Finite Group Databasebeta

← C31×#18134 #21474893

Subgroups — C31×#18134 #21474893

Order 7936.

1202 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#539C211yesmaximal, characteristic, complemented
2C31×#3007C211yesmaximal, characteristic, complemented
3C31×#3006C211yesmaximal, characteristic
4C31×#2406C211yesmaximal, characteristic, complemented
5C31×#1603C214yesmaximal, complemented
6C31×#1597C214yesmaximal, complemented
7C31×#1511C211yesmaximal, characteristic, complemented
8C31×#1510C211yesmaximal, characteristic, complemented
9C31×#1463C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C2×C31×#137—21no—
12C2×C31×AGL(1,ℤ/8)—21no—
13C31×#539—22no—
14C31×#539—22no—
15C31×#539—22no—
16C31×#539C2214yes—
17C2×C31×#126—21no—
18C2×C31×#125—21no—
19C2×C31×#123—21no—
20C2×C31×#123—21no—
21C2×C31×#123—21no—
22C2×C31×C22⋊Q8—21no—
23C2×C31×C22⋊Q8C2211yescharacteristic
24C2×C31×#121C2211yescharacteristic
25C2×C4×D4×C31—21no—
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#452—22no—
28C31×#451C2212yescomplemented
29C31×#450C2212yescomplemented
30C31×#449—22no—
31C31×#448C2212yes—
32C31×#447C2212yes—
33C31×#417—21no—
34C31×#417—21no—
35C31×#407C2211yescharacteristic
36C31×#407C2211yescharacteristic
37C31×#395C2212yescomplemented
38C31×#394C2212yescomplemented
39C31×#386—21no—
40C31×#385—22no—
41C31×#381C2211yescharacteristic
42C31×#328C2214yescomplemented
43C31×#327C2214yes—
44C31×#324C2214yescomplemented
45C2×#41×C31—21no—
46C2×#41×C31—21no—
47C2×#41×C31—21no—
48C2×#41×C31—21no—
49C2×#41×C31—44no—
50C22×Q8×C31D411yescharacteristic
51C22×Q8×C31C2311yescharacteristic
52C22×D4×C31—21no—
53C22×D4×C31—21no—
54C22×D4×C31D411yescharacteristic
55C22×D4×C31C2311yescharacteristic
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C23×C4×C31D411yescharacteristic
59C23×C4×C31D411yescharacteristic
60C23×C4×C31C2311yescharacteristic
61C31×#137—44no—
62C31×AGL(1,ℤ/8)—44no—
63C2×Q16×C31—22no—
64C2×SD16×C31—22no—
65C2×SD16×C31—22no—
66C2×D8×C31—22no—
67C2×M16×C31—21no—
68C2×M16×C31—21no—
69C31×#126—24no—
70C31×#126—24no—
71C31×#125—24no—
72C31×#125—24no—
73C31×#123—24no—
74C31×#123—24no—
75C31×#123—24no—
76C31×#123—24no—
77C31×#123—44no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—21no—
82C31×C22⋊Q8D412yes—
83C31×C22⋊Q8C2312yes—
84C31×#121—21no—
85C31×#121—21no—
86C31×#121D412yes—
87C31×#121C2312yes—
88C4×D4×C31—24no—
89C4×D4×C31—24no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—21no—
94C2×C4⋊C4×C31—21no—
95C2×C4⋊C4×C31—21no—
96C2×C4⋊C4×C31—21no—
97C2×C4⋊C4×C31—21no—
98C2×C4⋊C4×C31—22no—
99C2×C4⋊C4×C31D412yes—
100C2×C4⋊C4×C31D411yescharacteristic
101C2×C4⋊C4×C31D411yescharacteristic
102C2×C4⋊C4×C31C2312yes—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31—21no—
105C2×C22⋊C4×C31—21no—
106C2×C22⋊C4×C31—21no—
107C2×C22⋊C4×C31—21no—
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—22no—
110C2×C22⋊C4×C31—22no—
111C2×C42×C31—21no—
112C31×#103—22no—
113C31×#103—22no—
114C31×#102—22no—
115C31×#102—22no—
116C31×C22⋊C8—21no—
