Finite Group Databasebeta

← C31×#14715 #21471474

Subgroups — C31×#14715 #21471474

Order 7936.

7490 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×D4×C31C211yesmaximal, characteristic, complemented
2C22×C31×#120C212yesmaximal, complemented
3C2×C31×#392C214yesmaximal, complemented
4C2×C31×#386C211yesmaximal, characteristic, complemented
5C2×C31×#379C212yesmaximal, complemented
6C31×#1438C214yesmaximal, complemented
7C31×C22⋊#95C211yesmaximal, characteristic, complemented
8C26×C31C2212yes—
9C23×D4×C31—28no—
10C23×D4×C31—216no—
11C23×D4×C31—24no—
12C23×D4×C31C2214yes—
13C23×D4×C31C2211yescharacteristic
14C23×D4×C31C2214yescomplemented
15C24×C4×C31C2211yescharacteristic
16C2×C31×#120—28no—
17C2×C31×#120—28no—
18C2×C31×#120—28no—
19C22×C22⋊C4×C31C2211yescharacteristic
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2212yescomplemented
22C22×C22⋊C4×C31C2214yescomplemented
23C22×C22⋊C4×C31C2218yescomplemented
24C31×#392—216no—
25C31×#386—24no—
26C31×#379—28no—
27C2×C31×#95C2214yescomplemented
28C2×C31×#95C2212yescomplemented
29C25×C31—216no—
30C25×C31—24no—
31C25×C31—24no—
32C25×C31—24no—
33C25×C31—24no—
34C25×C31—24no—
35C25×C31—24no—
36C25×C31—48no—
37C25×C31D418yescomplemented
38C25×C31C2311yescharacteristic
39C25×C31C2314yes—
40C25×C31C2311yescharacteristic
41C22×D4×C31—2128no—
42C22×D4×C31—232no—
43C22×D4×C31—264no—
44C22×D4×C31—216no—
45C22×D4×C31—216no—
46C22×D4×C31—24no—
47C22×D4×C31—24no—
48C22×D4×C31—28no—
49C22×D4×C31—22no—
50C22×D4×C31—28no—
51C22×D4×C31—22no—
52C22×D4×C31—216no—
53C22×D4×C31—28no—
54C22×D4×C31—28no—
55C22×D4×C31—28no—
56C22×D4×C31—48no—
57C22×D4×C31D4116yescomplemented
58C22×D4×C31D414yescomplemented
59C23×C4×C31—24no—
60C23×C4×C31—28no—
61C23×C4×C31D414yescomplemented
62C23×C4×C31C2312yes—
63C23×C4×C31C2314yes—
64C23×C4×C31C2311yescharacteristic
65C23×C4×C31C2312yes—
66C31×#120—264no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—28no—
70C2×C22⋊C4×C31—28no—
71C2×C22⋊C4×C31—28no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—216no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31—28no—
77C2×C22⋊C4×C31—216no—
78C2×C22⋊C4×C31—48no—
79C2×C22⋊C4×C31D418yescomplemented
80C31×#95—24no—
81C31×#95—28no—
82C24×C31—24no—
83C24×C31—24no—
84C24×C31—232no—
85C24×C31—232no—
86C24×C31—216no—
87C24×C31—232no—
88C24×C31—24no—
89C24×C31—24no—
90C24×C31—24no—
91C24×C31—24no—
92C24×C31—24no—
93C24×C31—28no—
94C24×C31—24no—
95C24×C31—28no—
96C24×C31—28no—
97C24×C31—216no—
98C24×C31—216no—
99C24×C31—21no—
100C24×C31—24no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—21no—
104C24×C31—24no—
105C24×C31—416no—
106C24×C31—432no—
107C24×C31—432no—
108C24×C31—432no—
109C24×C31—432no—
110C24×C31—416no—
111C24×C31—416no—
112C24×C31—48no—
