Finite Group Databasebeta

← C31×#32935 #21489694

Subgroups — C31×#32935 #21489694

Order 7936.

1580 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C31C211yesmaximal, characteristic, complemented
2C2×C31×#468C212yesmaximal, complemented
3C2×C31×#465C211yesmaximal, characteristic, complemented
4C31×#2848C211yesmaximal, characteristic, complemented
5C31×#2847C211yesmaximal, characteristic
6C31×#1895C212yesmaximal, complemented
7C31×#1894C212yesmaximal, complemented
8C31×#1893C211yesmaximal, characteristic, complemented
9C31×#1892C211yesmaximal, characteristic, complemented
10C31×#1809C212yesmaximal, complemented
11C31×#1808C211yesmaximal, characteristic, complemented
12C24×C4×C31C2211yescharacteristic
13C22×SD16×C31C2211yescharacteristic, complemented
14C22×M16×C31—28no—
15C22×M16×C31—24no—
16C22×M16×C31C2212yes—
17C22×M16×C31C2212yes—
18C23×C8×C31C2211yescharacteristic
19C23×C8×C31C2211yescharacteristic
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C31×C22⋊Q8C2212yes—
22C2×C31×#121C2211yescharacteristic
23C2×C31×#121C2212yes—
24C2×C31×#120—21no—
25C31×#468—28no—
26C31×#465—24no—
27C2×C31×#107C2212yescomplemented
28C2×C31×#106C2211yescharacteristic, complemented
29C2×C31×#103C2212yescomplemented
30C2×C31×#103C2211yescharacteristic, complemented
31C2×C31×#102C2212yescomplemented
32C2×C31×#102C2211yescharacteristic, complemented
33C31×#386—22no—
34C31×#386—21no—
35C31×#385—22no—
36C31×#385—21no—
37C31×#380C2211yescharacteristic
38C31×#380C2211yescharacteristic
39C31×#380C2212yes—
40C31×#380C2212yes—
41C31×#379—21no—
42C31×#340C2212yescomplemented
43C31×#340C2212yescomplemented
44C31×#340C2212yescomplemented
45C25×C31D411yescharacteristic
46C22×Q8×C31C2311yescharacteristic
47C22×D4×C31—22no—
48C22×D4×C31—21no—
49C22×D4×C31C2311yescharacteristic
50C23×C4×C31—24no—
51C23×C4×C31—22no—
52C23×C4×C31—24no—
53C23×C4×C31—22no—
54C23×C4×C31D412yes—
55C23×C4×C31D411yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C23×C4×C31C2312yes—
58C2×SD16×C31—22no—
59C2×SD16×C31—24no—
60C2×M16×C31—232no—
61C2×M16×C31—24no—
62C2×M16×C31—24no—
63C2×M16×C31—24no—
64C2×M16×C31—24no—
65C2×M16×C31—24no—
66C2×M16×C31D418yescomplemented
67C22×C8×C31—22no—
68C22×C8×C31—24no—
69C22×C8×C31—24no—
70C22×C8×C31D414yescomplemented
71C22×C8×C31C2312yes—
72C22×C8×C31C2311yescharacteristic
73C22×C8×C31C2311yescharacteristic
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—24no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—24no—
78C31×#121—22no—
79C31×#121—24no—
80C31×#121—22no—
81C31×#121—24no—
82C31×#120—44no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2312yes—
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2312yes—
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—22no—
97C31×#107—24no—
98C31×#106—22no—
99C31×#103—22no—
100C31×#103—24no—
101C31×#102—22no—
102C31×#102—24no—
103C31×#95—21no—
104C31×#95—22no—
105C31×#95—21no—
106C31×#95—22no—
107C24×C31—24no—
108C24×C31—22no—
109C24×C31—22no—
110C24×C31—22no—
111C24×C31—21no—
112C24×C31—42no—
113C24×C31C2×D412yes—
114C24×C31C2×D411yescharacteristic
115C2×Q8×C31—21no—
116C2×Q8×C31—22no—
117C2×Q8×C31—21no—
118C2×Q8×C31—22no—
