Finite Group Databasebeta

← C31×#16024 #21472783

Subgroups — C31×#16024 #21472783

Order 7936.

1690 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#126C211yesmaximal, characteristic
2C2×C31×#385C211yesmaximal, characteristic
3C2×C31×#385C211yesmaximal, characteristic
4C2×C31×#381C211yesmaximal, characteristic
5C31×#1444C212yesmaximal, complemented
6C31×#1442C212yesmaximal, complemented
7C31×#1441C212yesmaximal, complemented
8C31×#1440C211yesmaximal, characteristic
9C31×#1440C211yesmaximal, characteristic
10C31×#1440C211yesmaximal, characteristic
11C31×#1436C212yesmaximal, complemented
12C24×C4×C31C2211yescharacteristic
13C2×C31×#126—24no—
14C2×C31×#126—24no—
15C2×C31×#126—24no—
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C22×C4⋊C4×C31C2212yes—
20C22×C4⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2211yescharacteristic
22C22×C22⋊C4×C31C2211yescharacteristic
23C22×C22⋊C4×C31C2211yescharacteristic
24C22×C22⋊C4×C31C2211yescharacteristic
25C22×C22⋊C4×C31C2211yescharacteristic
26C22×C22⋊C4×C31C2211yescharacteristic
27C22×C42×C31C2211yescharacteristic
28C31×#385—24no—
29C31×#385—24no—
30C31×#381—24no—
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2212yes—
36C2×C31×#95C2212yes—
37C2×C31×#95C2212yes—
38C2×C31×#95C2212yes—
39C2×C31×#95C2212yes—
40C2×C31×#95C2212yes—
41C25×C31C2311yescharacteristic
42C23×C4×C31—22no—
43C23×C4×C31—24no—
44C23×C4×C31—22no—
45C23×C4×C31—22no—
46C23×C4×C31—42no—
47C23×C4×C31C2312yes—
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C23×C4×C31C2312yes—
55C23×C4×C31C2312yes—
56C23×C4×C31C2312yes—
57C31×#126—232no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—22no—
66C2×C4⋊C4×C31—22no—
67C2×C4⋊C4×C31—24no—
68C2×C4⋊C4×C31—24no—
69C2×C4⋊C4×C31—24no—
70C2×C4⋊C4×C31—24no—
71C2×C4⋊C4×C31—42no—
72C2×C4⋊C4×C31—42no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—22no—
75C2×C22⋊C4×C31—22no—
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—22no—
78C2×C22⋊C4×C31—22no—
79C2×C22⋊C4×C31—22no—
80C2×C22⋊C4×C31—22no—
81C2×C22⋊C4×C31—24no—
82C2×C22⋊C4×C31—24no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—42no—
85C2×C22⋊C4×C31D414yes—
86C2×C22⋊C4×C31D414yes—
87C2×C22⋊C4×C31D414yes—
88C2×C42×C31—22no—
89C2×C42×C31—22no—
90C2×C42×C31—22no—
91C31×#95—24no—
92C31×#95—24no—
93C31×#95—24no—
94C31×#95—24no—
95C31×#95—24no—
96C31×#95—24no—
97C31×#95—24no—
98C31×#95—24no—
99C31×#95—42no—
100C31×#95—42no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—22no—
104C24×C31—22no—
105C24×C31—22no—
106C24×C31—22no—
107C24×C31C2×D412yes—
108C24×C31C2×D412yes—
109C24×C31C2×D412yes—
110C24×C31C2411yescharacteristic
111C22×C4×C31—28no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—48no—
162C22×C4×C31—48no—
163C22×C4×C31—44no—
164C22×C4×C31—44no—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D412yes—
168C22×C4×C31C2×D412yes—
169C22×C4×C31C2×D412yes—
170C22×C4×C31C2×D412yes—
171C22×C4×C31#4114yes—
172C22×C4×C31#4114yes—
173C22×C4×C31#4114yes—
174C4⋊C4×C31—28no—
175C4⋊C4×C31—28no—
176C4⋊C4×C31—28no—
177C4⋊C4×C31—48no—
178C4⋊C4×C31—48no—
179C22⋊C4×C31—28no—
180C22⋊C4×C31—28no—
181C22⋊C4×C31—28no—
182C22⋊C4×C31—28no—
183C22⋊C4×C31—28no—
184C22⋊C4×C31—44no—
185C42×C31—28no—
