Finite Group Databasebeta

← C31×#18663 #21475422

Subgroups — C31×#18663 #21475422

Order 7936.

1206 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3496C211yesmaximal, characteristic
2C31×#3246C211yesmaximal, characteristic, complemented
3C31×#3167C211yesmaximal, characteristic, complemented
4C31×#3077C211yesmaximal, characteristic, complemented
5C31×#1654C212yesmaximal, complemented
6C31×#1639C212yesmaximal, complemented
7C31×#1635C212yesmaximal, complemented
8C31×#1629C212yesmaximal, complemented
9C31×#1547C211yesmaximal, characteristic, complemented
10C31×#1492C211yesmaximal, characteristic, complemented
11C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×C31×H(5,𝔽2)—21no—
13C31×#547—21no—
14C31×#547C2212yes—
15C31×#546—22no—
16C31×#546—22no—
17D42×C31—21no—
18D42×C31C2212yes—
19C31×#537—21no—
20C31×#534—22no—
21C31×#533C2211yescharacteristic, complemented
22C31×#532C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C2×C31×#121—22no—
25C2×C4×D4×C31C2211yescharacteristic
26C31×#498C2211yescharacteristic, complemented
27C31×#492C2211yescharacteristic, complemented
28C31×#487C2211yescharacteristic, complemented
29C31×#486C2211yescharacteristic, complemented
30C31×#478—22no—
31C31×#476C2212yescomplemented
32C31×#476C2212yescomplemented
33C31×#475—21no—
34C31×#474C2212yescomplemented
35C31×#465—22no—
36C31×#465—22no—
37C31×#461C2212yescomplemented
38C31×#460—21no—
39C4×SD16×C31—22no—
40C31×#428—21no—
41C31×#423C2211yescharacteristic, complemented
42C2×C31×C4⋊C8C2211yescharacteristic, complemented
43C31×#418—21no—
44C2×C4×C8×C31C2211yescharacteristic, complemented
45C31×#332C2212yescomplemented
46C31×#331C2212yescomplemented
47C31×D4⋊C8C2212yescomplemented
48C31×D4⋊C8C2212yescomplemented
49C31×D4⋊C8C2212yescomplemented
50C31×D4⋊C8C2212yescomplemented
51C31×H(5,𝔽2)—44no—
52C31×H(5,𝔽2)—44no—
53C2×#41×C31—22no—
54C2×#41×C31—22no—
55C2×#41×C31—21no—
56C2×#41×C31—21no—
57C22×D4×C31—24no—
58C22×D4×C31—22no—
59C22×D4×C31—22no—
60C22×D4×C31—22no—
61C22×D4×C31D411yescharacteristic
62C23×C4×C31—21no—
63C2×SD16×C31—22no—
64C2×SD16×C31—22no—
65C2×M16×C31—21no—
66C22×C8×C31—21no—
67C22×C8×C31—21no—
68C31×C4⋊Q8D411yescharacteristic
69C31×C4⋊Q8C2311yescharacteristic
70C31×C4⋊Q8C2311yescharacteristic
71C31×C4⋊Q8C2311yescharacteristic
72C31×#127D411yescharacteristic
73C31×#127C2311yescharacteristic
74C31×#125—21no—
75C31×#124—21no—
76C31×#124—21no—
77C31×#123—24no—
78C31×#123—24no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—21no—
82C31×C22⋊Q8—21no—
83C31×#121—24no—
84C31×#121—24no—
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—21no—
88C31×#121—21no—
89C31×#121—22no—
90C31×#121—22no—
91C31×#121D412yes—
92C31×#121D412yes—
93C31×#120—24no—
94C31×#120—24no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31D412yes—
102C4×D4×C31D412yes—
103C4×D4×C31C2312yes—
104C4×D4×C31C2312yes—
105C31×#117—21no—
106C2×C4⋊C4×C31—22no—
107C2×C4⋊C4×C31D411yescharacteristic
108C2×C22⋊C4×C31—24no—
109C2×C22⋊C4×C31—21no—
110C2×C42×C31C2311yescharacteristic
111C31×#107—22no—
112C31×#106—22no—
113C31×#106—22no—
114C31×#106—22no—
115C31×C4⋊C8—21no—
116C31×C4⋊C8—21no—
117C31×C4⋊C8C2312yes—
118C31×C4⋊C8C2312yes—
119C31×#103—22no—
120C31×#103—22no—
121C31×#102—22no—
122C31×#102—22no—
123C31×#102—22no—
124C31×#102—22no—
125C4×C8×C31—21no—
126C4×C8×C31C2311yescharacteristic
127C4×C8×C31C2311yescharacteristic
128C24×C31—21no—
129C24×C31—24no—
130#41×C31—44no—
131#41×C31—48no—
132#41×C31—44no—
133#41×C31—42no—
134#41×C31—42no—
135#41×C31—42no—
136#41×C31—42no—
