Finite Group Databasebeta

← C31×#25605 #21482364

Subgroups — C31×#25605 #21482364

Order 7936.

1318 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#529C211yesmaximal, characteristic, complemented
2C2×C31×#496C211yesmaximal, characteristic, complemented
3C2×C31×#490C211yesmaximal, characteristic, complemented
4C31×#2856C211yesmaximal, characteristic, complemented
5C31×#2000C212yesmaximal, complemented
6C31×#1999C212yesmaximal, complemented
7C31×#1992C211yesmaximal, characteristic, complemented
8C31×#1959C211yesmaximal, characteristic, complemented
9C31×#1880C212yesmaximal, complemented
10C31×#1879C212yesmaximal, complemented
11C31×#1749C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C22×D8×C31C2212yes—
14C2×C31×#127C2211yescharacteristic
15C31×#529—24no—
16C31×#529—24no—
17C2×C31×#126—21no—
18C2×C31×#125C2211yescharacteristic
19C2×C31×#124C2211yescharacteristic
20C2×C31×#123—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×#121—21no—
23C2×C31×#121—21no—
24C2×C31×#121C2211yescharacteristic
25C2×C31×#121C2212yes—
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×D4×C31—21no—
28C2×C4×D4×C31C2211yescharacteristic
29C2×C31×#117—21no—
30C22×C42×C31C2211yescharacteristic
31C31×#496—24no—
32C31×#490—24no—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×#102C2212yes—
37C2×C31×C22⋊C8C2212yes—
38C2×C31×#97C2211yescharacteristic
39C2×C31×#97C2211yescharacteristic
40C31×#390—21no—
41C31×#390—21no—
42C31×#390C2212yes—
43C31×#384C2211yescharacteristic
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31C2311yescharacteristic
48C22×D4×C31—21no—
49C22×D4×C31—22no—
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C22×D4×C31C2312yes—
53C22×D4×C31C2311yescharacteristic
54C23×C4×C31—21no—
55C23×C4×C31D411yescharacteristic
56C23×C4×C31D411yescharacteristic
57C23×C4×C31C2311yescharacteristic
58C2×SD16×C31—24no—
59C2×SD16×C31—24no—
60C2×SD16×C31—42no—
61C2×D8×C31—24no—
62C2×D8×C31—24no—
63C2×D8×C31—42no—
64C22×C8×C31C2312yes—
65C22×C8×C31C2312yes—
66C31×#127—22no—
67C31×#127—22no—
68C31×#126—28no—
69C31×#125—22no—
70C31×#125D414yescomplemented
71C31×#124—22no—
72C31×#124D414yescomplemented
73C31×#123—28no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—42no—
79C31×#121—42no—
80C31×#121—44no—
81C31×#121—44no—
82C4×Q8×C31—22no—
83C4×Q8×C31—22no—
84C4×D4×C31—28no—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C31×#117—28no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—21no—
94C2×C4⋊C4×C31C2312yes—
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C4⋊C4×C31C2311yescharacteristic
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—22no—
105C2×C42×C31—21no—
106C2×C42×C31—22no—
107C2×C42×C31—22no—
108C2×C42×C31D411yescharacteristic
109C2×C42×C31D411yescharacteristic
110C2×C42×C31C2311yescharacteristic
111C2×C42×C31C2311yescharacteristic
112C31×#103—42no—
113C31×#102—24no—
114C31×#102—24no—
115C31×#102—42no—
116C31×C22⋊C8—24no—
117C31×#97—22no—
118C31×#97—22no—
119C24×C31—21no—
120C24×C31—22no—
121C24×C31C2×D411yescharacteristic
122C2×Q8×C31—21no—
123C2×Q8×C31—22no—
124C2×Q8×C31—22no—
125C2×Q8×C31C2×D412yes—
