Finite Group Databasebeta

← C31×#18634 #21475393

Subgroups — C31×#18634 #21475393

Order 7936.

1024 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3508C211yesmaximal, characteristic
2C31×#3229C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic, complemented
4C31×#3039C211yesmaximal, characteristic, complemented
5C31×#1642C214yesmaximal, complemented
6C31×#1640C214yesmaximal, complemented
7C31×#1545C211yesmaximal, characteristic, complemented
8C31×#1493C211yesmaximal, characteristic, complemented
9C31×#1476C211yesmaximal, characteristic, complemented
10C2×C31×#137—21no—
11C31×#559—21no—
12C31×#559C2212yes—
13C31×#550—21no—
14C31×#550C2212yes—
15C31×#546—22no—
16C31×#546—22no—
17C31×#540—22no—
18C31×#535—21no—
19C31×#533C2211yescharacteristic
20C31×#532—21no—
21C31×#532—21no—
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×#127C2211yescharacteristic
24C31×#518—21no—
25C4×#41×C31C2211yescharacteristic
26C31×#498—21no—
27C31×#487—21no—
28C31×#486—22no—
29C31×#486C2214yes—
30C31×#483—22no—
31C31×#475C2214yescomplemented
32C31×#462—21no—
33C31×#462C2212yescomplemented
34C31×#461—21no—
35C31×#461C2212yescomplemented
36C31×#431C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C31×#419—21no—
39C2×C31×#102—22no—
40C2×C4×C8×C31C2211yescharacteristic
41C31×#331C2214yes—
42C31×Q8⋊C8C2212yescomplemented
43C31×Q8⋊C8C2214yescomplemented
44C31×D4⋊C8C2212yescomplemented
45C2×#41×C31—22no—
46C2×#41×C31—22no—
47C2×#41×C31D411yescharacteristic
48C22×Q8×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31D412yes—
52C31×#137—44no—
53C2×Q16×C31—22no—
54C2×SD16×C31—22no—
55C2×M16×C31—21no—
56C22×C8×C31—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8D411yescharacteristic
61C31×C4⋊Q8D411yescharacteristic
62C31×C4⋊Q8C2312yes—
63C31×#127—22no—
64C31×#127—21no—
65C31×#127—21no—
66C31×#127D412yes—
67C31×#127C2312yes—
68C31×#125—21no—
69C31×#124—22no—
70C31×#124—22no—
71C31×#124—22no—
72C31×#123—24no—
73C31×#123—24no—
74C31×C22⋊Q8—21no—
75C31×C22⋊Q8D412yes—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—22no—
79C31×#121—24no—
80C31×#121—24no—
81C31×#121—22no—
82C31×#120—24no—
83C4×Q8×C31—21no—
84C4×Q8×C31D411yescharacteristic
85C4×Q8×C31C2311yescharacteristic
86C4×Q8×C31C2312yes—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—22no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31D411yescharacteristic
96C4×D4×C31C2311yescharacteristic
97C31×#117—21no—
98C31×#117—21no—
99C31×#117D411yescharacteristic
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—21no—
102C2×C22⋊C4×C31—24no—
103C2×C42×C31—21no—
104C2×C42×C31C2311yescharacteristic
105C31×#106—24no—
106C31×C4⋊C8—21no—
107C31×C4⋊C8C2312yes—
108C31×C4⋊C8C2312yes—
109C31×#103—24no—
110C31×#103—22no—
111C31×#103—22no—
112C31×#102—28no—
113C31×C22⋊C8—21no—
114C4×C8×C31—21no—
115C4×C8×C31—21no—
116C4×C8×C31C2312yes—
117C24×C31—22no—
118#41×C31—22no—
119#41×C31—22no—
120#41×C31—84no—
121#41×C31—84no—
122C2×Q8×C31—21no—
123C2×Q8×C31—22no—
124C2×Q8×C31—41no—
125C2×Q8×C31—42no—
126C2×Q8×C31—42no—
127C2×Q8×C31C2×D412yes—
128C2×Q8×C31C2×D411yescharacteristic
129C2×D4×C31—21no—
130C2×D4×C31—24no—
131C2×D4×C31—24no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—22no—
137C2×D4×C31—44no—
138C2×D4×C31—42no—
139C2×D4×C31—42no—
140C2×D4×C31—82no—
141C2×D4×C31C2×D411yescharacteristic
142C2×D4×C31C2×D414yes—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—41no—
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152Q16×C31—44no—
153SD16×C31—44no—
154M16×C31—42no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—42no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—41no—
169C4⋊C4×C31—41no—
170C4⋊C4×C31C2×D411yescharacteristic
171C4⋊C4×C31C2×D411yescharacteristic
172C4⋊C4×C31C2×D412yes—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—21no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—21no—
