Finite Group Databasebeta

← C31×#24265 #21481024

Subgroups — C31×#24265 #21481024

Order 7936.

1478 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#135C211yesmaximal, characteristic, complemented
2C2×C31×#465C212yesmaximal, complemented
3C2×C8×D4×C31C211yesmaximal, characteristic, complemented
4C31×#2407C212yesmaximal, complemented
5C31×#2056C211yesmaximal, characteristic, complemented
6C31×#2054C211yesmaximal, characteristic, complemented
7C31×#2001C211yesmaximal, characteristic, complemented
8C31×#1999C211yesmaximal, characteristic, complemented
9C31×#1968C212yesmaximal, complemented
10C31×#1807C211yesmaximal, characteristic, complemented
11C31×#1788C212yesmaximal, complemented
12C22×#41×C31C2212yes—
13C2×C31×#135—28no—
14C2×C31×#135—24no—
15C22×Q16×C31C2211yescharacteristic
16C22×SD16×C31C2212yes—
17C22×D8×C31C2211yescharacteristic
18C23×C8×C31C2211yescharacteristic
19C23×C8×C31C2211yescharacteristic
20C2×C31×C22⋊Q8C2212yes—
21C2×C31×#121C2212yes—
22C2×C4×D4×C31C2211yescharacteristic
23C31×#465—28no—
24C8×D4×C31—24no—
25C2×C31×#106C2212yes—
26C2×C31×C4⋊C8C2211yescharacteristic
27C2×C31×#103C2212yes—
28C2×C31×#103C2212yes—
29C2×C31×#102C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×C22⋊C8C2211yescharacteristic
32C2×C31×C22⋊C8C2211yescharacteristic
33C2×C4×C8×C31C2211yescharacteristic
34C31×#394—22no—
35C31×#392—22no—
36C31×#391C2212yes—
37C31×#390C2212yes—
38C31×#386—22no—
39C31×#385—22no—
40C31×#380C2212yes—
41C31×#340C2211yescharacteristic
42C31×#340C2211yescharacteristic
43C2×#41×C31—24no—
44C2×#41×C31—24no—
45C2×#41×C31—24no—
46C2×#41×C31—24no—
47C2×#41×C31—42no—
48C2×#41×C31—42no—
49C22×Q8×C31C2312yes—
50C22×D4×C31—22no—
51C22×D4×C31—22no—
52C22×D4×C31D411yescharacteristic
53C22×D4×C31C2312yes—
54C23×C4×C31—22no—
55C23×C4×C31C2311yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C31×#135—232no—
58C2×Q16×C31—24no—
59C2×Q16×C31—22no—
60C2×SD16×C31—24no—
61C2×SD16×C31—24no—
62C2×SD16×C31D418yescomplemented
63C2×D8×C31—24no—
64C2×D8×C31—22no—
65C22×C8×C31—24no—
66C22×C8×C31—22no—
67C22×C8×C31—24no—
68C22×C8×C31D414yescomplemented
69C22×C8×C31C2311yescharacteristic
70C22×C8×C31C2311yescharacteristic
71C22×C8×C31C2311yescharacteristic
72C22×C8×C31C2311yescharacteristic
73C31×C22⋊Q8—24no—
74C31×C22⋊Q8—42no—
75C31×#121—24no—
76C31×#121—42no—
77C4×D4×C31—22no—
78C4×D4×C31—22no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31D411yescharacteristic
81C2×C4⋊C4×C31C2312yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31D411yescharacteristic
90C2×C42×C31C2311yescharacteristic
91C31×#106—24no—
92C31×C4⋊C8—22no—
93C31×#103—24no—
94C31×#103—42no—
95C31×#102—24no—
96C31×#102—42no—
97C31×C22⋊C8—22no—
98C31×C22⋊C8—22no—
99C4×C8×C31—22no—
100C31×#95—22no—
101C31×#95—22no—
102C31×#95—22no—
103C24×C31—22no—
104C24×C31C2×D411yescharacteristic
105C24×C31C2×D411yescharacteristic
106#41×C31—216no—
107#41×C31—48no—
108#41×C31—48no—
109#41×C31—48no—
110C2×Q8×C31—22no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—22no—
114C2×Q8×C31—22no—
115C2×Q8×C31C2×D414yes—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31—22no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—44no—
125C2×D4×C31—44no—
126C2×D4×C31—44no—
127C2×D4×C31—44no—
128C2×D4×C31—44no—
129C2×D4×C31—82no—
130C2×D4×C31C2×D414yes—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—44no—
141C22×C4×C31—44no—
142C22×C4×C31—44no—
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31#4112yes—
149C22×C4×C31C2411yescharacteristic
150Q16×C31—28no—
151SD16×C31—216no—
152D8×C31—28no—
153C2×C8×C31—22no—
154C2×C8×C31—28no—
155C2×C8×C31—21no—
156C2×C8×C31—21no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C2×C8×C31—44no—
160C2×C8×C31C2×D412yes—
161C2×C8×C31C2×D412yes—
162C2×C8×C31C2×D414yes—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—44no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—44no—
