Finite Group Databasebeta

← C31×#18356 #21475115

Subgroups — C31×#18356 #21475115

Order 7936.

1054 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3491C211yesmaximal, characteristic, complemented
2C31×#3217C212yesmaximal, complemented
3C31×#2619C211yesmaximal, characteristic, complemented
4C31×#1597C214yesmaximal, complemented
5C31×#1596C212yesmaximal, complemented
6C31×#1595C212yesmaximal, complemented
7C31×#1512C212yesmaximal, complemented
8C31×#1464C211yesmaximal, characteristic, complemented
9C31×#552—22no—
10C31×#552—22no—
11C31×#551—22no—
12C31×#551—22no—
13C31×#543C2211yescharacteristic
14C31×#539—21no—
15C31×#539C2212yes—
16C31×#534C2211yescharacteristic
17C2×C31×#127C2211yescharacteristic, complemented
18C31×#529—21no—
19C31×#529C2212yes—
20C2×C31×#123—21no—
21C2×C31×#121—21no—
22C31×#518—21no—
23C2×C31×#117C2211yescharacteristic
24C31×#496—22no—
25C31×#490—22no—
26C31×#460—22no—
27C31×#459—22no—
28C31×#450C2214yescomplemented
29C31×#447C2214yes—
30C31×#432C2212yescomplemented
31C31×#394—21no—
32C31×#394C2212yescomplemented
33C31×#392—21no—
34C31×#392C2212yes—
35C31×#390—21no—
36C31×#382C2211yescharacteristic, complemented
37C31×#328C2214yescomplemented
38C31×#327C2214yes—
39C31×#324C2212yescomplemented
40C31×#323C2212yes—
41C22×D4×C31—21no—
42C22×D4×C31—22no—
43C22×D4×C31—21no—
44C22×D4×C31—21no—
45C22×D4×C31C2312yes—
46C23×C4×C31D411yescharacteristic
47C2×SD16×C31—24no—
48C2×D8×C31—24no—
49C31×#127—22no—
50C31×#127—22no—
51C31×#126—22no—
52C31×#126—22no—
53C31×#126—22no—
54C31×#126—22no—
55C31×#125—21no—
56C31×#125—22no—
57C31×#125—22no—
58C31×#125D412yescomplemented
59C31×#124—21no—
60C31×#124D412yescomplemented
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—21no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8C2312yes—
74C31×#121—21no—
75C31×#121—44no—
76C31×#121C2312yes—
77C31×#120—22no—
78C4×Q8×C31—21no—
79C4×Q8×C31D412yescomplemented
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C4×D4×C31—21no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31D412yescomplemented
86C31×#117—22no—
87C31×#117—22no—
88C31×#117—22no—
89C31×#117—22no—
90C31×#117—22no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—21no—
94C2×C4⋊C4×C31D411yescharacteristic
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—22no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31D411yescharacteristic
104C2×C22⋊C4×C31C2311yescharacteristic
105C2×C42×C31D411yescharacteristic
106C2×C42×C31C2311yescharacteristic
107C31×C4⋊C8—22no—
108C31×#102—24no—
109C31×#102—24no—
110C31×C22⋊C8C2314yes—
111C31×#97—22no—
112C31×#95—21no—
113C31×#95C2312yes—
114C24×C31—21no—
115C24×C31—22no—
116C24×C31C2×D411yescharacteristic
117C2×Q8×C31—21no—
118C2×Q8×C31C2×D412yes—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—21no—
122C2×D4×C31—24no—
123C2×D4×C31—42no—
124C2×D4×C31—42no—
125C2×D4×C31—42no—
126C2×D4×C31—42no—
127C2×D4×C31—44no—
128C2×D4×C31—41no—
129C2×D4×C31—41no—
130C2×D4×C31—42no—
131C2×D4×C31—82no—
132C2×D4×C31—82no—
133C2×D4×C31—82no—
134C2×D4×C31—82no—
135C2×D4×C31C2×D412yes—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—41no—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2411yescharacteristic
152SD16×C31—84no—
153D8×C31—84no—
154C2×C8×C31—24no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—42no—
168C4⋊C4×C31C2×D412yes—
169C4⋊C4×C31C2×D412yes—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—44no—
