Finite Group Databasebeta

← C31×#18132 #21474891

Subgroups — C31×#18132 #21474891

Order 7936.

1654 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#534C211yesmaximal, characteristic, complemented
2C31×#3006C212yesmaximal, complemented
3C31×#2404C211yesmaximal, characteristic, complemented
4C31×#1601C214yesmaximal, complemented
5C31×#1595C214yesmaximal, complemented
6C31×#1510C212yesmaximal, complemented
7C31×#1462C211yesmaximal, characteristic, complemented
8C22×#41×C31C2211yescharacteristic, complemented
9C23×D4×C31—21no—
10C2×C31×AGL(1,ℤ/8)—22no—
11C31×#534—24no—
12C31×#534—22no—
13C31×#534C2214yes—
14C2×C31×#123—22no—
15C2×C31×#121—21no—
16C2×C31×#121C2212yes—
17C2×C31×#120—22no—
18C2×C31×#120—21no—
19C22×C22⋊C4×C31C2211yescharacteristic
20C31×#449—24no—
21C31×#448C2214yescomplemented
22C31×#447C2214yescomplemented
23C31×#417—22no—
24C31×#407C2212yescomplemented
25C31×#393C2212yescomplemented
26C31×#392C2212yescomplemented
27C31×#386—21no—
28C31×#386—22no—
29C31×#381C2211yescharacteristic, complemented
30C31×#327C2218yescomplemented
31C31×#323C2214yescomplemented
32C25×C31—21no—
33C25×C31D411yescharacteristic
34C2×#41×C31—22no—
35C2×#41×C31—22no—
36C2×#41×C31—44no—
37C22×Q8×C31C2311yescharacteristic, complemented
38C22×D4×C31—22no—
39C22×D4×C31—24no—
40C22×D4×C31—21no—
41C22×D4×C31—22no—
42C22×D4×C31—24no—
43C22×D4×C31—21no—
44C22×D4×C31—22no—
45C22×D4×C31—48no—
46C22×D4×C31D412yescomplemented
47C22×D4×C31C2311yescharacteristic, complemented
48C23×C4×C31—21no—
49C23×C4×C31—21no—
50C23×C4×C31D411yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C31×AGL(1,ℤ/8)—48no—
53C2×SD16×C31—24no—
54C2×D8×C31—24no—
55C2×M16×C31—22no—
56C31×#123—28no—
57C31×#123—28no—
58C31×#121—24no—
59C31×#121—24no—
60C31×#121—22no—
61C31×#121—22no—
62C31×#121D414yescomplemented
63C31×#121C2314yes—
64C31×#120—28no—
65C31×#120—28no—
66C31×#120—44no—
67C2×C4⋊C4×C31—21no—
68C2×C4⋊C4×C31D412yescomplemented
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×C31—22no—
76C2×C22⋊C4×C31D412yes—
77C2×C22⋊C4×C31C2312yes—
78C31×#103—24no—
79C31×#102—24no—
80C31×C22⋊C8—22no—
81C31×C22⋊C8C2314yescomplemented
82C31×#95—21no—
83C31×#95—22no—
84C31×#95C2312yescomplemented
85C24×C31—21no—
86C24×C31—21no—
87C24×C31—24no—
88C24×C31—24no—
89C24×C31—22no—
90C24×C31—21no—
91C24×C31—44no—
92C24×C31—42no—
93C24×C31—81no—
94C24×C31—82no—
95C24×C31—82no—
96C24×C31C2×D411yescharacteristic
97C24×C31C2×D412yes—
98#41×C31—28no—
99#41×C31—44no—
100#41×C31—44no—
101#41×C31—44no—
102C2×Q8×C31—22no—
103C2×Q8×C31—42no—
104C2×D4×C31—28no—
105C2×D4×C31—24no—
106C2×D4×C31—24no—
107C2×D4×C31—24no—
108C2×D4×C31—24no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—44no—
112C2×D4×C31—42no—
113C2×D4×C31—48no—
114C2×D4×C31—416no—
115C2×D4×C31—48no—
116C2×D4×C31—48no—
117C2×D4×C31—42no—
118C2×D4×C31—41no—
119C2×D4×C31—41no—
