Finite Group Databasebeta

← C31×#23898 #21480657

Subgroups — C31×#23898 #21480657

Order 7936.

1718 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3487C211yesmaximal, characteristic, complemented
2C31×#3485C211yesmaximal, characteristic, complemented
3C31×#3020C212yesmaximal, complemented
4C31×#3019C212yesmaximal, complemented
5C31×#2010C214yesmaximal, complemented
6C31×#1912C212yesmaximal, complemented
7C31×#1911C212yesmaximal, complemented
8C31×#1785C211yesmaximal, characteristic, complemented
9C2×C31×2−1+4C2211yescharacteristic, complemented
10C2×C31×H(5,𝔽2)C2211yescharacteristic
11C22×#41×C31C2211yescharacteristic
12C31×#575C2214yes—
13C31×#550—24no—
14C31×#548—24no—
15C31×#547—24no—
16C31×#546—24no—
17C31×#536C2212yes—
18C31×#535—22no—
19C31×#535C2212yes—
20C31×#534—22no—
21C31×#525—22no—
22C31×#525C2212yes—
23C31×#525C2212yescomplemented
24C2×C31×#123—22no—
25C31×#519C2211yescharacteristic, complemented
26C31×#518C2211yescharacteristic
27C31×#456—24no—
28C31×#455—24no—
29C31×#454—24no—
30C31×#453—24no—
31C31×#421C2214yes—
32C31×#421C2214yescomplemented
33C2×C31×#101C2212yescomplemented
34C2×C31×#100C2212yes—
35C31×#407C2212yescomplemented
36C31×H(3,ℤ/4)C2214yescomplemented
37C31×2−1+4—24no—
38C31×2−1+4—42no—
39C31×H(5,𝔽2)—24no—
40C31×H(5,𝔽2)—42no—
41C2×#41×C31—24no—
42C2×#41×C31—22no—
43C2×#41×C31—24no—
44C2×#41×C31—22no—
45C2×#41×C31—22no—
46C2×#41×C31—42no—
47C2×#41×C31C2312yes—
48C2×#41×C31C2312yes—
49C22×Q8×C31—22no—
50C22×Q8×C31C2311yescharacteristic
51C22×D4×C31—22no—
52C22×D4×C31—22no—
53C22×D4×C31D412yes—
54C22×D4×C31C2311yescharacteristic
55C23×C4×C31D412yes—
56C23×C4×C31C2311yescharacteristic
57C31×#137—24no—
58C31×#137—24no—
59C31×AGL(1,ℤ/8)—24no—
60C31×AGL(1,ℤ/8)—24no—
61C31×#135—44no—
62C2×M16×C31C2314yes—
63C31×C4⋊Q8D414yescomplemented
64C31×#127—24no—
65C31×#127D414yescomplemented
66C31×#124—24no—
67C31×#124D414yescomplemented
68C31×#124D414yescomplemented
69C31×#123—28no—
70C31×#123—24no—
71C31×#123—22no—
72C31×#123—24no—
73C31×#123—22no—
74C31×C22⋊Q8—24no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—24no—
78C31×#121—24no—
79C31×#121—28no—
80C31×#121—28no—
81C31×#121—22no—
82C31×#121—22no—
83C31×#121—24no—
84C31×#120—24no—
85C31×#120—24no—
86C31×#120—22no—
87C31×#120—22no—
88C4×Q8×C31—24no—
89C4×D4×C31—28no—
90C4×D4×C31—24no—
91C4×D4×C31—24no—
92C4×D4×C31—24no—
93C31×#117—22no—
94C31×#117C2312yes—
95C31×#117C2312yes—
96C2×C4⋊C4×C31—24no—
97C2×C22⋊C4×C31—24no—
98C2×C22⋊C4×C31—22no—
99C31×#104—28no—
100C31×#104—28no—
101C31×#101—24no—
102C31×#100—24no—
103C31×C22⋊C8—42no—
104C24×C31—22no—
105C24×C31C2×D411yescharacteristic
106C24×C31C2×D412yes—
107#41×C31—24no—
108#41×C31—24no—
109#41×C31—24no—
110#41×C31—24no—
111#41×C31—44no—
112#41×C31—416no—
113#41×C31—48no—
114#41×C31—48no—
115#41×C31—44no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—48no—
120C2×Q8×C31C2×D412yes—
121C2×Q8×C31C2×D412yes—
122C2×Q8×C31C2×D414yes—
123C2×D4×C31—24no—
124C2×D4×C31—22no—
125C2×D4×C31—24no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—24no—
129C2×D4×C31—24no—
130C2×D4×C31—28no—
131C2×D4×C31—24no—
132C2×D4×C31—22no—
133C2×D4×C31—22no—
134C2×D4×C31—42no—
135C2×D4×C31—48no—
136C2×D4×C31—44no—
137C2×D4×C31—44no—
138C2×D4×C31C2×D412yes—
139C2×D4×C31C2×D412yes—
140C2×D4×C31C2×D414yes—
141C22×C4×C31—22no—
142C22×C4×C31—24no—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—24no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—42no—
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2411yescharacteristic
155Q16×C31—44no—
156SD16×C31—44no—
157SD16×C31—44no—
158D8×C31—44no—
159M16×C31—24no—
160M16×C31—24no—
