Finite Group Databasebeta

← C31×#9916 #21466675

Subgroups — C31×#9916 #21466675

Order 7936.

968 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#448C211yesmaximal, characteristic, complemented
2C31×#2617C211yesmaximal, characteristic
3C2×C31×#393C211yesmaximal, characteristic, complemented
4C31×#1787C211yesmaximal, characteristic, complemented
5C2×C31×#327C211yesmaximal, characteristic, complemented
6C2×C31×#323C211yesmaximal, characteristic, complemented
7C31×#1294C211yesmaximal, characteristic, complemented
8C23×Q8×C31C2211yescharacteristic
9C22×SD16×C31—21no—
10C2×C31×#121C2211yescharacteristic
11C2×C31×#120—21no—
12C22×C22⋊C4×C31—21no—
13C22×C22⋊C4×C31C2211yescharacteristic
14C31×#448—22no—
15C31×#448—22no—
16C2×C31×#103—21no—
17C2×C31×#103—22no—
18C2×C31×C22⋊C8C2211yescharacteristic
19C2×C31×C22⋊C8C2211yescharacteristic
20C31×#397—21no—
21C31×#394—22no—
22C31×#393—22no—
23C31×#393—22no—
24C31×#392—21no—
25C31×#386—21no—
26C31×#380C2211yescharacteristic
27C2×C31×#95—21no—
28C2×C31×#95C2211yescharacteristic
29C31×#327—22no—
30C31×#323—22no—
31C25×C31D411yescharacteristic
32C22×Q8×C31—21no—
33C22×Q8×C31—21no—
34C22×Q8×C31—21no—
35C22×Q8×C31—21no—
36C22×Q8×C31—21no—
37C22×Q8×C31—44no—
38C22×Q8×C31D412yescomplemented
39C22×D4×C31—21no—
40C22×D4×C31—22no—
41C22×D4×C31D411yescharacteristic
42C23×C4×C31—21no—
43C23×C4×C31—21no—
44C23×C4×C31D411yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C2×SD16×C31—24no—
47C2×SD16×C31—44no—
48C22×C8×C31—21no—
49C22×C8×C31—21no—
50C31×#121—21no—
51C31×#121—21no—
52C31×#121—21no—
53C31×#121—21no—
54C31×#120—42no—
55C31×#120—42no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D412yes—
58C2×C22⋊C4×C31—22no—
59C2×C22⋊C4×C31—24no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—21no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—21no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—44no—
70C31×#103—24no—
71C31×#103—44no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8D412yescomplemented
76C31×#95—22no—
77C31×#95—24no—
78C31×#95—21no—
79C31×#95—21no—
80C31×#95D412yescomplemented
81C24×C31—21no—
82C24×C31—21no—
83C24×C31—21no—
84C24×C31—21no—
85C24×C31—21no—
86C24×C31—21no—
87C24×C31—21no—
88C24×C31—42no—
89C24×C31—42no—
90C24×C31SD1612yescomplemented
91C24×C31C2×D411yescharacteristic
92C2×Q8×C31—24no—
93C2×Q8×C31—24no—
94C2×Q8×C31—22no—
95C2×Q8×C31—22no—
96C2×Q8×C31—44no—
97C2×Q8×C31—42no—
98C2×Q8×C31—42no—
99C2×Q8×C31—44no—
100C2×Q8×C31—44no—
101C2×Q8×C31—44no—
102C2×Q8×C31—41no—
103C2×Q8×C31—41no—
104C2×D4×C31—21no—
105C2×D4×C31—21no—
106C2×D4×C31—42no—
107C2×D4×C31—44no—
108C2×D4×C31—41no—
109C2×D4×C31—41no—
110C2×D4×C31—41no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—41no—
114C2×D4×C31—82no—
115C2×D4×C31—82no—
116C22×C4×C31—24no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—42no—
125C22×C4×C31—42no—
126C22×C4×C31—41no—
127C22×C4×C31—41no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31—41no—
132C22×C4×C31—41no—
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31#4111yescharacteristic
139SD16×C31—48no—
140C2×C8×C31—22no—
141C2×C8×C31—42no—
142C2×C8×C31—42no—
143C2×C8×C31—41no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—42no—
147C22⋊C4×C31—48no—
148C22⋊C4×C31—44no—
149C22⋊C4×C31—41no—
150C22⋊C4×C31—41no—
151C22⋊C4×C31—41no—
152C22⋊C4×C31—41no—
153C22⋊C4×C31—41no—
154C22⋊C4×C31—41no—
155C22⋊C4×C31—41no—
156C22⋊C4×C31—41no—
157C22⋊C4×C31—82no—
158#9916C3111yesmaximal, characteristic, direct product factor, Sylow
159C23×C31—22no—
160C23×C31—22no—
161C23×C31—22no—
