×#126—21no—
18C2×C31×#125C2211yescharacteristic, complemented
19C2×C31×#123—21no—
20C2×C31×#121—21no—
21C2×C31×#121C2212yes—
22C2×C31×#121C2211yescharacteristic
23C2×C31×#121C2211yescharacteristic, complemented
24C2×C31×#120—21no—
25C2×C4×D4×C31—21no—
26C2×C4×D4×C31C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic, complemented
28C2×C31×#117C2211yescharacteristic
29C22×C4⋊C4×C31C2211yescharacteristic, complemented
30C31×#447—24no—
31C31×#418—24no—
32C2×C31×#102C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#102C2212yescomplemented
36C2×C31×C22⋊C8C2211yescharacteristic
37C2×C31×C22⋊C8C2212yescomplemented
38C31×#394—21no—
39C31×#392—21no—
40C31×#390C2212yes—
41C31×#380C2211yescharacteristic
42C31×#379—21no—
43C31×#340C2212yescomplemented
44C31×#340C2212yescomplemented
45C31×#340C2212yescomplemented
46C31×#336C2212yescomplemented
47C25×C31—21no—
48C22×D4×C31—24no—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31—22no—
52C22×D4×C31—22no—
53C22×D4×C31—22no—
54C22×D4×C31—21no—
55C22×D4×C31—42no—
56C22×D4×C31D411yescharacteristic
57C22×D4×C31C2311yescharacteristic
58C22×D4×C31C2312yes—
59C23×C4×C31—21no—
60C23×C4×C31—21no—
61C23×C4×C31C2311yescharacteristic
62C23×C4×C31C2311yescharacteristic
63C2×D8×C31—24no—
64C2×D8×C31—24no—
65C2×D8×C31—42no—
66C2×M16×C31—22no—
67C2×M16×C31—42no—
68C22×C8×C31C2312yes—
69C22×C8×C31C2312yes—
70C31×#126—28no—
71C31×#125—22no—
72C31×#125—22no—
73C31×#123—28no—
74C31×#121—28no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—24no—
79C31×#121—42no—
80C31×#121D414yescomplemented
81C31×#120—44no—
82C4×D4×C31—28no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C31×#117—22no—
88C31×#117D414yescomplemented
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—22no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31D411yescharacteristic
95C2×C4⋊C4×C31C2312yes—
96C2×C4⋊C4×C31C2311yescharacteristic
97C2×C4⋊C4×C31C2311yescharacteristic
98C2×C4⋊C4×C31C2311yescharacteristic
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31—22no—
106C2×C22⋊C4×C31—22no—
107C2×C22⋊C4×C31D411yescharacteristic
108C2×C22⋊C4×C31D411yescharacteristic
109C2×C42×C31—21no—
110C2×C42×C31C2311yescharacteristic
111C31×#102—24no—
112C31×#102—24no—
113C31×#102—24no—
114C31×#102—42no—
115C31×C22⋊C8—22no—
116C31×C22⋊C8—42no—
117C31×#95—21no—
118C24×C31—21no—
119C24×C31—22no—
120C24×C31—44no—
121C24×C31—42no—
122C24×C31—42no—
123C24×C31—42no—
124C24×C31—82no—
125C24×C31C2×D411yescharacteristic
126C24×C31C2×D411yescharacteristic
127C2×D4×C31—28no—
128C2×D4×C31—24no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—24no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—21no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—48no—
141C2×D4×C31—41no—
142C2×D4×C31—44no—
143C2×D4×C31—44no—
144C2×D4×C31—42no—
145C2×D4×C31—42no—
146C2×D4×C31—44no—
147C2×D4×C31—44no—
148C2×D4×C31—42no—
149C2×D4×C31—82no—
150C2×D4×C31—82no—
151C2×D4×C31C2×D412yes—
152C2×D4×C31#4114yes—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—24no—
159C22×C4×C31—24no—
160C22×C4×C31—21no—
161C22×C4×C31—21no—
162C22×C4×C31—22no—
163C22×C4×C31—22no—
164C22×C4×C31—22no—
165C22×C4×C31—82no—
