Finite Group Databasebeta

← C31×#23970 #21480729

Subgroups — C31×#23970 #21480729

Order 7936.

1278 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#554C211yesmaximal, characteristic
2C2×C31×#451C211yesmaximal, characteristic, complemented
3C22×C31×#103C211yesmaximal, characteristic, complemented
4C31×#2409C211yesmaximal, characteristic
5C31×#2024C212yesmaximal, complemented
6C31×#2001C212yesmaximal, complemented
7C31×#1892C211yesmaximal, characteristic, complemented
8C31×#1878C212yesmaximal, complemented
9C31×#1872C212yesmaximal, complemented
10C31×#1869C211yesmaximal, characteristic
11C31×#1824C211yesmaximal, characteristic, complemented
12C23×Q8×C31C2211yescharacteristic
13C22×Q16×C31C2212yes—
14C23×C8×C31C2211yescharacteristic
15C31×#554—24no—
16C31×#554—24no—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×#125—21no—
19C2×C31×C22⋊Q8—21no—
20C2×C31×C22⋊Q8—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×C22⋊Q8C2212yes—
23C2×C31×C22⋊Q8C2211yescharacteristic
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×D4×C31—21no—
26C2×C4×D4×C31C2211yescharacteristic
27C22×C4⋊C4×C31C2211yescharacteristic
28C22×C4⋊C4×C31C2211yescharacteristic
29C31×#451—24no—
30C2×C31×#103—24no—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#103C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2212yes—
37C31×#395—21no—
38C31×#393—21no—
39C31×#391C2212yes—
40C31×#386—21no—
41C31×#385—21no—
42C31×#380C2211yescharacteristic
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31—21no—
49C22×Q8×C31—24no—
50C22×Q8×C31—22no—
51C22×Q8×C31—22no—
52C22×Q8×C31—42no—
53C22×Q8×C31C2311yescharacteristic
54C22×Q8×C31C2312yes—
55C22×D4×C31D411yescharacteristic
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C23×C4×C31—21no—
59C23×C4×C31C2311yescharacteristic
60C23×C4×C31C2311yescharacteristic
61C2×Q16×C31—24no—
62C2×Q16×C31—24no—
63C2×Q16×C31—42no—
64C22×C8×C31—22no—
65C22×C8×C31—42no—
66C22×C8×C31C2312yes—
67C22×C8×C31C2312yes—
68C31×C4⋊Q8—22no—
69C31×C4⋊Q8—22no—
70C31×#125—28no—
71C31×C22⋊Q8—28no—
72C31×C22⋊Q8—28no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—42no—
78C31×C22⋊Q8D414yes—
79C4×Q8×C31—22no—
80C4×Q8×C31—22no—
81C4×D4×C31—28no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—22no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—22no—
92C2×C4⋊C4×C31—22no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31D411yescharacteristic
95C2×C4⋊C4×C31D414yes—
96C2×C4⋊C4×C31C2312yes—
97C2×C4⋊C4×C31C2311yescharacteristic
98C2×C4⋊C4×C31C2311yescharacteristic
99C2×C4⋊C4×C31C2311yescharacteristic
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31D411yescharacteristic
105C2×C22⋊C4×C31D411yescharacteristic
106C2×C42×C31—21no—
107C2×C42×C31C2311yescharacteristic
108C31×#103—24no—
109C31×#103—24no—
110C31×#103—24no—
111C31×#103—42no—
112C31×C22⋊C8—22no—
113C31×C22⋊C8—42no—
114C31×#95—21no—
115C31×#95—22no—
116C31×#95—22no—
117C24×C31C2×D411yescharacteristic
118C24×C31C2×D411yescharacteristic
119C2×Q8×C31—24no—
120C2×Q8×C31—21no—
121C2×Q8×C31—22no—
122C2×Q8×C31—28no—
123C2×Q8×C31—22no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31—44no—
130C2×Q8×C31—48no—
131C2×Q8×C31—44no—
132C2×Q8×C31—42no—
133C2×Q8×C31—82no—
134C2×Q8×C31C2×D412yes—
135C2×Q8×C31#4114yes—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31Q1614yescomplemented
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—24no—
147C22×C4×C31—24no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—44no—
154C22×C4×C31—42no—
155C22×C4×C31—82no—
156C22×C4×C31—82no—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D412yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2411yescharacteristic
164Q16×C31—48no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31—22no—
171C2×C8×C31—44no—
172C4⋊C4×C31—24no—
173C4⋊C4×C31—24no—
174C4⋊C4×C31—24no—
175C4⋊C4×C31—24no—
