Finite Group Databasebeta

← C31×#25876 #21482635

Subgroups — C31×#25876 #21482635

Order 7936.

1222 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3499C211yesmaximal, characteristic
2C31×#3141C211yesmaximal, characteristic, complemented
3C31×#3079C211yesmaximal, characteristic, complemented
4C31×#2862C211yesmaximal, characteristic, complemented
5C31×#2038C214yesmaximal, complemented
6C31×#1966C211yesmaximal, characteristic, complemented
7C31×#1962C211yesmaximal, characteristic, complemented
8C31×#1793C214yesmaximal, complemented
9C4×C31×#104C211yesmaximal, characteristic, complemented
10C2×C31×2−1+4C2212yes—
11D4×Q8×C31—21no—
12D4×Q8×C31—21no—
13C31×#547—21no—
14C31×#547—21no—
15C31×#547—22no—
16C31×#547—22no—
17C31×#536—22no—
18C31×#536C2212yes—
19C31×#532—21no—
20C31×#532C2211yescharacteristic
21C2×C31×C4⋊Q8—21no—
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C4×#41×C31C2211yescharacteristic
25C31×#489—22no—
26C31×#487—21no—
27C31×#486—21no—
28C31×#477—21no—
29C31×#476—21no—
30C31×#475—21no—
31C31×#474—21no—
32C31×#428C2212yes—
33C31×#423C2211yescharacteristic
34C2×C31×#104C2212yes—
35C2×C31×#104C2214yes—
36C31×#417C2212yes—
37C31×#417C2214yes—
38C2×C31×#101C2214yes—
39C4×M16×C31C2211yescharacteristic
40C31×#391—22no—
41C31×#391C2214yes—
42C31×#382—22no—
43C43×C31C2211yescharacteristic
44C31×#339C2212yes—
45C31×2−1+4—28no—
46C31×2−1+4—44no—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—24no—
50C2×#41×C31C2312yes—
51C2×#41×C31C2311yescharacteristic
52C22×Q8×C31—22no—
53C22×Q8×C31—22no—
54C22×Q8×C31—21no—
55C22×Q8×C31C2312yes—
56C2×M16×C31C2312yes—
57C2×M16×C31C2312yes—
58C31×C4⋊Q8—22no—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8D411yescharacteristic
70C31×#127D411yescharacteristic
71C31×#124—22no—
72C31×#124D412yes—
73C31×#123—24no—
74C31×#123—24no—
75C31×#123—24no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—24no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×#121—22no—
81C31×#121—22no—
82C31×#121—21no—
83C31×#121—21no—
84C4×Q8×C31D411yescharacteristic
85C4×Q8×C31D411yescharacteristic
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31D411yescharacteristic
93C4×D4×C31D411yescharacteristic
94C31×#117—21no—
95C31×#117C2311yescharacteristic
96C2×C4⋊C4×C31—22no—
97C2×C4⋊C4×C31—22no—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31D412yes—
100C2×C4⋊C4×C31D412yes—
101C2×C4⋊C4×C31C2312yes—
102C2×C42×C31—21no—
103C2×C42×C31—21no—
104C2×C42×C31—21no—
105C2×C42×C31C2312yes—
106C2×C42×C31C2311yescharacteristic
107C31×#107—22no—
108C31×#106—22no—
109C31×C4⋊C8—21no—
110C31×C4⋊C8—21no—
111C31×#104—24no—
112C31×#104—44no—
113C31×#103—22no—
114C31×#103—24no—
115C31×#102—22no—
116C31×#102—24no—
117C31×#101—44no—
118C31×#97—21no—
119C4×C8×C31—21no—
120C31×#95—28no—
121#41×C31—22no—
122#41×C31—24no—
123#41×C31—24no—
124#41×C31—48no—
125#41×C31—416no—
126#41×C31—44no—
127#41×C31—44no—
128#41×C31#4114yes—
129C2×Q8×C31—24no—
130C2×Q8×C31—22no—
131C2×Q8×C31—22no—
132C2×Q8×C31—24no—
133C2×Q8×C31—24no—
134C2×Q8×C31—48no—
135C2×Q8×C31—44no—
136C2×Q8×C31—44no—
137C2×Q8×C31—44no—
138C2×Q8×C31—41no—
139C2×Q8×C31—41no—
140C2×Q8×C31C2×D412yes—
141C2×Q8×C31C2×D411yescharacteristic
142C2×D4×C31—21no—
143C2×D4×C31—24no—
144C2×D4×C31—22no—
145C2×D4×C31—21no—
146C2×D4×C31—21no—
147C2×D4×C31C2×D412yes—
148C2×D4×C31C2×D411yescharacteristic
149C22×C4×C31—21no—
150C22×C4×C31—24no—
151C22×C4×C31—24no—
152C22×C4×C31—22no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31C2×D414yes—
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2411yescharacteristic
161M16×C31—22no—
162M16×C31—22no—
163M16×C31—24no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C4⋊C4×C31—24no—
167C4⋊C4×C31—24no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—41no—
