Finite Group Databasebeta

← C31×#24180 #21480939

Subgroups — C31×#24180 #21480939

Order 7936.

1146 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic, complemented
2C31×#3077C211yesmaximal, characteristic, complemented
3C31×#3036C211yesmaximal, characteristic, complemented
4C31×#2328C211yesmaximal, characteristic, complemented
5C31×#2037C212yesmaximal, complemented
6C31×#2002C212yesmaximal, complemented
7C31×#1995C211yesmaximal, characteristic, complemented
8C31×#1963C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1860C212yesmaximal, complemented
11C31×#1800C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)C2212yes—
13C31×#559—22no—
14C31×#556—21no—
15C31×#555—21no—
16C31×#553—22no—
17C31×#550—21no—
18C31×#547—21no—
19C31×#538—22no—
20C31×#535C2212yes—
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C2×C31×#127C2211yescharacteristic
24C31×#525—22no—
25C31×#520C2212yes—
26C2×C4×D4×C31C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C31×#478—22no—
29C31×#476—21no—
30C31×#474—21no—
31C31×#463—22no—
32C31×#461—21no—
33C31×#459—21no—
34C31×#428C2212yes—
35C31×#423C2211yescharacteristic
36C31×#423C2211yescharacteristic
37C2×C31×#104C2212yes—
38C2×C31×#104C2212yes—
39C31×#418C2212yes—
40C31×#417C2212yes—
41C2×C31×#100C2212yes—
42C31×#390C2211yescharacteristic
43C31×#390C2211yescharacteristic
44C31×#386—22no—
45C31×#386—22no—
46C31×#382C2211yescharacteristic
47C31×#376C2211yescharacteristic
48C31×#343C2212yes—
49C31×#339C2212yes—
50C31×#339C2212yes—
51C2×#41×C31—22no—
52C2×#41×C31C2312yes—
53C22×D4×C31—22no—
54C22×D4×C31C2311yescharacteristic
55C22×D4×C31C2311yescharacteristic
56C23×C4×C31—21no—
57C23×C4×C31D412yes—
58C31×AGL(1,ℤ/8)—24no—
59C31×AGL(1,ℤ/8)—42no—
60C2×SD16×C31—22no—
61C2×D8×C31—22no—
62C2×M16×C31C2312yes—
63C2×M16×C31C2312yes—
64C31×C4⋊Q8D411yescharacteristic, complemented
65C31×C4⋊Q8D411yescharacteristic, complemented
66C31×C4⋊Q8D411yescharacteristic, complemented
67C31×#127—22no—
68C31×#127—22no—
69C31×#127—22no—
70C31×#127D411yescharacteristic, complemented
71C31×#126—24no—
72C31×#125D412yescomplemented
73C31×#124—24no—
74C31×#124—22no—
75C31×#124D412yescomplemented
76C31×#123—24no—
77C31×#123—24no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—21no—
81C31×#121—22no—
82C31×#121—24no—
83C31×#121—21no—
84C31×#121—21no—
85C31×#121—22no—
86C31×#121—22no—
87C31×#120—24no—
88C4×Q8×C31—22no—
89C4×Q8×C31—21no—
90C4×Q8×C31—21no—
91C4×D4×C31—24no—
92C4×D4×C31—22no—
93C4×D4×C31—24no—
94C4×D4×C31—21no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31—21no—
104C31×#117—22no—
105C31×#117—22no—
106C31×#117C2312yes—
107C2×C4⋊C4×C31—22no—
108C2×C4⋊C4×C31C2311yescharacteristic
109C2×C4⋊C4×C31C2311yescharacteristic
110C2×C22⋊C4×C31—21no—
111C2×C22⋊C4×C31—22no—
112C2×C22⋊C4×C31—22no—
113C2×C22⋊C4×C31D412yes—
114C2×C42×C31C2311yescharacteristic
115C2×C42×C31C2311yescharacteristic
116C2×C42×C31C2311yescharacteristic
117C31×#107—22no—
118C31×#106—22no—
119C31×C4⋊C8—21no—
120C31×C4⋊C8—21no—
121C31×C4⋊C8—21no—
122C31×C4⋊C8—21no—
123C31×#104—24no—
124C31×#104—42no—
125C31×#103—22no—
126C31×#102—22no—
127C31×#102—22no—
128C31×#102—22no—
129C31×#100—42no—
130C31×#95—21no—
131C31×#95—21no—
132C31×#95—22no—
133C24×C31—21no—
134C24×C31C2×D412yes—
135#41×C31—22no—
136#41×C31—22no—
137#41×C31—22no—
138#41×C31—22no—
139#41×C31—84no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31C2×D412yes—
