Finite Group Databasebeta

← C31×#33559 #21490318

Subgroups — C31×#33559 #21490318

Order 7936.

1248 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3549C211yesmaximal, characteristic, complemented
2C31×#3027C212yesmaximal, complemented
3C31×#3018C211yesmaximal, characteristic, complemented
4C31×#2199C214yesmaximal, complemented
5C31×#2196C214yesmaximal, complemented
6C31×#2126C211yesmaximal, characteristic, complemented
7C31×#2122C212yesmaximal, complemented
8C31×#585C2211yescharacteristic, complemented
9C31×#575—21no—
10C31×H(3,𝔽4)—21no—
11C31×H(3,𝔽4)C2212yescomplemented
12C31×#555—21no—
13C31×#555—22no—
14D42×C31—22no—
15D42×C31C2212yescomplemented
16C31×#535—21no—
17C31×#535—22no—
18C31×#535—24no—
19C31×#535C2212yes—
20C31×#532—21no—
21C31×#532C2211yescharacteristic
22C31×B(4,𝔽2)—24no—
23C31×B(4,𝔽2)C2218yescomplemented
24C31×#456—21no—
25C31×#455C2212yescomplemented
26C31×#454—21no—
27C31×#453C2212yes—
28C2×C31×#104—21no—
29C31×#413C2211yescharacteristic, complemented
30C31×#410—22no—
31C31×#410C2212yescomplemented
32C31×#354C2212yescomplemented
33C31×#353C2212yescomplemented
34C31×AΣL(1,𝔽16)C2218yescomplemented
35C31×2−1+4—21no—
36C31×2−1+4—41no—
37C31×H(5,𝔽2)—22no—
38C31×H(5,𝔽2)—41no—
39C31×H(5,𝔽2)C2312yes—
40C2×#41×C31—22no—
41C2×#41×C31—22no—
42C2×#41×C31—21no—
43C2×#41×C31—42no—
44C2×#41×C31D411yescharacteristic, complemented
45C2×#41×C31C2311yescharacteristic
46C22×D4×C31—21no—
47C22×D4×C31—24no—
48C22×D4×C31D412yescomplemented
49C31×#137—22no—
50C31×#137—41no—
51C31×AGL(1,ℤ/8)—22no—
52C31×AGL(1,ℤ/8)—41no—
53C31×#135—42no—
54C2×M16×C31—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8D411yescharacteristic, complemented
57C31×#127—21no—
58C31×#127—22no—
59C31×#127—22no—
60C31×#127—42no—
61C31×#127C2311yescharacteristic
62C31×#124—24no—
63C31×#124—22no—
64C31×#124—22no—
65C31×#124—24no—
66C31×#124—22no—
67C31×#124—42no—
68C31×#124D411yescharacteristic, complemented
69C31×#124C2311yescharacteristic
70C31×#121—22no—
71C31×#121—22no—
72C31×#121—24no—
73C31×#121—22no—
74C31×#121—22no—
75C31×#121—22no—
76C31×#121—24no—
77C31×#121—44no—
78C31×#120—22no—
79C31×#120—24no—
80C31×#120—24no—
81C31×#120—22no—
82C31×#120—28no—
83C31×#120D414yescomplemented
84C31×#120C2314yes—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C31×#117—21no—
89C31×#117—24no—
90C31×#117—22no—
91C31×#117D412yescomplemented
92C2×C42×C31D411yescharacteristic, complemented
93C31×#104—22no—
94C31×#104—22no—
95C31×#101—21no—
96C31×#100C2312yes—
97C31×#99—22no—
98C31×#99—28no—
99C31×#99C2314yes—
100C24×C31—22no—
101C24×C31C2×D414yes—
102#41×C31—22no—
103#41×C31—22no—
104#41×C31—41no—
105#41×C31—41no—
106#41×C31—42no—
107#41×C31—42no—
108#41×C31—42no—
109#41×C31—41no—
110#41×C31—44no—
111#41×C31—41no—
112#41×C31—41no—
113#41×C31—42no—
114#41×C31—42no—
115#41×C31—41no—
116#41×C31—84no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—21no—
120C2×Q8×C31—42no—
