Finite Group Databasebeta

← C31×#24151 #21480910

Subgroups — C31×#24151 #21480910

Order 7936.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#533C211yesmaximal, characteristic, complemented
2C2×C31×#476C211yesmaximal, characteristic, complemented
3C2×C31×#461C211yesmaximal, characteristic, complemented
4C4×C31×#121C211yesmaximal, characteristic
5C31×#2035C212yesmaximal, complemented
6C31×#2000C212yesmaximal
7C31×#1987C211yesmaximal, characteristic, complemented
8C31×#1981C211yesmaximal, characteristic, complemented
9C31×#1880C212yesmaximal, complemented
10C31×#1865C212yesmaximal, complemented
11C31×#1797C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C31×#533—24no—
14C31×#533—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#125—21no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×C22⋊Q8C2212yes—
20C2×C31×#121C2211yescharacteristic
21C2×C4×Q8×C31—21no—
22C2×C4×Q8×C31C2212yes—
23C2×C4×D4×C31—21no—
24C2×C4×D4×C31C2211yescharacteristic
25C2×C31×#117—21no—
26C22×C42×C31C2211yescharacteristic
27C31×#476—24no—
28C31×#461—24no—
29C2×C31×#106C2212yes—
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×C4⋊C8C2211yescharacteristic
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#103C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×C22⋊C8C2212yes—
37C31×#390C2211yescharacteristic
38C31×#389C2211yescharacteristic
39C31×#388—21no—
40C31×#386—21no—
41C31×#380C2211yescharacteristic
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C22⋊C4×C31—21no—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31C2312yes—
49C22×D4×C31—21no—
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31D411yescharacteristic
53C23×C4×C31D411yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C2×SD16×C31—24no—
56C2×SD16×C31—24no—
57C2×SD16×C31—42no—
58C22×C8×C31C2312yes—
59C22×C8×C31C2312yes—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8D414yescomplemented
63C31×C4⋊Q8D414yescomplemented
64C31×#125—28no—
65C31×C22⋊Q8—28no—
66C31×C22⋊Q8—24no—
67C31×C22⋊Q8—24no—
68C31×#121—22no—
69C31×#121—22no—
70C4×Q8×C31—28no—
71C4×Q8×C31—24no—
72C4×Q8×C31—24no—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31—44no—
76C31×#117—28no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2312yes—
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C42×C31—22no—
94C2×C42×C31—22no—
95C2×C42×C31—22no—
96C2×C42×C31—21no—
97C2×C42×C31D411yescharacteristic
98C2×C42×C31D411yescharacteristic
99C2×C42×C31C2311yescharacteristic
100C2×C42×C31C2311yescharacteristic
101C31×#106—24no—
102C31×C4⋊C8—22no—
103C31×C4⋊C8—22no—
104C31×#103—24no—
105C31×#103—24no—
106C31×#103—42no—
107C31×#102—24no—
108C31×C22⋊C8—42no—
109C31×#95—21no—
110C31×#95—21no—
111C31×#95—21no—
112C24×C31—21no—
113C24×C31C2×D411yescharacteristic
114C2×Q8×C31—24no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—22no—
120C2×Q8×C31—42no—
121C2×Q8×C31—42no—
122C2×Q8×C31C2×D414yes—
123C2×D4×C31—24no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—21no—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—42no—
130C2×D4×C31—42no—
131C2×D4×C31#4112yes—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—41no—
147C22×C4×C31—41no—
148C22×C4×C31—42no—
149C22×C4×C31—42no—
150C22×C4×C31—42no—
151C22×C4×C31SD1614yescomplemented
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2411yescharacteristic
158SD16×C31—48no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C2×C8×C31—22no—
162C2×C8×C31—22no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C4⋊C4×C31—28no—
166C4⋊C4×C31—28no—
167C4⋊C4×C31—24no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—42no—
177C4⋊C4×C31C2×D414yes—
178C4⋊C4×C31#4112yes—
179C22⋊C4×C31—28no—
180C22⋊C4×C31—24no—
181C22⋊C4×C31—42no—
182C22⋊C4×C31—42no—
183C22⋊C4×C31—42no—
184C22⋊C4×C31—42no—
185C42×C31—28no—
186C42×C31—22no—
187C42×C31—22no—
188C42×C31—21no—
189C42×C31—21no—
190C42×C31—42no—
191C42×C31C2×D412yes—
192C42×C31C2×D412yes—
193#24151C3111yesmaximal, characteristic, direct product factor, Sylow
