Finite Group Databasebeta

← C31×#24107 #21480866

Subgroups — C31×#24107 #21480866

Order 7936.

1082 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#556C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic, complemented
3C2×C31×#428C211yesmaximal, characteristic, complemented
4C31×#2486C211yesmaximal, characteristic
5C31×#2057C212yesmaximal, complemented
6C31×#2036C212yesmaximal, complemented
7C31×#1894C211yesmaximal, characteristic, complemented
8C31×#1870C211yesmaximal, characteristic
9C31×#1865C212yesmaximal, 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×#123—21no—
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8×C31—21no—
23C2×C4×Q8×C31C2211yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C2×C31×#117C2211yescharacteristic
27C22×C4⋊C4×C31C2211yescharacteristic
28C31×#484—24no—
29C31×#428—24no—
30C2×C31×#107C2212yes—
31C2×C31×#107C2212yes—
32C2×C31×#106C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#103C2212yes—
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×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×#123—44no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8D414yes—
67C4×Q8×C31—28no—
68C4×Q8×C31—22no—
69C4×Q8×C31—22no—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C31×#117—22no—
73C31×#117—22no—
74C31×#117—22no—
75C31×#117D414yes—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
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—22no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C42×C31—21no—
99C2×C42×C31—21no—
100C2×C42×C31—21no—
101C2×C42×C31C2311yescharacteristic
102C31×#107—24no—
103C31×#107—24no—
104C31×#106—24no—
105C31×#103—24no—
106C31×#103—24no—
107C31×C22⋊C8—22no—
108C31×C22⋊C8—24no—
109C31×#95—21no—
110C31×#95—22no—
111C31×#95—22no—
112C24×C31C2×D411yescharacteristic
113C24×C31C2×D411yescharacteristic
114C2×Q8×C31—24no—
115C2×Q8×C31—21no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—42no—
119C2×Q8×C31—42no—
120C2×Q8×C31C2×D412yes—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—41no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—42no—
137C22×C4×C31—42no—
138C22×C4×C31—42no—
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2411yescharacteristic
146M16×C31—28no—
147C2×C8×C31—22no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C2×C8×C31#4114yes—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—42no—
166C4⋊C4×C31—44no—
167C4⋊C4×C31C2×D412yes—
168C4⋊C4×C31C2×D412yes—
169C4⋊C4×C31C2×D412yes—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—42no—
177C22⋊C4×C31—44no—
178C42×C31—24no—
179C42×C31—24no—
180C42×C31—24no—
181C42×C31—21no—
182C42×C31—21no—
183#24107C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—22no—
185C23×C31—21no—
186C23×C31—21no—
187C23×C31—22no—
188C23×C31—21no—
189C23×C31—22no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—22no—
193C23×C31#12012yes—
194C23×C31C22×D411yescharacteristic
195Q8×C31—24no—
196Q8×C31—48no—
197Q8×C31—41no—
198Q8×C31—41no—
199D4×C31—44no—
200C2×C4×C31—21no—
201C2×C4×C31—22no—
202C2×C4×C31—21no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—24no—
206C2×C4×C31—22no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—42no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—42no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
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—42no—
229C2×C4×C31—84no—
230C2×C4×C31#12012yes—
231C2×C4×C31AGL(1,ℤ/8)12yes—
232C2×C4×C31C22×D411yescharacteristic
233C2×C4×C31C22×D411yescharacteristic
234C2×C4×C31C2×#4112yes—
235C2×C4×C312−1+412yes—
236C8×C31—24no—
237C8×C31—42no—
238C2×#556C2×C3111yescharacteristic
239C2×#484C2×C3111yescharacteristic, complemented
240C2×#428C2×C3111yescharacteristic, complemented
241#2486C2×C3111yescharacteristic
242#2057C2×C3112yescomplemented
243#2036C2×C3112yescomplemented
244#1894C2×C3111yescharacteristic, complemented
245#1870C2×C3111yescharacteristic
246#1865C2×C3112yescomplemented
247#1854C2×C3112yescomplemented
248#1825C2×C3111yescharacteristic, complemented
249C22×C31—22no—
250C22×C31—22no—
251C22×C31—22no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—41no—
257C22×C31—41no—
258C22×C31—41no—
259C22×C31C2×#12011yescharacteristic
260C22×C31#54712yes—
261C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C31#57511yescharacteristic
263C22×C31#57812yes—
264C4×C31—22no—
265C4×C31—22no—
266C4×C31—41no—
267C4×C31—42no—
268C4×C31—42no—
269C4×C31—42no—
270C4×C31—41no—
271C4×C31—42no—
272C4×C31—82no—
273C4×C31—82no—
274C4×C31#52511yescharacteristic
275C4×C31#53611yescharacteristic
276C4×C31#54712yes—