117C31×C22⋊C8—21no—
118C31×C22⋊C8C2312yes—
119C31×C22⋊C8C2312yes—
120C31×#95—22no—
121C31×#95—21no—
122C31×#95C2312yes—
123C24×C31—21no—
124C24×C31—21no—
125C24×C31C2×D411yescharacteristic
126#41×C31—24no—
127#41×C31—24no—
128#41×C31—42no—
129#41×C31—42no—
130#41×C31—42no—
131#41×C31—42no—
132#41×C31—42no—
133#41×C31—42no—
134C2×Q8×C31—21no—
135C2×Q8×C31—21no—
136C2×Q8×C31—21no—
137C2×Q8×C31—42no—
138C2×Q8×C31—42no—
139C2×Q8×C31C2×D412yes—
140C2×D4×C31—24no—
141C2×D4×C31—22no—
142C2×D4×C31—22no—
143C2×D4×C31—21no—
144C2×D4×C31—21no—
145C2×D4×C31—21no—
146C2×D4×C31—42no—
147C2×D4×C31—42no—
148C2×D4×C31—44no—
149C2×D4×C31—42no—
150C2×D4×C31—41no—
151C2×D4×C31—41no—
152C2×D4×C31C2×D412yes—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—21no—
159C22×C4×C31—21no—
160C22×C4×C31—24no—
161C22×C4×C31—21no—
162C22×C4×C31—22no—
163C22×C4×C31—42no—
164C22×C4×C31—42no—
165C22×C4×C31—42no—
166C22×C4×C31—42no—
167C22×C4×C31—41no—
168C22×C4×C31—41no—
169C22×C4×C31—42no—
170C22×C4×C31—82no—
171C22×C4×C31C2×D412yes—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D412yes—
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2411yescharacteristic
179Q16×C31—44no—
180SD16×C31—44no—
181SD16×C31—44no—
182D8×C31—44no—
183M16×C31—42no—
184M16×C31—42no—
185C2×C8×C31—22no—
186C2×C8×C31—22no—
187C4⋊C4×C31—24no—
188C4⋊C4×C31—24no—
189C4⋊C4×C31—24no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—44no—
199C4⋊C4×C31C2×D412yes—
200C4⋊C4×C31C2×D412yes—
201C22⋊C4×C31—24no—
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—44no—
209C22⋊C4×C31—44no—
210C22⋊C4×C31—42no—
211C42×C31—24no—
212#18134C3111yesmaximal, characteristic, direct product factor, Sylow
213C23×C31—22no—
214C23×C31—22no—
215C23×C31—22no—
216C23×C31—22no—
217C23×C31—41no—
218C23×C31—41no—
219C23×C31—42no—
220C23×C31—42no—
221C23×C31—42no—
222C23×C31—41no—
223C23×C31—42no—
224C23×C31—42no—
225C23×C31#12011yescharacteristic
226C23×C31#12011yescharacteristic
227C23×C31C22×D411yescharacteristic
228Q8×C31—22no—
229Q8×C31—22no—
230Q8×C31—42no—
231Q8×C31—41no—
232Q8×C31—41no—
233Q8×C31—81no—
234D4×C31—22no—
235D4×C31—22no—
236D4×C31—42no—
237D4×C31—41no—
238D4×C31—41no—
239D4×C31—44no—
240D4×C31—44no—
241D4×C31—44no—
242D4×C31—44no—
243D4×C31—81no—
244C2×C4×C31—22no—
245C2×C4×C31—22no—
246C2×C4×C31—22no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—22no—
254C2×C4×C31—41no—
255C2×C4×C31—41no—
256C2×C4×C31—41no—
257C2×C4×C31—41no—
258C2×C4×C31—44no—
259C2×C4×C31—44no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—41no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—81no—
273C2×C4×C31—84no—
274C2×C4×C31#12011yescharacteristic
275C2×C4×C31#12011yescharacteristic
276C2×C4×C31C22×D411yescharacteristic
277C2×C4×C31C22×D411yescharacteristic
278C8×C31—42no—
279C8×C31—42no—
280C2×#539C2×C3111yescharacteristic, complemented
281#3007C2×C3111yescharacteristic, complemented
282#3006C2×C3111yescharacteristic
283#2406C2×C3111yescharacteristic, complemented
284#1603C2×C3114yescomplemented