172C22⋊C4×C31—44no—
173C22⋊C4×C31—44no—
174C22⋊C4×C31—82no—
175C22⋊C4×C31—82no—
176C42×C31—21no—
177C42×C31—21no—
178#24265C3111yesmaximal, characteristic, direct product factor, Sylow
179C23×C31—21no—
180C23×C31—22no—
181C23×C31—22no—
182C23×C31—21no—
183C23×C31—21no—
184C23×C31—21no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—24no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31#12112yes—
195C23×C31#13512yescomplemented
196C23×C31C22×D411yescharacteristic
197Q8×C31—24no—
198Q8×C31—24no—
199Q8×C31—42no—
200Q8×C31—42no—
201D4×C31—28no—
202D4×C31—24no—
203D4×C31—24no—
204D4×C31—416no—
205D4×C31—42no—
206D4×C31—42no—
207D4×C31—88no—
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—21no—
216C2×C4×C31—22no—
217C2×C4×C31—24no—
218C2×C4×C31—22no—
219C2×C4×C31—416no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31#12112yes—
232C2×C4×C31#13512yescomplemented
233C2×C4×C31C22×D411yescharacteristic
234C2×C4×C31C22×D412yes—
235C2×C4×C31C2×#4111yescharacteristic
236C2×C4×C31H(5,𝔽2)12yes—
237C8×C31—22no—
238C8×C31—41no—
239C8×C31—41no—
240C8×C31#12014yes—
241C22×#135C2×C3111yescharacteristic, complemented
242C2×#465C2×C3112yescomplemented
243C2×C8×D4C2×C3111yescharacteristic, complemented
244#2407C2×C3112yescomplemented
245#2056C2×C3111yescharacteristic, complemented
246#2054C2×C3111yescharacteristic, complemented
247#2001C2×C3111yescharacteristic, complemented
248#1999C2×C3111yescharacteristic, complemented
249#1968C2×C3112yescomplemented
250#1807C2×C3111yescharacteristic, complemented
251#1788C2×C3112yescomplemented
252C22×C31—22no—
253C22×C31—24no—
254C22×C31—22no—
255C22×C31—22no—
256C22×C31—22no—
257C22×C31—41no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—41no—
262C22×C31—44no—
263C22×C31—44no—
264C22×C31—44no—
265C22×C31—82no—
266C22×C31—82no—
267C22×C31—82no—
268C22×C31—82no—
269C22×C31C2×#12111yescharacteristic
270C22×C31D4212yes—
271C22×C31C2×#13511yescharacteristic
272C22×C31C2×#13511yescharacteristic
273C22×C31#57712yes—
274C4×C31—22no—
275C4×C31—22no—
276C4×C31—41no—
277C4×C31—44no—
278C4×C31—41no—
279C4×C31—82no—
280C4×C31—82no—
281C4×C31C2×#12011yescharacteristic
282C4×C31#53511yescharacteristic
283C4×C31#54612yes—
284C22×#41C22×C3112yes—
285C2×#135—28no—
286C2×#135—24no—
287C22×Q16C22×C3111yescharacteristic
288C22×SD16C22×C3112yes—
289C22×D8C22×C3111yescharacteristic
290C23×C8C22×C3111yescharacteristic
291C23×C8C22×C3111yescharacteristic
292C2×C22⋊Q8C22×C3112yes—
293C2×#121C22×C3112yes—
294C2×C4×D4C22×C3111yescharacteristic
295#465—28no—
296C8×D4—24no—
297C2×#106C22×C3112yes—
298C2×C4⋊C8C22×C3111yescharacteristic
299C2×#103C22×C3112yes—
300C2×#103C22×C3112yes—
301C2×#102C22×C3112yes—
302C2×#102C22×C3112yes—
303C2×C22⋊C8C22×C3111yescharacteristic
304C2×C22⋊C8C22×C3111yescharacteristic
305C2×C4×C8C22×C3111yescharacteristic
306#394—22no—
307#392—22no—
308#391C22×C3112yes—
309#390C22×C3112yes—
310#386—22no—
311#385—22no—
312#380C22×C3112yes—
313#340C22×C3111yescharacteristic
314#340C22×C3111yescharacteristic
315C2×C31—42no—
316C2×C31—42no—
317C2×C31—84no—
318C2×C31#240811yescharacteristic
319C2×C31#305011yescharacteristic
320C2×C31#306011yescharacteristic
321C2×C31#328212yes—
322C2×C31#328412yes—
323C2×#41—24no—
324C2×#41—24no—
325C2×#41—24no—
326C2×#41—24no—
327C2×#41—42no—
328C2×#41—42no—
329C22×Q8C23×C3112yes—
330C22×D4—22no—
331C22×D4—22no—
332C22×D4D4×C3111yescharacteristic
333C22×D4C23×C3112yes—
334C23×C4—22no—
335C23×C4C23×C3111yescharacteristic
336C23×C4C23×C3111yescharacteristic
337#135—232no—
338C2×Q16—24no—
339C2×Q16—22no—
340C2×SD16—24no—
341C2×SD16—24no—
342C2×SD16D4×C3118yescomplemented
343C2×D8—24no—
344C2×D8—22no—
345C22×C8—24no—
346C22×C8—22no—
347C22×C8—24no—
348C22×C8D4×C3114yescomplemented
349C22×C8C23×C3111yescharacteristic
350C22×C8C23×C3111yescharacteristic
