Finite Group Databasebeta

← C31×#18457 #21475216

Subgroups — C31×#18457 #21475216

Order 7936.

954 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3536C211yesmaximal, characteristic
2C31×#3196C211yesmaximal, characteristic, complemented
3C31×#3187C211yesmaximal, characteristic, complemented
4C31×#2817C211yesmaximal, characteristic, complemented
5C31×#1616C212yesmaximal, complemented
6C31×#1615C212yesmaximal, complemented
7C31×#1604C212yesmaximal, complemented
8C31×#1603C212yesmaximal, complemented
9C31×#1516C211yesmaximal, characteristic, complemented
10C31×#1510C211yesmaximal, characteristic, complemented
11C31×#1472C211yesmaximal, characteristic, complemented
12C31×#562—22no—
13C31×#554—21no—
14C31×#548—21no—
15C31×#547—21no—
16C31×#546—21no—
17C31×#546—22no—
18C31×#543—21no—
19C31×#542—21no—
20C31×#539C2212yes—
21C31×#538—21no—
22C31×#537—21no—
23C31×#537C2212yes—
24C31×#525C2211yescharacteristic
25C2×C31×C22⋊Q8C2211yescharacteristic
26C2×C31×C22⋊Q8C2211yescharacteristic
27C2×C31×#121C2211yescharacteristic
28C31×#485—21no—
29C31×#484C2211yescharacteristic, complemented
30C31×#483—21no—
31C31×#481C2212yes—
32C31×#481C2211yescharacteristic, complemented
33C31×#470—21no—
34C31×#468—21no—
35C31×#465—22no—
36C31×#452—21no—
37C31×#451C2211yescharacteristic, complemented
38C31×#450—21no—
39C31×#448C2212yes—
40C31×#448C2211yescharacteristic, complemented
41C31×#407C2211yescharacteristic
42C2×C31×C22⋊C8C2211yescharacteristic
43C31×#400—21no—
44C31×#400C2212yescomplemented
45C31×#397—22no—
46C31×#395—21no—
47C31×#395C2212yescomplemented
48C31×#381C2211yescharacteristic
49C31×#328C2212yescomplemented
50C31×#327C2212yes—
51C31×#327C2212yes—
52C31×#327C2212yescomplemented
53C31×#324C2212yescomplemented
54C31×#324C2212yescomplemented
55C2×#41×C31—21no—
56C2×#41×C31—22no—
57C2×#41×C31D411yescharacteristic
58C22×Q8×C31D411yescharacteristic
59C22×Q8×C31C2311yescharacteristic
60C22×D4×C31—21no—
61C22×D4×C31D411yescharacteristic
62C23×C4×C31C2311yescharacteristic
63C2×Q16×C31—21no—
64C2×SD16×C31—22no—
65C2×SD16×C31—21no—
66C2×M16×C31—21no—
67C22×C8×C31—21no—
68C31×C4⋊Q8—21no—
69C31×#126—22no—
70C31×#125—22no—
71C31×#124—22no—
72C31×#124—21no—
73C31×#124—22no—
74C31×#123—22no—
75C31×#123—22no—
76C31×#123—22no—
77C31×#123—21no—
78C31×#123—22no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—21no—
93C31×C22⋊Q8—21no—
94C31×C22⋊Q8D411yescharacteristic
95C31×C22⋊Q8C2311yescharacteristic
96C31×#121—21no—
97C31×#121—22no—
98C31×#121—22no—
99C31×#121—22no—
100C31×#121—22no—
101C31×#121—21no—
102C31×#121—21no—
103C31×#121—22no—
104C31×#121D412yes—
105C31×#121D411yescharacteristic
106C31×#121C2312yes—
107C31×#121C2311yescharacteristic
108C31×#120—22no—
109C31×#120—21no—
110C4×Q8×C31—21no—
111C4×D4×C31—22no—
112C4×D4×C31—21no—
113C4×D4×C31—21no—
114C4×D4×C31—21no—
115C4×D4×C31—22no—
116C31×#117D411yescharacteristic
