Finite Group Databasebeta

← C31×#16682 #21473441

Subgroups — C31×#16682 #21473441

Order 7936.

1534 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#125C211yesmaximal, characteristic, complemented
2C2×C31×#399C211yesmaximal, characteristic
3C2×C31×#397C211yesmaximal, characteristic
4C2×C31×#388C211yesmaximal, characteristic
5C31×#1445C211yesmaximal, characteristic, complemented
6C31×#1445C211yesmaximal, characteristic, complemented
7C31×#1444C212yesmaximal, complemented
8C31×#1441C212yesmaximal, complemented
9C31×#1439C212yesmaximal
10C31×#1439C212yesmaximal
11C31×#1434C211yesmaximal, characteristic, complemented
12C2×C31×#125—24no—
13C2×C31×#125—24no—
14C2×C31×#125—24no—
15C22×C4⋊C4×C31C2212yes—
16C22×C4⋊C4×C31C2212yes—
17C22×C4⋊C4×C31C2212yes—
18C22×C4⋊C4×C31C2211yescharacteristic
19C22×C4⋊C4×C31C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C22⋊C4×C31C2211yescharacteristic
22C22×C22⋊C4×C31C2211yescharacteristic
23C22×C22⋊C4×C31C2212yes—
24C22×C22⋊C4×C31C2212yes—
25C22×C22⋊C4×C31C2211yescharacteristic
26C22×C42×C31C2211yescharacteristic
27C31×#399—24no—
28C31×#397—24no—
29C31×#388—24no—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2212yes—
32C2×C31×#95C2211yescharacteristic
33C2×C31×#95C2212yes—
34C2×C31×#95C2212yes—
35C2×C31×#95C2211yescharacteristic
36C2×C31×#95C2211yescharacteristic
37C2×C31×#95C2212yes—
38C2×C31×#95C2212yes—
39C2×C31×#95C2212yes—
40C2×C31×#95C2212yes—
41C25×C31C2311yescharacteristic
42C23×C4×C31C2311yescharacteristic
43C23×C4×C31C2312yes—
44C23×C4×C31C2311yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2312yes—
48C23×C4×C31C2312yes—
49C23×C4×C31C2312yes—
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C31×#125—232no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31—24no—
56C2×C4⋊C4×C31—24no—
57C2×C4⋊C4×C31—24no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—24no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—24no—
66C2×C4⋊C4×C31—42no—
67C2×C4⋊C4×C31—42no—
68C2×C4⋊C4×C31D414yes—
69C2×C4⋊C4×C31D414yes—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—22no—
76C2×C22⋊C4×C31—24no—
77C2×C22⋊C4×C31—22no—
78C2×C22⋊C4×C31—22no—
79C2×C22⋊C4×C31—22no—
80C2×C22⋊C4×C31—24no—
81C2×C22⋊C4×C31—42no—
82C2×C22⋊C4×C31—42no—
83C2×C22⋊C4×C31—42no—
84C2×C42×C31—22no—
85C2×C42×C31—22no—
86C2×C42×C31D414yes—
87C31×#95—24no—
88C31×#95—22no—
89C31×#95—22no—
90C31×#95—22no—
91C31×#95—22no—
92C31×#95—24no—
93C31×#95—24no—
94C31×#95—42no—
95C31×#95—42no—
96C31×#95—42no—
97C31×#95—42no—
98C24×C31—21no—
99C24×C31—21no—
100C24×C31—21no—
101C24×C31—21no—
102C24×C31—22no—
103C24×C31—22no—
104C24×C31—21no—
105C24×C31—22no—
106C24×C31—21no—
107C24×C31—22no—
108C24×C31#4112yes—
109C24×C31C2411yescharacteristic
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—21no—
113C22×C4×C31—22no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
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—21no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—21no—
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—21no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
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—21no—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D414yes—
163C22×C4×C31C2×D412yes—
164C22×C4×C31C2×D414yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D412yes—
168C22×C4×C31#4112yes—
169C22×C4×C31#4114yes—
170C4⋊C4×C31—216no—
171C4⋊C4×C31—216no—
172C4⋊C4×C31—28no—
173C4⋊C4×C31—28no—
174C4⋊C4×C31—48no—
175C4⋊C4×C31—44no—
176C22⋊C4×C31—28no—
177C22⋊C4×C31—48no—
178C22⋊C4×C31—48no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—44no—
181C42×C31—28no—
