Finite Group Databasebeta

← C31×#18624 #21475383

Subgroups — C31×#18624 #21475383

Order 7936.

1152 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3508C211yesmaximal, characteristic, complemented
2C31×#3208C211yesmaximal, characteristic, complemented
3C2×C31×#493C211yesmaximal, characteristic, complemented
4C31×#3038C211yesmaximal, characteristic, complemented
5C31×#1621C214yesmaximal, complemented
6C31×#1620C214yesmaximal, complemented
7C31×#1545C211yesmaximal, characteristic, complemented
8C31×#1493C211yesmaximal, characteristic, complemented
9C31×#1476C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)—21no—
11C22×D8×C31—22no—
12C31×#559—21no—
13C31×#559C2212yescomplemented
14C31×#550—21no—
15C31×#550C2212yescomplemented
16C31×#546—22no—
17C31×#546—22no—
18C31×#540—22no—
19C31×#535—21no—
20C31×#533C2211yescharacteristic
21C31×#532—21no—
22C31×#532—21no—
23C2×C31×#127C2211yescharacteristic
24C2×C31×#127C2211yescharacteristic
25C31×#518—21no—
26C4×#41×C31C2211yescharacteristic
27C31×#493—22no—
28C31×#493C2214yes—
29C31×#492—21no—
30C31×#486—21no—
31C31×#460—21no—
32C31×#460C2212yescomplemented
33C31×#460C2214yescomplemented
34C31×#459—21no—
35C31×#459C2212yescomplemented
36C31×#450—22no—
37C31×#431C2211yescharacteristic
38C31×#423C2211yescharacteristic
39C31×#418—21no—
40C2×C4×C8×C31C2211yescharacteristic
41C31×#331C2214yescomplemented
42C31×Q8⋊C8C2212yescomplemented
43C31×Q8⋊C8C2214yescomplemented
44C31×D4⋊C8C2212yescomplemented
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31D411yescharacteristic
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C22×D4×C31—21no—
53C22×D4×C31D412yes—
54C31×AGL(1,ℤ/8)—44no—
55C2×SD16×C31—22no—
56C2×SD16×C31—24no—
57C2×D8×C31—22no—
58C2×D8×C31—28no—
59C2×D8×C31—48no—
60C2×M16×C31—21no—
61C22×C8×C31—21no—
62C31×C4⋊Q8D411yescharacteristic
63C31×C4⋊Q8D411yescharacteristic
64C31×#127—22no—
65C31×#127—21no—
66C31×#127—21no—
67C31×#127—21no—
68C31×#127—21no—
69C31×#127—21no—
70C31×#127D412yes—
71C31×#127C2312yes—
72C31×#127C2312yes—
73C31×#125—21no—
74C31×#124—22no—
75C31×#124—22no—
76C31×#124—22no—
77C31×#123—24no—
78C31×#123—24no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8D412yes—
81C31×#121—22no—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121—24no—
85C31×#121—24no—
86C31×#121—22no—
87C31×#120—24no—
88C4×Q8×C31—21no—
89C4×Q8×C31D411yescharacteristic
90C4×Q8×C31C2311yescharacteristic
91C4×Q8×C31C2312yes—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C4×D4×C31—22no—
98C4×D4×C31—22no—
99C4×D4×C31—22no—
100C4×D4×C31D411yescharacteristic
101C4×D4×C31C2311yescharacteristic
102C31×#117—21no—
103C31×#117—21no—
104C31×#117D411yescharacteristic
105C2×C22⋊C4×C31—24no—
106C2×C42×C31—21no—
107C2×C42×C31C2311yescharacteristic
108C31×C4⋊C8—21no—
109C31×C4⋊C8C2312yes—
110C31×C4⋊C8C2312yes—
111C31×#102—22no—
112C31×#102—22no—
113C31×#102—24no—
114C31×C22⋊C8—21no—
115C4×C8×C31—21no—
116C4×C8×C31—21no—
117C4×C8×C31C2312yes—
118C24×C31—22no—
119C24×C31—41no—
120#41×C31—22no—
121#41×C31—22no—
122#41×C31—84no—
123#41×C31—84no—
124C2×Q8×C31—21no—
125C2×Q8×C31C2×D411yescharacteristic
126C2×Q8×C31C2×D412yes—
127C2×D4×C31—21no—
128C2×D4×C31—24no—
129C2×D4×C31—24no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—44no—
139C2×D4×C31—44no—
140C2×D4×C31—41no—
141C2×D4×C31—41no—
142C2×D4×C31—41no—
143C2×D4×C31—42no—
144C2×D4×C31—42no—
145C2×D4×C31—42no—
146C2×D4×C31—42no—
147C2×D4×C31—82no—
148C2×D4×C31—82no—
149C2×D4×C31C2×D411yescharacteristic
150C2×D4×C31C2×D414yes—
151C22×C4×C31—24no—
152C22×C4×C31—24no—
153C22×C4×C31—21no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159SD16×C31—44no—
160SD16×C31—84no—
161D8×C31—416no—
162D8×C31—44no—
163M16×C31—42no—
164C2×C8×C31—24no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31C2×D411yescharacteristic
176C4⋊C4×C31C2×D411yescharacteristic
177C4⋊C4×C31C2×D412yes—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—24no—
