Finite Group Databasebeta

← C31×#25624 #21482383

Subgroups — C31×#25624 #21482383

Order 7936.

1126 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3507C211yesmaximal, characteristic, complemented
2C31×#3151C211yesmaximal, characteristic
3C31×#3138C211yesmaximal, characteristic, complemented
4C31×#2548C211yesmaximal, characteristic, complemented
5C31×#2003C214yesmaximal, complemented
6C31×#1986C211yesmaximal, characteristic, complemented
7C31×#1980C211yesmaximal, characteristic, complemented
8C31×#1883C214yesmaximal, complemented
9C4×C31×#101C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)C2214yes—
11C31×#559—21no—
12C31×#557—21no—
13C31×#556—21no—
14C31×#556—22no—
15C31×#554—21no—
16C31×#552—22no—
17C31×#539—22no—
18C31×#536C2212yes—
19C31×#533C2211yescharacteristic
20C31×#532C2211yescharacteristic
21C2×C31×#127C2211yescharacteristic
22C31×#529—22no—
23C2×C31×#124C2212yes—
24C31×#520C2212yes—
25C2×C4×Q8×C31C2211yescharacteristic
26C31×#496—22no—
27C31×#493—21no—
28C31×#492—21no—
29C31×#490—22no—
30C31×#486—21no—
31C31×#486—21no—
32C2×C31×#104C2214yes—
33C31×#418C2214yes—
34C2×C31×#101C2212yes—
35C4×M16×C31C2211yescharacteristic
36C4×M16×C31C2211yescharacteristic
37C31×#394—22no—
38C31×#390—22no—
39C31×#390C2212yes—
40C4×C4⋊C4×C31C2211yescharacteristic
41C31×#343C2212yes—
42C31×#339C2214yes—
43C2×#41×C31C2312yes—
44C22×Q8×C31C2311yescharacteristic
45C22×D4×C31—21no—
46C22×D4×C31—21no—
47C22×D4×C31C2312yes—
48C31×AGL(1,ℤ/8)—28no—
49C31×AGL(1,ℤ/8)—44no—
50C2×SD16×C31—24no—
51C2×D8×C31—24no—
52C2×M16×C31C2312yes—
53C2×M16×C31C2312yes—
54C31×C4⋊Q8—22no—
55C31×C4⋊Q8D411yescharacteristic
56C31×C4⋊Q8D411yescharacteristic
57C31×C4⋊Q8D411yescharacteristic
58C31×#127—21no—
59C31×#127—21no—
60C31×#127—21no—
61C31×#127—21no—
62C31×#127D411yescharacteristic
63C31×#126—24no—
64C31×#126—24no—
65C31×#125—22no—
66C31×#125—24no—
67C31×#125D412yes—
68C31×#124—22no—
69C31×#124—22no—
70C31×#124—44no—
71C31×#124D412yes—
72C31×#123—24no—
73C31×#123—24no—
74C31×C22⋊Q8—24no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—21no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×#121—21no—
80C31×#121—21no—
81C4×Q8×C31—22no—
82C4×Q8×C31—22no—
83C4×Q8×C31—21no—
84C4×Q8×C31—21no—
85C4×Q8×C31—21no—
86C4×Q8×C31—21no—
87C4×Q8×C31—21no—
88C4×Q8×C31—21no—
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—24no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C31×#117—24no—
95C31×#117—22no—
96C31×#117C2312yes—
97C2×C4⋊C4×C31—22no—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31D412yes—
100C2×C4⋊C4×C31C2311yescharacteristic
101C2×C22⋊C4×C31—24no—
102C2×C22⋊C4×C31—24no—
103C2×C42×C31D412yes—
104C2×C42×C31C2312yes—
105C2×C42×C31C2311yescharacteristic
106C31×#104—44no—
107C31×#102—24no—
108C31×#102—24no—
109C31×#101—24no—
110C31×#97—21no—
111C31×#97—21no—
112C4×C8×C31—21no—
113C4×C8×C31—21no—
114C31×#95—21no—
115C24×C31—22no—
116#41×C31—24no—
117#41×C31—24no—
118C2×Q8×C31—21no—
119C2×Q8×C31—21no—
120C2×Q8×C31—21no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31C2×D412yes—
125C2×D4×C31—22no—
126C2×D4×C31—21no—
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—24no—
131C2×D4×C31—24no—
132C2×D4×C31—44no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—41no—
136C2×D4×C31—41no—
137C2×D4×C31—82no—
138C2×D4×C31C2×D412yes—
139C22×C4×C31—24no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2411yescharacteristic
147SD16×C31—48no—
148D8×C31—48no—
149M16×C31—22no—
150M16×C31—22no—
151M16×C31—22no—