182#16682C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—22no—
184C23×C31—24no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—24no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—41no—
196C23×C31—42no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—41no—
201C23×C31—41no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31C22×D411yescharacteristic
207C23×C31C22×D411yescharacteristic
208C23×C31C22×D411yescharacteristic
209C23×C31C2×#4111yescharacteristic
210C23×C31C2×#4111yescharacteristic
211C23×C31C2×#4112yes—
212C23×C31H(5,𝔽2)12yes—
213C23×C31H(5,𝔽2)11yescharacteristic
214C23×C312−1+412yes—
215C23×C312−1+412yes—
216C23×C312−1+411yescharacteristic
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
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—42no—
239C2×C4×C31—42no—
240C2×C4×C31—41no—
241C2×C4×C31—42no—
242C2×C4×C31—41no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—41no—
248C2×C4×C31—42no—
249C2×C4×C31—41no—
250C2×C4×C31—41no—
251C2×C4×C31—42no—
252C2×C4×C31—41no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—41no—
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—
276C2×C4×C31#12014yes—
277C22×#125C2×C3111yescharacteristic, complemented
278C2×#399C2×C3111yescharacteristic
279C2×#397C2×C3111yescharacteristic
280C2×#388C2×C3111yescharacteristic
281#1445C2×C3111yescharacteristic, complemented
282#1445C2×C3111yescharacteristic, complemented
283#1444C2×C3112yescomplemented
284#1441C2×C3112yescomplemented
285#1439C2×C3112yes—
286#1439C2×C3112yes—
287#1434C2×C3111yescharacteristic, complemented
288C22×C31—42no—
289C22×C31—42no—
290C22×C31—42no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—81no—
294C22×C31—81no—
295C22×C31—81no—
296C22×C31—81no—
297C22×C31—82no—
298C22×C31—82no—
299C22×C31C2×#12011yescharacteristic
300C22×C31#52511yescharacteristic
301C22×C31#53211yescharacteristic
302C22×C31#53511yescharacteristic
303C22×C31#53611yescharacteristic
304C22×C31#53611yescharacteristic
305C22×C31#53612yes—
306C22×C31#53612yes—
307C22×C31#53711yescharacteristic
308C22×C31#53912yes—
309C22×C31#53912yes—
310C22×C31#53912yes—
311C22×C31#54112yes—
312C22×C31#54611yescharacteristic
313C22×C31#54612yes—
314C22×C31#54711yescharacteristic
315C22×C31#54811yescharacteristic
316C22×C31#55011yescharacteristic
317C22×C31#55012yes—
318C22×C31#55212yes—
319C22×C31#55212yes—
320C22×C31#56212yes—
321C22×C31#56312yes—
322C4×C31—24no—
323C4×C31—42no—
324C4×C31—42no—
325C4×C31—44no—
326C4×C31—42no—
327C4×C31—44no—
328C4×C31—44no—
329C4×C31—81no—
330C4×C31—81no—
331C4×C31—82no—
332C2×#125—24no—
333C2×#125—24no—
334C2×#125—24no—
335C22×C4⋊C4C22×C3112yes—
336C22×C4⋊C4C22×C3112yes—
337C22×C4⋊C4C22×C3112yes—
338C22×C4⋊C4C22×C3111yescharacteristic
339C22×C4⋊C4C22×C3111yescharacteristic
340C22×C4⋊C4C22×C3111yescharacteristic
341C22×C22⋊C4C22×C3111yescharacteristic
342C22×C22⋊C4C22×C3111yescharacteristic
343C22×C22⋊C4C22×C3112yes—
344C22×C22⋊C4C22×C3112yes—
345C22×C22⋊C4C22×C3111yescharacteristic
346C22×C42C22×C3111yescharacteristic
347#399—24no—
348#397—24no—
349#388—24no—
350C2×#95C22×C3111yescharacteristic
351C2×#95C22×C3112yes—
352C2×#95C22×C3111yescharacteristic
353C2×#95C22×C3112yes—
354C2×#95C22×C3112yes—
355C2×#95C22×C3111yescharacteristic
356C2×#95C22×C3111yescharacteristic
357C2×#95C22×C3112yes—
358C2×#95C22×C3112yes—
359C2×#95C22×C3112yes—
360C2×#95C22×C3112yes—
361C2×C31—82no—
362C2×C31#240711yescharacteristic
363C2×C31#243111yescharacteristic
364C2×C31#261111yescharacteristic
365C2×C31#261411yescharacteristic
366C2×C31#263311yescharacteristic
367C2×C31#266411yescharacteristic
368C2×C31#269012yes—
