Finite Group Databasebeta

← C31×#18316 #21475075

Subgroups — C31×#18316 #21475075

Order 7936.

1516 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#534C211yesmaximal, characteristic, complemented
2C31×#3192C212yesmaximal, complemented
3C31×#2539C211yesmaximal, characteristic, complemented
4C31×#1613C214yesmaximal, complemented
5C31×#1595C214yesmaximal, complemented
6C31×#1507C212yesmaximal, complemented
7C31×#1462C211yesmaximal, characteristic, complemented
8C23×D4×C31—21no—
9C31×#534—24no—
10C31×#534—22no—
11C31×#534C2214yes—
12C2×C31×#123—21no—
13C2×C31×#123—22no—
14C2×C31×#121—21no—
15C2×C31×#121C2212yes—
16C2×C31×#121C2211yescharacteristic, complemented
17C2×C31×#120—22no—
18C2×C31×#120—21no—
19C22×C22⋊C4×C31C2211yescharacteristic
20C31×#481—22no—
21C31×#481C2214yescomplemented
22C31×#469—24no—
23C31×#447—22no—
24C31×#447C2214yescomplemented
25C31×#407C2212yescomplemented
26C31×#399C2212yescomplemented
27C31×#392C2212yescomplemented
28C31×#390—22no—
29C4×C22⋊C4×C31C2211yescharacteristic, complemented
30C31×#327C2218yescomplemented
31C31×#323C2214yescomplemented
32C25×C31—21no—
33C25×C31D411yescharacteristic
34C22×D4×C31—22no—
35C22×D4×C31—24no—
36C22×D4×C31—22no—
37C22×D4×C31—24no—
38C22×D4×C31—21no—
39C22×D4×C31—21no—
40C22×D4×C31—22no—
41C22×D4×C31—48no—
42C22×D4×C31D412yescomplemented
43C22×D4×C31C2311yescharacteristic, complemented
44C23×C4×C31—21no—
45C23×C4×C31D411yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C2×D8×C31—24no—
48C2×M16×C31—22no—
49C31×#123—28no—
50C31×#123—28no—
51C31×#123—44no—
52C31×#121—24no—
53C31×#121—24no—
54C31×#121—22no—
55C31×#121—22no—
56C31×#121—22no—
57C31×#121—22no—
58C31×#121D414yescomplemented
59C31×#121C2314yes—
60C31×#120—28no—
61C31×#120—28no—
62C31×#120—44no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31D412yescomplemented
66C2×C4⋊C4×C31C2311yescharacteristic, complemented
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—21no—
74C2×C22⋊C4×C31—22no—
75C2×C22⋊C4×C31D412yes—
76C2×C22⋊C4×C31C2312yes—
77C2×C42×C31—21no—
78C31×#106—24no—
79C31×#102—24no—
80C31×#102—24no—
81C31×C22⋊C8—22no—
82C31×C22⋊C8C2314yescomplemented
83C31×#95C2312yescomplemented
84C24×C31—21no—
85C24×C31—21no—
86C24×C31—24no—
87C24×C31—24no—
88C24×C31—22no—
89C24×C31—21no—
90C24×C31—44no—
91C24×C31—42no—
92C24×C31—81no—
93C24×C31—82no—
94C24×C31—82no—
95C24×C31C2×D411yescharacteristic
96C24×C31C2×D412yes—
97C2×D4×C31—28no—
98C2×D4×C31—24no—
99C2×D4×C31—24no—
100C2×D4×C31—24no—
101C2×D4×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31—44no—
104C2×D4×C31—48no—
105C2×D4×C31—416no—
106C2×D4×C31—48no—
107C2×D4×C31—48no—
108C2×D4×C31—48no—
109C2×D4×C31—42no—
110C2×D4×C31—42no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—44no—
114C2×D4×C31—81no—
115C2×D4×C31—82no—
116C2×D4×C31—82no—
117C2×D4×C31—82no—
118C2×D4×C31C2×D414yes—
119C22×C4×C31—24no—
120C22×C4×C31—21no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—44no—
126C22×C4×C31—42no—
127C22×C4×C31—81no—
128C22×C4×C31C2×D412yes—
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D412yes—
131C22×C4×C31C2×D412yes—
132C22×C4×C31C2411yescharacteristic
133D8×C31—84no—
134M16×C31—82no—
135C2×C8×C31—24no—
136C4⋊C4×C31—24no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—44no—
140C4⋊C4×C31C2×D414yes—
141C22⋊C4×C31—28no—
142C22⋊C4×C31—24no—
143C22⋊C4×C31—24no—
144C22⋊C4×C31—24no—
145C22⋊C4×C31—24no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—44no—
148C22⋊C4×C31—44no—
149C22⋊C4×C31—42no—
150C42×C31—81no—
151#18316C3111yesmaximal, characteristic, direct product factor, Sylow
152C23×C31—24no—
153C23×C31—24no—
154C23×C31—22no—
155C23×C31—22no—
156C23×C31—22no—
157C23×C31—22no—
