Finite Group Databasebeta

← C31×#18681 #21475440

Subgroups — C31×#18681 #21475440

Order 7936.

946 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic
2C31×#3246C211yesmaximal, characteristic, complemented
3C31×#3170C211yesmaximal, characteristic, complemented
4C31×#3077C211yesmaximal, characteristic, complemented
5C31×#1647C212yesmaximal, complemented
6C31×#1646C212yesmaximal, complemented
7C31×#1635C212yesmaximal, complemented
8C31×#1634C212yesmaximal, complemented
9C31×#1550C211yesmaximal, characteristic, complemented
10C31×#1492C211yesmaximal, characteristic, complemented
11C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
12C31×#559—22no—
13C31×#556—21no—
14C31×#555C2212yes—
15C31×#553—22no—
16C31×#550—21no—
17C31×#547C2212yes—
18C31×#538—22no—
19C31×#535—21no—
20C31×#533C2211yescharacteristic
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C31×#525—22no—
24C31×#520—21no—
25C2×C4×D4×C31C2211yescharacteristic
26C31×#500C2211yescharacteristic, complemented
27C31×#494C2211yescharacteristic, complemented
28C31×#487C2211yescharacteristic, complemented
29C31×#487C2211yescharacteristic, complemented
30C31×#478—22no—
31C31×#476C2212yes—
32C31×#476C2212yescomplemented
33C31×#475—21no—
34C31×#474C2212yes—
35C31×#467—22no—
36C31×#465—22no—
37C31×#461C2212yescomplemented
38C31×#460—21no—
39C4×Q16×C31—22no—
40C31×#428—21no—
41C31×#423C2211yescharacteristic
42C2×C31×C4⋊C8C2211yescharacteristic
43C31×#418—21no—
44C2×C4×C8×C31C2211yescharacteristic
45C31×#333C2212yescomplemented
46C31×#332C2212yescomplemented
47C31×D4⋊C8C2212yes—
48C31×D4⋊C8C2212yes—
49C31×D4⋊C8C2212yescomplemented
50C31×D4⋊C8C2212yescomplemented
51C2×#41×C31—22no—
52C2×#41×C31—21no—
53C22×D4×C31D411yescharacteristic
54C23×C4×C31—21no—
55C2×Q16×C31—22no—
56C2×SD16×C31—22no—
57C2×M16×C31—21no—
58C22×C8×C31—21no—
59C22×C8×C31—21no—
60C31×C4⋊Q8D411yescharacteristic
61C31×C4⋊Q8C2311yescharacteristic
62C31×C4⋊Q8C2311yescharacteristic
63C31×C4⋊Q8C2311yescharacteristic
64C31×C4⋊Q8C2311yescharacteristic
65C31×#127—22no—
66C31×#127D411yescharacteristic
67C31×#126—24no—
68C31×#125—21no—
69C31×#125—21no—
70C31×#124—22no—
71C31×#124—24no—
72C31×#124—21no—
73C31×#123—24no—
74C31×#123—24no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—21no—
78C31×C22⋊Q8—21no—
79C31×C22⋊Q8D412yes—
80C31×#121—24no—
81C31×#121—22no—
82C31×#121—22no—
83C31×#121—22no—
84C31×#121—21no—
85C31×#121D412yes—
86C31×#120—24no—
87C4×Q8×C31—22no—
88C4×Q8×C31—21no—
89C4×Q8×C31—21no—
90C4×Q8×C31—21no—
91C4×Q8×C31D412yes—
92C4×D4×C31—24no—
93C4×D4×C31—22no—
94C4×D4×C31—24no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31D412yes—
99C4×D4×C31C2312yes—
100C4×D4×C31C2312yes—
101C31×#117—21no—
102C31×#117—22no—
103C31×#117—22no—
104C31×#117—21no—
105C2×C4⋊C4×C31—22no—
106C2×C4⋊C4×C31D411yescharacteristic
107C2×C22⋊C4×C31—21no—
108C2×C42×C31C2311yescharacteristic
109C31×#107—22no—
110C31×#107—22no—
111C31×#106—22no—
112C31×#106—22no—
113C31×C4⋊C8—21no—
114C31×C4⋊C8—21no—
115C31×C4⋊C8C2312yes—