137C2×Q8×C31—21no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31—21no—
141C2×D4×C31—24no—
142C2×D4×C31—24no—
143C2×D4×C31—22no—
144C2×D4×C31—22no—
145C2×D4×C31—24no—
146C2×D4×C31—21no—
147C2×D4×C31—21no—
148C2×D4×C31—21no—
149C2×D4×C31—22no—
150C2×D4×C31—21no—
151C2×D4×C31—21no—
152C2×D4×C31—44no—
153C2×D4×C31—48no—
154C2×D4×C31—416no—
155C2×D4×C31—44no—
156C2×D4×C31—82no—
157C2×D4×C31—82no—
158C2×D4×C31SD1614yescomplemented
159C2×D4×C31C2×D412yes—
160C2×D4×C31C2×D412yes—
161C2×D4×C31C2×D412yes—
162C22×C4×C31—24no—
163C22×C4×C31—21no—
164C22×C4×C31—22no—
165C22×C4×C31—22no—
166C22×C4×C31—22no—
167C22×C4×C31—22no—
168C22×C4×C31—81no—
169C22×C4×C31—81no—
170C22×C4×C31C2×D411yescharacteristic
171C22×C4×C31C2×D411yescharacteristic
172C22×C4×C31C2×D411yescharacteristic
173SD16×C31—44no—
174SD16×C31—82no—
175M16×C31—42no—
176C2×C8×C31—21no—
177C2×C8×C31—21no—
178C2×C8×C31—22no—
179C2×C8×C31—22no—
180C2×C8×C31—42no—
181C2×C8×C31—81no—
182C4⋊C4×C31—24no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31—21no—
187C4⋊C4×C31—21no—
188C4⋊C4×C31—21no—
189C4⋊C4×C31—21no—
190C4⋊C4×C31—41no—
191C4⋊C4×C31—41no—
192C4⋊C4×C31C2×D412yes—
193C4⋊C4×C31C2×D412yes—
194C4⋊C4×C31C2×D412yes—
195C22⋊C4×C31—24no—
196C22⋊C4×C31—24no—
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—41no—
204C42×C31—41no—
205C42×C31C2×D411yescharacteristic
206C42×C31C2×D411yescharacteristic
207C42×C31C2×D411yescharacteristic
208C42×C31C2411yescharacteristic
209#18663C3111yesmaximal, characteristic, direct product factor, Sylow
210C23×C31—22no—
211C23×C31—24no—
212C23×C31—22no—
213C23×C31—22no—
214C23×C31—48no—
215C23×C31—81no—
216C23×C31—81no—
217C23×C31—84no—
218C23×C31—84no—
219C23×C31#12011yescharacteristic
220Q8×C31—42no—
221Q8×C31—41no—
222Q8×C31—41no—
223Q8×C31—42no—
224Q8×C31—41no—
225Q8×C31—41no—
226D4×C31—22no—
227D4×C31—22no—
228D4×C31—22no—
229D4×C31—24no—
230D4×C31—42no—
231D4×C31—48no—
232D4×C31—48no—
233D4×C31—44no—
234D4×C31—44no—
235D4×C31—41no—
236D4×C31—41no—
237D4×C31—41no—
238D4×C31—41no—
239D4×C31—42no—
240D4×C31—81no—
241D4×C31—84no—
242D4×C31C2×SD1614yes—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—41no—
249C2×C4×C31—48no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—44no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—82no—
258C2×C4×C31—81no—
259C2×C4×C31#12011yescharacteristic
260C2×C4×C31#12011yescharacteristic
261C2×C4×C31#12011yescharacteristic
262C2×C4×C31C2×SD1612yes—
263C2×C4×C31AGL(1,ℤ/8)12yes—
264C2×C4×C31C22×D411yescharacteristic
265C2×C4×C31C22×D411yescharacteristic
266C2×C4×C31C22×D411yescharacteristic
267C8×C31—22no—
268C8×C31—41no—
269C8×C31—42no—
270C8×C31—42no—
271#3496C2×C3111yescharacteristic
272#3246C2×C3111yescharacteristic, complemented
273#3167C2×C3111yescharacteristic, complemented
274#3077C2×C3111yescharacteristic, complemented
275#1654C2×C3112yescomplemented
276#1639C2×C3112yescomplemented
277#1635C2×C3112yescomplemented
278#1629C2×C3112yescomplemented
279#1547C2×C3111yescharacteristic, complemented
280#1492C2×C3111yescharacteristic, complemented
281C2×D4⋊C8C2×C3111yescharacteristic, complemented
282C22×C31—21no—
283C22×C31—21no—
284C22×C31—24no—
285C22×C31—42no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—42no—
292C22×C31—42no—
293C22×C31—84no—
294C22×C31—168no—
295C22×C31#45012yes—
296C22×C31C2×#12011yescharacteristic
297C4×C31—21no—
298C4×C31—21no—
299C4×C31—21no—
300C4×C31—42no—
301C4×C31—42no—
302C4×C31—42no—
303C4×C31—81no—
304C4×C31—81no—
305C4×C31#45012yes—
306C4×C31C22×SD1611yescharacteristic