120C2×D4×C31—48no—
121C2×D4×C31—44no—
122C2×D4×C31—44no—
123C2×D4×C31—82no—
124C2×D4×C31—82no—
125C2×D4×C31C2×D414yes—
126C22×C4×C31—24no—
127C22×C4×C31—21no—
128C22×C4×C31—22no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
132C22×C4×C31—42no—
133C22×C4×C31—42no—
134C22×C4×C31—44no—
135C22×C4×C31—42no—
136C22×C4×C31—42no—
137C22×C4×C31—81no—
138C22×C4×C31C2×D412yes—
139C22×C4×C31C2×D412yes—
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2411yescharacteristic
143SD16×C31—48no—
144D8×C31—48no—
145M16×C31—44no—
146C2×C8×C31—24no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31C2×D414yes—
150C22⋊C4×C31—28no—
151C22⋊C4×C31—24no—
152C22⋊C4×C31—24no—
153C22⋊C4×C31—24no—
154C22⋊C4×C31—24no—
155C22⋊C4×C31—44no—
156C22⋊C4×C31—42no—
157C22⋊C4×C31—82no—
158#18132C3111yesmaximal, characteristic, direct product factor, Sylow
159C23×C31—24no—
160C23×C31—24no—
161C23×C31—22no—
162C23×C31—22no—
163C23×C31—22no—
164C23×C31—22no—
165C23×C31—22no—
166C23×C31—24no—
167C23×C31—41no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—42no—
171C23×C31—42no—
172C23×C31—41no—
173C23×C31—44no—
174C23×C31—48no—
175C23×C31—44no—
176C23×C31—42no—
177C23×C31—41no—
178C23×C31—42no—
179C23×C31—44no—
180C23×C31—81no—
181C23×C31—81no—
182C23×C31—88no—
183C23×C31—84no—
184C23×C31—84no—
185C23×C31—816no—
186C23×C31#12011yescharacteristic
187C23×C31#12011yescharacteristic
188C23×C31AGL(1,ℤ/8)14yescomplemented
189C23×C31C22×D411yescharacteristic
190Q8×C31—24no—
191Q8×C31—42no—
192D4×C31—24no—
193D4×C31—44no—
194D4×C31—42no—
195D4×C31—416no—
196D4×C31—48no—
197D4×C31—44no—
198D4×C31—48no—
199D4×C31—44no—
200D4×C31—82no—
201D4×C31—84no—
202D4×C31—88no—
203C2×C4×C31—24no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—24no—
207C2×C4×C31—41no—
208C2×C4×C31—42no—
209C2×C4×C31—41no—
210C2×C4×C31—48no—
211C2×C4×C31—44no—
212C2×C4×C31—48no—
213C2×C4×C31—41no—
214C2×C4×C31—42no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—81no—
221C2×C4×C31#12012yes—
222C2×C4×C31C22×D412yes—
223C8×C31—44no—
224C2×#534C2×C3111yescharacteristic, complemented
225#3006C2×C3112yescomplemented
226#2404C2×C3111yescharacteristic, complemented
227#1601C2×C3114yescomplemented
228#1595C2×C3114yescomplemented
229#1510C2×C3112yescomplemented
230#1462C2×C3111yescharacteristic, complemented
231C22×C31—21no—
232C22×C31—21no—
233C22×C31—21no—
234C22×C31—21no—
235C22×C31—24no—
236C22×C31—21no—
237C22×C31—21no—
238C22×C31—21no—
239C22×C31—44no—
240C22×C31—44no—
241C22×C31—48no—
242C22×C31—44no—
243C22×C31—42no—
244C22×C31—42no—
245C22×C31—44no—
246C22×C31—44no—
247C22×C31—42no—
248C22×C31—44no—
249C22×C31—82no—
250C22×C31—82no—
251C22×C31—88no—
252C22×C31—84no—
253C22×C31—816no—
254C22×C31—81no—
255C22×C31—81no—
256C22×C31B(4,𝔽2)12yes—
257C22×C31C2×#12011yescharacteristic
258C22×C31C2×AGL(1,ℤ/8)12yes—
259C4×C31—22no—
260C4×C31—22no—
