Finite Group Databasebeta

← C31×#32998 #21489757

Subgroups — C31×#32998 #21489757

Order 7936.

1284 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3428C211yesmaximal, characteristic, complemented
2C31×#3094C211yesmaximal, characteristic, complemented
3C31×#3006C211yesmaximal, characteristic
4C31×#2500C211yesmaximal, characteristic, complemented
5C31×#1883C212yesmaximal, complemented
6C31×#1883C212yesmaximal, complemented
7C31×#1860C212yesmaximal, complemented
8C31×#1859C212yesmaximal, complemented
9C31×#1826C211yesmaximal, characteristic, complemented
10C31×#1822C211yesmaximal, characteristic, complemented
11C31×#1789C211yesmaximal, characteristic, complemented
12C22×#41×C31C2211yescharacteristic
13C2×C31×AGL(1,ℤ/8)C2212yes—
14C2×C31×#121C2211yescharacteristic
15C31×#520—21no—
16C31×#520—21no—
17C31×#520C2212yes—
18C31×#520C2212yes—
19C31×#519—21no—
20C31×#518—21no—
21C31×#518—22no—
22C4×#41×C31—22no—
23C4×#41×C31—22no—
24C2×C31×#117—22no—
25C2×C31×#117C2211yescharacteristic
26C2×C31×#117C2211yescharacteristic
27C31×#482—22no—
28C31×#481—21no—
29C31×#480—21no—
30C31×#449—22no—
31C31×#448—21no—
32C31×#447—21no—
33C31×#428C2212yescomplemented
34C2×C31×#104C2212yes—
35C2×C31×#104C2212yes—
36C31×#418C2212yescomplemented
37C31×#417C2212yes—
38C31×#407C2211yescharacteristic
39C31×#407C2211yescharacteristic
40C31×#394—22no—
41C31×#390C2211yescharacteristic, complemented
42C31×#390C2211yescharacteristic, complemented
43C31×#388—22no—
44C31×#384C2211yescharacteristic, complemented
45C31×#382C2211yescharacteristic, complemented
46C4×C22⋊C4×C31C2211yescharacteristic
47C31×#343C2212yescomplemented
48C31×#343C2212yescomplemented
49C31×#339C2212yescomplemented
50C31×#339C2212yescomplemented
51C2×#41×C31—24no—
52C2×#41×C31—24no—
53C2×#41×C31—21no—
54C2×#41×C31—21no—
55C2×#41×C31D412yes—
56C2×#41×C31C2312yes—
57C22×Q8×C31D411yescharacteristic
58C22×D4×C31—21no—
59C22×D4×C31—21no—
60C22×D4×C31D411yescharacteristic
61C22×D4×C31C2311yescharacteristic
62C23×C4×C31—21no—
63C23×C4×C31C2311yescharacteristic
64C31×AGL(1,ℤ/8)—24no—
65C31×AGL(1,ℤ/8)—24no—
66C2×SD16×C31—22no—
67C2×D8×C31—22no—
68C2×M16×C31C2312yes—
69C2×M16×C31C2312yes—
70C31×#121—21no—
71C31×#121—21no—
72C31×#121—21no—
73C31×#121—21no—
74C4×Q8×C31—22no—
75C4×Q8×C31—22no—
76C4×Q8×C31—22no—
77C4×Q8×C31—22no—
78C4×D4×C31—24no—
79C4×D4×C31—24no—
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—24no—
85C4×D4×C31—24no—
86C31×#117—24no—
87C31×#117—24no—
88C31×#117—21no—
89C31×#117—21no—
90C31×#117—21no—
91C31×#117—21no—
92C31×#117—21no—
93C31×#117—21no—
94C31×#117—22no—
95C31×#117—22no—
96C31×#117D412yes—
97C31×#117C2312yes—
98C2×C4⋊C4×C31—22no—
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31D411yescharacteristic
102C2×C4⋊C4×C31D411yescharacteristic
103C2×C4⋊C4×C31C2311yescharacteristic
104C2×C4⋊C4×C31C2311yescharacteristic
105C2×C4⋊C4×C31C2311yescharacteristic
106C2×C22⋊C4×C31—24no—
107C2×C22⋊C4×C31—21no—
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31—22no—
111C2×C22⋊C4×C31—22no—
112C2×C22⋊C4×C31D412yes—
113C2×C42×C31—24no—
114C2×C42×C31—21no—
115C2×C42×C31D412yes—
116C2×C42×C31C2311yescharacteristic
117C2×C42×C31C2311yescharacteristic
118C31×#107—22no—
119C31×#106—22no—
120C31×#104—24no—
121C31×#104—24no—
122C31×#103—22no—
123C31×#102—22no—
124C31×#102—22no—
125C31×#102—22no—
126C31×C22⋊C8—21no—
127C31×C22⋊C8—21no—
128C31×C22⋊C8—21no—
129C31×C22⋊C8—21no—
130C31×#95—21no—
131C31×#95—22no—
132C24×C31—21no—
