Finite Group Databasebeta

← C31×#24176 #21480935

Subgroups — C31×#24176 #21480935

Order 7936.

1238 conjugacy classes of subgroups.

The subgroup lattice has 670 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
4C4×C31×#127C211yesmaximal, characteristic, complemented
5C31×#2037C212yesmaximal, complemented
6C31×#2002C212yesmaximal, complemented
7C31×#1994C211yesmaximal, characteristic, complemented
8C31×#1962C211yesmaximal, characteristic, complemented
9C31×#1883C212yesmaximal, complemented
10C31×#1859C212yesmaximal, 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×C31—22no—
27C2×C4×D4×C31C2211yescharacteristic
28C2×C4×D4×C31C2211yescharacteristic
29C31×#478—22no—
30C31×#476—21no—
31C31×#474—21no—
32C31×#463—22no—
33C31×#461—21no—
34C31×#459—21no—
35C31×#428C2212yes—
36C31×#423C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C2×C31×#104C2212yes—
39C2×C31×#104C2212yes—
40C31×#418C2212yes—
41C31×#417C2212yes—
42C2×C31×#100C2212yes—
43C31×#390—22no—
44C31×#390C2211yescharacteristic
45C31×#390C2211yescharacteristic
46C31×#384C2211yescharacteristic
47C43×C31C2211yescharacteristic
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—
104C4×D4×C31—48no—
105C31×#117—22no—
106C31×#117—22no—
107C31×#117C2312yes—
108C2×C4⋊C4×C31—22no—
109C2×C4⋊C4×C31—22no—
110C2×C4⋊C4×C31C2311yescharacteristic
111C2×C4⋊C4×C31C2311yescharacteristic
112C2×C22⋊C4×C31—22no—
113C2×C22⋊C4×C31—22no—
114C2×C22⋊C4×C31—21no—
115C2×C22⋊C4×C31D412yes—
116C2×C42×C31—21no—
117C2×C42×C31—21no—
118C2×C42×C31C2311yescharacteristic
119C2×C42×C31C2311yescharacteristic
120C2×C42×C31C2311yescharacteristic
121C31×#107—22no—
122C31×#106—22no—
123C31×C4⋊C8—21no—
124C31×C4⋊C8—21no—
125C31×C4⋊C8—21no—
126C31×C4⋊C8—21no—
127C31×#104—24no—
128C31×#104—42no—
129C31×#103—22no—
130C31×#102—22no—
131C31×#102—22no—
132C31×#102—22no—
133C31×#100—42no—
134C24×C31—21no—
135C24×C31C2×D412yes—
136#41×C31—22no—
137#41×C31—22no—
138#41×C31—22no—
139#41×C31—22no—
140#41×C31—84no—
141C2×Q8×C31—21no—
142C2×Q8×C31—21no—
143C2×Q8×C31C2×D412yes—
144C2×D4×C31—28no—
145C2×D4×C31—22no—
146C2×D4×C31—24no—
147C2×D4×C31—24no—
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—21no—
161C2×D4×C31—44no—
162C2×D4×C31—44no—
163C2×D4×C31C2×D412yes—
164C2×D4×C31#4112yescomplemented
165C22×C4×C31—21no—
166C22×C4×C31—21no—
167C22×C4×C31—24no—
168C22×C4×C31—22no—
169C22×C4×C31—22no—
170C22×C4×C31—22no—
171C22×C4×C31—21no—
172C22×C4×C31—44no—
173C22×C4×C31—44no—
174C22×C4×C31—41no—
175C22×C4×C31—41no—
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D412yes—
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2411yescharacteristic
180SD16×C31—44no—
181D8×C31—44no—
182M16×C31—22no—
183M16×C31—22no—
184M16×C31—22no—
185M16×C31—22no—
186C2×C8×C31—22no—
187C2×C8×C31—22no—
188C4⋊C4×C31—24no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—44no—
200C4⋊C4×C31C2×D412yes—
201C4⋊C4×C31C2×D412yes—
202C22⋊C4×C31—24no—
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—22no—
210C22⋊C4×C31—44no—
211C22⋊C4×C31—44no—
