Finite Group Databasebeta

← C31×#24104 #21480863

Subgroups — C31×#24104 #21480863

Order 7936.

1146 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#555C211yesmaximal, characteristic
2C2×C31×#481C211yesmaximal, characteristic, complemented
3C2×C31×#428C211yesmaximal, characteristic, complemented
4C31×#2486C211yesmaximal, characteristic, complemented
5C31×#2058C212yesmaximal, complemented
6C31×#2034C212yesmaximal, complemented
7C31×#1895C211yesmaximal, characteristic, complemented
8C31×#1870C211yesmaximal, characteristic, complemented
9C31×#1862C212yesmaximal, 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×#123—21no—
19C2×C31×#121—21no—
20C2×C31×#121C2211yescharacteristic
21C2×C31×#121C2211yescharacteristic
22C2×C4×D4×C31C2211yescharacteristic
23C2×C4×D4×C31C2211yescharacteristic
24C2×C31×#117—21no—
25C2×C31×#117C2211yescharacteristic
26C2×C31×#117C2211yescharacteristic
27C22×C4⋊C4×C31C2211yescharacteristic
28C31×#481—24no—
29C31×#428—24no—
30C2×C31×#107C2212yes—
31C2×C31×#106C2212yes—
32C2×C31×#106C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2212yes—
37C31×#398—21no—
38C31×#397—21no—
39C31×#389C2212yes—
40C31×#381—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×#123—44no—
65C31×#121—28no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121D414yes—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C31×#117—28no—
75C31×#117—22no—
76C31×#117—22no—
77C31×#117—22no—
78C31×#117D414yes—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31D411yescharacteristic
85C2×C4⋊C4×C31C2312yes—
86C2×C4⋊C4×C31C2312yes—
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2311yescharacteristic
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×C31—21no—
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C22⋊C4×C31D411yescharacteristic
99C2×C42×C31—21no—
100C2×C42×C31—21no—
101C2×C42×C31C2311yescharacteristic
102C31×#107—24no—
103C31×#106—24no—
104C31×#106—24no—
105C31×#102—24no—
106C31×#102—24no—
107C31×C22⋊C8—22no—
108C31×C22⋊C8—24no—
109C31×#95—21no—
110C31×#95—22no—
111C31×#95—22no—
112C24×C31—21no—
113C24×C31C2×D411yescharacteristic
114C24×C31C2×D411yescharacteristic
115C2×Q8×C31—24no—
116C2×Q8×C31—82no—
117C2×D4×C31—22no—
118C2×D4×C31—24no—
119C2×D4×C31—24no—
120C2×D4×C31—22no—
121C2×D4×C31—21no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—41no—
125C2×D4×C31—82no—
126C2×D4×C31—82no—
127C2×D4×C31C2×D412yes—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—24no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—42no—
141C22×C4×C31—42no—
142C22×C4×C31—42no—
143C22×C4×C31—42no—
144C22×C4×C31—42no—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2411yescharacteristic
152M16×C31—28no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31#4114yes—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—42no—
169C4⋊C4×C31—44no—
170C4⋊C4×C31C2×D412yes—
171C4⋊C4×C31C2×D412yes—
172C4⋊C4×C31C2×D412yes—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—24no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—24no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—42no—
181C22⋊C4×C31—44no—
182C42×C31—24no—
183C42×C31—24no—
184C42×C31—21no—
185C42×C31—21no—
186#24104C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—22no—
188C23×C31—21no—
189C23×C31—21no—
190C23×C31—22no—
191C23×C31—21no—
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—21no—
195C23×C31—21no—
196C23×C31—22no—