133C24×C31—21no—
134C24×C31C2×D411yescharacteristic
135#41×C31—24no—
136#41×C31—24no—
137#41×C31—28no—
138#41×C31—24no—
139#41×C31—24no—
140#41×C31—22no—
141#41×C31—22no—
142#41×C31#4114yes—
143#41×C31#4114yes—
144C2×Q8×C31—24no—
145C2×Q8×C31—21no—
146C2×Q8×C31C2×D412yes—
147C2×D4×C31—24no—
148C2×D4×C31—24no—
149C2×D4×C31—22no—
150C2×D4×C31—22no—
151C2×D4×C31—24no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—22no—
156C2×D4×C31—22no—
157C2×D4×C31—41no—
158C2×D4×C31—41no—
159C2×D4×C31—82no—
160C2×D4×C31C2×D412yes—
161C22×C4×C31—28no—
162C22×C4×C31—21no—
163C22×C4×C31—22no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—24no—
169C22×C4×C31—24no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31—21no—
173C22×C4×C31—21no—
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D412yes—
180C22×C4×C31C2×D412yes—
181C22×C4×C31C2411yescharacteristic
182SD16×C31—44no—
183D8×C31—44no—
184M16×C31—22no—
185M16×C31—22no—
186M16×C31—22no—
187M16×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—22no—
190C4⋊C4×C31—24no—
191C4⋊C4×C31—24no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—82no—
201C4⋊C4×C31C2×D412yes—
202C4⋊C4×C31C2×D412yes—
203C22⋊C4×C31—24no—
204C22⋊C4×C31—24no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—44no—
210C22⋊C4×C31—42no—
211C22⋊C4×C31—41no—
212C22⋊C4×C31—41no—
213C22⋊C4×C31—82no—
214C42×C31—24no—
215C42×C31—24no—
216C42×C31—22no—
217C42×C31—22no—
218C42×C31—22no—
219C42×C31—22no—
220C42×C31—22no—
221C42×C31—22no—
222#32998C3111yesmaximal, characteristic, direct product factor, Sylow
223C23×C31—24no—
224C23×C31—22no—
225C23×C31—21no—
226C23×C31—21no—
227C23×C31—41no—
228C23×C31—41no—
229C23×C31—42no—
230C23×C31—42no—
231C23×C31—41no—
232C23×C31—41no—
233C23×C31—41no—
234C23×C31—81no—
235C23×C31—81no—
236C23×C31#12011yescharacteristic
237C23×C31#12011yescharacteristic
238C23×C31C22×D411yescharacteristic
239Q8×C31—24no—
240Q8×C31—22no—
241Q8×C31#12314yes—
242D4×C31—28no—
243D4×C31—24no—
244D4×C31—24no—
245D4×C31—24no—
246D4×C31—22no—
247D4×C31—42no—
248D4×C31—42no—
249D4×C31—82no—
250D4×C31—82no—
251D4×C31#12314yes—
252C2×C4×C31—24no—
253C2×C4×C31—28no—
254C2×C4×C31—22no—
255C2×C4×C31—21no—
256C2×C4×C31—21no—
257C2×C4×C31—22no—
258C2×C4×C31—22no—
259C2×C4×C31—22no—
260C2×C4×C31—24no—
261C2×C4×C31—24no—
262C2×C4×C31—22no—
263C2×C4×C31—22no—
264C2×C4×C31—22no—
265C2×C4×C31—22no—
266C2×C4×C31—22no—
267C2×C4×C31—22no—
268C2×C4×C31—41no—
269C2×C4×C31—41no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—42no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—81no—
278C2×C4×C31—81no—
279C2×C4×C31—84no—
280C2×C4×C31—84no—
281C2×C4×C31#12011yescharacteristic
282C2×C4×C31#12011yescharacteristic
283C2×C4×C31C22×D411yescharacteristic
284C2×C4×C31C22×D411yescharacteristic
285C2×C4×C31C2×#4112yes—
286C2×C4×C31C2×#4112yes—
287C8×C31—42no—
288C8×C31—42no—
289#3428C2×C3111yescharacteristic, complemented
290#3094C2×C3111yescharacteristic, complemented
291#3006C2×C3111yescharacteristic
292#2500C2×C3111yescharacteristic, complemented
293#1883C2×C3112yescomplemented
294#1883C2×C3112yescomplemented
295#1860C2×C3112yescomplemented
296#1859C2×C3112yescomplemented
297#1826C2×C3111yescharacteristic, complemented
298#1822C2×C3111yescharacteristic, complemented
299#1789C2×C3111yescharacteristic, complemented
300C22×C31—21no—
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—21no—
304C22×C31—24no—
305C22×C31—21no—
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—21no—