277C22×M16C22×C3111yescharacteristic
278#556—24no—
279#556—24no—
280C2×C4⋊Q8C22×C3111yescharacteristic
281C2×#126—21no—
282C2×#126—21no—
283C2×#124—21no—
284C2×#123—21no—
285C2×C22⋊Q8C22×C3111yescharacteristic
286C2×C22⋊Q8C22×C3111yescharacteristic
287C2×C4×Q8—21no—
288C2×C4×Q8C22×C3111yescharacteristic
289C2×C4×D4C22×C3111yescharacteristic
290C2×#117C22×C3111yescharacteristic
291C2×#117C22×C3111yescharacteristic
292C22×C4⋊C4C22×C3111yescharacteristic
293#484—24no—
294#428—24no—
295C2×#107C22×C3112yes—
296C2×#107C22×C3112yes—
297C2×#106C22×C3112yes—
298C2×#103C22×C3112yes—
299C2×#103C22×C3112yes—
300C2×C22⋊C8C22×C3111yescharacteristic
301C2×C22⋊C8C22×C3112yes—
302#398—21no—
303#397—21no—
304#389C22×C3112yes—
305#381—21no—
306#380C22×C3111yescharacteristic
307#380C22×C3112yes—
308#340C22×C3112yes—
309#340C22×C3112yes—
310#340C22×C3112yes—
311#336C22×C3112yes—
312C2×C31—42no—
313C2×C31—81no—
314C2×C31#240611yescharacteristic
315C2×C31#300611yescharacteristic
316C2×C31#301111yescharacteristic
317C2×C31#334912yes—
318C2×C31#335312yes—
319C22×Q8—21no—
320C22×Q8C23×C3111yescharacteristic
321C22×D4D4×C3111yescharacteristic
322C23×C4—21no—
323C23×C4C23×C3111yescharacteristic
324C23×C4C23×C3111yescharacteristic
325C2×M16—22no—
326C2×M16—24no—
327C22×C8C23×C3112yes—
328C22×C8C23×C3112yes—
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331#126—28no—
332#126—28no—
333#124—28no—
334#123—44no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8D4×C3114yes—
339C4×Q8—28no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×D4—22no—
343C4×D4—22no—
344#117—22no—
345#117—22no—
346#117—22no—
347#117D4×C3114yes—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4D4×C3111yescharacteristic
357C2×C4⋊C4C23×C3112yes—
358C2×C4⋊C4C23×C3112yes—
359C2×C4⋊C4C23×C3111yescharacteristic
360C2×C4⋊C4C23×C3111yescharacteristic
361C2×C4⋊C4C23×C3111yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4D4×C3111yescharacteristic
369C2×C22⋊C4D4×C3111yescharacteristic
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42C23×C3111yescharacteristic
374#107—24no—
375#107—24no—
376#106—24no—
377#103—24no—
378#103—24no—
379C22⋊C8—22no—
380C22⋊C8—24no—
381#95—22no—
382#95—21no—
383#95—22no—
384C31#2410711yescharacteristic, direct product factor, Sylow
385C24C2×D4×C3111yescharacteristic
386C24C2×D4×C3111yescharacteristic
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C2×D4×C3112yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—41no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3112yes—
416C22×C4C2×D4×C3112yes—
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C24×C3111yescharacteristic
419M16—28no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8#41×C3114yes—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4C2×D4×C3112yes—
441C4⋊C4C2×D4×C3112yes—
442C4⋊C4C2×D4×C3112yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—21no—
455C42—21no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23C31×#12012yes—
466C23C22×D4×C3111yescharacteristic
467Q8—24no—
468Q8—48no—
469Q8—41no—
470Q8—41no—
471D4—44no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4C31×#12012yes—
503C2×C4C31×AGL(1,ℤ/8)12yes—
504C2×C4C22×D4×C3111yescharacteristic
505C2×C4C22×D4×C3111yescharacteristic
506C2×C4C2×#41×C3112yes—
507C2×C4C31×2−1+412yes—
508C8—24no—
509C8—42no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22C2×C31×#12011yescharacteristic
521C22C31×#54712yes—
522C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
523C22C31×#57511yescharacteristic
524C22C31×#57812yes—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—82no—
534C4—82no—
535C4C31×#52511yescharacteristic
536C4C31×#53611yescharacteristic
537C4C31×#54712yes—
538C2—42no—
539C2—81no—
540C2C31×#240611yescharacteristic
541C2C31×#300611yescharacteristic
542C2C31×#301111yescharacteristic
543C2C31×#334912yes—
544C2C31×#335312yes—
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×#484C2semidirect
3. C2×C31×#428C2semidirect
5. C31×#2057C2semidirect
6. C31×#2036C2semidirect
7. C31×#1894C2semidirect
9. C31×#1865C2semidirect
10. C31×#1854C2semidirect
11. C31×#1825C2semidirect
183. #24107C31direct (N × H)
239. C2×#484C2×C31semidirect
240. C2×#428C2×C31semidirect
242. #2057C2×C31semidirect
243. #2036C2×C31semidirect
244. #1894C2×C31semidirect
246. #1865C2×C31semidirect
247. #1854C2×C31semidirect
248. #1825C2×C31semidirect
384. C31#24107direct (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×#4842
C2×C31×#4282
C31×#24862
C31×#20572
C31×#20362
C31×#18942
C31×#18702
C31×#18652
C31×#18542
C31×#18252
#2410731