152M16×C31#4114yes—
153C2×C8×C31—22no—
154C2×C8×C31—22no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—42no—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D412yes—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—48no—
178C22⋊C4×C31—84no—
179C42×C31—24no—
180C42×C31—22no—
181C42×C31—22no—
182C42×C31—22no—
183C42×C31—24no—
184C42×C31C2×D411yescharacteristic
185C42×C31C2×D411yescharacteristic
186C42×C31C2×D411yescharacteristic
187C42×C31C2×D411yescharacteristic
188#25624C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—22no—
190C23×C31—44no—
191C23×C31—44no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31C22×D411yescharacteristic
196Q8×C31—24no—
197Q8×C31—42no—
198Q8×C31—42no—
199Q8×C31—42no—
200Q8×C31—81no—
201D4×C31—24no—
202D4×C31—42no—
203D4×C31—42no—
204D4×C31—44no—
205D4×C31—44no—
206D4×C31—42no—
207D4×C31—81no—
208D4×C31—82no—
209D4×C31—82no—
210D4×C31—82no—
211D4×C31—82no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—41no—
221C2×C4×C31—44no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—84no—
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C22×D411yescharacteristic
233C2×C4×C31C2×#4112yes—
234C2×C4×C31H(5,𝔽2)12yes—
235C8×C31—24no—
236C8×C31—42no—
237#3507C2×C3111yescharacteristic, complemented
238#3151C2×C3111yescharacteristic
239#3138C2×C3111yescharacteristic, complemented
240#2548C2×C3111yescharacteristic, complemented
241#2003C2×C3114yescomplemented
242#1986C2×C3111yescharacteristic, complemented
243#1980C2×C3111yescharacteristic, complemented
244#1883C2×C3114yescomplemented
245C4×#101C2×C3111yescharacteristic, complemented
246C22×C31—21no—
247C22×C31—21no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—84no—
252C22×C31—84no—
253C22×C31—82no—
254C22×C31—82no—
255C22×C31—82no—
256C22×C31C2×#12011yescharacteristic
257C22×C31#53211yescharacteristic
258C22×C31#53511yescharacteristic
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—21no—
262C4×C31—42no—
263C4×C31—42no—
264C4×C31—42no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—82no—
268C4×C31#45412yes—
269C4×C31#54612yes—
270C4×C31#55012yes—
271C2×AGL(1,ℤ/8)C22×C3114yes—
272#559—21no—
273#557—21no—
274#556—21no—
275#556—22no—
276#554—21no—
277#552—22no—
278#539—22no—
279#536C22×C3112yes—
280#533C22×C3111yescharacteristic
281#532C22×C3111yescharacteristic
282C2×#127C22×C3111yescharacteristic
283#529—22no—
284C2×#124C22×C3112yes—
285#520C22×C3112yes—
286C2×C4×Q8C22×C3111yescharacteristic
287#496—22no—
288#493—21no—
289#492—21no—
290#490—22no—
291#486—21no—
292#486—21no—
293C2×#104C22×C3114yes—
294#418C22×C3114yes—
295C2×#101C22×C3112yes—
296C4×M16C22×C3111yescharacteristic
297C4×M16C22×C3111yescharacteristic
298#394—22no—
299#390—22no—
300#390C22×C3112yes—
301C4×C4⋊C4C22×C3111yescharacteristic
302#343C22×C3112yes—
303#339C22×C3114yes—
304C2×C31—21no—
305C2×C31—21no—
306C2×C31—82no—
307C2×C31—162no—
308C2×C31#243111yescharacteristic
309C2×C31C2×#45411yescharacteristic
310C2×C31#302111yescharacteristic
311C2×#41C23×C3112yes—
312C22×Q8C23×C3111yescharacteristic
313C22×D4—21no—
314C22×D4—21no—
315C22×D4C23×C3112yes—
316AGL(1,ℤ/8)—28no—
317AGL(1,ℤ/8)—44no—
318C2×SD16—24no—
319C2×D8—24no—
320C2×M16C23×C3112yes—
321C2×M16C23×C3112yes—
322C4⋊Q8—22no—
323C4⋊Q8D4×C3111yescharacteristic
324C4⋊Q8D4×C3111yescharacteristic
325C4⋊Q8D4×C3111yescharacteristic
326#127—21no—
327#127—21no—
328#127—21no—
329#127—21no—
330#127D4×C3111yescharacteristic
331#126—24no—
332#126—24no—
333#125—22no—
334#125—24no—
335#125D4×C3112yes—
