Finite Group Databasebeta

← C31×#24305 #21481064

Subgroups — C31×#24305 #21481064

Order 7936.

1162 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#125C211yesmaximal, characteristic, complemented
2C2×C31×#478C212yesmaximal, complemented
3C4×C31×#121C211yesmaximal, characteristic
4C31×#2059C212yesmaximal, complemented
5C31×#2025C212yesmaximal, complemented
6C31×#1989C212yesmaximal, complemented
7C31×#1874C211yesmaximal, characteristic
8C31×#1874C211yesmaximal, characteristic
9C31×#1853C211yesmaximal, characteristic, complemented
10C31×#1852C211yesmaximal, characteristic, complemented
11C31×#1797C211yesmaximal, characteristic, complemented
12C2×C31×#125—28no—
13C2×C31×#125—21no—
14C2×C31×#125—21no—
15C2×C31×#125C2212yes—
16C2×C31×#125C2212yes—
17C2×C31×#121C2211yescharacteristic
18C2×C4×D4×C31—21no—
19C2×C4×D4×C31C2211yescharacteristic
20C22×C4⋊C4×C31—21no—
21C22×C4⋊C4×C31—21no—
22C22×C4⋊C4×C31C2211yescharacteristic
23C22×C4⋊C4×C31C2211yescharacteristic
24C22×C42×C31C2211yescharacteristic
25C31×#478—28no—
26C2×C31×#107C2212yes—
27C2×C31×#106C2212yes—
28C2×C31×C4⋊C8C2212yes—
29C2×C31×#102C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×C22⋊C8C2211yescharacteristic
32C2×C31×C22⋊C8C2211yescharacteristic
33C31×#390C2211yescharacteristic
34C31×#389C2211yescharacteristic
35C31×#388—21no—
36C31×#386—21no—
37C31×#380C2211yescharacteristic
38C4×C4⋊C4×C31C2211yescharacteristic
39C4×C22⋊C4×C31—21no—
40C31×#340C2212yes—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#336C2212yes—
45C22×D4×C31—21no—
46C22×D4×C31C2311yescharacteristic
47C23×C4×C31—21no—
48C23×C4×C31—21no—
49C23×C4×C31—21no—
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C22×C8×C31C2312yes—
54C22×C8×C31C2312yes—
55C31×#125—28no—
56C31×#125—28no—
57C31×#125—24no—
58C31×#125—24no—
59C31×#125—24no—
60C31×#125D418yescomplemented
61C31×#121—22no—
62C31×#121Q814yes—
63C4×D4×C31—24no—
64C4×D4×C31—44no—
65C2×C4⋊C4×C31—28no—
66C2×C4⋊C4×C31—28no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—22no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—22no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—24no—
74C2×C4⋊C4×C31—24no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31C2311yescharacteristic
77C2×C4⋊C4×C31C2311yescharacteristic
78C2×C4⋊C4×C31C2312yes—
79C2×C4⋊C4×C31C2312yes—
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C22⋊C4×C31—21no—
82C2×C22⋊C4×C31—21no—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—21no—
86C2×C42×C31—24no—
87C2×C42×C31—21no—
88C2×C42×C31D411yescharacteristic
89C2×C42×C31D411yescharacteristic
90C2×C42×C31C2311yescharacteristic
91C2×C42×C31C2311yescharacteristic
92C31×#107—24no—
93C31×#106—24no—
94C31×C4⋊C8—24no—
95C31×#102—24no—
96C31×#102—24no—
97C31×C22⋊C8—22no—
98C31×C22⋊C8—22no—
99C31×#95—21no—
100C31×#95—21no—
101C31×#95—21no—
102C24×C31—21no—
103C24×C31C2×D411yescharacteristic
104C2×D4×C31—24no—
105C2×D4×C31—22no—
106C2×D4×C31—21no—
107C2×D4×C31—22no—
108C2×D4×C31—42no—
109C2×D4×C31—42no—
110C2×D4×C31C2×Q812yes—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—24no—
114C22×C4×C31—24no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—41no—
125C22×C4×C31—41no—
126C22×C4×C31—42no—
127C22×C4×C31—42no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31—42no—
132C22×C4×C31—42no—
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×Q812yes—
139C22×C4×C31#4112yes—
140C22×C4×C31C2411yescharacteristic
141C2×C8×C31—22no—
142C2×C8×C31—22no—
143C2×C8×C31—22no—
144C2×C8×C31—22no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—24no—
149C4⋊C4×C31—28no—
150C4⋊C4×C31—28no—
