Finite Group Databasebeta

← C31×#18835 #21475594

Subgroups — C31×#18835 #21475594

Order 7936.

940 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3514C211yesmaximal, characteristic
2C31×#3238C211yesmaximal, characteristic, complemented
3C31×#3206C211yesmaximal, characteristic, complemented
4C31×#3079C211yesmaximal, characteristic, complemented
5C31×#1638C212yesmaximal, complemented
6C31×#1637C212yesmaximal, complemented
7C31×#1630C212yesmaximal, complemented
8C31×#1629C212yesmaximal, complemented
9C31×#1557C211yesmaximal, characteristic, complemented
10C31×#1495C211yesmaximal, characteristic, complemented
11C31×#1488C211yesmaximal, characteristic, complemented
12C31×#559C2211yescharacteristic
13C31×#556C2211yescharacteristic
14C31×#555—21no—
15C31×#551—22no—
16D4×Q8×C31C2211yescharacteristic
17C31×#548—21no—
18C31×#546—21no—
19C31×#546—21no—
20D42×C31C2211yescharacteristic
21C31×#542—21no—
22C31×#538—22no—
23C31×#532C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic, complemented
25C31×#529—21no—
26C31×#529C2211yescharacteristic
27C2×C31×#124—21no—
28C31×#518—21no—
29C4×#41×C31C2211yescharacteristic
30C31×#501—21no—
31C31×#492C2212yes—
32C31×#489—21no—
33C31×#487C2212yescomplemented
34C31×#483—21no—
35C31×#480—21no—
36C31×#477C2212yescomplemented
37C31×#477C2212yescomplemented
38C31×#476C2212yes—
39C31×#476C2212yes—
40C31×#475—21no—
41C31×#474—21no—
42C31×#461—22no—
43C31×#448—22no—
44C31×#432C2211yescharacteristic, complemented
45C31×#431C2211yescharacteristic, complemented
46C31×#428—21no—
47C31×#423C2211yescharacteristic, complemented
48C31×#417—21no—
49C31×#332C2212yescomplemented
50C31×#331C2212yes—
51C31×Q8⋊C8C2212yescomplemented
52C31×Q8⋊C8C2212yescomplemented
53C31×D4⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31D411yescharacteristic
57C22×Q8×C31—21no—
58C22×Q8×C31—21no—
59C22×D4×C31—21no—
60C22×D4×C31—21no—
61C2×SD16×C31—24no—
62C2×M16×C31—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8C2312yes—
68C31×C4⋊Q8C2311yescharacteristic
69C31×#127C2311yescharacteristic
70C31×#126—22no—
71C31×#126—22no—
72C31×#126—21no—
73C31×#125D411yescharacteristic
74C31×#124—22no—
75C31×#124—22no—
76C31×#124—22no—
77C31×#124—21no—
78C31×#124—21no—
79C31×#124—21no—
80C31×#124—21no—
81C31×#124D411yescharacteristic
82C31×#123—22no—
83C31×#123—22no—
84C31×#123—22no—
85C31×#123—21no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8D411yescharacteristic
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—21no—
94C31×#121—21no—
95C31×#121—21no—
96C31×#121D412yes—
97C31×#121D411yescharacteristic
98C31×#120—22no—
99C31×#120—22no—
100C4×Q8×C31—21no—
101C4×Q8×C31D411yescharacteristic
102C4×Q8×C31C2311yescharacteristic
103C4×Q8×C31C2311yescharacteristic
104C4×D4×C31—22no—
105C4×D4×C31—22no—
106C4×D4×C31—21no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31—21no—
110C4×D4×C31—21no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31D412yes—
114C4×D4×C31D411yescharacteristic
115C4×D4×C31C2311yescharacteristic
116C4×D4×C31C2311yescharacteristic
117C31×#117—22no—
118C31×#117—21no—
119C31×#117—21no—
120C31×#117—21no—
121C31×#117D411yescharacteristic
122C2×C4⋊C4×C31—21no—
123C2×C4⋊C4×C31—21no—
124C2×C22⋊C4×C31—22no—
125C2×C22⋊C4×C31—22no—
126C2×C22⋊C4×C31—22no—
127C2×C42×C31—21no—
128C2×C42×C31C2311yescharacteristic
129C31×#107—22no—
130C31×#107—22no—
131C31×#106—22no—
132C31×#106—22no—
133C31×C4⋊C8—21no—
134C31×C4⋊C8—21no—
135C31×C4⋊C8C2312yes—