143C2×D4×C31—22no—
144C2×D4×C31—24no—
145C2×D4×C31—24no—
146C2×D4×C31—28no—
147C2×D4×C31—21no—
148C2×D4×C31—21no—
149C2×D4×C31—21no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—21no—
153C2×D4×C31—21no—
154C2×D4×C31—21no—
155C2×D4×C31—21no—
156C2×D4×C31—21no—
157C2×D4×C31—21no—
158C2×D4×C31—21no—
159C2×D4×C31—21no—
160C2×D4×C31—82no—
161C2×D4×C31—82no—
162C2×D4×C31C2×D412yes—
163C2×D4×C31#4112yescomplemented
164C22×C4×C31—21no—
165C22×C4×C31—24no—
166C22×C4×C31—22no—
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—41no—
172C22×C4×C31—41no—
173C22×C4×C31—82no—
174C22×C4×C31—82no—
175C22×C4×C31C2×D412yes—
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2411yescharacteristic
179SD16×C31—44no—
180D8×C31—44no—
181M16×C31—22no—
182M16×C31—22no—
183M16×C31—22no—
184M16×C31—22no—
185C2×C8×C31—22no—
186C2×C8×C31—22no—
187C4⋊C4×C31—24no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31C2×D412yes—
199C4⋊C4×C31C2×D412yes—
200C22⋊C4×C31—24no—
201C22⋊C4×C31—24no—
202C22⋊C4×C31—24no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—41no—
209C22⋊C4×C31—41no—
210C22⋊C4×C31—82no—
211C22⋊C4×C31—82no—
212C42×C31—24no—
213C42×C31—22no—
214C42×C31—21no—
215C42×C31—21no—
216C42×C31—21no—
217C42×C31C2×D411yescharacteristic
218C42×C31C2×D411yescharacteristic
219C42×C31C2×D411yescharacteristic
220C42×C31C2×D411yescharacteristic
221C42×C31#4112yescomplemented
222#24180C3111yesmaximal, characteristic, direct product factor, Sylow
223C23×C31—22no—
224C23×C31—22no—
225C23×C31—22no—
226C23×C31—22no—
227C23×C31—22no—
228C23×C31—42no—
229C23×C31—42no—
230C23×C31—42no—
231C23×C31—42no—
232C23×C31—41no—
233C23×C31—41no—
234C23×C31C22×D411yescharacteristic
235Q8×C31—22no—
236Q8×C31—22no—
237Q8×C31—42no—
238Q8×C31—81no—
239D4×C31—22no—
240D4×C31—22no—
241D4×C31—22no—
242D4×C31—22no—
243D4×C31—42no—
244D4×C31—42no—
245D4×C31—42no—
246D4×C31—41no—
247D4×C31—41no—
248D4×C31—41no—
249D4×C31—41no—
250D4×C31—41no—
251D4×C31—41no—
252D4×C31—81no—
253D4×C31—84no—
254D4×C31—88no—
255D4×C31—84no—
256C2×C4×C31—22no—
257C2×C4×C31—22no—
258C2×C4×C31—22no—
259C2×C4×C31—22no—
260C2×C4×C31—22no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—44no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—41no—
270C2×C4×C31—41no—
271C2×C4×C31—42no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—82no—
275C2×C4×C31—82no—
276C2×C4×C31—82no—
277C2×C4×C31—84no—
278C2×C4×C31#12011yescharacteristic
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31#12011yescharacteristic
281C2×C4×C31#12011yescharacteristic
282C2×C4×C31C22×D411yescharacteristic
283C2×C4×C31C22×D411yescharacteristic
284C2×C4×C31C2×#4111yescharacteristic
285C2×C4×C31C2×#4111yescharacteristic
286C2×C4×C31H(5,𝔽2)11yescharacteristic
287C2×C4×C312−1+411yescharacteristic
288C8×C31—42no—
289C8×C31—42no—
290#3506C2×C3111yescharacteristic, complemented
291#3077C2×C3111yescharacteristic, complemented
292#3036C2×C3111yescharacteristic, complemented
293#2328C2×C3111yescharacteristic, complemented
294#2037C2×C3112yescomplemented
295#2002C2×C3112yescomplemented
296#1995C2×C3111yescharacteristic, complemented
297#1963C2×C3111yescharacteristic, complemented
298#1883C2×C3112yescomplemented
299#1860C2×C3112yescomplemented
300#1800C2×C3111yescharacteristic, complemented
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—44no—
304C22×C31—44no—
305C22×C31—42no—
306C22×C31—42no—
