Finite Group Databasebeta

← C31×#18803 #21475562

Subgroups — C31×#18803 #21475562

Order 7936.

958 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×#3480C211yesmaximal, characteristic
2C31×#3227C211yesmaximal, characteristic, complemented
3C31×#3079C211yesmaximal, characteristic, complemented
4C2×C31×#462C211yesmaximal, characteristic, complemented
5C31×#1653C212yesmaximal, complemented
6C31×#1652C212yesmaximal, complemented
7C31×#1641C212yesmaximal, complemented
8C31×#1640C212yesmaximal, complemented
9C31×#1547C211yesmaximal, characteristic, complemented
10C31×#1487C211yesmaximal, characteristic, complemented
11C31×#1477C211yesmaximal, characteristic, complemented
12C22×Q16×C31—21no—
13C31×#559—21no—
14C31×#559C2211yescharacteristic
15C31×#557C2211yescharacteristic
16C31×#556—21no—
17C31×#556C2211yescharacteristic
18C31×#553—22no—
19C31×#553—21no—
20C31×#550C2211yescharacteristic
21C31×#548—21no—
22C31×#532—21no—
23C31×#532C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#529—21no—
26C31×#529—22no—
27C4×#41×C31—21no—
28C4×#41×C31—21no—
29C4×#41×C31C2211yescharacteristic
30C2×C4×Q8×C31C2211yescharacteristic
31C31×#498C2212yes—
32C31×#486C2212yes—
33C31×#481—22no—
34C31×#477—21no—
35C31×#476—21no—
36C31×#476—22no—
37C31×#475C2212yes—
38C31×#474C2212yes—
39C31×#462—21no—
40C31×#462—21no—
41C31×#462C2212yes—
42C31×#462C2212yes—
43C31×#431C2211yescharacteristic
44C31×#428—21no—
45C31×#423C2211yescharacteristic
46C2×C31×C4⋊C8C2211yescharacteristic
47C31×#417—21no—
48C2×C31×#103—21no—
49C31×#332C2212yes—
50C31×#331C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×Q8⋊C8C2212yes—
53C31×Q8⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C2×#41×C31D411yescharacteristic
58C22×Q8×C31—21no—
59C22×Q8×C31D411yescharacteristic
60C2×Q16×C31—22no—
61C2×Q16×C31—22no—
62C2×Q16×C31—44no—
63C2×M16×C31—21no—
64C22×C8×C31—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8C2312yes—
70C31×C4⋊Q8C2311yescharacteristic
71C31×#127—21no—
72C31×#127C2311yescharacteristic
73C31×#126—22no—
74C31×#126—22no—
75C31×#126—22no—
76C31×#125—22no—
77C31×#125—21no—
78C31×#124—22no—
79C31×#124—21no—
80C31×#124—21no—
81C31×#124—21no—
82C31×#124—21no—
83C31×#123—22no—
84C31×#123—22no—
85C31×#123—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—21no—
88C31×#121—22no—
89C31×#121—22no—
90C31×#121—22no—
91C31×#121—21no—
92C31×#121—21no—
93C31×#121D412yes—
94C4×Q8×C31—21no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31D411yescharacteristic
100C4×Q8×C31D411yescharacteristic
101C4×Q8×C31D411yescharacteristic
102C4×Q8×C31C2311yescharacteristic
103C4×Q8×C31C2311yescharacteristic
104C4×Q8×C31C2311yescharacteristic
105C4×D4×C31—22no—
106C4×D4×C31—22no—
107C4×D4×C31—22no—
108C4×D4×C31—22no—
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31—21no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31—21no—
116C4×D4×C31D411yescharacteristic
117C4×D4×C31D412yes—
118C4×D4×C31C2311yescharacteristic
119C31×#117—22no—
120C31×#117—22no—
121C31×#117—21no—
122C31×#117—21no—
123C31×#117D411yescharacteristic
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31—21no—
126C2×C4⋊C4×C31—21no—
127C2×C4⋊C4×C31D411yescharacteristic
128C2×C42×C31—22no—
129C2×C42×C31—21no—
130C2×C42×C31—21no—
131C2×C42×C31C2311yescharacteristic
132C31×#107—22no—
133C31×#106—22no—
134C31×#106—24no—
135C31×C4⋊C8—21no—
136C31×C4⋊C8—21no—
137C31×C4⋊C8—21no—
138C31×C4⋊C8C2312yes—
