Finite Group Databasebeta

← C31×#14777 #21471536

Subgroups — C31×#14777 #21471536

Order 7936.

2506 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C31C211yesmaximal, characteristic, complemented
2C2×C31×#399C212yesmaximal, complemented
3C2×C31×#386C211yesmaximal, characteristic
4C2×C31×#386C212yesmaximal
5C2×C31×#385C212yesmaximal, complemented
6C31×#1445C212yesmaximal, complemented
7C31×#1444C212yesmaximal, complemented
8C31×#1439C212yesmaximal, complemented
9C31×C22⋊#95C211yesmaximal, characteristic, complemented
10C23×D4×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic
12C24×C4×C31C2212yes—
13C22×C4⋊C4×C31—24no—
14C22×C4⋊C4×C31—28no—
15C22×C4⋊C4×C31C2212yes—
16C22×C4⋊C4×C31C2214yes—
17C22×C4⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2211yescharacteristic
19C22×C22⋊C4×C31C2211yescharacteristic
20C22×C22⋊C4×C31C2211yescharacteristic
21C22×C22⋊C4×C31C2214yes—
22C22×C22⋊C4×C31C2212yes—
23C31×#399—28no—
24C31×#386—24no—
25C31×#386—28no—
26C31×#385—28no—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2214yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2214yes—
32C2×C31×#95C2211yescharacteristic
33C25×C31C2311yescharacteristic
34C25×C31C2311yescharacteristic
35C22×D4×C31—24no—
36C22×D4×C31—22no—
37C22×D4×C31—22no—
38C22×D4×C31—24no—
39C22×D4×C31Q814yes—
40C23×C4×C31—24no—
41C23×C4×C31—24no—
42C23×C4×C31—24no—
43C23×C4×C31—24no—
44C23×C4×C31—28no—
45C23×C4×C31—24no—
46C23×C4×C31—24no—
47C23×C4×C31—44no—
48C23×C4×C31D414yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2314yes—
52C23×C4×C31C2314yes—
53C23×C4×C31C2311yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C23×C4×C31C2311yescharacteristic
56C2×C4⋊C4×C31—232no—
57C2×C4⋊C4×C31—28no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—24no—
60C2×C4⋊C4×C31—24no—
61C2×C4⋊C4×C31—28no—
62C2×C4⋊C4×C31—28no—
63C2×C4⋊C4×C31—44no—
64C2×C4⋊C4×C31D418yescomplemented
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—28no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—28no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—44no—
76C2×C22⋊C4×C31D418yes—
77C31×#95—24no—
78C31×#95—28no—
79C31×#95—28no—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95—24no—
83C24×C31—22no—
84C24×C31—22no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—21no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—21no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31C2×D414yes—
96C24×C31C2×D412yes—
97C24×C31C2×Q812yes—
98C24×C31C2×Q812yes—
99C24×C31#4112yes—
100C24×C31#4114yes—
101C24×C31C2411yescharacteristic
102C2×D4×C31—28no—
103C2×D4×C31—216no—
104C2×D4×C31—48no—
105C2×D4×C31—44no—
106C2×D4×C31—44no—
107C22×C4×C31—232no—
108C22×C4×C31—28no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—28no—
114C22×C4×C31—28no—
115C22×C4×C31—216no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—24no—
119C22×C4×C31—24no—
120C22×C4×C31—24no—
121C22×C4×C31—24no—
122C22×C4×C31—24no—
123C22×C4×C31—24no—
124C22×C4×C31—24no—
125C22×C4×C31—24no—
126C22×C4×C31—22no—
127C22×C4×C31—21no—
128C22×C4×C31—24no—
129C22×C4×C31—24no—
130C22×C4×C31—24no—
131C22×C4×C31—24no—
132C22×C4×C31—22no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—416no—
141C22×C4×C31—416no—
142C22×C4×C31—48no—
143C22×C4×C31—48no—
144C22×C4×C31C2×D418yes—
145C22×C4×C31C2×D418yes—
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D414yes—
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×Q812yes—
150C22×C4×C31#4114yes—
151C4⋊C4×C31—216no—
152C4⋊C4×C31—216no—
153C4⋊C4×C31—416no—
154C22⋊C4×C31—216no—
155C22⋊C4×C31—416no—
156C22⋊C4×C31—44no—
157C22⋊C4×C31—44no—
158C22⋊C4×C31—44no—
159#14777C3111yesmaximal, characteristic, direct product factor, Sylow
160C23×C31—22no—
161C23×C31—22no—
162C23×C31—28no—
163C23×C31—22no—
164C23×C31—22no—
165C23×C31—28no—
166C23×C31—28no—
