Finite Group Databasebeta

← C31×#18664 #21475423

Subgroups — C31×#18664 #21475423

Order 7936.

980 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×#3501C211yesmaximal, characteristic, complemented
2C31×#3246C211yesmaximal, characteristic, complemented
3C31×#3167C211yesmaximal, characteristic, complemented
4C31×#3036C211yesmaximal, characteristic
5C31×#1651C212yesmaximal, complemented
6C31×#1643C212yesmaximal, complemented
7C31×#1627C212yesmaximal, complemented
8C31×#1622C212yesmaximal, complemented
9C31×#1550C211yesmaximal, characteristic, complemented
10C31×#1493C211yesmaximal, characteristic, complemented
11C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×C31×AGL(1,ℤ/8)—21no—
13C31×#554—21no—
14C31×#554—22no—
15C31×#554—22no—
16C31×#554C2212yes—
17D4×Q8×C31—21no—
18D4×Q8×C31C2212yes—
19C31×#544—21no—
20C31×#543—22no—
21C31×#533C2211yescharacteristic
22C31×#533C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C2×C31×C22⋊Q8—22no—
25C31×#518—21no—
26C2×C4×D4×C31C2211yescharacteristic
27C31×#498C2211yescharacteristic, complemented
28C31×#492C2211yescharacteristic, complemented
29C31×#487C2211yescharacteristic, complemented
30C31×#486C2211yescharacteristic, complemented
31C31×#476—21no—
32C31×#475C2212yescomplemented
33C31×#465—22no—
34C31×#465—22no—
35C31×#463—22no—
36C31×#461—21no—
37C31×#461C2212yes—
38C31×#460C2212yescomplemented
39C31×#459C2212yes—
40C4×SD16×C31—22no—
41C31×#423C2211yescharacteristic
42C2×C31×C4⋊C8C2211yescharacteristic
43C31×#417—21no—
44C2×C4×C8×C31C2211yescharacteristic
45C31×#333C2212yescomplemented
46C31×#332C2212yescomplemented
47C31×Q8⋊C8C2212yescomplemented
48C31×Q8⋊C8C2212yescomplemented
49C31×D4⋊C8C2212yes—
50C31×D4⋊C8C2212yes—
51C2×#41×C31—21no—
52C22×Q8×C31—22no—
53C22×D4×C31D411yescharacteristic
54C23×C4×C31—21no—
55C31×AGL(1,ℤ/8)—44no—
56C2×SD16×C31—22no—
57C2×SD16×C31—22no—
58C2×SD16×C31—22no—
59C2×D8×C31—22no—
60C2×M16×C31—21no—
61C22×C8×C31—21no—
62C22×C8×C31—21no—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8D411yescharacteristic
67C31×C4⋊Q8C2311yescharacteristic
68C31×C4⋊Q8C2311yescharacteristic
69C31×C4⋊Q8C2311yescharacteristic
70C31×#127C2311yescharacteristic
71C31×#125—22no—
72C31×#125—22no—
73C31×#125—21no—
74C31×#125—21no—
75C31×#124—21no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—24no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—24no—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—24no—
86C31×C22⋊Q8D412yes—
87C31×C22⋊Q8D412yes—
88C31×#121—21no—
89C31×#121—21no—
90C4×Q8×C31—21no—
91C4×Q8×C31—21no—
92C4×Q8×C31D412yes—
93C4×Q8×C31C2312yes—
94C4×D4×C31—22no—
95C4×D4×C31—22no—
96C4×D4×C31—22no—
97C4×D4×C31—22no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31D412yes—
103C4×D4×C31C2312yes—
104C31×#117—21no—
105C31×#117—21no—
106C2×C4⋊C4×C31—24no—
107C2×C4⋊C4×C31—22no—
108C2×C4⋊C4×C31D411yescharacteristic
109C2×C22⋊C4×C31—24no—
110C2×C22⋊C4×C31—21no—
111C2×C42×C31C2311yescharacteristic
112C31×#106—22no—
113C31×#106—22no—
114C31×C4⋊C8—21no—
115C31×C4⋊C8—21no—
116C31×C4⋊C8C2312yes—
117C31×C4⋊C8C2312yes—
118C31×#103—22no—
119C31×#103—22no—
120C31×#103—22no—
121C31×#102—22no—