176C4⋊C4×C31—24no—
177C4⋊C4×C31—24no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—44no—
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D412yes—
187C4⋊C4×C31C2×D412yes—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—24no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—44no—
193C22⋊C4×C31—42no—
194C42×C31—24no—
195C42×C31—21no—
196C42×C31—21no—
197#23970C3111yesmaximal, characteristic, direct product factor, Sylow
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—21no—
201C23×C31—22no—
202C23×C31—21no—
203C23×C31—22no—
204C23×C31#12012yes—
205C23×C31C2×Q1612yes—
206C23×C31C2×Q1612yes—
207C23×C31#13712yes—
208C23×C31C22×D411yescharacteristic
209Q8×C31—24no—
210Q8×C31—24no—
211Q8×C31—24no—
212Q8×C31—48no—
213Q8×C31—41no—
214Q8×C31—41no—
215Q8×C31—42no—
216Q8×C31—42no—
217Q8×C31—84no—
218D4×C31—28no—
219C2×C4×C31—21no—
220C2×C4×C31—22no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—24no—
225C2×C4×C31—22no—
226C2×C4×C31—21no—
227C2×C4×C31—22no—
228C2×C4×C31—21no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—24no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—48no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—84no—
248C2×C4×C31—84no—
249C2×C4×C31#12012yes—
250C2×C4×C31C2×Q1612yes—
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C2×#4112yes—
254C2×C4×C31H(5,𝔽2)12yes—
255C8×C31—42no—
256C8×C31—42no—
257C2×#554C2×C3111yescharacteristic
258C2×#451C2×C3111yescharacteristic, complemented
259C22×#103C2×C3111yescharacteristic, complemented
260#2409C2×C3111yescharacteristic
261#2024C2×C3112yescomplemented
262#2001C2×C3112yescomplemented
263#1892C2×C3111yescharacteristic, complemented
264#1878C2×C3112yescomplemented
265#1872C2×C3112yescomplemented
266#1869C2×C3111yescharacteristic
267#1824C2×C3111yescharacteristic, complemented
268C22×C31—22no—
269C22×C31—22no—
270C22×C31—24no—
271C22×C31—24no—
272C22×C31—22no—
273C22×C31—24no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—42no—
277C22×C31#45114yes—
278C22×C31C2×#12011yescharacteristic
279C22×C31#54612yes—
280C22×C31C22×Q1611yescharacteristic
281C22×C31C2×#13711yescharacteristic
282C22×C31#57712yes—
283C4×C31—22no—
284C4×C31—22no—
285C4×C31—41no—
286C4×C31—44no—
287C4×C31—42no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—42no—
291C4×C31—42no—
292C4×C31—82no—
293C4×C31#52511yescharacteristic
294C4×C31#53511yescharacteristic
295C4×C31#54612yes—
296C23×Q8C22×C3111yescharacteristic
297C22×Q16C22×C3112yes—
298C23×C8C22×C3111yescharacteristic
299#554—24no—
300#554—24no—
301C2×C4⋊Q8C22×C3111yescharacteristic
302C2×#125—21no—
303C2×C22⋊Q8—21no—
304C2×C22⋊Q8—21no—
305C2×C22⋊Q8C22×C3111yescharacteristic
306C2×C22⋊Q8C22×C3112yes—
307C2×C22⋊Q8C22×C3111yescharacteristic
308C2×C4×Q8C22×C3111yescharacteristic
309C2×C4×D4—21no—
310C2×C4×D4C22×C3111yescharacteristic
311C22×C4⋊C4C22×C3111yescharacteristic
312C22×C4⋊C4C22×C3111yescharacteristic
313#451—24no—
314C2×#103—24no—
315C2×#103C22×C3112yes—
316C2×#103C22×C3112yes—
317C2×#103C22×C3112yes—
318C2×#103C22×C3112yes—
319C2×C22⋊C8C22×C3111yescharacteristic
320C2×C22⋊C8C22×C3112yes—
321#395—21no—
322#393—21no—
323#391C22×C3112yes—
324#386—21no—
325#385—21no—
326#380C22×C3111yescharacteristic
327#340C22×C3112yes—
328#340C22×C3112yes—
329#340C22×C3112yes—
330#336C22×C3112yes—
331C2×C31—24no—
332C2×C31—42no—
333C2×C31#240411yescharacteristic
334C2×C31C2×#45111yescharacteristic
335C2×C31#300911yescharacteristic
336C2×C31#329912yes—
337C2×C31#331612yes—
338C22×Q8—21no—
339C22×Q8—21no—
340C22×Q8—24no—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—42no—
344C22×Q8C23×C3111yescharacteristic
345C22×Q8C23×C3112yes—
346C22×D4D4×C3111yescharacteristic
347C23×C4—21no—
348C23×C4—21no—
349C23×C4—21no—
350C23×C4C23×C3111yescharacteristic
351C23×C4C23×C3111yescharacteristic
352C2×Q16—24no—
