Finite Group Databasebeta

← C31×#24108 #21480867

Subgroups — C31×#24108 #21480867

Order 7936.

1062 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#556C211yesmaximal, characteristic
2C2×C31×#483C211yesmaximal, characteristic, complemented
3C2×C31×#428C211yesmaximal, characteristic, complemented
4C31×#2487C211yesmaximal, characteristic
5C31×#2058C212yesmaximal, complemented
6C31×#2035C212yesmaximal, complemented
7C31×#1894C211yesmaximal, characteristic, complemented
8C31×#1870C211yesmaximal, characteristic
9C31×#1863C212yesmaximal, complemented
10C31×#1854C212yesmaximal, complemented
11C31×#1825C211yesmaximal, characteristic, complemented
12C22×M16×C31C2211yescharacteristic
13C31×#556—24no—
14C31×#556—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#126—21no—
17C2×C31×#126—21no—
18C2×C31×#124—21no—
19C2×C31×C22⋊Q8C2211yescharacteristic
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8×C31—21no—
22C2×C4×Q8×C31C2211yescharacteristic
23C2×C4×D4×C31C2211yescharacteristic
24C2×C31×#117C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#483—24no—
28C31×#428—24no—
29C2×C31×#107C2212yes—
30C2×C31×#106C2212yes—
31C2×C31×#106C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×C22⋊C8C2211yescharacteristic
35C2×C31×C22⋊C8C2212yes—
36C31×#400—21no—
37C31×#399—21no—
38C31×#389C2212yes—
39C31×#386—21no—
40C31×#385—21no—
41C31×#380C2211yescharacteristic
42C31×#380C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31D411yescharacteristic
50C23×C4×C31—21no—
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C2×M16×C31—22no—
54C2×M16×C31—24no—
55C22×C8×C31C2312yes—
56C22×C8×C31C2312yes—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—22no—
59C31×#126—28no—
60C31×#126—28no—
61C31×#124—28no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8D414yes—
66C4×Q8×C31—28no—
67C4×Q8×C31—22no—
68C4×Q8×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31—22no—
71C31×#117—22no—
72C31×#117—22no—
73C31×#117—22no—
74C31×#117D414yes—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31D411yescharacteristic
84C2×C4⋊C4×C31C2312yes—
85C2×C4⋊C4×C31C2312yes—
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31D411yescharacteristic
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C42×C31—21no—
98C2×C42×C31—21no—
99C2×C42×C31—21no—
100C2×C42×C31C2311yescharacteristic
101C31×#107—24no—
102C31×#106—24no—
103C31×#106—24no—
104C31×#103—24no—
105C31×#103—24no—
106C31×C22⋊C8—22no—
107C31×C22⋊C8—24no—
108C31×#95—21no—
109C31×#95—22no—
110C31×#95—22no—
111C24×C31C2×D411yescharacteristic
112C24×C31C2×D411yescharacteristic
113C2×Q8×C31—24no—
114C2×Q8×C31—21no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—42no—
118C2×Q8×C31—42no—
119C2×Q8×C31C2×D412yes—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—41no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—42no—
136C22×C4×C31—82no—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2411yescharacteristic
144M16×C31—28no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C2×C8×C31—22no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31#4114yes—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—42no—
164C4⋊C4×C31—82no—
165C4⋊C4×C31C2×D412yes—
166C4⋊C4×C31C2×D412yes—
167C4⋊C4×C31C2×D412yes—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—42no—
175C22⋊C4×C31—82no—
176C42×C31—24no—
177C42×C31—24no—
178C42×C31—24no—
179C42×C31—21no—
180C42×C31—21no—
181#24108C3111yesmaximal, characteristic, direct product factor, Sylow
182C23×C31—22no—
183C23×C31—21no—
184C23×C31—21no—
185C23×C31—22no—
186C23×C31—21no—
187C23×C31—22no—
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—22no—
191C23×C31#12012yes—
192C23×C31C22×D411yescharacteristic
193Q8×C31—24no—
194Q8×C31—48no—
195Q8×C31—41no—
196Q8×C31—41no—
197D4×C31—44no—
198C2×C4×C31—21no—
199C2×C4×C31—22no—
200C2×C4×C31—21no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—24no—
204C2×C4×C31—22no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—42no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—42no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
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—84no—
228C2×C4×C31#12012yes—
229C2×C4×C31#13712yes—
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C2×#4112yes—
233C2×C4×C312−1+412yes—
234C8×C31—24no—
235C8×C31—42no—
236C2×#556C2×C3111yescharacteristic
237C2×#483C2×C3111yescharacteristic, complemented
238C2×#428C2×C3111yescharacteristic, complemented
239#2487C2×C3111yescharacteristic
240#2058C2×C3112yescomplemented
241#2035C2×C3112yescomplemented
242#1894C2×C3111yescharacteristic, complemented
243#1870C2×C3111yescharacteristic
244#1863C2×C3112yescomplemented
245#1854C2×C3112yescomplemented
246#1825C2×C3111yescharacteristic, complemented
247C22×C31—22no—
248C22×C31—22no—
249C22×C31—22no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—41no—
255C22×C31—41no—
256C22×C31—41no—
257C22×C31C2×#12011yescharacteristic
258C22×C31#54712yes—
259C22×C31C2×#13711yescharacteristic
260C22×C31#57511yescharacteristic
261C22×C31#57712yes—
262C4×C31—22no—
263C4×C31—22no—
