Finite Group Databasebeta

← C31×#18836 #21475595

Subgroups — C31×#18836 #21475595

Order 7936.

972 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3515C211yesmaximal, characteristic, complemented
2C31×#3217C211yesmaximal, characteristic, complemented
3C31×#3208C211yesmaximal, characteristic, complemented
4C31×#3038C211yesmaximal, characteristic, complemented
5C31×#1626C212yesmaximal, complemented
6C31×#1625C212yesmaximal, complemented
7C31×#1622C212yesmaximal, complemented
8C31×#1622C212yesmaximal, complemented
9C31×#1559C211yesmaximal, characteristic, complemented
10C31×#1495C211yesmaximal, characteristic, complemented
11C31×#1488C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)—21no—
13C31×#557C2211yescharacteristic, complemented
14C31×#554C2211yescharacteristic, complemented
15C31×#554C2211yescharacteristic, complemented
16C31×#553—21no—
17C31×#552—21no—
18C31×#552—22no—
19C31×#552—21no—
20C31×#547—21no—
21C31×#547C2211yescharacteristic, complemented
22C31×#541—21no—
23C31×#539—22no—
24C31×#533C2211yescharacteristic
25C2×C31×#127C2211yescharacteristic
26C31×#529—21no—
27C31×#529C2211yescharacteristic
28C2×C31×#125—21no—
29C31×#520—21no—
30C4×#41×C31C2211yescharacteristic
31C31×#496—21no—
32C31×#492C2212yes—
33C31×#490—21no—
34C31×#486C2212yes—
35C31×#460—22no—
36C31×#460—21no—
37C31×#460C2212yes—
38C31×#460C2212yes—
39C31×#459—21no—
40C31×#459C2212yes—
41C31×#459C2212yes—
42C31×#450—22no—
43C31×#450—21no—
44C31×#447—21no—
45C31×#432C2211yescharacteristic
46C31×#431C2211yescharacteristic
47C31×#423C2211yescharacteristic
48C31×#418—21no—
49C31×#333C2212yescomplemented
50C31×#332C2212yescomplemented
51C31×Q8⋊C8C2212yescomplemented
52C31×Q8⋊C8C2212yescomplemented
53C31×D4⋊C8C2212yescomplemented
54C31×D4⋊C8C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31D411yescharacteristic
57C22×D4×C31—21no—
58C22×D4×C31—21no—
59C22×D4×C31—21no—
60C31×AGL(1,ℤ/8)—44no—
61C2×SD16×C31—22no—
62C2×SD16×C31—24no—
63C2×SD16×C31—22no—
64C2×D8×C31—22no—
65C2×D8×C31—22no—
66C2×M16×C31—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8C2311yescharacteristic
69C31×C4⋊Q8C2311yescharacteristic
70C31×#127—21no—
71C31×#127—21no—
72C31×#127—21no—
73C31×#127C2312yes—
74C31×#126—22no—
75C31×#126—22no—
76C31×#126—22no—
77C31×#126—21no—
78C31×#125—22no—
79C31×#125—22no—
80C31×#125—22no—
81C31×#125—21no—
82C31×#125—21no—
83C31×#125—22no—
84C31×#125—21no—
85C31×#125—21no—
86C31×#125D411yescharacteristic
87C31×#124D411yescharacteristic
88C31×#123—22no—
89C31×#123—22no—
90C31×#123—22no—
91C31×#123—21no—
92C31×#123—22no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8—22no—
95C31×C22⋊Q8—22no—
96C31×C22⋊Q8—21no—
97C31×C22⋊Q8—21no—
98C31×C22⋊Q8—21no—
99C31×C22⋊Q8D411yescharacteristic
100C31×C22⋊Q8D412yes—
101C31×#121—22no—
102C31×#121—21no—
103C31×#121D411yescharacteristic
