Finite Group Databasebeta

← C31×#18166 #21474925

Subgroups — C31×#18166 #21474925

Order 7936.

1498 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3496C211yesmaximal, characteristic, complemented
2C31×#3006C212yesmaximal, complemented
3C31×#2388C211yesmaximal, characteristic, complemented
4C31×#1604C212yesmaximal, complemented
5C31×#1601C212yesmaximal, complemented
6C31×#1598C212yesmaximal, complemented
7C31×#1595C212yesmaximal, complemented
8C31×#1507C212yesmaximal, complemented
9C31×#1465C211yesmaximal, characteristic, complemented
10C2×C31×H(5,𝔽2)—21no—
11C22×#41×C31C2211yescharacteristic, complemented
12C2×C31×AGL(1,ℤ/8)—22no—
13C31×#547—22no—
14C31×#546—24no—
15D42×C31—22no—
16C31×#537C2212yes—
17C31×#534—21no—
18C31×#534C2212yes—
19C31×#532—21no—
20C2×C31×#121—21no—
21C2×C31×#121C2212yes—
22C2×C4×D4×C31—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C31×#449—24no—
25C31×#448C2214yescomplemented
26C31×#447C2214yescomplemented
27C31×#417—22no—
28C31×#407C2212yescomplemented
29C31×#395C2211yescharacteristic, complemented
30C31×#394—21no—
31C31×#394C2211yescharacteristic, complemented
32C31×#393C2211yescharacteristic, complemented
33C31×#392—21no—
34C31×#392C2211yescharacteristic, complemented
35C4×C22⋊C4×C31—21no—
36C4×C22⋊C4×C31C2211yescharacteristic, complemented
37C31×#327C2214yescomplemented
38C31×#327C2214yescomplemented
39C31×#324C2212yescomplemented
40C31×#323C2212yescomplemented
41C31×H(5,𝔽2)—44no—
42C31×H(5,𝔽2)—44no—
43C2×#41×C31—24no—
44C2×#41×C31—22no—
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31—44no—
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31—24no—
50C22×D4×C31—22no—
51C22×D4×C31—22no—
52C22×D4×C31—21no—
53C22×D4×C31—21no—
54C22×D4×C31—21no—
55C22×D4×C31D411yescharacteristic
56C22×D4×C31D412yescomplemented
57C22×D4×C31C2311yescharacteristic
58C23×C4×C31—21no—
59C23×C4×C31D411yescharacteristic
60C23×C4×C31C2311yescharacteristic
61C31×AGL(1,ℤ/8)—48no—
62C2×SD16×C31—24no—
63C2×D8×C31—24no—
64C2×M16×C31—22no—
65C31×C4⋊Q8—21no—
66C31×#127—21no—
67C31×#124—22no—
68C31×#123—24no—
69C31×#123—24no—
70C31×C22⋊Q8—24no—
71C31×#121—22no—
72C31×#121—22no—
73C31×#121—24no—
74C31×#121—22no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121D414yescomplemented
79C31×#121C2314yes—
80C31×#120—24no—
81C31×#120—24no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31—22no—
86C4×D4×C31—44no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31D412yescomplemented
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—22no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31D411yescharacteristic
99C2×C22⋊C4×C31C2311yescharacteristic
100C2×C42×C31—21no—
101C2×C42×C31D411yescharacteristic
102C31×#103—24no—
103C31×#102—24no—
104C31×C22⋊C8—22no—
105C31×C22⋊C8C2314yes—
106C31×#95—21no—
107C31×#95—21no—
108C31×#95C2311yescharacteristic
109C31×#95C2311yescharacteristic
110C24×C31—22no—
111C24×C31—21no—
112C24×C31—22no—
113C24×C31C2×D411yescharacteristic
114C24×C31C2×D411yescharacteristic
115#41×C31—28no—
116#41×C31—44no—
117#41×C31—44no—
118#41×C31—44no—
119#41×C31—416no—
120#41×C31—44no—
121#41×C31—44no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—42no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—24no—
128C2×D4×C31—22no—
