Finite Group Databasebeta

← C31×#18644 #21475403

Subgroups — C31×#18644 #21475403

Order 7936.

968 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3509C211yesmaximal, characteristic, complemented
2C31×#3208C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic, complemented
4C31×#3038C211yesmaximal, characteristic
5C31×#1628C214yesmaximal, complemented
6C31×#1626C214yesmaximal, complemented
7C31×#1546C211yesmaximal, characteristic, complemented
8C31×#1493C211yesmaximal, characteristic, complemented
9C31×#1476C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)—21no—
11C31×#557—21no—
12C31×#557C2212yescomplemented
13C31×#556—21no—
14C31×#556C2212yescomplemented
15C31×#552—22no—
16C31×#552—22no—
17C31×#541—22no—
18C31×#536—21no—
19C31×#533—21no—
20C31×#533—21no—
21C31×#533C2211yescharacteristic
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×#127C2211yescharacteristic
24C31×#519—21no—
25C4×#41×C31C2211yescharacteristic
26C31×#492—21no—
27C31×#486—22no—
28C31×#486—21no—
29C31×#486C2214yes—
30C31×#460—21no—
31C31×#460C2212yes—
32C31×#460C2214yes—
33C31×#459—21no—
34C31×#459C2212yes—
35C31×#450—22no—
36C31×#431C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C31×#418—21no—
39C2×C31×#102—22no—
40C2×C4×C8×C31C2211yescharacteristic
41C31×#332C2214yescomplemented
42C31×Q8⋊C8C2212yescomplemented
43C31×Q8⋊C8C2214yescomplemented
44C31×D4⋊C8C2212yescomplemented
45C2×#41×C31D411yescharacteristic
46C22×Q8×C31—21no—
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C31×AGL(1,ℤ/8)—44no—
51C2×SD16×C31—22no—
52C2×SD16×C31—24no—
53C2×D8×C31—22no—
54C2×M16×C31—21no—
55C22×C8×C31—21no—
56C31×C4⋊Q8—22no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8D411yescharacteristic
60C31×C4⋊Q8D412yes—
61C31×C4⋊Q8D411yescharacteristic
62C31×C4⋊Q8C2312yes—
63C31×#127—21no—
64C31×#127—21no—
65C31×#127—21no—
66C31×#127C2312yes—
67C31×#126—24no—
68C31×#126—24no—
69C31×#125—22no—
70C31×#125—24no—
71C31×#125—22no—
72C31×#125—24no—
73C31×#125—21no—
74C31×#124—22no—
75C31×#123—24no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8D412yes—
82C4×Q8×C31—22no—
83C4×Q8×C31—22no—
84C4×Q8×C31—22no—
85C4×Q8×C31—22no—
86C4×Q8×C31—22no—
87C4×Q8×C31—22no—
88C4×Q8×C31—21no—
89C4×Q8×C31D411yescharacteristic
90C4×Q8×C31C2311yescharacteristic
91C4×Q8×C31C2312yes—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31D411yescharacteristic
95C4×D4×C31C2311yescharacteristic
96C31×#117—24no—
97C31×#117—22no—
98C31×#117—21no—
99C31×#117—22no—
100C31×#117—21no—
101C31×#117D411yescharacteristic
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31D412yes—
104C2×C42×C31—21no—
105C2×C42×C31C2311yescharacteristic
106C31×C4⋊C8—21no—
107C31×C4⋊C8C2312yes—
108C31×C4⋊C8C2312yes—
109C31×#102—28no—
110C31×#102—22no—
111C31×#102—22no—
112C31×#102—24no—
113C31×C22⋊C8—21no—
114C4×C8×C31—21no—
115C4×C8×C31—21no—
116C4×C8×C31C2312yes—
117C24×C31—41no—
118#41×C31—22no—
119#41×C31—22no—
120C2×Q8×C31—21no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—42no—
124C2×Q8×C31—42no—
125C2×Q8×C31C2×D412yes—
126C2×Q8×C31C2×D411yescharacteristic
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—44no—
132C2×D4×C31—41no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—42no—
136C2×D4×C31—42no—
137C2×D4×C31—82no—
138C2×D4×C31C2×D411yescharacteristic
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146SD16×C31—44no—
147SD16×C31—84no—
148D8×C31—44no—
149M16×C31—42no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C2×C8×C31—42no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31C2×D414yes—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D411yescharacteristic
171C4⋊C4×C31C2×D411yescharacteristic
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—21no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—21no—
177C22⋊C4×C31—21no—
178C42×C31—24no—
