Finite Group Databasebeta

← C31×#24105 #21480864

Subgroups — C31×#24105 #21480864

Order 7936.

1126 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#555C211yesmaximal, characteristic
2C2×C31×#480C211yesmaximal, characteristic, complemented
3C2×C31×#428C211yesmaximal, characteristic, complemented
4C31×#2487C211yesmaximal, characteristic, complemented
5C31×#2057C212yesmaximal, complemented
6C31×#2033C212yesmaximal, complemented
7C31×#1895C211yesmaximal, characteristic, complemented
8C31×#1870C211yesmaximal, characteristic, complemented
9C31×#1864C212yesmaximal, complemented
10C31×#1854C212yesmaximal, complemented
11C31×#1825C211yesmaximal, characteristic, complemented
12C22×M16×C31C2211yescharacteristic
13C31×#555—24no—
14C31×#555—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#124—21no—
17C2×C31×#124—21no—
18C2×C31×#121—21no—
19C2×C31×#121C2211yescharacteristic
20C2×C31×#121C2211yescharacteristic
21C2×C4×D4×C31C2211yescharacteristic
22C2×C4×D4×C31C2211yescharacteristic
23C2×C31×#117—21no—
24C2×C31×#117C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#480—24no—
28C31×#428—24no—
29C2×C31×#107C2212yes—
30C2×C31×#107C2212yes—
31C2×C31×#106C2212yes—
32C2×C31×#102C2212yes—
33C2×C31×#102C2212yes—
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×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31D411yescharacteristic
51C22×D4×C31C2311yescharacteristic
52C23×C4×C31—21no—
53C23×C4×C31—21no—
54C23×C4×C31C2311yescharacteristic
55C23×C4×C31C2311yescharacteristic
56C2×M16×C31—22no—
57C2×M16×C31—24no—
58C22×C8×C31C2312yes—
59C22×C8×C31C2312yes—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—22no—
62C31×#124—28no—
63C31×#124—28no—
64C31×#121—28no—
65C31×#121—22no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121D414yes—
69C4×D4×C31—22no—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C31×#117—28no—
74C31×#117—22no—
75C31×#117—22no—
76C31×#117—22no—
77C31×#117D414yes—
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—21no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C42×C31—21no—
99C2×C42×C31—21no—
100C2×C42×C31C2311yescharacteristic
101C31×#107—24no—
102C31×#107—24no—
103C31×#106—24no—
104C31×#102—24no—
105C31×#102—24no—
106C31×C22⋊C8—22no—
107C31×C22⋊C8—24no—
108C31×#95—21no—
109C31×#95—22no—
110C31×#95—22no—
111C24×C31—21no—
112C24×C31C2×D411yescharacteristic
113C24×C31C2×D411yescharacteristic
114C2×Q8×C31—24no—
115C2×Q8×C31—82no—
116C2×D4×C31—22no—
117C2×D4×C31—24no—
118C2×D4×C31—24no—
119C2×D4×C31—22no—
120C2×D4×C31—21no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—41no—
124C2×D4×C31—82no—
125C2×D4×C31—82no—
126C2×D4×C31C2×D412yes—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—24no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31—42no—
142C22×C4×C31—82no—
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D412yes—
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2411yescharacteristic
150M16×C31—28no—
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31#4114yes—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—42no—
167C4⋊C4×C31—82no—
168C4⋊C4×C31C2×D412yes—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D412yes—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—42no—
179C22⋊C4×C31—82no—
180C42×C31—24no—
181C42×C31—24no—
182C42×C31—21no—
183C42×C31—21no—
184#24105C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—22no—
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—22no—