167C23×C31—28no—
168C23×C31—24no—
169C23×C31—24no—
170C23×C31—24no—
171C23×C31—24no—
172C23×C31—22no—
173C23×C31—24no—
174C23×C31—24no—
175C23×C31—24no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—42no—
180C23×C31—44no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—44no—
185C23×C31—44no—
186C23×C31—44no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31C22⋊Q814yes—
190C23×C31C22⋊Q818yes—
191C23×C31#12318yes—
192C23×C31#12314yes—
193C23×C31#12714yes—
194C23×C31C22×D411yescharacteristic
195C23×C31C22×D412yes—
196C23×C31C22×D412yes—
197C23×C31C22×Q811yescharacteristic
198C23×C31C2×#4112yes—
199C23×C31C2×#4111yescharacteristic
200C23×C31C2×#4112yes—
201C23×C31H(5,𝔽2)12yes—
202C23×C312−1+412yes—
203D4×C31—416no—
204C2×C4×C31—232no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—28no—
210C2×C4×C31—28no—
211C2×C4×C31—28no—
212C2×C4×C31—22no—
213C2×C4×C31—24no—
214C2×C4×C31—28no—
215C2×C4×C31—28no—
216C2×C4×C31—28no—
217C2×C4×C31—28no—
218C2×C4×C31—28no—
219C2×C4×C31—432no—
220C2×C4×C31—41no—
221C2×C4×C31—42no—
222C2×C4×C31—44no—
223C2×C4×C31—44no—
224C2×C4×C31—41no—
225C2×C4×C31—44no—
226C2×C4×C31—44no—
227C2×C4×C31—44no—
228C2×C4×C31—44no—
229C2×C4×C31—44no—
230C2×C4×C31—44no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—44no—
238C2×C4×C31—44no—
239C2×C4×C31#12014yes—
240C2×C4×C31#12314yes—
241C23×C4⋊C4C2×C3111yescharacteristic, complemented
242C2×#399C2×C3112yescomplemented
243C2×#386C2×C3111yescharacteristic
244C2×#386C2×C3112yes—
245C2×#385C2×C3112yescomplemented
246#1445C2×C3112yescomplemented
247#1444C2×C3112yescomplemented
248#1439C2×C3112yescomplemented
249C22⋊#95C2×C3111yescharacteristic, complemented
250C22×C31—24no—
251C22×C31—24no—
252C22×C31—24no—
253C22×C31—24no—
254C22×C31—216no—
255C22×C31—216no—
256C22×C31—24no—
257C22×C31—24no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—48no—
261C22×C31—42no—
262C22×C31—81no—
263C22×C31—81no—
264C22×C31—81no—
265C22×C31—84no—
266C22×C31—81no—
267C22×C31#39918yescomplemented
268C22×C31C2×#12011yescharacteristic
269C22×C31C2×C22⋊Q811yescharacteristic
270C22×C31C2×C22⋊Q812yes—
271C22×C31C2×#12311yescharacteristic
272C22×C31C2×#12311yescharacteristic
273C22×C31C2×#12312yes—
274C22×C31#52514yes—
275C22×C31C2×#12711yescharacteristic
276C22×C31#53511yescharacteristic
277C22×C31#53611yescharacteristic
278C22×C31#53914yes—
279C22×C31D4214yes—
280C22×C31#54714yes—
281C22×C31#54812yes—
282C22×C31D4×Q812yes—
283C22×C31#55312yes—
284C22×C31#55412yes—
285C4×C31—24no—
286C4×C31—24no—
287C4×C31—42no—
288C4×C31—48no—
289C4×C31—48no—
290C4×C31—81no—
291C4×C31—81no—
292C23×D4C22×C3111yescharacteristic
293C24×C4C22×C3111yescharacteristic
294C24×C4C22×C3112yes—
295C22×C4⋊C4—24no—
296C22×C4⋊C4—28no—
297C22×C4⋊C4C22×C3112yes—
298C22×C4⋊C4C22×C3114yes—
299C22×C4⋊C4C22×C3112yes—
300C22×C22⋊C4C22×C3111yescharacteristic
301C22×C22⋊C4C22×C3111yescharacteristic
302C22×C22⋊C4C22×C3111yescharacteristic
303C22×C22⋊C4C22×C3114yes—
304C22×C22⋊C4C22×C3112yes—
305#399—28no—
306#386—24no—
307#386—28no—
308#385—28no—
309C2×#95C22×C3112yes—
310C2×#95C22×C3112yes—
311C2×#95C22×C3114yes—
312C2×#95C22×C3111yescharacteristic
313C2×#95C22×C3114yes—
314C2×#95C22×C3111yescharacteristic
315C2×C31—28no—
316C2×C31—82no—
317C2×C31C2×#39911yescharacteristic
318C2×C31#240911yescharacteristic
319C2×C31#243311yescharacteristic
320C2×C31#244314yes—
321C2×C31#244914yes—
322C2×C31#248711yescharacteristic
323C2×C31#250111yescharacteristic
324C2×C31#262311yescharacteristic
325C2×C31#284911yescharacteristic
326C25C23×C3111yescharacteristic
327C25C23×C3111yescharacteristic
328C22×D4—24no—
329C22×D4—22no—
330C22×D4—22no—
331C22×D4—24no—
332C22×D4Q8×C3114yes—
333C23×C4—24no—
334C23×C4—24no—
335C23×C4—24no—
336C23×C4—24no—
337C23×C4—28no—
338C23×C4—24no—
339C23×C4—24no—
340C23×C4—44no—
341C23×C4D4×C3114yes—
