Finite Group Databasebeta

← C31×#23979 #21480738

Subgroups — C31×#23979 #21480738

Order 7936.

1418 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#556C211yesmaximal, characteristic, complemented
2C2×C31×#450C211yesmaximal, characteristic, complemented
3C2×C31×#418C211yesmaximal, characteristic, complemented
4C31×#2408C211yesmaximal, characteristic
5C31×#2023C212yesmaximal, complemented
6C31×#2000C212yesmaximal, complemented
7C31×#1894C211yesmaximal, characteristic, complemented
8C31×#1880C212yesmaximal, complemented
9C31×#1875C212yesmaximal, complemented
10C31×#1870C211yesmaximal, characteristic, complemented
11C31×#1825C211yesmaximal, characteristic, complemented
12C23×D4×C31C2211yescharacteristic
13C22×SD16×C31C2212yes—
14C22×M16×C31C2211yescharacteristic
15C31×#556—24no—
16C31×#556—24no—
17C2×C31×C4⋊Q8C2211yescharacteristic, complemented
18C2×C31×#126—21no—
19C2×C31×#126—21no—
20C2×C31×#124—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×C22⋊Q8C2211yescharacteristic, complemented
23C2×C31×#121C2212yes—
24C2×C31×#120—21no—
25C2×C4×Q8×C31—21no—
26C2×C4×Q8×C31C2211yescharacteristic, complemented
27C2×C4×D4×C31C2211yescharacteristic
28C2×C31×#117C2211yescharacteristic
29C2×C31×#117C2211yescharacteristic, complemented
30C31×#450—24no—
31C31×#418—24no—
32C2×C31×#103C2212yescomplemented
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#102C2212yes—
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×Q8×C31—21no—
49C22×Q8×C31C2311yescharacteristic
50C22×D4×C31—24no—
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×C31C2312yes—
58C23×C4×C31C2311yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C2×SD16×C31—24no—
61C2×SD16×C31—24no—
62C2×SD16×C31—42no—
63C2×M16×C31—22no—
64C2×M16×C31—42no—
65C22×C8×C31C2312yes—
66C22×C8×C31C2312yes—
67C31×C4⋊Q8—22no—
68C31×C4⋊Q8—22no—
69C31×#126—28no—
70C31×#126—28no—
71C31×#124—28no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8D414yescomplemented
76C31×#121—24no—
77C31×#121—42no—
78C31×#120—44no—
79C4×Q8×C31—28no—
80C4×Q8×C31—22no—
81C4×Q8×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C31×#117—22no—
85C31×#117—22no—
86C31×#117—22no—
87C31×#117D414yescomplemented
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31D411yescharacteristic
93C2×C4⋊C4×C31C2312yes—
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C4⋊C4×C31C2311yescharacteristic
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—22no—
105C2×C22⋊C4×C31D411yescharacteristic
106C2×C22⋊C4×C31D411yescharacteristic
107C2×C42×C31—21no—
108C2×C42×C31—21no—
109C2×C42×C31—21no—
110C2×C42×C31C2311yescharacteristic
111C31×#103—42no—
112C31×#102—24no—
113C31×#102—24no—
114C31×#102—24no—
115C31×C22⋊C8—22no—
116C31×C22⋊C8—42no—
117C31×#95—21no—
118C24×C31—22no—
119C24×C31—44no—
120C24×C31—42no—
121C24×C31—42no—
122C24×C31—42no—
123C24×C31—82no—
124C24×C31C2×D411yescharacteristic
125C24×C31C2×D411yescharacteristic
126C2×Q8×C31—24no—
127C2×Q8×C31—21no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—42no—
131C2×Q8×C31—42no—
132C2×Q8×C31C2×D412yes—
133C2×D4×C31—28no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—22no—
139C2×D4×C31—22no—
140C2×D4×C31—22no—
141C2×D4×C31—48no—