113C24×C31—48no—
114C24×C31—48no—
115C24×C31—48no—
116C24×C31C2×D418yes—
117C24×C31C2×D4116yes—
118C24×C31C2×D412yes—
119C24×C31C2×D414yes—
120C24×C31C2×D414yes—
121C24×C31C2×D412yes—
122C24×C31C2×D412yescomplemented
123C24×C31#4112yes—
124C24×C31C2411yescharacteristic
125C2×D4×C31—2256no—
126C2×D4×C31—232no—
127C2×D4×C31—264no—
128C2×D4×C31—232no—
129C2×D4×C31—264no—
130C2×D4×C31—232no—
131C2×D4×C31—216no—
132C2×D4×C31—28no—
133C2×D4×C31—232no—
134C2×D4×C31—432no—
135C2×D4×C31—432no—
136C2×D4×C31—416no—
137C2×D4×C31—44no—
138C2×D4×C31—416no—
139C2×D4×C31—48no—
140C2×D4×C31—44no—
141C22×C4×C31—232no—
142C22×C4×C31—28no—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—216no—
146C22×C4×C31—21no—
147C22×C4×C31—28no—
148C22×C4×C31—22no—
149C22×C4×C31—24no—
150C22×C4×C31—22no—
151C22×C4×C31—24no—
152C22×C4×C31—21no—
153C22×C4×C31—28no—
154C22×C4×C31—22no—
155C22×C4×C31—24no—
156C22×C4×C31—24no—
157C22×C4×C31—24no—
158C22×C4×C31C2×D418yes—
159C22×C4×C31C2×D414yes—
160C22×C4×C31C2×D412yescomplemented
161C22×C4×C31C2×D418yescomplemented
162C22⋊C4×C31—232no—
163C22⋊C4×C31—216no—
164C22⋊C4×C31—432no—
165C22⋊C4×C31—44no—
166C22⋊C4×C31—48no—
167#14715C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—28no—
169C23×C31—24no—
170C23×C31—28no—
171C23×C31—24no—
172C23×C31—28no—
173C23×C31—216no—
174C23×C31—28no—
175C23×C31—264no—
176C23×C31—28no—
177C23×C31—28no—
178C23×C31—28no—
179C23×C31—216no—
180C23×C31—216no—
181C23×C31—28no—
182C23×C31—28no—
183C23×C31—28no—
184C23×C31—216no—
185C23×C31—216no—
186C23×C31—216no—
187C23×C31—216no—
188C23×C31—24no—
189C23×C31—22no—
190C23×C31—28no—
191C23×C31—464no—
192C23×C31—464no—
193C23×C31—432no—
194C23×C31—464no—
195C23×C31—464no—
196C23×C31—464no—
197C23×C31—464no—
198C23×C31—44no—
199C23×C31—44no—
200C23×C31—44no—
201C23×C31—44no—
202C23×C31—48no—
203C23×C31—48no—
204C23×C31—48no—
205C23×C31—48no—
206C23×C31—48no—
207C23×C31—48no—
208C23×C31—432no—
209C23×C31—42no—
210C23×C31—41no—
211C23×C31—44no—
212C23×C31—42no—
213C23×C31—41no—
214C23×C31—44no—
215C23×C31—42no—
216C23×C31—48no—
217C23×C31—44no—
218C23×C31#12018yes—
219C23×C31#12014yes—
220C23×C31#120116yescomplemented
221C23×C31#12114yes—
222C23×C31#12118yes—
223C23×C31#12714yes—
224C23×C31C22×D412yes—
225C23×C31C22×D411yescharacteristic
226C23×C31C22×D414yes—
227C23×C31C22×D411yescharacteristic
228C23×C31C22×D412yes—
229C23×C31C2×#4111yescharacteristic
230C23×C31H(5,𝔽2)14yes—
231D4×C31—2128no—
232D4×C31—464no—
233D4×C31—416no—
234C2×C4×C31—232no—
235C2×C4×C31—24no—
236C2×C4×C31—216no—
237C2×C4×C31—24no—
238C2×C4×C31—24no—
239C2×C4×C31—28no—
240C2×C4×C31—22no—
241C2×C4×C31—216no—
242C2×C4×C31—41no—
243C2×C4×C31—44no—
244C2×C4×C31—42no—
245C2×C4×C31—41no—
246C2×C4×C31—44no—