197C23×C31—42no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31—42no—
201C23×C31#12012yes—
202C23×C31C22×D411yescharacteristic
203Q8×C31—84no—
204D4×C31—24no—
205D4×C31—44no—
206D4×C31—41no—
207D4×C31—41no—
208D4×C31—84no—
209D4×C31—84no—
210C2×C4×C31—21no—
211C2×C4×C31—22no—
212C2×C4×C31—21no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—21no—
217C2×C4×C31—22no—
218C2×C4×C31—21no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—48no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
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—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—84no—
238C2×C4×C31#12012yes—
239C2×C4×C31AGL(1,ℤ/8)12yes—
240C2×C4×C31C22×D411yescharacteristic
241C2×C4×C31C22×D411yescharacteristic
242C2×C4×C31C2×#4112yes—
243C2×C4×C312−1+412yes—
244C8×C31—24no—
245C8×C31—42no—
246C2×#555C2×C3111yescharacteristic
247C2×#481C2×C3111yescharacteristic, complemented
248C2×#428C2×C3111yescharacteristic, complemented
249#2486C2×C3111yescharacteristic, complemented
250#2058C2×C3112yescomplemented
251#2034C2×C3112yescomplemented
252#1895C2×C3111yescharacteristic, complemented
253#1870C2×C3111yescharacteristic, complemented
254#1862C2×C3112yescomplemented
255#1854C2×C3112yescomplemented
256#1825C2×C3111yescharacteristic, complemented
257C22×C31—22no—
258C22×C31—22no—
259C22×C31—22no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—41no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—41no—
268C22×C31—41no—
269C22×C31—42no—
270C22×C31—82no—
271C22×C31—82no—
272C22×C31C2×#12011yescharacteristic
273C22×C31#54712yes—
274C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C31#57511yescharacteristic
276C22×C31#57712yes—
277C4×C31—22no—
278C4×C31—22no—
279C4×C31—41no—
280C4×C31—42no—
281C4×C31—41no—
282C4×C31—42no—
283C4×C31—42no—
284C4×C31—82no—
285C4×C31—82no—
286C4×C31#52511yescharacteristic
287C4×C31#53611yescharacteristic
288C4×C31#54712yes—
289C22×M16C22×C3111yescharacteristic
290#555—24no—
291#555—24no—
292C2×C4⋊Q8C22×C3111yescharacteristic
293C2×#124—21no—
294C2×#124—21no—
295C2×#123—21no—
296C2×#121—21no—
297C2×#121C22×C3111yescharacteristic
298C2×#121C22×C3111yescharacteristic
299C2×C4×D4C22×C3111yescharacteristic
300C2×C4×D4C22×C3111yescharacteristic
301C2×#117—21no—
302C2×#117C22×C3111yescharacteristic
303C2×#117C22×C3111yescharacteristic
304C22×C4⋊C4C22×C3111yescharacteristic
305#481—24no—
306#428—24no—
307C2×#107C22×C3112yes—
308C2×#106C22×C3112yes—
309C2×#106C22×C3112yes—
310C2×#102C22×C3112yes—
311C2×#102C22×C3112yes—
312C2×C22⋊C8C22×C3111yescharacteristic
313C2×C22⋊C8C22×C3112yes—
314#398—21no—
315#397—21no—
316#389C22×C3112yes—
317#381—21no—
318#380C22×C3111yescharacteristic
319#380C22×C3112yes—
320#340C22×C3112yes—
321#340C22×C3112yes—
322#340C22×C3112yes—
323#336C22×C3112yes—
324C2×C31—42no—
325C2×C31—81no—
326C2×C31—82no—
327C2×C31#240611yescharacteristic
328C2×C31#300611yescharacteristic
329C2×C31#300911yescharacteristic
330C2×C31#334812yes—
331C2×C31#335412yes—
332C22×Q8—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4D4×C3111yescharacteristic
336C22×D4C23×C3111yescharacteristic
337C23×C4—21no—
338C23×C4—21no—
339C23×C4C23×C3111yescharacteristic
340C23×C4C23×C3111yescharacteristic
341C2×M16—22no—
342C2×M16—24no—
343C22×C8C23×C3112yes—
344C22×C8C23×C3112yes—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347#124—28no—
348#124—28no—
349#123—44no—
350#121—28no—
351#121—22no—
352#121—22no—
353#121—22no—