142C2×D4×C31—41no—
143C2×D4×C31—44no—
144C2×D4×C31—44no—
145C2×D4×C31—44no—
146C2×D4×C31—44no—
147C2×D4×C31—42no—
148C2×D4×C31—82no—
149C2×D4×C31#4114yes—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31C2×D411yescharacteristic
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2411yescharacteristic
168SD16×C31—48no—
169M16×C31—44no—
170C2×C8×C31—22no—
171C2×C8×C31—22no—
172C2×C8×C31—22no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C2×C8×C31—22no—
176C4⋊C4×C31—24no—
177C4⋊C4×C31—24no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31C2×D412yes—
187C4⋊C4×C31C2×D412yes—
188C4⋊C4×C31C2×D412yes—
189C22⋊C4×C31—24no—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—24no—
192C22⋊C4×C31—24no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—44no—
196C22⋊C4×C31—42no—
197C22⋊C4×C31—44no—
198C22⋊C4×C31—82no—
199C42×C31—24no—
200C42×C31—24no—
201C42×C31—24no—
202C42×C31—21no—
203C42×C31—21no—
204#23979C3111yesmaximal, characteristic, direct product factor, Sylow
205C23×C31—22no—
206C23×C31—21no—
207C23×C31—21no—
208C23×C31—22no—
209C23×C31—21no—
210C23×C31—24no—
211C23×C31—22no—
212C23×C31—21no—
213C23×C31—21no—
214C23×C31—22no—
215C23×C31—48no—
216C23×C31—42no—
217C23×C31—42no—
218C23×C31—42no—
219C23×C31—42no—
220C23×C31—42no—
221C23×C31—42no—
222C23×C31—84no—
223C23×C31—84no—
224C23×C31—88no—
225C23×C31—88no—
226C23×C31#12012yes—
227C23×C31C22×D411yescharacteristic
228Q8×C31—24no—
229Q8×C31—48no—
230Q8×C31—41no—
231Q8×C31—41no—
232D4×C31—24no—
233D4×C31—24no—
234D4×C31—44no—
235D4×C31—48no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—88no—
239C2×C4×C31—21no—
240C2×C4×C31—22no—
241C2×C4×C31—21no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—24no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—42no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—42no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31#12012yes—
266C2×C4×C31AGL(1,ℤ/8)12yes—
267C2×C4×C31C22×D411yescharacteristic
268C2×C4×C31C22×D411yescharacteristic
269C2×C4×C31C2×#4112yes—
270C2×C4×C31H(5,𝔽2)12yes—
271C8×C31—42no—
272C8×C31—42no—
273C2×#556C2×C3111yescharacteristic, complemented
274C2×#450C2×C3111yescharacteristic, complemented
275C2×#418C2×C3111yescharacteristic, complemented
276#2408C2×C3111yescharacteristic
277#2023C2×C3112yescomplemented
278#2000C2×C3112yescomplemented
279#1894C2×C3111yescharacteristic, complemented
280#1880C2×C3112yescomplemented
281#1875C2×C3112yescomplemented
282#1870C2×C3111yescharacteristic, complemented
283#1825C2×C3111yescharacteristic, complemented
284C22×C31—22no—
285C22×C31—22no—
286C22×C31—22no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—44no—
291C22×C31—41no—
292C22×C31—41no—
293C22×C31—42no—
294C22×C31—44no—
295C22×C31—42no—
296C22×C31—44no—
297C22×C31—816no—
298C22×C31—82no—
299C22×C31—82no—
300C22×C31—82no—
301C22×C31—82no—
302C22×C31C2×#12011yescharacteristic
303C22×C31#54612yes—
304C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C31#57511yescharacteristic
306C22×C31#57712yes—
307C4×C31—22no—
308C4×C31—22no—
309C4×C31—41no—
310C4×C31—42no—
311C4×C31—42no—
312C4×C31—42no—
313C4×C31—41no—
314C4×C31—42no—
315C4×C31—82no—
316C4×C31#52511yescharacteristic