129C2×D4×C31—24no—
130C2×D4×C31—24no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—42no—
137C2×D4×C31—42no—
138C2×D4×C31—44no—
139C2×D4×C31—42no—
140C2×D4×C31—416no—
141C2×D4×C31—42no—
142C2×D4×C31—42no—
143C2×D4×C31—44no—
144C2×D4×C31—44no—
145C2×D4×C31—44no—
146C2×D4×C31—42no—
147C2×D4×C31—42no—
148C2×D4×C31—44no—
149C2×D4×C31—82no—
150C2×D4×C31C2×D414yes—
151C22×C4×C31—22no—
152C22×C4×C31—24no—
153C22×C4×C31—24no—
154C22×C4×C31—22no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—42no—
161C22×C4×C31—42no—
162C22×C4×C31—42no—
163C22×C4×C31—41no—
164C22×C4×C31—41no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2×D412yes—
171C22×C4×C31C2411yescharacteristic
172SD16×C31—48no—
173D8×C31—48no—
174M16×C31—44no—
175C2×C8×C31—24no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—42no—
181C4⋊C4×C31C2×D414yes—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—42no—
189C22⋊C4×C31—42no—
190C22⋊C4×C31—42no—
191C22⋊C4×C31—42no—
192C22⋊C4×C31—42no—
193C42×C31—21no—
194C42×C31—21no—
195C42×C31—42no—
196#18166C3111yesmaximal, characteristic, direct product factor, Sylow
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—21no—
201C23×C31—22no—
202C23×C31—24no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31—41no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31—44no—
211C23×C31—44no—
212C23×C31—41no—
213C23×C31—41no—
214C23×C31—41no—
215C23×C31—82no—
216C23×C31—82no—
217C23×C31—82no—
218C23×C31—82no—
219C23×C31#12012yes—
220C23×C31AGL(1,ℤ/8)12yescomplemented
221C23×C31C22×D411yescharacteristic
222Q8×C31—24no—
223Q8×C31—42no—
224Q8×C31—44no—
225D4×C31—28no—
226D4×C31—24no—
227D4×C31—44no—
228D4×C31—44no—
229D4×C31—42no—
230D4×C31—48no—
231D4×C31—48no—
232D4×C31—48no—
233D4×C31—48no—
234D4×C31—44no—
235D4×C31—44no—
236D4×C31—44no—
237D4×C31—82no—
238D4×C31—82no—
239C2×C4×C31—21no—
240C2×C4×C31—21no—
241C2×C4×C31—22no—
242C2×C4×C31—24no—
243C2×C4×C31—22no—
244C2×C4×C31—22no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—24no—
248C2×C4×C31—42no—
249C2×C4×C31—41no—
250C2×C4×C31—44no—
251C2×C4×C31—48no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—44no—
255C2×C4×C31—44no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—44no—
259C2×C4×C31—42no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—81no—
266C2×C4×C31#12011yescharacteristic
267C2×C4×C31#12011yescharacteristic
268C2×C4×C31AGL(1,ℤ/8)12yescomplemented
269C2×C4×C31C22×D412yes—
270C8×C31—44no—
271#3496C2×C3111yescharacteristic, complemented
272#3006C2×C3112yescomplemented
273#2388C2×C3111yescharacteristic, complemented
274#1604C2×C3112yescomplemented
275#1601C2×C3112yescomplemented
276#1598C2×C3112yescomplemented
277#1595C2×C3112yescomplemented
278#1507C2×C3112yescomplemented
279#1465C2×C3111yescharacteristic, complemented
280C22×C31—21no—
281C22×C31—21no—
282C22×C31—22no—
283C22×C31—22no—
284C22×C31—22no—
285C22×C31—22no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—41no—
290C22×C31—41no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—42no—
294C22×C31—44no—
295C22×C31—81no—
296C22×C31—81no—
297C22×C31—81no—
298C22×C31—81no—
299C22×C31—81no—
300C22×C31—81no—
301C22×C31—81no—
302C22×C31—164no—
303C22×C31—164no—
304C22×C31C2×#12011yescharacteristic
