Finite Group Databasebeta

← C31×#18139 #21474898

Subgroups — C31×#18139 #21474898

Order 7936.

962 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#539C211yesmaximal, characteristic
2C31×#3095C211yesmaximal, characteristic, complemented
3C31×#3094C211yesmaximal, characteristic
4C31×#2484C211yesmaximal, characteristic, complemented
5C31×#1615C214yesmaximal, complemented
6C31×#1609C214yesmaximal, complemented
7C31×#1511C211yesmaximal, characteristic, complemented
8C31×#1510C211yesmaximal, characteristic
9C31×#1463C211yesmaximal, characteristic, complemented
10C31×#539—22no—
11C31×#539—22no—
12C31×#539—22no—
13C31×#539C2214yes—
14C2×C31×#126—21no—
15C2×C31×#125—21no—
16C2×C31×#123—21no—
17C2×C31×#123—21no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×C22⋊Q8C2211yescharacteristic
20C2×C31×#121C2211yescharacteristic
21C2×C4×D4×C31—21no—
22C2×C31×#117C2211yescharacteristic
23C22×C4⋊C4×C31C2211yescharacteristic
24C31×#485—22no—
25C31×#484C2212yescomplemented
26C31×#483C2212yescomplemented
27C31×#482—22no—
28C31×#481C2212yes—
29C31×#480C2212yes—
30C31×#428—21no—
31C31×#428—21no—
32C31×#419—21no—
33C31×#418—21no—
34C31×#407C2211yescharacteristic
35C31×#407C2211yescharacteristic
36C31×#400C2212yescomplemented
37C31×#398C2212yescomplemented
38C31×#387—22no—
39C31×#385—21no—
40C31×#381—21no—
41C31×#381C2211yescharacteristic
42C31×#328C2214yescomplemented
43C31×#327C2214yes—
44C31×#324C2214yescomplemented
45C22×Q8×C31D411yescharacteristic
46C22×D4×C31—21no—
47C22×D4×C31D411yescharacteristic
48C23×C4×C31—21no—
49C23×C4×C31D411yescharacteristic
50C23×C4×C31D411yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C2×M16×C31—21no—
53C2×M16×C31—21no—
54C31×#126—24no—
55C31×#126—24no—
56C31×#125—24no—
57C31×#125—24no—
58C31×#123—24no—
59C31×#123—24no—
60C31×#123—24no—
61C31×#123—24no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—24no—
64C31×C22⋊Q8—21no—
65C31×C22⋊Q8—21no—
66C31×C22⋊Q8D412yes—
67C31×C22⋊Q8C2312yes—
68C31×#121—21no—
69C31×#121—21no—
70C31×#121D412yes—
71C31×#121C2312yes—
72C4×D4×C31—24no—
73C4×D4×C31—24no—
74C31×#117—21no—
75C31×#117—21no—
76C31×#117—21no—
77C31×#117—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—22no—
87C2×C4⋊C4×C31D412yes—
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C4⋊C4×C31C2312yes—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C42×C31—21no—
101C2×C42×C31—21no—
102C31×#107—22no—
103C31×#107—22no—
104C31×#106—22no—
105C31×#106—22no—
106C31×#103—22no—
107C31×#103—22no—
108C31×#102—22no—
109C31×#102—22no—
110C31×C22⋊C8—21no—
111C31×C22⋊C8—21no—
112C31×C22⋊C8C2312yes—
113C31×C22⋊C8C2312yes—
114C31×#95—22no—
115C31×#95—22no—
116C31×#95—22no—
117C31×#95—21no—
118C31×#95C2312yes—
119C24×C31—21no—
120C24×C31C2×D411yescharacteristic
121C2×Q8×C31—21no—
122C2×Q8×C31—42no—
123C2×Q8×C31C2×D412yes—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—21no—
127C2×D4×C31—42no—
128C2×D4×C31—44no—
129C2×D4×C31C2×D412yes—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—24no—
137C22×C4×C31—21no—
138C22×C4×C31—22no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—42no—
142C22×C4×C31—42no—
143C22×C4×C31—42no—
144C22×C4×C31—82no—
145C22×C4×C31C2×D412yes—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2411yescharacteristic
153M16×C31—42no—
154M16×C31—42no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—44no—
173C4⋊C4×C31C2×D412yes—
174C4⋊C4×C31C2×D412yes—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—24no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—42no—
183C22⋊C4×C31—41no—
184C22⋊C4×C31—41no—
185C42×C31—24no—
186C42×C31—41no—
187C42×C31—41no—
188#18139C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—41no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31#12011yescharacteristic
197C23×C31#12011yescharacteristic
198C23×C31C22×D411yescharacteristic
199Q8×C31—42no—
200Q8×C31—81no—
201D4×C31—42no—
202D4×C31—44no—
203D4×C31—44no—
204D4×C31—81no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—81no—
233C2×C4×C31—84no—
234C2×C4×C31#12011yescharacteristic
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31C22×D411yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C8×C31—42no—
239C8×C31—42no—
240C2×#539C2×C3111yescharacteristic
241#3095C2×C3111yescharacteristic, complemented
242#3094C2×C3111yescharacteristic
243#2484C2×C3111yescharacteristic, complemented
244#1615C2×C3114yescomplemented
245#1609C2×C3114yescomplemented
246#1511C2×C3111yescharacteristic, complemented
247#1510C2×C3111yescharacteristic
248#1463C2×C3111yescharacteristic, complemented
249C22×C31—21no—
250C22×C31—21no—
251C22×C31—21no—
252C22×C31—21no—
253C22×C31—21no—
254C22×C31—21no—
255C22×C31—21no—
256C22×C31—44no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—84no—
261C22×C31#45812yes—
262C22×C31C2×#12011yescharacteristic
263C22×C31#57512yes—
264C4×C31—21no—
265C4×C31—21no—
