Finite Group Databasebeta

← C31×#24304 #21481063

Subgroups — C31×#24304 #21481063

Order 7936.

1150 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#533C211yesmaximal, characteristic, complemented
2C2×C31×#478C212yesmaximal, complemented
3C4×C31×#121C211yesmaximal, characteristic
4C31×#2058C211yesmaximal, characteristic, complemented
5C31×#2057C211yesmaximal, characteristic, complemented
6C31×#2023C211yesmaximal, characteristic, complemented
7C31×#2023C211yesmaximal, characteristic, complemented
8C31×#1989C212yesmaximal, complemented
9C31×#1875C212yesmaximal
10C31×#1854C212yesmaximal, complemented
11C31×#1797C211yesmaximal, characteristic, complemented
12C31×#533—28no—
13C2×C31×C4⋊Q8C2211yescharacteristic
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×#125C2212yes—
16C2×C31×C22⋊Q8—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×#121C2211yescharacteristic
19C2×C4×Q8×C31—21no—
20C2×C4×Q8×C31—21no—
21C2×C4×D4×C31—21no—
22C2×C4×D4×C31C2211yescharacteristic
23C2×C31×#117C2212yes—
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⋊C8C2212yes—
32C31×#390C2211yescharacteristic
33C31×#389C2211yescharacteristic
34C31×#388—21no—
35C31×#386—21no—
36C31×#380C2211yescharacteristic
37C4×C4⋊C4×C31C2211yescharacteristic
38C4×C22⋊C4×C31—21no—
39C31×#340C2212yes—
40C31×#340C2212yes—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#336C2211yescharacteristic
44C31×#336C2211yescharacteristic
45C22×Q8×C31—21no—
46C22×Q8×C31—21no—
47C22×D4×C31—21no—
48C22×D4×C31C2311yescharacteristic
49C23×C4×C31—21no—
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C22×C8×C31C2312yes—
54C22×C8×C31C2312yes—
55C31×C4⋊Q8—24no—
56C31×C4⋊Q8—24no—
57C31×#125—24no—
58C31×#125D418yescomplemented
59C31×C22⋊Q8—28no—
60C31×C22⋊Q8—28no—
61C31×#121—22no—
62C31×#121Q814yescomplemented
63C4×Q8×C31—28no—
64C4×Q8×C31—28no—
65C4×D4×C31—24no—
66C4×D4×C31—44no—
67C31×#117—24no—
68C31×#117—24no—
69C2×C4⋊C4×C31—22no—
70C2×C4⋊C4×C31—22no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31C2311yescharacteristic
75C2×C4⋊C4×C31C2311yescharacteristic
76C2×C4⋊C4×C31C2312yes—
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2311yescharacteristic
79C2×C22⋊C4×C31—22no—
80C2×C22⋊C4×C31—21no—
81C2×C22⋊C4×C31—21no—
82C2×C22⋊C4×C31—21no—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—21no—
85C2×C42×C31—22no—
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—24no—
98C31×#95—21no—
99C31×#95—21no—
100C31×#95—21no—
101C24×C31—21no—
102C24×C31C2×D411yescharacteristic
103C2×Q8×C31—24no—
104C2×Q8×C31—24no—
105C2×Q8×C31—42no—
106C2×Q8×C31—42no—
107C2×Q8×C31—42no—
108C2×Q8×C31—42no—
109C2×D4×C31—24no—
110C2×D4×C31—22no—
111C2×D4×C31—21no—
112C2×D4×C31—22no—
113C2×D4×C31—42no—
114C2×D4×C31—42no—
115C2×D4×C31C2×Q812yes—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—41no—
128C22×C4×C31—41no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31—42no—
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×Q812yes—
138C22×C4×C31#4112yes—
139C22×C4×C31C2411yescharacteristic
140C2×C8×C31—22no—
141C2×C8×C31—22no—
142C2×C8×C31—22no—
143C2×C8×C31—22no—
144C2×C8×C31—22no—
145C2×C8×C31—22no—
146C4⋊C4×C31—24no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—28no—
149C4⋊C4×C31—28no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—42no—
156C4⋊C4×C31C2×D414yes—
157C4⋊C4×C31C2×D414yes—
158C4⋊C4×C31C2×Q812yes—
159C22⋊C4×C31—28no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—42no—
162C22⋊C4×C31—42no—
163C22⋊C4×C31—42no—
164C22⋊C4×C31—42no—
165C42×C31—28no—
166C42×C31—22no—
167C42×C31—22no—
168C42×C31—22no—
169C42×C31—42no—