307C22×C31—42no—
308C22×C31—42no—
309C22×C31—42no—
310C22×C31—42no—
311C22×C31—42no—
312C22×C31—82no—
313C22×C31—82no—
314C22×C31—81no—
315C22×C31—81no—
316C22×C31C2×#12011yescharacteristic
317C22×C31#52511yescharacteristic
318C22×C31#53711yescharacteristic
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—21no—
322C4×C31—21no—
323C4×C31—42no—
324C4×C31—42no—
325C4×C31—42no—
326C4×C31—42no—
327C4×C31—42no—
328C4×C31—81no—
329C4×C31—81no—
330C4×C31#54611yescharacteristic
331C4×C31#54711yescharacteristic
332C4×C31#54811yescharacteristic
333C4×C31#55011yescharacteristic
334C2×AGL(1,ℤ/8)C22×C3112yes—
335#559—22no—
336#556—21no—
337#555—21no—
338#553—22no—
339#550—21no—
340#547—21no—
341#538—22no—
342#535C22×C3112yes—
343#533C22×C3111yescharacteristic
344#532C22×C3111yescharacteristic
345C2×#127C22×C3111yescharacteristic
346#525—22no—
347#520C22×C3112yes—
348C2×C4×D4C22×C3111yescharacteristic
349C2×C4×D4C22×C3111yescharacteristic
350#478—22no—
351#476—21no—
352#474—21no—
353#463—22no—
354#461—21no—
355#459—21no—
356#428C22×C3112yes—
357#423C22×C3111yescharacteristic
358#423C22×C3111yescharacteristic
359C2×#104C22×C3112yes—
360C2×#104C22×C3112yes—
361#418C22×C3112yes—
362#417C22×C3112yes—
363C2×#100C22×C3112yes—
364#390C22×C3111yescharacteristic
365#390C22×C3111yescharacteristic
366#386—22no—
367#386—22no—
368#382C22×C3111yescharacteristic
369#376C22×C3111yescharacteristic
370#343C22×C3112yes—
371#339C22×C3112yes—
372#339C22×C3112yes—
373C2×C31—21no—
374C2×C31—21no—
375C2×C31—82no—
376C2×C31—82no—
377C2×C31—82no—
378C2×C31#240711yescharacteristic
379C2×C31#301911yescharacteristic
380C2×C31#302111yescharacteristic
381C2×#41—22no—
382C2×#41C23×C3112yes—
383C22×D4—22no—
384C22×D4C23×C3111yescharacteristic
385C22×D4C23×C3111yescharacteristic
386C23×C4—21no—
387C23×C4D4×C3112yes—
388AGL(1,ℤ/8)—24no—
389AGL(1,ℤ/8)—42no—
390C2×SD16—22no—
391C2×D8—22no—
392C2×M16C23×C3112yes—
393C2×M16C23×C3112yes—
394C4⋊Q8D4×C3111yescharacteristic, complemented
395C4⋊Q8D4×C3111yescharacteristic, complemented
396C4⋊Q8D4×C3111yescharacteristic, complemented
397#127—22no—
398#127—22no—
399#127—22no—
400#127D4×C3111yescharacteristic, complemented
401#126—24no—
402#125D4×C3112yescomplemented
403#124—24no—
404#124—22no—
405#124D4×C3112yescomplemented
406#123—24no—
407#123—24no—
408C22⋊Q8—22no—
409C22⋊Q8—21no—
410C22⋊Q8—21no—
411#121—22no—
412#121—24no—
413#121—21no—
414#121—21no—
415#121—22no—
416#121—22no—
417#120—24no—
418C4×Q8—22no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×D4—24no—
422C4×D4—22no—
423C4×D4—24no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434#117—22no—
435#117—22no—
436#117C23×C3112yes—
437C2×C4⋊C4—22no—
438C2×C4⋊C4C23×C3111yescharacteristic
439C2×C4⋊C4C23×C3111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—22no—
442C2×C22⋊C4—22no—
443C2×C22⋊C4D4×C3112yes—
444C2×C42C23×C3111yescharacteristic
445C2×C42C23×C3111yescharacteristic
446C2×C42C23×C3111yescharacteristic
447#107—22no—
448#106—22no—
449C4⋊C8—21no—
450C4⋊C8—21no—
451C4⋊C8—21no—
452C4⋊C8—21no—
453#104—24no—
454#104—42no—
455#103—22no—
456#102—22no—
457#102—22no—
458#102—22no—
459#100—42no—
460#95—21no—
461#95—21no—
462#95—22no—
463C31#2418011yescharacteristic, direct product factor, Sylow
464C24—21no—
465C24C2×D4×C3112yes—
466#41—22no—
467#41—22no—
468#41—22no—
469#41—22no—
470#41—84no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8C2×D4×C3112yes—