121C2×Q8×C31C2×D411yescharacteristic
122C2×Q8×C31C2×D411yescharacteristic
123C2×D4×C31—21no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—24no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—24no—
131C2×D4×C31—21no—
132C2×D4×C31—24no—
133C2×D4×C31—24no—
134C2×D4×C31—22no—
135C2×D4×C31—22no—
136C2×D4×C31—42no—
137C2×D4×C31—42no—
138C2×D4×C31—42no—
139C2×D4×C31—44no—
140C2×D4×C31—41no—
141C2×D4×C31—48no—
142C2×D4×C31—88no—
143C2×D4×C31—81no—
144C2×D4×C31—81no—
145C2×D4×C31C2×D412yes—
146C2×D4×C31C2×D411yescharacteristic
147C2×D4×C31C2411yescharacteristic
148C22×C4×C31—24no—
149C22×C4×C31—24no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—41no—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D411yescharacteristic
155Q16×C31—42no—
156SD16×C31—42no—
157SD16×C31—42no—
158D8×C31—42no—
159M16×C31—22no—
160M16×C31—41no—
161C2×C8×C31—41no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—42no—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—48no—
175C22⋊C4×C31C2×D414yes—
176C42×C31—22no—
177C42×C31—22no—
178C42×C31—21no—
179C42×C31—42no—
180C42×C31C2×D411yescharacteristic
181C42×C31C2×D411yescharacteristic
182#33559C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—21no—
186C23×C31—41no—
187C23×C31—44no—
188C23×C31—84no—
189C23×C31—84no—
190C23×C31—82no—
191C23×C31#12011yescharacteristic
192C23×C31C22×D412yes—
193Q8×C31—22no—
194Q8×C31—41no—
195Q8×C31—41no—
196Q8×C31—42no—
197Q8×C31—41no—
198Q8×C31—82no—
199D4×C31—22no—
200D4×C31—42no—
201D4×C31—41no—
202D4×C31—41no—
203D4×C31—42no—
204D4×C31—44no—
205D4×C31—44no—
206D4×C31—44no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—41no—
210D4×C31—42no—
211D4×C31—42no—
212D4×C31—82no—
213D4×C31—81no—
214D4×C31—84no—
215D4×C31—84no—
216D4×C31—82no—
217D4×C31—84no—
218D4×C31—82no—
219D4×C31—168no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—24no—
226C2×C4×C31—21no—
227C2×C4×C31—24no—
228C2×C4×C31—21no—
229C2×C4×C31—44no—
230C2×C4×C31—48no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—44no—
238C2×C4×C31#12012yes—
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31C22×D411yescharacteristic
241C8×C31—42no—
242#3549C2×C3111yescharacteristic, complemented
243#3027C2×C3112yescomplemented
244#3018C2×C3111yescharacteristic, complemented
245#2199C2×C3114yescomplemented
246#2196C2×C3114yescomplemented
247#2126C2×C3111yescharacteristic, complemented
248#2122C2×C3112yescomplemented
249C22×C31—22no—
250C22×C31—21no—
251C22×C31—41no—
252C22×C31—42no—
253C22×C31—44no—
254C22×C31—44no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—41no—
258C22×C31—44no—
259C22×C31—84no—
260C22×C31—88no—
261C22×C31—161no—
262C22×C31—164no—
263C22×C31C2×#12011yescharacteristic
264C4×C31—22no—
265C4×C31—21no—
266C4×C31—22no—
267C4×C31—44no—
268C4×C31—41no—
269C4×C31—42no—
270C4×C31—41no—
271C4×C31—84no—
272C4×C31B(4,𝔽2)11yescharacteristic
273C4×C31B(4,𝔽2)11yescharacteristic
274#585C22×C3111yescharacteristic, complemented