180C22⋊C4×C31—21no—
181C42×C31—21no—
182C42×C31—21no—
183C42×C31—21no—
184C42×C31—22no—
185C42×C31C2×D411yescharacteristic
186C42×C31C2×D411yescharacteristic
187C42×C31C2×D411yescharacteristic
188C42×C31C2411yescharacteristic
189#18634C3111yesmaximal, characteristic, direct product factor, Sylow
190C23×C31—24no—
191C23×C31—21no—
192C23×C31—44no—
193C23×C31—82no—
194C23×C31—82no—
195C23×C31#12011yescharacteristic
196Q8×C31—21no—
197Q8×C31—21no—
198Q8×C31—24no—
199Q8×C31—42no—
200Q8×C31—42no—
201Q8×C31—42no—
202Q8×C31—82no—
203D4×C31—21no—
204D4×C31—21no—
205D4×C31—44no—
206D4×C31—42no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—44no—
210D4×C31—44no—
211D4×C31—44no—
212D4×C31—82no—
213D4×C31—84no—
214D4×C31—84no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—21no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—84no—
227C2×C4×C31—84no—
228C2×C4×C31—82no—
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31#12011yescharacteristic
232C2×C4×C31#13512yes—
233C2×C4×C31AGL(1,ℤ/8)12yes—
234C2×C4×C31C22×D411yescharacteristic
235C2×C4×C31C22×D411yescharacteristic
236C2×C4×C31C22×D411yescharacteristic
237C8×C31—42no—
238C8×C31—42no—
239C8×C31—42no—
240#3508C2×C3111yescharacteristic
241#3229C2×C3111yescharacteristic, complemented
242C2×#486C2×C3111yescharacteristic, complemented
243#3039C2×C3111yescharacteristic, complemented
244#1642C2×C3114yescomplemented
245#1640C2×C3114yescomplemented
246#1545C2×C3111yescharacteristic, complemented
247#1493C2×C3111yescharacteristic, complemented
248#1476C2×C3111yescharacteristic, complemented
249C22×C31—21no—
250C22×C31—21no—
251C22×C31—41no—
252C22×C31—44no—
253C22×C31—41no—
254C22×C31—44no—
255C22×C31—41no—
256C22×C31—44no—
257C22×C31—164no—
258C22×C31#44912yes—
259C22×C31C2×#12011yescharacteristic
260C4×C31—21no—
261C4×C31—21no—
262C4×C31—41no—
263C4×C31—41no—
264C4×C31—42no—
265C4×C31—41no—
266C4×C31—42no—
267C4×C31—82no—
268C4×C31#44912yes—
269C4×C31#45512yes—
270C4×C31C2×#13511yescharacteristic
271C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
272C4×C31#57712yes—
273C2×#137—21no—
274#559—21no—
275#559C22×C3112yes—
276#550—21no—
277#550C22×C3112yes—
278#546—22no—
279#546—22no—
280#540—22no—
281#535—21no—
282#533C22×C3111yescharacteristic
283#532—21no—
284#532—21no—
285C2×C4⋊Q8C22×C3111yescharacteristic
286C2×#127C22×C3111yescharacteristic
287#518—21no—
288C4×#41C22×C3111yescharacteristic
289#498—21no—
290#487—21no—
291#486—22no—
292#486C22×C3114yes—
293#483—22no—
294#475C22×C3114yescomplemented
295#462—21no—
296#462C22×C3112yescomplemented
297#461—21no—
298#461C22×C3112yescomplemented
299#431C22×C3111yescharacteristic
300#423C22×C3111yescharacteristic
301#419—21no—
302C2×#102—22no—
303C2×C4×C8C22×C3111yescharacteristic
304#331C22×C3114yes—
305Q8⋊C8C22×C3112yescomplemented
306Q8⋊C8C22×C3114yescomplemented
307D4⋊C8C22×C3112yescomplemented
308C2×C31—22no—
309C2×C31—81no—
310C2×C31—84no—
311C2×C31C2×#44911yescharacteristic
312C2×C31#300911yescharacteristic
313C2×C31C2×#45511yescharacteristic
314C2×#41—22no—
315C2×#41—22no—
316C2×#41D4×C3111yescharacteristic
317C22×Q8—21no—
318C22×D4—21no—
319C22×D4—21no—
320C22×D4D4×C3112yes—
321#137—44no—
322C2×Q16—22no—
323C2×SD16—22no—
324C2×M16—21no—
325C22×C8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8D4×C3111yescharacteristic
330C4⋊Q8D4×C3111yescharacteristic
331C4⋊Q8C23×C3112yes—
332#127—22no—
333#127—21no—
334#127—21no—
335#127D4×C3112yes—
336#127C23×C3112yes—
337#125—21no—
338#124—22no—
339#124—22no—
340#124—22no—
341#123—24no—
342#123—24no—
343C22⋊Q8—21no—
344C22⋊Q8D4×C3112yes—
345#121—22no—
346#121—22no—
347#121—22no—
348#121—24no—
349#121—24no—
350#121—22no—
351#120—24no—
352C4×Q8—21no—
353C4×Q8D4×C3111yescharacteristic
354C4×Q8C23×C3111yescharacteristic
355C4×Q8C23×C3112yes—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4D4×C3111yescharacteristic
