Finite Group Databasebeta

← C31×#25599 #21482358

Subgroups — C31×#25599 #21482358

Order 7936.

1578 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#532C211yesmaximal, characteristic, complemented
2C2×C31×#493C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic, complemented
4C31×#2856C211yesmaximal, characteristic, complemented
5C31×#1999C214yesmaximal, complemented
6C31×#1990C211yesmaximal, characteristic, complemented
7C31×#1957C211yesmaximal, characteristic, complemented
8C31×#1879C214yesmaximal, complemented
9C4×C31×C22⋊C8C211yesmaximal, characteristic, complemented
10C22×#41×C31—21no—
11C22×D8×C31C2214yes—
12C31×#532—24no—
13C31×#532—24no—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×#127C2211yescharacteristic
16C2×C31×#127C2211yescharacteristic
17C2×C31×#124—21no—
18C2×C31×#121—21no—
19C2×C31×#121—21no—
20C2×C31×#121—21no—
21C2×C31×#121C2212yes—
22C2×C31×#121C2212yes—
23C2×C4×D4×C31—21no—
24C2×C4×D4×C31C2212yes—
25C22×C42×C31C2211yescharacteristic
26C31×#493—24no—
27C31×#486—24no—
28C2×C31×#102C2214yes—
29C2×C31×#102C2214yes—
30C2×C31×C22⋊C8C2212yes—
31C2×C4×C8×C31C2211yescharacteristic
32C2×C4×C8×C31C2211yescharacteristic
33C31×#390—21no—
34C31×#390—21no—
35C31×#390C2212yes—
36C31×#384C2211yescharacteristic
37C31×#340C2214yes—
38C31×#336C2212yes—
39C2×#41×C31—28no—
40C2×#41×C31—44no—
41C2×#41×C31—44no—
42C22×Q8×C31—21no—
43C22×D4×C31—22no—
44C22×D4×C31—22no—
45C22×D4×C31—21no—
46C22×D4×C31—21no—
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31C2312yes—
50C22×D4×C31C2312yes—
51C23×C4×C31—22no—
52C23×C4×C31D411yescharacteristic
53C23×C4×C31D411yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C2×D8×C31—28no—
56C2×D8×C31—28no—
57C2×D8×C31—44no—
58C22×C8×C31C2312yes—
59C22×C8×C31C2312yes—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8D414yescomplemented
62C31×#127—22no—
63C31×#127—22no—
64C31×#127—22no—
65C31×#127D414yescomplemented
66C31×#124—28no—
67C31×#121—28no—
68C31×#121—24no—
69C31×#121—24no—
70C31×#121—44no—
71C31×#121—44no—
72C31×#121—44no—
73C4×D4×C31—28no—
74C4×D4×C31—24no—
75C4×D4×C31—24no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31C2312yes—
80C2×C4⋊C4×C31C2312yes—
81C2×C22⋊C4×C31—22no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C42×C31—22no—
87C2×C42×C31—22no—
88C2×C42×C31D411yescharacteristic
89C2×C42×C31D411yescharacteristic
90C2×C42×C31C2311yescharacteristic
91C2×C42×C31C2311yescharacteristic
92C31×#102—28no—
93C31×#102—44no—
94C31×C22⋊C8—24no—
95C4×C8×C31—22no—
96C4×C8×C31—22no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31C2×D411yescharacteristic
100#41×C31—432no—
101C2×Q8×C31—24no—
102C2×Q8×C31—42no—
103C2×Q8×C31—42no—
104C2×D4×C31—28no—
105C2×D4×C31—24no—
106C2×D4×C31—24no—
107C2×D4×C31—22no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31—24no—
111C2×D4×C31—24no—
112C2×D4×C31—24no—
113C2×D4×C31—24no—
114C2×D4×C31—44no—
115C2×D4×C31—44no—
116C2×D4×C31—44no—
117C2×D4×C31—44no—
118C2×D4×C31—44no—
119C2×D4×C31—42no—
120C2×D4×C31—42no—
121C2×D4×C31—42no—
122C2×D4×C31—42no—
123C2×D4×C31—42no—
124C2×D4×C31—42no—
125C2×D4×C31—42no—
126C2×D4×C31—42no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31C2×D414yes—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—28no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—44no—
143C22×C4×C31—44no—
144C22×C4×C31D814yescomplemented
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2411yescharacteristic
151D8×C31—416no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31#4114yescomplemented
158C4⋊C4×C31—24no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—24no—
161C4⋊C4×C31—42no—
162C4⋊C4×C31—42no—
163C4⋊C4×C31C2×D414yes—
164C22⋊C4×C31—28no—
165C22⋊C4×C31—28no—
166C22⋊C4×C31—44no—
