Finite Group Databasebeta

← C31×#23948 #21480707

Subgroups — C31×#23948 #21480707

Order 7936.

1474 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×#3492C211yesmaximal, characteristic, complemented
2C31×#3003C212yesmaximal, complemented
3C2×C31×#390C211yesmaximal, characteristic, complemented
4C31×#2027C212yesmaximal, complemented
5C31×#2006C212yesmaximal, complemented
6C31×#1896C212yesmaximal, complemented
7C31×#1883C214yesmaximal, complemented
8C31×#1821C211yesmaximal, characteristic, complemented
9C23×D4×C31C2211yescharacteristic
10C2×C31×AGL(1,ℤ/8)C2214yes—
11C31×#554—22no—
12C31×#553—22no—
13C31×#552—22no—
14C31×#547—22no—
15C31×#544C2211yescharacteristic, complemented
16C31×#539—22no—
17C31×#537C2211yescharacteristic, complemented
18C31×#525—22no—
19C31×#525C2212yes—
20C31×#520C2212yescomplemented
21C22×C4⋊C4×C31C2211yescharacteristic
22C22×C22⋊C4×C31—22no—
23C22×C42×C31C2211yescharacteristic
24C31×#450—22no—
25C31×#450—22no—
26C31×#447—22no—
27C31×#447—22no—
28C2×C31×#104C2214yescomplemented
29C31×#418C2214yes—
30C31×#407C2212yes—
31C31×#390—22no—
32C31×#390C2212yes—
33C31×#390C2212yes—
34C31×#343C2212yescomplemented
35C31×#343C2212yescomplemented
36C31×#339C2214yescomplemented
37C25×C31—21no—
38C2×#41×C31—22no—
39C2×#41×C31C2312yes—
40C22×D4×C31—24no—
41C22×D4×C31—21no—
42C22×D4×C31—22no—
43C22×D4×C31—22no—
44C22×D4×C31—42no—
45C22×D4×C31C2312yes—
46C23×C4×C31—21no—
47C23×C4×C31D412yes—
48C23×C4×C31C2311yescharacteristic
49C31×AGL(1,ℤ/8)—28no—
50C31×AGL(1,ℤ/8)—44no—
51C2×SD16×C31—24no—
52C2×D8×C31—24no—
53C2×M16×C31C2314yes—
54C31×C4⋊Q8—22no—
55C31×#126—24no—
56C31×#125—24no—
57C31×#125—22no—
58C31×#123—24no—
59C31×#123—24no—
60C31×#123—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8D412yes—
66C31×C22⋊Q8D412yes—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121D412yes—
71C31×#121D412yes—
72C31×#120—22no—
73C4×Q8×C31—22no—
74C4×D4×C31—24no—
75C4×D4×C31—22no—
76C4×D4×C31—24no—
77C4×D4×C31—22no—
78C4×D4×C31—22no—
79C31×#117—22no—
80C31×#117—22no—
81C31×#117C2312yes—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31D412yes—
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C22⋊C4×C31—24no—
89C2×C22⋊C4×C31—24no—
90C2×C22⋊C4×C31—48no—
91C2×C42×C31—21no—
92C2×C42×C31—42no—
93C2×C42×C31C2312yes—
94C31×#104—44no—
95C31×#102—24no—
96C31×#102—24no—
97C31×C22⋊C8—22no—
98C31×C22⋊C8—22no—
99C24×C31—24no—
100C24×C31—22no—
101C24×C31—44no—
102C24×C31—42no—
103C24×C31—44no—
104C24×C31—41no—
105C24×C31—41no—
106C24×C31C2×D411yescharacteristic
107#41×C31—24no—
108#41×C31—24no—
109#41×C31—84no—
110C2×Q8×C31—22no—
111C2×Q8×C31C2×D412yes—
112C2×D4×C31—24no—
113C2×D4×C31—22no—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—24no—
118C2×D4×C31—22no—
119C2×D4×C31—24no—
120C2×D4×C31—48no—
121C2×D4×C31—48no—
122C2×D4×C31—42no—
123C2×D4×C31—42no—
124C2×D4×C31C2×D412yes—
125C2×D4×C31#4114yescomplemented
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—24no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—24no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—42no—
135C22×C4×C31—42no—
136C22×C4×C31—44no—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2×D412yes—
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2411yescharacteristic
143SD16×C31—48no—
144D8×C31—48no—
145M16×C31—24no—
146M16×C31—24no—
147C2×C8×C31—24no—
148C4⋊C4×C31—24no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31C2×D412yes—
158C4⋊C4×C31C2×D412yes—
159C22⋊C4×C31—24no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—48no—
167C22⋊C4×C31—48no—
168C42×C31—24no—
169C42×C31—22no—