117C2×C4⋊C4×C31—21no—
118C2×C4⋊C4×C31—21no—
119C2×C4⋊C4×C31—22no—
120C2×C4⋊C4×C31D411yescharacteristic
121C2×C4⋊C4×C31D411yescharacteristic
122C2×C4⋊C4×C31C2311yescharacteristic
123C2×C4⋊C4×C31C2312yes—
124C2×C22⋊C4×C31—22no—
125C2×C22⋊C4×C31—21no—
126C2×C22⋊C4×C31—21no—
127C2×C22⋊C4×C31—21no—
128C2×C22⋊C4×C31D412yes—
129C31×#107—21no—
130C31×#106—22no—
131C31×#106—21no—
132C31×#103—21no—
133C31×#103—22no—
134C31×#103—21no—
135C31×#103—21no—
136C31×#102—22no—
137C31×#102—21no—
138C31×C22⋊C8—21no—
139C31×C22⋊C8—21no—
140C31×C22⋊C8C2312yes—
141C31×C22⋊C8C2311yescharacteristic
142C31×C22⋊C8C2311yescharacteristic
143C31×#95—22no—
144C31×#95—21no—
145C31×#95—22no—
146C31×#95—22no—
147C31×#95C2312yes—
148C24×C31—21no—
149C24×C31C2×D411yescharacteristic
150#41×C31—84no—
151#41×C31—81no—
152#41×C31—82no—
153C2×Q8×C31—22no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—21no—
157C2×Q8×C31—21no—
158C2×Q8×C31—81no—
159C2×Q8×C31—81no—
160C2×Q8×C31C2×D411yescharacteristic
161C2×D4×C31—21no—
162C2×D4×C31—21no—
163C2×D4×C31—22no—
164C2×D4×C31—22no—
165C2×D4×C31—22no—
166C2×D4×C31—21no—
167C2×D4×C31—22no—
168C2×D4×C31—21no—
169C2×D4×C31—81no—
170C2×D4×C31—82no—
171C2×D4×C31C2×D411yescharacteristic
172C2×D4×C31C2×D412yes—
173C22×C4×C31—22no—
174C22×C4×C31—22no—
175C22×C4×C31—22no—
176C22×C4×C31—21no—
177C22×C4×C31—21no—
178C22×C4×C31—22no—
179C22×C4×C31—21no—
180C22×C4×C31—21no—
181C22×C4×C31—81no—
182C22×C4×C31C2×D412yes—
183C22×C4×C31C2×D412yes—
184C22×C4×C31C2×D411yescharacteristic
185C22×C4×C31C2×D411yescharacteristic
186C22×C4×C31C2×D411yescharacteristic
187C22×C4×C31C2×D411yescharacteristic
188C22×C4×C31C2×D411yescharacteristic
189C22×C4×C31C2411yescharacteristic
190Q16×C31—81no—
191SD16×C31—82no—
192SD16×C31—81no—
193M16×C31—81no—
194C2×C8×C31—22no—
195C2×C8×C31—21no—
196C2×C8×C31—21no—
197C2×C8×C31—81no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—22no—
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—22no—
207C4⋊C4×C31—22no—
208C4⋊C4×C31—21no—
209C4⋊C4×C31—21no—
210C4⋊C4×C31—21no—
211C4⋊C4×C31—41no—
212C4⋊C4×C31—41no—
213C4⋊C4×C31—82no—
214C4⋊C4×C31C2×D412yes—
215C4⋊C4×C31C2×D411yescharacteristic
216C4⋊C4×C31C2×D411yescharacteristic
217C22⋊C4×C31—22no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—22no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—22no—
224C22⋊C4×C31—22no—
225C22⋊C4×C31—21no—
226C22⋊C4×C31—21no—
227C22⋊C4×C31—22no—
228C22⋊C4×C31—21no—
229C22⋊C4×C31—22no—
230C22⋊C4×C31—21no—
231C22⋊C4×C31—41no—
232C22⋊C4×C31—41no—
233C22⋊C4×C31—81no—
234C42×C31—22no—
235C42×C31—21no—
236#18457C3111yesmaximal, characteristic, direct product factor, Sylow
237C23×C31—22no—
238C23×C31—22no—
239C23×C31—21no—
240C23×C31—41no—
241C23×C31—41no—
242C23×C31—42no—
243C23×C31—81no—
244C23×C31—81no—
245C23×C31#12011yescharacteristic
246C23×C31#12011yescharacteristic
247C23×C31C22×D411yescharacteristic