183C22⋊C4×C31—82no—
184C22⋊C4×C31—82no—
185C42×C31—22no—
186C42×C31—21no—
187C42×C31—21no—
188C42×C31—22no—
189C42×C31C2×D412yes—
190C42×C31C2×D412yes—
191#18356C3111yesmaximal, characteristic, direct product factor, Sylow
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—81no—
206C23×C31—81no—
207C23×C31AGL(1,ℤ/8)12yes—
208C23×C31C22×D411yescharacteristic
209Q8×C31—42no—
210Q8×C31—81no—
211D4×C31—42no—
212D4×C31—44no—
213D4×C31—44no—
214D4×C31—44no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—82no—
218D4×C31—82no—
219D4×C31—84no—
220D4×C31—81no—
221D4×C31—88no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—82no—
243C2×C4×C31—81no—
244C2×C4×C31—82no—
245C2×C4×C31—81no—
246C2×C4×C31#12011yescharacteristic
247C2×C4×C31#12011yescharacteristic
248C2×C4×C31#12012yes—
249C2×C4×C31C22×D412yes—
250C8×C31—44no—
251#3491C2×C3111yescharacteristic, complemented
252#3217C2×C3112yescomplemented
253#2619C2×C3111yescharacteristic, complemented
254#1597C2×C3114yescomplemented
255#1596C2×C3112yescomplemented
256#1595C2×C3112yescomplemented
257#1512C2×C3112yescomplemented
258#1464C2×C3111yescharacteristic, complemented
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—22no—
262C22×C31—42no—
263C22×C31—41no—
264C22×C31—41no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—81no—
269C22×C31—82no—
270C22×C31—82no—
271C22×C31—82no—
272C22×C31—82no—
273C22×C31—82no—
274C22×C31—82no—
275C22×C31—82no—
276C22×C31—162no—
277C22×C31C2×#12011yescharacteristic
278C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C31#57511yescharacteristic
280C4×C31—21no—
281C4×C31—21no—
282C4×C31—22no—
283C4×C31—22no—
284C4×C31—41no—
285C4×C31—42no—
286C4×C31—41no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—82no—
291#552—22no—
292#552—22no—
293#551—22no—
294#551—22no—
295#543C22×C3111yescharacteristic
296#539—21no—
297#539C22×C3112yes—
298#534C22×C3111yescharacteristic
299C2×#127C22×C3111yescharacteristic, complemented
300#529—21no—
301#529C22×C3112yes—
302C2×#123—21no—
303C2×#121—21no—
304#518—21no—
305C2×#117C22×C3111yescharacteristic
306#496—22no—
307#490—22no—
308#460—22no—
309#459—22no—
310#450C22×C3114yescomplemented
311#447C22×C3114yes—
312#432C22×C3112yescomplemented
313#394—21no—
314#394C22×C3112yescomplemented
315#392—21no—
316#392C22×C3112yes—
317#390—21no—
318#382C22×C3111yescharacteristic, complemented
319#328C22×C3114yescomplemented
320#327C22×C3114yes—
321#324C22×C3112yescomplemented
322#323C22×C3112yes—
323C2×C31—21no—
324C2×C31—21no—
325C2×C31—42no—
326C2×C31—82no—
327C2×C31—162no—
328C2×C31#300611yescharacteristic
329C2×C31#302111yescharacteristic
330C2×C31#302711yescharacteristic
331C22×D4—21no—
332C22×D4—22no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4C23×C3112yes—
336C23×C4D4×C3111yescharacteristic
337C2×SD16—24no—
338C2×D8—24no—
339#127—22no—
340#127—22no—
341#126—22no—
342#126—22no—
343#126—22no—
344#126—22no—
345#125—21no—
346#125—22no—
347#125—22no—
348#125D4×C3112yescomplemented
349#124—21no—
350#124D4×C3112yescomplemented
351#123—22no—
352#123—22no—
353#123—22no—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—21no—
362C22⋊Q8—22no—
363C22⋊Q8C23×C3112yes—
364#121—21no—
365#121—44no—
366#121C23×C3112yes—
367#120—22no—
368C4×Q8—21no—
369C4×Q8D4×C3112yescomplemented
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—21no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4D4×C3112yescomplemented
