Finite Group Databasebeta

← C31×#23923 #21480682

Subgroups — C31×#23923 #21480682

Order 7936.

1948 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#525C212yesmaximal, complemented
2C2×C31×#452C212yesmaximal, complemented
3C2×C31×#449C212yesmaximal, complemented
4C31×#2001C211yesmaximal, characteristic, complemented
5C31×#2000C212yesmaximal, complemented
6C31×#1999C211yesmaximal, characteristic, complemented
7C31×#1875C212yesmaximal, complemented
8C31×#1873C212yesmaximal, complemented
9C31×#1779C211yesmaximal, characteristic, complemented
10C22×#41×C31C2212yes—
11C24×C4×C31C2211yescharacteristic
12C22×Q16×C31C2211yescharacteristic, complemented
13C22×SD16×C31C2212yescomplemented
14C22×D8×C31C2211yescharacteristic, complemented
15C31×#525—28no—
16C31×#525—28no—
17C2×C31×#123—22no—
18C2×C31×C22⋊Q8C2212yes—
19C2×C31×C22⋊Q8C2212yes—
20C2×C31×#121C2212yes—
21C2×C31×#121C2212yes—
22C2×C31×#120—22no—
23C2×C4×D4×C31—22no—
24C2×C4×D4×C31—22no—
25C2×C31×#117C2212yes—
26C31×#452—28no—
27C31×#449—28no—
28C2×C31×#103C2212yescomplemented
29C2×C31×#103C2212yescomplemented
30C2×C31×#102C2212yescomplemented
31C2×C31×#102C2212yescomplemented
32C2×C31×C22⋊C8C2212yes—
33C2×C31×C22⋊C8C2212yes—
34C2×C31×C22⋊C8C2211yescharacteristic
35C2×C31×C22⋊C8C2211yescharacteristic
36C31×#340C2211yescharacteristic, complemented
37C31×#340C2212yescomplemented
38C31×#340C2211yescharacteristic, complemented
39C25×C31D411yescharacteristic
40C2×#41×C31—24no—
41C2×#41×C31—24no—
42C2×#41×C31—24no—
43C2×#41×C31—24no—
44C2×#41×C31—42no—
45C2×#41×C31—42no—
46C22×Q8×C31C2312yes—
47C22×D4×C31—22no—
48C22×D4×C31—22no—
49C22×D4×C31—22no—
50C22×D4×C31C2312yes—
51C23×C4×C31—22no—
52C23×C4×C31—24no—
53C23×C4×C31—24no—
54C23×C4×C31—41no—
55C23×C4×C31—41no—
56C23×C4×C31D411yescharacteristic
57C23×C4×C31D412yes—
58C23×C4×C31C2311yescharacteristic
59C23×C4×C31C2312yes—
60C2×Q16×C31—24no—
61C2×Q16×C31—41no—
62C2×SD16×C31—24no—
63C2×SD16×C31—24no—
64C2×SD16×C31—42no—
65C2×D8×C31—24no—
66C2×D8×C31—41no—
67C22×C8×C31C2311yescharacteristic
68C22×C8×C31C2312yes—
69C22×C8×C31C2311yescharacteristic
70C31×#123—216no—
71C31×C22⋊Q8—24no—
72C31×C22⋊Q8—24no—
73C31×C22⋊Q8—24no—
74C31×C22⋊Q8D418yescomplemented
75C31×#121—24no—
76C31×#121—24no—
77C31×#121—24no—
78C31×#121D418yescomplemented
79C31×#120—216no—
80C4×D4×C31—216no—
81C4×D4×C31—216no—
82C31×#117—24no—
83C31×#117—24no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31C2312yes—
87C2×C4⋊C4×C31C2312yes—
88C2×C22⋊C4×C31—22no—
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—22no—
93C2×C42×C31—22no—
94C31×#103—24no—
95C31×#103—24no—
96C31×#102—24no—
97C31×#102—24no—
98C31×C22⋊C8—24no—
99C31×C22⋊C8—24no—
100C31×C22⋊C8—41no—
101C31×C22⋊C8—41no—
102C24×C31—24no—
103C24×C31—22no—
104C24×C31—22no—
105C24×C31—41no—
106C24×C31—42no—
107C24×C31—41no—
108C24×C31C2×D412yes—
109C24×C31C2×D411yescharacteristic
110#41×C31—48no—
111#41×C31—48no—
112#41×C31—48no—
113#41×C31—48no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31C2×D414yes—
120C2×D4×C31—28no—
121C2×D4×C31—28no—
122C2×D4×C31—28no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—44no—
129C2×D4×C31—44no—
130C2×D4×C31—44no—
131C2×D4×C31—44no—
132C2×D4×C31—44no—
133C2×D4×C31—44no—
134C2×D4×C31C2×D414yes—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—28no—
140C22×C4×C31—24no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—24no—
145C22×C4×C31—24no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—41no—
151C22×C4×C31—42no—
152C22×C4×C31—44no—
153C22×C4×C31—48no—
154C22×C4×C31—44no—
155C22×C4×C31—44no—
156C22×C4×C31—44no—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D414yes—
160C22×C4×C31C2×D414yes—
161C22×C4×C31C2411yescharacteristic
162Q16×C31—44no—
163SD16×C31—48no—
164D8×C31—44no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—21no—
170C2×C8×C31—22no—
171C2×C8×C31—21no—
172C4⋊C4×C31—28no—
173C4⋊C4×C31—28no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31C2×D414yes—
177C4⋊C4×C31C2×D414yes—