167C22⋊C4×C31—44no—
168C22⋊C4×C31—44no—
169C42×C31—22no—
170C42×C31—22no—
171C42×C31—21no—
172C42×C31—21no—
173C42×C31C2×D412yes—
174C42×C31C2×D412yes—
175#25599C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—21no—
181C23×C31—21no—
182C23×C31—24no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—42no—
196C23×C31C2×D812yes—
197C23×C31C22×D411yescharacteristic
198Q8×C31—48no—
199D4×C31—28no—
200D4×C31—416no—
201D4×C31—44no—
202D4×C31—44no—
203D4×C31—44no—
204D4×C31—48no—
205D4×C31—48no—
206D4×C31—88no—
207D4×C31—84no—
208D4×C31—84no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—28no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—24no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—416no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—44no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31#12012yes—
229C2×C4×C31#12012yes—
230C2×C4×C31C2×D812yes—
231C2×C4×C31C2×D812yes—
232C2×C4×C31#13512yes—
233C2×C4×C31AGL(1,ℤ/8)12yes—
234C2×C4×C31AGL(1,ℤ/8)12yes—
235C2×C4×C31C22×D411yescharacteristic
236C2×C4×C31C22×D411yescharacteristic
237C2×C4×C31C2×#4112yes—
238C2×C4×C31H(5,𝔽2)12yes—
239C8×C31—24no—
240C8×C31—42no—
241C2×#532C2×C3111yescharacteristic, complemented
242C2×#493C2×C3111yescharacteristic, complemented
243C2×#486C2×C3111yescharacteristic, complemented
244#2856C2×C3111yescharacteristic, complemented
245#1999C2×C3114yescomplemented
246#1990C2×C3111yescharacteristic, complemented
247#1957C2×C3111yescharacteristic, complemented
248#1879C2×C3114yescomplemented
249C4×C22⋊C8C2×C3111yescharacteristic, complemented
250C22×C31—22no—
251C22×C31—22no—
252C22×C31—22no—
253C22×C31—41no—
254C22×C31—41no—
255C22×C31—41no—
256C22×C31—41no—
257C22×C31—44no—
258C22×C31—44no—
259C22×C31—44no—
260C22×C31—82no—
261C22×C31—82no—
262C22×C31—82no—
263C22×C31—82no—
264C22×C31—82no—
265C22×C31—82no—
266C22×C31—82no—
267C22×C31—82no—
268C22×C31—82no—
269C22×C31—82no—
270C22×C31—82no—
271C22×C31C2×#12011yescharacteristic
272C22×C31#53211yescharacteristic
273C22×C31#53511yescharacteristic
274C22×C31C22×D811yescharacteristic
275C22×C31C2×#13511yescharacteristic
276C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
278C4×C31—22no—
279C4×C31—41no—
280C4×C31—44no—
281C4×C31—41no—
282C4×C31—82no—
283C4×C31#44714yes—
284C4×C31#44914yes—
285C4×C31#54612yes—
286C4×C31#55012yes—
287C22×#41—21no—
288C22×D8C22×C3114yes—
289#532—24no—
290#532—24no—
291C2×C4⋊Q8C22×C3111yescharacteristic
292C2×#127C22×C3111yescharacteristic
293C2×#127C22×C3111yescharacteristic
294C2×#124—21no—
295C2×#121—21no—
296C2×#121—21no—
297C2×#121—21no—
298C2×#121C22×C3112yes—
299C2×#121C22×C3112yes—
300C2×C4×D4—21no—
301C2×C4×D4C22×C3112yes—
302C22×C42C22×C3111yescharacteristic
303#493—24no—
304#486—24no—
305C2×#102C22×C3114yes—
306C2×#102C22×C3114yes—
307C2×C22⋊C8C22×C3112yes—
308C2×C4×C8C22×C3111yescharacteristic
309C2×C4×C8C22×C3111yescharacteristic
310#390—21no—
311#390—21no—
312#390C22×C3112yes—
313#384C22×C3111yescharacteristic
314#340C22×C3114yes—
315#336C22×C3112yes—
316C2×C31—42no—
317C2×C31—84no—
318C2×C31—162no—
319C2×C31#243111yescharacteristic
320C2×C31C2×#44711yescharacteristic
321C2×C31C2×#44911yescharacteristic
322C2×C31#316911yescharacteristic
323C2×C31#318411yescharacteristic
324C2×C31#320511yescharacteristic
325C2×C31#320711yescharacteristic
326C2×#41—28no—
327C2×#41—44no—
328C2×#41—44no—
329C22×Q8—21no—
330C22×D4—22no—
331C22×D4—22no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C23×C3112yes—
337C22×D4C23×C3112yes—
338C23×C4—22no—
339C23×C4D4×C3111yescharacteristic
340C23×C4D4×C3111yescharacteristic
341C23×C4C23×C3111yescharacteristic
342C2×D8—28no—
343C2×D8—28no—
344C2×D8—44no—
345C22×C8C23×C3112yes—
346C22×C8C23×C3112yes—
347C4⋊Q8—22no—