139C31×C4⋊C8C2312yes—
140C31×#103—22no—
141C31×#103—22no—
142C31×#103—22no—
143C31×#102—22no—
144C31×#102—24no—
145C31×C22⋊C8—21no—
146C4×C8×C31C2312yes—
147#41×C31—22no—
148#41×C31—22no—
149#41×C31—42no—
150#41×C31—42no—
151#41×C31—42no—
152#41×C31—42no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—22no—
156C2×Q8×C31—22no—
157C2×Q8×C31—41no—
158C2×Q8×C31—42no—
159C2×Q8×C31—42no—
160C2×Q8×C31—41no—
161C2×Q8×C31C2×D411yescharacteristic
162C2×Q8×C31C2×D411yescharacteristic
163C2×Q8×C31C2×D411yescharacteristic
164C2×D4×C31—22no—
165C2×D4×C31—21no—
166C2×D4×C31—22no—
167C2×D4×C31—22no—
168C2×D4×C31—21no—
169C2×D4×C31—21no—
170C2×D4×C31C2×D411yescharacteristic
171C2×D4×C31C2×D412yes—
172C22×C4×C31—22no—
173C22×C4×C31—21no—
174C22×C4×C31—22no—
175C22×C4×C31—21no—
176C22×C4×C31—21no—
177C22×C4×C31—22no—
178C22×C4×C31—21no—
179C22×C4×C31—21no—
180C22×C4×C31—41no—
181C22×C4×C31C2×D411yescharacteristic
182C22×C4×C31C2×D411yescharacteristic
183C22×C4×C31C2×D411yescharacteristic
184Q16×C31—44no—
185Q16×C31—44no—
186M16×C31—42no—
187C2×C8×C31—22no—
188C2×C8×C31—22no—
189C2×C8×C31—22no—
190C2×C8×C31—42no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—22no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—22no—
204C4⋊C4×C31—22no—
205C4⋊C4×C31—41no—
206C4⋊C4×C31—41no—
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D411yescharacteristic
209C4⋊C4×C31C2×D411yescharacteristic
210C4⋊C4×C31C2×D412yes—
211C4⋊C4×C31C2×D411yescharacteristic
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—22no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—21no—
223C42×C31—22no—
224C42×C31—22no—
225C42×C31—21no—
226C42×C31—21no—
227C42×C31—21no—
228C42×C31—21no—
229C42×C31—21no—
230C42×C31—21no—
231C42×C31—21no—
232C42×C31C2×D411yescharacteristic
233C42×C31C2×D411yescharacteristic
234C42×C31C2×D411yescharacteristic
235C42×C31C2411yescharacteristic
236#18803C3111yesmaximal, characteristic, direct product factor, Sylow
237C23×C31—21no—
238C23×C31—22no—
239C23×C31#12011yescharacteristic
240Q8×C31—21no—
241Q8×C31—21no—
242Q8×C31—21no—
243Q8×C31—21no—
244Q8×C31—21no—
245Q8×C31—42no—
246Q8×C31—41no—
247Q8×C31—41no—
248Q8×C31—42no—
249Q8×C31—42no—
250Q8×C31—82no—
251Q8×C31AGL(1,ℤ/8)12yes—
252D4×C31—21no—
253D4×C31—21no—
254D4×C31—42no—
255D4×C31—44no—
256D4×C31—42no—
257D4×C31—44no—
258D4×C31—41no—
259D4×C31—41no—
260D4×C31—41no—
261D4×C31—41no—
262D4×C31—82no—
263C2×C4×C31—21no—
264C2×C4×C31—21no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—21no—
268C2×C4×C31—21no—
269C2×C4×C31—21no—
270C2×C4×C31—21no—
271C2×C4×C31—22no—
272C2×C4×C31—21no—
273C2×C4×C31—41no—
274C2×C4×C31—42no—
275C2×C4×C31—44no—
276C2×C4×C31—41no—
277C2×C4×C31—41no—
278C2×C4×C31—42no—
279C2×C4×C31—44no—
280C2×C4×C31—42no—
281C2×C4×C31—81no—
282C2×C4×C31—81no—
283C2×C4×C31—82no—
284C2×C4×C31—82no—
285C2×C4×C31#12011yescharacteristic
286C2×C4×C31#12011yescharacteristic
287C2×C4×C31#12011yescharacteristic
288C2×C4×C31#13512yes—
289C2×C4×C31#13712yes—
290C2×C4×C31C22×D411yescharacteristic
291C2×C4×C31C22×D411yescharacteristic
292C2×C4×C31C22×D411yescharacteristic
293C8×C31—42no—
294C8×C31—42no—
295C8×C31—42no—
296#3480C2×C3111yescharacteristic
297#3227C2×C3111yescharacteristic, complemented
298#3079C2×C3111yescharacteristic, complemented
299C2×#462C2×C3111yescharacteristic, complemented
300#1653C2×C3112yescomplemented
301#1652C2×C3112yescomplemented