181C22⋊C4×C31—24no—
182C22⋊C4×C31—21no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—21no—
185C22⋊C4×C31—21no—
186C42×C31—21no—
187C42×C31—21no—
188C42×C31—21no—
189C42×C31—22no—
190C42×C31C2×D411yescharacteristic
191C42×C31C2×D411yescharacteristic
192C42×C31C2×D411yescharacteristic
193C42×C31C2411yescharacteristic
194#18624C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—24no—
196C23×C31—21no—
197C23×C31—44no—
198C23×C31—42no—
199C23×C31—82no—
200C23×C31—82no—
201C23×C31—82no—
202C23×C31—82no—
203C23×C31—82no—
204C23×C31#12011yescharacteristic
205Q8×C31—21no—
206Q8×C31—21no—
207Q8×C31—24no—
208Q8×C31—42no—
209D4×C31—21no—
210D4×C31—21no—
211D4×C31—44no—
212D4×C31—44no—
213D4×C31—42no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—42no—
218D4×C31—44no—
219D4×C31—44no—
220D4×C31—44no—
221D4×C31—44no—
222D4×C31—82no—
223D4×C31—82no—
224D4×C31—84no—
225D4×C31—84no—
226D4×C31—82no—
227D4×C31—82no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—84no—
239C2×C4×C31—84no—
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#13512yes—
244C2×C4×C31AGL(1,ℤ/8)12yes—
245C2×C4×C31C22×D411yescharacteristic
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C22×D411yescharacteristic
248C8×C31—24no—
249C8×C31—42no—
250C8×C31—42no—
251#3508C2×C3111yescharacteristic, complemented
252#3208C2×C3111yescharacteristic, complemented
253C2×#493C2×C3111yescharacteristic, complemented
254#3038C2×C3111yescharacteristic, complemented
255#1621C2×C3114yescomplemented
256#1620C2×C3114yescomplemented
257#1545C2×C3111yescharacteristic, complemented
258#1493C2×C3111yescharacteristic, complemented
259#1476C2×C3111yescharacteristic, complemented
260C22×C31—21no—
261C22×C31—21no—
262C22×C31—41no—
263C22×C31—44no—
264C22×C31—41no—
265C22×C31—44no—
266C22×C31—41no—
267C22×C31—44no—
268C22×C31—82no—
269C22×C31—82no—
270C22×C31—82no—
271C22×C31—164no—
272C22×C31—164no—
273C22×C31#44912yes—
274C22×C31C2×#12011yescharacteristic
275C4×C31—21no—
276C4×C31—21no—
277C4×C31—41no—
278C4×C31—41no—
279C4×C31—42no—
280C4×C31—41no—
281C4×C31—42no—
282C4×C31#44912yes—
283C4×C31#45312yes—
284C4×C31C2×#13511yescharacteristic
285C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
286C4×C31#57612yes—
287C2×AGL(1,ℤ/8)—21no—
288C22×D8—22no—
289#559—21no—
290#559C22×C3112yescomplemented
291#550—21no—
292#550C22×C3112yescomplemented
293#546—22no—
294#546—22no—
295#540—22no—
296#535—21no—
297#533C22×C3111yescharacteristic
298#532—21no—
299#532—21no—
300C2×#127C22×C3111yescharacteristic
301C2×#127C22×C3111yescharacteristic
302#518—21no—
303C4×#41C22×C3111yescharacteristic
304#493—22no—
305#493C22×C3114yes—
306#492—21no—
307#486—21no—
308#460—21no—
309#460C22×C3112yescomplemented
310#460C22×C3114yescomplemented
311#459—21no—
312#459C22×C3112yescomplemented
313#450—22no—
314#431C22×C3111yescharacteristic
315#423C22×C3111yescharacteristic
316#418—21no—
317C2×C4×C8C22×C3111yescharacteristic
318#331C22×C3114yescomplemented
319Q8⋊C8C22×C3112yescomplemented
320Q8⋊C8C22×C3114yescomplemented
321D4⋊C8C22×C3112yescomplemented
322C2×C31—22no—
323C2×C31—81no—
324C2×C31—84no—
325C2×C31—162no—
326C2×C31C2×#44911yescharacteristic
327C2×C31#300811yescharacteristic
328C2×C31C2×#45311yescharacteristic
329C2×#41—22no—
330C2×#41—22no—
331C2×#41D4×C3111yescharacteristic
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—21no—
337C22×D4D4×C3112yes—
338AGL(1,ℤ/8)—44no—
339C2×SD16—22no—
340C2×SD16—24no—
341C2×D8—28no—
342C2×D8—22no—
343C2×D8—48no—
344C2×M16—21no—
345C22×C8—21no—
346C4⋊Q8D4×C3111yescharacteristic
347C4⋊Q8D4×C3111yescharacteristic
348#127—22no—
349#127—21no—
350#127—21no—
351#127—21no—
352#127—21no—
353#127—21no—
354#127D4×C3112yes—
355#127C23×C3112yes—
356#127C23×C3112yes—
357#125—21no—
358#124—22no—
359#124—22no—
360#124—22no—
361#123—24no—
362#123—24no—
363C22⋊Q8—21no—
364C22⋊Q8D4×C3112yes—
365#121—22no—
366#121—22no—
367#121—22no—
368#121—24no—
369#121—24no—
370#121—22no—
371#120—24no—