179C42×C31—24no—
180C42×C31—21no—
181C42×C31—21no—
182C42×C31—21no—
183C42×C31—22no—
184C42×C31C2×D411yescharacteristic
185C42×C31C2×D411yescharacteristic
186C42×C31C2×D411yescharacteristic
187C42×C31C2411yescharacteristic
188#18644C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—21no—
190C23×C31—42no—
191C23×C31—82no—
192C23×C31—82no—
193C23×C31—82no—
194C23×C31#12011yescharacteristic
195Q8×C31—21no—
196Q8×C31—21no—
197Q8×C31—24no—
198Q8×C31—42no—
199Q8×C31—42no—
200Q8×C31—44no—
201Q8×C31—42no—
202D4×C31—21no—
203D4×C31—21no—
204D4×C31—44no—
205D4×C31—42no—
206D4×C31—42no—
207D4×C31—42no—
208D4×C31—44no—
209D4×C31—82no—
210D4×C31—82no—
211D4×C31—82no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—24no—
219C2×C4×C31—21no—
220C2×C4×C31—44no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31#12011yescharacteristic
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31#13512yes—
228C2×C4×C31AGL(1,ℤ/8)12yes—
229C2×C4×C31C22×D411yescharacteristic
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C8×C31—42no—
233C8×C31—42no—
234C8×C31—42no—
235#3509C2×C3111yescharacteristic, complemented
236#3208C2×C3111yescharacteristic, complemented
237C2×#486C2×C3111yescharacteristic, complemented
238#3038C2×C3111yescharacteristic
239#1628C2×C3114yescomplemented
240#1626C2×C3114yescomplemented
241#1546C2×C3111yescharacteristic, complemented
242#1493C2×C3111yescharacteristic, complemented
243#1476C2×C3111yescharacteristic, complemented
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—41no—
247C22×C31—41no—
248C22×C31—41no—
249C22×C31—82no—
250C22×C31—82no—
251C22×C31—82no—
252C22×C31—164no—
253C22×C31#44912yes—
254C22×C31C2×#12011yescharacteristic
255C4×C31—21no—
256C4×C31—21no—
257C4×C31—41no—
258C4×C31—44no—
259C4×C31—41no—
260C4×C31—42no—
261C4×C31—41no—
262C4×C31—42no—
263C4×C31#44912yes—
264C4×C31#45412yes—
265C4×C31C2×#13511yescharacteristic
266C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
267C4×C31#57712yes—
268C2×AGL(1,ℤ/8)—21no—
269#557—21no—
270#557C22×C3112yescomplemented
271#556—21no—
272#556C22×C3112yescomplemented
273#552—22no—
274#552—22no—
275#541—22no—
276#536—21no—
277#533—21no—
278#533—21no—
279#533C22×C3111yescharacteristic
280C2×C4⋊Q8C22×C3111yescharacteristic
281C2×#127C22×C3111yescharacteristic
282#519—21no—
283C4×#41C22×C3111yescharacteristic
284#492—21no—
285#486—22no—
286#486—21no—
287#486C22×C3114yes—
288#460—21no—
289#460C22×C3112yes—
290#460C22×C3114yes—
291#459—21no—
292#459C22×C3112yes—
293#450—22no—
294#431C22×C3111yescharacteristic
295#423C22×C3111yescharacteristic
296#418—21no—
297C2×#102—22no—
298C2×C4×C8C22×C3111yescharacteristic
299#332C22×C3114yescomplemented
300Q8⋊C8C22×C3112yescomplemented
301Q8⋊C8C22×C3114yescomplemented
302D4⋊C8C22×C3112yescomplemented
303C2×C31—22no—
304C2×C31—81no—
305C2×C31—162no—
306C2×C31C2×#44911yescharacteristic
307C2×C31#301011yescharacteristic
308C2×C31C2×#45411yescharacteristic
309C2×#41D4×C3111yescharacteristic
310C22×Q8—21no—
311C22×D4—21no—
312C22×D4—21no—
313C22×D4—21no—
314AGL(1,ℤ/8)—44no—
315C2×SD16—22no—
316C2×SD16—24no—
317C2×D8—22no—
318C2×M16—21no—
319C22×C8—21no—
320C4⋊Q8—22no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8D4×C3111yescharacteristic
324C4⋊Q8D4×C3112yes—
325C4⋊Q8D4×C3111yescharacteristic
326C4⋊Q8C23×C3112yes—
327#127—21no—
328#127—21no—
329#127—21no—
330#127C23×C3112yes—
331#126—24no—
332#126—24no—
333#125—22no—
334#125—24no—
335#125—22no—
336#125—24no—
337#125—21no—
338#124—22no—
339#123—24no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—21no—
345C22⋊Q8D4×C3112yes—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—21no—
353C4×Q8D4×C3111yescharacteristic
354C4×Q8C23×C3111yescharacteristic
355C4×Q8C23×C3112yes—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4D4×C3111yescharacteristic