186#16024C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—22no—
192C23×C31—22no—
193C23×C31—24no—
194C23×C31—24no—
195C23×C31—24no—
196C23×C31—24no—
197C23×C31—24no—
198C23×C31—24no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31#12014yes—
208C23×C31C22×D411yescharacteristic
209C23×C31C22×D411yescharacteristic
210C23×C31C22×D411yescharacteristic
211C23×C31C2×#4112yes—
212C23×C31C2×#4112yes—
213C23×C31C2×#4112yes—
214C23×C31H(5,𝔽2)12yes—
215C23×C312−1+412yes—
216C23×C312−1+412yes—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—24no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—24no—
230C2×C4×C31—24no—
231C2×C4×C31—24no—
232C2×C4×C31—24no—
233C2×C4×C31—24no—
234C2×C4×C31—24no—
235C2×C4×C31—24no—
236C2×C4×C31—24no—
237C2×C4×C31—24no—
238C2×C4×C31—416no—
239C2×C4×C31—42no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—42no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—42no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—41no—
253C2×C4×C31—41no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—42no—
260C2×C4×C31—42no—
261C2×C4×C31—42no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C22×#126C2×C3111yescharacteristic
277C2×#385C2×C3111yescharacteristic
278C2×#385C2×C3111yescharacteristic
279C2×#381C2×C3111yescharacteristic
280#1444C2×C3112yescomplemented
281#1442C2×C3112yescomplemented
282#1441C2×C3112yescomplemented
283#1440C2×C3111yescharacteristic
284#1440C2×C3111yescharacteristic
285#1440C2×C3111yescharacteristic
286#1436C2×C3112yescomplemented
287C22×C31—24no—
288C22×C31—24no—
289C22×C31—24no—
290C22×C31—44no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—44no—
295C22×C31—44no—
296C22×C31C2×#12011yescharacteristic
297C22×C31#52511yescharacteristic
298C22×C31#52511yescharacteristic
299C22×C31#52511yescharacteristic
300C22×C31#52912yes—
301C22×C31#53511yescharacteristic
302C22×C31#53611yescharacteristic
303C22×C31#53611yescharacteristic
304C22×C31#53912yes—
305C22×C31#54112yes—
306C22×C31#54212yes—
307C22×C31#54612yes—
308C22×C31#54612yes—
309C22×C31#54712yes—
310C22×C31#54712yes—
311C22×C31#54712yes—
312C22×C31#54712yes—
313C22×C31#55212yes—
314C22×C31#55312yes—
315C22×C31#55612yes—
316C22×C31#56312yes—
317C4×C31—42no—
318C4×C31—42no—
319C4×C31—44no—
320C4×C31—42no—
321C4×C31—42no—
322C4×C31—42no—
323C4×C31—42no—
324C4×C31—44no—
325C4×C31—44no—
326C4×C31—82no—
327C24×C4C22×C3111yescharacteristic
328C2×#126—24no—
329C2×#126—24no—
330C2×#126—24no—
331C22×C4⋊C4C22×C3111yescharacteristic
332C22×C4⋊C4C22×C3111yescharacteristic
333C22×C4⋊C4C22×C3111yescharacteristic
334C22×C4⋊C4C22×C3112yes—
335C22×C4⋊C4C22×C3112yes—
336C22×C22⋊C4C22×C3111yescharacteristic
337C22×C22⋊C4C22×C3111yescharacteristic
338C22×C22⋊C4C22×C3111yescharacteristic
339C22×C22⋊C4C22×C3111yescharacteristic
340C22×C22⋊C4C22×C3111yescharacteristic
341C22×C22⋊C4C22×C3111yescharacteristic
342C22×C42C22×C3111yescharacteristic
343#385—24no—
344#385—24no—
345#381—24no—
346C2×#95C22×C3112yes—
347C2×#95C22×C3112yes—
348C2×#95C22×C3112yes—
349C2×#95C22×C3112yes—
350C2×#95C22×C3112yes—
351C2×#95C22×C3112yes—
352C2×#95C22×C3112yes—
353C2×#95C22×C3112yes—
354C2×#95C22×C3112yes—
355C2×#95C22×C3112yes—
356C2×C31—44no—
357C2×C31#240411yescharacteristic