212C22⋊C4×C31—41no—
213C22⋊C4×C31—41no—
214C42×C31—24no—
215C42×C31—24no—
216C42×C31—22no—
217C42×C31—21no—
218C42×C31—21no—
219C42×C31—21no—
220C42×C31—42no—
221C42×C31C2×D411yescharacteristic
222C42×C31C2×D411yescharacteristic
223C42×C31C2×D411yescharacteristic
224C42×C31C2×D411yescharacteristic
225C42×C31#4112yescomplemented
226#24176C3111yesmaximal, characteristic, direct product factor, Sylow
227C23×C31—22no—
228C23×C31—22no—
229C23×C31—22no—
230C23×C31—22no—
231C23×C31—22no—
232C23×C31—42no—
233C23×C31—42no—
234C23×C31—42no—
235C23×C31—42no—
236C23×C31—41no—
237C23×C31—41no—
238C23×C31C22×D411yescharacteristic
239Q8×C31—22no—
240Q8×C31—22no—
241Q8×C31—42no—
242Q8×C31—81no—
243D4×C31—22no—
244D4×C31—22no—
245D4×C31—22no—
246D4×C31—22no—
247D4×C31—416no—
248D4×C31—42no—
249D4×C31—42no—
250D4×C31—41no—
251D4×C31—41no—
252D4×C31—41no—
253D4×C31—41no—
254D4×C31—41no—
255D4×C31—41no—
256D4×C31—42no—
257D4×C31—81no—
258D4×C31—84no—
259D4×C31—84no—
260C2×C4×C31—22no—
261C2×C4×C31—22no—
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—22no—
269C2×C4×C31—41no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—44no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—41no—
278C2×C4×C31—82no—
279C2×C4×C31—82no—
280C2×C4×C31—84no—
281C2×C4×C31—82no—
282C2×C4×C31#12011yescharacteristic
283C2×C4×C31#12011yescharacteristic
284C2×C4×C31#12011yescharacteristic
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31C22×D411yescharacteristic
287C2×C4×C31C22×D411yescharacteristic
288C2×C4×C31C2×#4111yescharacteristic
289C2×C4×C31C2×#4111yescharacteristic
290C2×C4×C31H(5,𝔽2)11yescharacteristic
291C2×C4×C312−1+411yescharacteristic
292C8×C31—42no—
293C8×C31—42no—
294#3506C2×C3111yescharacteristic, complemented
295#3077C2×C3111yescharacteristic, complemented
296#3036C2×C3111yescharacteristic, complemented
297C4×#127C2×C3111yescharacteristic, complemented
298#2037C2×C3112yescomplemented
299#2002C2×C3112yescomplemented
300#1994C2×C3111yescharacteristic, complemented
301#1962C2×C3111yescharacteristic, complemented
302#1883C2×C3112yescomplemented
303#1859C2×C3112yescomplemented
304#1800C2×C3111yescharacteristic, complemented
305C22×C31—21no—
306C22×C31—21no—
307C22×C31—44no—
308C22×C31—44no—
309C22×C31—42no—
310C22×C31—42no—
311C22×C31—42no—
312C22×C31—42no—
313C22×C31—42no—
314C22×C31—42no—
315C22×C31—42no—
316C22×C31—82no—
317C22×C31—82no—
318C22×C31—81no—
319C22×C31—81no—
320C22×C31C2×#12011yescharacteristic
321C22×C31#52511yescharacteristic
322C22×C31#53711yescharacteristic
323C4×C31—24no—
324C4×C31—21no—
325C4×C31—21no—
326C4×C31—21no—
327C4×C31—42no—
328C4×C31—42no—
329C4×C31—42no—
330C4×C31—42no—
331C4×C31—42no—
332C4×C31—81no—
333C4×C31#45312yes—
334C4×C31#54611yescharacteristic
335C4×C31#54711yescharacteristic
336C4×C31#54811yescharacteristic
337C4×C31#55011yescharacteristic
338C2×AGL(1,ℤ/8)C22×C3112yes—
339#559—22no—
340#556—21no—
341#555—21no—
342#553—22no—
343#550—21no—
344#547—21no—
345#538—22no—
346#535C22×C3112yes—
347#533C22×C3111yescharacteristic