104C4×Q8×C31—21no—
105C4×Q8×C31—21no—
106C4×Q8×C31—21no—
107C4×Q8×C31D411yescharacteristic
108C4×Q8×C31D412yes—
109C4×Q8×C31C2311yescharacteristic
110C4×Q8×C31C2311yescharacteristic
111C4×D4×C31—22no—
112C4×D4×C31—22no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31—21no—
116C4×D4×C31—21no—
117C4×D4×C31—21no—
118C4×D4×C31D411yescharacteristic
119C4×D4×C31C2311yescharacteristic
120C4×D4×C31C2311yescharacteristic
121C31×#117—22no—
122C31×#117—22no—
123C31×#117—21no—
124C31×#117—21no—
125C31×#117—21no—
126C31×#117D411yescharacteristic
127C2×C4⋊C4×C31—22no—
128C2×C4⋊C4×C31—22no—
129C2×C4⋊C4×C31—21no—
130C2×C4⋊C4×C31—21no—
131C2×C4⋊C4×C31—21no—
132C2×C42×C31—21no—
133C2×C42×C31C2311yescharacteristic
134C31×C4⋊C8—21no—
135C31×C4⋊C8—21no—
136C31×C4⋊C8C2312yes—
137C31×C4⋊C8C2312yes—
138C31×#102—22no—
139C31×#102—22no—
140C31×#102—22no—
141C31×#102—22no—
142C31×#102—24no—
143C31×C22⋊C8—21no—
144C31×C22⋊C8—21no—
145C31×#97—21no—
146C4×C8×C31C2312yes—
147C24×C31—41no—
148#41×C31—22no—
149#41×C31—41no—
150#41×C31—42no—
151#41×C31—42no—
152C2×Q8×C31—21no—
153C2×Q8×C31C2×D411yescharacteristic
154C2×Q8×C31C2×D411yescharacteristic
155C2×Q8×C31C2×D412yes—
156C2×D4×C31—22no—
157C2×D4×C31—21no—
158C2×D4×C31—21no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—42no—
163C2×D4×C31—42no—
164C2×D4×C31—41no—
165C2×D4×C31—41no—
166C2×D4×C31—41no—
167C2×D4×C31—82no—
168C2×D4×C31—82no—
169C2×D4×C31C2×D411yescharacteristic
170C2×D4×C31C2×D411yescharacteristic
171C22×C4×C31—22no—
172C22×C4×C31—22no—
173C22×C4×C31—22no—
174C22×C4×C31—21no—
175C22×C4×C31—21no—
176C22×C4×C31—21no—
177C22×C4×C31—21no—
178C22×C4×C31C2×D411yescharacteristic
179C22×C4×C31C2×D411yescharacteristic
180C22×C4×C31C2×D411yescharacteristic
181SD16×C31—44no—
182SD16×C31—82no—
183SD16×C31—84no—
184D8×C31—44no—
185D8×C31—82no—
186M16×C31—42no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31—21no—
207C4⋊C4×C31—21no—
208C4⋊C4×C31—21no—
209C4⋊C4×C31C2×D411yescharacteristic
210C4⋊C4×C31C2×D411yescharacteristic
211C4⋊C4×C31C2×D411yescharacteristic
212C4⋊C4×C31C2×D411yescharacteristic
213C4⋊C4×C31C2×D412yes—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—21no—
224C22⋊C4×C31—21no—
225C22⋊C4×C31—21no—
226C42×C31—22no—
227C42×C31—21no—
228C42×C31—21no—
229C42×C31—22no—
230C42×C31—21no—
231C42×C31—21no—
232C42×C31C2×D411yescharacteristic
233C42×C31C2×D411yescharacteristic
234C42×C31C2×D411yescharacteristic
235C42×C31C2411yescharacteristic
236#18836C3111yesmaximal, characteristic, direct product factor, Sylow
237C23×C31—21no—
238C23×C31—21no—
239C23×C31—42no—
240C23×C31—82no—
241C23×C31—82no—
242C23×C31—82no—
243C23×C31#12011yescharacteristic
244Q8×C31—21no—
245Q8×C31—21no—
246Q8×C31—22no—
247Q8×C31—42no—
248Q8×C31—41no—
249Q8×C31—41no—
250D4×C31—21no—
251D4×C31—22no—
252D4×C31—44no—
253D4×C31—42no—
254D4×C31—44no—