261C4×C31—44no—
262C4×C31—42no—
263C4×C31—42no—
264C4×C31—82no—
265C22×#41C22×C3111yescharacteristic, complemented
266C23×D4—21no—
267C2×AGL(1,ℤ/8)—22no—
268#534—24no—
269#534—22no—
270#534C22×C3114yes—
271C2×#123—22no—
272C2×#121—21no—
273C2×#121C22×C3112yes—
274C2×#120—22no—
275C2×#120—21no—
276C22×C22⋊C4C22×C3111yescharacteristic
277#449—24no—
278#448C22×C3114yescomplemented
279#447C22×C3114yescomplemented
280#417—22no—
281#407C22×C3112yescomplemented
282#393C22×C3112yescomplemented
283#392C22×C3112yescomplemented
284#386—21no—
285#386—22no—
286#381C22×C3111yescharacteristic, complemented
287#327C22×C3118yescomplemented
288#323C22×C3114yescomplemented
289C2×C31—22no—
290C2×C31—22no—
291C2×C31—41no—
292C2×C31—44no—
293C2×C31—82no—
294C2×C31—84no—
295C2×C31#300311yescharacteristic
296C2×C31#301911yescharacteristic
297C2×C31C2×B(4,𝔽2)11yescharacteristic
298C25—21no—
299C25D4×C3111yescharacteristic
300C2×#41—22no—
301C2×#41—22no—
302C2×#41—44no—
303C22×Q8C23×C3111yescharacteristic, complemented
304C22×D4—22no—
305C22×D4—24no—
306C22×D4—21no—
307C22×D4—22no—
308C22×D4—24no—
309C22×D4—21no—
310C22×D4—22no—
311C22×D4—48no—
312C22×D4D4×C3112yescomplemented
313C22×D4C23×C3111yescharacteristic, complemented
314C23×C4—21no—
315C23×C4—21no—
316C23×C4D4×C3111yescharacteristic
317C23×C4C23×C3111yescharacteristic
318AGL(1,ℤ/8)—48no—
319C2×SD16—24no—
320C2×D8—24no—
321C2×M16—22no—
322#123—28no—
323#123—28no—
324#121—24no—
325#121—24no—
326#121—22no—
327#121—22no—
328#121D4×C3114yescomplemented
329#121C23×C3114yes—
330#120—28no—
331#120—28no—
332#120—44no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4D4×C3112yescomplemented
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4D4×C3112yes—
343C2×C22⋊C4C23×C3112yes—
344#103—24no—
345#102—24no—
346C22⋊C8—22no—
347C22⋊C8C23×C3114yescomplemented
348#95—21no—
349#95—22no—
350#95C23×C3112yescomplemented
351C31#1813211yescharacteristic, direct product factor, Sylow
352C24—21no—
353C24—21no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—21no—
358C24—44no—
359C24—42no—
360C24—81no—
361C24—82no—
362C24—82no—
363C24C2×D4×C3111yescharacteristic
364C24C2×D4×C3112yes—
365#41—28no—
366#41—44no—
367#41—44no—
368#41—44no—
369C2×Q8—22no—
370C2×Q8—42no—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—44no—
379C2×D4—42no—
380C2×D4—48no—
381C2×D4—416no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—42no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4C2×D4×C3114yes—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—81no—
405C22×C4C2×D4×C3112yes—
406C22×C4C2×D4×C3112yes—
407C22×C4C2×D4×C3111yescharacteristic
408C22×C4C2×D4×C3112yes—
409C22×C4C24×C3111yescharacteristic
410SD16—48no—
411D8—48no—
412M16—44no—
413C2×C8—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4C2×D4×C3114yes—
417C22⋊C4—28no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—44no—