178C22⋊C4×C31—28no—
179C22⋊C4×C31—28no—
180C22⋊C4×C31—28no—
181C22⋊C4×C31—28no—
182C22⋊C4×C31—28no—
183C42×C31—28no—
184#23923C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—21no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—22no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—24no—
196C23×C31—44no—
197C23×C31—48no—
198C23×C31—44no—
199C23×C31—48no—
200C23×C31—44no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31#12014yes—
208C23×C31#13512yes—
209C23×C31C22×D411yescharacteristic
210Q8×C31—24no—
211Q8×C31—24no—
212Q8×C31—42no—
213Q8×C31—42no—
214D4×C31—24no—
215D4×C31—24no—
216D4×C31—416no—
217D4×C31—416no—
218D4×C31—416no—
219D4×C31—42no—
220D4×C31—42no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—22no—
230C2×C4×C31—24no—
231C2×C4×C31—21no—
232C2×C4×C31—44no—
233C2×C4×C31—48no—
234C2×C4×C31—416no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—41no—
242C2×C4×C31—42no—
243C2×C4×C31—41no—
244C2×C4×C31#12014yes—
245C2×C4×C31C22×D412yes—
246C2×C4×C31C22×D412yes—
247C2×C4×C31H(5,𝔽2)11yescharacteristic
248C2×C4×C31H(5,𝔽2)11yescharacteristic
249C8×C31—41no—
250C8×C31—42no—
251C8×C31—41no—
252C2×#525C2×C3112yescomplemented
253C2×#452C2×C3112yescomplemented
254C2×#449C2×C3112yescomplemented
255#2001C2×C3111yescharacteristic, complemented
256#2000C2×C3112yescomplemented
257#1999C2×C3111yescharacteristic, complemented
258#1875C2×C3112yescomplemented
259#1873C2×C3112yescomplemented
260#1779C2×C3111yescharacteristic, complemented
261C22×C31—24no—
262C22×C31—22no—
263C22×C31—24no—
264C22×C31—24no—
265C22×C31—24no—
266C22×C31—416no—
267C22×C31—41no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—41no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—44no—
275C22×C31—44no—
276C22×C31—44no—
277C22×C31—82no—
278C22×C31—82no—
279C22×C31—82no—
280C22×C31—82no—
281C22×C31C2×#12012yes—
282C22×C31#53411yescharacteristic
283C22×C31C2×#13511yescharacteristic
284C22×C31#57511yescharacteristic
285C22×C31#57512yes—
286C4×C31—24no—
287C4×C31—41no—
288C4×C31—44no—
289C4×C31—44no—
290C4×C31—44no—
291C4×C31D4211yescharacteristic
292C4×C31D4212yes—
293C4×C31D4211yescharacteristic
294C22×#41C22×C3112yes—
295C24×C4C22×C3111yescharacteristic
296C22×Q16C22×C3111yescharacteristic, complemented
297C22×SD16C22×C3112yescomplemented
298C22×D8C22×C3111yescharacteristic, complemented
299#525—28no—
300#525—28no—
301C2×#123—22no—
302C2×C22⋊Q8C22×C3112yes—
303C2×C22⋊Q8C22×C3112yes—
304C2×#121C22×C3112yes—
305C2×#121C22×C3112yes—
306C2×#120—22no—
307C2×C4×D4—22no—
308C2×C4×D4—22no—
309C2×#117C22×C3112yes—
310#452—28no—
311#449—28no—
312C2×#103C22×C3112yescomplemented
313C2×#103C22×C3112yescomplemented
314C2×#102C22×C3112yescomplemented
315C2×#102C22×C3112yescomplemented
316C2×C22⋊C8C22×C3112yes—
317C2×C22⋊C8C22×C3112yes—
318C2×C22⋊C8C22×C3111yescharacteristic
319C2×C22⋊C8C22×C3111yescharacteristic
320#340C22×C3111yescharacteristic, complemented
321#340C22×C3112yescomplemented
322#340C22×C3111yescharacteristic, complemented
323C2×C31—44no—
324C2×C31—42no—
325C2×C31—84no—
326C2×C31#240311yescharacteristic
327C2×C31#300212yes—
328C2×C31#300512yes—
329C2×C31#319111yescharacteristic
330C2×C31#319711yescharacteristic
331C25D4×C3111yescharacteristic
332C2×#41—24no—
333C2×#41—24no—
334C2×#41—24no—
335C2×#41—24no—
336C2×#41—42no—
337C2×#41—42no—
338C22×Q8C23×C3112yes—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4C23×C3112yes—
343C23×C4—22no—
344C23×C4—24no—
345C23×C4—24no—
346C23×C4—41no—
347C23×C4—41no—
348C23×C4D4×C3111yescharacteristic
349C23×C4D4×C3112yes—
350C23×C4C23×C3111yescharacteristic
351C23×C4C23×C3112yes—
352C2×Q16—24no—
353C2×Q16—41no—
354C2×SD16—24no—
355C2×SD16—24no—
356C2×SD16—42no—
357C2×D8—24no—
358C2×D8—41no—
359C22×C8C23×C3111yescharacteristic
360C22×C8C23×C3112yes—
361C22×C8C23×C3111yescharacteristic
362#123—216no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—24no—
366C22⋊Q8D4×C3118yescomplemented
367#121—24no—
368#121—24no—
369#121—24no—
370#121D4×C3118yescomplemented