317C4×C31#53511yescharacteristic
318C4×C31#54612yes—
319C23×D4C22×C3111yescharacteristic
320C22×SD16C22×C3112yes—
321C22×M16C22×C3111yescharacteristic
322#556—24no—
323#556—24no—
324C2×C4⋊Q8C22×C3111yescharacteristic, complemented
325C2×#126—21no—
326C2×#126—21no—
327C2×#124—21no—
328C2×C22⋊Q8C22×C3111yescharacteristic
329C2×C22⋊Q8C22×C3111yescharacteristic, complemented
330C2×#121C22×C3112yes—
331C2×#120—21no—
332C2×C4×Q8—21no—
333C2×C4×Q8C22×C3111yescharacteristic, complemented
334C2×C4×D4C22×C3111yescharacteristic
335C2×#117C22×C3111yescharacteristic
336C2×#117C22×C3111yescharacteristic, complemented
337#450—24no—
338#418—24no—
339C2×#103C22×C3112yescomplemented
340C2×#102C22×C3112yes—
341C2×#102C22×C3112yes—
342C2×#102C22×C3112yes—
343C2×C22⋊C8C22×C3111yescharacteristic
344C2×C22⋊C8C22×C3112yescomplemented
345#394—21no—
346#392—21no—
347#390C22×C3112yes—
348#380C22×C3111yescharacteristic
349#379—21no—
350#340C22×C3112yescomplemented
351#340C22×C3112yescomplemented
352#340C22×C3112yescomplemented
353#336C22×C3112yescomplemented
354C2×C31—42no—
355C2×C31—81no—
356C2×C31—84no—
357C2×C31#240411yescharacteristic
358C2×C31#300611yescharacteristic
359C2×C31#301011yescharacteristic
360C2×C31#331112yes—
361C2×C31#332412yes—
362C25—21no—
363C22×Q8—21no—
364C22×Q8C23×C3111yescharacteristic
365C22×D4—24no—
366C22×D4—22no—
367C22×D4—22no—
368C22×D4—22no—
369C22×D4—21no—
370C22×D4—42no—
371C22×D4D4×C3111yescharacteristic
372C22×D4C23×C3112yes—
373C23×C4C23×C3111yescharacteristic
374C23×C4C23×C3111yescharacteristic
375C2×SD16—24no—
376C2×SD16—24no—
377C2×SD16—42no—
378C2×M16—22no—
379C2×M16—42no—
380C22×C8C23×C3112yes—
381C22×C8C23×C3112yes—
382C4⋊Q8—22no—
383C4⋊Q8—22no—
384#126—28no—
385#126—28no—
386#124—28no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8D4×C3114yescomplemented
391#121—24no—
392#121—42no—
393#120—44no—
394C4×Q8—28no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×D4—22no—
398C4×D4—22no—
399#117—22no—
400#117—22no—
401#117—22no—
402#117D4×C3114yescomplemented
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4D4×C3111yescharacteristic
408C2×C4⋊C4C23×C3112yes—
409C2×C4⋊C4C23×C3111yescharacteristic
410C2×C4⋊C4C23×C3111yescharacteristic
411C2×C4⋊C4C23×C3111yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4D4×C3111yescharacteristic
421C2×C22⋊C4D4×C3111yescharacteristic
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42C23×C3111yescharacteristic
426#103—42no—
427#102—24no—
428#102—24no—
429#102—24no—
430C22⋊C8—22no—
431C22⋊C8—42no—
432#95—21no—
433C31#2397911yescharacteristic, direct product factor, Sylow
434C24—22no—
435C24—44no—
436C24—42no—
437C24—42no—
438C24—42no—
439C24—82no—
440C24C2×D4×C3111yescharacteristic
441C24C2×D4×C3111yescharacteristic
442C2×Q8—24no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8C2×D4×C3112yes—
449C2×D4—28no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—48no—
458C2×D4—41no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—82no—
465C2×D4#41×C3114yes—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479C22×C4C2×D4×C3111yescharacteristic
480C22×C4C2×D4×C3112yes—
481C22×C4C2×D4×C3112yes—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C24×C3111yescharacteristic
