Finite Group Databasebeta

← C31×#33550 #21490309

Subgroups — C31×#33550 #21490309

Order 7936.

1144 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3529C211yesmaximal, characteristic, complemented
2C31×#3025C212yesmaximal, complemented
3C31×#3018C211yesmaximal, characteristic, complemented
4C31×#2201C212yesmaximal, complemented
5C31×#2200C214yesmaximal, complemented
6C31×#2199C212yesmaximal, complemented
7C31×#2126C211yesmaximal, characteristic, complemented
8C31×#2120C212yesmaximal, complemented
9C31×#585C2211yescharacteristic
10C31×#575—21no—
11C31×#563—21no—
12C31×#563C2211yescharacteristic, complemented
13C31×#562—21no—
14C31×#562—22no—
15C31×H(3,𝔽4)C2211yescharacteristic, complemented
16C31×#560—22no—
17C31×#560C2212yescomplemented
18C31×#532—21no—
19C31×#532C2211yescharacteristic
20C31×#529—22no—
21C31×#529—21no—
22C31×#525—22no—
23C31×#525—22no—
24C31×#525C2212yes—
25C31×#458—22no—
26C31×#458C2214yes—
27C31×B(4,𝔽2)—22no—
28C31×B(4,𝔽2)C2214yes—
29C31×#456—21no—
30C31×#455C2212yes—
31C31×#454—21no—
32C31×#453C2212yes—
33C2×C31×#104—21no—
34C31×#413C2211yescharacteristic
35C31×#410—22no—
36C31×#410C2212yes—
37C31×#354C2212yescomplemented
38C31×#353C2212yescomplemented
39C31×#352C2214yescomplemented
40C31×AΣL(1,𝔽16)C2214yescomplemented
41C31×2−1+4—21no—
42C31×2−1+4—41no—
43C31×H(5,𝔽2)—22no—
44C31×H(5,𝔽2)—41no—
45C31×H(5,𝔽2)C2312yes—
46C2×#41×C31—21no—
47C2×#41×C31—22no—
48C2×#41×C31—21no—
49C2×#41×C31—42no—
50C2×#41×C31D411yescharacteristic, complemented
51C2×#41×C31C2311yescharacteristic
52C23×C4×C31D412yescomplemented
53C31×#137—22no—
54C31×#137—41no—
55C31×AGL(1,ℤ/8)—22no—
56C31×AGL(1,ℤ/8)—41no—
57C31×#135—42no—
58C2×M16×C31—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8D411yescharacteristic, complemented
61C31×#127—21no—
62C31×#127C2311yescharacteristic
63C31×#126—22no—
64C31×#126—22no—
65C31×#126—22no—
66C31×#125—22no—
67C31×#125—21no—
68C31×#124—21no—
69C31×#124—21no—
70C31×#124—22no—
71C31×#124—21no—
72C31×#124D411yescharacteristic, complemented
73C31×#124C2311yescharacteristic
74C31×#123—22no—
75C31×#123—22no—
76C31×#123—22no—
77C31×#123—22no—
78C31×#123—22no—
79C31×#123—24no—
80C31×#123D412yescomplemented
81C31×#123C2312yes—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—42no—
88C31×#121—22no—
89C31×#121—21no—
90C31×#121—22no—
91C31×#121—22no—
92C31×#121—22no—
93C31×#121—22no—
94C31×#121—22no—
95C31×#121—22no—
96C31×#121—42no—
97C31×#120—22no—
98C31×#120—22no—
99C31×#120—24no—
100C31×#120D412yescomplemented
101C31×#120C2312yes—
102C4×Q8×C31—21no—
103C4×D4×C31—21no—
104C4×D4×C31—22no—
105C4×D4×C31—22no—
106C4×D4×C31—22no—
107C4×D4×C31—22no—
108C4×D4×C31—44no—
109C31×#117—22no—
110C31×#117—22no—
111C31×#117D412yescomplemented
112C2×C42×C31—21no—
113C2×C42×C31D411yescharacteristic, complemented
114C31×#104—22no—
115C31×#104—22no—
116C31×#101—21no—
117C31×#100C2312yes—
118C31×#99—22no—
119C31×#99—24no—
120C31×#99—24no—
121C31×#99C2314yes—
122C24×C31C2×D412yes—
123#41×C31—22no—
124#41×C31—22no—
125#41×C31—41no—
126#41×C31—41no—
127#41×C31—42no—
128#41×C31—42no—
129#41×C31—42no—
130#41×C31—41no—
131#41×C31—44no—
132#41×C31—41no—
133#41×C31—41no—
134#41×C31—42no—
135#41×C31—42no—
136#41×C31—41no—
137#41×C31—81no—
138#41×C31—81no—