423C22⋊C4—42no—
424C22⋊C4—82no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—44no—
440C23—48no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—41no—
445C23—42no—
446C23—81no—
447C23—81no—
448C23—88no—
449C23—84no—
450C23—84no—
451C23—816no—
452C23C31×#12011yescharacteristic
453C23C31×#12011yescharacteristic
454C23C31×AGL(1,ℤ/8)14yescomplemented
455C23C22×D4×C3111yescharacteristic
456Q8—24no—
457Q8—42no—
458D4—24no—
459D4—44no—
460D4—42no—
461D4—416no—
462D4—48no—
463D4—44no—
464D4—44no—
465D4—48no—
466D4—82no—
467D4—84no—
468D4—88no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—81no—
487C2×C4C31×#12012yes—
488C2×C4C22×D4×C3112yes—
489C8—44no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—24no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—44no—
499C22—44no—
500C22—48no—
501C22—44no—
502C22—42no—
503C22—42no—
504C22—44no—
505C22—44no—
506C22—42no—
507C22—44no—
508C22—82no—
509C22—88no—
510C22—82no—
511C22—84no—
512C22—816no—
513C22—81no—
514C22—81no—
515C22C31×B(4,𝔽2)12yes—
516C22C2×C31×#12011yescharacteristic
517C22C2×C31×AGL(1,ℤ/8)12yes—
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—82no—
524C2—22no—
525C2—22no—
526C2—41no—
527C2—44no—
528C2—84no—
529C2—82no—
530C2C31×#300311yescharacteristic
531C2C31×#301911yescharacteristic
532C2C2×C31×B(4,𝔽2)11yescharacteristic
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×#3006C2semidirect
3. C31×#2404C2semidirect
4. C31×#1601C2semidirect
5. C31×#1595C2semidirect
6. C31×#1510C2semidirect
7. C31×#1462C2semidirect
8. C22×#41×C31C22semidirect
21. C31×#448C22semidirect
22. C31×#447C22semidirect
24. C31×#407C22semidirect
25. C31×#393C22semidirect
26. C31×#392C22semidirect
29. C31×#381C22semidirect
30. C31×#327C22semidirect
31. C31×#323C22semidirect
37. C22×Q8×C31C23semidirect
46. C22×D4×C31D4semidirect
47. C22×D4×C31C23semidirect
62. C31×#121D4semidirect
68. C2×C4⋊C4×C31D4semidirect
81. C31×C22⋊C8C23semidirect
84. C31×#95C23semidirect
158. #18132C31direct (N × H)
188. C23×C31AGL(1,ℤ/8)semidirect
224. C2×#534C2×C31semidirect
225. #3006C2×C31semidirect
226. #2404C2×C31semidirect
227. #1601C2×C31semidirect
228. #1595C2×C31semidirect
229. #1510C2×C31semidirect
230. #1462C2×C31semidirect
265. C22×#41C22×C31semidirect
278. #448C22×C31semidirect
279. #447C22×C31semidirect
281. #407C22×C31semidirect
282. #393C22×C31semidirect
283. #392C22×C31semidirect
286. #381C22×C31semidirect
287. #327C22×C31semidirect
288. #323C22×C31semidirect
303. C22×Q8C23×C31semidirect
312. C22×D4D4×C31semidirect
313. C22×D4C23×C31semidirect
328. #121D4×C31semidirect
334. C2×C4⋊C4D4×C31semidirect
347. C22⋊C8C23×C31semidirect
350. #95C23×C31semidirect
351. C31#18132direct (N × H)
454. C23C31×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5342
C31×#30062
C31×#24042
C31×#16012
C31×#15952
C31×#15102
C31×#14622
#1813231