336#124—22no—
337#124—22no—
338#124—44no—
339#124D4×C3112yes—
340#123—24no—
341#123—24no—
342C22⋊Q8—24no—
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347#121—21no—
348#121—21no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—24no—
360C4×D4—21no—
361C4×D4—21no—
362#117—24no—
363#117—22no—
364#117C23×C3112yes—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4D4×C3112yes—
368C2×C4⋊C4C23×C3111yescharacteristic
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C42D4×C3112yes—
372C2×C42C23×C3112yes—
373C2×C42C23×C3111yescharacteristic
374#104—44no—
375#102—24no—
376#102—24no—
377#101—24no—
378#97—21no—
379#97—21no—
380C4×C8—21no—
381C4×C8—21no—
382#95—21no—
383C31#2562411yescharacteristic, direct product factor, Sylow
384C24—22no—
385#41—24no—
386#41—24no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8C2×D4×C3112yes—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—44no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—82no—
407C2×D4C2×D4×C3112yes—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4C2×D4×C3112yes—
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3112yes—
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C24×C3111yescharacteristic
416SD16—48no—
417D8—48no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16#41×C3114yes—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—42no—
438C4⋊C4C2×D4×C3112yes—
439C4⋊C4C2×D4×C3112yes—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—48no—
447C22⋊C4—84no—
448C42—24no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42C2×D4×C3111yescharacteristic
454C42C2×D4×C3111yescharacteristic
455C42C2×D4×C3111yescharacteristic
456C42C2×D4×C3111yescharacteristic
457C23—22no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C22×D4×C3111yescharacteristic
464Q8—24no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—81no—
469D4—24no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—44no—
474D4—44no—
475D4—81no—
476D4—82no—
477D4—82no—
478D4—82no—
479D4—82no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4C31×#12011yescharacteristic
496C2×C4C31×#12011yescharacteristic
497C2×C4C31×#12011yescharacteristic
498C2×C4C31×#12011yescharacteristic
499C2×C4C22×D4×C3111yescharacteristic
500C2×C4C22×D4×C3111yescharacteristic
501C2×C4C2×#41×C3112yes—
502C2×C4C31×H(5,𝔽2)12yes—
503C8—24no—
504C8—42no—
505C22—21no—
506C22—21no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—84no—
511C22—84no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22C2×C31×#12011yescharacteristic
516C22C31×#53211yescharacteristic
517C22C31×#53511yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4C31×#45412yes—
528C4C31×#54612yes—
529C4C31×#55012yes—
530C2—21no—
531C2—21no—
532C2—82no—
533C2—162no—
534C2C31×#243111yescharacteristic
535C2C2×C31×#45411yescharacteristic
536C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C31×#3507C2semidirect
3. C31×#3138C2semidirect
4. C31×#2548C2semidirect
5. C31×#2003C2semidirect
6. C31×#1986C2semidirect
7. C31×#1980C2semidirect
8. C31×#1883C2semidirect
9. C4×C31×#101C2semidirect
188. #25624C31direct (N × H)
237. #3507C2×C31semidirect
239. #3138C2×C31semidirect
240. #2548C2×C31semidirect
241. #2003C2×C31semidirect
242. #1986C2×C31semidirect
243. #1980C2×C31semidirect
244. #1883C2×C31semidirect
245. C4×#101C2×C31semidirect
383. C31#25624direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35072
C31×#31512
C31×#31382
C31×#25482
C31×#20032
C31×#19862
C31×#19802
C31×#18832
C4×C31×#1012
#2562431