302#1641C2×C3112yescomplemented
303#1640C2×C3112yescomplemented
304#1547C2×C3111yescharacteristic, complemented
305#1487C2×C3111yescharacteristic, complemented
306#1477C2×C3111yescharacteristic, complemented
307C22×C31—21no—
308C22×C31—21no—
309C22×C31—41no—
310C22×C31—42no—
311C22×C31—42no—
312C22×C31—41no—
313C22×C31—42no—
314C22×C31—41no—
315C22×C31#45212yes—
316C22×C31C2×#12011yescharacteristic
317C4×C31—21no—
318C4×C31—22no—
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—41no—
324C4×C31—41no—
325C4×C31—41no—
326C4×C31—41no—
327C4×C31—41no—
328C4×C31—82no—
329C4×C31#45212yes—
330C4×C31C2×#13511yescharacteristic
331C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
332C4×C31C2×#13711yescharacteristic
333C4×C31#57511yescharacteristic
334C22×Q16—21no—
335#559—21no—
336#559C22×C3111yescharacteristic
337#557C22×C3111yescharacteristic
338#556—21no—
339#556C22×C3111yescharacteristic
340#553—22no—
341#553—21no—
342#550C22×C3111yescharacteristic
343#548—21no—
344#532—21no—
345#532C22×C3111yescharacteristic
346C2×C4⋊Q8C22×C3111yescharacteristic
347#529—21no—
348#529—22no—
349C4×#41—21no—
350C4×#41—21no—
351C4×#41C22×C3111yescharacteristic
352C2×C4×Q8C22×C3111yescharacteristic
353#498C22×C3112yes—
354#486C22×C3112yes—
355#481—22no—
356#477—21no—
357#476—21no—
358#476—22no—
359#475C22×C3112yes—
360#474C22×C3112yes—
361#462—21no—
362#462—21no—
363#462C22×C3112yes—
364#462C22×C3112yes—
365#431C22×C3111yescharacteristic
366#428—21no—
367#423C22×C3111yescharacteristic
368C2×C4⋊C8C22×C3111yescharacteristic
369#417—21no—
370C2×#103—21no—
371#332C22×C3112yes—
372#331C22×C3112yes—
373Q8⋊C8C22×C3112yes—
374Q8⋊C8C22×C3112yes—
375Q8⋊C8C22×C3112yes—
376D4⋊C8C22×C3112yes—
377C2×C31—22no—
378C2×C31—81no—
379C2×C31—82no—
380C2×C31C2×#45211yescharacteristic
381C2×C31#300611yescharacteristic
382C2×C31#302211yescharacteristic
383C2×#41—21no—
384C2×#41—21no—
385C2×#41D4×C3111yescharacteristic
386C22×Q8—21no—
387C22×Q8D4×C3111yescharacteristic
388C2×Q16—22no—
389C2×Q16—22no—
390C2×Q16—44no—
391C2×M16—21no—
392C22×C8—21no—
393C4⋊Q8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8C23×C3112yes—
398C4⋊Q8C23×C3111yescharacteristic
399#127—21no—
400#127C23×C3111yescharacteristic
401#126—22no—
402#126—22no—
403#126—22no—
404#125—22no—
405#125—21no—
406#124—22no—
407#124—21no—
408#124—21no—
409#124—21no—
410#124—21no—
411#123—22no—
412#123—22no—
413#123—22no—
414C22⋊Q8—22no—
415C22⋊Q8—21no—
416#121—22no—
417#121—22no—
418#121—22no—
419#121—21no—
420#121—21no—
421#121D4×C3112yes—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8D4×C3111yescharacteristic
428C4×Q8D4×C3111yescharacteristic
429C4×Q8D4×C3111yescharacteristic
430C4×Q8C23×C3111yescharacteristic
431C4×Q8C23×C3111yescharacteristic
432C4×Q8C23×C3111yescharacteristic
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—22no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4D4×C3111yescharacteristic
445C4×D4D4×C3112yes—
446C4×D4C23×C3111yescharacteristic
447#117—22no—
448#117—22no—
449#117—21no—
450#117—21no—
451#117D4×C3111yescharacteristic
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4D4×C3111yescharacteristic
456C2×C42—22no—
457C2×C42—21no—
458C2×C42—21no—
459C2×C42C23×C3111yescharacteristic
460#107—22no—
461#106—22no—
462#106—24no—
463C4⋊C8—21no—
464C4⋊C8—21no—
465C4⋊C8—21no—
466C4⋊C8C23×C3112yes—
467C4⋊C8C23×C3112yes—
468#103—22no—
469#103—22no—
470#103—22no—
471#102—22no—
472#102—24no—