162C23×C31—22no—
163C23×C31—22no—
164C23×C31—44no—
165C23×C31—42no—
166C23×C31—44no—
167C23×C31—42no—
168C23×C31—41no—
169C23×C31—41no—
170C23×C31—41no—
171C23×C31—42no—
172C23×C31—42no—
173C23×C31—41no—
174C23×C31—41no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—82no—
180C23×C31—82no—
181C23×C31—81no—
182C23×C31—81no—
183C23×C31#12011yescharacteristic
184C23×C31#12011yescharacteristic
185C23×C31#12111yescharacteristic
186C23×C31C2×SD1611yescharacteristic
187C23×C31#13511yescharacteristic
188C23×C31AGL(1,ℤ/8)11yescharacteristic
189C23×C31#13711yescharacteristic
190Q8×C31—24no—
191Q8×C31—24no—
192Q8×C31—42no—
193Q8×C31—44no—
194Q8×C31—44no—
195Q8×C31—42no—
196D4×C31—42no—
197D4×C31—81no—
198D4×C31—88no—
199D4×C31—81no—
200D4×C31—81no—
201D4×C31—81no—
202D4×C31—81no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—24no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—42no—
211C2×C4×C31—44no—
212C2×C4×C31—42no—
213C2×C4×C31—44no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—82no—
227C2×C4×C31—82no—
228C2×C4×C31—81no—
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#12111yescharacteristic
231C2×C4×C31#12111yescharacteristic
232C2×C4×C31#12711yescharacteristic
233C8×C31—42no—
234C8×C31—81no—
235C2×#448C2×C3111yescharacteristic, complemented
236#2617C2×C3111yescharacteristic
237C2×#393C2×C3111yescharacteristic, complemented
238#1787C2×C3111yescharacteristic, complemented
239C2×#327C2×C3111yescharacteristic, complemented
240C2×#323C2×C3111yescharacteristic, complemented
241#1294C2×C3111yescharacteristic, complemented
242C22×C31—22no—
243C22×C31—22no—
244C22×C31—22no—
245C22×C31—22no—
246C22×C31—21no—
247C22×C31—21no—
248C22×C31—21no—
249C22×C31—21no—
250C22×C31—44no—
251C22×C31—44no—
252C22×C31—42no—
253C22×C31—82no—
254C22×C31—82no—
255C22×C31—82no—
256C22×C31—82no—
257C22×C31—81no—
258C22×C31—81no—
259C22×C31—82no—
260C22×C31—82no—
261C22×C31—81no—
262C22×C31—81no—
263C22×C31—81no—
264C22×C31#39211yescharacteristic
265C22×C31#44811yescharacteristic
266C22×C31#44911yescharacteristic
267C22×C31#45011yescharacteristic
268C22×C31#45211yescharacteristic
269C22×C31#45512yes—
270C22×C31B(4,𝔽2)12yes—
271C22×C31#46511yescharacteristic
272C22×C31#46811yescharacteristic
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—21no—
277C4×C31—44no—
278C4×C31—82no—
279C4×C31—82no—
280C4×C31—81no—
281C4×C31—81no—
282C23×Q8C22×C3111yescharacteristic
283C22×SD16—21no—
284C2×#121C22×C3111yescharacteristic
285C2×#120—21no—
286C22×C22⋊C4—21no—
287C22×C22⋊C4C22×C3111yescharacteristic
288#448—22no—
289#448—22no—
290C2×#103—21no—
291C2×#103—22no—
292C2×C22⋊C8C22×C3111yescharacteristic
293C2×C22⋊C8C22×C3111yescharacteristic
294#397—21no—
295#394—22no—
296#393—22no—
297#393—22no—
298#392—21no—
299#386—21no—
300#380C22×C3111yescharacteristic
301C2×#95—21no—
302C2×#95C22×C3111yescharacteristic
303#327—22no—
304#323—22no—
305C2×C31—21no—
306C2×C31—21no—
307C2×C31—21no—
308C2×C31—21no—
309C2×C31—82no—
310C2×C31—161no—
311C2×C31#160112yes—
312C2×C31#160612yes—
313C2×C31#200011yescharacteristic
314C2×C31#200411yescharacteristic
315C2×C31#202111yescharacteristic
316C25D4×C3111yescharacteristic
317C22×Q8—21no—
318C22×Q8—21no—
319C22×Q8—21no—
320C22×Q8—21no—
321C22×Q8—21no—
322C22×Q8—44no—
323C22×Q8D4×C3112yescomplemented
324C22×D4—21no—
325C22×D4—22no—
326C22×D4D4×C3111yescharacteristic
327C23×C4—21no—
328C23×C4—21no—
329C23×C4D4×C3111yescharacteristic
330C23×C4C23×C3111yescharacteristic
331C2×SD16—24no—
332C2×SD16—44no—
333C22×C8—21no—
334C22×C8—21no—
335#121—21no—