116C31×C4⋊C8C2312yes—
117C31×#103—22no—
118C31×#103—22no—
119C31×#103—22no—
120C31×#102—22no—
121C31×#102—22no—
122C31×#102—22no—
123C4×C8×C31—21no—
124C4×C8×C31C2311yescharacteristic
125C4×C8×C31C2311yescharacteristic
126C24×C31—21no—
127#41×C31—42no—
128#41×C31—42no—
129#41×C31—82no—
130#41×C31—82no—
131C2×Q8×C31—21no—
132C2×Q8×C31—21no—
133C2×Q8×C31—21no—
134C2×Q8×C31—21no—
135C2×Q8×C31C2×D412yes—
136C2×D4×C31—22no—
137C2×D4×C31—24no—
138C2×D4×C31—24no—
139C2×D4×C31—21no—
140C2×D4×C31—21no—
141C2×D4×C31—21no—
142C2×D4×C31—22no—
143C2×D4×C31—22no—
144C2×D4×C31C2×D412yes—
145C2×D4×C31C2×D412yes—
146C22×C4×C31—24no—
147C22×C4×C31—21no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—81no—
152C22×C4×C31—81no—
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156Q16×C31—44no—
157SD16×C31—82no—
158M16×C31—42no—
159C2×C8×C31—21no—
160C2×C8×C31—21no—
161C2×C8×C31—22no—
162C2×C8×C31—22no—
163C2×C8×C31—42no—
164C2×C8×C31—81no—
165C4⋊C4×C31—24no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—41no—
178C4⋊C4×C31—41no—
179C4⋊C4×C31C2×D412yes—
180C4⋊C4×C31C2×D412yes—
181C4⋊C4×C31C2×D412yes—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—24no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—41no—
191C42×C31—24no—
192C42×C31—22no—
193C42×C31—41no—
194C42×C31C2×D411yescharacteristic
195C42×C31C2×D411yescharacteristic
196C42×C31C2×D411yescharacteristic
197C42×C31C2411yescharacteristic
198#18681C3111yesmaximal, characteristic, direct product factor, Sylow
199C23×C31—22no—
200C23×C31—22no—
201C23×C31—42no—
202C23×C31—81no—
203C23×C31—81no—
204C23×C31#12011yescharacteristic
205Q8×C31—42no—
206Q8×C31—42no—
207Q8×C31—41no—
208Q8×C31—41no—
209Q8×C31—42no—
210Q8×C31—41no—
211Q8×C31—41no—
212D4×C31—22no—
213D4×C31—22no—
214D4×C31—22no—
215D4×C31—22no—
216D4×C31—44no—
217D4×C31—42no—
218D4×C31—42no—
219D4×C31—81no—
220D4×C31—84no—
221D4×C31—84no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—82no—
236C2×C4×C31—81no—
237C2×C4×C31—84no—
238C2×C4×C31#12011yescharacteristic
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#13512yes—
242C2×C4×C31#13712yes—
243C2×C4×C31C22×D411yescharacteristic
244C2×C4×C31C22×D411yescharacteristic
245C2×C4×C31C22×D411yescharacteristic
246C8×C31—22no—
247C8×C31—41no—
248C8×C31—42no—
249C8×C31—42no—
250#3506C2×C3111yescharacteristic
251#3246C2×C3111yescharacteristic, complemented
252#3170C2×C3111yescharacteristic, complemented
253#3077C2×C3111yescharacteristic, complemented
254#1647C2×C3112yescomplemented
255#1646C2×C3112yescomplemented
256#1635C2×C3112yescomplemented
257#1634C2×C3112yescomplemented
258#1550C2×C3111yescharacteristic, complemented
259#1492C2×C3111yescharacteristic, complemented
260C2×D4⋊C8C2×C3111yescharacteristic, complemented
261C22×C31—21no—
262C22×C31—21no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—42no—
269C22×C31—84no—
270C22×C31#45212yes—
271C22×C31C2×#12011yescharacteristic
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—42no—
278C4×C31—42no—
279C4×C31—81no—