248Q8×C31—41no—
249Q8×C31—42no—
250Q8×C31—42no—
251Q8×C31—41no—
252Q8×C31—41no—
253Q8×C31—81no—
254Q8×C31—82no—
255D4×C31—42no—
256D4×C31—41no—
257D4×C31—42no—
258D4×C31—42no—
259D4×C31—44no—
260D4×C31—42no—
261D4×C31—82no—
262D4×C31—81no—
263D4×C31—82no—
264D4×C31—82no—
265C2×C4×C31—21no—
266C2×C4×C31—22no—
267C2×C4×C31—21no—
268C2×C4×C31—22no—
269C2×C4×C31—21no—
270C2×C4×C31—22no—
271C2×C4×C31—24no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—41no—
278C2×C4×C31—42no—
279C2×C4×C31—42no—
280C2×C4×C31—42no—
281C2×C4×C31—42no—
282C2×C4×C31—42no—
283C2×C4×C31—42no—
284C2×C4×C31—41no—
285C2×C4×C31—41no—
286C2×C4×C31—42no—
287C2×C4×C31—81no—
288C2×C4×C31—82no—
289C2×C4×C31—82no—
290C2×C4×C31—82no—
291C2×C4×C31#12011yescharacteristic
292C2×C4×C31#12011yescharacteristic
293C2×C4×C31C22×D411yescharacteristic
294C2×C4×C31C22×D411yescharacteristic
295C8×C31—42no—
296C8×C31—41no—
297C8×C31—41no—
298#3536C2×C3111yescharacteristic
299#3196C2×C3111yescharacteristic, complemented
300#3187C2×C3111yescharacteristic, complemented
301#2817C2×C3111yescharacteristic, complemented
302#1616C2×C3112yescomplemented
303#1615C2×C3112yescomplemented
304#1604C2×C3112yescomplemented
305#1603C2×C3112yescomplemented
306#1516C2×C3111yescharacteristic, complemented
307#1510C2×C3111yescharacteristic, complemented
308#1472C2×C3111yescharacteristic, complemented
309C22×C31—21no—
310C22×C31—21no—
311C22×C31—21no—
312C22×C31—21no—
313C22×C31—21no—
314C22×C31—21no—
315C22×C31—21no—
316C22×C31—21no—
317C22×C31—41no—
318C22×C31—42no—
319C22×C31—41no—
320C22×C31—42no—
321C22×C31—41no—
322C22×C31—42no—
323C22×C31—82no—
324C22×C31—162no—
325C22×C31C2×#12011yescharacteristic
326C22×C31#57712yes—
327C4×C31—21no—
328C4×C31—21no—
329C4×C31—21no—
330C4×C31—21no—
331C4×C31—42no—
332C4×C31—42no—
333C4×C31—41no—
334C4×C31—41no—
335C4×C31—41no—
336C4×C31—42no—
337C4×C31—82no—
338C4×C31—81no—
339C4×C31—81no—
340#562—22no—
341#554—21no—
342#548—21no—
343#547—21no—
344#546—21no—
345#546—22no—
346#543—21no—
347#542—21no—
348#539C22×C3112yes—
349#538—21no—
350#537—21no—
351#537C22×C3112yes—
352#525C22×C3111yescharacteristic
353C2×C22⋊Q8C22×C3111yescharacteristic
354C2×C22⋊Q8C22×C3111yescharacteristic
355C2×#121C22×C3111yescharacteristic
356#485—21no—
357#484C22×C3111yescharacteristic, complemented
358#483—21no—
359#481C22×C3112yes—
360#481C22×C3111yescharacteristic, complemented
361#470—21no—
362#468—21no—
363#465—22no—
364#452—21no—
365#451C22×C3111yescharacteristic, complemented
366#450—21no—
367#448C22×C3112yes—
368#448C22×C3111yescharacteristic, complemented
369#407C22×C3111yescharacteristic
370C2×C22⋊C8C22×C3111yescharacteristic
371#400—21no—
372#400C22×C3112yescomplemented
373#397—22no—
374#395—21no—
375#395C22×C3112yescomplemented
376#381C22×C3111yescharacteristic
377#328C22×C3112yescomplemented
378#327C22×C3112yes—
379#327C22×C3112yes—