372C4×Q8—21no—
373C4×Q8D4×C3111yescharacteristic
374C4×Q8C23×C3111yescharacteristic
375C4×Q8C23×C3112yes—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4D4×C3111yescharacteristic
385C4×D4C23×C3111yescharacteristic
386#117—21no—
387#117—21no—
388#117D4×C3111yescharacteristic
389C2×C22⋊C4—24no—
390C2×C42—21no—
391C2×C42C23×C3111yescharacteristic
392C4⋊C8—21no—
393C4⋊C8C23×C3112yes—
394C4⋊C8C23×C3112yes—
395#102—22no—
396#102—22no—
397#102—24no—
398C22⋊C8—21no—
399C4×C8—21no—
400C4×C8—21no—
401C4×C8C23×C3112yes—
402C31#1862411yescharacteristic, direct product factor, Sylow
403C24—22no—
404C24—41no—
405#41—22no—
406#41—22no—
407#41—84no—
408#41—84no—
409C2×Q8—21no—
410C2×Q8C2×D4×C3111yescharacteristic
411C2×Q8C2×D4×C3112yes—
412C2×D4—21no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4C2×D4×C3111yescharacteristic
435C2×D4C2×D4×C3114yes—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444SD16—44no—
445SD16—84no—
446D8—416no—
447D8—44no—
448M16—42no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4C2×D4×C3111yescharacteristic
461C4⋊C4C2×D4×C3111yescharacteristic
462C4⋊C4C2×D4×C3112yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42C2×D4×C3111yescharacteristic
476C42C2×D4×C3111yescharacteristic
477C42C2×D4×C3111yescharacteristic
478C42C24×C3111yescharacteristic
479C23—24no—
480C23—21no—
481C23—44no—
482C23—42no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23C31×#12011yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—24no—
492Q8—42no—
493D4—21no—
494D4—21no—
495D4—44no—
496D4—44no—
497D4—42no—
498D4—44no—
499D4—44no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—44no—
505D4—44no—
506D4—82no—
507D4—82no—
508D4—84no—
509D4—84no—
510D4—82no—
511D4—82no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4C31×#12011yescharacteristic
525C2×C4C31×#12011yescharacteristic
526C2×C4C31×#12011yescharacteristic
527C2×C4C31×#13512yes—
528C2×C4C31×AGL(1,ℤ/8)12yes—
529C2×C4C22×D4×C3111yescharacteristic
530C2×C4C22×D4×C3111yescharacteristic
531C2×C4C22×D4×C3111yescharacteristic
532C8—24no—
533C8—42no—
534C8—42no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—44no—
539C22—41no—
540C22—44no—
541C22—41no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—164no—
547C22—164no—
548C22C31×#44912yes—
549C22C2×C31×#12011yescharacteristic
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4C31×#44912yes—
558C4C31×#45312yes—
559C4C2×C31×#13511yescharacteristic
560C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
561C4C31×#57612yes—
562C2—22no—
563C2—84no—
564C2—81no—
565C2—162no—
566C2C2×C31×#44911yescharacteristic
567C2C31×#300811yescharacteristic
568C2C2×C31×#45311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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
1. C31×#3508C2semidirect
2. C31×#3208C2semidirect
3. C2×C31×#493C2semidirect
4. C31×#3038C2semidirect
5. C31×#1621C2semidirect
6. C31×#1620C2semidirect
7. C31×#1545C2semidirect
8. C31×#1493C2semidirect
9. C31×#1476C2semidirect
13. C31×#559C22semidirect
15. C31×#550C22semidirect
32. C31×#460C22semidirect
33. C31×#460C22semidirect
35. C31×#459C22semidirect
41. C31×#331C22semidirect
42. C31×Q8⋊C8C22semidirect
43. C31×Q8⋊C8C22semidirect
44. C31×D4⋊C8C22semidirect
194. #18624C31direct (N × H)
251. #3508C2×C31semidirect
252. #3208C2×C31semidirect
253. C2×#493C2×C31semidirect
254. #3038C2×C31semidirect
255. #1621C2×C31semidirect
256. #1620C2×C31semidirect
257. #1545C2×C31semidirect
258. #1493C2×C31semidirect
259. #1476C2×C31semidirect
290. #559C22×C31semidirect
292. #550C22×C31semidirect
309. #460C22×C31semidirect
310. #460C22×C31semidirect
312. #459C22×C31semidirect
318. #331C22×C31semidirect
319. Q8⋊C8C22×C31semidirect
320. Q8⋊C8C22×C31semidirect
321. D4⋊C8C22×C31semidirect
402. C31#18624direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35082
C31×#32082
C2×C31×#4932
C31×#30382
C31×#16212
C31×#16202
C31×#15452
C31×#14932
C31×#14762
#1862431