170C42×C31C2×D414yes—
171#24304C3111yesmaximal, characteristic, direct product factor, Sylow
172C23×C31—21no—
173C23×C31—21no—
174C23×C31—21no—
175C23×C31—21no—
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—22no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31C22⋊Q812yes—
186C23×C31C22×D411yescharacteristic
187Q8×C31—48no—
188Q8×C31—48no—
189D4×C31—24no—
190D4×C31—48no—
191D4×C31—42no—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31—21no—
196C2×C4×C31—21no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—21no—
201C2×C4×C31—21no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—22no—
207C2×C4×C31—44no—
208C2×C4×C31—44no—
209C2×C4×C31—42no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—84no—
222C2×C4×C31#12012yes—
223C2×C4×C31#12012yes—
224C2×C4×C31C22⋊Q812yes—
225C2×C4×C31#13512yes—
226C2×C4×C31AGL(1,ℤ/8)12yes—
227C2×C4×C31C22×D412yes—
228C2×C4×C31C22×Q811yescharacteristic
229C2×C4×C31C2×#4111yescharacteristic
230C2×C4×C312−1+412yes—
231C8×C31—42no—
232C8×C31—42no—
233C2×#533C2×C3111yescharacteristic, complemented
234C2×#478C2×C3112yescomplemented
235C4×#121C2×C3111yescharacteristic
236#2058C2×C3111yescharacteristic, complemented
237#2057C2×C3111yescharacteristic, complemented
238#2023C2×C3111yescharacteristic, complemented
239#2023C2×C3111yescharacteristic, complemented
240#1989C2×C3112yescomplemented
241#1875C2×C3112yes—
242#1854C2×C3112yescomplemented
243#1797C2×C3111yescharacteristic, complemented
244C22×C31—22no—
245C22×C31—22no—
246C22×C31—22no—
247C22×C31—41no—
248C22×C31—41no—
249C22×C31—41no—
250C22×C31—41no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—81no—
255C22×C31—81no—
256C22×C31—81no—
257C22×C31—81no—
258C22×C31C2×#12011yescharacteristic
259C22×C31C2×C22⋊Q811yescharacteristic
260C22×C31#53611yescharacteristic
261C22×C31C2×#13511yescharacteristic
262C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C31#57511yescharacteristic
264C22×C31#57511yescharacteristic
265C4×C31—22no—
266C4×C31—22no—
267C4×C31—22no—
268C4×C31—22no—
269C4×C31—22no—
270C4×C31—42no—
271C4×C31—44no—
272C4×C31—44no—
273C4×C31—81no—
274C4×C31—81no—
275C4×C31#54812yes—
276C4×C31D4×Q812yes—
277#533—28no—
278C2×C4⋊Q8C22×C3111yescharacteristic
279C2×C4⋊Q8C22×C3111yescharacteristic
280C2×#125C22×C3112yes—
281C2×C22⋊Q8—21no—
282C2×C22⋊Q8—21no—
283C2×#121C22×C3111yescharacteristic
284C2×C4×Q8—21no—
285C2×C4×Q8—21no—
286C2×C4×D4—21no—
287C2×C4×D4C22×C3111yescharacteristic
288C2×#117C22×C3112yes—
289C22×C42C22×C3111yescharacteristic
290#478—28no—
291C2×#107C22×C3112yes—
292C2×#106C22×C3112yes—
293C2×C4⋊C8C22×C3112yes—
294C2×#102C22×C3112yes—
295C2×#102C22×C3112yes—
296C2×C22⋊C8C22×C3112yes—
297#390C22×C3111yescharacteristic
298#389C22×C3111yescharacteristic
299#388—21no—
300#386—21no—
301#380C22×C3111yescharacteristic
302C4×C4⋊C4C22×C3111yescharacteristic
303C4×C22⋊C4—21no—
304#340C22×C3112yes—
305#340C22×C3112yes—
306#340C22×C3112yes—
307#340C22×C3112yes—
308#336C22×C3111yescharacteristic
309#336C22×C3111yescharacteristic
310C2×C31—42no—
311C2×C31—82no—
312C2×C31#240911yescharacteristic
313C2×C31#300211yescharacteristic
314C2×C31#300611yescharacteristic
315C2×C31#307611yescharacteristic
316C2×C31#307711yescharacteristic
317C2×C31#322811yescharacteristic
318C2×C31#324011yescharacteristic
319C22×Q8—21no—
320C22×Q8—21no—
321C22×D4—21no—
322C22×D4C23×C3111yescharacteristic
323C23×C4—21no—
324C23×C4D4×C3111yescharacteristic
325C23×C4D4×C3111yescharacteristic
326C23×C4C23×C3111yescharacteristic
327C22×C8C23×C3112yes—
328C22×C8C23×C3112yes—
329C4⋊Q8—24no—
330C4⋊Q8—24no—
331#125—24no—