255D4×C31—42no—
256D4×C31—42no—
257D4×C31—42no—
258D4×C31—42no—
259D4×C31—82no—
260D4×C31—82no—
261D4×C31—82no—
262D4×C31—81no—
263D4×C31AGL(1,ℤ/8)12yes—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—22no—
267C2×C4×C31—21no—
268C2×C4×C31—21no—
269C2×C4×C31—21no—
270C2×C4×C31—21no—
271C2×C4×C31—22no—
272C2×C4×C31—22no—
273C2×C4×C31—21no—
274C2×C4×C31—21no—
275C2×C4×C31—41no—
276C2×C4×C31—42no—
277C2×C4×C31—44no—
278C2×C4×C31—41no—
279C2×C4×C31—41no—
280C2×C4×C31—42no—
281C2×C4×C31—81no—
282C2×C4×C31—82no—
283C2×C4×C31—82no—
284C2×C4×C31#12011yescharacteristic
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31#12011yescharacteristic
287C2×C4×C31C22×D411yescharacteristic
288C2×C4×C31C22×D411yescharacteristic
289C2×C4×C31C22×D411yescharacteristic
290C8×C31—42no—
291C8×C31—42no—
292C8×C31—42no—
293#3515C2×C3111yescharacteristic, complemented
294#3217C2×C3111yescharacteristic, complemented
295#3208C2×C3111yescharacteristic, complemented
296#3038C2×C3111yescharacteristic, complemented
297#1626C2×C3112yescomplemented
298#1625C2×C3112yescomplemented
299#1622C2×C3112yescomplemented
300#1622C2×C3112yescomplemented
301#1559C2×C3111yescharacteristic, complemented
302#1495C2×C3111yescharacteristic, complemented
303#1488C2×C3111yescharacteristic, complemented
304C22×C31—21no—
305C22×C31—22no—
306C22×C31—21no—
307C22×C31—21no—
308C22×C31—41no—
309C22×C31—41no—
310C22×C31—41no—
311C22×C31—41no—
312C22×C31—41no—
313C22×C31—82no—
314C22×C31—82no—
315C22×C31—82no—
316C22×C31—164no—
317C22×C31C2×#12011yescharacteristic
318C4×C31—21no—
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—21no—
322C4×C31—41no—
323C4×C31—41no—
324C4×C31—42no—
325C4×C31—41no—
326C4×C31—41no—
327C4×C31—41no—
328C4×C31—41no—
329C4×C31—42no—
330C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
331C4×C31#57511yescharacteristic
332C4×C31#57712yes—
333C2×AGL(1,ℤ/8)—21no—
334#557C22×C3111yescharacteristic, complemented
335#554C22×C3111yescharacteristic, complemented
336#554C22×C3111yescharacteristic, complemented
337#553—21no—
338#552—21no—
339#552—22no—
340#552—21no—
341#547—21no—
342#547C22×C3111yescharacteristic, complemented
343#541—21no—
344#539—22no—
345#533C22×C3111yescharacteristic
346C2×#127C22×C3111yescharacteristic
347#529—21no—
348#529C22×C3111yescharacteristic
349C2×#125—21no—
350#520—21no—
351C4×#41C22×C3111yescharacteristic
352#496—21no—
353#492C22×C3112yes—
354#490—21no—
355#486C22×C3112yes—
356#460—22no—
357#460—21no—
358#460C22×C3112yes—
359#460C22×C3112yes—
360#459—21no—
361#459C22×C3112yes—
362#459C22×C3112yes—
363#450—22no—
364#450—21no—
365#447—21no—
366#432C22×C3111yescharacteristic
367#431C22×C3111yescharacteristic
368#423C22×C3111yescharacteristic
369#418—21no—
370#333C22×C3112yescomplemented
371#332C22×C3112yescomplemented
372Q8⋊C8C22×C3112yescomplemented