307C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C31#57611yescharacteristic
309C4×C31#57711yescharacteristic
310C2×H(5,𝔽2)—21no—
311#547—21no—
312#547C22×C3112yes—
313#546—22no—
314#546—22no—
315D42—21no—
316D42C22×C3112yes—
317#537—21no—
318#534—22no—
319#533C22×C3111yescharacteristic, complemented
320#532C22×C3111yescharacteristic
321#532C22×C3111yescharacteristic
322C2×#121—22no—
323C2×C4×D4C22×C3111yescharacteristic
324#498C22×C3111yescharacteristic, complemented
325#492C22×C3111yescharacteristic, complemented
326#487C22×C3111yescharacteristic, complemented
327#486C22×C3111yescharacteristic, complemented
328#478—22no—
329#476C22×C3112yescomplemented
330#476C22×C3112yescomplemented
331#475—21no—
332#474C22×C3112yescomplemented
333#465—22no—
334#465—22no—
335#461C22×C3112yescomplemented
336#460—21no—
337C4×SD16—22no—
338#428—21no—
339#423C22×C3111yescharacteristic, complemented
340C2×C4⋊C8C22×C3111yescharacteristic, complemented
341#418—21no—
342C2×C4×C8C22×C3111yescharacteristic, complemented
343#332C22×C3112yescomplemented
344#331C22×C3112yescomplemented
345D4⋊C8C22×C3112yescomplemented
346D4⋊C8C22×C3112yescomplemented
347D4⋊C8C22×C3112yescomplemented
348D4⋊C8C22×C3112yescomplemented
349C2×C31—22no—
350C2×C31—44no—
351C2×C31—82no—
352C2×C31—82no—
353C2×C31C2×#45011yescharacteristic
354C2×C31#301211yescharacteristic
355C2×C31#301811yescharacteristic
356H(5,𝔽2)—44no—
357H(5,𝔽2)—44no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—21no—
361C2×#41—21no—
362C22×D4—24no—
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4D4×C3111yescharacteristic
367C23×C4—21no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×M16—21no—
371C22×C8—21no—
372C22×C8—21no—
373C4⋊Q8D4×C3111yescharacteristic
374C4⋊Q8C23×C3111yescharacteristic
375C4⋊Q8C23×C3111yescharacteristic
376C4⋊Q8C23×C3111yescharacteristic
377#127D4×C3111yescharacteristic
378#127C23×C3111yescharacteristic
379#125—21no—
380#124—21no—
381#124—21no—
382#123—24no—
383#123—24no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388#121—24no—
389#121—24no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—21no—
394#121—22no—
395#121—22no—
396#121D4×C3112yes—
397#121D4×C3112yes—
398#120—24no—
399#120—24no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4D4×C3112yes—
407C4×D4D4×C3112yes—
408C4×D4C23×C3112yes—
409C4×D4C23×C3112yes—
410#117—21no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4D4×C3111yescharacteristic
413C2×C22⋊C4—24no—
414C2×C22⋊C4—21no—
415C2×C42C23×C3111yescharacteristic
416#107—22no—
417#106—22no—
418#106—22no—
419#106—22no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8C23×C3112yes—
423C4⋊C8C23×C3112yes—
424#103—22no—
425#103—22no—
426#102—22no—
427#102—22no—
428#102—22no—
429#102—22no—
430C4×C8—21no—
431C4×C8C23×C3111yescharacteristic
432C4×C8C23×C3111yescharacteristic
433C31#1866311yescharacteristic, direct product factor, Sylow
434C24—21no—
435C24—24no—
436#41—48no—
437#41—44no—
438#41—44no—
439#41—42no—
440#41—42no—
441#41—42no—
442#41—42no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—48no—
459C2×D4—44no—
460C2×D4—416no—
461C2×D4—44no—
462C2×D4—82no—
463C2×D4—82no—
464C2×D4SD16×C3114yescomplemented
465C2×D4C2×D4×C3112yes—
466C2×D4C2×D4×C3112yes—
467C2×D4C2×D4×C3112yes—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4C2×D4×C3111yescharacteristic
477C22×C4C2×D4×C3111yescharacteristic
478C22×C4C2×D4×C3111yescharacteristic
479SD16—44no—
480SD16—82no—
481M16—42no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—42no—