166C22×C4×C31—82no—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D412yes—
172C22×C4×C31C2×D412yes—
173C22×C4×C31C2411yescharacteristic
174D8×C31—48no—
175M16×C31—44no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C2×C8×C31—22no—
181C2×C8×C31—22no—
182C4⋊C4×C31—24no—
183C4⋊C4×C31—24no—
184C4⋊C4×C31—24no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31C2×D412yes—
192C4⋊C4×C31C2×D412yes—
193C4⋊C4×C31C2×D412yes—
194C22⋊C4×C31—24no—
195C22⋊C4×C31—24no—
196C22⋊C4×C31—24no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—24no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—44no—
201C22⋊C4×C31—44no—
202C22⋊C4×C31—42no—
203C22⋊C4×C31—82no—
204C42×C31—24no—
205C42×C31—21no—
206C42×C31—21no—
207#23973C3111yesmaximal, characteristic, direct product factor, Sylow
208C23×C31—21no—
209C23×C31—22no—
210C23×C31—22no—
211C23×C31—21no—
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—22no—
215C23×C31—22no—
216C23×C31—22no—
217C23×C31—24no—
218C23×C31—48no—
219C23×C31—42no—
220C23×C31—42no—
221C23×C31—42no—
222C23×C31—42no—
223C23×C31—41no—
224C23×C31—41no—
225C23×C31—42no—
226C23×C31—42no—
227C23×C31—42no—
228C23×C31—42no—
229C23×C31—84no—
230C23×C31—84no—
231C23×C31—88no—
232C23×C31—88no—
233C23×C31#12012yes—
234C23×C31AGL(1,ℤ/8)12yes—
235C23×C31C22×D411yescharacteristic
236D4×C31—24no—
237D4×C31—24no—
238D4×C31—24no—
239D4×C31—48no—
240D4×C31—44no—
241D4×C31—48no—
242D4×C31—41no—
243D4×C31—41no—
244D4×C31—42no—
245D4×C31—42no—
246D4×C31—84no—
247D4×C31—88no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—24no—
252C2×C4×C31—21no—
253C2×C4×C31—22no—
254C2×C4×C31—21no—
255C2×C4×C31—22no—
256C2×C4×C31—21no—
257C2×C4×C31—22no—
258C2×C4×C31—22no—
259C2×C4×C31—21no—
260C2×C4×C31—21no—
261C2×C4×C31—22no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—84no—
273C2×C4×C31—84no—
274C2×C4×C31#12012yes—
275C2×C4×C31C22×D411yescharacteristic
276C2×C4×C31C22×D411yescharacteristic
277C2×C4×C31C2×#4112yes—
278C2×C4×C31H(5,𝔽2)12yes—
279C8×C31—42no—
280C8×C31—42no—
281C2×#553C2×C3111yescharacteristic, complemented
282C2×#447C2×C3111yescharacteristic, complemented
283C2×#418C2×C3111yescharacteristic, complemented
284#2408C2×C3111yescharacteristic, complemented
285#2025C2×C3112yescomplemented
286#1999C2×C3112yescomplemented
287#1895C2×C3111yescharacteristic, complemented
288#1879C2×C3112yescomplemented
289#1874C2×C3112yescomplemented
290#1870C2×C3111yescharacteristic, complemented
291#1825C2×C3111yescharacteristic, complemented
292C22×C31—22no—
293C22×C31—22no—
294C22×C31—22no—
295C22×C31—22no—
296C22×C31—24no—
297C22×C31—41no—
298C22×C31—41no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—42no—
302C22×C31—44no—
303C22×C31—42no—
304C22×C31—42no—
305C22×C31—42no—
306C22×C31—44no—
307C22×C31—44no—
308C22×C31—816no—
309C22×C31—82no—
310C22×C31—82no—
311C22×C31—82no—
312C22×C31—82no—
313C22×C31—82no—
314C22×C31—82no—
315C22×C31C2×#12011yescharacteristic
316C22×C31#54612yes—
317C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
318C22×C31#57511yescharacteristic
319C22×C31#57612yes—
320C4×C31—22no—
321C4×C31—41no—
322C4×C31—41no—
323C4×C31—42no—
324C4×C31—41no—
325C4×C31—42no—
326C4×C31—42no—