305C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
307C4×C31—22no—
308C4×C31—22no—
309C4×C31—21no—
310C4×C31—42no—
311C4×C31—44no—
312C4×C31—41no—
313C4×C31—42no—
314C4×C31—41no—
315C4×C31—81no—
316C4×C31B(4,𝔽2)12yes—
317C2×H(5,𝔽2)—21no—
318C22×#41C22×C3111yescharacteristic, complemented
319C2×AGL(1,ℤ/8)—22no—
320#547—22no—
321#546—24no—
322D42—22no—
323#537C22×C3112yes—
324#534—21no—
325#534C22×C3112yes—
326#532—21no—
327C2×#121—21no—
328C2×#121C22×C3112yes—
329C2×C4×D4—21no—
330C2×C4×D4C22×C3111yescharacteristic
331#449—24no—
332#448C22×C3114yescomplemented
333#447C22×C3114yescomplemented
334#417—22no—
335#407C22×C3112yescomplemented
336#395C22×C3111yescharacteristic, complemented
337#394—21no—
338#394C22×C3111yescharacteristic, complemented
339#393C22×C3111yescharacteristic, complemented
340#392—21no—
341#392C22×C3111yescharacteristic, complemented
342C4×C22⋊C4—21no—
343C4×C22⋊C4C22×C3111yescharacteristic, complemented
344#327C22×C3114yescomplemented
345#327C22×C3114yescomplemented
346#324C22×C3112yescomplemented
347#323C22×C3112yescomplemented
348C2×C31—21no—
349C2×C31—21no—
350C2×C31—42no—
351C2×C31—42no—
352C2×C31—82no—
353C2×C31—84no—
354C2×C31#300311yescharacteristic
355C2×C31#301811yescharacteristic
356C2×C31C2×B(4,𝔽2)11yescharacteristic
357H(5,𝔽2)—44no—
358H(5,𝔽2)—44no—
359C2×#41—24no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—44no—
364C22×Q8C23×C3111yescharacteristic
365C22×D4—24no—
366C22×D4—22no—
367C22×D4—22no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4D4×C3111yescharacteristic
372C22×D4D4×C3112yescomplemented
373C22×D4C23×C3111yescharacteristic
374C23×C4—21no—
375C23×C4D4×C3111yescharacteristic
376C23×C4C23×C3111yescharacteristic
377AGL(1,ℤ/8)—48no—
378C2×SD16—24no—
379C2×D8—24no—
380C2×M16—22no—
381C4⋊Q8—21no—
382#127—21no—
383#124—22no—
384#123—24no—
385#123—24no—
386C22⋊Q8—24no—
387#121—22no—
388#121—22no—
389#121—24no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—22no—
394#121D4×C3114yescomplemented
395#121C23×C3114yes—
396#120—24no—
397#120—24no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—44no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4D4×C3112yescomplemented
406C2×C4⋊C4C23×C3111yescharacteristic
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4D4×C3111yescharacteristic
415C2×C22⋊C4C23×C3111yescharacteristic
416C2×C42—21no—
417C2×C42D4×C3111yescharacteristic
418#103—24no—
419#102—24no—
420C22⋊C8—22no—
421C22⋊C8C23×C3114yes—
422#95—21no—
423#95—21no—
424#95C23×C3111yescharacteristic
425#95C23×C3111yescharacteristic
426C31#1816611yescharacteristic, direct product factor, Sylow
427C24—22no—
428C24—21no—
429C24—22no—
430C24C2×D4×C3111yescharacteristic
431C24C2×D4×C3111yescharacteristic
432#41—28no—
433#41—44no—
434#41—44no—
435#41—44no—
436#41—416no—
437#41—44no—
438#41—44no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—42no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—416no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—82no—
467C2×D4C2×D4×C3114yes—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4C2×D4×C3111yescharacteristic
483C22×C4C2×D4×C3112yes—
484C22×C4C2×D4×C3111yescharacteristic
485C22×C4C2×D4×C3111yescharacteristic
486C22×C4C2×D4×C3111yescharacteristic
487C22×C4C2×D4×C3112yes—