354#121D4×C3114yes—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359#117—28no—
360#117—22no—
361#117—22no—
362#117—22no—
363#117D4×C3114yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4D4×C3111yescharacteristic
370C2×C4⋊C4C23×C3112yes—
371C2×C4⋊C4C23×C3112yes—
372C2×C4⋊C4C23×C3111yescharacteristic
373C2×C4⋊C4C23×C3111yescharacteristic
374C2×C4⋊C4C23×C3111yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4D4×C3111yescharacteristic
383C2×C22⋊C4D4×C3111yescharacteristic
384C2×C42—21no—
385C2×C42—21no—
386C2×C42C23×C3111yescharacteristic
387#107—24no—
388#106—24no—
389#106—24no—
390#102—24no—
391#102—24no—
392C22⋊C8—22no—
393C22⋊C8—24no—
394#95—22no—
395#95—21no—
396#95—22no—
397C31#2410411yescharacteristic, direct product factor, Sylow
398C24—21no—
399C24C2×D4×C3111yescharacteristic
400C24C2×D4×C3111yescharacteristic
401C2×Q8—24no—
402C2×Q8—82no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—41no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4C2×D4×C3112yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3112yes—
435C22×C4C2×D4×C3112yes—
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C24×C3111yescharacteristic
438M16—28no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8#41×C3114yes—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4C2×D4×C3112yes—
457C4⋊C4C2×D4×C3112yes—
458C4⋊C4C2×D4×C3112yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—42no—
467C22⋊C4—44no—
468C42—24no—
469C42—24no—
470C42—21no—
471C42—21no—
472C23—22no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23—22no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23C31×#12012yes—
487C23C22×D4×C3111yescharacteristic
488Q8—84no—
489D4—24no—
490D4—44no—
491D4—41no—
492D4—41no—
493D4—84no—
494D4—84no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—48no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4C31×#12012yes—
524C2×C4C31×AGL(1,ℤ/8)12yes—
525C2×C4C22×D4×C3111yescharacteristic
526C2×C4C22×D4×C3111yescharacteristic
527C2×C4C2×#41×C3112yes—
528C2×C4C31×2−1+412yes—
529C8—24no—
530C8—42no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22C2×C31×#12011yescharacteristic
547C22C31×#54712yes—
548C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
549C22C31×#57511yescharacteristic
550C22C31×#57712yes—
551C4—22no—
552C4—22no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4C31×#52511yescharacteristic
561C4C31×#53611yescharacteristic
562C4C31×#54712yes—
563C2—42no—
564C2—81no—
565C2—82no—
566C2C31×#240611yescharacteristic
567C2C31×#300611yescharacteristic
568C2C31×#300911yescharacteristic
569C2C31×#334812yes—
570C2C31×#335412yes—
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×#481C2semidirect
3. C2×C31×#428C2semidirect
4. C31×#2486C2semidirect
5. C31×#2058C2semidirect
6. C31×#2034C2semidirect
7. C31×#1895C2semidirect
8. C31×#1870C2semidirect
9. C31×#1862C2semidirect
10. C31×#1854C2semidirect
11. C31×#1825C2semidirect
186. #24104C31direct (N × H)
247. C2×#481C2×C31semidirect
248. C2×#428C2×C31semidirect
249. #2486C2×C31semidirect
250. #2058C2×C31semidirect
251. #2034C2×C31semidirect
252. #1895C2×C31semidirect
253. #1870C2×C31semidirect
254. #1862C2×C31semidirect
255. #1854C2×C31semidirect
256. #1825C2×C31semidirect
397. C31#24104direct (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×#4812
C2×C31×#4282
C31×#24862
C31×#20582
C31×#20342
C31×#18952
C31×#18702
C31×#18622
C31×#18542
C31×#18252
#2410431