474C2×D4—22no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—28no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—82no—
492C2×D4—82no—
493C2×D4C2×D4×C3112yes—
494C2×D4#41×C3112yescomplemented
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4C2×D4×C3112yes—
507C22×C4C2×D4×C3111yescharacteristic
508C22×C4C2×D4×C3111yescharacteristic
509C22×C4C24×C3111yescharacteristic
510SD16—44no—
511D8—44no—
512M16—22no—
513M16—22no—
514M16—22no—
515M16—22no—
516C2×C8—22no—
517C2×C8—22no—
518C4⋊C4—24no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4C2×D4×C3112yes—
530C4⋊C4C2×D4×C3112yes—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C42—24no—
544C42—22no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42C2×D4×C3111yescharacteristic
549C42C2×D4×C3111yescharacteristic
550C42C2×D4×C3111yescharacteristic
551C42C2×D4×C3111yescharacteristic
552C42#41×C3112yescomplemented
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23C22×D4×C3111yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—42no—
568Q8—81no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—81no—
583D4—84no—
584D4—88no—
585D4—84no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—84no—
608C2×C4C31×#12011yescharacteristic
609C2×C4C31×#12011yescharacteristic
610C2×C4C31×#12011yescharacteristic
611C2×C4C31×#12011yescharacteristic
612C2×C4C22×D4×C3111yescharacteristic
613C2×C4C22×D4×C3111yescharacteristic
614C2×C4C2×#41×C3111yescharacteristic
615C2×C4C2×#41×C3111yescharacteristic
616C2×C4C31×H(5,𝔽2)11yescharacteristic
617C2×C4C31×2−1+411yescharacteristic
618C8—42no—
619C8—42no—
620C22—21no—
621C22—21no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—82no—
632C22—82no—
633C22—81no—
634C22—81no—
635C22C2×C31×#12011yescharacteristic
636C22C31×#52511yescharacteristic
637C22C31×#53711yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4C31×#54611yescharacteristic
650C4C31×#54711yescharacteristic
651C4C31×#54811yescharacteristic
652C4C31×#55011yescharacteristic
653C2—21no—
654C2—21no—
655C2—82no—
656C2—82no—
657C2—82no—
658C2C31×#240711yescharacteristic
659C2C31×#301911yescharacteristic
660C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3506C2semidirect
2. C31×#3077C2semidirect
3. C31×#3036C2semidirect
4. C31×#2328C2semidirect
5. C31×#2037C2semidirect
6. C31×#2002C2semidirect
7. C31×#1995C2semidirect
8. C31×#1963C2semidirect
9. C31×#1883C2semidirect
10. C31×#1860C2semidirect
11. C31×#1800C2semidirect
64. C31×C4⋊Q8D4semidirect
65. C31×C4⋊Q8D4semidirect
66. C31×C4⋊Q8D4semidirect
70. C31×#127D4semidirect
72. C31×#125D4semidirect
75. C31×#124D4semidirect
163. C2×D4×C31#41semidirect
221. C42×C31#41semidirect
222. #24180C31direct (N × H)
290. #3506C2×C31semidirect
291. #3077C2×C31semidirect
292. #3036C2×C31semidirect
293. #2328C2×C31semidirect
294. #2037C2×C31semidirect
295. #2002C2×C31semidirect
296. #1995C2×C31semidirect
297. #1963C2×C31semidirect
298. #1883C2×C31semidirect
299. #1860C2×C31semidirect
300. #1800C2×C31semidirect
394. C4⋊Q8D4×C31semidirect
395. C4⋊Q8D4×C31semidirect
396. C4⋊Q8D4×C31semidirect
400. #127D4×C31semidirect
402. #125D4×C31semidirect
405. #124D4×C31semidirect
463. C31#24180direct (N × H)
494. C2×D4#41×C31semidirect
552. C42#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#30772
C31×#30362
C31×#23282
C31×#20372
C31×#20022
C31×#19952
C31×#19632
C31×#18832
C31×#18602
C31×#18002
#2418031