158C23×C31—24no—
159C23×C31—41no—
160C23×C31—42no—
161C23×C31—42no—
162C23×C31—42no—
163C23×C31—44no—
164C23×C31—48no—
165C23×C31—44no—
166C23×C31—41no—
167C23×C31—41no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—44no—
171C23×C31—41no—
172C23×C31—42no—
173C23×C31—81no—
174C23×C31—81no—
175C23×C31—88no—
176C23×C31—84no—
177C23×C31—84no—
178C23×C31—816no—
179C23×C31#12011yescharacteristic
180C23×C31#12011yescharacteristic
181C23×C31AGL(1,ℤ/8)14yes—
182C23×C31C22×D411yescharacteristic
183D4×C31—44no—
184D4×C31—44no—
185D4×C31—416no—
186D4×C31—48no—
187D4×C31—48no—
188D4×C31—82no—
189D4×C31—84no—
190D4×C31—88no—
191D4×C31—82no—
192D4×C31—82no—
193C2×C4×C31—24no—
194C2×C4×C31—22no—
195C2×C4×C31—24no—
196C2×C4×C31—41no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—41no—
200C2×C4×C31—42no—
201C2×C4×C31—44no—
202C2×C4×C31—48no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—42no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—81no—
210C2×C4×C31#12012yes—
211C2×C4×C31C22×D412yes—
212C8×C31—44no—
213C2×#534C2×C3111yescharacteristic, complemented
214#3192C2×C3112yescomplemented
215#2539C2×C3111yescharacteristic, complemented
216#1613C2×C3114yescomplemented
217#1595C2×C3114yescomplemented
218#1507C2×C3112yescomplemented
219#1462C2×C3111yescharacteristic, complemented
220C22×C31—21no—
221C22×C31—21no—
222C22×C31—21no—
223C22×C31—21no—
224C22×C31—24no—
225C22×C31—21no—
226C22×C31—21no—
227C22×C31—21no—
228C22×C31—44no—
229C22×C31—44no—
230C22×C31—48no—
231C22×C31—42no—
232C22×C31—42no—
233C22×C31—44no—
234C22×C31—44no—
235C22×C31—44no—
236C22×C31—82no—
237C22×C31—88no—
238C22×C31—82no—
239C22×C31—82no—
240C22×C31—84no—
241C22×C31—816no—
242C22×C31—81no—
243C22×C31—81no—
244C22×C31—81no—
245C22×C31#45312yes—
246C22×C31C2×#12011yescharacteristic
247C22×C31C2×AGL(1,ℤ/8)12yes—
248C4×C31—22no—
249C4×C31—22no—
250C4×C31—44no—
251C4×C31—44no—
252C4×C31—42no—
253C4×C31—81no—
254C23×D4—21no—
255#534—24no—
256#534—22no—
257#534C22×C3114yes—
258C2×#123—21no—
259C2×#123—22no—
260C2×#121—21no—
261C2×#121C22×C3112yes—
262C2×#121C22×C3111yescharacteristic, complemented
263C2×#120—22no—
264C2×#120—21no—
265C22×C22⋊C4C22×C3111yescharacteristic
266#481—22no—
267#481C22×C3114yescomplemented
268#469—24no—
269#447—22no—
270#447C22×C3114yescomplemented
271#407C22×C3112yescomplemented
272#399C22×C3112yescomplemented
273#392C22×C3112yescomplemented
274#390—22no—
275C4×C22⋊C4C22×C3111yescharacteristic, complemented
276#327C22×C3118yescomplemented
277#323C22×C3114yescomplemented
278C2×C31—22no—
279C2×C31—22no—
280C2×C31—41no—
281C2×C31—44no—
282C2×C31—84no—
283C2×C31—161no—
284C2×C31#300311yescharacteristic
285C2×C31C2×#45311yescharacteristic
286C2×C31#302611yescharacteristic
287C25—21no—
288C25D4×C3111yescharacteristic
289C22×D4—22no—
290C22×D4—24no—
291C22×D4—22no—
292C22×D4—24no—
293C22×D4—21no—
294C22×D4—21no—
295C22×D4—22no—
296C22×D4—48no—
297C22×D4D4×C3112yescomplemented
298C22×D4C23×C3111yescharacteristic, complemented
299C23×C4—21no—
300C23×C4D4×C3111yescharacteristic
301C23×C4C23×C3111yescharacteristic
302C2×D8—24no—
303C2×M16—22no—
304#123—28no—
305#123—28no—
306#123—44no—
307#121—24no—
308#121—24no—
309#121—22no—
310#121—22no—
311#121—22no—
312#121—22no—
313#121D4×C3114yescomplemented
314#121C23×C3114yes—
315#120—28no—
316#120—28no—
317#120—44no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4D4×C3112yescomplemented
321C2×C4⋊C4C23×C3111yescharacteristic, complemented
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4D4×C3112yes—
331C2×C22⋊C4C23×C3112yes—
332C2×C42—21no—
333#106—24no—
334#102—24no—
335#102—24no—
336C22⋊C8—22no—
337C22⋊C8C23×C3114yescomplemented