348#532C22×C3111yescharacteristic
349C2×#127C22×C3111yescharacteristic
350#525—22no—
351#520C22×C3112yes—
352C2×C4×D4—22no—
353C2×C4×D4C22×C3111yescharacteristic
354C2×C4×D4C22×C3111yescharacteristic
355#478—22no—
356#476—21no—
357#474—21no—
358#463—22no—
359#461—21no—
360#459—21no—
361#428C22×C3112yes—
362#423C22×C3111yescharacteristic
363#423C22×C3111yescharacteristic
364C2×#104C22×C3112yes—
365C2×#104C22×C3112yes—
366#418C22×C3112yes—
367#417C22×C3112yes—
368C2×#100C22×C3112yes—
369#390—22no—
370#390C22×C3111yescharacteristic
371#390C22×C3111yescharacteristic
372#384C22×C3111yescharacteristic
373C43C22×C3111yescharacteristic
374#343C22×C3112yes—
375#339C22×C3112yes—
376#339C22×C3112yes—
377C2×C31—21no—
378C2×C31—21no—
379C2×C31—82no—
380C2×C31—82no—
381C2×C31—82no—
382C2×C31#240711yescharacteristic
383C2×C31C2×#45311yescharacteristic
384C2×C31#301811yescharacteristic
385C2×#41—22no—
386C2×#41C23×C3112yes—
387C22×D4—22no—
388C22×D4C23×C3111yescharacteristic
389C22×D4C23×C3111yescharacteristic
390C23×C4—21no—
391C23×C4D4×C3112yes—
392AGL(1,ℤ/8)—24no—
393AGL(1,ℤ/8)—42no—
394C2×SD16—22no—
395C2×D8—22no—
396C2×M16C23×C3112yes—
397C2×M16C23×C3112yes—
398C4⋊Q8D4×C3111yescharacteristic, complemented
399C4⋊Q8D4×C3111yescharacteristic, complemented
400C4⋊Q8D4×C3111yescharacteristic, complemented
401#127—22no—
402#127—22no—
403#127—22no—
404#127D4×C3111yescharacteristic, complemented
405#126—24no—
406#125D4×C3112yescomplemented
407#124—24no—
408#124—22no—
409#124D4×C3112yescomplemented
410#123—24no—
411#123—24no—
412C22⋊Q8—22no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415#121—22no—
416#121—24no—
417#121—21no—
418#121—21no—
419#121—22no—
420#121—22no—
421#120—24no—
422C4×Q8—22no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×D4—24no—
426C4×D4—22no—
427C4×D4—24no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—48no—
439#117—22no—
440#117—22no—
441#117C23×C3112yes—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4C23×C3111yescharacteristic
445C2×C4⋊C4C23×C3111yescharacteristic
446C2×C22⋊C4—22no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4D4×C3112yes—
450C2×C42—21no—
451C2×C42—21no—
452C2×C42C23×C3111yescharacteristic
453C2×C42C23×C3111yescharacteristic
454C2×C42C23×C3111yescharacteristic
455#107—22no—
456#106—22no—
457C4⋊C8—21no—
458C4⋊C8—21no—
459C4⋊C8—21no—
460C4⋊C8—21no—
461#104—24no—
462#104—42no—
463#103—22no—
464#102—22no—
465#102—22no—
466#102—22no—
467#100—42no—
468C31#2417611yescharacteristic, direct product factor, Sylow
469C24—21no—
470C24C2×D4×C3112yes—
471#41—22no—
472#41—22no—
473#41—22no—
474#41—22no—
475#41—84no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8C2×D4×C3112yes—
479C2×D4—28no—
480C2×D4—22no—
481C2×D4—24no—
482C2×D4—24no—
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—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—44no—
497C2×D4—44no—
498C2×D4C2×D4×C3112yes—
499C2×D4#41×C3112yescomplemented
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—41no—
510C22×C4—41no—