119C2×D4×C31—21no—
120C2×D4×C31—22no—
121C2×D4×C31—21no—
122C2×D4×C31—22no—
123C2×D4×C31—44no—
124C2×D4×C31—44no—
125C2×D4×C31—44no—
126C2×D4×C31—42no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C22×C4×C31—28no—
130C22×C4×C31—28no—
131C22×C4×C31—216no—
132C22×C4×C31—24no—
133C22×C4×C31—22no—
134C22×C4×C31—24no—
135C22×C4×C31—24no—
136C22×C4×C31—24no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D414yes—
152C22×C4×C31#4114yes—
153C22×C4×C31#4112yes—
154C22×C4×C31C2411yescharacteristic
155SD16×C31—44no—
156M16×C31—216no—
157M16×C31—216no—
158C2×C8×C31—28no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C2×C8×C31—22no—
162C2×C8×C31—28no—
163C2×C8×C31C2×D414yes—
164C2×C8×C31C2×D412yes—
165C2×C8×C31C2×D412yes—
166C2×C8×C31C2×D414yes—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—42no—
176C4⋊C4×C31—44no—
177C22⋊C4×C31—44no—
178C22⋊C4×C31—44no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—42no—
181C22⋊C4×C31—42no—
182C22⋊C4×C31—42no—
183C22⋊C4×C31—42no—
184C22⋊C4×C31—44no—
185#32935C3111yesmaximal, characteristic, direct product factor, Sylow
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—28no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—48no—
196C23×C31—48no—
197C23×C31—44no—
198C23×C31—44no—
199C23×C31—41no—
200C23×C31—42no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—42no—
204C23×C31#12114yes—
205C23×C31#12112yes—
206C23×C31AGL(1,ℤ/8)12yes—
207C23×C31#13712yes—
208C23×C31C22×D411yescharacteristic
209Q8×C31—41no—
210Q8×C31—41no—
211Q8×C31—42no—
212D4×C31—41no—
213D4×C31—41no—
214D4×C31—42no—
215D4×C31—88no—
216D4×C31—84no—
217C2×C4×C31—28no—
218C2×C4×C31—216no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—24no—
224C2×C4×C31—22no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—41no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—41no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31#12112yes—
243C2×C4×C31#12114yes—
244C2×C4×C31C22×D411yescharacteristic
245C2×C4×C31C22×D412yes—
246C2×C4×C31C2×#4111yescharacteristic
247C2×C4×C31C2×#4112yes—
248C8×C31—24no—
249C8×C31—22no—
250C8×C31#12014yescomplemented
251C23×M16C2×C3111yescharacteristic, complemented
252C2×#468C2×C3112yescomplemented
253C2×#465C2×C3111yescharacteristic, complemented
254#2848C2×C3111yescharacteristic, complemented
255#2847C2×C3111yescharacteristic
256#1895C2×C3112yescomplemented
257#1894C2×C3112yescomplemented
258#1893C2×C3111yescharacteristic, complemented
259#1892C2×C3111yescharacteristic, complemented
260#1809C2×C3112yescomplemented
261#1808C2×C3111yescharacteristic, complemented
262C22×C31—22no—
263C22×C31—22no—
264C22×C31—24no—
265C22×C31—24no—
266C22×C31—24no—
267C22×C31—24no—
268C22×C31—24no—
269C22×C31—416no—
270C22×C31—42no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—81no—
275C22×C31—81no—
276C22×C31—81no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31#46814yescomplemented
280C22×C31C2×#12111yescharacteristic
281C22×C31C2×#12112yes—
282C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C31C2×#13711yescharacteristic
284C22×C31#57512yes—
285C4×C31—22no—
286C4×C31—24no—
287C4×C31—81no—
288C4×C31—81no—