136C31×C4⋊C8C2312yes—
137C31×#103—22no—
138C31×#103—22no—
139C31×#103—24no—
140C31×#102—22no—
141C31×#102—22no—
142C31×C22⋊C8—21no—
143C31×C22⋊C8—21no—
144C31×#97—21no—
145C4×C8×C31C2312yes—
146C24×C31—22no—
147#41×C31—22no—
148#41×C31—41no—
149#41×C31—82no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—22no—
153C2×Q8×C31—41no—
154C2×Q8×C31—82no—
155C2×Q8×C31—82no—
156C2×Q8×C31C2×D411yescharacteristic
157C2×Q8×C31C2×D411yescharacteristic
158C2×D4×C31—22no—
159C2×D4×C31—21no—
160C2×D4×C31—22no—
161C2×D4×C31—21no—
162C2×D4×C31—22no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31—42no—
166C2×D4×C31—42no—
167C2×D4×C31—82no—
168C2×D4×C31C2×D411yescharacteristic
169C2×D4×C31C2×D411yescharacteristic
170C2×D4×C31C2×D412yes—
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—21no—
174C22×C4×C31—21no—
175C22×C4×C31—21no—
176C22×C4×C31—21no—
177C22×C4×C31—22no—
178C22×C4×C31—41no—
179C22×C4×C31C2×D411yescharacteristic
180C22×C4×C31C2×D411yescharacteristic
181C22×C4×C31C2×D411yescharacteristic
182SD16×C31—84no—
183M16×C31—42no—
184C2×C8×C31—22no—
185C2×C8×C31—22no—
186C2×C8×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—41no—
199C4⋊C4×C31—41no—
200C4⋊C4×C31C2×D412yes—
201C4⋊C4×C31C2×D411yescharacteristic
202C4⋊C4×C31C2×D411yescharacteristic
203C4⋊C4×C31C2×D411yescharacteristic
204C4⋊C4×C31C2×D411yescharacteristic
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—22no—
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—21no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—21no—
222C42×C31—22no—
223C42×C31—21no—
224C42×C31—21no—
225C42×C31—21no—
226C42×C31—21no—
227C42×C31C2×D411yescharacteristic
228C42×C31C2×D411yescharacteristic
229C42×C31C2×D411yescharacteristic
230C42×C31C2411yescharacteristic
231#18835C3111yesmaximal, characteristic, direct product factor, Sylow
232C23×C31—21no—
233C23×C31—21no—
234C23×C31—22no—
235C23×C31—44no—
236C23×C31—82no—
237C23×C31—82no—
238C23×C31#12011yescharacteristic
239Q8×C31—21no—
240Q8×C31—21no—
241Q8×C31—22no—
242Q8×C31—44no—
243Q8×C31—42no—
244Q8×C31—82no—
245Q8×C31—82no—
246D4×C31—21no—
247D4×C31—22no—
248D4×C31—42no—
249D4×C31—42no—
250D4×C31—44no—
251D4×C31—41no—
252D4×C31—41no—
253D4×C31—42no—
254D4×C31—81no—
255D4×C31—82no—
256D4×C31—82no—
257D4×C31#13712yes—
258C2×C4×C31—21no—
259C2×C4×C31—21no—
260C2×C4×C31—22no—
261C2×C4×C31—21no—
262C2×C4×C31—21no—
263C2×C4×C31—21no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—22no—
268C2×C4×C31—41no—
269C2×C4×C31—42no—
270C2×C4×C31—41no—
271C2×C4×C31—41no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—81no—
275C2×C4×C31—82no—
276C2×C4×C31—82no—
277C2×C4×C31—82no—
278C2×C4×C31—82no—
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31#12011yescharacteristic
281C2×C4×C31#12011yescharacteristic
282C2×C4×C31C22×D411yescharacteristic
283C2×C4×C31C22×D411yescharacteristic
284C2×C4×C31C22×D411yescharacteristic
285C8×C31—42no—
286C8×C31—42no—
287C8×C31—42no—
288#3514C2×C3111yescharacteristic
289#3238C2×C3111yescharacteristic, complemented
290#3206C2×C3111yescharacteristic, complemented
291#3079C2×C3111yescharacteristic, complemented
292#1638C2×C3112yescomplemented
293#1637C2×C3112yescomplemented
294#1630C2×C3112yescomplemented
295#1629C2×C3112yescomplemented
296#1557C2×C3111yescharacteristic, complemented
297#1495C2×C3111yescharacteristic, complemented
298#1488C2×C3111yescharacteristic, complemented
299C22×C31—21no—
300C22×C31—22no—