189C23×C31—21no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—42no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—42no—
199C23×C31#12012yes—
200C23×C31C22×D411yescharacteristic
201Q8×C31—84no—
202D4×C31—24no—
203D4×C31—44no—
204D4×C31—41no—
205D4×C31—41no—
206D4×C31—84no—
207D4×C31—84no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—21no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—24no—
214C2×C4×C31—21no—
215C2×C4×C31—22no—
216C2×C4×C31—21no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—48no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—84no—
236C2×C4×C31#12012yes—
237C2×C4×C31#13712yes—
238C2×C4×C31C22×D411yescharacteristic
239C2×C4×C31C22×D411yescharacteristic
240C2×C4×C31C2×#4112yes—
241C2×C4×C312−1+412yes—
242C8×C31—24no—
243C8×C31—42no—
244C2×#555C2×C3111yescharacteristic
245C2×#480C2×C3111yescharacteristic, complemented
246C2×#428C2×C3111yescharacteristic, complemented
247#2487C2×C3111yescharacteristic, complemented
248#2057C2×C3112yescomplemented
249#2033C2×C3112yescomplemented
250#1895C2×C3111yescharacteristic, complemented
251#1870C2×C3111yescharacteristic, complemented
252#1864C2×C3112yescomplemented
253#1854C2×C3112yescomplemented
254#1825C2×C3111yescharacteristic, complemented
255C22×C31—22no—
256C22×C31—22no—
257C22×C31—22no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—41no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—41no—
266C22×C31—41no—
267C22×C31—42no—
268C22×C31—82no—
269C22×C31—82no—
270C22×C31C2×#12011yescharacteristic
271C22×C31#54712yes—
272C22×C31C2×#13711yescharacteristic
273C22×C31#57511yescharacteristic
274C22×C31#57612yes—
275C4×C31—22no—
276C4×C31—22no—
277C4×C31—41no—
278C4×C31—42no—
279C4×C31—41no—
280C4×C31—42no—
281C4×C31—42no—
282C4×C31—82no—
283C4×C31—82no—
284C4×C31#52511yescharacteristic
285C4×C31#53611yescharacteristic
286C4×C31#54712yes—
287C22×M16C22×C3111yescharacteristic
288#555—24no—
289#555—24no—
290C2×C4⋊Q8C22×C3111yescharacteristic
291C2×#124—21no—
292C2×#124—21no—
293C2×#121—21no—
294C2×#121C22×C3111yescharacteristic
295C2×#121C22×C3111yescharacteristic
296C2×C4×D4C22×C3111yescharacteristic
297C2×C4×D4C22×C3111yescharacteristic
298C2×#117—21no—
299C2×#117C22×C3111yescharacteristic
300C2×#117C22×C3111yescharacteristic
301C22×C4⋊C4C22×C3111yescharacteristic
302#480—24no—
303#428—24no—
304C2×#107C22×C3112yes—
305C2×#107C22×C3112yes—
306C2×#106C22×C3112yes—
307C2×#102C22×C3112yes—
308C2×#102C22×C3112yes—
309C2×C22⋊C8C22×C3111yescharacteristic
310C2×C22⋊C8C22×C3112yes—
311#400—21no—
312#399—21no—
313#389C22×C3112yes—
314#386—21no—
315#385—21no—
316#380C22×C3111yescharacteristic
317#380C22×C3112yes—
318#340C22×C3112yes—
319#340C22×C3112yes—
320#340C22×C3112yes—
321#336C22×C3112yes—
322C2×C31—42no—
323C2×C31—81no—
324C2×C31—82no—
325C2×C31#240611yescharacteristic
326C2×C31#300711yescharacteristic
327C2×C31#300811yescharacteristic
328C2×C31#334712yes—
329C2×C31#335712yes—
330C22×Q8—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4D4×C3111yescharacteristic
334C22×D4C23×C3111yescharacteristic
335C23×C4—21no—
336C23×C4—21no—
337C23×C4C23×C3111yescharacteristic
338C23×C4C23×C3111yescharacteristic
339C2×M16—22no—
340C2×M16—24no—
341C22×C8C23×C3112yes—
342C22×C8C23×C3112yes—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345#124—28no—
346#124—28no—