473C22⋊C8—21no—
474C4×C8C23×C3112yes—
475C31#1880311yescharacteristic, direct product factor, Sylow
476#41—22no—
477#41—22no—
478#41—42no—
479#41—42no—
480#41—42no—
481#41—42no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—41no—
487C2×Q8—42no—
488C2×Q8—42no—
489C2×Q8—41no—
490C2×Q8C2×D4×C3111yescharacteristic
491C2×Q8C2×D4×C3111yescharacteristic
492C2×Q8C2×D4×C3111yescharacteristic
493C2×D4—22no—
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4C2×D4×C3111yescharacteristic
500C2×D4C2×D4×C3112yes—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—41no—
510C22×C4C2×D4×C3111yescharacteristic
511C22×C4C2×D4×C3111yescharacteristic
512C22×C4C2×D4×C3111yescharacteristic
513Q16—44no—
514Q16—44no—
515M16—42no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—42no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4C2×D4×C3111yescharacteristic
537C4⋊C4C2×D4×C3111yescharacteristic
538C4⋊C4C2×D4×C3111yescharacteristic
539C4⋊C4C2×D4×C3112yes—
540C4⋊C4C2×D4×C3111yescharacteristic
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C42—22no—
553C42—22no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42C2×D4×C3111yescharacteristic
562C42C2×D4×C3111yescharacteristic
563C42C2×D4×C3111yescharacteristic
564C42C24×C3111yescharacteristic
565C23—21no—
566C23—22no—
567C23C31×#12011yescharacteristic
568Q8—21no—
569Q8—21no—
570Q8—21no—
571Q8—21no—
572Q8—21no—
573Q8—42no—
574Q8—41no—
575Q8—41no—
576Q8—42no—
577Q8—42no—
578Q8—82no—
579Q8C31×AGL(1,ℤ/8)12yes—
580D4—21no—
581D4—21no—
582D4—42no—
583D4—44no—
584D4—42no—
585D4—44no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—82no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4C31×#12011yescharacteristic
614C2×C4C31×#12011yescharacteristic
615C2×C4C31×#12011yescharacteristic
616C2×C4C31×#13512yes—
617C2×C4C31×#13712yes—
618C2×C4C22×D4×C3111yescharacteristic
619C2×C4C22×D4×C3111yescharacteristic
620C2×C4C22×D4×C3111yescharacteristic
621C8—42no—
622C8—42no—
623C8—42no—
624C22—21no—
625C22—21no—
626C22—41no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—42no—
631C22—41no—
632C22C31×#45212yes—
633C22C2×C31×#12011yescharacteristic
634C4—21no—
635C4—22no—
636C4—21no—
637C4—21no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—82no—
646C4C31×#45212yes—
647C4C2×C31×#13511yescharacteristic
648C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
649C4C2×C31×#13711yescharacteristic
650C4C31×#57511yescharacteristic
651C2—22no—
652C2—81no—
653C2—82no—
654C2C2×C31×#45211yescharacteristic
655C2C31×#300611yescharacteristic
656C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3480C2
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×#3227C2semidirect
3. C31×#3079C2semidirect
4. C2×C31×#462C2semidirect
5. C31×#1653C2semidirect
6. C31×#1652C2semidirect
7. C31×#1641C2semidirect
8. C31×#1640C2semidirect
9. C31×#1547C2semidirect
10. C31×#1487C2semidirect
11. C31×#1477C2semidirect
236. #18803C31direct (N × H)
297. #3227C2×C31semidirect
298. #3079C2×C31semidirect
299. C2×#462C2×C31semidirect
300. #1653C2×C31semidirect
301. #1652C2×C31semidirect
302. #1641C2×C31semidirect
303. #1640C2×C31semidirect
304. #1547C2×C31semidirect
305. #1487C2×C31semidirect
306. #1477C2×C31semidirect
475. C31#18803direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34802
C31×#32272
C31×#30792
C2×C31×#4622
C31×#16532
C31×#16522
C31×#16412
C31×#16402
C31×#15472
C31×#14872
C31×#14772
#1880331