309C22×C31—42no—
310C22×C31—42no—
311C22×C31—81no—
312C22×C31—81no—
313C22×C31—82no—
314C22×C31—82no—
315C22×C31—84no—
316C22×C31—81no—
317C22×C31—81no—
318C22×C31—81no—
319C22×C31C2×#12011yescharacteristic
320C22×C31#52511yescharacteristic
321C22×C31#52511yescharacteristic
322C4×C31—21no—
323C4×C31—24no—
324C4×C31—21no—
325C4×C31—42no—
326C4×C31—42no—
327C4×C31—42no—
328C4×C31—42no—
329C4×C31—42no—
330C4×C31—81no—
331C4×C31—81no—
332C4×C31C2×#12312yes—
333C4×C31#52512yes—
334C22×#41C22×C3111yescharacteristic
335C2×AGL(1,ℤ/8)C22×C3112yes—
336C2×#121C22×C3111yescharacteristic
337#520—21no—
338#520—21no—
339#520C22×C3112yes—
340#520C22×C3112yes—
341#519—21no—
342#518—21no—
343#518—22no—
344C4×#41—22no—
345C4×#41—22no—
346C2×#117—22no—
347C2×#117C22×C3111yescharacteristic
348C2×#117C22×C3111yescharacteristic
349#482—22no—
350#481—21no—
351#480—21no—
352#449—22no—
353#448—21no—
354#447—21no—
355#428C22×C3112yescomplemented
356C2×#104C22×C3112yes—
357C2×#104C22×C3112yes—
358#418C22×C3112yescomplemented
359#417C22×C3112yes—
360#407C22×C3111yescharacteristic
361#407C22×C3111yescharacteristic
362#394—22no—
363#390C22×C3111yescharacteristic, complemented
364#390C22×C3111yescharacteristic, complemented
365#388—22no—
366#384C22×C3111yescharacteristic, complemented
367#382C22×C3111yescharacteristic, complemented
368C4×C22⋊C4C22×C3111yescharacteristic
369#343C22×C3112yescomplemented
370#343C22×C3112yescomplemented
371#339C22×C3112yescomplemented
372#339C22×C3112yescomplemented
373C2×C31—21no—
374C2×C31—21no—
375C2×C31—41no—
376C2×C31—41no—
377C2×C31—44no—
378C2×C31—161no—
379C2×C31#284911yescharacteristic
380C2×C31#301811yescharacteristic
381C2×C31#302111yescharacteristic
382C2×#41—24no—
383C2×#41—24no—
384C2×#41—21no—
385C2×#41—21no—
386C2×#41D4×C3112yes—
387C2×#41C23×C3112yes—
388C22×Q8D4×C3111yescharacteristic
389C22×D4—21no—
390C22×D4—21no—
391C22×D4D4×C3111yescharacteristic
392C22×D4C23×C3111yescharacteristic
393C23×C4—21no—
394C23×C4C23×C3111yescharacteristic
395AGL(1,ℤ/8)—24no—
396AGL(1,ℤ/8)—24no—
397C2×SD16—22no—
398C2×D8—22no—
399C2×M16C23×C3112yes—
400C2×M16C23×C3112yes—
401#121—21no—
402#121—21no—
403#121—21no—
404#121—21no—
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—22no—
408C4×Q8—22no—
409C4×D4—24no—
410C4×D4—24no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—24no—
416C4×D4—24no—
417#117—24no—
418#117—24no—
419#117—21no—
420#117—21no—
421#117—21no—
422#117—21no—
423#117—21no—
424#117—21no—
425#117—22no—
426#117—22no—
427#117D4×C3112yes—
428#117C23×C3112yes—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4D4×C3111yescharacteristic
433C2×C4⋊C4D4×C3111yescharacteristic
434C2×C4⋊C4C23×C3111yescharacteristic
435C2×C4⋊C4C23×C3111yescharacteristic
436C2×C4⋊C4C23×C3111yescharacteristic
437C2×C22⋊C4—24no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—22no—
442C2×C22⋊C4—22no—
443C2×C22⋊C4D4×C3112yes—
444C2×C42—24no—
445C2×C42—21no—
446C2×C42D4×C3112yes—
447C2×C42C23×C3111yescharacteristic
448C2×C42C23×C3111yescharacteristic
449#107—22no—
450#106—22no—
451#104—24no—
452#104—24no—
453#103—22no—
454#102—22no—
455#102—22no—
456#102—22no—
457C22⋊C8—21no—
458C22⋊C8—21no—
459C22⋊C8—21no—
460C22⋊C8—21no—
461#95—21no—
462#95—22no—
463C31#3299811yescharacteristic, direct product factor, Sylow
464C24—21no—
465C24—21no—
466C24C2×D4×C3111yescharacteristic
467#41—24no—
468#41—24no—
469#41—28no—
470#41—24no—
471#41—24no—
472#41—22no—
473#41—22no—
