Finite Group Databasebeta

← C31×#24661 #21481420

Subgroups — C31×#24661 #21481420

Order 7936.

1016 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#460C211yesmaximal, characteristic, complemented
2C2×C31×#432C211yesmaximal, characteristic, complemented
3C31×#2856C211yesmaximal, characteristic
4C4×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
5C31×#2033C212yesmaximal, complemented
6C31×#2000C212yesmaximal, complemented
7C31×#1982C211yesmaximal, characteristic, complemented
8C31×#1973C212yesmaximal, complemented
9C31×#1947C211yesmaximal, characteristic, complemented
10C31×#1936C212yesmaximal, complemented
11C31×#1803C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C2×C31×#127C2211yescharacteristic
14C2×C31×C22⋊Q8C2211yescharacteristic
15C2×C31×#121—21no—
16C2×C31×#121—21no—
17C2×C31×#121C2212yes—
18C2×C4×Q8×C31C2211yescharacteristic
19C22×C42×C31C2211yescharacteristic
20C31×#460—24no—
21C31×#432—24no—
22C2×C31×#107C2212yes—
23C2×C31×C4⋊C8C2211yescharacteristic
24C2×C31×C4⋊C8C2211yescharacteristic
25C2×C31×#103C2212yes—
26C2×C31×#102C2212yes—
27C2×C31×#102C2212yes—
28C2×C31×#102C2212yes—
29C2×C31×C22⋊C8C2212yes—
30C2×C31×#97C2212yes—
31C31×#391C2211yescharacteristic
32C31×#390—21no—
33C31×#390—21no—
34C31×#390C2212yes—
35C31×#389C2211yescharacteristic
36C31×#387—21no—
37C31×#385—21no—
38C31×#384C2211yescharacteristic
39C31×#380C2211yescharacteristic
40C4×C4⋊C4×C31—21no—
41C4×C4⋊C4×C31C2211yescharacteristic
42C4×C22⋊C4×C31—21no—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C22×Q8×C31C2311yescharacteristic
46C22×D4×C31—22no—
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31C2312yes—
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C2×SD16×C31—24no—
54C2×SD16×C31—24no—
55C2×SD16×C31—42no—
56C22×C8×C31C2312yes—
57C22×C8×C31C2312yes—
58C31×#127—22no—
59C31×#127—22no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×#121—42no—
63C31×#121—42no—
64C31×#121—44no—
65C31×#121—44no—
66C4×Q8×C31—22no—
67C4×Q8×C31—22no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31C2311yescharacteristic
74C2×C4⋊C4×C31C2311yescharacteristic
75C2×C4⋊C4×C31C2312yes—
76C2×C4⋊C4×C31C2311yescharacteristic
77C2×C22⋊C4×C31—21no—
78C2×C22⋊C4×C31—21no—
79C2×C22⋊C4×C31—22no—
80C2×C22⋊C4×C31—22no—
81C2×C22⋊C4×C31—22no—
82C2×C42×C31—22no—
83C2×C42×C31—22no—
84C2×C42×C31—21no—
85C2×C42×C31—21no—
86C2×C42×C31D411yescharacteristic
87C2×C42×C31D411yescharacteristic
88C2×C42×C31C2311yescharacteristic
89C2×C42×C31C2311yescharacteristic
90C31×#107—42no—
91C31×C4⋊C8—22no—
92C31×C4⋊C8D414yescomplemented
93C31×#103—42no—
94C31×#102—24no—
95C31×#102—24no—
96C31×#102—42no—
97C31×C22⋊C8—24no—
98C31×#97—24no—
99C31×#95—21no—
100C31×#95—21no—
101C31×#95—21no—
102C31×#95—21no—
103C31×#95—21no—
104C24×C31—22no—
105C24×C31C2×D411yescharacteristic
106C2×Q8×C31—21no—
107C2×Q8×C31—21no—
108C2×Q8×C31—21no—
109C2×Q8×C31—21no—
110C2×Q8×C31—21no—
111C2×Q8×C31#4112yes—
112C2×D4×C31—22no—
113C2×D4×C31—22no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31—42no—
119C2×D4×C31—42no—
120C2×D4×C31—44no—
121C2×D4×C31—44no—
122C2×D4×C31—44no—
123C2×D4×C31—42no—
124C2×D4×C31—42no—
125C2×D4×C31—42no—
126C2×D4×C31—42no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—22no—
143C22×C4×C31—41no—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2411yescharacteristic
150SD16×C31—48no—
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31C2×D414yes—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—42no—
166C4⋊C4×C31—42no—
167C4⋊C4×C31—42no—
168C4⋊C4×C31—42no—
169C4⋊C4×C31#4112yes—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—44no—