126C2×D4×C31—24no—
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—42no—
137C2×D4×C31—42no—
138C2×D4×C31—42no—
139C2×D4×C31—42no—
140C2×D4×C31—42no—
141C2×D4×C31—42no—
142C2×D4×C31—44no—
143C2×D4×C31—44no—
144C2×D4×C31—44no—
145C2×D4×C31—42no—
146C2×D4×C31—42no—
147C2×D4×C31C2×D412yes—
148C22×C4×C31—24no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—21no—
157C22×C4×C31—21no—
158C22×C4×C31—22no—
159C22×C4×C31—22no—
160C22×C4×C31—22no—
161C22×C4×C31—41no—
162C22×C4×C31—41no—
163C22×C4×C31—42no—
164C22×C4×C31—42no—
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31C2×D411yescharacteristic
170C22×C4×C31C2411yescharacteristic
171SD16×C31—48no—
172D8×C31—48no—
173C2×C8×C31—22no—
174C2×C8×C31—22no—
175C2×C8×C31—22no—
176C2×C8×C31—22no—
177C2×C8×C31—22no—
178C2×C8×C31#4114yes—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—24no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—42no—
188C4⋊C4×C31—42no—
189C4⋊C4×C31C2×D412yes—
190C4⋊C4×C31C2×D412yes—
191C22⋊C4×C31—24no—
192C22⋊C4×C31—24no—
193C22⋊C4×C31—24no—
194C22⋊C4×C31—24no—
195C22⋊C4×C31—24no—
196C22⋊C4×C31—44no—
197C22⋊C4×C31—44no—
198C22⋊C4×C31—44no—
199C42×C31—24no—
200C42×C31—22no—
201C42×C31—22no—
202C42×C31—21no—
203C42×C31—21no—
204C42×C31C2×D412yes—
205C42×C31C2×D412yes—
206#25605C3111yesmaximal, characteristic, direct product factor, Sylow
207C23×C31—21no—
208C23×C31—21no—
209C23×C31—21no—
210C23×C31—21no—
211C23×C31—21no—
212C23×C31—21no—
213C23×C31—22no—
214C23×C31—41no—
215C23×C31—41no—
216C23×C31—41no—
217C23×C31—41no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31—42no—
221C23×C31—42no—
222C23×C31—42no—
223C23×C31—42no—
224C23×C31—42no—
225C23×C31—42no—
226C23×C31—42no—
227C23×C31AGL(1,ℤ/8)12yes—
228C23×C31C22×D411yescharacteristic
229Q8×C31—24no—
230Q8×C31—42no—
231D4×C31—24no—
232D4×C31—48no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—42no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—84no—
239D4×C31—84no—
240D4×C31—88no—
241C2×C4×C31—21no—
242C2×C4×C31—21no—
243C2×C4×C31—21no—
244C2×C4×C31—21no—
245C2×C4×C31—21no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—21no—
249C2×C4×C31—22no—
250C2×C4×C31—22no—
251C2×C4×C31—22no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—44no—
257C2×C4×C31—44no—
258C2×C4×C31—48no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31#12012yes—
271C2×C4×C31#12012yes—
272C2×C4×C31AGL(1,ℤ/8)12yes—
273C2×C4×C31C22×D411yescharacteristic
274C2×C4×C31C22×D411yescharacteristic
275C2×C4×C31C2×#4112yes—
276C2×C4×C31H(5,𝔽2)12yes—
277C8×C31—24no—
278C8×C31—42no—
279C2×#529C2×C3111yescharacteristic, complemented
280C2×#496C2×C3111yescharacteristic, complemented
281C2×#490C2×C3111yescharacteristic, complemented
282#2856C2×C3111yescharacteristic, complemented
283#2000C2×C3112yescomplemented
284#1999C2×C3112yescomplemented
285#1992C2×C3111yescharacteristic, complemented
286#1959C2×C3111yescharacteristic, complemented
287#1880C2×C3112yescomplemented
288#1879C2×C3112yescomplemented
289#1749C2×C3111yescharacteristic, complemented
290C22×C31—22no—
291C22×C31—22no—
292C22×C31—22no—