376#117—22no—
377#117—22no—
378#117—22no—
379#117—22no—
380#117—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4D4×C3111yescharacteristic
385C2×C4⋊C4C23×C3111yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4D4×C3111yescharacteristic
394C2×C22⋊C4C23×C3111yescharacteristic
395C2×C42D4×C3111yescharacteristic
396C2×C42C23×C3111yescharacteristic
397C4⋊C8—22no—
398#102—24no—
399#102—24no—
400C22⋊C8C23×C3114yes—
401#97—22no—
402#95—21no—
403#95C23×C3112yes—
404C31#1835611yescharacteristic, direct product factor, Sylow
405C24—21no—
406C24—22no—
407C24C2×D4×C3111yescharacteristic
408C2×Q8—21no—
409C2×Q8C2×D4×C3112yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—24no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4C2×D4×C3112yes—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C2×D4×C3111yescharacteristic
439C22×C4C2×D4×C3111yescharacteristic
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C24×C3111yescharacteristic
443SD16—84no—
444D8—84no—
445C2×C8—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4C2×D4×C3112yes—
460C4⋊C4C2×D4×C3112yes—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42C2×D4×C3112yes—
481C42C2×D4×C3112yes—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23C31×AGL(1,ℤ/8)12yes—
498C23C22×D4×C3111yescharacteristic
499Q8—42no—
500Q8—81no—
501D4—42no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—42no—
506D4—42no—
507D4—82no—
508D4—82no—
509D4—84no—
510D4—81no—
511D4—88no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4C31×#12011yescharacteristic
537C2×C4C31×#12011yescharacteristic
538C2×C4C31×#12012yes—
539C2×C4C22×D4×C3112yes—
540C8—44no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—81no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—162no—
559C22C2×C31×#12011yescharacteristic
560C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
561C22C31×#57511yescharacteristic
562C4—21no—
563C4—21no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—82no—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—82no—
577C2—162no—
578C2C31×#300611yescharacteristic
579C2C31×#302111yescharacteristic
580C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ 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×#3491C2semidirect
2. C31×#3217C2semidirect
3. C31×#2619C2semidirect
4. C31×#1597C2semidirect
5. C31×#1596C2semidirect
6. C31×#1595C2semidirect
7. C31×#1512C2semidirect
8. C31×#1464C2semidirect
17. C2×C31×#127C22semidirect
28. C31×#450C22semidirect
30. C31×#432C22semidirect
32. C31×#394C22semidirect
36. C31×#382C22semidirect
37. C31×#328C22semidirect
39. C31×#324C22semidirect
58. C31×#125D4semidirect
60. C31×#124D4semidirect
79. C4×Q8×C31D4semidirect
85. C4×D4×C31D4semidirect
191. #18356C31direct (N × H)
251. #3491C2×C31semidirect
252. #3217C2×C31semidirect
253. #2619C2×C31semidirect
254. #1597C2×C31semidirect
255. #1596C2×C31semidirect
256. #1595C2×C31semidirect
257. #1512C2×C31semidirect
258. #1464C2×C31semidirect
299. C2×#127C22×C31semidirect
310. #450C22×C31semidirect
312. #432C22×C31semidirect
314. #394C22×C31semidirect
318. #382C22×C31semidirect
319. #328C22×C31semidirect
321. #324C22×C31semidirect
348. #125D4×C31semidirect
350. #124D4×C31semidirect
369. C4×Q8D4×C31semidirect
375. C4×D4D4×C31semidirect
404. C31#18356direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34912
C31×#32172
C31×#26192
C31×#15972
C31×#15962
C31×#15952
C31×#15122
C31×#14642
#1835631