139C2×Q8×C31—22no—
140C2×Q8×C31—21no—
141C2×Q8×C31—21no—
142C2×Q8×C31—21no—
143C2×Q8×C31—42no—
144C2×Q8×C31C2×D411yescharacteristic
145C2×Q8×C31C2×D411yescharacteristic
146C2×D4×C31—21no—
147C2×D4×C31—22no—
148C2×D4×C31—24no—
149C2×D4×C31—21no—
150C2×D4×C31—22no—
151C2×D4×C31—21no—
152C2×D4×C31—22no—
153C2×D4×C31—22no—
154C2×D4×C31—21no—
155C2×D4×C31—22no—
156C2×D4×C31—22no—
157C2×D4×C31—42no—
158C2×D4×C31—44no—
159C2×D4×C31—41no—
160C2×D4×C31—44no—
161C2×D4×C31C2×D412yes—
162C2×D4×C31C2×D411yescharacteristic
163C2×D4×C31C2411yescharacteristic
164C22×C4×C31—24no—
165C22×C4×C31—22no—
166C22×C4×C31—21no—
167C22×C4×C31—21no—
168C22×C4×C31—22no—
169C22×C4×C31—22no—
170C22×C4×C31—41no—
171C22×C4×C31—42no—
172C22×C4×C31—42no—
173C22×C4×C31C2×D412yes—
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D412yes—
176Q16×C31—42no—
177SD16×C31—42no—
178SD16×C31—42no—
179D8×C31—42no—
180M16×C31—22no—
181M16×C31—41no—
182C2×C8×C31—41no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—42no—
194C4⋊C4×C31—44no—
195C22⋊C4×C31—22no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—22no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—22no—
200C22⋊C4×C31—22no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—22no—
203C22⋊C4×C31—22no—
204C22⋊C4×C31—22no—
205C22⋊C4×C31—22no—
206C22⋊C4×C31—24no—
207C22⋊C4×C31—44no—
208C22⋊C4×C31—44no—
209C22⋊C4×C31C2×D412yes—
210C22⋊C4×C31C2×D412yes—
211C42×C31—21no—
212C42×C31—21no—
213C42×C31—22no—
214C42×C31—21no—
215C42×C31—42no—
216C42×C31C2×D411yescharacteristic
217C42×C31C2×D411yescharacteristic
218#33550C3111yesmaximal, characteristic, direct product factor, Sylow
219C23×C31—24no—
220C23×C31—22no—
221C23×C31—21no—
222C23×C31—42no—
223C23×C31—41no—
224C23×C31—82no—
225C23×C31#12011yescharacteristic
226C23×C31C22×D412yes—
227Q8×C31—22no—
228Q8×C31—41no—
229Q8×C31—41no—
230Q8×C31—42no—
231Q8×C31—41no—
232Q8×C31—81no—
233Q8×C31—81no—
234D4×C31—22no—
235D4×C31—42no—
236D4×C31—41no—
237D4×C31—41no—
238D4×C31—42no—
239D4×C31—44no—
240D4×C31—42no—
241D4×C31—42no—
242D4×C31—41no—
243D4×C31—81no—
244D4×C31—82no—
245D4×C31—82no—
246D4×C31—82no—
247D4×C31—82no—
248D4×C31—81no—
249D4×C31—82no—
250D4×C31—81no—
251D4×C31—88no—
252C2×C4×C31—22no—
253C2×C4×C31—21no—
254C2×C4×C31—22no—
255C2×C4×C31—21no—
256C2×C4×C31—24no—
257C2×C4×C31—22no—
258C2×C4×C31—21no—
259C2×C4×C31—22no—
260C2×C4×C31—21no—
261C2×C4×C31—21no—
262C2×C4×C31—22no—
263C2×C4×C31—44no—
264C2×C4×C31—42no—
265C2×C4×C31—41no—
266C2×C4×C31—42no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—41no—
271C2×C4×C31—42no—
272C2×C4×C31—44no—
273C2×C4×C31—82no—
274C2×C4×C31—81no—
275C2×C4×C31—82no—
276C2×C4×C31#12012yes—
277C2×C4×C31#12011yescharacteristic
278C2×C4×C31C22×D411yescharacteristic
279C8×C31—42no—
280#3529C2×C3111yescharacteristic, complemented
281#3025C2×C3112yescomplemented
282#3018C2×C3111yescharacteristic, complemented
283#2201C2×C3112yescomplemented
284#2200C2×C3114yescomplemented
285#2199C2×C3112yescomplemented
286#2126C2×C3111yescharacteristic, complemented
287#2120C2×C3112yescomplemented
288C22×C31—22no—
289C22×C31—21no—
290C22×C31—41no—
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—42no—
294C22×C31—42no—
295C22×C31—44no—
296C22×C31—42no—