280C4×C31—81no—
281C4×C31#45212yes—
282C4×C31C2×#13511yescharacteristic
283C4×C31C2×#13711yescharacteristic
284C4×C31#57611yescharacteristic
285C4×C31#57711yescharacteristic
286#559—22no—
287#556—21no—
288#555C22×C3112yes—
289#553—22no—
290#550—21no—
291#547C22×C3112yes—
292#538—22no—
293#535—21no—
294#533C22×C3111yescharacteristic
295#533C22×C3111yescharacteristic
296#532C22×C3111yescharacteristic
297#525—22no—
298#520—21no—
299C2×C4×D4C22×C3111yescharacteristic
300#500C22×C3111yescharacteristic, complemented
301#494C22×C3111yescharacteristic, complemented
302#487C22×C3111yescharacteristic, complemented
303#487C22×C3111yescharacteristic, complemented
304#478—22no—
305#476C22×C3112yes—
306#476C22×C3112yescomplemented
307#475—21no—
308#474C22×C3112yes—
309#467—22no—
310#465—22no—
311#461C22×C3112yescomplemented
312#460—21no—
313C4×Q16—22no—
314#428—21no—
315#423C22×C3111yescharacteristic
316C2×C4⋊C8C22×C3111yescharacteristic
317#418—21no—
318C2×C4×C8C22×C3111yescharacteristic
319#333C22×C3112yescomplemented
320#332C22×C3112yescomplemented
321D4⋊C8C22×C3112yes—
322D4⋊C8C22×C3112yes—
323D4⋊C8C22×C3112yescomplemented
324D4⋊C8C22×C3112yescomplemented
325C2×C31—22no—
326C2×C31—82no—
327C2×C31—82no—
328C2×C31C2×#45211yescharacteristic
329C2×C31#301211yescharacteristic
330C2×C31#301811yescharacteristic
331C2×#41—22no—
332C2×#41—21no—
333C22×D4D4×C3111yescharacteristic
334C23×C4—21no—
335C2×Q16—22no—
336C2×SD16—22no—
337C2×M16—21no—
338C22×C8—21no—
339C22×C8—21no—
340C4⋊Q8D4×C3111yescharacteristic
341C4⋊Q8C23×C3111yescharacteristic
342C4⋊Q8C23×C3111yescharacteristic
343C4⋊Q8C23×C3111yescharacteristic
344C4⋊Q8C23×C3111yescharacteristic
345#127—22no—
346#127D4×C3111yescharacteristic
347#126—24no—
348#125—21no—
349#125—21no—
350#124—22no—
351#124—24no—
352#124—21no—
353#123—24no—
354#123—24no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8D4×C3112yes—
360#121—24no—
361#121—22no—
362#121—22no—
363#121—22no—
364#121—21no—
365#121D4×C3112yes—
366#120—24no—
367C4×Q8—22no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8D4×C3112yes—
372C4×D4—24no—
373C4×D4—22no—
374C4×D4—24no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4D4×C3112yes—
379C4×D4C23×C3112yes—
380C4×D4C23×C3112yes—
381#117—21no—
382#117—22no—
383#117—22no—
384#117—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4D4×C3111yescharacteristic
387C2×C22⋊C4—21no—
388C2×C42C23×C3111yescharacteristic
389#107—22no—
390#107—22no—
391#106—22no—
392#106—22no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8C23×C3112yes—
396C4⋊C8C23×C3112yes—
397#103—22no—
398#103—22no—
399#103—22no—
400#102—22no—
401#102—22no—
402#102—22no—
403C4×C8—21no—
404C4×C8C23×C3111yescharacteristic
405C4×C8C23×C3111yescharacteristic
406C31#1868111yescharacteristic, direct product factor, Sylow
407C24—21no—
408#41—42no—
409#41—42no—
410#41—82no—
411#41—82no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8C2×D4×C3112yes—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4C2×D4×C3112yes—