358C2×C31#240611yescharacteristic
359C2×C31#240611yescharacteristic
360C2×C31#246612yes—
361C2×C31#246812yes—
362C2×C31#247812yes—
363C2×C31#268912yes—
364C2×C31#269812yes—
365C2×C31#271812yes—
366C25C23×C3111yescharacteristic
367C23×C4—22no—
368C23×C4—24no—
369C23×C4—22no—
370C23×C4—22no—
371C23×C4—42no—
372C23×C4C23×C3112yes—
373C23×C4C23×C3111yescharacteristic
374C23×C4C23×C3111yescharacteristic
375C23×C4C23×C3111yescharacteristic
376C23×C4C23×C3111yescharacteristic
377C23×C4C23×C3111yescharacteristic
378C23×C4C23×C3111yescharacteristic
379C23×C4C23×C3112yes—
380C23×C4C23×C3112yes—
381C23×C4C23×C3112yes—
382#126—232no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—42no—
397C2×C4⋊C4—42no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—24no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—42no—
410C2×C22⋊C4D4×C3114yes—
411C2×C22⋊C4D4×C3114yes—
412C2×C22⋊C4D4×C3114yes—
413C2×C42—22no—
414C2×C42—22no—
415C2×C42—22no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—42no—
425#95—42no—
426C31#1602411yescharacteristic, direct product factor, Sylow
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24C2×D4×C3112yes—
434C24C2×D4×C3112yes—
435C24C2×D4×C3112yes—
436C24C24×C3111yescharacteristic
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4C2×D4×C3112yes—
492C22×C4C2×D4×C3112yes—
493C22×C4C2×D4×C3112yes—
494C22×C4C2×D4×C3112yes—
495C22×C4C2×D4×C3112yes—
496C22×C4C2×D4×C3112yes—
497C22×C4#41×C3114yes—
498C22×C4#41×C3114yes—
499C22×C4#41×C3114yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C42—28no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C31×#12014yes—
533C23C22×D4×C3111yescharacteristic
534C23C22×D4×C3111yescharacteristic
535C23C22×D4×C3111yescharacteristic
536C23C2×#41×C3112yes—
537C23C2×#41×C3112yes—
538C23C2×#41×C3112yes—
539C23C31×H(5,𝔽2)12yes—
540C23C31×2−1+412yes—
541C23C31×2−1+412yes—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22C2×C31×#12011yescharacteristic
611C22C31×#52511yescharacteristic
612C22C31×#52511yescharacteristic
613C22C31×#52511yescharacteristic
614C22C31×#52912yes—
615C22C31×#53511yescharacteristic
616C22C31×#53611yescharacteristic
617C22C31×#53611yescharacteristic
618C22C31×#53912yes—
619C22C31×#54112yes—
620C22C31×#54212yes—
621C22C31×#54612yes—
622C22C31×#54612yes—
623C22C31×#54712yes—
624C22C31×#54712yes—
625C22C31×#54712yes—
626C22C31×#54712yes—
627C22C31×#55212yes—
628C22C31×#55312yes—
629C22C31×#55612yes—
630C22C31×#56312yes—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—44no—
640C4—82no—
641C2—44no—
642C2C31×#240411yescharacteristic
643C2C31×#240611yescharacteristic
644C2C31×#240611yescharacteristic
645C2C31×#246612yes—
646C2C31×#246812yes—
647C2C31×#247812yes—
648C2C31×#268912yes—
649C2C31×#269812yes—
650C2C31×#271812yes—
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
5. C31×#1444C2semidirect
6. C31×#1442C2semidirect
7. C31×#1441C2semidirect
11. C31×#1436C2semidirect
186. #16024C31direct (N × H)
280. #1444C2×C31semidirect
281. #1442C2×C31semidirect
282. #1441C2×C31semidirect
286. #1436C2×C31semidirect
426. C31#16024direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1262
C2×C31×#3852
C2×C31×#3812
C31×#14442
C31×#14422
C31×#14412
C31×#14402
C31×#14362
#1602431