373Q8⋊C8C22×C3112yescomplemented
374D4⋊C8C22×C3112yescomplemented
375D4⋊C8C22×C3112yescomplemented
376C2×C31—41no—
377C2×C31—42no—
378C2×C31—81no—
379C2×C31—162no—
380C2×C31#300611yescharacteristic
381C2×C31#301011yescharacteristic
382C2×C31#302111yescharacteristic
383C2×#41—21no—
384C2×#41D4×C3111yescharacteristic
385C22×D4—21no—
386C22×D4—21no—
387C22×D4—21no—
388AGL(1,ℤ/8)—44no—
389C2×SD16—22no—
390C2×SD16—24no—
391C2×SD16—22no—
392C2×D8—22no—
393C2×D8—22no—
394C2×M16—21no—
395C4⋊Q8—21no—
396C4⋊Q8C23×C3111yescharacteristic
397C4⋊Q8C23×C3111yescharacteristic
398#127—21no—
399#127—21no—
400#127—21no—
401#127C23×C3112yes—
402#126—22no—
403#126—22no—
404#126—22no—
405#126—21no—
406#125—22no—
407#125—22no—
408#125—22no—
409#125—21no—
410#125—21no—
411#125—22no—
412#125—21no—
413#125—21no—
414#125D4×C3111yescharacteristic
415#124D4×C3111yescharacteristic
416#123—22no—
417#123—22no—
418#123—22no—
419#123—21no—
420#123—22no—
421C22⋊Q8—21no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—21no—
425C22⋊Q8—21no—
426C22⋊Q8—21no—
427C22⋊Q8D4×C3111yescharacteristic
428C22⋊Q8D4×C3112yes—
429#121—22no—
430#121—21no—
431#121D4×C3111yescharacteristic
432C4×Q8—21no—
433C4×Q8—21no—
434C4×Q8—21no—
435C4×Q8D4×C3111yescharacteristic
436C4×Q8D4×C3112yes—
437C4×Q8C23×C3111yescharacteristic
438C4×Q8C23×C3111yescharacteristic
439C4×D4—22no—
440C4×D4—22no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4—21no—
446C4×D4D4×C3111yescharacteristic
447C4×D4C23×C3111yescharacteristic
448C4×D4C23×C3111yescharacteristic
449#117—22no—
450#117—22no—
451#117—21no—
452#117—21no—
453#117—21no—
454#117D4×C3111yescharacteristic
455C2×C4⋊C4—22no—
456C2×C4⋊C4—22no—
457C2×C4⋊C4—21no—
458C2×C4⋊C4—21no—
459C2×C4⋊C4—21no—
460C2×C42—21no—
461C2×C42C23×C3111yescharacteristic
462C4⋊C8—21no—
463C4⋊C8—21no—
464C4⋊C8C23×C3112yes—
465C4⋊C8C23×C3112yes—
466#102—22no—
467#102—22no—
468#102—22no—
469#102—22no—
470#102—24no—
471C22⋊C8—21no—
472C22⋊C8—21no—
473#97—21no—
474C4×C8C23×C3112yes—
475C31#1883611yescharacteristic, direct product factor, Sylow
476C24—41no—
477#41—22no—
478#41—41no—
479#41—42no—
480#41—42no—
481C2×Q8—21no—
482C2×Q8C2×D4×C3111yescharacteristic
483C2×Q8C2×D4×C3111yescharacteristic
484C2×Q8C2×D4×C3112yes—
485C2×D4—22no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—41no—
494C2×D4—41no—
495C2×D4—41no—
496C2×D4—82no—
497C2×D4—82no—
498C2×D4C2×D4×C3111yescharacteristic
499C2×D4C2×D4×C3111yescharacteristic
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4C2×D4×C3111yescharacteristic
508C22×C4C2×D4×C3111yescharacteristic
509C22×C4C2×D4×C3111yescharacteristic
510SD16—44no—
511SD16—82no—
512SD16—84no—
513D8—44no—
514D8—82no—
515M16—42no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4C2×D4×C3111yescharacteristic
539C4⋊C4C2×D4×C3111yescharacteristic
540C4⋊C4C2×D4×C3111yescharacteristic