122C31×#102—22no—
123C31×#102—22no—
124C4×C8×C31—21no—
125C4×C8×C31C2311yescharacteristic
126C4×C8×C31C2311yescharacteristic
127C24×C31—21no—
128#41×C31—42no—
129#41×C31—42no—
130C2×Q8×C31—24no—
131C2×Q8×C31—21no—
132C2×Q8×C31—21no—
133C2×Q8×C31—21no—
134C2×Q8×C31—82no—
135C2×Q8×C31—82no—
136C2×Q8×C31C2×D412yes—
137C2×Q8×C31C2×D412yes—
138C2×D4×C31—21no—
139C2×D4×C31—22no—
140C2×D4×C31—21no—
141C2×D4×C31—21no—
142C2×D4×C31—21no—
143C2×D4×C31—41no—
144C2×D4×C31SD1614yescomplemented
145C2×D4×C31C2×D412yes—
146C22×C4×C31—24no—
147C22×C4×C31—24no—
148C22×C4×C31—21no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—41no—
152C22×C4×C31—81no—
153C22×C4×C31—81no—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157SD16×C31—44no—
158SD16×C31—44no—
159SD16×C31—82no—
160D8×C31—44no—
161M16×C31—42no—
162C2×C8×C31—21no—
163C2×C8×C31—21no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—42no—
167C2×C8×C31—81no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31C2×D412yes—
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D412yes—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—41no—
194C22⋊C4×C31—41no—
195C42×C31—22no—
196C42×C31—22no—
197C42×C31C2×D411yescharacteristic
198C42×C31C2×D411yescharacteristic
199C42×C31C2×D411yescharacteristic
200C42×C31C2411yescharacteristic
201#18664C3111yesmaximal, characteristic, direct product factor, Sylow
202C23×C31—22no—
203C23×C31—24no—
204C23×C31—41no—
205C23×C31—81no—
206C23×C31—81no—
207C23×C31#12011yescharacteristic
208Q8×C31—22no—
209Q8×C31—22no—
210Q8×C31—42no—
211Q8×C31—41no—
212Q8×C31—41no—
213Q8×C31—41no—
214Q8×C31—41no—
215Q8×C31—81no—
216Q8×C31—84no—
217D4×C31—22no—
218D4×C31—24no—
219D4×C31—42no—
220D4×C31—41no—
221D4×C31—41no—
222D4×C31—41no—
223D4×C31—41no—
224D4×C31—82no—
225D4×C31C2×SD1614yes—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—82no—
238C2×C4×C31—81no—
239C2×C4×C31—84no—
240C2×C4×C31—84no—
241C2×C4×C31—82no—
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31#12011yescharacteristic
245C2×C4×C31C2×SD1612yes—
246C2×C4×C31#13712yes—
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C22×D411yescharacteristic
249C2×C4×C31C22×D411yescharacteristic
250C8×C31—22no—
251C8×C31—41no—
252C8×C31—42no—
253C8×C31—42no—
254#3501C2×C3111yescharacteristic, complemented
255#3246C2×C3111yescharacteristic, complemented
256#3167C2×C3111yescharacteristic, complemented
257#3036C2×C3111yescharacteristic
258#1651C2×C3112yescomplemented
259#1643C2×C3112yescomplemented
260#1627C2×C3112yescomplemented
261#1622C2×C3112yescomplemented
262#1550C2×C3111yescharacteristic, complemented
263#1493C2×C3111yescharacteristic, complemented
264C2×D4⋊C8C2×C3111yescharacteristic, complemented
265C22×C31—21no—
266C22×C31—21no—
267C22×C31—24no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—84no—
271C22×C31—81no—
272C22×C31—81no—
273C22×C31—81no—
274C22×C31#44812yes—
275C22×C31C2×#12011yescharacteristic
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—21no—
279C4×C31—42no—
280C4×C31—42no—
281C4×C31—42no—
282C4×C31—42no—
283C4×C31—42no—
284C4×C31—81no—
285C4×C31#44812yes—
286C4×C31C22×SD1611yescharacteristic