297C22×C31—42no—
298C22×C31—41no—
299C22×C31—84no—
300C22×C31—82no—
301C22×C31—161no—
302C22×C31C2×#12011yescharacteristic
303C4×C31—21no—
304C4×C31—22no—
305C4×C31—42no—
306C4×C31—41no—
307C4×C31—42no—
308C4×C31—41no—
309C4×C31—42no—
310C4×C31—42no—
311C4×C31—41no—
312C4×C31—84no—
313C4×C31B(4,𝔽2)11yescharacteristic
314C4×C31B(4,𝔽2)11yescharacteristic
315#585C22×C3111yescharacteristic
316#575—21no—
317#563—21no—
318#563C22×C3111yescharacteristic, complemented
319#562—21no—
320#562—22no—
321H(3,𝔽4)C22×C3111yescharacteristic, complemented
322#560—22no—
323#560C22×C3112yescomplemented
324#532—21no—
325#532C22×C3111yescharacteristic
326#529—22no—
327#529—21no—
328#525—22no—
329#525—22no—
330#525C22×C3112yes—
331#458—22no—
332#458C22×C3114yes—
333B(4,𝔽2)—22no—
334B(4,𝔽2)C22×C3114yes—
335#456—21no—
336#455C22×C3112yes—
337#454—21no—
338#453C22×C3112yes—
339C2×#104—21no—
340#413C22×C3111yescharacteristic
341#410—22no—
342#410C22×C3112yes—
343#354C22×C3112yescomplemented
344#353C22×C3112yescomplemented
345#352C22×C3114yescomplemented
346AΣL(1,𝔽16)C22×C3114yescomplemented
347C2×C31—21no—
348C2×C31—42no—
349C2×C31—41no—
350C2×C31—81no—
351C2×C31—82no—
352C2×C31—82no—
353C2×C31C2×B(4,𝔽2)11yescharacteristic
3542−1+4—21no—
3552−1+4—41no—
356H(5,𝔽2)—22no—
357H(5,𝔽2)—41no—
358H(5,𝔽2)C23×C3112yes—
359C2×#41—21no—
360C2×#41—22no—
361C2×#41—21no—
362C2×#41—42no—
363C2×#41D4×C3111yescharacteristic, complemented
364C2×#41C23×C3111yescharacteristic
365C23×C4D4×C3112yescomplemented
366#137—22no—
367#137—41no—
368AGL(1,ℤ/8)—22no—
369AGL(1,ℤ/8)—41no—
370#135—42no—
371C2×M16—21no—
372C4⋊Q8—21no—
373C4⋊Q8D4×C3111yescharacteristic, complemented
374#127—21no—
375#127C23×C3111yescharacteristic
376#126—22no—
377#126—22no—
378#126—22no—
379#125—22no—
380#125—21no—
381#124—21no—
382#124—21no—
383#124—22no—
384#124—21no—
385#124D4×C3111yescharacteristic, complemented
386#124C23×C3111yescharacteristic
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—24no—
393#123D4×C3112yescomplemented
394#123C23×C3112yes—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—42no—
401#121—22no—
402#121—21no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121—22no—
408#121—22no—
409#121—42no—
410#120—22no—
411#120—22no—
412#120—24no—
413#120D4×C3112yescomplemented
414#120C23×C3112yes—
415C4×Q8—21no—
416C4×D4—21no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—44no—
422#117—22no—
423#117—22no—
424#117D4×C3112yescomplemented
425C2×C42—21no—
426C2×C42D4×C3111yescharacteristic, complemented
427#104—22no—
428#104—22no—
429#101—21no—
430#100C23×C3112yes—
431#99—22no—
432#99—24no—
433#99—24no—
434#99C23×C3114yes—
435C31#3355011yescharacteristic, direct product factor, Sylow
436C24C2×D4×C3112yes—
437#41—22no—
438#41—22no—
439#41—41no—
440#41—41no—
441#41—42no—
442#41—42no—
443#41—42no—
444#41—41no—
445#41—44no—
446#41—41no—
447#41—41no—
448#41—42no—
449#41—41no—
450#41—42no—
451#41—81no—
452#41—81no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—42no—
458C2×Q8C2×D4×C3111yescharacteristic
459C2×Q8C2×D4×C3111yescharacteristic
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—42no—
472C2×D4—44no—
473C2×D4—41no—
474C2×D4—44no—
475C2×D4C2×D4×C3112yes—
476C2×D4C2×D4×C3111yescharacteristic