541C4⋊C4C2×D4×C3111yescharacteristic
542C4⋊C4C2×D4×C3112yes—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42C2×D4×C3111yescharacteristic
562C42C2×D4×C3111yescharacteristic
563C42C2×D4×C3111yescharacteristic
564C42C24×C3111yescharacteristic
565C23—21no—
566C23—21no—
567C23—42no—
568C23—82no—
569C23—82no—
570C23—82no—
571C23C31×#12011yescharacteristic
572Q8—21no—
573Q8—21no—
574Q8—22no—
575Q8—42no—
576Q8—41no—
577Q8—41no—
578D4—21no—
579D4—22no—
580D4—44no—
581D4—42no—
582D4—44no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—82no—
588D4—82no—
589D4—82no—
590D4—81no—
591D4C31×AGL(1,ℤ/8)12yes—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—81no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4C31×#12011yescharacteristic
613C2×C4C31×#12011yescharacteristic
614C2×C4C31×#12011yescharacteristic
615C2×C4C22×D4×C3111yescharacteristic
616C2×C4C22×D4×C3111yescharacteristic
617C2×C4C22×D4×C3111yescharacteristic
618C8—42no—
619C8—42no—
620C8—42no—
621C22—21no—
622C22—22no—
623C22—21no—
624C22—21no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—164no—
634C22C2×C31×#12011yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—42no—
647C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
648C4C31×#57511yescharacteristic
649C4C31×#57712yes—
650C2—41no—
651C2—42no—
652C2—81no—
653C2—162no—
654C2C31×#300611yescharacteristic
655C2C31×#301011yescharacteristic
656C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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
1. C31×#3515C2semidirect
2. C31×#3217C2semidirect
3. C31×#3208C2semidirect
4. C31×#3038C2semidirect
5. C31×#1626C2semidirect
6. C31×#1625C2semidirect
7. C31×#1622C2semidirect
8. C31×#1622C2semidirect
9. C31×#1559C2semidirect
10. C31×#1495C2semidirect
11. C31×#1488C2semidirect
13. C31×#557C22semidirect
14. C31×#554C22semidirect
15. C31×#554C22semidirect
21. C31×#547C22semidirect
49. C31×#333C22semidirect
50. C31×#332C22semidirect
51. C31×Q8⋊C8C22semidirect
52. C31×Q8⋊C8C22semidirect
53. C31×D4⋊C8C22semidirect
54. C31×D4⋊C8C22semidirect
236. #18836C31direct (N × H)
293. #3515C2×C31semidirect
294. #3217C2×C31semidirect
295. #3208C2×C31semidirect
296. #3038C2×C31semidirect
297. #1626C2×C31semidirect
298. #1625C2×C31semidirect
299. #1622C2×C31semidirect
300. #1622C2×C31semidirect
301. #1559C2×C31semidirect
302. #1495C2×C31semidirect
303. #1488C2×C31semidirect
334. #557C22×C31semidirect
335. #554C22×C31semidirect
336. #554C22×C31semidirect
342. #547C22×C31semidirect
370. #333C22×C31semidirect
371. #332C22×C31semidirect
372. Q8⋊C8C22×C31semidirect
373. Q8⋊C8C22×C31semidirect
374. D4⋊C8C22×C31semidirect
375. D4⋊C8C22×C31semidirect
475. C31#18836direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35152
C31×#32172
C31×#32082
C31×#30382
C31×#16262
C31×#16252
C31×#16222
C31×#15592
C31×#14952
C31×#14882
#1883631