Finite Group Databasebeta

← C31×#18153 #21474912

Subgroups — C31×#18153 #21474912

Order 7936.

1184 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3449C211yesmaximal, characteristic
2C31×#3112C212yesmaximal, complemented
3C31×#2599C211yesmaximal, characteristic, complemented
4C31×#1610C212yesmaximal, complemented
5C31×#1607C212yesmaximal, complemented
6C31×#1604C212yesmaximal, complemented
7C31×#1601C212yesmaximal, complemented
8C31×#1514C212yesmaximal, complemented
9C31×#1464C211yesmaximal, 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×#480C2214yescomplemented
23C31×#477—22no—
24C31×#460—22no—
25C31×#448C2214yescomplemented
26C31×#440—24no—
27C31×#432C2212yescomplemented
28C31×#398C2211yescharacteristic, complemented
29C31×#397C2211yescharacteristic
30C31×#395C2211yescharacteristic, complemented
31C31×#393C2211yescharacteristic
32C31×#388—21no—
33C31×#387—21no—
34C31×#387—21no—
35C31×#385—21no—
36C31×#383C2211yescharacteristic, complemented
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×C31D411yescharacteristic
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31C2311yescharacteristic
90C2×C42×C31—21no—
91C2×C42×C31D411yescharacteristic
92C2×C42×C31C2311yescharacteristic
93C31×#107—24no—
94C31×C4⋊C8—22no—
95C31×#103—24no—
96C31×#102—24no—
97C31×C22⋊C8C2314yes—
98C31×#97—22no—
99C31×#95—21no—
100C31×#95—21no—
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—48no—
110#41×C31—48no—
111#41×C31—44no—
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—22no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—22no—
135C22×C4×C31—24no—
136C22×C4×C31—22no—
137C22×C4×C31—21no—
138C22×C4×C31—24no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—48no—
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—41no—
159C4⋊C4×C31—41no—
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—42no—
173C42×C31C2×D412yes—
174C42×C31C2×D412yes—
175#18153C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—24no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—44no—
182C23×C31—82no—
183C23×C31—82no—
184C23×C31AGL(1,ℤ/8)12yes—
185C23×C31C22×D411yescharacteristic
186Q8×C31—24no—
187Q8×C31—42no—
188Q8×C31—44no—
189Q8×C31—44no—
190D4×C31—44no—
191D4×C31—44no—
192D4×C31—44no—
193D4×C31—48no—
194D4×C31—44no—
195D4×C31—44no—
196D4×C31—82no—
197D4×C31—82no—
198D4×C31—82no—
199C2×C4×C31—24no—
200C2×C4×C31—24no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—22no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—24no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—48no—
212C2×C4×C31—48no—
213C2×C4×C31—44no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
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#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12012yes—
230C2×C4×C31AGL(1,ℤ/8)12yes—
231C2×C4×C31C22×D412yes—
232C8×C31—44no—
233#3449C2×C3111yescharacteristic
234#3112C2×C3112yescomplemented
235#2599C2×C3111yescharacteristic, complemented
236#1610C2×C3112yescomplemented
237#1607C2×C3112yescomplemented
238#1604C2×C3112yescomplemented
239#1601C2×C3112yescomplemented
240#1514C2×C3112yescomplemented
241#1464C2×C3111yescharacteristic, complemented
242C22×C31—21no—
243C22×C31—21no—
244C22×C31—22no—
245C22×C31—42no—
246C22×C31—41no—
247C22×C31—41no—
248C22×C31—44no—
249C22×C31—44no—
250C22×C31—44no—
251C22×C31—81no—
252C22×C31—88no—
253C22×C31C2×#12011yescharacteristic
254C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
256C4×C31—21no—
257C4×C31—22no—
258C4×C31—21no—
259C4×C31—22no—
260C4×C31—22no—
261C4×C31—44no—
262C4×C31—41no—
263C4×C31—42no—
264C4×C31—41no—
265C4×C31—81no—
266C4×C31—81no—
267C4×C31—81no—
268C22×#41—21no—
269#537—21no—
270#537C22×C3112yes—
271#536—22no—
272#535—22no—
273#534—21no—
274#534C22×C3112yes—
275#532—21no—
276#532C22×C3112yes—
277#525—24no—
278C2×C4×Q8C22×C3111yescharacteristic, complemented
279C2×#117C22×C3111yescharacteristic
280#480C22×C3114yescomplemented
281#477—22no—
282#460—22no—
283#448C22×C3114yescomplemented
284#440—24no—
285#432C22×C3112yescomplemented
286#398C22×C3111yescharacteristic, complemented
287#397C22×C3111yescharacteristic
288#395C22×C3111yescharacteristic, complemented
289#393C22×C3111yescharacteristic
290#388—21no—
291#387—21no—
292#387—21no—
293#385—21no—
294#383C22×C3111yescharacteristic, complemented
295#327C22×C3114yescomplemented
296#327C22×C3114yescomplemented
