Finite Group Databasebeta

← C31×#33001 #21489760

Subgroups — C31×#33001 #21489760

Order 7936.

1242 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3428C211yesmaximal, characteristic
2C31×#3095C211yesmaximal, characteristic, complemented
3C31×#3007C211yesmaximal, characteristic
4C31×#2499C211yesmaximal, characteristic, complemented
5C31×#1882C212yesmaximal, complemented
6C31×#1882C212yesmaximal, complemented
7C31×#1860C212yesmaximal, complemented
8C31×#1859C212yesmaximal, complemented
9C31×#1826C211yesmaximal, characteristic, complemented
10C31×#1822C211yesmaximal, characteristic, complemented
11C31×#1789C211yesmaximal, characteristic
12C22×#41×C31C2211yescharacteristic
13C2×C31×#137C2212yes—
14C2×C31×C22⋊Q8C2211yescharacteristic
15C31×#520—21no—
16C31×#520—21no—
17C31×#520C2212yes—
18C31×#520C2212yes—
19C31×#519—21no—
20C31×#518—21no—
21C31×#518—22no—
22C4×#41×C31—22no—
23C4×#41×C31—22no—
24C2×C31×#117—22no—
25C2×C31×#117C2211yescharacteristic
26C2×C31×#117C2211yescharacteristic
27C31×#485—22no—
28C31×#484—21no—
29C31×#483—21no—
30C31×#452—22no—
31C31×#451—21no—
32C31×#450—21no—
33C31×#428C2212yescomplemented
34C2×C31×#104C2212yes—
35C2×C31×#104C2212yes—
36C31×#419C2212yescomplemented
37C31×#417C2212yes—
38C31×#407C2211yescharacteristic
39C31×#407C2211yescharacteristic
40C31×#397—22no—
41C31×#391C2211yescharacteristic, complemented
42C31×#391C2211yescharacteristic, complemented
43C31×#387—22no—
44C31×#384C2211yescharacteristic, complemented
45C31×#382C2211yescharacteristic, complemented
46C4×C22⋊C4×C31C2211yescharacteristic
47C31×#343C2212yescomplemented
48C31×#343C2212yescomplemented
49C31×#339C2212yescomplemented
50C31×#339C2212yescomplemented
51C2×#41×C31—24no—
52C2×#41×C31—24no—
53C2×#41×C31—21no—
54C2×#41×C31—21no—
55C2×#41×C31D412yes—
56C2×#41×C31C2312yes—
57C22×Q8×C31D411yescharacteristic
58C22×Q8×C31C2311yescharacteristic
59C22×D4×C31—21no—
60C22×D4×C31D411yescharacteristic
61C23×C4×C31—21no—
62C23×C4×C31C2311yescharacteristic
63C31×#137—24no—
64C31×#137—24no—
65C2×Q16×C31—22no—
66C2×SD16×C31—22no—
67C2×M16×C31C2312yes—
68C2×M16×C31C2312yes—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—21no—
73C4×Q8×C31—22no—
74C4×Q8×C31—22no—
75C4×Q8×C31—22no—
76C4×Q8×C31—22no—
77C4×D4×C31—24no—
78C4×D4×C31—24no—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C4×D4×C31—22no—
83C4×D4×C31—24no—
84C4×D4×C31—24no—
85C31×#117—24no—
86C31×#117—24no—
87C31×#117—21no—
88C31×#117—21no—
89C31×#117—21no—
90C31×#117—21no—
91C31×#117—21no—
92C31×#117—21no—
93C31×#117—22no—
94C31×#117—22no—
95C31×#117D412yes—
96C31×#117C2312yes—
97C2×C4⋊C4×C31—22no—
98C2×C4⋊C4×C31—22no—
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31D411yescharacteristic
102C2×C4⋊C4×C31D411yescharacteristic
103C2×C4⋊C4×C31C2311yescharacteristic
104C2×C4⋊C4×C31C2311yescharacteristic
105C2×C4⋊C4×C31C2311yescharacteristic
106C2×C22⋊C4×C31—24no—
107C2×C22⋊C4×C31—21no—
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C22⋊C4×C31D412yes—
111C2×C42×C31—24no—
112C2×C42×C31—21no—
113C2×C42×C31D412yes—
114C2×C42×C31C2311yescharacteristic
115C2×C42×C31C2311yescharacteristic
116C31×#107—22no—
117C31×#106—22no—
118C31×#104—24no—
119C31×#104—24no—
120C31×#103—22no—
121C31×#103—22no—
122C31×#103—22no—
123C31×#102—22no—
124C31×C22⋊C8—21no—
125C31×C22⋊C8—21no—
