Finite Group Databasebeta

← C31×#18123 #21474882

Subgroups — C31×#18123 #21474882

Order 7936.

1208 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3449C211yesmaximal, characteristic
2C31×#3103C212yesmaximal, complemented
3C31×#2396C211yesmaximal, characteristic, complemented
4C31×#1616C212yesmaximal, complemented
5C31×#1613C212yesmaximal, complemented
6C31×#1604C212yesmaximal, complemented
7C31×#1601C212yesmaximal, complemented
8C31×#1502C212yesmaximal, complemented
9C31×#1458C211yesmaximal, characteristic, complemented
10C22×#41×C31—21no—
11C31×#537—21no—
12C31×#537C2212yes—
13C31×#536—22no—
14C31×#535—22no—
15C31×#534—21no—
16C31×#534C2212yes—
17C31×#532—21no—
18C31×#532C2212yes—
19C31×#525—24no—
20C2×C4×Q8×C31C2211yescharacteristic, complemented
21C2×C31×#117C2211yescharacteristic
22C31×#481C2214yescomplemented
23C31×#475—22no—
24C31×#460—22no—
25C31×#448C2214yescomplemented
26C4×SD16×C31—24no—
27C31×#431C2212yescomplemented
28C31×#400C2211yescharacteristic, complemented
29C31×#399C2211yescharacteristic
30C31×#398—21no—
31C31×#396—21no—
32C31×#395C2211yescharacteristic, complemented
33C31×#393C2211yescharacteristic
34C4×C4⋊C4×C31—21no—
35C4×C4⋊C4×C31C2211yescharacteristic, complemented
36C4×C22⋊C4×C31—21no—
37C31×#327C2214yescomplemented
38C31×#327C2214yescomplemented
39C31×#324C2212yescomplemented
40C31×#323C2212yes—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31—44no—
46C2×#41×C31—44no—
47C22×Q8×C31—21no—
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31—21no—
50C22×D4×C31—22no—
51C23×C4×C31—22no—
52C23×C4×C31D411yescharacteristic
53C2×SD16×C31—24no—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8D412yes—
56C31×#127—21no—
57C31×#127D412yes—
58C31×#124—22no—
59C31×#124—22no—
60C31×#123—24no—
61C31×#123—24no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—24no—
64C31×#121—24no—
65C31×#121—24no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121C2314yes—
70C31×#120—24no—
71C31×#120—24no—
72C4×Q8×C31—22no—
73C4×Q8×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31—22no—
77C4×D4×C31D414yes—
78C31×#117—22no—
79C31×#117—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31D411yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31D411yescharacteristic
90C2×C22⋊C4×C31C2311yescharacteristic
91C2×C42×C31—21no—
92C2×C42×C31—21no—
93C2×C42×C31D411yescharacteristic
94C2×C42×C31C2311yescharacteristic
95C31×#106—24no—
96C31×C4⋊C8—22no—
97C31×#103—24no—
98C31×#102—24no—
99C31×C22⋊C8C2314yes—
100C4×C8×C31—22no—
101C31×#95—21no—
102C31×#95—21no—
103C31×#95—21no—
104C31×#95—21no—
105C31×#95C2311yescharacteristic
106C31×#95C2311yescharacteristic
107C24×C31—22no—
108C24×C31C2×D411yescharacteristic
109#41×C31—44no—
110#41×C31—48no—
111#41×C31—48no—
112#41×C31—44no—
113#41×C31—44no—
114#41×C31—44no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—42no—
119C2×Q8×C31—44no—
120C2×D4×C31—24no—
121C2×D4×C31—22no—
122C2×D4×C31—24no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—44no—
128C2×D4×C31—44no—
129C2×D4×C31—44no—
130C2×D4×C31C2×D414yes—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—24no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—24no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—48no—
142C22×C4×C31SD1614yescomplemented
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2411yescharacteristic
149SD16×C31—48no—
150C2×C8×C31—24no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—42no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31—42no—
160C4⋊C4×C31—42no—
161C4⋊C4×C31C2×D414yes—
162C22⋊C4×C31—24no—
163C22⋊C4×C31—24no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—24no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—42no—
169C22⋊C4×C31—42no—