289C4×C31—81no—
290C4×C31—82no—
291C4×C31—82no—
292C4×C31C2×#12011yescharacteristic
293C4×C31#52511yescharacteristic
294C4×C31#52512yes—
295C24×C4C22×C3111yescharacteristic
296C22×SD16C22×C3111yescharacteristic, complemented
297C22×M16—28no—
298C22×M16—24no—
299C22×M16C22×C3112yes—
300C22×M16C22×C3112yes—
301C23×C8C22×C3111yescharacteristic
302C23×C8C22×C3111yescharacteristic
303C2×C22⋊Q8C22×C3111yescharacteristic
304C2×C22⋊Q8C22×C3112yes—
305C2×#121C22×C3111yescharacteristic
306C2×#121C22×C3112yes—
307C2×#120—21no—
308#468—28no—
309#465—24no—
310C2×#107C22×C3112yescomplemented
311C2×#106C22×C3111yescharacteristic, complemented
312C2×#103C22×C3112yescomplemented
313C2×#103C22×C3111yescharacteristic, complemented
314C2×#102C22×C3112yescomplemented
315C2×#102C22×C3111yescharacteristic, complemented
316#386—22no—
317#386—21no—
318#385—22no—
319#385—21no—
320#380C22×C3111yescharacteristic
321#380C22×C3111yescharacteristic
322#380C22×C3112yes—
323#380C22×C3112yes—
324#379—21no—
325#340C22×C3112yescomplemented
326#340C22×C3112yescomplemented
327#340C22×C3112yescomplemented
328C2×C31—24no—
329C2×C31—44no—
330C2×C31—161no—
331C2×C31#284711yescharacteristic
332C2×C31C2×#46811yescharacteristic
333C2×C31#305411yescharacteristic
334C2×C31#305512yes—
335C2×C31#305612yes—
336C25D4×C3111yescharacteristic
337C22×Q8C23×C3111yescharacteristic
338C22×D4—22no—
339C22×D4—21no—
340C22×D4C23×C3111yescharacteristic
341C23×C4—24no—
342C23×C4—22no—
343C23×C4—24no—
344C23×C4—22no—
345C23×C4D4×C3112yes—
346C23×C4D4×C3111yescharacteristic
347C23×C4C23×C3111yescharacteristic
348C23×C4C23×C3112yes—
349C2×SD16—22no—
350C2×SD16—24no—
351C2×M16—232no—
352C2×M16—24no—
353C2×M16—24no—
354C2×M16—24no—
355C2×M16—24no—
356C2×M16—24no—
357C2×M16D4×C3118yescomplemented
358C22×C8—22no—
359C22×C8—24no—
360C22×C8—24no—
361C22×C8D4×C3114yescomplemented
362C22×C8C23×C3112yes—
363C22×C8C23×C3111yescharacteristic
364C22×C8C23×C3111yescharacteristic
365C22⋊Q8—22no—
366C22⋊Q8—24no—
367C22⋊Q8—22no—
368C22⋊Q8—24no—
369#121—22no—
370#121—24no—
371#121—22no—
372#121—24no—
373#120—44no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4C23×C3111yescharacteristic
377C2×C4⋊C4C23×C3112yes—
378C2×C4⋊C4C23×C3111yescharacteristic
379C2×C4⋊C4C23×C3112yes—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—22no—
388#107—24no—
389#106—22no—
390#103—22no—
391#103—24no—
392#102—22no—
393#102—24no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—22no—
398C31#3293511yescharacteristic, direct product factor, Sylow
399C24—24no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—42no—
405C24C2×D4×C3112yes—
406C24C2×D4×C3111yescharacteristic
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—216no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3112yes—
442C22×C4C2×D4×C3112yes—
443C22×C4C2×D4×C3114yes—
444C22×C4#41×C3114yes—
445C22×C4#41×C3112yes—
446C22×C4C24×C3111yescharacteristic
447SD16—44no—
448M16—216no—
449M16—216no—
450C2×C8—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—28no—
455C2×C8C2×D4×C3114yes—
456C2×C8C2×D4×C3112yes—
457C2×C8C2×D4×C3112yes—
458C2×C8C2×D4×C3114yes—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—44no—