126C31×C22⋊C8—21no—
127C31×C22⋊C8—21no—
128C31×#95—22no—
129C31×#95—22no—
130C31×#95—22no—
131C31×#95—21no—
132C24×C31—21no—
133C24×C31C2×D411yescharacteristic
134#41×C31—24no—
135#41×C31—24no—
136#41×C31—28no—
137#41×C31—24no—
138#41×C31—24no—
139#41×C31—22no—
140#41×C31—22no—
141#41×C31#4114yes—
142#41×C31#4114yes—
143C2×Q8×C31—24no—
144C2×Q8×C31—21no—
145C2×Q8×C31—21no—
146C2×Q8×C31—21no—
147C2×Q8×C31—22no—
148C2×Q8×C31—22no—
149C2×Q8×C31C2×D412yes—
150C2×D4×C31—24no—
151C2×D4×C31—24no—
152C2×D4×C31—22no—
153C2×D4×C31—22no—
154C2×D4×C31—24no—
155C2×D4×C31—21no—
156C2×D4×C31C2×D412yes—
157C22×C4×C31—28no—
158C22×C4×C31—21no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—22no—
162C22×C4×C31—22no—
163C22×C4×C31—21no—
164C22×C4×C31—24no—
165C22×C4×C31—24no—
166C22×C4×C31—21no—
167C22×C4×C31—21no—
168C22×C4×C31—21no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D412yes—
177C22×C4×C31C2×D412yes—
178C22×C4×C31C2411yescharacteristic
179Q16×C31—44no—
180SD16×C31—44no—
181M16×C31—22no—
182M16×C31—22no—
183M16×C31—22no—
184M16×C31—22no—
185C2×C8×C31—22no—
186C2×C8×C31—22no—
187C4⋊C4×C31—24no—
188C4⋊C4×C31—24no—
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—21no—
197C4⋊C4×C31—44no—
198C4⋊C4×C31—41no—
199C4⋊C4×C31—41no—
200C4⋊C4×C31C2×D412yes—
201C4⋊C4×C31C2×D412yes—
202C22⋊C4×C31—24no—
203C22⋊C4×C31—24no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—22no—
207C22⋊C4×C31—22no—
208C22⋊C4×C31—42no—
209C22⋊C4×C31—41no—
210C22⋊C4×C31—41no—
211C42×C31—24no—
212C42×C31—24no—
213C42×C31—22no—
214C42×C31—22no—
215C42×C31—22no—
216C42×C31—22no—
217C42×C31—22no—
218C42×C31—22no—
219#33001C3111yesmaximal, characteristic, direct product factor, Sylow
220C23×C31—24no—
221C23×C31—22no—
222C23×C31—21no—
223C23×C31—21no—
224C23×C31—41no—
225C23×C31—41no—
226C23×C31—81no—
227C23×C31—81no—
228C23×C31#12011yescharacteristic
229C23×C31#12011yescharacteristic
230C23×C31C22×D411yescharacteristic
231Q8×C31—24no—
232Q8×C31—22no—
233Q8×C31—42no—
234Q8×C31—42no—
235Q8×C31#12314yes—
236D4×C31—28no—
237D4×C31—24no—
238D4×C31—24no—
239D4×C31—24no—
240D4×C31—22no—
241D4×C31#12314yes—
242C2×C4×C31—24no—
243C2×C4×C31—28no—
244C2×C4×C31—22no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—24no—
251C2×C4×C31—24no—
252C2×C4×C31—22no—
253C2×C4×C31—22no—
254C2×C4×C31—22no—
255C2×C4×C31—22no—
256C2×C4×C31—22no—
257C2×C4×C31—22no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—41no—
263C2×C4×C31—42no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—41no—
269C2×C4×C31—41no—
270C2×C4×C31—81no—
271C2×C4×C31—81no—
272C2×C4×C31—84no—
273C2×C4×C31—84no—
274C2×C4×C31#12011yescharacteristic
275C2×C4×C31#12011yescharacteristic
276C2×C4×C31C22×D411yescharacteristic
277C2×C4×C31C22×D411yescharacteristic
278C2×C4×C31C2×#4112yes—
279C2×C4×C31C2×#4112yes—
280C8×C31—42no—
281C8×C31—42no—
282#3428C2×C3111yescharacteristic
283#3095C2×C3111yescharacteristic, complemented
284#3007C2×C3111yescharacteristic
285#2499C2×C3111yescharacteristic, complemented
286#1882C2×C3112yescomplemented
287#1882C2×C3112yescomplemented
288#1860C2×C3112yescomplemented
289#1859C2×C3112yescomplemented
290#1826C2×C3111yescharacteristic, complemented
291#1822C2×C3111yescharacteristic, complemented
292#1789C2×C3111yescharacteristic
293C22×C31—21no—
294C22×C31—21no—
295C22×C31—21no—
296C22×C31—21no—