332#125D4×C3118yescomplemented
333C22⋊Q8—28no—
334C22⋊Q8—28no—
335#121—22no—
336#121Q8×C3114yescomplemented
337C4×Q8—28no—
338C4×Q8—28no—
339C4×D4—24no—
340C4×D4—44no—
341#117—24no—
342#117—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4C23×C3111yescharacteristic
349C2×C4⋊C4C23×C3111yescharacteristic
350C2×C4⋊C4C23×C3112yes—
351C2×C4⋊C4C23×C3112yes—
352C2×C4⋊C4C23×C3111yescharacteristic
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42D4×C3111yescharacteristic
363C2×C42D4×C3111yescharacteristic
364C2×C42C23×C3111yescharacteristic
365C2×C42C23×C3111yescharacteristic
366#107—24no—
367#106—24no—
368C4⋊C8—24no—
369#102—24no—
370#102—24no—
371C22⋊C8—24no—
372#95—21no—
373#95—21no—
374#95—21no—
375C31#2430411yescharacteristic, direct product factor, Sylow
376C24—21no—
377C24C2×D4×C3111yescharacteristic
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4C2×Q8×C3112yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4C2×D4×C3111yescharacteristic
408C22×C4C2×D4×C3111yescharacteristic
409C22×C4C2×D4×C3111yescharacteristic
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×Q8×C3112yes—
413C22×C4#41×C3112yes—
414C22×C4C24×C3111yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4C2×D4×C3114yes—
432C4⋊C4C2×D4×C3114yes—
433C4⋊C4C2×Q8×C3112yes—
434C22⋊C4—28no—
435C22⋊C4—24no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C42—28no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—42no—
445C42C2×D4×C3114yes—
446C23—21no—
447C23—21no—
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—
460C23C22×D4×C3111yescharacteristic
461Q8—48no—
462Q8—48no—
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—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—84no—
496C2×C4C31×#12012yes—
497C2×C4C31×#12012yes—
498C2×C4C31×C22⋊Q812yes—
499C2×C4C31×#13512yes—
500C2×C4C31×AGL(1,ℤ/8)12yes—
501C2×C4C22×D4×C3112yes—
502C2×C4C22×Q8×C3111yescharacteristic
503C2×C4C2×#41×C3111yescharacteristic
504C2×C4C31×2−1+412yes—
505C8—42no—
506C8—42no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22C2×C31×#12011yescharacteristic
522C22C2×C31×C22⋊Q811yescharacteristic
523C22C31×#53611yescharacteristic
524C22C2×C31×#13511yescharacteristic
525C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
526C22C31×#57511yescharacteristic
527C22C31×#57511yescharacteristic
528C4—22no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—42no—
534C4—44no—
535C4—44no—
536C4—81no—
537C4—81no—
538C4C31×#54812yes—
539C4D4×Q8×C3112yes—
540C2—42no—
541C2—82no—
542C2C31×#240911yescharacteristic
543C2C31×#300211yescharacteristic
544C2C31×#300611yescharacteristic
545C2C31×#307611yescharacteristic
546C2C31×#307711yescharacteristic
547C2C31×#322811yescharacteristic
548C2C31×#324011yescharacteristic
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. C2×C31×#533C2semidirect
2. C2×C31×#478C2semidirect
4. C31×#2058C2semidirect
5. C31×#2057C2semidirect
6. C31×#2023C2semidirect
7. C31×#2023C2semidirect
8. C31×#1989C2semidirect
10. C31×#1854C2semidirect
11. C31×#1797C2semidirect
58. C31×#125D4semidirect
62. C31×#121Q8semidirect
171. #24304C31direct (N × H)
233. C2×#533C2×C31semidirect
234. C2×#478C2×C31semidirect
236. #2058C2×C31semidirect
237. #2057C2×C31semidirect
238. #2023C2×C31semidirect
239. #2023C2×C31semidirect
240. #1989C2×C31semidirect
242. #1854C2×C31semidirect
243. #1797C2×C31semidirect
332. #125D4×C31semidirect
336. #121Q8×C31semidirect
375. C31#24304direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5332
C2×C31×#4782
C4×C31×#1212
C31×#20582
C31×#20572
C31×#20232
C31×#19892
C31×#18752
C31×#18542
C31×#17972
#2430431