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—41no—
304C22×C31—41no—
305C22×C31—41no—
306C22×C31—42no—
307C22×C31—42no—
308C22×C31—41no—
309C22×C31—42no—
310C22×C31—41no—
311C22×C31—164no—
312C22×C31C2×#12011yescharacteristic
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—21no—
316C4×C31—21no—
317C4×C31—41no—
318C4×C31—41no—
319C4×C31—41no—
320C4×C31—42no—
321C4×C31—41no—
322C4×C31—41no—
323C4×C31—41no—
324C4×C31—82no—
325C4×C31C2×#13711yescharacteristic
326C4×C31#57511yescharacteristic
327C4×C31#57712yes—
328#559C22×C3111yescharacteristic
329#556C22×C3111yescharacteristic
330#555—21no—
331#551—22no—
332D4×Q8C22×C3111yescharacteristic
333#548—21no—
334#546—21no—
335#546—21no—
336D42C22×C3111yescharacteristic
337#542—21no—
338#538—22no—
339#532C22×C3111yescharacteristic
340C2×C4⋊Q8C22×C3111yescharacteristic, complemented
341#529—21no—
342#529C22×C3111yescharacteristic
343C2×#124—21no—
344#518—21no—
345C4×#41C22×C3111yescharacteristic
346#501—21no—
347#492C22×C3112yes—
348#489—21no—
349#487C22×C3112yescomplemented
350#483—21no—
351#480—21no—
352#477C22×C3112yescomplemented
353#477C22×C3112yescomplemented
354#476C22×C3112yes—
355#476C22×C3112yes—
356#475—21no—
357#474—21no—
358#461—22no—
359#448—22no—
360#432C22×C3111yescharacteristic, complemented
361#431C22×C3111yescharacteristic, complemented
362#428—21no—
363#423C22×C3111yescharacteristic, complemented
364#417—21no—
365#332C22×C3112yescomplemented
366#331C22×C3112yes—
367Q8⋊C8C22×C3112yescomplemented
368Q8⋊C8C22×C3112yescomplemented
369D4⋊C8C22×C3112yes—
370D4⋊C8C22×C3112yes—
371C2×C31—41no—
372C2×C31—42no—
373C2×C31—82no—
374C2×C31—81no—
375C2×C31#300711yescharacteristic
376C2×C31#300911yescharacteristic
377C2×C31#302111yescharacteristic
378C2×#41—21no—
379C2×#41D4×C3111yescharacteristic
380C22×Q8—21no—
381C22×Q8—21no—
382C22×D4—21no—
383C22×D4—21no—
384C2×SD16—24no—
385C2×M16—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8C23×C3112yes—
391C4⋊Q8C23×C3111yescharacteristic
392#127C23×C3111yescharacteristic
393#126—22no—
394#126—22no—
395#126—21no—
396#125D4×C3111yescharacteristic
397#124—22no—
398#124—22no—
399#124—22no—
400#124—21no—
401#124—21no—
402#124—21no—
403#124—21no—
404#124D4×C3111yescharacteristic
405#123—22no—
406#123—22no—
407#123—22no—
408#123—21no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—22no—
413C22⋊Q8D4×C3111yescharacteristic
414#121—22no—
415#121—22no—
416#121—21no—
417#121—21no—
418#121—21no—
419#121D4×C3111yescharacteristic
420#121D4×C3112yes—
421#120—22no—
422#120—22no—
423C4×Q8—21no—
424C4×Q8D4×C3111yescharacteristic
425C4×Q8C23×C3111yescharacteristic
426C4×Q8C23×C3111yescharacteristic
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4D4×C3111yescharacteristic
437C4×D4D4×C3112yes—
438C4×D4C23×C3111yescharacteristic
439C4×D4C23×C3111yescharacteristic
440#117—22no—
441#117—21no—
442#117—21no—
443#117—21no—
444#117D4×C3111yescharacteristic
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—22no—
450C2×C42—21no—
451C2×C42C23×C3111yescharacteristic
452#107—22no—
453#107—22no—
454#106—22no—
455#106—22no—
456C4⋊C8—21no—
457C4⋊C8—21no—
458C4⋊C8C23×C3112yes—
459C4⋊C8C23×C3112yes—
460#103—22no—
461#103—22no—
462#103—24no—
463#102—22no—
464#102—22no—
465C22⋊C8—21no—
466C22⋊C8—21no—
467#97—21no—
468C4×C8C23×C3112yes—
469C31#1883511yescharacteristic, direct product factor, Sylow
470C24—22no—
471#41—22no—
472#41—41no—
473#41—82no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—22no—