161C2×C8×C31—24no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—24no—
165C4⋊C4×C31—28no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—28no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—28no—
177C42×C31—24no—
178C42×C31C2×D414yes—
179C42×C31C2×D414yes—
180#23898C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—22no—
187C23×C31—21no—
188C23×C31—44no—
189C23×C31—81no—
190C23×C31—82no—
191C23×C31—84no—
192C23×C31#12014yes—
193C23×C31C22×D411yescharacteristic
194Q8×C31—28no—
195Q8×C31—24no—
196Q8×C31—24no—
197Q8×C31—44no—
198D4×C31—28no—
199D4×C31—24no—
200D4×C31—24no—
201D4×C31—48no—
202D4×C31—44no—
203D4×C31—48no—
204D4×C31—42no—
205D4×C31—42no—
206D4×C31—42no—
207D4×C31—42no—
208D4×C31—84no—
209D4×C31—88no—
210D4×C31—84no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—24no—
214C2×C4×C31—24no—
215C2×C4×C31—22no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—22no—
219C2×C4×C31—21no—
220C2×C4×C31—48no—
221C2×C4×C31—48no—
222C2×C4×C31—44no—
223C2×C4×C31—44no—
224C2×C4×C31—44no—
225C2×C4×C31—44no—
226C2×C4×C31—84no—
227C2×C4×C31#12014yes—
228C2×C4×C31C22×D412yes—
229C2×C4×C31C22×D412yes—
230C2×C4×C31H(5,𝔽2)12yes—
231C8×C31—44no—
232#3487C2×C3111yescharacteristic, complemented
233#3485C2×C3111yescharacteristic, complemented
234#3020C2×C3112yescomplemented
235#3019C2×C3112yescomplemented
236#2010C2×C3114yescomplemented
237#1912C2×C3112yescomplemented
238#1911C2×C3112yescomplemented
239#1785C2×C3111yescharacteristic, complemented
240C22×C31—24no—
241C22×C31—41no—
242C22×C31—44no—
243C22×C31—44no—
244C22×C31—44no—
245C22×C31—42no—
246C22×C31—42no—
247C22×C31—44no—
248C22×C31—42no—
249C22×C31—42no—
250C22×C31—41no—
251C22×C31—42no—
252C22×C31—82no—
253C22×C31—164no—
254C22×C31C2×#12011yescharacteristic
255C22×C31C2×#12011yescharacteristic
256C22×C31#53411yescharacteristic
257C4×C31—24no—
258C4×C31—44no—
259C4×C31—44no—
260C4×C31—41no—
261C4×C31—42no—
262C4×C31—42no—
263C4×C31D4214yes—
264C2×2−1+4C22×C3111yescharacteristic, complemented
265C2×H(5,𝔽2)C22×C3111yescharacteristic
266C22×#41C22×C3111yescharacteristic
267#575C22×C3114yes—
268#550—24no—
269#548—24no—
270#547—24no—
271#546—24no—
272#536C22×C3112yes—
273#535—22no—
274#535C22×C3112yes—
275#534—22no—
276#525—22no—
277#525C22×C3112yes—
278#525C22×C3112yescomplemented
279C2×#123—22no—
280#519C22×C3111yescharacteristic, complemented
281#518C22×C3111yescharacteristic
282#456—24no—
283#455—24no—
284#454—24no—
285#453—24no—
286#421C22×C3114yes—
287#421C22×C3114yescomplemented
288C2×#101C22×C3112yescomplemented
289C2×#100C22×C3112yes—
290#407C22×C3112yescomplemented
291H(3,ℤ/4)C22×C3114yescomplemented
292C2×C31—21no—
293C2×C31—21no—
294C2×C31—21no—
295C2×C31—44no—
296C2×C31—82no—
297C2×C31—81no—
298C2×C31—82no—
299C2×C31#240311yescharacteristic
3002−1+4—24no—
3012−1+4—42no—
302H(5,𝔽2)—24no—
303H(5,𝔽2)—42no—
304C2×#41—24no—
305C2×#41—22no—
306C2×#41—24no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—42no—
310C2×#41C23×C3112yes—
311C2×#41C23×C3112yes—
312C22×Q8—22no—
313C22×Q8C23×C3111yescharacteristic
314C22×D4—22no—
315C22×D4—22no—
316C22×D4D4×C3112yes—
317C22×D4C23×C3111yescharacteristic
318C23×C4D4×C3112yes—
319C23×C4C23×C3111yescharacteristic
320#137—24no—
321#137—24no—
322AGL(1,ℤ/8)—24no—
323AGL(1,ℤ/8)—24no—
324#135—44no—
325C2×M16C23×C3114yes—
326C4⋊Q8D4×C3114yescomplemented
327#127—24no—
328#127D4×C3114yescomplemented
329#124—24no—
330#124D4×C3114yescomplemented
331#124D4×C3114yescomplemented
332#123—28no—
333#123—24no—
334#123—22no—
335#123—24no—
336#123—22no—
337C22⋊Q8—24no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—24no—
341#121—24no—
342#121—28no—