287C4×C31C2×#13711yescharacteristic
288C4×C31#57611yescharacteristic
289C4×C31#57711yescharacteristic
290C2×AGL(1,ℤ/8)—21no—
291#554—21no—
292#554—22no—
293#554—22no—
294#554C22×C3112yes—
295D4×Q8—21no—
296D4×Q8C22×C3112yes—
297#544—21no—
298#543—22no—
299#533C22×C3111yescharacteristic
300#533C22×C3111yescharacteristic
301#532C22×C3111yescharacteristic
302C2×C22⋊Q8—22no—
303#518—21no—
304C2×C4×D4C22×C3111yescharacteristic
305#498C22×C3111yescharacteristic, complemented
306#492C22×C3111yescharacteristic, complemented
307#487C22×C3111yescharacteristic, complemented
308#486C22×C3111yescharacteristic, complemented
309#476—21no—
310#475C22×C3112yescomplemented
311#465—22no—
312#465—22no—
313#463—22no—
314#461—21no—
315#461C22×C3112yes—
316#460C22×C3112yescomplemented
317#459C22×C3112yes—
318C4×SD16—22no—
319#423C22×C3111yescharacteristic
320C2×C4⋊C8C22×C3111yescharacteristic
321#417—21no—
322C2×C4×C8C22×C3111yescharacteristic
323#333C22×C3112yescomplemented
324#332C22×C3112yescomplemented
325Q8⋊C8C22×C3112yescomplemented
326Q8⋊C8C22×C3112yescomplemented
327D4⋊C8C22×C3112yes—
328D4⋊C8C22×C3112yes—
329C2×C31—22no—
330C2×C31—44no—
331C2×C31—161no—
332C2×C31C2×#44811yescharacteristic
333C2×C31#301211yescharacteristic
334C2×C31#301811yescharacteristic
335C2×#41—21no—
336C22×Q8—22no—
337C22×D4D4×C3111yescharacteristic
338C23×C4—21no—
339AGL(1,ℤ/8)—44no—
340C2×SD16—22no—
341C2×SD16—22no—
342C2×SD16—22no—
343C2×D8—22no—
344C2×M16—21no—
345C22×C8—21no—
346C22×C8—21no—
347C4⋊Q8—22no—
348C4⋊Q8—22no—
349C4⋊Q8D4×C3111yescharacteristic
350C4⋊Q8D4×C3111yescharacteristic
351C4⋊Q8C23×C3111yescharacteristic
352C4⋊Q8C23×C3111yescharacteristic
353C4⋊Q8C23×C3111yescharacteristic
354#127C23×C3111yescharacteristic
355#125—22no—
356#125—22no—
357#125—21no—
358#125—21no—
359#124—21no—
360C22⋊Q8—24no—
361C22⋊Q8—24no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364C22⋊Q8—22no—
365C22⋊Q8—24no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—22no—
369C22⋊Q8—24no—
370C22⋊Q8D4×C3112yes—
371C22⋊Q8D4×C3112yes—
372#121—21no—
373#121—21no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8D4×C3112yes—
377C4×Q8C23×C3112yes—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4D4×C3112yes—
387C4×D4C23×C3112yes—
388#117—21no—
389#117—21no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4D4×C3111yescharacteristic
393C2×C22⋊C4—24no—
394C2×C22⋊C4—21no—
395C2×C42C23×C3111yescharacteristic
396#106—22no—
397#106—22no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8C23×C3112yes—
401C4⋊C8C23×C3112yes—
402#103—22no—
403#103—22no—
404#103—22no—
405#102—22no—
406#102—22no—
407#102—22no—
408C4×C8—21no—
409C4×C8C23×C3111yescharacteristic
410C4×C8C23×C3111yescharacteristic
411C31#1866411yescharacteristic, direct product factor, Sylow
412C24—21no—
413#41—42no—
414#41—42no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8C2×D4×C3112yes—
422C2×Q8C2×D4×C3112yes—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—41no—
429C2×D4SD16×C3114yescomplemented
430C2×D4C2×D4×C3112yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4C2×D4×C3111yescharacteristic
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3111yescharacteristic