371#120—216no—
372C4×D4—216no—
373C4×D4—216no—
374#117—24no—
375#117—24no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4C23×C3112yes—
379C2×C4⋊C4C23×C3112yes—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C42—22no—
386#103—24no—
387#103—24no—
388#102—24no—
389#102—24no—
390C22⋊C8—24no—
391C22⋊C8—24no—
392C22⋊C8—41no—
393C22⋊C8—41no—
394C31#2392311yescharacteristic, direct product factor, Sylow
395C24—24no—
396C24—22no—
397C24—22no—
398C24—41no—
399C24—42no—
400C24—41no—
401C24C2×D4×C3112yes—
402C24C2×D4×C3111yescharacteristic
403#41—48no—
404#41—48no—
405#41—48no—
406#41—48no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8C2×D4×C3114yes—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4C2×D4×C3114yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4C2×D4×C3111yescharacteristic
451C22×C4C2×D4×C3112yes—
452C22×C4C2×D4×C3114yes—
453C22×C4C2×D4×C3114yes—
454C22×C4C24×C3111yescharacteristic
455Q16—44no—
456SD16—48no—
457D8—44no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—21no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4C2×D4×C3114yes—
470C4⋊C4C2×D4×C3114yes—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C42—28no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—21no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—21no—
486C23—22no—
487C23—24no—
488C23—44no—
489C23—48no—
490C23—44no—
491C23—48no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C31×#12014yes—
500C23C31×#13512yes—
501C23C22×D4×C3111yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506D4—24no—
507D4—24no—
508D4—416no—
509D4—416no—
510D4—416no—
511D4—42no—
512D4—42no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—416no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4C31×#12014yes—
537C2×C4C22×D4×C3112yes—
538C2×C4C22×D4×C3112yes—
539C2×C4C31×H(5,𝔽2)11yescharacteristic
540C2×C4C31×H(5,𝔽2)11yescharacteristic
541C8—41no—
542C8—42no—
543C8—41no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—416no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22C2×C31×#12012yes—
565C22C31×#53411yescharacteristic
566C22C2×C31×#13511yescharacteristic
567C22C31×#57511yescharacteristic
568C22C31×#57512yes—
569C4—24no—
570C4—41no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4D42×C3111yescharacteristic
575C4D42×C3112yes—
576C4D42×C3111yescharacteristic
577C2—44no—
578C2—42no—
579C2—84no—
580C2C31×#240311yescharacteristic
581C2C31×#300212yes—
582C2C31×#300512yes—
583C2C31×#319111yescharacteristic
584C2C31×#319711yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
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. C2×C31×#525C2semidirect
2. C2×C31×#452C2semidirect
3. C2×C31×#449C2semidirect
4. C31×#2001C2semidirect
5. C31×#2000C2semidirect
6. C31×#1999C2semidirect
7. C31×#1875C2semidirect
8. C31×#1873C2semidirect
9. C31×#1779C2semidirect
12. C22×Q16×C31C22semidirect
13. C22×SD16×C31C22semidirect
14. C22×D8×C31C22semidirect
28. C2×C31×#103C22semidirect
29. C2×C31×#103C22semidirect
30. C2×C31×#102C22semidirect
31. C2×C31×#102C22semidirect
36. C31×#340C22semidirect
37. C31×#340C22semidirect
38. C31×#340C22semidirect
74. C31×C22⋊Q8D4semidirect
78. C31×#121D4semidirect
184. #23923C31direct (N × H)
252. C2×#525C2×C31semidirect
253. C2×#452C2×C31semidirect
254. C2×#449C2×C31semidirect
255. #2001C2×C31semidirect
256. #2000C2×C31semidirect
257. #1999C2×C31semidirect
258. #1875C2×C31semidirect
259. #1873C2×C31semidirect
260. #1779C2×C31semidirect
296. C22×Q16C22×C31semidirect
297. C22×SD16C22×C31semidirect
298. C22×D8C22×C31semidirect
312. C2×#103C22×C31semidirect
313. C2×#103C22×C31semidirect
314. C2×#102C22×C31semidirect
315. C2×#102C22×C31semidirect
320. #340C22×C31semidirect
321. #340C22×C31semidirect
322. #340C22×C31semidirect
366. C22⋊Q8D4×C31semidirect
370. #121D4×C31semidirect
394. C31#23923direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5252
C2×C31×#4522
C2×C31×#4492
C31×#20012
C31×#20002
C31×#19992
C31×#18752
C31×#18732
C31×#17792
#2392331