487C2×C8—81no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4C2×D4×C3112yes—
499C4⋊C4C2×D4×C3112yes—
500C4⋊C4C2×D4×C3112yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—41no—
510C42—41no—
511C42C2×D4×C3111yescharacteristic
512C42C2×D4×C3111yescharacteristic
513C42C2×D4×C3111yescharacteristic
514C42C24×C3111yescharacteristic
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—48no—
520C23—81no—
521C23—81no—
522C23—84no—
523C23—84no—
524C23C31×#12011yescharacteristic
525Q8—42no—
526Q8—42no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531D4—24no—
532D4—22no—
533D4—22no—
534D4—22no—
535D4—42no—
536D4—48no—
537D4—48no—
538D4—44no—
539D4—44no—
540D4—42no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—81no—
546D4—84no—
547D4C2×SD16×C3114yes—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4C31×#12011yescharacteristic
565C2×C4C31×#12011yescharacteristic
566C2×C4C31×#12011yescharacteristic
567C2×C4C2×SD16×C3112yes—
568C2×C4C31×AGL(1,ℤ/8)12yes—
569C2×C4C22×D4×C3111yescharacteristic
570C2×C4C22×D4×C3111yescharacteristic
571C2×C4C22×D4×C3111yescharacteristic
572C8—22no—
573C8—41no—
574C8—42no—
575C8—42no—
576C22—21no—
577C22—21no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—84no—
588C22—168no—
589C22C31×#45012yes—
590C22C2×C31×#12011yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4C31×#45012yes—
600C4C22×SD16×C3111yescharacteristic
601C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
602C4C31×#57611yescharacteristic
603C4C31×#57711yescharacteristic
604C2—22no—
605C2—44no—
606C2—82no—
607C2—82no—
608C2C2×C31×#45011yescharacteristic
609C2C31×#301211yescharacteristic
610C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3496C2
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×#3246C2semidirect
3. C31×#3167C2semidirect
4. C31×#3077C2semidirect
5. C31×#1654C2semidirect
6. C31×#1639C2semidirect
7. C31×#1635C2semidirect
8. C31×#1629C2semidirect
9. C31×#1547C2semidirect
10. C31×#1492C2semidirect
11. C2×C31×D4⋊C8C2semidirect
21. C31×#533C22semidirect
26. C31×#498C22semidirect
27. C31×#492C22semidirect
28. C31×#487C22semidirect
29. C31×#486C22semidirect
31. C31×#476C22semidirect
32. C31×#476C22semidirect
34. C31×#474C22semidirect
37. C31×#461C22semidirect
41. C31×#423C22semidirect
42. C2×C31×C4⋊C8C22semidirect
44. C2×C4×C8×C31C22semidirect
45. C31×#332C22semidirect
46. C31×#331C22semidirect
47. C31×D4⋊C8C22semidirect
48. C31×D4⋊C8C22semidirect
49. C31×D4⋊C8C22semidirect
50. C31×D4⋊C8C22semidirect
158. C2×D4×C31SD16semidirect
209. #18663C31direct (N × H)
272. #3246C2×C31semidirect
273. #3167C2×C31semidirect
274. #3077C2×C31semidirect
275. #1654C2×C31semidirect
276. #1639C2×C31semidirect
277. #1635C2×C31semidirect
278. #1629C2×C31semidirect
279. #1547C2×C31semidirect
280. #1492C2×C31semidirect
281. C2×D4⋊C8C2×C31semidirect
319. #533C22×C31semidirect
324. #498C22×C31semidirect
325. #492C22×C31semidirect
326. #487C22×C31semidirect
327. #486C22×C31semidirect
329. #476C22×C31semidirect
330. #476C22×C31semidirect
332. #474C22×C31semidirect
335. #461C22×C31semidirect
339. #423C22×C31semidirect
340. C2×C4⋊C8C22×C31semidirect
342. C2×C4×C8C22×C31semidirect
343. #332C22×C31semidirect
344. #331C22×C31semidirect
345. D4⋊C8C22×C31semidirect
346. D4⋊C8C22×C31semidirect
347. D4⋊C8C22×C31semidirect
348. D4⋊C8C22×C31semidirect
433. C31#18663direct (N × H)
464. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34962
C31×#32462
C31×#31672
C31×#30772
C31×#16542
C31×#16392
C31×#16352
C31×#16292
C31×#15472
C31×#14922
C2×C31×D4⋊C82
#1866331