264C4×C31—41no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—42no—
268C4×C31—41no—
269C4×C31—42no—
270C4×C31—82no—
271C4×C31—82no—
272C4×C31#52511yescharacteristic
273C4×C31#53611yescharacteristic
274C4×C31#54712yes—
275C22×M16C22×C3111yescharacteristic
276#556—24no—
277#556—24no—
278C2×C4⋊Q8C22×C3111yescharacteristic
279C2×#126—21no—
280C2×#126—21no—
281C2×#124—21no—
282C2×C22⋊Q8C22×C3111yescharacteristic
283C2×C22⋊Q8C22×C3111yescharacteristic
284C2×C4×Q8—21no—
285C2×C4×Q8C22×C3111yescharacteristic
286C2×C4×D4C22×C3111yescharacteristic
287C2×#117C22×C3111yescharacteristic
288C2×#117C22×C3111yescharacteristic
289C22×C4⋊C4C22×C3111yescharacteristic
290#483—24no—
291#428—24no—
292C2×#107C22×C3112yes—
293C2×#106C22×C3112yes—
294C2×#106C22×C3112yes—
295C2×#103C22×C3112yes—
296C2×#103C22×C3112yes—
297C2×C22⋊C8C22×C3111yescharacteristic
298C2×C22⋊C8C22×C3112yes—
299#400—21no—
300#399—21no—
301#389C22×C3112yes—
302#386—21no—
303#385—21no—
304#380C22×C3111yescharacteristic
305#380C22×C3112yes—
306#340C22×C3112yes—
307#340C22×C3112yes—
308#340C22×C3112yes—
309#336C22×C3112yes—
310C2×C31—42no—
311C2×C31—81no—
312C2×C31#240611yescharacteristic
313C2×C31#300711yescharacteristic
314C2×C31#301011yescharacteristic
315C2×C31#335012yes—
316C2×C31#335612yes—
317C22×Q8—21no—
318C22×Q8C23×C3111yescharacteristic
319C22×D4D4×C3111yescharacteristic
320C23×C4—21no—
321C23×C4C23×C3111yescharacteristic
322C23×C4C23×C3111yescharacteristic
323C2×M16—22no—
324C2×M16—24no—
325C22×C8C23×C3112yes—
326C22×C8C23×C3112yes—
327C4⋊Q8—22no—
328C4⋊Q8—22no—
329#126—28no—
330#126—28no—
331#124—28no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8D4×C3114yes—
336C4×Q8—28no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×D4—22no—
340C4×D4—22no—
341#117—22no—
342#117—22no—
343#117—22no—
344#117D4×C3114yes—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4D4×C3111yescharacteristic
354C2×C4⋊C4C23×C3112yes—
355C2×C4⋊C4C23×C3112yes—
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C4⋊C4C23×C3111yescharacteristic
358C2×C4⋊C4C23×C3111yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4D4×C3111yescharacteristic
366C2×C22⋊C4D4×C3111yescharacteristic
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42C23×C3111yescharacteristic
371#107—24no—
372#106—24no—
373#106—24no—
374#103—24no—
375#103—24no—
376C22⋊C8—22no—
377C22⋊C8—24no—
378#95—22no—
379#95—21no—
380#95—22no—
381C31#2410811yescharacteristic, direct product factor, Sylow
382C24C2×D4×C3111yescharacteristic
383C24C2×D4×C3111yescharacteristic
384C2×Q8—24no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8C2×D4×C3112yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—82no—
408C22×C4C2×D4×C3111yescharacteristic
409C22×C4C2×D4×C3111yescharacteristic
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3112yes—
412C22×C4C2×D4×C3112yes—
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C24×C3111yescharacteristic
415M16—28no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8#41×C3114yes—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—82no—
436C4⋊C4C2×D4×C3112yes—
437C4⋊C4C2×D4×C3112yes—
438C4⋊C4C2×D4×C3112yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—42no—
446C22⋊C4—82no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—21no—
451C42—21no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—21no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23C31×#12012yes—
462C23C22×D4×C3111yescharacteristic
463Q8—24no—
464Q8—48no—
465Q8—41no—
466Q8—41no—
467D4—44no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—84no—
498C2×C4C31×#12012yes—
499C2×C4C31×#13712yes—
500C2×C4C22×D4×C3111yescharacteristic
501C2×C4C22×D4×C3111yescharacteristic
502C2×C4C2×#41×C3112yes—
503C2×C4C31×2−1+412yes—
504C8—24no—
505C8—42no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22C2×C31×#12011yescharacteristic
517C22C31×#54712yes—
518C22C2×C31×#13711yescharacteristic
519C22C31×#57511yescharacteristic
520C22C31×#57712yes—
521C4—22no—
522C4—22no—
523C4—41no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4C31×#52511yescharacteristic
532C4C31×#53611yescharacteristic
533C4C31×#54712yes—
534C2—42no—
535C2—81no—
536C2C31×#240611yescharacteristic
537C2C31×#300711yescharacteristic
538C2C31×#301011yescharacteristic
539C2C31×#335012yes—
540C2C31×#335612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D4×C31D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#483C2semidirect
3. C2×C31×#428C2semidirect
5. C31×#2058C2semidirect
6. C31×#2035C2semidirect
7. C31×#1894C2semidirect
9. C31×#1863C2semidirect
10. C31×#1854C2semidirect
11. C31×#1825C2semidirect
181. #24108C31direct (N × H)
237. C2×#483C2×C31semidirect
238. C2×#428C2×C31semidirect
240. #2058C2×C31semidirect
241. #2035C2×C31semidirect
242. #1894C2×C31semidirect
244. #1863C2×C31semidirect
245. #1854C2×C31semidirect
246. #1825C2×C31semidirect
381. C31#24108direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5562
C2×C31×#4832
C2×C31×#4282
C31×#24872
C31×#20582
C31×#20352
C31×#18942
C31×#18702
C31×#18632
C31×#18542
C31×#18252
#2410831