194C23×C31—21no—
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—22no—
201C23×C31—41no—
202C23×C31—41no—
203C23×C31—41no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31C2×SD1612yes—
208C23×C31C22×D411yescharacteristic
209Q8×C31—24no—
210Q8×C31—24no—
211Q8×C31—48no—
212Q8×C31—42no—
213Q8×C31—42no—
214D4×C31—22no—
215D4×C31—22no—
216D4×C31—48no—
217D4×C31—41no—
218D4×C31—41no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—28no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—21no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—44no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—42no—
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—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—41no—
250C2×C4×C31—84no—
251C2×C4×C31#12012yes—
252C2×C4×C31#12012yes—
253C2×C4×C31C2×SD1612yes—
254C2×C4×C31C2×SD1612yes—
255C2×C4×C31#13712yes—
256C2×C4×C31C22×D411yescharacteristic
257C2×C4×C31C22×D411yescharacteristic
258C2×C4×C31C2×#4111yescharacteristic
259C2×C4×C31C2×#4111yescharacteristic
260C2×C4×C31H(5,𝔽2)11yescharacteristic
261C2×C4×C312−1+411yescharacteristic
262C8×C31—42no—
263C8×C31—42no—
264C2×#533C2×C3111yescharacteristic, complemented
265C2×#476C2×C3111yescharacteristic, complemented
266C2×#461C2×C3111yescharacteristic, complemented
267C4×#121C2×C3111yescharacteristic
268#2035C2×C3112yescomplemented
269#2000C2×C3112yes—
270#1987C2×C3111yescharacteristic, complemented
271#1981C2×C3111yescharacteristic, complemented
272#1880C2×C3112yescomplemented
273#1865C2×C3112yescomplemented
274#1797C2×C3111yescharacteristic, complemented
275C22×C31—22no—
276C22×C31—22no—
277C22×C31—22no—
278C22×C31—41no—
279C22×C31—41no—
280C22×C31—41no—
281C22×C31—41no—
282C22×C31—42no—
283C22×C31—42no—
284C22×C31—42no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31—81no—
288C22×C31—81no—
289C22×C31C2×#12011yescharacteristic
290C22×C31#52511yescharacteristic
291C22×C31#53711yescharacteristic
292C22×C31C22×SD1611yescharacteristic
293C22×C31C2×#13711yescharacteristic
294C22×C31#57511yescharacteristic
295C22×C31#57511yescharacteristic
296C4×C31—22no—
297C4×C31—22no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—44no—
301C4×C31—41no—
302C4×C31—44no—
303C4×C31—81no—
304C4×C31—81no—
305C4×C31#44814yes—
306C4×C31#54611yescharacteristic
307C4×C31#54711yescharacteristic
308C4×C31#54811yescharacteristic
309C4×C31#55011yescharacteristic
310C22×SD16C22×C3112yes—
311#533—24no—
312#533—24no—
313C2×C4⋊Q8C22×C3111yescharacteristic
314C2×C4⋊Q8C22×C3111yescharacteristic
315C2×#125—21no—
316C2×C22⋊Q8—21no—
317C2×C22⋊Q8C22×C3112yes—
318C2×#121C22×C3111yescharacteristic
319C2×C4×Q8—21no—
320C2×C4×Q8C22×C3112yes—
321C2×C4×D4—21no—
322C2×C4×D4C22×C3111yescharacteristic
323C2×#117—21no—
324C22×C42C22×C3111yescharacteristic
325#476—24no—
326#461—24no—
327C2×#106C22×C3112yes—
328C2×C4⋊C8C22×C3111yescharacteristic
329C2×C4⋊C8C22×C3111yescharacteristic
330C2×#103C22×C3112yes—
331C2×#103C22×C3112yes—
332C2×#103C22×C3112yes—
333C2×#102C22×C3112yes—
334C2×C22⋊C8C22×C3112yes—
335#390C22×C3111yescharacteristic
336#389C22×C3111yescharacteristic
337#388—21no—
338#386—21no—
339#380C22×C3111yescharacteristic
340C4×C4⋊C4C22×C3111yescharacteristic
341C4×C22⋊C4—21no—
342#340C22×C3112yes—
343#340C22×C3112yes—
344#336C22×C3112yes—
345C2×C31—42no—
346C2×C31—82no—
347C2×C31#240711yescharacteristic
348C2×C31C2×#44811yescharacteristic
349C2×C31#300511yescharacteristic
350C2×C31#310111yescharacteristic
351C2×C31#311611yescharacteristic
352C2×C31#324611yescharacteristic
353C2×C31#325211yescharacteristic
354C22×Q8—21no—
355C22×Q8C23×C3112yes—
356C22×D4—21no—
357C22×D4C23×C3111yescharacteristic
358C23×C4—21no—
359C23×C4D4×C3111yescharacteristic
360C23×C4D4×C3111yescharacteristic
361C23×C4C23×C3111yescharacteristic
362C2×SD16—24no—
363C2×SD16—24no—
364C2×SD16—42no—
365C22×C8C23×C3112yes—
366C22×C8C23×C3112yes—
367C4⋊Q8—22no—
368C4⋊Q8—22no—
369C4⋊Q8D4×C3114yescomplemented
370C4⋊Q8D4×C3114yescomplemented
371#125—28no—