351C22×C8C23×C3111yescharacteristic
352C22×C8C23×C3111yescharacteristic
353C22⋊Q8—24no—
354C22⋊Q8—42no—
355#121—24no—
356#121—42no—
357C4×D4—22no—
358C4×D4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4D4×C3111yescharacteristic
361C2×C4⋊C4C23×C3112yes—
362C2×C4⋊C4C23×C3112yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4D4×C3111yescharacteristic
369C2×C22⋊C4D4×C3111yescharacteristic
370C2×C42C23×C3111yescharacteristic
371#106—24no—
372C4⋊C8—22no—
373#103—24no—
374#103—42no—
375#102—24no—
376#102—42no—
377C22⋊C8—22no—
378C22⋊C8—22no—
379C4×C8—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C31#2426511yescharacteristic, direct product factor, Sylow
384C24—22no—
385C24C2×D4×C3111yescharacteristic
386C24C2×D4×C3111yescharacteristic
387#41—216no—
388#41—48no—
389#41—48no—
390#41—48no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8C2×D4×C3114yes—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—82no—
411C2×D4C2×D4×C3114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4C2×D4×C3111yescharacteristic
426C22×C4C2×D4×C3111yescharacteristic
427C22×C4C2×D4×C3112yes—
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4#41×C3112yes—
430C22×C4C24×C3111yescharacteristic
431Q16—28no—
432SD16—216no—
433D8—28no—
434C2×C8—22no—
435C2×C8—28no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—44no—
441C2×C8C2×D4×C3112yes—
442C2×C8C2×D4×C3112yes—
443C2×C8C2×D4×C3114yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C42—21no—
458C42—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23C31×#12112yes—
475C23C31×#13512yescomplemented
476C23C22×D4×C3111yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—42no—
480Q8—42no—
481D4—28no—
482D4—24no—
483D4—24no—
484D4—416no—
485D4—42no—
486D4—42no—
487D4—88no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4C31×#12112yes—
512C2×C4C31×#13512yescomplemented
513C2×C4C22×D4×C3111yescharacteristic
514C2×C4C22×D4×C3112yes—
515C2×C4C2×#41×C3111yescharacteristic
516C2×C4C31×H(5,𝔽2)12yes—
517C8—22no—
518C8—41no—
519C8—41no—
520C8C31×#12014yes—
521C22—22no—
522C22—24no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—41no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22C2×C31×#12111yescharacteristic
539C22D42×C3112yes—
540C22C2×C31×#13511yescharacteristic
541C22C2×C31×#13511yescharacteristic
542C22C31×#57712yes—
543C4—22no—
544C4—22no—
545C4—41no—
546C4—44no—
547C4—41no—
548C4—82no—
549C4—82no—
550C4C2×C31×#12011yescharacteristic
551C4C31×#53511yescharacteristic
552C4C31×#54612yes—
553C2—42no—
554C2—42no—
555C2—84no—
556C2C31×#240811yescharacteristic
557C2C31×#305011yescharacteristic
558C2C31×#306011yescharacteristic
559C2C31×#328212yes—
560C2C31×#328412yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
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
1. C22×C31×#135C2semidirect
2. C2×C31×#465C2semidirect
3. C2×C8×D4×C31C2semidirect
4. C31×#2407C2semidirect
5. C31×#2056C2semidirect
6. C31×#2054C2semidirect
7. C31×#2001C2semidirect
8. C31×#1999C2semidirect
9. C31×#1968C2semidirect
10. C31×#1807C2semidirect
11. C31×#1788C2semidirect
62. C2×SD16×C31D4semidirect
68. C22×C8×C31D4semidirect
178. #24265C31direct (N × H)
195. C23×C31#135semidirect
232. C2×C4×C31#135semidirect
241. C22×#135C2×C31semidirect
242. C2×#465C2×C31semidirect
243. C2×C8×D4C2×C31semidirect
244. #2407C2×C31semidirect
245. #2056C2×C31semidirect
246. #2054C2×C31semidirect
247. #2001C2×C31semidirect
248. #1999C2×C31semidirect
249. #1968C2×C31semidirect
250. #1807C2×C31semidirect
251. #1788C2×C31semidirect
342. C2×SD16D4×C31semidirect
348. C22×C8D4×C31semidirect
383. C31#24265direct (N × H)
475. C23C31×#135semidirect
512. C2×C4C31×#135semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1352
C2×C31×#4652
C2×C8×D4×C312
C31×#24072
C31×#20562
C31×#20542
C31×#20012
C31×#19992
C31×#19682
C31×#18072
C31×#17882
#2426531