297C22×C31—24no—
298C22×C31—21no—
299C22×C31—21no—
300C22×C31—21no—
301C22×C31—21no—
302C22×C31—42no—
303C22×C31—42no—
304C22×C31—81no—
305C22×C31—81no—
306C22×C31—84no—
307C22×C31C2×#12011yescharacteristic
308C22×C31#52511yescharacteristic
309C22×C31#52511yescharacteristic
310C4×C31—21no—
311C4×C31—24no—
312C4×C31—21no—
313C4×C31—42no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—42no—
317C4×C31—42no—
318C4×C31—81no—
319C4×C31—81no—
320C4×C31—81no—
321C4×C31C2×#12312yes—
322C4×C31#52512yes—
323C22×#41C22×C3111yescharacteristic
324C2×#137C22×C3112yes—
325C2×C22⋊Q8C22×C3111yescharacteristic
326#520—21no—
327#520—21no—
328#520C22×C3112yes—
329#520C22×C3112yes—
330#519—21no—
331#518—21no—
332#518—22no—
333C4×#41—22no—
334C4×#41—22no—
335C2×#117—22no—
336C2×#117C22×C3111yescharacteristic
337C2×#117C22×C3111yescharacteristic
338#485—22no—
339#484—21no—
340#483—21no—
341#452—22no—
342#451—21no—
343#450—21no—
344#428C22×C3112yescomplemented
345C2×#104C22×C3112yes—
346C2×#104C22×C3112yes—
347#419C22×C3112yescomplemented
348#417C22×C3112yes—
349#407C22×C3111yescharacteristic
350#407C22×C3111yescharacteristic
351#397—22no—
352#391C22×C3111yescharacteristic, complemented
353#391C22×C3111yescharacteristic, complemented
354#387—22no—
355#384C22×C3111yescharacteristic, complemented
356#382C22×C3111yescharacteristic, complemented
357C4×C22⋊C4C22×C3111yescharacteristic
358#343C22×C3112yescomplemented
359#343C22×C3112yescomplemented
360#339C22×C3112yescomplemented
361#339C22×C3112yescomplemented
362C2×C31—21no—
363C2×C31—21no—
364C2×C31—41no—
365C2×C31—41no—
366C2×C31—44no—
367C2×C31#284911yescharacteristic
368C2×C31#301811yescharacteristic
369C2×C31#302211yescharacteristic
370C2×#41—24no—
371C2×#41—24no—
372C2×#41—21no—
373C2×#41—21no—
374C2×#41D4×C3112yes—
375C2×#41C23×C3112yes—
376C22×Q8D4×C3111yescharacteristic
377C22×Q8C23×C3111yescharacteristic
378C22×D4—21no—
379C22×D4D4×C3111yescharacteristic
380C23×C4—21no—
381C23×C4C23×C3111yescharacteristic
382#137—24no—
383#137—24no—
384C2×Q16—22no—
385C2×SD16—22no—
386C2×M16C23×C3112yes—
387C2×M16C23×C3112yes—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C4×Q8—22no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—22no—
396C4×D4—24no—
397C4×D4—24no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—24no—
403C4×D4—24no—
404#117—24no—
405#117—24no—
406#117—21no—
407#117—21no—
408#117—21no—
409#117—21no—
410#117—21no—
411#117—21no—
412#117—22no—
413#117—22no—
414#117D4×C3112yes—
415#117C23×C3112yes—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4D4×C3111yescharacteristic
421C2×C4⋊C4D4×C3111yescharacteristic
422C2×C4⋊C4C23×C3111yescharacteristic
423C2×C4⋊C4C23×C3111yescharacteristic
424C2×C4⋊C4C23×C3111yescharacteristic
425C2×C22⋊C4—24no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4D4×C3112yes—
430C2×C42—24no—
431C2×C42—21no—
432C2×C42D4×C3112yes—
433C2×C42C23×C3111yescharacteristic
434C2×C42C23×C3111yescharacteristic
435#107—22no—
436#106—22no—
437#104—24no—
438#104—24no—
439#103—22no—
440#103—22no—
441#103—22no—
442#102—22no—
443C22⋊C8—21no—
444C22⋊C8—21no—
445C22⋊C8—21no—
446C22⋊C8—21no—
447#95—21no—
448#95—22no—
449#95—22no—
450#95—22no—
451C31#3300111yescharacteristic, direct product factor, Sylow
452C24—21no—
453C24C2×D4×C3111yescharacteristic
454#41—24no—