372C22⋊Q8—28no—
373C22⋊Q8—24no—
374C22⋊Q8—24no—
375#121—22no—
376#121—22no—
377C4×Q8—28no—
378C4×Q8—24no—
379C4×Q8—24no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—44no—
383#117—28no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4C23×C3111yescharacteristic
391C2×C4⋊C4C23×C3111yescharacteristic
392C2×C4⋊C4C23×C3112yes—
393C2×C4⋊C4C23×C3111yescharacteristic
394C2×C22⋊C4—22no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42D4×C3111yescharacteristic
405C2×C42D4×C3111yescharacteristic
406C2×C42C23×C3111yescharacteristic
407C2×C42C23×C3111yescharacteristic
408#106—24no—
409C4⋊C8—22no—
410C4⋊C8—22no—
411#103—24no—
412#103—24no—
413#103—42no—
414#102—24no—
415C22⋊C8—42no—
416#95—21no—
417#95—21no—
418#95—21no—
419C31#2415111yescharacteristic, direct product factor, Sylow
420C24—21no—
421C24C2×D4×C3111yescharacteristic
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8C2×D4×C3114yes—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4#41×C3112yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
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—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4SD16×C3114yescomplemented
460C22×C4C2×D4×C3111yescharacteristic
461C22×C4C2×D4×C3111yescharacteristic
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C2×D4×C3111yescharacteristic
464C22×C4C2×D4×C3111yescharacteristic
465C22×C4C24×C3111yescharacteristic
466SD16—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4C2×D4×C3114yes—
486C4⋊C4#41×C3112yes—
487C22⋊C4—28no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C42—28no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—42no—
499C42C2×D4×C3112yes—
500C42C2×D4×C3112yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×SD16×C3112yes—
515C23C22×D4×C3111yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—42no—
520Q8—42no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4C31×#12012yes—
559C2×C4C31×#12012yes—
560C2×C4C2×SD16×C3112yes—
561C2×C4C2×SD16×C3112yes—
562C2×C4C31×#13712yes—
563C2×C4C22×D4×C3111yescharacteristic
564C2×C4C22×D4×C3111yescharacteristic
565C2×C4C2×#41×C3111yescharacteristic
566C2×C4C2×#41×C3111yescharacteristic
567C2×C4C31×H(5,𝔽2)11yescharacteristic
568C2×C4C31×2−1+411yescharacteristic
569C8—42no—
570C8—42no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×C31×#12011yescharacteristic
586C22C31×#52511yescharacteristic
587C22C31×#53711yescharacteristic
588C22C22×SD16×C3111yescharacteristic
589C22C2×C31×#13711yescharacteristic
590C22C31×#57511yescharacteristic
591C22C31×#57511yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—41no—
596C4—44no—
597C4—41no—
598C4—44no—
599C4—81no—
600C4—81no—
601C4C31×#44814yes—
602C4C31×#54611yescharacteristic
603C4C31×#54711yescharacteristic
604C4C31×#54811yescharacteristic
605C4C31×#55011yescharacteristic
606C2—42no—
607C2—82no—
608C2C31×#240711yescharacteristic
609C2C2×C31×#44811yescharacteristic
610C2C31×#300511yescharacteristic
611C2C31×#310111yescharacteristic
612C2C31×#311611yescharacteristic
613C2C31×#324611yescharacteristic
614C2C31×#325211yescharacteristic
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×#476C2semidirect
3. C2×C31×#461C2semidirect
5. C31×#2035C2semidirect
7. C31×#1987C2semidirect
8. C31×#1981C2semidirect
9. C31×#1880C2semidirect
10. C31×#1865C2semidirect
11. C31×#1797C2semidirect
62. C31×C4⋊Q8D4semidirect
63. C31×C4⋊Q8D4semidirect
151. C22×C4×C31SD16semidirect
193. #24151C31direct (N × H)
264. C2×#533C2×C31semidirect
265. C2×#476C2×C31semidirect
266. C2×#461C2×C31semidirect
268. #2035C2×C31semidirect
270. #1987C2×C31semidirect
271. #1981C2×C31semidirect
272. #1880C2×C31semidirect
273. #1865C2×C31semidirect
274. #1797C2×C31semidirect
369. C4⋊Q8D4×C31semidirect
370. C4⋊Q8D4×C31semidirect
419. C31#24151direct (N × H)
459. C22×C4SD16×C31semidirect

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×#4762
C2×C31×#4612
C4×C31×#1212
C31×#20352
C31×#20002
C31×#19872
C31×#19812
C31×#18802
C31×#18652
C31×#17972
#2415131