297#324C22×C3112yescomplemented
298#323C22×C3112yes—
299C2×C31—21no—
300C2×C31—21no—
301C2×C31—42no—
302C2×C31—84no—
303C2×C31#300311yescharacteristic
304C2×C31#302211yescharacteristic
305C2×C31#302511yescharacteristic
306C2×#41—22no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—44no—
311C2×#41—44no—
312C22×Q8—21no—
313C22×Q8C23×C3111yescharacteristic
314C22×D4—21no—
315C22×D4—22no—
316C23×C4—22no—
317C23×C4D4×C3111yescharacteristic
318C2×SD16—24no—
319C4⋊Q8—21no—
320C4⋊Q8D4×C3112yes—
321#127—21no—
322#127D4×C3112yes—
323#124—22no—
324#124—22no—
325#123—24no—
326#123—24no—
327C22⋊Q8—24no—
328C22⋊Q8—24no—
329#121—24no—
330#121—24no—
331#121—22no—
332#121—22no—
333#121—22no—
334#121C23×C3114yes—
335#120—24no—
336#120—24no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4D4×C3114yes—
343#117—22no—
344#117—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4D4×C3111yescharacteristic
349C2×C4⋊C4C23×C3111yescharacteristic
350C2×C4⋊C4C23×C3111yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4D4×C3111yescharacteristic
354C2×C22⋊C4C23×C3111yescharacteristic
355C2×C42—21no—
356C2×C42D4×C3111yescharacteristic
357C2×C42C23×C3111yescharacteristic
358#107—24no—
359C4⋊C8—22no—
360#103—24no—
361#102—24no—
362C22⋊C8C23×C3114yes—
363#97—22no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95C23×C3111yescharacteristic
371#95C23×C3111yescharacteristic
372C31#1815311yescharacteristic, direct product factor, Sylow
373C24—22no—
374C24C2×D4×C3111yescharacteristic
375#41—48no—
376#41—48no—
377#41—44no—
378#41—44no—
379#41—44no—
380#41—44no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4C2×D4×C3114yes—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—48no—
409C22×C4C2×D4×C3111yescharacteristic
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C24×C3111yescharacteristic
415SD16—48no—
416C2×C8—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4C2×D4×C3114yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C42—21no—
437C42—21no—
438C42—42no—
439C42C2×D4×C3112yes—
440C42C2×D4×C3112yes—
441C23—21no—
442C23—21no—
443C23—24no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—82no—
448C23—82no—
449C23C31×AGL(1,ℤ/8)12yes—
450C23C22×D4×C3111yescharacteristic
451Q8—24no—
452Q8—42no—
453Q8—44no—
454Q8—44no—
455D4—44no—
456D4—44no—
457D4—44no—
458D4—48no—
459D4—44no—
460D4—44no—
461D4—82no—
462D4—82no—
463D4—82no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4C31×#12011yescharacteristic
493C2×C4C31×#12011yescharacteristic
494C2×C4C31×#12012yes—
495C2×C4C31×AGL(1,ℤ/8)12yes—
496C2×C4C22×D4×C3112yes—
497C8—44no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—81no—
508C22—88no—
509C22C2×C31×#12011yescharacteristic
510C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
511C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
512C4—21no—
513C4—22no—
514C4—21no—
515C4—22no—
516C4—22no—
517C4—44no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—81no—
522C4—81no—
523C4—81no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—84no—
528C2C31×#300311yescharacteristic
529C2C31×#302211yescharacteristic
530C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3449C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3112C2semidirect
3. C31×#2599C2semidirect
4. C31×#1610C2semidirect
5. C31×#1607C2semidirect
6. C31×#1604C2semidirect
7. C31×#1601C2semidirect
8. C31×#1514C2semidirect
9. C31×#1464C2semidirect
20. C2×C4×Q8×C31C22semidirect
22. C31×#480C22semidirect
25. C31×#448C22semidirect
27. C31×#432C22semidirect
28. C31×#398C22semidirect
30. C31×#395C22semidirect
36. C31×#383C22semidirect
37. C31×#327C22semidirect
38. C31×#327C22semidirect
39. C31×#324C22semidirect
175. #18153C31direct (N × H)
234. #3112C2×C31semidirect
235. #2599C2×C31semidirect
236. #1610C2×C31semidirect
237. #1607C2×C31semidirect
238. #1604C2×C31semidirect
239. #1601C2×C31semidirect
240. #1514C2×C31semidirect
241. #1464C2×C31semidirect
278. C2×C4×Q8C22×C31semidirect
280. #480C22×C31semidirect
283. #448C22×C31semidirect
285. #432C22×C31semidirect
286. #398C22×C31semidirect
288. #395C22×C31semidirect
294. #383C22×C31semidirect
295. #327C22×C31semidirect
296. #327C22×C31semidirect
297. #324C22×C31semidirect
372. C31#18153direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34492
C31×#31122
C31×#25992
C31×#16102
C31×#16072
C31×#16042
C31×#16012
C31×#15142
C31×#14642
#1815331