484SD16—48no—
485M16—44no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4C2×D4×C3112yes—
503C4⋊C4C2×D4×C3112yes—
504C4⋊C4C2×D4×C3112yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—21no—
519C42—21no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—21no—
525C23—24no—
526C23—22no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—48no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—84no—
538C23—84no—
539C23—88no—
540C23—88no—
541C23C31×#12012yes—
542C23C22×D4×C3111yescharacteristic
543Q8—24no—
544Q8—48no—
545Q8—41no—
546Q8—41no—
547D4—24no—
548D4—24no—
549D4—44no—
550D4—48no—
551D4—42no—
552D4—42no—
553D4—88no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4C31×#12012yes—
581C2×C4C31×AGL(1,ℤ/8)12yes—
582C2×C4C22×D4×C3111yescharacteristic
583C2×C4C22×D4×C3111yescharacteristic
584C2×C4C2×#41×C3112yes—
585C2×C4C31×H(5,𝔽2)12yes—
586C8—42no—
587C8—42no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—42no—
592C22—42no—
593C22—41no—
594C22—44no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—816no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C2×C31×#12011yescharacteristic
607C22C31×#54612yes—
608C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
609C22C31×#57511yescharacteristic
610C22C31×#57712yes—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—82no—
620C4C31×#52511yescharacteristic
621C4C31×#53511yescharacteristic
622C4C31×#54612yes—
623C2—42no—
624C2—81no—
625C2—84no—
626C2C31×#240411yescharacteristic
627C2C31×#300611yescharacteristic
628C2C31×#301011yescharacteristic
629C2C31×#331112yes—
630C2C31×#332412yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2408C2
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×#556C2semidirect
2. C2×C31×#450C2semidirect
3. C2×C31×#418C2semidirect
5. C31×#2023C2semidirect
6. C31×#2000C2semidirect
7. C31×#1894C2semidirect
8. C31×#1880C2semidirect
9. C31×#1875C2semidirect
10. C31×#1870C2semidirect
11. C31×#1825C2semidirect
17. C2×C31×C4⋊Q8C22semidirect
22. C2×C31×C22⋊Q8C22semidirect
26. C2×C4×Q8×C31C22semidirect
29. C2×C31×#117C22semidirect
32. C2×C31×#103C22semidirect
37. C2×C31×C22⋊C8C22semidirect
43. C31×#340C22semidirect
44. C31×#340C22semidirect
45. C31×#340C22semidirect
46. C31×#336C22semidirect
75. C31×C22⋊Q8D4semidirect
87. C31×#117D4semidirect
204. #23979C31direct (N × H)
273. C2×#556C2×C31semidirect
274. C2×#450C2×C31semidirect
275. C2×#418C2×C31semidirect
277. #2023C2×C31semidirect
278. #2000C2×C31semidirect
279. #1894C2×C31semidirect
280. #1880C2×C31semidirect
281. #1875C2×C31semidirect
282. #1870C2×C31semidirect
283. #1825C2×C31semidirect
324. C2×C4⋊Q8C22×C31semidirect
329. C2×C22⋊Q8C22×C31semidirect
333. C2×C4×Q8C22×C31semidirect
336. C2×#117C22×C31semidirect
339. C2×#103C22×C31semidirect
344. C2×C22⋊C8C22×C31semidirect
350. #340C22×C31semidirect
351. #340C22×C31semidirect
352. #340C22×C31semidirect
353. #336C22×C31semidirect
390. C22⋊Q8D4×C31semidirect
402. #117D4×C31semidirect
433. C31#23979direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5562
C2×C31×#4502
C2×C31×#4182
C31×#24082
C31×#20232
C31×#20002
C31×#18942
C31×#18802
C31×#18752
C31×#18702
C31×#18252
#2397931