488C22×C4C24×C3111yescharacteristic
489SD16—48no—
490D8—48no—
491M16—44no—
492C2×C8—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4C2×D4×C3114yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C42—21no—
511C42—21no—
512C42—42no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—24no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—44no—
527C23—44no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23C31×#12012yes—
536C23C31×AGL(1,ℤ/8)12yescomplemented
537C23C22×D4×C3111yescharacteristic
538Q8—24no—
539Q8—42no—
540Q8—44no—
541D4—28no—
542D4—24no—
543D4—44no—
544D4—44no—
545D4—42no—
546D4—48no—
547D4—48no—
548D4—48no—
549D4—48no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—82no—
554D4—82no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—81no—
582C2×C4C31×#12011yescharacteristic
583C2×C4C31×#12011yescharacteristic
584C2×C4C31×AGL(1,ℤ/8)12yescomplemented
585C2×C4C22×D4×C3112yes—
586C8—44no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—164no—
610C22—164no—
611C22C2×C31×#12011yescharacteristic
612C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
613C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
614C4—22no—
615C4—22no—
616C4—21no—
617C4—42no—
618C4—44no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—81no—
623C4C31×B(4,𝔽2)12yes—
624C2—21no—
625C2—21no—
626C2—42no—
627C2—42no—
628C2—82no—
629C2—84no—
630C2C31×#300311yescharacteristic
631C2C31×#301811yescharacteristic
632C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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
1. C31×#3496C2semidirect
2. C31×#3006C2semidirect
3. C31×#2388C2semidirect
4. C31×#1604C2semidirect
5. C31×#1601C2semidirect
6. C31×#1598C2semidirect
7. C31×#1595C2semidirect
8. C31×#1507C2semidirect
9. C31×#1465C2semidirect
11. C22×#41×C31C22semidirect
25. C31×#448C22semidirect
26. C31×#447C22semidirect
28. C31×#407C22semidirect
29. C31×#395C22semidirect
31. C31×#394C22semidirect
32. C31×#393C22semidirect
34. C31×#392C22semidirect
36. C4×C22⋊C4×C31C22semidirect
37. C31×#327C22semidirect
38. C31×#327C22semidirect
39. C31×#324C22semidirect
40. C31×#323C22semidirect
56. C22×D4×C31D4semidirect
78. C31×#121D4semidirect
89. C2×C4⋊C4×C31D4semidirect
196. #18166C31direct (N × H)
220. C23×C31AGL(1,ℤ/8)semidirect
268. C2×C4×C31AGL(1,ℤ/8)semidirect
271. #3496C2×C31semidirect
272. #3006C2×C31semidirect
273. #2388C2×C31semidirect
274. #1604C2×C31semidirect
275. #1601C2×C31semidirect
276. #1598C2×C31semidirect
277. #1595C2×C31semidirect
278. #1507C2×C31semidirect
279. #1465C2×C31semidirect
318. C22×#41C22×C31semidirect
332. #448C22×C31semidirect
333. #447C22×C31semidirect
335. #407C22×C31semidirect
336. #395C22×C31semidirect
338. #394C22×C31semidirect
339. #393C22×C31semidirect
341. #392C22×C31semidirect
343. C4×C22⋊C4C22×C31semidirect
344. #327C22×C31semidirect
345. #327C22×C31semidirect
346. #324C22×C31semidirect
347. #323C22×C31semidirect
372. C22×D4D4×C31semidirect
394. #121D4×C31semidirect
405. C2×C4⋊C4D4×C31semidirect
426. C31#18166direct (N × H)
536. C23C31×AGL(1,ℤ/8)semidirect
584. C2×C4C31×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C31×#34962
C31×#30062
C31×#23882
C31×#16042
C31×#16012
C31×#15982
C31×#15952
C31×#15072
C31×#14652
#1816631