293C22×C31—41no—
294C22×C31—41no—
295C22×C31—41no—
296C22×C31—41no—
297C22×C31—42no—
298C22×C31—42no—
299C22×C31—42no—
300C22×C31—81no—
301C22×C31—81no—
302C22×C31—81no—
303C22×C31—81no—
304C22×C31—82no—
305C22×C31—82no—
306C22×C31—82no—
307C22×C31—82no—
308C22×C31—82no—
309C22×C31—82no—
310C22×C31—82no—
311C22×C31C2×#12011yescharacteristic
312C22×C31#53211yescharacteristic
313C22×C31#53511yescharacteristic
314C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
316C22×C31#57511yescharacteristic
317C22×C31#57511yescharacteristic
318C4×C31—22no—
319C4×C31—22no—
320C4×C31—22no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—41no—
324C4×C31—42no—
325C4×C31—42no—
326C4×C31—82no—
327C4×C31#54612yes—
328C4×C31#55012yes—
329C22×SD16C22×C3112yes—
330C22×D8C22×C3112yes—
331C2×#127C22×C3111yescharacteristic
332#529—24no—
333#529—24no—
334C2×#126—21no—
335C2×#125C22×C3111yescharacteristic
336C2×#124C22×C3111yescharacteristic
337C2×#123—21no—
338C2×C22⋊Q8C22×C3111yescharacteristic
339C2×#121—21no—
340C2×#121—21no—
341C2×#121C22×C3111yescharacteristic
342C2×#121C22×C3112yes—
343C2×C4×Q8C22×C3111yescharacteristic
344C2×C4×D4—21no—
345C2×C4×D4C22×C3111yescharacteristic
346C2×#117—21no—
347C22×C42C22×C3111yescharacteristic
348#496—24no—
349#490—24no—
350C2×#103C22×C3112yes—
351C2×#102C22×C3112yes—
352C2×#102C22×C3112yes—
353C2×#102C22×C3112yes—
354C2×C22⋊C8C22×C3112yes—
355C2×#97C22×C3111yescharacteristic
356C2×#97C22×C3111yescharacteristic
357#390—21no—
358#390—21no—
359#390C22×C3112yes—
360#384C22×C3111yescharacteristic
361#340C22×C3112yes—
362#340C22×C3112yes—
363#336C22×C3112yes—
364C2×C31—42no—
365C2×C31—82no—
366C2×C31—162no—
367C2×C31#243111yescharacteristic
368C2×C31#300611yescharacteristic
369C2×C31#300611yescharacteristic
370C2×C31#317111yescharacteristic
371C2×C31#318011yescharacteristic
372C2×C31#321711yescharacteristic
373C2×C31#321911yescharacteristic
374C22×Q8C23×C3111yescharacteristic
375C22×D4—21no—
376C22×D4—22no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4C23×C3112yes—
380C22×D4C23×C3111yescharacteristic
381C23×C4—21no—
382C23×C4D4×C3111yescharacteristic
383C23×C4D4×C3111yescharacteristic
384C23×C4C23×C3111yescharacteristic
385C2×SD16—24no—
386C2×SD16—24no—
387C2×SD16—42no—
388C2×D8—24no—
389C2×D8—24no—
390C2×D8—42no—
391C22×C8C23×C3112yes—
392C22×C8C23×C3112yes—
393#127—22no—
394#127—22no—
395#126—28no—
396#125—22no—
397#125D4×C3114yescomplemented
398#124—22no—
399#124D4×C3114yescomplemented
400#123—28no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403#121—22no—
404#121—22no—
405#121—42no—
406#121—42no—
407#121—44no—
408#121—44no—
409C4×Q8—22no—
410C4×Q8—22no—
411C4×D4—28no—
412C4×D4—22no—
413C4×D4—22no—
414#117—28no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4C23×C3112yes—
422C2×C4⋊C4C23×C3111yescharacteristic
423C2×C4⋊C4C23×C3111yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C42—21no—
433C2×C42—22no—
434C2×C42—22no—
435C2×C42D4×C3111yescharacteristic
436C2×C42D4×C3111yescharacteristic
437C2×C42C23×C3111yescharacteristic
438C2×C42C23×C3111yescharacteristic
439#103—42no—
440#102—24no—