151C4⋊C4×C31—28no—
152C4⋊C4×C31—28no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31C2×D414yes—
160C4⋊C4×C31C2×D414yes—
161C4⋊C4×C31C2×Q812yes—
162C22⋊C4×C31—24no—
163C22⋊C4×C31—42no—
164C22⋊C4×C31—42no—
165C22⋊C4×C31—42no—
166C22⋊C4×C31—42no—
167C42×C31—22no—
168C42×C31—22no—
169C42×C31—22no—
170C42×C31—42no—
171C42×C31C2×D414yes—
172#24305C3111yesmaximal, characteristic, direct product factor, Sylow
173C23×C31—21no—
174C23×C31—21no—
175C23×C31—21no—
176C23×C31—21no—
177C23×C31—22no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31C22⋊Q812yes—
185C23×C31#13512yes—
186C23×C31AGL(1,ℤ/8)12yes—
187C23×C31C22×D411yescharacteristic
188D4×C31—24no—
189D4×C31—48no—
190D4×C31—42no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31—21no—
196C2×C4×C31—21no—
197C2×C4×C31—24no—
198C2×C4×C31—24no—
199C2×C4×C31—21no—
200C2×C4×C31—21no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—22no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—44no—
209C2×C4×C31—44no—
210C2×C4×C31—48no—
211C2×C4×C31—48no—
212C2×C4×C31—42no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—84no—
225C2×C4×C31#12012yes—
226C2×C4×C31#12012yes—
227C2×C4×C31C22⋊Q812yes—
228C2×C4×C31C22×D412yes—
229C2×C4×C31C22×Q811yescharacteristic
230C2×C4×C31C2×#4111yescharacteristic
231C2×C4×C312−1+412yes—
232C8×C31—42no—
233C8×C31—42no—
234C22×#125C2×C3111yescharacteristic, complemented
235C2×#478C2×C3112yescomplemented
236C4×#121C2×C3111yescharacteristic
237#2059C2×C3112yescomplemented
238#2025C2×C3112yescomplemented
239#1989C2×C3112yescomplemented
240#1874C2×C3111yescharacteristic
241#1874C2×C3111yescharacteristic
242#1853C2×C3111yescharacteristic, complemented
243#1852C2×C3111yescharacteristic, complemented
244#1797C2×C3111yescharacteristic, complemented
245C22×C31—22no—
246C22×C31—22no—
247C22×C31—22no—
248C22×C31—22no—
249C22×C31—22no—
250C22×C31—22no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—81no—
255C22×C31—81no—
256C22×C31—81no—
257C22×C31—81no—
258C22×C31#47814yescomplemented
259C22×C31C2×#12011yescharacteristic
260C22×C31C2×C22⋊Q811yescharacteristic
261C22×C31#53611yescharacteristic
262C22×C31C2×#13511yescharacteristic
263C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C31#57511yescharacteristic
265C22×C31#57511yescharacteristic
266C4×C31—22no—
267C4×C31—22no—
268C4×C31—22no—
269C4×C31—42no—
270C4×C31—41no—
271C4×C31—44no—
272C4×C31—41no—
273C4×C31—44no—
274C4×C31—81no—
275C4×C31—81no—
276C4×C31#54812yes—
277C4×C31D4×Q812yes—
278C2×#125—28no—
279C2×#125—21no—
280C2×#125—21no—
281C2×#125C22×C3112yes—
282C2×#125C22×C3112yes—
283C2×#121C22×C3111yescharacteristic
284C2×C4×D4—21no—
285C2×C4×D4C22×C3111yescharacteristic
286C22×C4⋊C4—21no—
287C22×C4⋊C4—21no—
288C22×C4⋊C4C22×C3111yescharacteristic
289C22×C4⋊C4C22×C3111yescharacteristic
290C22×C42C22×C3111yescharacteristic
291#478—28no—
292C2×#107C22×C3112yes—
293C2×#106C22×C3112yes—
294C2×C4⋊C8C22×C3112yes—
295C2×#102C22×C3112yes—
296C2×#102C22×C3112yes—
297C2×C22⋊C8C22×C3111yescharacteristic
298C2×C22⋊C8C22×C3111yescharacteristic
299#390C22×C3111yescharacteristic
300#389C22×C3111yescharacteristic
301#388—21no—
302#386—21no—
303#380C22×C3111yescharacteristic
304C4×C4⋊C4C22×C3111yescharacteristic
305C4×C22⋊C4—21no—
306#340C22×C3112yes—
307#340C22×C3112yes—
308#340C22×C3112yes—
309#340C22×C3112yes—
310#336C22×C3112yes—
311C2×C31—24no—
312C2×C31—82no—
313C2×C31#240911yescharacteristic
314C2×C31#300211yescharacteristic
315C2×C31#300611yescharacteristic
316C2×C31C2×#47811yescharacteristic
317C2×C31#308011yescharacteristic
318C2×C31#323211yescharacteristic