338#95C23×C3112yescomplemented
339C31#1831611yescharacteristic, direct product factor, Sylow
340C24—21no—
341C24—21no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—21no—
346C24—44no—
347C24—42no—
348C24—81no—
349C24—82no—
350C24—82no—
351C24C2×D4×C3111yescharacteristic
352C24C2×D4×C3112yes—
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—416no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—44no—
370C2×D4—81no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4C2×D4×C3114yes—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—81no—
384C22×C4C2×D4×C3112yes—
385C22×C4C2×D4×C3111yescharacteristic
386C22×C4C2×D4×C3112yes—
387C22×C4C2×D4×C3112yes—
388C22×C4C24×C3111yescharacteristic
389D8—84no—
390M16—82no—
391C2×C8—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—44no—
396C4⋊C4C2×D4×C3114yes—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—42no—
406C42—81no—
407C23—24no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—48no—
420C23—44no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—42no—
425C23—44no—
426C23—41no—
427C23—42no—
428C23—81no—
429C23—81no—
430C23—88no—
431C23—84no—
432C23—84no—
433C23—816no—
434C23C31×#12011yescharacteristic
435C23C31×#12011yescharacteristic
436C23C31×AGL(1,ℤ/8)14yes—
437C23C22×D4×C3111yescharacteristic
438D4—44no—
439D4—44no—
440D4—416no—
441D4—48no—
442D4—48no—
443D4—82no—
444D4—84no—
445D4—88no—
446D4—82no—
447D4—82no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—81no—
465C2×C4C31×#12012yes—
466C2×C4C22×D4×C3112yes—
467C8—44no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—24no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—44no—
477C22—44no—
478C22—48no—
479C22—42no—
480C22—42no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—82no—
485C22—88no—
486C22—82no—
487C22—82no—
488C22—84no—
489C22—816no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22C31×#45312yes—
494C22C2×C31×#12011yescharacteristic
495C22C2×C31×AGL(1,ℤ/8)12yes—
496C4—22no—
497C4—22no—
498C4—44no—
499C4—44no—
500C4—42no—
501C4—81no—
502C2—22no—
503C2—22no—
504C2—41no—
505C2—44no—
506C2—84no—
507C2—161no—
508C2C31×#300311yescharacteristic
509C2C2×C31×#45311yescharacteristic
510C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#534C2semidirect
2. C31×#3192C2semidirect
3. C31×#2539C2semidirect
4. C31×#1613C2semidirect
5. C31×#1595C2semidirect
6. C31×#1507C2semidirect
7. C31×#1462C2semidirect
16. C2×C31×#121C22semidirect
21. C31×#481C22semidirect
24. C31×#447C22semidirect
25. C31×#407C22semidirect
26. C31×#399C22semidirect
27. C31×#392C22semidirect
29. C4×C22⋊C4×C31C22semidirect
30. C31×#327C22semidirect
31. C31×#323C22semidirect
42. C22×D4×C31D4semidirect
43. C22×D4×C31C23semidirect
58. C31×#121D4semidirect
65. C2×C4⋊C4×C31D4semidirect
66. C2×C4⋊C4×C31C23semidirect
82. C31×C22⋊C8C23semidirect
83. C31×#95C23semidirect
151. #18316C31direct (N × H)
213. C2×#534C2×C31semidirect
214. #3192C2×C31semidirect
215. #2539C2×C31semidirect
216. #1613C2×C31semidirect
217. #1595C2×C31semidirect
218. #1507C2×C31semidirect
219. #1462C2×C31semidirect
262. C2×#121C22×C31semidirect
267. #481C22×C31semidirect
270. #447C22×C31semidirect
271. #407C22×C31semidirect
272. #399C22×C31semidirect
273. #392C22×C31semidirect
275. C4×C22⋊C4C22×C31semidirect
276. #327C22×C31semidirect
277. #323C22×C31semidirect
297. C22×D4D4×C31semidirect
298. C22×D4C23×C31semidirect
313. #121D4×C31semidirect
320. C2×C4⋊C4D4×C31semidirect
321. C2×C4⋊C4C23×C31semidirect
337. C22⋊C8C23×C31semidirect
338. #95C23×C31semidirect
339. C31#18316direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5342
C31×#31922
C31×#25392
C31×#16132
C31×#15952
C31×#15072
C31×#14622
#1831631