441#102—24no—
442#102—42no—
443C22⋊C8—24no—
444#97—22no—
445#97—22no—
446C31#2560511yescharacteristic, direct product factor, Sylow
447C24—21no—
448C24—22no—
449C24C2×D4×C3111yescharacteristic
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8C2×D4×C3112yes—
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×D4×C3112yes—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4C2×D4×C3111yescharacteristic
494C22×C4C2×D4×C3111yescharacteristic
495C22×C4C2×D4×C3111yescharacteristic
496C22×C4C2×D4×C3111yescharacteristic
497C22×C4C2×D4×C3111yescharacteristic
498C22×C4C24×C3111yescharacteristic
499SD16—48no—
500D8—48no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8#41×C3114yes—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4C2×D4×C3112yes—
518C4⋊C4C2×D4×C3112yes—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42C2×D4×C3112yes—
533C42C2×D4×C3112yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C31×AGL(1,ℤ/8)12yes—
555C23C22×D4×C3111yescharacteristic
556Q8—24no—
557Q8—42no—
558D4—24no—
559D4—48no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—84no—
566D4—84no—
567D4—88no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4C31×#12012yes—
598C2×C4C31×#12012yes—
599C2×C4C31×AGL(1,ℤ/8)12yes—
600C2×C4C22×D4×C3111yescharacteristic
601C2×C4C22×D4×C3111yescharacteristic
602C2×C4C2×#41×C3112yes—
603C2×C4C31×H(5,𝔽2)12yes—
604C8—24no—
605C8—42no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22C2×C31×#12011yescharacteristic
628C22C31×#53211yescharacteristic
629C22C31×#53511yescharacteristic
630C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
631C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
632C22C31×#57511yescharacteristic
633C22C31×#57511yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—82no—
643C4C31×#54612yes—
644C4C31×#55012yes—
645C2—42no—
646C2—82no—
647C2—162no—
648C2C31×#243111yescharacteristic
649C2C31×#300611yescharacteristic
650C2C31×#300611yescharacteristic
651C2C31×#317111yescharacteristic
652C2C31×#318011yescharacteristic
653C2C31×#321711yescharacteristic
654C2C31×#321911yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
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×#529C2semidirect
2. C2×C31×#496C2semidirect
3. C2×C31×#490C2semidirect
4. C31×#2856C2semidirect
5. C31×#2000C2semidirect
6. C31×#1999C2semidirect
7. C31×#1992C2semidirect
8. C31×#1959C2semidirect
9. C31×#1880C2semidirect
10. C31×#1879C2semidirect
11. C31×#1749C2semidirect
70. C31×#125D4semidirect
72. C31×#124D4semidirect
206. #25605C31direct (N × H)
279. C2×#529C2×C31semidirect
280. C2×#496C2×C31semidirect
281. C2×#490C2×C31semidirect
282. #2856C2×C31semidirect
283. #2000C2×C31semidirect
284. #1999C2×C31semidirect
285. #1992C2×C31semidirect
286. #1959C2×C31semidirect
287. #1880C2×C31semidirect
288. #1879C2×C31semidirect
289. #1749C2×C31semidirect
397. #125D4×C31semidirect
399. #124D4×C31semidirect
446. C31#25605direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5292
C2×C31×#4962
C2×C31×#4902
C31×#28562
C31×#20002
C31×#19992
C31×#19922
C31×#19592
C31×#18802
C31×#18792
C31×#17492
#2560531