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—42no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—42no—
272C4×C31—42no—
273C4×C31—82no—
274C4×C31—81no—
275C4×C31—81no—
276#539—22no—
277#539—22no—
278#539—22no—
279#539C22×C3114yes—
280C2×#126—21no—
281C2×#125—21no—
282C2×#123—21no—
283C2×#123—21no—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8C22×C3111yescharacteristic
286C2×#121C22×C3111yescharacteristic
287C2×C4×D4—21no—
288C2×#117C22×C3111yescharacteristic
289C22×C4⋊C4C22×C3111yescharacteristic
290#485—22no—
291#484C22×C3112yescomplemented
292#483C22×C3112yescomplemented
293#482—22no—
294#481C22×C3112yes—
295#480C22×C3112yes—
296#428—21no—
297#428—21no—
298#419—21no—
299#418—21no—
300#407C22×C3111yescharacteristic
301#407C22×C3111yescharacteristic
302#400C22×C3112yescomplemented
303#398C22×C3112yescomplemented
304#387—22no—
305#385—21no—
306#381—21no—
307#381C22×C3111yescharacteristic
308#328C22×C3114yescomplemented
309#327C22×C3114yes—
310#324C22×C3114yescomplemented
311C2×C31—22no—
312C2×C31—22no—
313C2×C31—41no—
314C2×C31—82no—
315C2×C31#300511yescharacteristic
316C2×C31#302011yescharacteristic
317C2×C31C2×#45811yescharacteristic
318C22×Q8D4×C3111yescharacteristic
319C22×D4—21no—
320C22×D4D4×C3111yescharacteristic
321C23×C4—21no—
322C23×C4D4×C3111yescharacteristic
323C23×C4D4×C3111yescharacteristic
324C23×C4C23×C3111yescharacteristic
325C2×M16—21no—
326C2×M16—21no—
327#126—24no—
328#126—24no—
329#125—24no—
330#125—24no—
331#123—24no—
332#123—24no—
333#123—24no—
334#123—24no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8D4×C3112yes—
340C22⋊Q8C23×C3112yes—
341#121—21no—
342#121—21no—
343#121D4×C3112yes—
344#121C23×C3112yes—
345C4×D4—24no—
346C4×D4—24no—
347#117—21no—
348#117—21no—
349#117—21no—
350#117—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4D4×C3112yes—
361C2×C4⋊C4D4×C3111yescharacteristic
362C2×C4⋊C4D4×C3111yescharacteristic
363C2×C4⋊C4C23×C3111yescharacteristic
364C2×C4⋊C4C23×C3111yescharacteristic
365C2×C4⋊C4C23×C3112yes—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C42—21no—
374C2×C42—21no—
375#107—22no—
376#107—22no—
377#106—22no—
378#106—22no—
379#103—22no—
380#103—22no—
381#102—22no—
382#102—22no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385C22⋊C8C23×C3112yes—
386C22⋊C8C23×C3112yes—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95C23×C3112yes—
392C31#1813911yescharacteristic, direct product factor, Sylow
393C24—21no—
394C24C2×D4×C3111yescharacteristic
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8C2×D4×C3112yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4C2×D4×C3112yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—82no—
419C22×C4C2×D4×C3112yes—
420C22×C4C2×D4×C3111yescharacteristic
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C2×D4×C3112yes—
423C22×C4C2×D4×C3111yescharacteristic
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4C2×D4×C3111yescharacteristic
426C22×C4C24×C3111yescharacteristic
427M16—42no—
428M16—42no—
429C2×C8—22no—
430C2×C8—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—44no—
447C4⋊C4C2×D4×C3112yes—
448C4⋊C4C2×D4×C3112yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C42—24no—
460C42—41no—
461C42—41no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23C31×#12011yescharacteristic
470C23C31×#12011yescharacteristic
471C23C22×D4×C3111yescharacteristic
472Q8—42no—
473Q8—81no—
474D4—42no—
475D4—44no—
476D4—44no—
477D4—81no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—81no—
506C2×C4—84no—
507C2×C4C31×#12011yescharacteristic
508C2×C4C31×#12011yescharacteristic
509C2×C4C22×D4×C3111yescharacteristic
510C2×C4C22×D4×C3111yescharacteristic
511C8—42no—
512C8—42no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—84no—
525C22C31×#45812yes—
526C22C2×C31×#12011yescharacteristic
527C22C31×#57512yes—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—82no—
538C4—81no—
539C4—81no—
540C2—22no—
541C2—22no—
542C2—41no—
543C2—82no—
544C2C31×#300511yescharacteristic
545C2C31×#302011yescharacteristic
546C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C31×#3095C2semidirect
4. C31×#2484C2semidirect
5. C31×#1615C2semidirect
6. C31×#1609C2semidirect
7. C31×#1511C2semidirect
9. C31×#1463C2semidirect
25. C31×#484C22semidirect
26. C31×#483C22semidirect
36. C31×#400C22semidirect
37. C31×#398C22semidirect
42. C31×#328C22semidirect
44. C31×#324C22semidirect
188. #18139C31direct (N × H)
241. #3095C2×C31semidirect
243. #2484C2×C31semidirect
244. #1615C2×C31semidirect
245. #1609C2×C31semidirect
246. #1511C2×C31semidirect
248. #1463C2×C31semidirect
291. #484C22×C31semidirect
292. #483C22×C31semidirect
302. #400C22×C31semidirect
303. #398C22×C31semidirect
308. #328C22×C31semidirect
310. #324C22×C31semidirect
392. C31#18139direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5392
C31×#30952
C31×#30942
C31×#24842
C31×#16152
C31×#16092
C31×#15112
C31×#15102
C31×#14632
#1813931