327C4×C31—82no—
328C4×C31#52511yescharacteristic
329C4×C31#53511yescharacteristic
330C4×C31#54612yes—
331C23×D4C22×C3111yescharacteristic
332C22×D8C22×C3112yes—
333C22×M16C22×C3111yescharacteristic
334#553—24no—
335#553—24no—
336C2×#126—21no—
337C2×#125C22×C3111yescharacteristic, complemented
338C2×#123—21no—
339C2×#121—21no—
340C2×#121C22×C3112yes—
341C2×#121C22×C3111yescharacteristic
342C2×#121C22×C3111yescharacteristic, complemented
343C2×#120—21no—
344C2×C4×D4—21no—
345C2×C4×D4C22×C3111yescharacteristic
346C2×C4×D4C22×C3111yescharacteristic, complemented
347C2×#117C22×C3111yescharacteristic
348C22×C4⋊C4C22×C3111yescharacteristic, complemented
349#447—24no—
350#418—24no—
351C2×#102C22×C3112yes—
352C2×#102C22×C3112yes—
353C2×#102C22×C3112yes—
354C2×#102C22×C3112yescomplemented
355C2×C22⋊C8C22×C3111yescharacteristic
356C2×C22⋊C8C22×C3112yescomplemented
357#394—21no—
358#392—21no—
359#390C22×C3112yes—
360#380C22×C3111yescharacteristic
361#379—21no—
362#340C22×C3112yescomplemented
363#340C22×C3112yescomplemented
364#340C22×C3112yescomplemented
365#336C22×C3112yescomplemented
366C2×C31—42no—
367C2×C31—42no—
368C2×C31—82no—
369C2×C31—84no—
370C2×C31#240411yescharacteristic
371C2×C31#300611yescharacteristic
372C2×C31#300811yescharacteristic
373C2×C31#331812yes—
374C2×C31#332212yes—
375C25—21no—
376C22×D4—24no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—22no—
380C22×D4—22no—
381C22×D4—22no—
382C22×D4—21no—
383C22×D4—42no—
384C22×D4D4×C3111yescharacteristic
385C22×D4C23×C3111yescharacteristic
386C22×D4C23×C3112yes—
387C23×C4—21no—
388C23×C4—21no—
389C23×C4C23×C3111yescharacteristic
390C23×C4C23×C3111yescharacteristic
391C2×D8—24no—
392C2×D8—24no—
393C2×D8—42no—
394C2×M16—22no—
395C2×M16—42no—
396C22×C8C23×C3112yes—
397C22×C8C23×C3112yes—
398#126—28no—
399#125—22no—
400#125—22no—
401#123—28no—
402#121—28no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—24no—
407#121—42no—
408#121D4×C3114yescomplemented
409#120—44no—
410C4×D4—28no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415#117—22no—
416#117D4×C3114yescomplemented
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4D4×C3111yescharacteristic
423C2×C4⋊C4C23×C3112yes—
424C2×C4⋊C4C23×C3111yescharacteristic
425C2×C4⋊C4C23×C3111yescharacteristic
426C2×C4⋊C4C23×C3111yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4D4×C3111yescharacteristic
436C2×C22⋊C4D4×C3111yescharacteristic
437C2×C42—21no—
438C2×C42C23×C3111yescharacteristic
439#102—24no—
440#102—24no—
441#102—24no—
442#102—42no—
443C22⋊C8—22no—
444C22⋊C8—42no—
445#95—21no—
446C31#2397311yescharacteristic, direct product factor, Sylow
447C24—21no—
448C24—22no—
449C24—44no—
450C24—42no—
451C24—42no—
452C24—42no—
453C24—82no—
454C24C2×D4×C3111yescharacteristic
455C24C2×D4×C3111yescharacteristic
456C2×D4—28no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—48no—
470C2×D4—41no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—42no—
478C2×D4—82no—
479C2×D4—82no—
480C2×D4C2×D4×C3112yes—
481C2×D4#41×C3114yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4C2×D4×C3111yescharacteristic
497C22×C4C2×D4×C3111yescharacteristic
498C22×C4C2×D4×C3111yescharacteristic