426C2×D4C2×D4×C3112yes—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3111yescharacteristic
437Q16—44no—
438SD16—82no—
439M16—42no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—42no—
445C2×C8—81no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4C2×D4×C3112yes—
461C4⋊C4C2×D4×C3112yes—
462C4⋊C4C2×D4×C3112yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C42—24no—
473C42—22no—
474C42—41no—
475C42C2×D4×C3111yescharacteristic
476C42C2×D4×C3111yescharacteristic
477C42C2×D4×C3111yescharacteristic
478C42C24×C3111yescharacteristic
479C23—22no—
480C23—22no—
481C23—42no—
482C23—81no—
483C23—81no—
484C23C31×#12011yescharacteristic
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492D4—22no—
493D4—22no—
494D4—22no—
495D4—22no—
496D4—44no—
497D4—42no—
498D4—42no—
499D4—81no—
500D4—84no—
501D4—84no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—84no—
518C2×C4C31×#12011yescharacteristic
519C2×C4C31×#12011yescharacteristic
520C2×C4C31×#12011yescharacteristic
521C2×C4C31×#13512yes—
522C2×C4C31×#13712yes—
523C2×C4C22×D4×C3111yescharacteristic
524C2×C4C22×D4×C3111yescharacteristic
525C2×C4C22×D4×C3111yescharacteristic
526C8—22no—
527C8—41no—
528C8—42no—
529C8—42no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—84no—
539C22C31×#45212yes—
540C22C2×C31×#12011yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—81no—
549C4—81no—
550C4C31×#45212yes—
551C4C2×C31×#13511yescharacteristic
552C4C2×C31×#13711yescharacteristic
553C4C31×#57611yescharacteristic
554C4C31×#57711yescharacteristic
555C2—22no—
556C2—82no—
557C2—82no—
558C2C2×C31×#45211yescharacteristic
559C2C31×#301211yescharacteristic
560C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3506C2
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×#3170C2semidirect
4. C31×#3077C2semidirect
5. C31×#1647C2semidirect
6. C31×#1646C2semidirect
7. C31×#1635C2semidirect
8. C31×#1634C2semidirect
9. C31×#1550C2semidirect
10. C31×#1492C2semidirect
11. C2×C31×D4⋊C8C2semidirect
26. C31×#500C22semidirect
27. C31×#494C22semidirect
28. C31×#487C22semidirect
29. C31×#487C22semidirect
32. C31×#476C22semidirect
37. C31×#461C22semidirect
45. C31×#333C22semidirect
46. C31×#332C22semidirect
49. C31×D4⋊C8C22semidirect
50. C31×D4⋊C8C22semidirect
198. #18681C31direct (N × H)
251. #3246C2×C31semidirect
252. #3170C2×C31semidirect
253. #3077C2×C31semidirect
254. #1647C2×C31semidirect
255. #1646C2×C31semidirect
256. #1635C2×C31semidirect
257. #1634C2×C31semidirect
258. #1550C2×C31semidirect
259. #1492C2×C31semidirect
260. C2×D4⋊C8C2×C31semidirect
300. #500C22×C31semidirect
301. #494C22×C31semidirect
302. #487C22×C31semidirect
303. #487C22×C31semidirect
306. #476C22×C31semidirect
311. #461C22×C31semidirect
319. #333C22×C31semidirect
320. #332C22×C31semidirect
323. D4⋊C8C22×C31semidirect
324. D4⋊C8C22×C31semidirect
406. C31#18681direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#32462
C31×#31702
C31×#30772
C31×#16472
C31×#16462
C31×#16352
C31×#16342
C31×#15502
C31×#14922
C2×C31×D4⋊C82
#1868131