275#575—21no—
276H(3,𝔽4)—21no—
277H(3,𝔽4)C22×C3112yescomplemented
278#555—21no—
279#555—22no—
280D42—22no—
281D42C22×C3112yescomplemented
282#535—21no—
283#535—22no—
284#535—24no—
285#535C22×C3112yes—
286#532—21no—
287#532C22×C3111yescharacteristic
288B(4,𝔽2)—24no—
289B(4,𝔽2)C22×C3118yescomplemented
290#456—21no—
291#455C22×C3112yescomplemented
292#454—21no—
293#453C22×C3112yes—
294C2×#104—21no—
295#413C22×C3111yescharacteristic, complemented
296#410—22no—
297#410C22×C3112yescomplemented
298#354C22×C3112yescomplemented
299#353C22×C3112yescomplemented
300AΣL(1,𝔽16)C22×C3118yescomplemented
301C2×C31—21no—
302C2×C31—42no—
303C2×C31—41no—
304C2×C31—84no—
305C2×C31—81no—
306C2×C31—82no—
307C2×C31C2×B(4,𝔽2)11yescharacteristic
3082−1+4—21no—
3092−1+4—41no—
310H(5,𝔽2)—22no—
311H(5,𝔽2)—41no—
312H(5,𝔽2)C23×C3112yes—
313C2×#41—22no—
314C2×#41—22no—
315C2×#41—21no—
316C2×#41—42no—
317C2×#41D4×C3111yescharacteristic, complemented
318C2×#41C23×C3111yescharacteristic
319C22×D4—21no—
320C22×D4—24no—
321C22×D4D4×C3112yescomplemented
322#137—22no—
323#137—41no—
324AGL(1,ℤ/8)—22no—
325AGL(1,ℤ/8)—41no—
326#135—42no—
327C2×M16—21no—
328C4⋊Q8—21no—
329C4⋊Q8D4×C3111yescharacteristic, complemented
330#127—21no—
331#127—22no—
332#127—22no—
333#127—42no—
334#127C23×C3111yescharacteristic
335#124—24no—
336#124—22no—
337#124—22no—
338#124—24no—
339#124—22no—
340#124—42no—
341#124D4×C3111yescharacteristic, complemented
342#124C23×C3111yescharacteristic
343#121—22no—
344#121—22no—
345#121—24no—
346#121—22no—
347#121—22no—
348#121—22no—
349#121—24no—
350#121—44no—
351#120—22no—
352#120—24no—
353#120—24no—
354#120—22no—
355#120—28no—
356#120D4×C3114yescomplemented
357#120C23×C3114yes—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361#117—21no—
362#117—24no—
363#117—22no—
364#117D4×C3112yescomplemented
365C2×C42D4×C3111yescharacteristic, complemented
366#104—22no—
367#104—22no—
368#101—21no—
369#100C23×C3112yes—
370#99—22no—
371#99—28no—
372#99C23×C3114yes—
373C31#3355911yescharacteristic, direct product factor, Sylow
374C24—22no—
375C24C2×D4×C3114yes—
376#41—22no—
377#41—22no—
378#41—41no—
379#41—41no—
380#41—42no—
381#41—42no—
382#41—42no—
383#41—41no—
384#41—44no—
385#41—41no—
386#41—41no—
387#41—42no—
388#41—41no—
389#41—42no—
390#41—84no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—42no—
395C2×Q8C2×D4×C3111yescharacteristic
396C2×Q8C2×D4×C3111yescharacteristic
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—41no—
415C2×D4—48no—
416C2×D4—88no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4C2×D4×C3112yes—
420C2×D4C2×D4×C3111yescharacteristic
421C2×D4C24×C3111yescharacteristic
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4C2×D4×C3112yes—
428C22×C4C2×D4×C3111yescharacteristic
429Q16—42no—
430SD16—42no—
431SD16—42no—
432D8—42no—
433M16—22no—
434M16—41no—
435C2×C8—41no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—42no—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—48no—