359C4×D4C23×C3111yescharacteristic
360#117—24no—
361#117—22no—
362#117—21no—
363#117—22no—
364#117—21no—
365#117D4×C3111yescharacteristic
366C2×C4⋊C4—21no—
367C2×C4⋊C4D4×C3112yes—
368C2×C42—21no—
369C2×C42C23×C3111yescharacteristic
370C4⋊C8—21no—
371C4⋊C8C23×C3112yes—
372C4⋊C8C23×C3112yes—
373#102—28no—
374#102—22no—
375#102—22no—
376#102—24no—
377C22⋊C8—21no—
378C4×C8—21no—
379C4×C8—21no—
380C4×C8C23×C3112yes—
381C31#1864411yescharacteristic, direct product factor, Sylow
382C24—41no—
383#41—22no—
384#41—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8C2×D4×C3112yes—
391C2×Q8C2×D4×C3111yescharacteristic
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—44no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—82no—
403C2×D4C2×D4×C3111yescharacteristic
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4C2×D4×C3111yescharacteristic
409C22×C4C2×D4×C3111yescharacteristic
410C22×C4C2×D4×C3111yescharacteristic
411SD16—44no—
412SD16—84no—
413D8—44no—
414M16—42no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—42no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4C2×D4×C3114yes—
434C4⋊C4C2×D4×C3112yes—
435C4⋊C4C2×D4×C3111yescharacteristic
436C4⋊C4C2×D4×C3111yescharacteristic
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—21no—
440C22⋊C4—22no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C42—24no—
444C42—24no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—22no—
449C42C2×D4×C3111yescharacteristic
450C42C2×D4×C3111yescharacteristic
451C42C2×D4×C3111yescharacteristic
452C42C24×C3111yescharacteristic
453C23—21no—
454C23—42no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23C31×#12011yescharacteristic
459Q8—21no—
460Q8—21no—
461Q8—24no—
462Q8—44no—
463Q8—42no—
464Q8—42no—
465Q8—42no—
466D4—21no—
467D4—21no—
468D4—44no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—44no—
473D4—82no—
474D4—82no—
475D4—82no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4C31×#12011yescharacteristic
489C2×C4C31×#12011yescharacteristic
490C2×C4C31×#12011yescharacteristic
491C2×C4C31×#13512yes—
492C2×C4C31×AGL(1,ℤ/8)12yes—
493C2×C4C22×D4×C3111yescharacteristic
494C2×C4C22×D4×C3111yescharacteristic
495C2×C4C22×D4×C3111yescharacteristic
496C8—42no—
497C8—42no—
498C8—42no—
499C22—21no—
500C22—21no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—164no—
508C22C31×#44912yes—
509C22C2×C31×#12011yescharacteristic
510C4—21no—
511C4—21no—
512C4—41no—
513C4—44no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4C31×#44912yes—
519C4C31×#45412yes—
520C4C2×C31×#13511yescharacteristic
521C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
522C4C31×#57712yes—
523C2—22no—
524C2—81no—
525C2—162no—
526C2C2×C31×#44911yescharacteristic
527C2C31×#301011yescharacteristic
528C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3038C2
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×#3509C2semidirect
2. C31×#3208C2semidirect
3. C2×C31×#486C2semidirect
5. C31×#1628C2semidirect
6. C31×#1626C2semidirect
7. C31×#1546C2semidirect
8. C31×#1493C2semidirect
9. C31×#1476C2semidirect
12. C31×#557C22semidirect
14. C31×#556C22semidirect
41. C31×#332C22semidirect
42. C31×Q8⋊C8C22semidirect
43. C31×Q8⋊C8C22semidirect
44. C31×D4⋊C8C22semidirect
188. #18644C31direct (N × H)
235. #3509C2×C31semidirect
236. #3208C2×C31semidirect
237. C2×#486C2×C31semidirect
239. #1628C2×C31semidirect
240. #1626C2×C31semidirect
241. #1546C2×C31semidirect
242. #1493C2×C31semidirect
243. #1476C2×C31semidirect
270. #557C22×C31semidirect
272. #556C22×C31semidirect
299. #332C22×C31semidirect
300. Q8⋊C8C22×C31semidirect
301. Q8⋊C8C22×C31semidirect
302. D4⋊C8C22×C31semidirect
381. C31#18644direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35092
C31×#32082
C2×C31×#4862
C31×#30382
C31×#16282
C31×#16262
C31×#15462
C31×#14932
C31×#14762
#1864431