380#327C22×C3112yescomplemented
381#324C22×C3112yescomplemented
382#324C22×C3112yescomplemented
383C2×C31—22no—
384C2×C31—41no—
385C2×C31—41no—
386C2×C31—81no—
387C2×C31—82no—
388C2×C31#300911yescharacteristic
389C2×C31#302011yescharacteristic
390C2×C31#302611yescharacteristic
391C2×#41—21no—
392C2×#41—22no—
393C2×#41D4×C3111yescharacteristic
394C22×Q8D4×C3111yescharacteristic
395C22×Q8C23×C3111yescharacteristic
396C22×D4—21no—
397C22×D4D4×C3111yescharacteristic
398C23×C4C23×C3111yescharacteristic
399C2×Q16—21no—
400C2×SD16—22no—
401C2×SD16—21no—
402C2×M16—21no—
403C22×C8—21no—
404C4⋊Q8—21no—
405#126—22no—
406#125—22no—
407#124—22no—
408#124—21no—
409#124—22no—
410#123—22no—
411#123—22no—
412#123—22no—
413#123—21no—
414#123—22no—
415C22⋊Q8—21no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—21no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—21no—
430C22⋊Q8D4×C3111yescharacteristic
431C22⋊Q8C23×C3111yescharacteristic
432#121—21no—
433#121—22no—
434#121—22no—
435#121—22no—
436#121—22no—
437#121—21no—
438#121—21no—
439#121—22no—
440#121D4×C3112yes—
441#121D4×C3111yescharacteristic
442#121C23×C3112yes—
443#121C23×C3111yescharacteristic
444#120—22no—
445#120—21no—
446C4×Q8—21no—
447C4×D4—22no—
448C4×D4—21no—
449C4×D4—21no—
450C4×D4—21no—
451C4×D4—22no—
452#117D4×C3111yescharacteristic
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—22no—
456C2×C4⋊C4D4×C3111yescharacteristic
457C2×C4⋊C4D4×C3111yescharacteristic
458C2×C4⋊C4C23×C3111yescharacteristic
459C2×C4⋊C4C23×C3112yes—
460C2×C22⋊C4—22no—
461C2×C22⋊C4—21no—
462C2×C22⋊C4—21no—
463C2×C22⋊C4—21no—
464C2×C22⋊C4D4×C3112yes—
465#107—21no—
466#106—22no—
467#106—21no—
468#103—21no—
469#103—22no—
470#103—21no—
471#103—21no—
472#102—22no—
473#102—21no—
474C22⋊C8—21no—
475C22⋊C8—21no—
476C22⋊C8C23×C3111yescharacteristic
477C22⋊C8C23×C3111yescharacteristic
478C22⋊C8C23×C3112yes—
479#95—21no—
480#95—22no—
481#95—22no—
482#95—22no—
483#95C23×C3112yes—
484C31#1845711yescharacteristic, direct product factor, Sylow
485C24—21no—
486C24C2×D4×C3111yescharacteristic
487#41—84no—
488#41—81no—
489#41—82no—
490C2×Q8—22no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—81no—
496C2×Q8—81no—
497C2×Q8C2×D4×C3111yescharacteristic
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—21no—
504C2×D4—22no—
505C2×D4—21no—
506C2×D4—81no—
507C2×D4—82no—
508C2×D4C2×D4×C3111yescharacteristic
509C2×D4C2×D4×C3112yes—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—81no—
519C22×C4C2×D4×C3112yes—
520C22×C4C2×D4×C3112yes—
521C22×C4C2×D4×C3111yescharacteristic
522C22×C4C2×D4×C3111yescharacteristic
523C22×C4C2×D4×C3111yescharacteristic
524C22×C4C2×D4×C3111yescharacteristic
525C22×C4C2×D4×C3111yescharacteristic
526C22×C4C24×C3111yescharacteristic
527Q16—81no—
528SD16—82no—
529SD16—81no—
530M16—81no—
531C2×C8—22no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—81no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—82no—