347#121—28no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121D4×C3114yes—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356#117—28no—
357#117—22no—
358#117—22no—
359#117—22no—
360#117D4×C3114yes—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4D4×C3111yescharacteristic
367C2×C4⋊C4C23×C3112yes—
368C2×C4⋊C4C23×C3112yes—
369C2×C4⋊C4C23×C3111yescharacteristic
370C2×C4⋊C4C23×C3111yescharacteristic
371C2×C4⋊C4C23×C3111yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4D4×C3111yescharacteristic
380C2×C22⋊C4D4×C3111yescharacteristic
381C2×C42—21no—
382C2×C42—21no—
383C2×C42C23×C3111yescharacteristic
384#107—24no—
385#107—24no—
386#106—24no—
387#102—24no—
388#102—24no—
389C22⋊C8—22no—
390C22⋊C8—24no—
391#95—22no—
392#95—21no—
393#95—22no—
394C31#2410511yescharacteristic, direct product factor, Sylow
395C24—21no—
396C24C2×D4×C3111yescharacteristic
397C24C2×D4×C3111yescharacteristic
398C2×Q8—24no—
399C2×Q8—82no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—41no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4C2×D4×C3112yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—82no—
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C2×D4×C3112yes—
431C22×C4C2×D4×C3112yes—
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C24×C3111yescharacteristic
434M16—28no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8#41×C3114yes—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—82no—
452C4⋊C4C2×D4×C3112yes—
453C4⋊C4C2×D4×C3112yes—
454C4⋊C4C2×D4×C3112yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—42no—
463C22⋊C4—82no—
464C42—24no—
465C42—24no—
466C42—21no—
467C42—21no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23C31×#12012yes—
483C23C22×D4×C3111yescharacteristic
484Q8—84no—
485D4—24no—
486D4—44no—
487D4—41no—
488D4—41no—
489D4—84no—
490D4—84no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4C31×#12012yes—
520C2×C4C31×#13712yes—
521C2×C4C22×D4×C3111yescharacteristic
522C2×C4C22×D4×C3111yescharacteristic
523C2×C4C2×#41×C3112yes—
524C2×C4C31×2−1+412yes—
525C8—24no—
526C8—42no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22C2×C31×#12011yescharacteristic
543C22C31×#54712yes—
544C22C2×C31×#13711yescharacteristic
545C22C31×#57511yescharacteristic
546C22C31×#57612yes—
547C4—22no—
548C4—22no—
549C4—41no—
550C4—42no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4C31×#52511yescharacteristic
557C4C31×#53611yescharacteristic
558C4C31×#54712yes—
559C2—42no—
560C2—81no—
561C2—82no—
562C2C31×#240611yescharacteristic
563C2C31×#300711yescharacteristic
564C2C31×#300811yescharacteristic
565C2C31×#334712yes—
566C2C31×#335712yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#555C2
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×#480C2semidirect
3. C2×C31×#428C2semidirect
4. C31×#2487C2semidirect
5. C31×#2057C2semidirect
6. C31×#2033C2semidirect
7. C31×#1895C2semidirect
8. C31×#1870C2semidirect
9. C31×#1864C2semidirect
10. C31×#1854C2semidirect
11. C31×#1825C2semidirect
184. #24105C31direct (N × H)
245. C2×#480C2×C31semidirect
246. C2×#428C2×C31semidirect
247. #2487C2×C31semidirect
248. #2057C2×C31semidirect
249. #2033C2×C31semidirect
250. #1895C2×C31semidirect
251. #1870C2×C31semidirect
252. #1864C2×C31semidirect
253. #1854C2×C31semidirect
254. #1825C2×C31semidirect
394. C31#24105direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5552
C2×C31×#4802
C2×C31×#4282
C31×#24872
C31×#20572
C31×#20332
C31×#18952
C31×#18702
C31×#18642
C31×#18542
C31×#18252
#2410531