170C42×C31—24no—
171C42×C31—42no—
172#23948C3111yesmaximal, characteristic, direct product factor, Sylow
173C23×C31—21no—
174C23×C31—21no—
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—24no—
179C23×C31—44no—
180C23×C31—44no—
181C23×C31—44no—
182C23×C31—416no—
183C23×C31—44no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—88no—
187C23×C31—82no—
188C23×C31—82no—
189C23×C31—82no—
190C23×C31—82no—
191C23×C31#12011yescharacteristic
192C23×C31#12011yescharacteristic
193C23×C31C22×D411yescharacteristic
194Q8×C31—24no—
195Q8×C31—82no—
196D4×C31—24no—
197D4×C31—28no—
198D4×C31—48no—
199D4×C31—42no—
200D4×C31—42no—
201D4×C31—44no—
202D4×C31—44no—
203D4×C31—44no—
204D4×C31—82no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—24no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—48no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—44no—
219C2×C4×C31—44no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—82no—
223C2×C4×C31—84no—
224C2×C4×C31#12012yes—
225C2×C4×C31C22×D412yes—
226C2×C4×C31C2×#4112yes—
227C2×C4×C31H(5,𝔽2)12yes—
228C8×C31—44no—
229#3492C2×C3111yescharacteristic, complemented
230#3003C2×C3112yescomplemented
231C2×#390C2×C3111yescharacteristic, complemented
232#2027C2×C3112yescomplemented
233#2006C2×C3112yescomplemented
234#1896C2×C3112yescomplemented
235#1883C2×C3114yescomplemented
236#1821C2×C3111yescharacteristic, complemented
237C22×C31—24no—
238C22×C31—24no—
239C22×C31—21no—
240C22×C31—21no—
241C22×C31—21no—
242C22×C31—21no—
243C22×C31—21no—
244C22×C31—21no—
245C22×C31—48no—
246C22×C31—42no—
247C22×C31—44no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—84no—
251C22×C31—88no—
252C22×C31—88no—
253C22×C31—82no—
254C22×C31—82no—
255C22×C31#45312yes—
256C22×C31#45412yes—
257C22×C31C2×#12011yescharacteristic
258C22×C31#52511yescharacteristic
259C22×C31#53511yescharacteristic
260C4×C31—22no—
261C4×C31—44no—
262C4×C31—42no—
263C4×C31—42no—
264C4×C31—42no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31#54614yes—
268C23×D4C22×C3111yescharacteristic
269C2×AGL(1,ℤ/8)C22×C3114yes—
270#554—22no—
271#553—22no—
272#552—22no—
273#547—22no—
274#544C22×C3111yescharacteristic, complemented
275#539—22no—
276#537C22×C3111yescharacteristic, complemented
277#525—22no—
278#525C22×C3112yes—
279#520C22×C3112yescomplemented
280C22×C4⋊C4C22×C3111yescharacteristic
281C22×C22⋊C4—22no—
282C22×C42C22×C3111yescharacteristic
283#450—22no—
284#450—22no—
285#447—22no—
286#447—22no—
287C2×#104C22×C3114yescomplemented
288#418C22×C3114yes—
289#407C22×C3112yes—
290#390—22no—
291#390C22×C3112yes—
292#390C22×C3112yes—
293#343C22×C3112yescomplemented
294#343C22×C3112yescomplemented
295#339C22×C3114yescomplemented
296C2×C31—21no—
297C2×C31—21no—
298C2×C31—22no—
299C2×C31—22no—
300C2×C31—82no—
301C2×C31—84no—
302C2×C31#240411yescharacteristic
303C2×C31C2×#45311yescharacteristic
304C2×C31C2×#45411yescharacteristic
305C25—21no—
306C2×#41—22no—
307C2×#41C23×C3112yes—
308C22×D4—24no—
309C22×D4—21no—
310C22×D4—22no—
311C22×D4—22no—
312C22×D4—42no—
313C22×D4C23×C3112yes—
314C23×C4—21no—
315C23×C4D4×C3112yes—
316C23×C4C23×C3111yescharacteristic
317AGL(1,ℤ/8)—28no—
318AGL(1,ℤ/8)—44no—
319C2×SD16—24no—
320C2×D8—24no—
321C2×M16C23×C3114yes—
322C4⋊Q8—22no—
323#126—24no—
324#125—24no—
325#125—22no—
326#123—24no—
327#123—24no—
328#123—22no—
329C22⋊Q8—22no—
330C22⋊Q8—24no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8D4×C3112yes—
334C22⋊Q8D4×C3112yes—
335#121—22no—
336#121—22no—
337#121—22no—
338#121D4×C3112yes—
339#121D4×C3112yes—
340#120—22no—
341C4×Q8—22no—
342C4×D4—24no—
343C4×D4—22no—