551C4⋊C4C2×D4×C3112yes—
552C4⋊C4C2×D4×C3111yescharacteristic
553C4⋊C4C2×D4×C3111yescharacteristic
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—21no—
563C22⋊C4—21no—
564C22⋊C4—22no—
565C22⋊C4—21no—
566C22⋊C4—22no—
567C22⋊C4—21no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—81no—
571C42—22no—
572C42—21no—
573C23—22no—
574C23—22no—
575C23—21no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—81no—
580C23—81no—
581C23C31×#12011yescharacteristic
582C23C31×#12011yescharacteristic
583C23C22×D4×C3111yescharacteristic
584Q8—41no—
585Q8—42no—
586Q8—42no—
587Q8—41no—
588Q8—41no—
589Q8—81no—
590Q8—82no—
591D4—42no—
592D4—41no—
593D4—42no—
594D4—42no—
595D4—44no—
596D4—42no—
597D4—82no—
598D4—81no—
599D4—82no—
600D4—82no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—81no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4C31×#12011yescharacteristic
628C2×C4C31×#12011yescharacteristic
629C2×C4C22×D4×C3111yescharacteristic
630C2×C4C22×D4×C3111yescharacteristic
631C8—42no—
632C8—41no—
633C8—41no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—41no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—82no—
649C22—162no—
650C22C2×C31×#12011yescharacteristic
651C22C31×#57712yes—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—82no—
663C4—81no—
664C4—81no—
665C2—22no—
666C2—41no—
667C2—41no—
668C2—82no—
669C2—81no—
670C2C31×#300911yescharacteristic
671C2C31×#302011yescharacteristic
672C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3536C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3196C2semidirect
3. C31×#3187C2semidirect
4. C31×#2817C2semidirect
5. C31×#1616C2semidirect
6. C31×#1615C2semidirect
7. C31×#1604C2semidirect
8. C31×#1603C2semidirect
9. C31×#1516C2semidirect
10. C31×#1510C2semidirect
11. C31×#1472C2semidirect
29. C31×#484C22semidirect
32. C31×#481C22semidirect
37. C31×#451C22semidirect
40. C31×#448C22semidirect
44. C31×#400C22semidirect
47. C31×#395C22semidirect
49. C31×#328C22semidirect
52. C31×#327C22semidirect
53. C31×#324C22semidirect
54. C31×#324C22semidirect
236. #18457C31direct (N × H)
299. #3196C2×C31semidirect
300. #3187C2×C31semidirect
301. #2817C2×C31semidirect
302. #1616C2×C31semidirect
303. #1615C2×C31semidirect
304. #1604C2×C31semidirect
305. #1603C2×C31semidirect
306. #1516C2×C31semidirect
307. #1510C2×C31semidirect
308. #1472C2×C31semidirect
357. #484C22×C31semidirect
360. #481C22×C31semidirect
365. #451C22×C31semidirect
368. #448C22×C31semidirect
372. #400C22×C31semidirect
375. #395C22×C31semidirect
377. #328C22×C31semidirect
380. #327C22×C31semidirect
381. #324C22×C31semidirect
382. #324C22×C31semidirect
484. C31#18457direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35362
C31×#31962
C31×#31872
C31×#28172
C31×#16162
C31×#16152
C31×#16042
C31×#16032
C31×#15162
C31×#15102
C31×#14722
#1845731