477C23—22no—
478C23—22no—
479C23—28no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—48no—
487C23—48no—
488C23—44no—
489C23—44no—
490C23—41no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23C31×#12114yes—
496C23C31×#12112yes—
497C23C31×AGL(1,ℤ/8)12yes—
498C23C31×#13712yes—
499C23C22×D4×C3111yescharacteristic
500Q8—41no—
501Q8—41no—
502Q8—42no—
503D4—41no—
504D4—41no—
505D4—42no—
506D4—88no—
507D4—84no—
508C2×C4—28no—
509C2×C4—216no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4C31×#12112yes—
534C2×C4C31×#12114yes—
535C2×C4C22×D4×C3111yescharacteristic
536C2×C4C22×D4×C3112yes—
537C2×C4C2×#41×C3111yescharacteristic
538C2×C4C2×#41×C3112yes—
539C8—24no—
540C8—22no—
541C8C31×#12014yescomplemented
542C22—22no—
543C22—22no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—416no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—82no—
558C22—82no—
559C22C31×#46814yescomplemented
560C22C2×C31×#12111yescharacteristic
561C22C2×C31×#12112yes—
562C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
563C22C2×C31×#13711yescharacteristic
564C22C31×#57512yes—
565C4—22no—
566C4—24no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—82no—
571C4—82no—
572C4C2×C31×#12011yescharacteristic
573C4C31×#52511yescharacteristic
574C4C31×#52512yes—
575C2—24no—
576C2—44no—
577C2—161no—
578C2C31×#284711yescharacteristic
579C2C2×C31×#46811yescharacteristic
580C2C31×#305411yescharacteristic
581C2C31×#305512yes—
582C2C31×#305612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×M16×C31C2semidirect
2. C2×C31×#468C2semidirect
3. C2×C31×#465C2semidirect
4. C31×#2848C2semidirect
6. C31×#1895C2semidirect
7. C31×#1894C2semidirect
8. C31×#1893C2semidirect
9. C31×#1892C2semidirect
10. C31×#1809C2semidirect
11. C31×#1808C2semidirect
13. C22×SD16×C31C22semidirect
27. C2×C31×#107C22semidirect
28. C2×C31×#106C22semidirect
29. C2×C31×#103C22semidirect
30. C2×C31×#103C22semidirect
31. C2×C31×#102C22semidirect
32. C2×C31×#102C22semidirect
42. C31×#340C22semidirect
43. C31×#340C22semidirect
44. C31×#340C22semidirect
66. C2×M16×C31D4semidirect
70. C22×C8×C31D4semidirect
185. #32935C31direct (N × H)
250. C8×C31#120semidirect
251. C23×M16C2×C31semidirect
252. C2×#468C2×C31semidirect
253. C2×#465C2×C31semidirect
254. #2848C2×C31semidirect
256. #1895C2×C31semidirect
257. #1894C2×C31semidirect
258. #1893C2×C31semidirect
259. #1892C2×C31semidirect
260. #1809C2×C31semidirect
261. #1808C2×C31semidirect
279. C22×C31#468semidirect
296. C22×SD16C22×C31semidirect
310. C2×#107C22×C31semidirect
311. C2×#106C22×C31semidirect
312. C2×#103C22×C31semidirect
313. C2×#103C22×C31semidirect
314. C2×#102C22×C31semidirect
315. C2×#102C22×C31semidirect
325. #340C22×C31semidirect
326. #340C22×C31semidirect
327. #340C22×C31semidirect
357. C2×M16D4×C31semidirect
361. C22×C8D4×C31semidirect
398. C31#32935direct (N × H)
541. C8C31×#120semidirect
559. C22C31×#468semidirect

Maximal subgroups

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

subgroupindex
C23×M16×C312
C2×C31×#4682
C2×C31×#4652
C31×#28482
C31×#28472
C31×#18952
C31×#18942
C31×#18932
C31×#18922
C31×#18092
C31×#18082
#3293531