172C22⋊C4×C31—44no—
173C22⋊C4×C31—44no—
174C22⋊C4×C31—42no—
175C42×C31—22no—
176C42×C31—22no—
177C42×C31—21no—
178C42×C31—21no—
179C42×C31—42no—
180C42×C31—42no—
181C42×C31C2×D412yes—
182C42×C31#4112yes—
183#24661C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31AGL(1,ℤ/8)12yes—
198C23×C31C22×D411yescharacteristic
199Q8×C31—22no—
200Q8×C31—22no—
201Q8×C31—41no—
202Q8×C31—41no—
203D4×C31—42no—
204D4×C31—42no—
205D4×C31—42no—
206D4×C31—42no—
207D4×C31—84no—
208D4×C31—88no—
209D4×C31—84no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—44no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—44no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31#12112yes—
243C2×C4×C31#12112yes—
244C2×C4×C31AGL(1,ℤ/8)12yes—
245C2×C4×C31#13712yes—
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C2×#4111yescharacteristic
248C2×C4×C31C2×#4111yescharacteristic
249C2×C4×C31C2×#4111yescharacteristic
250C2×C4×C31H(5,𝔽2)11yescharacteristic
251C2×C4×C31H(5,𝔽2)11yescharacteristic
252C8×C31—24no—
253C8×C31—42no—
254C2×#460C2×C3111yescharacteristic, complemented
255C2×#432C2×C3111yescharacteristic, complemented
256#2856C2×C3111yescharacteristic
257C4×C22⋊Q8C2×C3111yescharacteristic, complemented
258#2033C2×C3112yescomplemented
259#2000C2×C3112yescomplemented
260#1982C2×C3111yescharacteristic, complemented
261#1973C2×C3112yescomplemented
262#1947C2×C3111yescharacteristic, complemented
263#1936C2×C3112yescomplemented
264#1803C2×C3111yescharacteristic, complemented
265C22×C31—22no—
266C22×C31—22no—
267C22×C31—22no—
268C22×C31—41no—
269C22×C31—41no—
270C22×C31—41no—
271C22×C31—41no—
272C22×C31—82no—
273C22×C31—82no—
274C22×C31—82no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31C2×#12111yescharacteristic
280C22×C31#52511yescharacteristic
281C22×C31#53511yescharacteristic
282C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C31C2×#13711yescharacteristic
285C22×C31#57511yescharacteristic
286C4×C31—22no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—41no—
291C4×C31—41no—
292C4×C31—81no—
293C4×C31—82no—
294C4×C31—81no—
295C4×C31#46814yes—
296C4×C31#54611yescharacteristic
297C4×C31#55011yescharacteristic
298C4×C31#55311yescharacteristic
299C4×C31#55511yescharacteristic
300C22×SD16C22×C3112yes—
301C2×#127C22×C3111yescharacteristic
302C2×C22⋊Q8C22×C3111yescharacteristic
303C2×#121—21no—
304C2×#121—21no—
305C2×#121C22×C3112yes—
306C2×C4×Q8C22×C3111yescharacteristic
307C22×C42C22×C3111yescharacteristic
308#460—24no—
309#432—24no—
310C2×#107C22×C3112yes—
311C2×C4⋊C8C22×C3111yescharacteristic
312C2×C4⋊C8C22×C3111yescharacteristic
313C2×#103C22×C3112yes—
314C2×#102C22×C3112yes—
315C2×#102C22×C3112yes—
316C2×#102C22×C3112yes—
317C2×C22⋊C8C22×C3112yes—
318C2×#97C22×C3112yes—
319#391C22×C3111yescharacteristic
320#390—21no—
321#390—21no—
322#390C22×C3112yes—
323#389C22×C3111yescharacteristic
324#387—21no—
325#385—21no—
326#384C22×C3111yescharacteristic
327#380C22×C3111yescharacteristic
328C4×C4⋊C4—21no—
329C4×C4⋊C4C22×C3111yescharacteristic
330C4×C22⋊C4—21no—
331#340C22×C3112yes—
332#340C22×C3112yes—
333C2×C31—42no—
334C2×C31—162no—
335C2×C31#241411yescharacteristic
336C2×C31C2×#46811yescharacteristic
337C2×C31#305511yescharacteristic
338C2×C31#311211yescharacteristic
339C2×C31#311511yescharacteristic
340C2×C31#321611yescharacteristic
341C2×C31#321711yescharacteristic
342C22×Q8C23×C3111yescharacteristic
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4C23×C3112yes—
347C23×C4D4×C3111yescharacteristic
348C23×C4D4×C3111yescharacteristic
349C23×C4C23×C3111yescharacteristic
350C2×SD16—24no—
351C2×SD16—24no—
352C2×SD16—42no—
353C22×C8C23×C3112yes—