285#1597C2×C3114yescomplemented
286#1511C2×C3111yescharacteristic, complemented
287#1510C2×C3111yescharacteristic, complemented
288#1463C2×C3111yescharacteristic, complemented
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—21no—
292C22×C31—21no—
293C22×C31—21no—
294C22×C31—21no—
295C22×C31—21no—
296C22×C31—44no—
297C22×C31—44no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—42no—
302C22×C31—82no—
303C22×C31—84no—
304C22×C31—81no—
305C22×C31—81no—
306C22×C31B(4,𝔽2)12yes—
307C22×C31C2×#12011yescharacteristic
308C22×C31#57512yes—
309C4×C31—21no—
310C4×C31—21no—
311C4×C31—21no—
312C4×C31—21no—
313C4×C31—42no—
314C4×C31—42no—
315C4×C31—42no—
316C4×C31—42no—
317C4×C31—42no—
318C4×C31—42no—
319C4×C31—82no—
320C22×#41C22×C3111yescharacteristic
321C2×#137—21no—
322C2×AGL(1,ℤ/8)—21no—
323#539—22no—
324#539—22no—
325#539—22no—
326#539C22×C3114yes—
327C2×#126—21no—
328C2×#125—21no—
329C2×#123—21no—
330C2×#123—21no—
331C2×#123—21no—
332C2×C22⋊Q8—21no—
333C2×C22⋊Q8C22×C3111yescharacteristic
334C2×#121C22×C3111yescharacteristic
335C2×C4×D4—21no—
336C22×C4⋊C4C22×C3111yescharacteristic
337#452—22no—
338#451C22×C3112yescomplemented
339#450C22×C3112yescomplemented
340#449—22no—
341#448C22×C3112yes—
342#447C22×C3112yes—
343#417—21no—
344#417—21no—
345#407C22×C3111yescharacteristic
346#407C22×C3111yescharacteristic
347#395C22×C3112yescomplemented
348#394C22×C3112yescomplemented
349#386—21no—
350#385—22no—
351#381C22×C3111yescharacteristic
352#328C22×C3114yescomplemented
353#327C22×C3114yes—
354#324C22×C3114yescomplemented
355C2×C31—22no—
356C2×C31—22no—
357C2×C31—41no—
358C2×C31—82no—
359C2×C31—82no—
360C2×C31#300511yescharacteristic
361C2×C31#302011yescharacteristic
362C2×C31C2×B(4,𝔽2)11yescharacteristic
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—44no—
368C22×Q8D4×C3111yescharacteristic
369C22×Q8C23×C3111yescharacteristic
370C22×D4—21no—
371C22×D4—21no—
372C22×D4D4×C3111yescharacteristic
373C22×D4C23×C3111yescharacteristic
374C23×C4—21no—
375C23×C4—21no—
376C23×C4D4×C3111yescharacteristic
377C23×C4D4×C3111yescharacteristic
378C23×C4C23×C3111yescharacteristic
379#137—44no—
380AGL(1,ℤ/8)—44no—
381C2×Q16—22no—
382C2×SD16—22no—
383C2×SD16—22no—
384C2×D8—22no—
385C2×M16—21no—
386C2×M16—21no—
387#126—24no—
388#126—24no—
389#125—24no—
390#125—24no—
391#123—24no—
392#123—24no—
393#123—24no—
394#123—24no—
395#123—44no—
396C22⋊Q8—24no—
397C22⋊Q8—24no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8D4×C3112yes—
401C22⋊Q8C23×C3112yes—
402#121—21no—
403#121—21no—
404#121D4×C3112yes—
405#121C23×C3112yes—
406C4×D4—24no—
407C4×D4—24no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4D4×C3112yes—
418C2×C4⋊C4D4×C3111yescharacteristic
419C2×C4⋊C4D4×C3111yescharacteristic
420C2×C4⋊C4C23×C3112yes—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C42—21no—
430#103—22no—
431#103—22no—
432#102—22no—
433#102—22no—
434C22⋊C8—21no—
435C22⋊C8—21no—
436C22⋊C8C23×C3112yes—
437C22⋊C8C23×C3112yes—
438#95—21no—
439#95—22no—
440#95C23×C3112yes—