449C22⋊C4C2×D4×C3114yes—
450C42—22no—
451C42—21no—
452C42—22no—
453C42—42no—
454C42C2×D4×C3111yescharacteristic
455C42C2×D4×C3111yescharacteristic
456C23—24no—
457C23—24no—
458C23—21no—
459C23—41no—
460C23—44no—
461C23—84no—
462C23—84no—
463C23—82no—
464C23C31×#12011yescharacteristic
465C23C22×D4×C3112yes—
466Q8—22no—
467Q8—41no—
468Q8—41no—
469Q8—42no—
470Q8—41no—
471Q8—82no—
472D4—22no—
473D4—42no—
474D4—41no—
475D4—41no—
476D4—42no—
477D4—44no—
478D4—44no—
479D4—44no—
480D4—42no—
481D4—42no—
482D4—41no—
483D4—42no—
484D4—42no—
485D4—82no—
486D4—81no—
487D4—84no—
488D4—84no—
489D4—82no—
490D4—84no—
491D4—82no—
492D4—168no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4C31×#12012yes—
512C2×C4C31×#12011yescharacteristic
513C2×C4C22×D4×C3111yescharacteristic
514C8—42no—
515C22—22no—
516C22—21no—
517C22—41no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—41no—
524C22—44no—
525C22—84no—
526C22—88no—
527C22—161no—
528C22—164no—
529C22C2×C31×#12011yescharacteristic
530C4—22no—
531C4—21no—
532C4—22no—
533C4—44no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—84no—
538C4C31×B(4,𝔽2)11yescharacteristic
539C4C31×B(4,𝔽2)11yescharacteristic
540C2—21no—
541C2—42no—
542C2—41no—
543C2—84no—
544C2—81no—
545C2—82no—
546C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C31C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C31
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3549C2semidirect
2. C31×#3027C2semidirect
3. C31×#3018C2semidirect
4. C31×#2199C2semidirect
5. C31×#2196C2semidirect
6. C31×#2126C2semidirect
7. C31×#2122C2semidirect
8. C31×#585C22semidirect
11. C31×H(3,𝔽4)C22semidirect
15. D42×C31C22semidirect
23. C31×B(4,𝔽2)C22semidirect
25. C31×#455C22semidirect
29. C31×#413C22semidirect
31. C31×#410C22semidirect
32. C31×#354C22semidirect
33. C31×#353C22semidirect
34. C31×AΣL(1,𝔽16)C22semidirect
44. C2×#41×C31D4semidirect
48. C22×D4×C31D4semidirect
56. C31×C4⋊Q8D4semidirect
68. C31×#124D4semidirect
83. C31×#120D4semidirect
91. C31×#117D4semidirect
92. C2×C42×C31D4semidirect
182. #33559C31direct (N × H)
242. #3549C2×C31semidirect
243. #3027C2×C31semidirect
244. #3018C2×C31semidirect
245. #2199C2×C31semidirect
246. #2196C2×C31semidirect
247. #2126C2×C31semidirect
248. #2122C2×C31semidirect
274. #585C22×C31semidirect
277. H(3,𝔽4)C22×C31semidirect
281. D42C22×C31semidirect
289. B(4,𝔽2)C22×C31semidirect
291. #455C22×C31semidirect
295. #413C22×C31semidirect
297. #410C22×C31semidirect
298. #354C22×C31semidirect
299. #353C22×C31semidirect
300. AΣL(1,𝔽16)C22×C31semidirect
317. C2×#41D4×C31semidirect
321. C22×D4D4×C31semidirect
329. C4⋊Q8D4×C31semidirect
341. #124D4×C31semidirect
356. #120D4×C31semidirect
364. #117D4×C31semidirect
365. C2×C42D4×C31semidirect
373. C31#33559direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35492
C31×#30272
C31×#30182
C31×#21992
C31×#21962
C31×#21262
C31×#21222
#3355931