369C2×C31#269412yes—
370C2×C31#269412yes—
371C2×C31#272112yes—
372C2×C31#282011yescharacteristic
373C25C23×C3111yescharacteristic
374C23×C4C23×C3111yescharacteristic
375C23×C4C23×C3112yes—
376C23×C4C23×C3111yescharacteristic
377C23×C4C23×C3112yes—
378C23×C4C23×C3112yes—
379C23×C4C23×C3111yescharacteristic
380C23×C4C23×C3111yescharacteristic
381C23×C4C23×C3112yes—
382C23×C4C23×C3111yescharacteristic
383C23×C4C23×C3111yescharacteristic
384#125—232no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—42no—
399C2×C4⋊C4—42no—
400C2×C4⋊C4D4×C3114yes—
401C2×C4⋊C4D4×C3114yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—42no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C42—22no—
417C2×C42—22no—
418C2×C42D4×C3114yes—
419#95—22no—
420#95—22no—
421#95—24no—
422#95—22no—
423#95—22no—
424#95—24no—
425#95—24no—
426#95—42no—
427#95—42no—
428#95—42no—
429#95—42no—
430C31#1668211yescharacteristic, direct product factor, Sylow
431C24—21no—
432C24—21no—
433C24—21no—
434C24—21no—
435C24—22no—
436C24—22no—
437C24—21no—
438C24—22no—
439C24—21no—
440C24—22no—
441C24#41×C3112yes—
442C24C24×C3111yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4C2×D4×C3112yes—
495C22×C4C2×D4×C3114yes—
496C22×C4C2×D4×C3112yes—
497C22×C4C2×D4×C3114yes—
498C22×C4C2×D4×C3112yes—
499C22×C4C2×D4×C3112yes—
500C22×C4C2×D4×C3112yes—
501C22×C4#41×C3112yes—
502C22×C4#41×C3114yes—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C42—28no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C22×D4×C3111yescharacteristic
539C23C22×D4×C3111yescharacteristic
540C23C22×D4×C3111yescharacteristic
541C23C2×#41×C3111yescharacteristic
542C23C2×#41×C3111yescharacteristic
543C23C2×#41×C3112yes—
544C23C31×H(5,𝔽2)12yes—
545C23C31×H(5,𝔽2)11yescharacteristic
546C23C31×2−1+412yes—
547C23C31×2−1+412yes—
548C23C31×2−1+411yescharacteristic
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4C31×#12014yes—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—82no—
619C22—82no—
620C22C2×C31×#12011yescharacteristic
621C22C31×#52511yescharacteristic
622C22C31×#53211yescharacteristic
623C22C31×#53511yescharacteristic
624C22C31×#53611yescharacteristic
625C22C31×#53611yescharacteristic
626C22C31×#53612yes—
627C22C31×#53612yes—
628C22C31×#53711yescharacteristic
629C22C31×#53912yes—
630C22C31×#53912yes—
631C22C31×#53912yes—
632C22C31×#54112yes—
633C22C31×#54611yescharacteristic
634C22C31×#54612yes—
635C22C31×#54711yescharacteristic
636C22C31×#54811yescharacteristic
637C22C31×#55011yescharacteristic
638C22C31×#55012yes—
639C22C31×#55212yes—
640C22C31×#55212yes—
641C22C31×#56212yes—
642C22C31×#56312yes—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—81no—
651C4—81no—
652C4—82no—
653C2—82no—
654C2C31×#240711yescharacteristic
655C2C31×#243111yescharacteristic
656C2C31×#261111yescharacteristic
657C2C31×#261411yescharacteristic
658C2C31×#263311yescharacteristic
659C2C31×#266411yescharacteristic
660C2C31×#269012yes—
661C2C31×#269412yes—
662C2C31×#269412yes—
663C2C31×#272112yes—
664C2C31×#282011yescharacteristic
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
1. C22×C31×#125C2semidirect
5. C31×#1445C2semidirect
6. C31×#1445C2semidirect
7. C31×#1444C2semidirect
8. C31×#1441C2semidirect
11. C31×#1434C2semidirect
182. #16682C31direct (N × H)
277. C22×#125C2×C31semidirect
281. #1445C2×C31semidirect
282. #1445C2×C31semidirect
283. #1444C2×C31semidirect
284. #1441C2×C31semidirect
287. #1434C2×C31semidirect
430. C31#16682direct (N × H)

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×#3992
C2×C31×#3972
C2×C31×#3882
C31×#14452
C31×#14442
C31×#14412
C31×#14392
C31×#14342
#1668231