441C31#1813411yescharacteristic, direct product factor, Sylow
442C24—21no—
443C24—21no—
444C24C2×D4×C3111yescharacteristic
445#41—24no—
446#41—24no—
447#41—42no—
448#41—42no—
449#41—42no—
450#41—42no—
451#41—42no—
452#41—42no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8C2×D4×C3112yes—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4C2×D4×C3112yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—42no—
489C22×C4—82no—
490C22×C4C2×D4×C3112yes—
491C22×C4C2×D4×C3111yescharacteristic
492C22×C4C2×D4×C3111yescharacteristic
493C22×C4C2×D4×C3112yes—
494C22×C4C2×D4×C3111yescharacteristic
495C22×C4C2×D4×C3111yescharacteristic
496C22×C4C2×D4×C3111yescharacteristic
497C22×C4C24×C3111yescharacteristic
498Q16—44no—
499SD16—44no—
500SD16—44no—
501D8—44no—
502M16—42no—
503M16—42no—
504C2×C8—22no—
505C2×C8—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—44no—
518C4⋊C4C2×D4×C3112yes—
519C4⋊C4C2×D4×C3112yes—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C42—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23C31×#12011yescharacteristic
544C23C31×#12011yescharacteristic
545C23C22×D4×C3111yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—42no—
549Q8—41no—
550Q8—41no—
551Q8—81no—
552D4—22no—
553D4—22no—
554D4—42no—
555D4—41no—
556D4—41no—
557D4—44no—
558D4—44no—
559D4—44no—
560D4—44no—
561D4—81no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—81no—
591C2×C4—84no—
592C2×C4C31×#12011yescharacteristic
593C2×C4C31×#12011yescharacteristic
594C2×C4C22×D4×C3111yescharacteristic
595C2×C4C22×D4×C3111yescharacteristic
596C8—42no—
597C8—42no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—82no—
612C22—84no—
613C22—81no—
614C22—81no—
615C22C31×B(4,𝔽2)12yes—
616C22C2×C31×#12011yescharacteristic
617C22C31×#57512yes—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—82no—
629C2—22no—
630C2—22no—
631C2—41no—
632C2—82no—
633C2—82no—
634C2C31×#300511yescharacteristic
635C2C31×#302011yescharacteristic
636C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3006C2
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. C2×C31×#539C2semidirect
2. C31×#3007C2semidirect
4. C31×#2406C2semidirect
5. C31×#1603C2semidirect
6. C31×#1597C2semidirect
7. C31×#1511C2semidirect
8. C31×#1510C2semidirect
9. C31×#1463C2semidirect
28. C31×#451C22semidirect
29. C31×#450C22semidirect
37. C31×#395C22semidirect
38. C31×#394C22semidirect
42. C31×#328C22semidirect
44. C31×#324C22semidirect
212. #18134C31direct (N × H)
280. C2×#539C2×C31semidirect
281. #3007C2×C31semidirect
283. #2406C2×C31semidirect
284. #1603C2×C31semidirect
285. #1597C2×C31semidirect
286. #1511C2×C31semidirect
287. #1510C2×C31semidirect
288. #1463C2×C31semidirect
338. #451C22×C31semidirect
339. #450C22×C31semidirect
347. #395C22×C31semidirect
348. #394C22×C31semidirect
352. #328C22×C31semidirect
354. #324C22×C31semidirect
441. C31#18134direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5392
C31×#30072
C31×#30062
C31×#24062
C31×#16032
C31×#15972
C31×#15112
C31×#15102
C31×#14632
#1813431