499C22×C4C2×D4×C3111yescharacteristic
500C22×C4C2×D4×C3112yes—
501C22×C4C2×D4×C3112yes—
502C22×C4C24×C3111yescharacteristic
503D8—48no—
504M16—44no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4C2×D4×C3112yes—
521C4⋊C4C2×D4×C3112yes—
522C4⋊C4C2×D4×C3112yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—22no—
527C22⋊C4—24no—
528C22⋊C4—22no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—82no—
533C42—24no—
534C42—21no—
535C42—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—48no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—88no—
560C23—88no—
561C23C31×#12012yes—
562C23C31×AGL(1,ℤ/8)12yes—
563C23C22×D4×C3111yescharacteristic
564D4—24no—
565D4—24no—
566D4—24no—
567D4—48no—
568D4—44no—
569D4—48no—
570D4—41no—
571D4—41no—
572D4—42no—
573D4—42no—
574D4—84no—
575D4—88no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4C31×#12012yes—
603C2×C4C22×D4×C3111yescharacteristic
604C2×C4C22×D4×C3111yescharacteristic
605C2×C4C2×#41×C3112yes—
606C2×C4C31×H(5,𝔽2)12yes—
607C8—42no—
608C8—42no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—24no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22C2×C31×#12011yescharacteristic
633C22C31×#54612yes—
634C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
635C22C31×#57511yescharacteristic
636C22C31×#57612yes—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—82no—
645C4C31×#52511yescharacteristic
646C4C31×#53511yescharacteristic
647C4C31×#54612yes—
648C2—42no—
649C2—42no—
650C2—84no—
651C2—82no—
652C2C31×#240411yescharacteristic
653C2C31×#300611yescharacteristic
654C2C31×#300811yescharacteristic
655C2C31×#331812yes—
656C2C31×#332212yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#553C2semidirect
2. C2×C31×#447C2semidirect
3. C2×C31×#418C2semidirect
4. C31×#2408C2semidirect
5. C31×#2025C2semidirect
6. C31×#1999C2semidirect
7. C31×#1895C2semidirect
8. C31×#1879C2semidirect
9. C31×#1874C2semidirect
10. C31×#1870C2semidirect
11. C31×#1825C2semidirect
18. C2×C31×#125C22semidirect
23. C2×C31×#121C22semidirect
27. C2×C4×D4×C31C22semidirect
29. C22×C4⋊C4×C31C22semidirect
35. C2×C31×#102C22semidirect
37. C2×C31×C22⋊C8C22semidirect
43. C31×#340C22semidirect
44. C31×#340C22semidirect
45. C31×#340C22semidirect
46. C31×#336C22semidirect
80. C31×#121D4semidirect
88. C31×#117D4semidirect
207. #23973C31direct (N × H)
281. C2×#553C2×C31semidirect
282. C2×#447C2×C31semidirect
283. C2×#418C2×C31semidirect
284. #2408C2×C31semidirect
285. #2025C2×C31semidirect
286. #1999C2×C31semidirect
287. #1895C2×C31semidirect
288. #1879C2×C31semidirect
289. #1874C2×C31semidirect
290. #1870C2×C31semidirect
291. #1825C2×C31semidirect
337. C2×#125C22×C31semidirect
342. C2×#121C22×C31semidirect
346. C2×C4×D4C22×C31semidirect
348. C22×C4⋊C4C22×C31semidirect
354. C2×#102C22×C31semidirect
356. C2×C22⋊C8C22×C31semidirect
362. #340C22×C31semidirect
363. #340C22×C31semidirect
364. #340C22×C31semidirect
365. #336C22×C31semidirect
408. #121D4×C31semidirect
416. #117D4×C31semidirect
446. C31#23973direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5532
C2×C31×#4472
C2×C31×#4182
C31×#24082
C31×#20252
C31×#19992
C31×#18952
C31×#18792
C31×#18742
C31×#18702
C31×#18252
#2397331