179C4⋊C4×C31—41no—
180C4⋊C4×C31C2×D411yescharacteristic
181C4⋊C4×C31C2×D411yescharacteristic
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—21no—
185C22⋊C4×C31—21no—
186C22⋊C4×C31—21no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C42×C31—24no—
190C42×C31—21no—
191C42×C31—21no—
192C42×C31—21no—
193C42×C31—22no—
194C42×C31—22no—
195C42×C31—42no—
196C42×C31C2×D411yescharacteristic
197C42×C31C2×D411yescharacteristic
198C42×C31C2×D411yescharacteristic
199C42×C31C2×D411yescharacteristic
200#25876C3111yesmaximal, characteristic, direct product factor, Sylow
201C23×C31—21no—
202C23×C31—22no—
203C23×C31C22×D411yescharacteristic
204Q8×C31—21no—
205Q8×C31—24no—
206Q8×C31—48no—
207Q8×C31—42no—
208Q8×C31—42no—
209Q8×C31—44no—
210Q8×C31—82no—
211Q8×C31—82no—
212Q8×C31#12412yes—
213D4×C31—24no—
214D4×C31—21no—
215D4×C31—24no—
216D4×C31—48no—
217D4×C31—44no—
218D4×C31—42no—
219D4×C31—42no—
220D4×C31#12412yes—
221C2×C4×C31—21no—
222C2×C4×C31—24no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—24no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—22no—
229C2×C4×C31—24no—
230C2×C4×C31—28no—
231C2×C4×C31—48no—
232C2×C4×C31—42no—
233C2×C4×C31—48no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—44no—
240C2×C4×C31—42no—
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31#12011yescharacteristic
245C2×C4×C31C22×D411yescharacteristic
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C2×#4112yes—
248C2×C4×C31H(5,𝔽2)12yes—
249C8×C31—42no—
250C8×C31—42no—
251#3499C2×C3111yescharacteristic
252#3141C2×C3111yescharacteristic, complemented
253#3079C2×C3111yescharacteristic, complemented
254#2862C2×C3111yescharacteristic, complemented
255#2038C2×C3114yescomplemented
256#1966C2×C3111yescharacteristic, complemented
257#1962C2×C3111yescharacteristic, complemented
258#1793C2×C3114yescomplemented
259C4×#104C2×C3111yescharacteristic, complemented
260C22×C31—21no—
261C22×C31—21no—
262C22×C31—22no—
263C22×C31—41no—
264C22×C31—41no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31C2×#12011yescharacteristic
269C22×C31#54612yes—
270C4×C31—22no—
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—44no—
274C4×C31—44no—
275C4×C31—42no—
276C4×C31—41no—
277C4×C31—41no—
278C4×C31—82no—
279C4×C31#45412yes—
280C4×C31#45612yes—
281C4×C31C2×#12411yescharacteristic
282C4×C31#53511yescharacteristic
283C4×C31#54612yes—
284C2×2−1+4C22×C3112yes—
285D4×Q8—21no—
286D4×Q8—21no—
287#547—21no—
288#547—21no—
289#547—22no—
290#547—22no—
291#536—22no—
292#536C22×C3112yes—
293#532—21no—
294#532C22×C3111yescharacteristic
295C2×C4⋊Q8—21no—
296C2×C4⋊Q8C22×C3111yescharacteristic
297C2×C4⋊Q8C22×C3111yescharacteristic
298C4×#41C22×C3111yescharacteristic
299#489—22no—
300#487—21no—
301#486—21no—
302#477—21no—
303#476—21no—
304#475—21no—
305#474—21no—
306#428C22×C3112yes—
307#423C22×C3111yescharacteristic
308C2×#104C22×C3112yes—
309C2×#104C22×C3114yes—
310#417C22×C3112yes—
311#417C22×C3114yes—
312C2×#101C22×C3114yes—
313C4×M16C22×C3111yescharacteristic
314#391—22no—
315#391C22×C3114yes—
316#382—22no—
317C43C22×C3111yescharacteristic
318#339C22×C3112yes—
319C2×C31—22no—
320C2×C31—42no—
321C2×C31—82no—
322C2×C31#243511yescharacteristic
323C2×C31C2×#45411yescharacteristic
324C2×C31C2×#45611yescharacteristic
3252−1+4—28no—
3262−1+4—44no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—24no—
330C2×#41C23×C3112yes—
331C2×#41C23×C3111yescharacteristic
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—21no—
335C22×Q8C23×C3112yes—
336C2×M16C23×C3112yes—
337C2×M16C23×C3112yes—
338C4⋊Q8—22no—
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—21no—
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8D4×C3111yescharacteristic
350#127D4×C3111yescharacteristic
351#124—22no—
352#124D4×C3112yes—