343#121—28no—
344#121—22no—
345#121—22no—
346#121—24no—
347#120—24no—
348#120—24no—
349#120—22no—
350#120—22no—
351C4×Q8—24no—
352C4×D4—28no—
353C4×D4—24no—
354C4×D4—24no—
355C4×D4—24no—
356#117—22no—
357#117C23×C3112yes—
358#117C23×C3112yes—
359C2×C4⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362#104—28no—
363#104—28no—
364#101—24no—
365#100—24no—
366C22⋊C8—42no—
367C31#2389811yescharacteristic, direct product factor, Sylow
368C24—22no—
369C24C2×D4×C3111yescharacteristic
370C24C2×D4×C3112yes—
371#41—24no—
372#41—24no—
373#41—24no—
374#41—24no—
375#41—44no—
376#41—416no—
377#41—48no—
378#41—48no—
379#41—44no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—48no—
384C2×Q8C2×D4×C3112yes—
385C2×Q8C2×D4×C3112yes—
386C2×Q8C2×D4×C3114yes—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—48no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4C2×D4×C3112yes—
403C2×D4C2×D4×C3112yes—
404C2×D4C2×D4×C3114yes—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—42no—
416C22×C4C2×D4×C3111yescharacteristic
417C22×C4C2×D4×C3112yes—
418C22×C4C24×C3111yescharacteristic
419Q16—44no—
420SD16—44no—
421SD16—44no—
422D8—44no—
423M16—24no—
424M16—24no—
425C2×C8—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—28no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—28no—
441C42—24no—
442C42C2×D4×C3114yes—
443C42C2×D4×C3114yes—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—44no—
452C23—81no—
453C23—82no—
454C23—84no—
455C23C31×#12014yes—
456C23C22×D4×C3111yescharacteristic
457Q8—28no—
458Q8—24no—
459Q8—24no—
460Q8—44no—
461D4—28no—
462D4—24no—
463D4—24no—
464D4—48no—
465D4—44no—
466D4—48no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—84no—
472D4—88no—
473D4—84no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—84no—
490C2×C4C31×#12014yes—
491C2×C4C22×D4×C3112yes—
492C2×C4C22×D4×C3112yes—
493C2×C4C31×H(5,𝔽2)12yes—
494C8—44no—
495C22—24no—
496C22—41no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—42no—
507C22—82no—
508C22—164no—
509C22C2×C31×#12011yescharacteristic
510C22C2×C31×#12011yescharacteristic
511C22C31×#53411yescharacteristic
512C4—24no—
513C4—44no—
514C4—44no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4D42×C3114yes—
519C2—21no—
520C2—21no—
521C2—21no—
522C2—44no—
523C2—81no—
524C2—82no—
525C2—82no—
526C2C31×#240311yescharacteristic
subgroupG/H
center, socleC2×C31#2403
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3487C2semidirect
2. C31×#3485C2semidirect
3. C31×#3020C2semidirect
4. C31×#3019C2semidirect
5. C31×#2010C2semidirect
6. C31×#1912C2semidirect
7. C31×#1911C2semidirect
8. C31×#1785C2semidirect
9. C2×C31×2−1+4C22semidirect
23. C31×#525C22semidirect
25. C31×#519C22semidirect
32. C31×#421C22semidirect
33. C2×C31×#101C22semidirect
35. C31×#407C22semidirect
36. C31×H(3,ℤ/4)C22semidirect
63. C31×C4⋊Q8D4semidirect
65. C31×#127D4semidirect
67. C31×#124D4semidirect
68. C31×#124D4semidirect
180. #23898C31direct (N × H)
232. #3487C2×C31semidirect
233. #3485C2×C31semidirect
234. #3020C2×C31semidirect
235. #3019C2×C31semidirect
236. #2010C2×C31semidirect
237. #1912C2×C31semidirect
238. #1911C2×C31semidirect
239. #1785C2×C31semidirect
264. C2×2−1+4C22×C31semidirect
278. #525C22×C31semidirect
280. #519C22×C31semidirect
287. #421C22×C31semidirect
288. C2×#101C22×C31semidirect
290. #407C22×C31semidirect
291. H(3,ℤ/4)C22×C31semidirect
326. C4⋊Q8D4×C31semidirect
328. #127D4×C31semidirect
330. #124D4×C31semidirect
331. #124D4×C31semidirect
367. C31#23898direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34872
C31×#34852
C31×#30202
C31×#30192
C31×#20102
C31×#19122
C31×#19112
C31×#17852
#2389831