247C2×C4×C31—48no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—48no—
252C2×C4×C31#12018yescomplemented
253C24×D4C2×C3111yescharacteristic, complemented
254C22×#120C2×C3112yescomplemented
255C2×#392C2×C3114yescomplemented
256C2×#386C2×C3111yescharacteristic, complemented
257C2×#379C2×C3112yescomplemented
258#1438C2×C3114yescomplemented
259C22⋊#95C2×C3111yescharacteristic, complemented
260C22×C31—24no—
261C22×C31—24no—
262C22×C31—216no—
263C22×C31—28no—
264C22×C31—216no—
265C22×C31—28no—
266C22×C31—216no—
267C22×C31—216no—
268C22×C31—216no—
269C22×C31—4128no—
270C22×C31—48no—
271C22×C31—48no—
272C22×C31—48no—
273C22×C31—416no—
274C22×C31—416no—
275C22×C31—48no—
276C22×C31—416no—
277C22×C31—416no—
278C22×C31—42no—
279C22×C31—48no—
280C22×C31—44no—
281C22×C31—81no—
282C22×C31—81no—
283C22×C31—84no—
284C22×C31—82no—
285C22×C31#39218yescomplemented
286C22×C31C2×#12011yescharacteristic
287C22×C31C2×#12014yes—
288C22×C31C2×#12012yes—
289C22×C31C2×#12012yes—
290C22×C31C2×#12111yescharacteristic
291C22×C31C2×#12112yes—
292C22×C31C2×#12711yescharacteristic
293C22×C31#53414yes—
294C22×C31#53512yes—
295C22×C31D4214yes—
296C22×C31D4218yes—
297C22×C31#54614yes—
298C4×C31—28no—
299C4×C31—42no—
300C4×C31—48no—
301C4×C31—82no—
302C26C22×C3112yes—
303C23×D4—28no—
304C23×D4—216no—
305C23×D4—24no—
306C23×D4C22×C3114yes—
307C23×D4C22×C3111yescharacteristic
308C23×D4C22×C3114yescomplemented
309C24×C4C22×C3111yescharacteristic
310C2×#120—28no—
311C2×#120—28no—
312C2×#120—28no—
313C22×C22⋊C4C22×C3111yescharacteristic
314C22×C22⋊C4C22×C3112yes—
315C22×C22⋊C4C22×C3112yescomplemented
316C22×C22⋊C4C22×C3114yescomplemented
317C22×C22⋊C4C22×C3118yescomplemented
318#392—216no—
319#386—24no—
320#379—28no—
321C2×#95C22×C3114yescomplemented
322C2×#95C22×C3112yescomplemented
323C2×C31—28no—
324C2×C31—416no—
325C2×C31—82no—
326C2×C31C2×#39211yescharacteristic
327C2×C31#240314yes—
328C2×C31#240812yes—
329C2×C31#241014yes—
330C2×C31#243312yes—
331C2×C31#261311yescharacteristic
332C2×C31#284611yescharacteristic
333C25—216no—
334C25—24no—
335C25—24no—
336C25—24no—
337C25—24no—
338C25—24no—
339C25—24no—
340C25—48no—
341C25D4×C3118yescomplemented
342C25C23×C3111yescharacteristic
343C25C23×C3114yes—
344C25C23×C3111yescharacteristic
345C22×D4—2128no—
346C22×D4—232no—
347C22×D4—264no—
348C22×D4—216no—
349C22×D4—216no—
350C22×D4—24no—
351C22×D4—24no—
352C22×D4—28no—
353C22×D4—22no—
354C22×D4—28no—
355C22×D4—22no—
356C22×D4—216no—
357C22×D4—28no—
358C22×D4—28no—
359C22×D4—28no—
360C22×D4—48no—
361C22×D4D4×C31116yescomplemented
362C22×D4D4×C3114yescomplemented
363C23×C4—24no—
364C23×C4—28no—
365C23×C4D4×C3114yescomplemented
366C23×C4C23×C3112yes—
367C23×C4C23×C3114yes—
368C23×C4C23×C3111yescharacteristic