336#121—21no—
337#121—21no—
338#121—21no—
339#120—42no—
340#120—42no—
341C2×C4⋊C4D4×C3111yescharacteristic
342C2×C4⋊C4D4×C3112yes—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—44no—
355#103—24no—
356#103—44no—
357C22⋊C8—21no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8D4×C3112yescomplemented
361#95—24no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95D4×C3112yescomplemented
366C31#991611yescharacteristic, direct product factor, Sylow
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—21no—
374C24—42no—
375C24—42no—
376C24SD16×C3112yescomplemented
377C24C2×D4×C3111yescharacteristic
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—82no—
401C2×D4—82no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4C2×D4×C3111yescharacteristic
420C22×C4C2×D4×C3111yescharacteristic
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C2×D4×C3111yescharacteristic
423C22×C4C2×D4×C3111yescharacteristic
424C22×C4#41×C3111yescharacteristic
425SD16—48no—
426C2×C8—22no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—41no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—82no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—44no—
450C23—42no—
451C23—44no—
452C23—42no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23—82no—
465C23—82no—
466C23—81no—
467C23—81no—
468C23C31×#12011yescharacteristic
469C23C31×#12011yescharacteristic
470C23C31×#12111yescharacteristic
471C23C2×SD16×C3111yescharacteristic
472C23C31×#13511yescharacteristic
473C23C31×AGL(1,ℤ/8)11yescharacteristic
474C23C31×#13711yescharacteristic
475Q8—24no—
476Q8—24no—
477Q8—42no—
478Q8—44no—
479Q8—44no—
480Q8—42no—
481D4—42no—
482D4—81no—
483D4—88no—
484D4—81no—
485D4—81no—
486D4—81no—
487D4—81no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4C31×#12011yescharacteristic
515C2×C4C31×#12111yescharacteristic
516C2×C4C31×#12111yescharacteristic
517C2×C4C31×#12711yescharacteristic
518C8—42no—
519C8—81no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—44no—
529C22—44no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—81no—
536C22—81no—
537C22—82no—
538C22—82no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C31×#39211yescharacteristic
543C22C31×#44811yescharacteristic
544C22C31×#44911yescharacteristic
545C22C31×#45011yescharacteristic
546C22C31×#45211yescharacteristic
547C22C31×#45512yes—
548C22C31×B(4,𝔽2)12yes—
549C22C31×#46511yescharacteristic
550C22C31×#46811yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—44no—
556C4—82no—
557C4—82no—
558C4—81no—
559C4—81no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—82no—
565C2—161no—
566C2C31×#160112yes—
567C2C31×#160612yes—
568C2C31×#200011yescharacteristic
569C2C31×#200411yescharacteristic
570C2C31×#202111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#2617C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#448C2semidirect
3. C2×C31×#393C2semidirect
4. C31×#1787C2semidirect
5. C2×C31×#327C2semidirect
6. C2×C31×#323C2semidirect
7. C31×#1294C2semidirect
38. C22×Q8×C31D4semidirect
75. C31×C22⋊C8D4semidirect
80. C31×#95D4semidirect
90. C24×C31SD16semidirect
158. #9916C31direct (N × H)
235. C2×#448C2×C31semidirect
237. C2×#393C2×C31semidirect
238. #1787C2×C31semidirect
239. C2×#327C2×C31semidirect
240. C2×#323C2×C31semidirect
241. #1294C2×C31semidirect
323. C22×Q8D4×C31semidirect
360. C22⋊C8D4×C31semidirect
365. #95D4×C31semidirect
366. C31#9916direct (N × H)
376. C24SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4482
C31×#26172
C2×C31×#3932
C31×#17872
C2×C31×#3272
C2×C31×#3232
C31×#12942
#991631