365C4×D4C23×C3111yescharacteristic
366#117—21no—
367#117—21no—
368#117D4×C3111yescharacteristic
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C22⋊C4—24no—
372C2×C42—21no—
373C2×C42C23×C3111yescharacteristic
374#106—24no—
375C4⋊C8—21no—
376C4⋊C8C23×C3112yes—
377C4⋊C8C23×C3112yes—
378#103—24no—
379#103—22no—
380#103—22no—
381#102—28no—
382C22⋊C8—21no—
383C4×C8—21no—
384C4×C8—21no—
385C4×C8C23×C3112yes—
386C31#1863411yescharacteristic, direct product factor, Sylow
387C24—22no—
388#41—22no—
389#41—22no—
390#41—84no—
391#41—84no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—41no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8C2×D4×C3112yes—
398C2×Q8C2×D4×C3111yescharacteristic
399C2×D4—21no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—82no—
411C2×D4C2×D4×C3111yescharacteristic
412C2×D4C2×D4×C3114yes—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4C2×D4×C3111yescharacteristic
420C22×C4C2×D4×C3111yescharacteristic
421C22×C4C2×D4×C3111yescharacteristic
422Q16—44no—
423SD16—44no—
424M16—42no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—42no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4C2×D4×C3111yescharacteristic
441C4⋊C4C2×D4×C3111yescharacteristic
442C4⋊C4C2×D4×C3112yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—21no—
448C22⋊C4—22no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—22no—
455C42C2×D4×C3111yescharacteristic
456C42C2×D4×C3111yescharacteristic
457C42C2×D4×C3111yescharacteristic
458C42C24×C3111yescharacteristic
459C23—24no—
460C23—21no—
461C23—44no—
462C23—82no—
463C23—82no—
464C23C31×#12011yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—24no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—82no—
472D4—21no—
473D4—21no—
474D4—44no—
475D4—42no—
476D4—44no—
477D4—44no—
478D4—42no—
479D4—42no—
480D4—44no—
481D4—82no—
482D4—84no—
483D4—84no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4C31×#12011yescharacteristic
499C2×C4C31×#12011yescharacteristic
500C2×C4C31×#12011yescharacteristic
501C2×C4C31×#13512yes—
502C2×C4C31×AGL(1,ℤ/8)12yes—
503C2×C4C22×D4×C3111yescharacteristic
504C2×C4C22×D4×C3111yescharacteristic
505C2×C4C22×D4×C3111yescharacteristic
506C8—42no—
507C8—42no—
508C8—42no—
509C22—21no—
510C22—21no—
511C22—41no—
512C22—44no—
513C22—41no—
514C22—44no—
515C22—41no—
516C22—44no—
517C22—164no—
518C22C31×#44912yes—
519C22C2×C31×#12011yescharacteristic
520C4—21no—
521C4—21no—
522C4—41no—
523C4—41no—
524C4—42no—
525C4—41no—
526C4—42no—
527C4—82no—
528C4C31×#44912yes—
529C4C31×#45512yes—
530C4C2×C31×#13511yescharacteristic
531C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
532C4C31×#57712yes—
533C2—22no—
534C2—84no—
535C2—81no—
536C2C2×C31×#44911yescharacteristic
537C2C31×#300911yescharacteristic
538C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3508C2
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×#3229C2semidirect
3. C2×C31×#486C2semidirect
4. C31×#3039C2semidirect
5. C31×#1642C2semidirect
6. C31×#1640C2semidirect
7. C31×#1545C2semidirect
8. C31×#1493C2semidirect
9. C31×#1476C2semidirect
31. C31×#475C22semidirect
33. C31×#462C22semidirect
35. C31×#461C22semidirect
42. C31×Q8⋊C8C22semidirect
43. C31×Q8⋊C8C22semidirect
44. C31×D4⋊C8C22semidirect
189. #18634C31direct (N × H)
241. #3229C2×C31semidirect
242. C2×#486C2×C31semidirect
243. #3039C2×C31semidirect
244. #1642C2×C31semidirect
245. #1640C2×C31semidirect
246. #1545C2×C31semidirect
247. #1493C2×C31semidirect
248. #1476C2×C31semidirect
294. #475C22×C31semidirect
296. #462C22×C31semidirect
298. #461C22×C31semidirect
305. Q8⋊C8C22×C31semidirect
306. Q8⋊C8C22×C31semidirect
307. D4⋊C8C22×C31semidirect
386. C31#18634direct (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×#32292
C2×C31×#4862
C31×#30392
C31×#16422
C31×#16402
C31×#15452
C31×#14932
C31×#14762
#1863431