170C42×C31—21no—
171C42×C31—21no—
172C42×C31—41no—
173C42×C31—41no—
174C42×C31—42no—
175C42×C31C2×D412yes—
176C42×C31C2×D412yes—
177#18123C3111yesmaximal, characteristic, direct product factor, Sylow
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—24no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—44no—
184C23×C31—82no—
185C23×C31—82no—
186C23×C31C2×SD1612yes—
187C23×C31C22×D411yescharacteristic
188Q8×C31—24no—
189Q8×C31—42no—
190Q8×C31—44no—
191Q8×C31—44no—
192D4×C31—44no—
193D4×C31—44no—
194D4×C31—48no—
195D4×C31—44no—
196D4×C31—44no—
197D4×C31—44no—
198D4×C31—82no—
199D4×C31—82no—
200D4×C31—82no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—24no—
204C2×C4×C31—24no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—24no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31—48no—
213C2×C4×C31—41no—
214C2×C4×C31—44no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—48no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31#12012yes—
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31C2×SD1612yes—
231C2×C4×C31C2×SD1612yes—
232C2×C4×C31AGL(1,ℤ/8)12yes—
233C2×C4×C31C22×D412yes—
234C8×C31—44no—
235#3449C2×C3111yescharacteristic
236#3103C2×C3112yescomplemented
237#2396C2×C3111yescharacteristic, complemented
238#1616C2×C3112yescomplemented
239#1613C2×C3112yescomplemented
240#1604C2×C3112yescomplemented
241#1601C2×C3112yescomplemented
242#1502C2×C3112yescomplemented
243#1458C2×C3111yescharacteristic, complemented
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—22no—
247C22×C31—42no—
248C22×C31—41no—
249C22×C31—41no—
250C22×C31—44no—
251C22×C31—44no—
252C22×C31—44no—
253C22×C31—81no—
254C22×C31—88no—
255C22×C31C2×#12011yescharacteristic
256C22×C31C22×SD1611yescharacteristic
257C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
258C4×C31—22no—
259C4×C31—22no—
260C4×C31—22no—
261C4×C31—42no—
262C4×C31—44no—
263C4×C31—41no—
264C4×C31—42no—
265C4×C31—41no—
266C4×C31—81no—
267C4×C31—81no—
268C4×C31#45012yes—
269C4×C31#45012yes—
270C22×#41—21no—
271#537—21no—
272#537C22×C3112yes—
273#536—22no—
274#535—22no—
275#534—21no—
276#534C22×C3112yes—
277#532—21no—
278#532C22×C3112yes—
279#525—24no—
280C2×C4×Q8C22×C3111yescharacteristic, complemented
281C2×#117C22×C3111yescharacteristic
282#481C22×C3114yescomplemented
283#475—22no—
284#460—22no—
285#448C22×C3114yescomplemented
286C4×SD16—24no—
287#431C22×C3112yescomplemented
288#400C22×C3111yescharacteristic, complemented
289#399C22×C3111yescharacteristic
290#398—21no—
291#396—21no—
292#395C22×C3111yescharacteristic, complemented
293#393C22×C3111yescharacteristic
294C4×C4⋊C4—21no—
295C4×C4⋊C4C22×C3111yescharacteristic, complemented
296C4×C22⋊C4—21no—
297#327C22×C3114yescomplemented
298#327C22×C3114yescomplemented
299#324C22×C3112yescomplemented
300#323C22×C3112yes—
301C2×C31—21no—
302C2×C31—21no—
303C2×C31—42no—
304C2×C31—84no—
305C2×C31C2×#45011yescharacteristic
306C2×C31#301811yescharacteristic
307C2×C31#302511yescharacteristic
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—44no—
313C2×#41—44no—
314C22×Q8—21no—
315C22×Q8C23×C3111yescharacteristic
316C22×D4—21no—
317C22×D4—22no—
318C23×C4—22no—
319C23×C4D4×C3111yescharacteristic
320C2×SD16—24no—
321C4⋊Q8—21no—
322C4⋊Q8D4×C3112yes—
323#127—21no—
324#127D4×C3112yes—
325#124—22no—
326#124—22no—
327#123—24no—
328#123—24no—
329C22⋊Q8—24no—
330C22⋊Q8—24no—
331#121—24no—
332#121—24no—
333#121—22no—
334#121—22no—
335#121—22no—
336#121C23×C3114yes—
337#120—24no—
338#120—24no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4D4×C3114yes—
345#117—22no—
346#117—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4D4×C3111yescharacteristic
352C2×C4⋊C4C23×C3111yescharacteristic
353C2×C4⋊C4C23×C3111yescharacteristic