477C2×Q8—41no—
478C2×Q8—82no—
479C2×Q8—82no—
480C2×Q8C2×D4×C3111yescharacteristic
481C2×Q8C2×D4×C3111yescharacteristic
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—22no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4—82no—
492C2×D4C2×D4×C3111yescharacteristic
493C2×D4C2×D4×C3111yescharacteristic
494C2×D4C2×D4×C3112yes—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—41no—
503C22×C4C2×D4×C3111yescharacteristic
504C22×C4C2×D4×C3111yescharacteristic
505C22×C4C2×D4×C3111yescharacteristic
506SD16—84no—
507M16—42no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4C2×D4×C3112yes—
525C4⋊C4C2×D4×C3111yescharacteristic
526C4⋊C4C2×D4×C3111yescharacteristic
527C4⋊C4C2×D4×C3111yescharacteristic
528C4⋊C4C2×D4×C3111yescharacteristic
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C42—22no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42C2×D4×C3111yescharacteristic
552C42C2×D4×C3111yescharacteristic
553C42C2×D4×C3111yescharacteristic
554C42C24×C3111yescharacteristic
555C23—21no—
556C23—21no—
557C23—22no—
558C23—44no—
559C23—82no—
560C23—82no—
561C23C31×#12011yescharacteristic
562Q8—21no—
563Q8—21no—
564Q8—22no—
565Q8—44no—
566Q8—42no—
567Q8—82no—
568Q8—82no—
569D4—21no—
570D4—22no—
571D4—42no—
572D4—42no—
573D4—44no—
574D4—41no—
575D4—41no—
576D4—42no—
577D4—81no—
578D4—82no—
579D4—82no—
580D4C31×#13712yes—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4C31×#12011yescharacteristic
603C2×C4C31×#12011yescharacteristic
604C2×C4C31×#12011yescharacteristic
605C2×C4C22×D4×C3111yescharacteristic
606C2×C4C22×D4×C3111yescharacteristic
607C2×C4C22×D4×C3111yescharacteristic
608C8—42no—
609C8—42no—
610C8—42no—
611C22—21no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—164no—
624C22C2×C31×#12011yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—82no—
637C4C2×C31×#13711yescharacteristic
638C4C31×#57511yescharacteristic
639C4C31×#57712yes—
640C2—41no—
641C2—42no—
642C2—81no—
643C2—82no—
644C2C31×#300711yescharacteristic
645C2C31×#300911yescharacteristic
646C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3514C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3238C2semidirect
3. C31×#3206C2semidirect
4. C31×#3079C2semidirect
5. C31×#1638C2semidirect
6. C31×#1637C2semidirect
7. C31×#1630C2semidirect
8. C31×#1629C2semidirect
9. C31×#1557C2semidirect
10. C31×#1495C2semidirect
11. C31×#1488C2semidirect
24. C2×C31×C4⋊Q8C22semidirect
33. C31×#487C22semidirect
36. C31×#477C22semidirect
37. C31×#477C22semidirect
44. C31×#432C22semidirect
45. C31×#431C22semidirect
47. C31×#423C22semidirect
49. C31×#332C22semidirect
51. C31×Q8⋊C8C22semidirect
52. C31×Q8⋊C8C22semidirect
231. #18835C31direct (N × H)
289. #3238C2×C31semidirect
290. #3206C2×C31semidirect
291. #3079C2×C31semidirect
292. #1638C2×C31semidirect
293. #1637C2×C31semidirect
294. #1630C2×C31semidirect
295. #1629C2×C31semidirect
296. #1557C2×C31semidirect
297. #1495C2×C31semidirect
298. #1488C2×C31semidirect
340. C2×C4⋊Q8C22×C31semidirect
349. #487C22×C31semidirect
352. #477C22×C31semidirect
353. #477C22×C31semidirect
360. #432C22×C31semidirect
361. #431C22×C31semidirect
363. #423C22×C31semidirect
365. #332C22×C31semidirect
367. Q8⋊C8C22×C31semidirect
368. Q8⋊C8C22×C31semidirect
469. C31#18835direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35142
C31×#32382
C31×#32062
C31×#30792
C31×#16382
C31×#16372
C31×#16302
C31×#16292
C31×#15572
C31×#14952
C31×#14882
#1883531