369C23×C4C23×C3112yes—
370#120—264no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—28no—
374C2×C22⋊C4—28no—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—216no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—28no—
381C2×C22⋊C4—216no—
382C2×C22⋊C4—48no—
383C2×C22⋊C4D4×C3118yescomplemented
384#95—24no—
385#95—28no—
386C31#1471511yescharacteristic, direct product factor, Sylow
387C24—24no—
388C24—24no—
389C24—232no—
390C24—232no—
391C24—216no—
392C24—232no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—28no—
399C24—24no—
400C24—28no—
401C24—28no—
402C24—216no—
403C24—216no—
404C24—21no—
405C24—24no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24—24no—
410C24—416no—
411C24—432no—
412C24—432no—
413C24—432no—
414C24—432no—
415C24—416no—
416C24—416no—
417C24—48no—
418C24—48no—
419C24—48no—
420C24—48no—
421C24C2×D4×C3118yes—
422C24C2×D4×C31116yes—
423C24C2×D4×C3112yes—
424C24C2×D4×C3114yes—
425C24C2×D4×C3114yes—
426C24C2×D4×C3112yes—
427C24C2×D4×C3112yescomplemented
428C24#41×C3112yes—
429C24C24×C3111yescharacteristic
430C2×D4—2256no—
431C2×D4—232no—
432C2×D4—264no—
433C2×D4—232no—
434C2×D4—264no—
435C2×D4—232no—
436C2×D4—216no—
437C2×D4—28no—
438C2×D4—232no—
439C2×D4—432no—
440C2×D4—432no—
441C2×D4—416no—
442C2×D4—44no—
443C2×D4—416no—
444C2×D4—48no—
445C2×D4—44no—
446C22×C4—232no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—216no—
451C22×C4—21no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4C2×D4×C3118yes—
464C22×C4C2×D4×C3114yes—
465C22×C4C2×D4×C3112yescomplemented
466C22×C4C2×D4×C3118yescomplemented
467C22⋊C4—232no—
468C22⋊C4—216no—
469C22⋊C4—432no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C23—28no—
473C23—24no—
474C23—28no—
475C23—24no—
476C23—28no—
477C23—216no—
478C23—28no—
479C23—264no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—216no—
484C23—216no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—216no—
489C23—216no—
490C23—216no—
491C23—216no—
492C23—24no—
493C23—22no—
494C23—28no—
495C23—464no—
496C23—464no—
497C23—432no—
498C23—464no—
499C23—464no—
500C23—464no—
501C23—464no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—432no—
513C23—42no—
514C23—41no—
515C23—44no—
516C23—42no—
517C23—41no—
518C23—44no—
519C23—42no—
520C23—48no—
521C23—44no—
522C23C31×#12018yes—
523C23C31×#12014yes—
524C23C31×#120116yescomplemented
525C23C31×#12114yes—
526C23C31×#12118yes—
527C23C31×#12714yes—
528C23C22×D4×C3112yes—
529C23C22×D4×C3111yescharacteristic
530C23C22×D4×C3114yes—
531C23C22×D4×C3111yescharacteristic
532C23C22×D4×C3112yes—
533C23C2×#41×C3111yescharacteristic
534C23C31×H(5,𝔽2)14yes—
535D4—2128no—
536D4—464no—
537D4—416no—
538C2×C4—232no—
539C2×C4—24no—