455#41—24no—
456#41—28no—
457#41—24no—
458#41—24no—
459#41—22no—
460#41—22no—
461#41#41×C3114yes—
462#41#41×C3114yes—
463C2×Q8—24no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8C2×D4×C3112yes—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—24no—
475C2×D4—21no—
476C2×D4C2×D4×C3112yes—
477C22×C4—28no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4C2×D4×C3111yescharacteristic
492C22×C4C2×D4×C3111yescharacteristic
493C22×C4C2×D4×C3111yescharacteristic
494C22×C4C2×D4×C3111yescharacteristic
495C22×C4C2×D4×C3111yescharacteristic
496C22×C4C2×D4×C3112yes—
497C22×C4C2×D4×C3112yes—
498C22×C4C24×C3111yescharacteristic
499Q16—44no—
500SD16—44no—
501M16—22no—
502M16—22no—
503M16—22no—
504M16—22no—
505C2×C8—22no—
506C2×C8—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—44no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4C2×D4×C3112yes—
521C4⋊C4C2×D4×C3112yes—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—42no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C23—24no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—81no—
546C23—81no—
547C23C31×#12011yescharacteristic
548C23C31×#12011yescharacteristic
549C23C22×D4×C3111yescharacteristic
550Q8—24no—
551Q8—22no—
552Q8—42no—
553Q8—42no—
554Q8C31×#12314yes—
555D4—28no—
556D4—24no—
557D4—24no—
558D4—24no—
559D4—22no—
560D4C31×#12314yes—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4C31×#12011yescharacteristic
594C2×C4C31×#12011yescharacteristic
595C2×C4C22×D4×C3111yescharacteristic
596C2×C4C22×D4×C3111yescharacteristic
597C2×C4C2×#41×C3112yes—
598C2×C4C2×#41×C3112yes—
599C8—42no—
600C8—42no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—24no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—84no—
615C22C2×C31×#12011yescharacteristic
616C22C31×#52511yescharacteristic
617C22C31×#52511yescharacteristic
618C4—21no—
619C4—24no—
620C4—21no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4C2×C31×#12312yes—
630C4C31×#52512yes—
631C2—21no—
632C2—21no—
633C2—41no—
634C2—41no—
635C2—44no—
636C2C31×#284911yescharacteristic
637C2C31×#301811yescharacteristic
638C2C31×#302211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3095C2semidirect
4. C31×#2499C2semidirect
5. C31×#1882C2semidirect
6. C31×#1882C2semidirect
7. C31×#1860C2semidirect
8. C31×#1859C2semidirect
9. C31×#1826C2semidirect
10. C31×#1822C2semidirect
33. C31×#428C22semidirect
36. C31×#419C22semidirect
41. C31×#391C22semidirect
42. C31×#391C22semidirect
44. C31×#384C22semidirect
45. C31×#382C22semidirect
47. C31×#343C22semidirect
48. C31×#343C22semidirect
49. C31×#339C22semidirect
50. C31×#339C22semidirect
219. #33001C31direct (N × H)
283. #3095C2×C31semidirect
285. #2499C2×C31semidirect
286. #1882C2×C31semidirect
287. #1882C2×C31semidirect
288. #1860C2×C31semidirect
289. #1859C2×C31semidirect
290. #1826C2×C31semidirect
291. #1822C2×C31semidirect
344. #428C22×C31semidirect
347. #419C22×C31semidirect
352. #391C22×C31semidirect
353. #391C22×C31semidirect
355. #384C22×C31semidirect
356. #382C22×C31semidirect
358. #343C22×C31semidirect
359. #343C22×C31semidirect
360. #339C22×C31semidirect
361. #339C22×C31semidirect
451. C31#33001direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34282
C31×#30952
C31×#30072
C31×#24992
C31×#18822
C31×#18602
C31×#18592
C31×#18262
C31×#18222
C31×#17892
#3300131