353C2×Q16—24no—
354C2×Q16—42no—
355C22×C8—22no—
356C22×C8—42no—
357C22×C8C23×C3112yes—
358C22×C8C23×C3112yes—
359C4⋊Q8—22no—
360C4⋊Q8—22no—
361#125—28no—
362C22⋊Q8—28no—
363C22⋊Q8—28no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—24no—
368C22⋊Q8—42no—
369C22⋊Q8D4×C3114yes—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×D4—28no—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4D4×C3111yescharacteristic
386C2×C4⋊C4D4×C3114yes—
387C2×C4⋊C4C23×C3112yes—
388C2×C4⋊C4C23×C3111yescharacteristic
389C2×C4⋊C4C23×C3111yescharacteristic
390C2×C4⋊C4C23×C3111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4D4×C3111yescharacteristic
396C2×C22⋊C4D4×C3111yescharacteristic
397C2×C42—21no—
398C2×C42C23×C3111yescharacteristic
399#103—24no—
400#103—24no—
401#103—24no—
402#103—42no—
403C22⋊C8—22no—
404C22⋊C8—42no—
405#95—21no—
406#95—22no—
407#95—22no—
408C31#2397011yescharacteristic, direct product factor, Sylow
409C24C2×D4×C3111yescharacteristic
410C24C2×D4×C3111yescharacteristic
411C2×Q8—24no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—44no—
422C2×Q8—48no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—82no—
426C2×Q8C2×D4×C3112yes—
427C2×Q8#41×C3114yes—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4Q16×C3114yescomplemented
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4C2×D4×C3111yescharacteristic
450C22×C4C2×D4×C3111yescharacteristic
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3112yes—
453C22×C4C2×D4×C3112yes—
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C24×C3111yescharacteristic
456Q16—48no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4C2×D4×C3112yes—
478C4⋊C4C2×D4×C3112yes—
479C4⋊C4C2×D4×C3112yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C42—24no—
487C42—21no—
488C42—21no—
489C23—22no—
490C23—22no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—22no—
495C23C31×#12012yes—
496C23C2×Q16×C3112yes—
497C23C2×Q16×C3112yes—
498C23C31×#13712yes—
499C23C22×D4×C3111yescharacteristic
500Q8—24no—
501Q8—24no—
502Q8—24no—
503Q8—48no—
504Q8—41no—
505Q8—41no—
506Q8—42no—
507Q8—42no—
508Q8—84no—
509D4—28no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4C31×#12012yes—
541C2×C4C2×Q16×C3112yes—
542C2×C4C22×D4×C3111yescharacteristic
543C2×C4C22×D4×C3111yescharacteristic
544C2×C4C2×#41×C3112yes—
545C2×C4C31×H(5,𝔽2)12yes—
546C8—42no—
547C8—42no—
548C22—22no—
549C22—22no—
550C22—24no—
551C22—24no—
552C22—22no—
553C22—24no—
554C22—41no—
555C22—41no—
556C22—42no—
557C22C31×#45114yes—
558C22C2×C31×#12011yescharacteristic
559C22C31×#54612yes—
560C22C22×Q16×C3111yescharacteristic
561C22C2×C31×#13711yescharacteristic
562C22C31×#57712yes—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—44no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4C31×#52511yescharacteristic
574C4C31×#53511yescharacteristic
575C4C31×#54612yes—
576C2—24no—
577C2—42no—
578C2C31×#240411yescharacteristic
579C2C2×C31×#45111yescharacteristic
580C2C31×#300911yescharacteristic
581C2C31×#329912yes—
582C2C31×#331612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D4×C31D4
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
2. C2×C31×#451C2semidirect
3. C22×C31×#103C2semidirect
5. C31×#2024C2semidirect
6. C31×#2001C2semidirect
7. C31×#1892C2semidirect
8. C31×#1878C2semidirect
9. C31×#1872C2semidirect
11. C31×#1824C2semidirect
138. C2×D4×C31Q16semidirect
197. #23970C31direct (N × H)
258. C2×#451C2×C31semidirect
259. C22×#103C2×C31semidirect
261. #2024C2×C31semidirect
262. #2001C2×C31semidirect
263. #1892C2×C31semidirect
264. #1878C2×C31semidirect
265. #1872C2×C31semidirect
267. #1824C2×C31semidirect
408. C31#23970direct (N × H)
430. C2×D4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5542
C2×C31×#4512
C22×C31×#1032
C31×#24092
C31×#20242
C31×#20012
C31×#18922
C31×#18782
C31×#18722
C31×#18692
C31×#18242
#2397031