477C2×D4C24×C3111yescharacteristic
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—41no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4C2×D4×C3112yes—
488C22×C4C2×D4×C3111yescharacteristic
489C22×C4C2×D4×C3112yes—
490Q16—42no—
491SD16—42no—
492SD16—42no—
493D8—42no—
494M16—22no—
495M16—41no—
496C2×C8—41no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—24no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4C2×D4×C3112yes—
524C22⋊C4C2×D4×C3112yes—
525C42—21no—
526C42—21no—
527C42—22no—
528C42—21no—
529C42—42no—
530C42C2×D4×C3111yescharacteristic
531C42C2×D4×C3111yescharacteristic
532C23—24no—
533C23—22no—
534C23—21no—
535C23—42no—
536C23—41no—
537C23—82no—
538C23C31×#12011yescharacteristic
539C23C22×D4×C3112yes—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—42no—
544Q8—41no—
545Q8—81no—
546Q8—81no—
547D4—22no—
548D4—42no—
549D4—41no—
550D4—41no—
551D4—42no—
552D4—44no—
553D4—42no—
554D4—42no—
555D4—41no—
556D4—81no—
557D4—82no—
558D4—82no—
559D4—82no—
560D4—82no—
561D4—81no—
562D4—82no—
563D4—81no—
564D4—88no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4C31×#12012yes—
590C2×C4C31×#12011yescharacteristic
591C2×C4C22×D4×C3111yescharacteristic
592C8—42no—
593C22—22no—
594C22—21no—
595C22—41no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—84no—
605C22—82no—
606C22—161no—
607C22C2×C31×#12011yescharacteristic
608C4—21no—
609C4—22no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—41no—
617C4—84no—
618C4C31×B(4,𝔽2)11yescharacteristic
619C4C31×B(4,𝔽2)11yescharacteristic
620C2—21no—
621C2—42no—
622C2—41no—
623C2—82no—
624C2—81no—
625C2—82no—
626C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
centerC4×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C31
socleC2×C31C2×B(4,𝔽2)
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3529C2semidirect
2. C31×#3025C2semidirect
3. C31×#3018C2semidirect
4. C31×#2201C2semidirect
5. C31×#2200C2semidirect
6. C31×#2199C2semidirect
7. C31×#2126C2semidirect
8. C31×#2120C2semidirect
12. C31×#563C22semidirect
15. C31×H(3,𝔽4)C22semidirect
17. C31×#560C22semidirect
37. C31×#354C22semidirect
38. C31×#353C22semidirect
39. C31×#352C22semidirect
40. C31×AΣL(1,𝔽16)C22semidirect
50. C2×#41×C31D4semidirect
52. C23×C4×C31D4semidirect
60. C31×C4⋊Q8D4semidirect
72. C31×#124D4semidirect
80. C31×#123D4semidirect
100. C31×#120D4semidirect
111. C31×#117D4semidirect
113. C2×C42×C31D4semidirect
218. #33550C31direct (N × H)
280. #3529C2×C31semidirect
281. #3025C2×C31semidirect
282. #3018C2×C31semidirect
283. #2201C2×C31semidirect
284. #2200C2×C31semidirect
285. #2199C2×C31semidirect
286. #2126C2×C31semidirect
287. #2120C2×C31semidirect
318. #563C22×C31semidirect
321. H(3,𝔽4)C22×C31semidirect
323. #560C22×C31semidirect
343. #354C22×C31semidirect
344. #353C22×C31semidirect
345. #352C22×C31semidirect
346. AΣL(1,𝔽16)C22×C31semidirect
363. C2×#41D4×C31semidirect
365. C23×C4D4×C31semidirect
373. C4⋊Q8D4×C31semidirect
385. #124D4×C31semidirect
393. #123D4×C31semidirect
413. #120D4×C31semidirect
424. #117D4×C31semidirect
426. C2×C42D4×C31semidirect
435. C31#33550direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35292
C31×#30252
C31×#30182
C31×#22012
C31×#22002
C31×#21992
C31×#21262
C31×#21202
#3355031