540C2×C4—216no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—216no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—48no—
556C2×C4C31×#12018yescomplemented
557C22—24no—
558C22—24no—
559C22—216no—
560C22—28no—
561C22—216no—
562C22—28no—
563C22—216no—
564C22—216no—
565C22—216no—
566C22—4128no—
567C22—48no—
568C22—48no—
569C22—48no—
570C22—416no—
571C22—416no—
572C22—48no—
573C22—416no—
574C22—416no—
575C22—42no—
576C22—48no—
577C22—44no—
578C22—81no—
579C22—81no—
580C22—84no—
581C22—82no—
582C22C31×#39218yescomplemented
583C22C2×C31×#12011yescharacteristic
584C22C2×C31×#12014yes—
585C22C2×C31×#12012yes—
586C22C2×C31×#12012yes—
587C22C2×C31×#12111yescharacteristic
588C22C2×C31×#12112yes—
589C22C2×C31×#12711yescharacteristic
590C22C31×#53414yes—
591C22C31×#53512yes—
592C22D42×C3114yes—
593C22D42×C3118yes—
594C22C31×#54614yes—
595C4—28no—
596C4—42no—
597C4—48no—
598C4—82no—
599C2—28no—
600C2—416no—
601C2—82no—
602C2C2×C31×#39211yescharacteristic
603C2C31×#240314yes—
604C2C31×#240812yes—
605C2C31×#241014yes—
606C2C31×#243312yes—
607C2C31×#261311yescharacteristic
608C2C31×#284611yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C24×D4×C31C2semidirect
2. C22×C31×#120C2semidirect
3. C2×C31×#392C2semidirect
4. C2×C31×#386C2semidirect
5. C2×C31×#379C2semidirect
6. C31×#1438C2semidirect
7. C31×C22⋊#95C2semidirect
14. C23×D4×C31C22semidirect
21. C22×C22⋊C4×C31C22semidirect
22. C22×C22⋊C4×C31C22semidirect
23. C22×C22⋊C4×C31C22semidirect
27. C2×C31×#95C22semidirect
28. C2×C31×#95C22semidirect
37. C25×C31D4semidirect
57. C22×D4×C31D4semidirect
58. C22×D4×C31D4semidirect
61. C23×C4×C31D4semidirect
79. C2×C22⋊C4×C31D4semidirect
122. C24×C31C2×D4semidirect
160. C22×C4×C31C2×D4semidirect
161. C22×C4×C31C2×D4semidirect
167. #14715C31direct (N × H)
220. C23×C31#120semidirect
252. C2×C4×C31#120semidirect
253. C24×D4C2×C31semidirect
254. C22×#120C2×C31semidirect
255. C2×#392C2×C31semidirect
256. C2×#386C2×C31semidirect
257. C2×#379C2×C31semidirect
258. #1438C2×C31semidirect
259. C22⋊#95C2×C31semidirect
285. C22×C31#392semidirect
308. C23×D4C22×C31semidirect
315. C22×C22⋊C4C22×C31semidirect
316. C22×C22⋊C4C22×C31semidirect
317. C22×C22⋊C4C22×C31semidirect
321. C2×#95C22×C31semidirect
322. C2×#95C22×C31semidirect
341. C25D4×C31semidirect
361. C22×D4D4×C31semidirect
362. C22×D4D4×C31semidirect
365. C23×C4D4×C31semidirect
383. C2×C22⋊C4D4×C31semidirect
386. C31#14715direct (N × H)
427. C24C2×D4×C31semidirect
465. C22×C4C2×D4×C31semidirect
466. C22×C4C2×D4×C31semidirect
524. C23C31×#120semidirect
556. C2×C4C31×#120semidirect
582. C22C31×#392semidirect

Maximal subgroups

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

subgroupindex
C24×D4×C312
C22×C31×#1202
C2×C31×#3922
C2×C31×#3862
C2×C31×#3792
C31×#14382
C31×C22⋊#952
#1471531