344C4×D4—24no—
345C4×D4—22no—
346C4×D4—22no—
347#117—22no—
348#117—22no—
349#117C23×C3112yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4D4×C3112yes—
354C2×C4⋊C4C23×C3111yescharacteristic
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—48no—
359C2×C42—21no—
360C2×C42—42no—
361C2×C42C23×C3112yes—
362#104—44no—
363#102—24no—
364#102—24no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C31#2394811yescharacteristic, direct product factor, Sylow
368C24—24no—
369C24—22no—
370C24—44no—
371C24—42no—
372C24—44no—
373C24—41no—
374C24—41no—
375C24C2×D4×C3111yescharacteristic
376#41—24no—
377#41—24no—
378#41—84no—
379C2×Q8—22no—
380C2×Q8C2×D4×C3112yes—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C2×D4×C3112yes—
394C2×D4#41×C3114yescomplemented
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4C2×D4×C3111yescharacteristic
407C22×C4C2×D4×C3112yes—
408C22×C4C2×D4×C3112yes—
409C22×C4C2×D4×C3112yes—
410C22×C4C2×D4×C3112yes—
411C22×C4C24×C3111yescharacteristic
412SD16—48no—
413D8—48no—
414M16—24no—
415M16—24no—
416C2×C8—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4C2×D4×C3112yes—
427C4⋊C4C2×D4×C3112yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C42—24no—
438C42—22no—
439C42—24no—
440C42—42no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—22no—
446C23—24no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—416no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—88no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23C31×#12011yescharacteristic
460C23C31×#12011yescharacteristic
461C23C22×D4×C3111yescharacteristic
462Q8—24no—
463Q8—82no—
464D4—24no—
465D4—28no—
466D4—48no—
467D4—44no—
468D4—44no—
469D4—42no—
470D4—42no—
471D4—44no—
472D4—82no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—84no—
492C2×C4C31×#12012yes—
493C2×C4C22×D4×C3112yes—
494C2×C4C2×#41×C3112yes—
495C2×C4C31×H(5,𝔽2)12yes—
496C8—44no—
497C22—24no—
498C22—24no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—48no—
506C22—42no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—84no—
511C22—88no—
512C22—88no—
513C22—82no—
514C22—82no—
515C22C31×#45312yes—
516C22C31×#45412yes—
517C22C2×C31×#12011yescharacteristic
518C22C31×#52511yescharacteristic
519C22C31×#53511yescharacteristic
520C4—22no—
521C4—44no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4C31×#54614yes—
528C2—21no—
529C2—21no—
530C2—22no—
531C2—22no—
532C2—84no—
533C2—82no—
534C2C31×#240411yescharacteristic
535C2C2×C31×#45311yescharacteristic
536C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3492C2semidirect
2. C31×#3003C2semidirect
3. C2×C31×#390C2semidirect
4. C31×#2027C2semidirect
5. C31×#2006C2semidirect
6. C31×#1896C2semidirect
7. C31×#1883C2semidirect
8. C31×#1821C2semidirect
15. C31×#544C22semidirect
17. C31×#537C22semidirect
20. C31×#520C22semidirect
28. C2×C31×#104C22semidirect
34. C31×#343C22semidirect
35. C31×#343C22semidirect
36. C31×#339C22semidirect
125. C2×D4×C31#41semidirect
172. #23948C31direct (N × H)
229. #3492C2×C31semidirect
230. #3003C2×C31semidirect
231. C2×#390C2×C31semidirect
232. #2027C2×C31semidirect
233. #2006C2×C31semidirect
234. #1896C2×C31semidirect
235. #1883C2×C31semidirect
236. #1821C2×C31semidirect
274. #544C22×C31semidirect
276. #537C22×C31semidirect
279. #520C22×C31semidirect
287. C2×#104C22×C31semidirect
293. #343C22×C31semidirect
294. #343C22×C31semidirect
295. #339C22×C31semidirect
367. C31#23948direct (N × H)
394. C2×D4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34922
C31×#30032
C2×C31×#3902
C31×#20272
C31×#20062
C31×#18962
C31×#18832
C31×#18212
#2394831