354C22×C8C23×C3112yes—
355#127—22no—
356#127—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359#121—42no—
360#121—42no—
361#121—44no—
362#121—44no—
363C4×Q8—22no—
364C4×Q8—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4C23×C3111yescharacteristic
371C2×C4⋊C4C23×C3112yes—
372C2×C4⋊C4C23×C3111yescharacteristic
373C2×C4⋊C4C23×C3111yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42D4×C3111yescharacteristic
384C2×C42D4×C3111yescharacteristic
385C2×C42C23×C3111yescharacteristic
386C2×C42C23×C3111yescharacteristic
387#107—42no—
388C4⋊C8—22no—
389C4⋊C8D4×C3114yescomplemented
390#103—42no—
391#102—24no—
392#102—24no—
393#102—42no—
394C22⋊C8—24no—
395#97—24no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401C31#2466111yescharacteristic, direct product factor, Sylow
402C24—22no—
403C24C2×D4×C3111yescharacteristic
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8#41×C3112yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C24×C3111yescharacteristic
448SD16—48no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8C2×D4×C3114yes—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4#41×C3112yes—
468C22⋊C4—24no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—42no—
478C42—42no—
479C42C2×D4×C3112yes—
480C42#41×C3112yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23C31×AGL(1,ℤ/8)12yes—
495C23C22×D4×C3111yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—41no—
499Q8—41no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—84no—
505D4—88no—
506D4—84no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4C31×#12112yes—
540C2×C4C31×#12112yes—
541C2×C4C31×AGL(1,ℤ/8)12yes—
542C2×C4C31×#13712yes—
543C2×C4C22×D4×C3111yescharacteristic
544C2×C4C2×#41×C3111yescharacteristic
545C2×C4C2×#41×C3111yescharacteristic
546C2×C4C2×#41×C3111yescharacteristic
547C2×C4C31×H(5,𝔽2)11yescharacteristic
548C2×C4C31×H(5,𝔽2)11yescharacteristic
549C8—24no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22C2×C31×#12111yescharacteristic
566C22C31×#52511yescharacteristic
567C22C31×#53511yescharacteristic
568C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
569C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
570C22C2×C31×#13711yescharacteristic
571C22C31×#57511yescharacteristic
572C4—22no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—41no—
578C4—82no—
579C4—81no—
580C4—81no—
581C4C31×#46814yes—
582C4C31×#54611yescharacteristic
583C4C31×#55011yescharacteristic
584C4C31×#55311yescharacteristic
585C4C31×#55511yescharacteristic
586C2—42no—
587C2—162no—
588C2C31×#241411yescharacteristic
589C2C2×C31×#46811yescharacteristic
590C2C31×#305511yescharacteristic
591C2C31×#311211yescharacteristic
592C2C31×#311511yescharacteristic
593C2C31×#321611yescharacteristic
594C2C31×#321711yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#460C2semidirect
2. C2×C31×#432C2semidirect
4. C4×C31×C22⋊Q8C2semidirect
5. C31×#2033C2semidirect
6. C31×#2000C2semidirect
7. C31×#1982C2semidirect
8. C31×#1973C2semidirect
9. C31×#1947C2semidirect
10. C31×#1936C2semidirect
11. C31×#1803C2semidirect
92. C31×C4⋊C8D4semidirect
183. #24661C31direct (N × H)
254. C2×#460C2×C31semidirect
255. C2×#432C2×C31semidirect
257. C4×C22⋊Q8C2×C31semidirect
258. #2033C2×C31semidirect
259. #2000C2×C31semidirect
260. #1982C2×C31semidirect
261. #1973C2×C31semidirect
262. #1947C2×C31semidirect
263. #1936C2×C31semidirect
264. #1803C2×C31semidirect
389. C4⋊C8D4×C31semidirect
401. C31#24661direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4602
C2×C31×#4322
C31×#28562
C4×C31×C22⋊Q82
C31×#20332
C31×#20002
C31×#19822
C31×#19732
C31×#19472
C31×#19362
C31×#18032
#2466131