348C4⋊Q8D4×C3114yescomplemented
349#127—22no—
350#127—22no—
351#127—22no—
352#127D4×C3114yescomplemented
353#124—28no—
354#121—28no—
355#121—24no—
356#121—24no—
357#121—44no—
358#121—44no—
359#121—44no—
360C4×D4—28no—
361C4×D4—24no—
362C4×D4—24no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4C23×C3112yes—
367C2×C4⋊C4C23×C3112yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42D4×C3111yescharacteristic
376C2×C42D4×C3111yescharacteristic
377C2×C42C23×C3111yescharacteristic
378C2×C42C23×C3111yescharacteristic
379#102—28no—
380#102—44no—
381C22⋊C8—24no—
382C4×C8—22no—
383C4×C8—22no—
384C31#2559911yescharacteristic, direct product factor, Sylow
385C24—22no—
386C24—22no—
387C24C2×D4×C3111yescharacteristic
388#41—432no—
389C2×Q8—24no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C2×D4×C3114yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4D8×C3114yescomplemented
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C24×C3111yescharacteristic
439D8—416no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8#41×C3114yescomplemented
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4C2×D4×C3114yes—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C42—22no—
458C42—22no—
459C42—21no—
460C42—21no—
461C42C2×D4×C3112yes—
462C42C2×D4×C3112yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23C2×D8×C3112yes—
484C23C22×D4×C3111yescharacteristic
485Q8—48no—
486D4—28no—
487D4—416no—
488D4—48no—
489D4—48no—
490D4—44no—
491D4—44no—
492D4—44no—
493D4—88no—
494D4—84no—
495D4—84no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—28no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—416no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4C31×#12012yes—
516C2×C4C31×#12012yes—
517C2×C4C2×D8×C3112yes—
518C2×C4C2×D8×C3112yes—
519C2×C4C31×#13512yes—
520C2×C4C31×AGL(1,ℤ/8)12yes—
521C2×C4C31×AGL(1,ℤ/8)12yes—
522C2×C4C22×D4×C3111yescharacteristic
523C2×C4C22×D4×C3111yescharacteristic
524C2×C4C2×#41×C3112yes—
525C2×C4C31×H(5,𝔽2)12yes—
526C8—24no—
527C8—42no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22C2×C31×#12011yescharacteristic
550C22C31×#53211yescharacteristic
551C22C31×#53511yescharacteristic
552C22C22×D8×C3111yescharacteristic
553C22C2×C31×#13511yescharacteristic
554C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
555C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
556C4—22no—
557C4—41no—
558C4—44no—
559C4—41no—
560C4—82no—
561C4C31×#44714yes—
562C4C31×#44914yes—
563C4C31×#54612yes—
564C4C31×#55012yes—
565C2—42no—
566C2—84no—
567C2—162no—
568C2C31×#243111yescharacteristic
569C2C2×C31×#44711yescharacteristic
570C2C2×C31×#44911yescharacteristic
571C2C31×#316911yescharacteristic
572C2C31×#318411yescharacteristic
573C2C31×#320511yescharacteristic
574C2C31×#320711yescharacteristic
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×#532C2semidirect
2. C2×C31×#493C2semidirect
3. C2×C31×#486C2semidirect
4. C31×#2856C2semidirect
5. C31×#1999C2semidirect
6. C31×#1990C2semidirect
7. C31×#1957C2semidirect
8. C31×#1879C2semidirect
9. C4×C31×C22⋊C8C2semidirect
61. C31×C4⋊Q8D4semidirect
65. C31×#127D4semidirect
144. C22×C4×C31D8semidirect
157. C2×C8×C31#41semidirect
175. #25599C31direct (N × H)
241. C2×#532C2×C31semidirect
242. C2×#493C2×C31semidirect
243. C2×#486C2×C31semidirect
244. #2856C2×C31semidirect
245. #1999C2×C31semidirect
246. #1990C2×C31semidirect
247. #1957C2×C31semidirect
248. #1879C2×C31semidirect
249. C4×C22⋊C8C2×C31semidirect
348. C4⋊Q8D4×C31semidirect
352. #127D4×C31semidirect
384. C31#25599direct (N × H)
432. C22×C4D8×C31semidirect
445. C2×C8#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5322
C2×C31×#4932
C2×C31×#4862
C31×#28562
C31×#19992
C31×#19902
C31×#19572
C31×#18792
C4×C31×C22⋊C82
#2559931