353#123—24no—
354#123—24no—
355#123—24no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360#121—22no—
361#121—22no—
362#121—21no—
363#121—21no—
364C4×Q8D4×C3111yescharacteristic
365C4×Q8D4×C3111yescharacteristic
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4D4×C3111yescharacteristic
373C4×D4D4×C3111yescharacteristic
374#117—21no—
375#117C23×C3111yescharacteristic
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4D4×C3112yes—
380C2×C4⋊C4D4×C3112yes—
381C2×C4⋊C4C23×C3112yes—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42C23×C3112yes—
386C2×C42C23×C3111yescharacteristic
387#107—22no—
388#106—22no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391#104—24no—
392#104—44no—
393#103—22no—
394#103—24no—
395#102—22no—
396#102—24no—
397#101—44no—
398#97—21no—
399C4×C8—21no—
400#95—28no—
401C31#2587611yescharacteristic, direct product factor, Sylow
402#41—22no—
403#41—24no—
404#41—24no—
405#41—48no—
406#41—416no—
407#41—44no—
408#41—44no—
409#41#41×C3114yes—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—48no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8C2×D4×C3112yes—
422C2×Q8C2×D4×C3111yescharacteristic
423C2×D4—21no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C2×D4×C3112yes—
429C2×D4C2×D4×C3111yescharacteristic
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4C2×D4×C3114yes—
439C22×C4C2×D4×C3111yescharacteristic
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C24×C3111yescharacteristic
442M16—22no—
443M16—22no—
444M16—24no—
445C2×C8—22no—
446C2×C8—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4C2×D4×C3111yescharacteristic
462C4⋊C4C2×D4×C3111yescharacteristic
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C42—24no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—22no—
476C42—42no—
477C42C2×D4×C3111yescharacteristic
478C42C2×D4×C3111yescharacteristic
479C42C2×D4×C3111yescharacteristic
480C42C2×D4×C3111yescharacteristic
481C23—21no—
482C23—22no—
483C23C22×D4×C3111yescharacteristic
484Q8—21no—
485Q8—24no—
486Q8—48no—
487Q8—42no—
488Q8—42no—
489Q8—44no—
490Q8—82no—
491Q8—82no—
492Q8C31×#12412yes—
493D4—24no—
494D4—21no—
495D4—24no—
496D4—48no—
497D4—44no—
498D4—42no—
499D4—42no—
500D4C31×#12412yes—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4C31×#12011yescharacteristic
522C2×C4C31×#12011yescharacteristic
523C2×C4C31×#12011yescharacteristic
524C2×C4C31×#12011yescharacteristic
525C2×C4C22×D4×C3111yescharacteristic
526C2×C4C22×D4×C3111yescharacteristic
527C2×C4C2×#41×C3112yes—
528C2×C4C31×H(5,𝔽2)12yes—
529C8—42no—
530C8—42no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22C2×C31×#12011yescharacteristic
540C22C31×#54612yes—
541C4—22no—
542C4—21no—
543C4—21no—
544C4—44no—
545C4—44no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—82no—
550C4C31×#45412yes—
551C4C31×#45612yes—
552C4C2×C31×#12411yescharacteristic
553C4C31×#53511yescharacteristic
554C4C31×#54612yes—
555C2—22no—
556C2—42no—
557C2—82no—
558C2C31×#243511yescharacteristic
559C2C2×C31×#45411yescharacteristic
560C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3499C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3141C2semidirect
3. C31×#3079C2semidirect
4. C31×#2862C2semidirect
5. C31×#2038C2semidirect
6. C31×#1966C2semidirect
7. C31×#1962C2semidirect
8. C31×#1793C2semidirect
9. C4×C31×#104C2semidirect
200. #25876C31direct (N × H)
252. #3141C2×C31semidirect
253. #3079C2×C31semidirect
254. #2862C2×C31semidirect
255. #2038C2×C31semidirect
256. #1966C2×C31semidirect
257. #1962C2×C31semidirect
258. #1793C2×C31semidirect
259. C4×#104C2×C31semidirect
401. C31#25876direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34992
C31×#31412
C31×#30792
C31×#28622
C31×#20382
C31×#19662
C31×#19622
C31×#17932
C4×C31×#1042
#2587631