442SD16—44no—
443SD16—44no—
444SD16—82no—
445D8—44no—
446M16—42no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—42no—
452C2×C8—81no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4C2×D4×C3112yes—
470C4⋊C4C2×D4×C3112yes—
471C4⋊C4C2×D4×C3112yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C42—22no—
481C42—22no—
482C42C2×D4×C3111yescharacteristic
483C42C2×D4×C3111yescharacteristic
484C42C2×D4×C3111yescharacteristic
485C42C24×C3111yescharacteristic
486C23—22no—
487C23—24no—
488C23—41no—
489C23—81no—
490C23—81no—
491C23C31×#12011yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—42no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—81no—
500Q8—84no—
501D4—24no—
502D4—22no—
503D4—42no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—82no—
509D4C2×SD16×C3114yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4C31×#12011yescharacteristic
527C2×C4C31×#12011yescharacteristic
528C2×C4C31×#12011yescharacteristic
529C2×C4C2×SD16×C3112yes—
530C2×C4C31×#13712yes—
531C2×C4C22×D4×C3111yescharacteristic
532C2×C4C22×D4×C3111yescharacteristic
533C2×C4C22×D4×C3111yescharacteristic
534C8—22no—
535C8—41no—
536C8—42no—
537C8—42no—
538C22—21no—
539C22—21no—
540C22—24no—
541C22—42no—
542C22—42no—
543C22—84no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C31×#44812yes—
548C22C2×C31×#12011yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—81no—
558C4C31×#44812yes—
559C4C22×SD16×C3111yescharacteristic
560C4C2×C31×#13711yescharacteristic
561C4C31×#57611yescharacteristic
562C4C31×#57711yescharacteristic
563C2—22no—
564C2—44no—
565C2—161no—
566C2C2×C31×#44811yescharacteristic
567C2C31×#301211yescharacteristic
568C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3036C2
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×#3501C2semidirect
2. C31×#3246C2semidirect
3. C31×#3167C2semidirect
5. C31×#1651C2semidirect
6. C31×#1643C2semidirect
7. C31×#1627C2semidirect
8. C31×#1622C2semidirect
9. C31×#1550C2semidirect
10. C31×#1493C2semidirect
11. C2×C31×D4⋊C8C2semidirect
27. C31×#498C22semidirect
28. C31×#492C22semidirect
29. C31×#487C22semidirect
30. C31×#486C22semidirect
32. C31×#475C22semidirect
38. C31×#460C22semidirect
45. C31×#333C22semidirect
46. C31×#332C22semidirect
47. C31×Q8⋊C8C22semidirect
48. C31×Q8⋊C8C22semidirect
144. C2×D4×C31SD16semidirect
201. #18664C31direct (N × H)
254. #3501C2×C31semidirect
255. #3246C2×C31semidirect
256. #3167C2×C31semidirect
258. #1651C2×C31semidirect
259. #1643C2×C31semidirect
260. #1627C2×C31semidirect
261. #1622C2×C31semidirect
262. #1550C2×C31semidirect
263. #1493C2×C31semidirect
264. C2×D4⋊C8C2×C31semidirect
305. #498C22×C31semidirect
306. #492C22×C31semidirect
307. #487C22×C31semidirect
308. #486C22×C31semidirect
310. #475C22×C31semidirect
316. #460C22×C31semidirect
323. #333C22×C31semidirect
324. #332C22×C31semidirect
325. Q8⋊C8C22×C31semidirect
326. Q8⋊C8C22×C31semidirect
411. C31#18664direct (N × H)
429. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#32462
C31×#31672
C31×#30362
C31×#16512
C31×#16432
C31×#16272
C31×#16222
C31×#15502
C31×#14932
C2×C31×D4⋊C82
#1866431