474#41#41×C3114yes—
475#41#41×C3114yes—
476C2×Q8—24no—
477C2×Q8—21no—
478C2×Q8C2×D4×C3112yes—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—24no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—41no—
490C2×D4—41no—
491C2×D4—82no—
492C2×D4C2×D4×C3112yes—
493C22×C4—28no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4C2×D4×C3111yescharacteristic
507C22×C4C2×D4×C3111yescharacteristic
508C22×C4C2×D4×C3111yescharacteristic
509C22×C4C2×D4×C3111yescharacteristic
510C22×C4C2×D4×C3111yescharacteristic
511C22×C4C2×D4×C3112yes—
512C22×C4C2×D4×C3112yes—
513C22×C4C24×C3111yescharacteristic
514SD16—44no—
515D8—44no—
516M16—22no—
517M16—22no—
518M16—22no—
519M16—22no—
520C2×C8—22no—
521C2×C8—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—82no—
533C4⋊C4C2×D4×C3112yes—
534C4⋊C4C2×D4×C3112yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—82no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C23—24no—
555C23—22no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—81no—
566C23—81no—
567C23C31×#12011yescharacteristic
568C23C31×#12011yescharacteristic
569C23C22×D4×C3111yescharacteristic
570Q8—24no—
571Q8—22no—
572Q8C31×#12314yes—
573D4—28no—
574D4—24no—
575D4—24no—
576D4—24no—
577D4—22no—
578D4—42no—
579D4—42no—
580D4—82no—
581D4—82no—
582D4C31×#12314yes—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4C31×#12011yescharacteristic
613C2×C4C31×#12011yescharacteristic
614C2×C4C22×D4×C3111yescharacteristic
615C2×C4C22×D4×C3111yescharacteristic
616C2×C4C2×#41×C3112yes—
617C2×C4C2×#41×C3112yes—
618C8—42no—
619C8—42no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—24no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—42no—
630C22—42no—
631C22—81no—
632C22—81no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C2×C31×#12011yescharacteristic
640C22C31×#52511yescharacteristic
641C22C31×#52511yescharacteristic
642C4—21no—
643C4—24no—
644C4—21no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4C2×C31×#12312yes—
653C4C31×#52512yes—
654C2—21no—
655C2—21no—
656C2—41no—
657C2—41no—
658C2—44no—
659C2—161no—
660C2C31×#284911yescharacteristic
661C2C31×#301811yescharacteristic
662C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3428C2semidirect
2. C31×#3094C2semidirect
4. C31×#2500C2semidirect
5. C31×#1883C2semidirect
6. C31×#1883C2semidirect
7. C31×#1860C2semidirect
8. C31×#1859C2semidirect
9. C31×#1826C2semidirect
10. C31×#1822C2semidirect
11. C31×#1789C2semidirect
33. C31×#428C22semidirect
36. C31×#418C22semidirect
41. C31×#390C22semidirect
42. C31×#390C22semidirect
44. C31×#384C22semidirect
45. C31×#382C22semidirect
47. C31×#343C22semidirect
48. C31×#343C22semidirect
49. C31×#339C22semidirect
50. C31×#339C22semidirect
222. #32998C31direct (N × H)
289. #3428C2×C31semidirect
290. #3094C2×C31semidirect
292. #2500C2×C31semidirect
293. #1883C2×C31semidirect
294. #1883C2×C31semidirect
295. #1860C2×C31semidirect
296. #1859C2×C31semidirect
297. #1826C2×C31semidirect
298. #1822C2×C31semidirect
299. #1789C2×C31semidirect
355. #428C22×C31semidirect
358. #418C22×C31semidirect
363. #390C22×C31semidirect
364. #390C22×C31semidirect
366. #384C22×C31semidirect
367. #382C22×C31semidirect
369. #343C22×C31semidirect
370. #343C22×C31semidirect
371. #339C22×C31semidirect
372. #339C22×C31semidirect
463. C31#32998direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34282
C31×#30942
C31×#30062
C31×#25002
C31×#18832
C31×#18602
C31×#18592
C31×#18262
C31×#18222
C31×#17892
#3299831