342C23×C4C23×C3111yescharacteristic
343C23×C4C23×C3111yescharacteristic
344C23×C4C23×C3114yes—
345C23×C4C23×C3114yes—
346C23×C4C23×C3111yescharacteristic
347C23×C4C23×C3111yescharacteristic
348C23×C4C23×C3111yescharacteristic
349C2×C4⋊C4—232no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—28no—
355C2×C4⋊C4—28no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4D4×C3118yescomplemented
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4D4×C3118yes—
370#95—28no—
371#95—28no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376C31#1477711yescharacteristic, direct product factor, Sylow
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—21no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—22no—
389C24C2×D4×C3114yes—
390C24C2×D4×C3112yes—
391C24C2×Q8×C3112yes—
392C24C2×Q8×C3112yes—
393C24#41×C3112yes—
394C24#41×C3114yes—
395C24C24×C3111yescharacteristic
396C2×D4—28no—
397C2×D4—216no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—44no—
401C22×C4—232no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4C2×D4×C3118yes—
439C22×C4C2×D4×C3118yes—
440C22×C4C2×D4×C3112yes—
441C22×C4C2×D4×C3114yes—
442C22×C4C2×D4×C3112yes—
443C22×C4C2×Q8×C3112yes—
444C22×C4#41×C3114yes—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—416no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—22no—
457C23—22no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—44no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23C31×C22⋊Q814yes—
483C23C31×C22⋊Q818yes—
484C23C31×#12318yes—
485C23C31×#12314yes—
486C23C31×#12714yes—
487C23C22×D4×C3111yescharacteristic
488C23C22×D4×C3112yes—
489C23C22×D4×C3112yes—
490C23C22×Q8×C3111yescharacteristic
491C23C2×#41×C3112yes—
492C23C2×#41×C3111yescharacteristic
493C23C2×#41×C3112yes—
494C23C31×H(5,𝔽2)12yes—
495C23C31×2−1+412yes—
496D4—416no—
497C2×C4—232no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—432no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4C31×#12014yes—
533C2×C4C31×#12314yes—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—216no—
539C22—216no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—48no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—84no—
550C22—81no—
551C22C31×#39918yescomplemented
552C22C2×C31×#12011yescharacteristic
553C22C2×C31×C22⋊Q811yescharacteristic
554C22C2×C31×C22⋊Q812yes—
555C22C2×C31×#12311yescharacteristic
556C22C2×C31×#12311yescharacteristic
557C22C2×C31×#12312yes—
558C22C31×#52514yes—
559C22C2×C31×#12711yescharacteristic
560C22C31×#53511yescharacteristic
561C22C31×#53611yescharacteristic
562C22C31×#53914yes—
563C22D42×C3114yes—
564C22C31×#54714yes—
565C22C31×#54812yes—
566C22D4×Q8×C3112yes—
567C22C31×#55312yes—
568C22C31×#55412yes—
569C4—24no—
570C4—24no—
571C4—42no—
572C4—48no—
573C4—48no—
574C4—81no—
575C4—81no—
576C2—28no—
577C2—82no—
578C2C2×C31×#39911yescharacteristic
579C2C31×#240911yescharacteristic
580C2C31×#243311yescharacteristic
581C2C31×#244314yes—
582C2C31×#244914yes—
583C2C31×#248711yescharacteristic
584C2C31×#250111yescharacteristic
585C2C31×#262311yescharacteristic
586C2C31×#284911yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊C4×C31C2semidirect
2. C2×C31×#399C2semidirect
5. C2×C31×#385C2semidirect
6. C31×#1445C2semidirect
7. C31×#1444C2semidirect
8. C31×#1439C2semidirect
9. C31×C22⋊#95C2semidirect
64. C2×C4⋊C4×C31D4semidirect
159. #14777C31direct (N × H)
241. C23×C4⋊C4C2×C31semidirect
242. C2×#399C2×C31semidirect
245. C2×#385C2×C31semidirect
246. #1445C2×C31semidirect
247. #1444C2×C31semidirect
248. #1439C2×C31semidirect
249. C22⋊#95C2×C31semidirect
267. C22×C31#399semidirect
357. C2×C4⋊C4D4×C31semidirect
376. C31#14777direct (N × H)
551. C22C31×#399semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C312
C2×C31×#3992
C2×C31×#3862
C2×C31×#3852
C31×#14452
C31×#14442
C31×#14392
C31×C22⋊#952
#1477731