511C22×C4C2×D4×C3111yescharacteristic
512C22×C4C2×D4×C3112yes—
513C22×C4C2×D4×C3111yescharacteristic
514C22×C4C24×C3111yescharacteristic
515SD16—44no—
516D8—44no—
517M16—22no—
518M16—22no—
519M16—22no—
520M16—22no—
521C2×C8—22no—
522C2×C8—22no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—44no—
535C4⋊C4C2×D4×C3112yes—
536C4⋊C4C2×D4×C3112yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42C2×D4×C3111yescharacteristic
557C42C2×D4×C3111yescharacteristic
558C42C2×D4×C3111yescharacteristic
559C42C2×D4×C3111yescharacteristic
560C42#41×C3112yescomplemented
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23C22×D4×C3111yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—42no—
576Q8—81no—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—416no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—81no—
592D4—84no—
593D4—84no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—84no—
615C2×C4—82no—
616C2×C4C31×#12011yescharacteristic
617C2×C4C31×#12011yescharacteristic
618C2×C4C31×#12011yescharacteristic
619C2×C4C31×#12011yescharacteristic
620C2×C4C22×D4×C3111yescharacteristic
621C2×C4C22×D4×C3111yescharacteristic
622C2×C4C2×#41×C3111yescharacteristic
623C2×C4C2×#41×C3111yescharacteristic
624C2×C4C31×H(5,𝔽2)11yescharacteristic
625C2×C4C31×2−1+411yescharacteristic
626C8—42no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22C2×C31×#12011yescharacteristic
644C22C31×#52511yescharacteristic
645C22C31×#53711yescharacteristic
646C4—24no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—81no—
656C4C31×#45312yes—
657C4C31×#54611yescharacteristic
658C4C31×#54711yescharacteristic
659C4C31×#54811yescharacteristic
660C4C31×#55011yescharacteristic
661C2—21no—
662C2—21no—
663C2—82no—
664C2—82no—
665C2—82no—
666C2C31×#240711yescharacteristic
667C2C2×C31×#45311yescharacteristic
668C2C31×#301811yescharacteristic
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. C4×C31×#127C2semidirect
5. C31×#2037C2semidirect
6. C31×#2002C2semidirect
7. C31×#1994C2semidirect
8. C31×#1962C2semidirect
9. C31×#1883C2semidirect
10. C31×#1859C2semidirect
11. C31×#1800C2semidirect
64. C31×C4⋊Q8D4semidirect
65. C31×C4⋊Q8D4semidirect
66. C31×C4⋊Q8D4semidirect
70. C31×#127D4semidirect
72. C31×#125D4semidirect
75. C31×#124D4semidirect
164. C2×D4×C31#41semidirect
225. C42×C31#41semidirect
226. #24176C31direct (N × H)
294. #3506C2×C31semidirect
295. #3077C2×C31semidirect
296. #3036C2×C31semidirect
297. C4×#127C2×C31semidirect
298. #2037C2×C31semidirect
299. #2002C2×C31semidirect
300. #1994C2×C31semidirect
301. #1962C2×C31semidirect
302. #1883C2×C31semidirect
303. #1859C2×C31semidirect
304. #1800C2×C31semidirect
398. C4⋊Q8D4×C31semidirect
399. C4⋊Q8D4×C31semidirect
400. C4⋊Q8D4×C31semidirect
404. #127D4×C31semidirect
406. #125D4×C31semidirect
409. #124D4×C31semidirect
468. C31#24176direct (N × H)
499. C2×D4#41×C31semidirect
560. 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
C4×C31×#1272
C31×#20372
C31×#20022
C31×#19942
C31×#19622
C31×#18832
C31×#18592
C31×#18002
#2417631