319C2×C31#323411yescharacteristic
320C22×D4—21no—
321C22×D4C23×C3111yescharacteristic
322C23×C4—21no—
323C23×C4—21no—
324C23×C4—21no—
325C23×C4D4×C3111yescharacteristic
326C23×C4D4×C3111yescharacteristic
327C23×C4C23×C3111yescharacteristic
328C22×C8C23×C3112yes—
329C22×C8C23×C3112yes—
330#125—28no—
331#125—28no—
332#125—24no—
333#125—24no—
334#125—24no—
335#125D4×C3118yescomplemented
336#121—22no—
337#121Q8×C3114yes—
338C4×D4—24no—
339C4×D4—44no—
340C2×C4⋊C4—28no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4C23×C3111yescharacteristic
352C2×C4⋊C4C23×C3111yescharacteristic
353C2×C4⋊C4C23×C3112yes—
354C2×C4⋊C4C23×C3112yes—
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C42—24no—
362C2×C42—21no—
363C2×C42D4×C3111yescharacteristic
364C2×C42D4×C3111yescharacteristic
365C2×C42C23×C3111yescharacteristic
366C2×C42C23×C3111yescharacteristic
367#107—24no—
368#106—24no—
369C4⋊C8—24no—
370#102—24no—
371#102—24no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377C31#2430511yescharacteristic, direct product factor, Sylow
378C24—21no—
379C24C2×D4×C3111yescharacteristic
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4C2×Q8×C3112yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4C2×D4×C3111yescharacteristic
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×Q8×C3112yes—
415C22×C4#41×C3112yes—
416C22×C4C24×C3111yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4C2×D4×C3114yes—
436C4⋊C4C2×D4×C3114yes—
437C4⋊C4C2×Q8×C3112yes—
438C22⋊C4—24no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—42no—
447C42C2×D4×C3114yes—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23C31×C22⋊Q812yes—
460C23C31×#13512yes—
461C23C31×AGL(1,ℤ/8)12yes—
462C23C22×D4×C3111yescharacteristic
463D4—24no—
464D4—48no—
465D4—42no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—84no—
500C2×C4C31×#12012yes—
501C2×C4C31×#12012yes—
502C2×C4C31×C22⋊Q812yes—
503C2×C4C22×D4×C3112yes—
504C2×C4C22×Q8×C3111yescharacteristic
505C2×C4C2×#41×C3111yescharacteristic
506C2×C4C31×2−1+412yes—
507C8—42no—
508C8—42no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22C31×#47814yescomplemented
523C22C2×C31×#12011yescharacteristic
524C22C2×C31×C22⋊Q811yescharacteristic
525C22C31×#53611yescharacteristic
526C22C2×C31×#13511yescharacteristic
527C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
528C22C31×#57511yescharacteristic
529C22C31×#57511yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—41no—
536C4—44no—
537C4—42no—
538C4—81no—
539C4—81no—
540C4C31×#54812yes—
541C4D4×Q8×C3112yes—
542C2—24no—
543C2—82no—
544C2C31×#240911yescharacteristic
545C2C31×#300211yescharacteristic
546C2C31×#300611yescharacteristic
547C2C2×C31×#47811yescharacteristic
548C2C31×#308011yescharacteristic
549C2C31×#323211yescharacteristic
550C2C31×#323411yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×#125C2semidirect
2. C2×C31×#478C2semidirect
4. C31×#2059C2semidirect
5. C31×#2025C2semidirect
6. C31×#1989C2semidirect
9. C31×#1853C2semidirect
10. C31×#1852C2semidirect
11. C31×#1797C2semidirect
60. C31×#125D4semidirect
172. #24305C31direct (N × H)
234. C22×#125C2×C31semidirect
235. C2×#478C2×C31semidirect
237. #2059C2×C31semidirect
238. #2025C2×C31semidirect
239. #1989C2×C31semidirect
242. #1853C2×C31semidirect
243. #1852C2×C31semidirect
244. #1797C2×C31semidirect
258. C22×C31#478semidirect
335. #125D4×C31semidirect
377. C31#24305direct (N × H)
522. C22C31×#478semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1252
C2×C31×#4782
C4×C31×#1212
C31×#20592
C31×#20252
C31×#19892
C31×#18742
C31×#18532
C31×#18522
C31×#17972
#2430531