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4D4×C3111yescharacteristic
357C2×C22⋊C4C23×C3111yescharacteristic
358C2×C42—21no—
359C2×C42—21no—
360C2×C42D4×C3111yescharacteristic
361C2×C42C23×C3111yescharacteristic
362#106—24no—
363C4⋊C8—22no—
364#103—24no—
365#102—24no—
366C22⋊C8C23×C3114yes—
367C4×C8—22no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95C23×C3111yescharacteristic
373#95C23×C3111yescharacteristic
374C31#1812311yescharacteristic, direct product factor, Sylow
375C24—22no—
376C24C2×D4×C3111yescharacteristic
377#41—44no—
378#41—48no—
379#41—48no—
380#41—44no—
381#41—44no—
382#41—44no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4C2×D4×C3114yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—48no—
410C22×C4SD16×C3114yescomplemented
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3111yescharacteristic
416C22×C4C24×C3111yescharacteristic
417SD16—48no—
418C2×C8—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4C2×D4×C3114yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—24no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42C2×D4×C3112yes—
444C42C2×D4×C3112yes—
445C23—21no—
446C23—21no—
447C23—24no—
448C23—41no—
449C23—41no—
450C23—44no—
451C23—82no—
452C23—82no—
453C23C2×SD16×C3112yes—
454C23C22×D4×C3111yescharacteristic
455Q8—24no—
456Q8—42no—
457Q8—44no—
458Q8—44no—
459D4—44no—
460D4—44no—
461D4—48no—
462D4—44no—
463D4—44no—
464D4—44no—
465D4—82no—
466D4—82no—
467D4—82no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—48no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—48no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4C31×#12012yes—
495C2×C4C31×#12011yescharacteristic
496C2×C4C31×#12011yescharacteristic
497C2×C4C2×SD16×C3112yes—
498C2×C4C2×SD16×C3112yes—
499C2×C4C31×AGL(1,ℤ/8)12yes—
500C2×C4C22×D4×C3112yes—
501C8—44no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—81no—
512C22—88no—
513C22C2×C31×#12011yescharacteristic
514C22C22×SD16×C3111yescharacteristic
515C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
516C4—22no—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—44no—
521C4—41no—
522C4—42no—
523C4—41no—
524C4—81no—
525C4—81no—
526C4C31×#45012yes—
527C4C31×#45012yes—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—84no—
532C2C2×C31×#45011yescharacteristic
533C2C31×#301811yescharacteristic
534C2C31×#302511yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
socleC22×C31C2×#120
prime radicalC31×#3449C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3103C2semidirect
3. C31×#2396C2semidirect
4. C31×#1616C2semidirect
5. C31×#1613C2semidirect
6. C31×#1604C2semidirect
7. C31×#1601C2semidirect
8. C31×#1502C2semidirect
9. C31×#1458C2semidirect
20. C2×C4×Q8×C31C22semidirect
22. C31×#481C22semidirect
25. C31×#448C22semidirect
27. C31×#431C22semidirect
28. C31×#400C22semidirect
32. C31×#395C22semidirect
35. C4×C4⋊C4×C31C22semidirect
37. C31×#327C22semidirect
38. C31×#327C22semidirect
39. C31×#324C22semidirect
142. C22×C4×C31SD16semidirect
177. #18123C31direct (N × H)
236. #3103C2×C31semidirect
237. #2396C2×C31semidirect
238. #1616C2×C31semidirect
239. #1613C2×C31semidirect
240. #1604C2×C31semidirect
241. #1601C2×C31semidirect
242. #1502C2×C31semidirect
243. #1458C2×C31semidirect
280. C2×C4×Q8C22×C31semidirect
282. #481C22×C31semidirect
285. #448C22×C31semidirect
287. #431C22×C31semidirect
288. #400C22×C31semidirect
292. #395C22×C31semidirect
295. C4×C4⋊C4C22×C31semidirect
297. #327C22×C31semidirect
298. #327C22×C31semidirect
299. #324C22×C31semidirect
374. C31#18123direct (N × H)
410. C22×C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34492
C31×#31032
C31×#23962
C31×#16162
C31×#16132
C31×#16042
C31×#16012
C31×#15022
C31×#14582
#1812331