Finite Group Databasebeta

← C31×#9996 #21466755

Subgroups — C31×#9996 #21466755

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3100C211yesmaximal, characteristic, complemented
2C31×#3036C211yesmaximal, characteristic, complemented
3C31×#2535C211yesmaximal, characteristic, complemented
4C31×#1517C211yesmaximal, characteristic, complemented
5C31×#1486C211yesmaximal, characteristic, complemented
6C31×#1478C211yesmaximal, characteristic, complemented
7C31×#1286C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#532C2211yescharacteristic
10C2×C31×#121—21no—
11C2×C4×D4×C31C2211yescharacteristic
12C2×C4×D4×C31C2211yescharacteristic
13C31×#480—21no—
14C31×#474—21no—
15C31×#463—22no—
16C31×#461—21no—
17C31×#459—21no—
18C31×#459—21no—
19C31×#447—21no—
20C4×D8×C31—22no—
21C31×#431C2211yescharacteristic
22C31×#431C2211yescharacteristic
23C31×#423C2211yescharacteristic
24C31×#417—21no—
25C31×#397—21no—
26C31×#392—21no—
27C31×#386—22no—
28C31×#382C2211yescharacteristic
29C31×#380—21no—
30C31×#327—22no—
31C31×D4⋊C8—22no—
32C31×D4⋊C8—22no—
33C2×#41×C31—21no—
34C22×D4×C31—22no—
35C22×D4×C31D411yescharacteristic
36C22×D4×C31D411yescharacteristic
37C23×C4×C31—21no—
38C23×C4×C31—21no—
39C31×AGL(1,ℤ/8)—42no—
40C31×AGL(1,ℤ/8)—42no—
41C2×SD16×C31—22no—
42C2×D8×C31—22no—
43C2×D8×C31—22no—
44C2×M16×C31—21no—
45C31×C4⋊Q8D411yescharacteristic, complemented
46C31×#127D411yescharacteristic, complemented
47C31×#124—21no—
48C31×#121—21no—
49C31×#121—21no—
50C31×#121—42no—
51C31×#121—42no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C4×D4×C31—21no—
58C4×D4×C31—21no—
59C4×D4×C31—21no—
60C4×D4×C31—21no—
61C4×D4×C31—21no—
62C2×C4⋊C4×C31—21no—
63C2×C4⋊C4×C31D411yescharacteristic
64C2×C4⋊C4×C31D411yescharacteristic
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—21no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—22no—
69C2×C42×C31C2311yescharacteristic
70C31×#107—22no—
71C31×C4⋊C8—21no—
72C31×C4⋊C8—21no—
73C31×C4⋊C8—21no—
74C31×C4⋊C8D412yescomplemented
75C31×#103—22no—
76C31×#102—22no—
77C31×#102—22no—
78C31×#102—22no—
79C31×C22⋊C8—21no—
80C31×C22⋊C8—21no—
81C4×C8×C31—21no—
82C4×C8×C31D412yescomplemented
83C31×#95—22no—
84C31×#95—21no—
85C31×#95D412yes—
86C24×C31—21no—
87C24×C31—21no—
88#41×C31—41no—
89#41×C31—41no—
90#41×C31—41no—
91#41×C31—41no—
92C2×Q8×C31—21no—
93C2×D4×C31—21no—
94C2×D4×C31—21no—
95C2×D4×C31—21no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—21no—
100C2×D4×C31—21no—
101C2×D4×C31—21no—
102C2×D4×C31—21no—
103C2×D4×C31—42no—
104C2×D4×C31—44no—
105C2×D4×C31—44no—
106C2×D4×C31—41no—
107C2×D4×C31—82no—
108C2×D4×C31D812yescomplemented
109C22×C4×C31—22no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—41no—
113C22×C4×C31—41no—
114C22×C4×C31—41no—
115C22×C4×C31—41no—
116C22×C4×C31—41no—
117C22×C4×C31—41no—
118C22×C4×C31—41no—
119C22×C4×C31—41no—
120C22×C4×C31—41no—
121C22×C4×C31—82no—
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D411yescharacteristic
124C22×C4×C31C2×D411yescharacteristic
125SD16×C31—44no—
126D8×C31—44no—
127D8×C31—44no—
128M16×C31—41no—
129M16×C31—41no—
130C2×C8×C31—22no—
131C2×C8×C31—22no—
132C2×C8×C31—41no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—21no—
136C4⋊C4×C31—21no—
137C4⋊C4×C31—21no—
138C4⋊C4×C31—21no—
139C4⋊C4×C31—41no—
140C4⋊C4×C31—41no—
141C22⋊C4×C31—44no—
142C22⋊C4×C31—41no—
143C22⋊C4×C31—41no—
144C22⋊C4×C31—41no—
145C22⋊C4×C31—41no—
146C22⋊C4×C31—41no—
147C22⋊C4×C31—41no—
148C22⋊C4×C31—82no—
149C42×C31C2×D411yescharacteristic
150C42×C31C2×D411yescharacteristic
151C42×C31C2×D411yescharacteristic
152C42×C31#4111yescharacteristic
153#9996C3111yesmaximal, characteristic, direct product factor, Sylow
154C23×C31—22no—
155C23×C31—22no—
156C23×C31—41no—
157C23×C31—42no—
158C23×C31—42no—
159C23×C31—41no—
160C23×C31—41no—
161C23×C31—41no—
162C23×C31—41no—
163C23×C31—41no—
164C23×C31—41no—
165C23×C31—41no—
166C23×C31—41no—
167C23×C31#12011yescharacteristic
168Q8×C31—41no—
169Q8×C31—41no—
170D4×C31—22no—
171D4×C31—22no—
172D4×C31—22no—
173D4×C31—22no—
174D4×C31—41no—
175D4×C31—41no—
176D4×C31—42no—
177D4×C31—41no—
178D4×C31—41no—
179D4×C31—41no—
180D4×C31—41no—
181D4×C31—81no—
182D4×C31—88no—
183D4×C31—81no—
184D4×C31—81no—
185C2×C4×C31—41no—
186C2×C4×C31—41no—
187C2×C4×C31—42no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—41no—
193C2×C4×C31—41no—
194C2×C4×C31—42no—
195C2×C4×C31—41no—
196C2×C4×C31—82no—
197C2×C4×C31—82no—
198C2×C4×C31—81no—
199C2×C4×C31—81no—
200C2×C4×C31—81no—
201C2×C4×C31—81no—
202C2×C4×C31—81no—
203C2×C4×C31—81no—
204C2×C4×C31#12011yescharacteristic
205C2×C4×C31#12011yescharacteristic
206C2×C4×C31#12111yescharacteristic
207C2×C4×C31#12111yescharacteristic
208C2×C4×C31#12111yescharacteristic
209C2×C4×C31#12711yescharacteristic
210C2×C4×C31C2×D811yescharacteristic
211C2×C4×C31#13511yescharacteristic
212C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
213C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
214C8×C31—42no—
215C8×C31—42no—
216C8×C31—81no—
217#3100C2×C3111yescharacteristic, complemented
218#3036C2×C3111yescharacteristic, complemented
219#2535C2×C3111yescharacteristic, complemented
220#1517C2×C3111yescharacteristic, complemented
221#1486C2×C3111yescharacteristic, complemented
222#1478C2×C3111yescharacteristic, complemented
223#1286C2×C3111yescharacteristic, complemented
224C22×C31—21no—
225C22×C31—21no—
226C22×C31—42no—
227C22×C31—44no—
228C22×C31—42no—
229C22×C31—42no—
230C22×C31—42no—
231C22×C31—81no—
232C22×C31—82no—
233C22×C31—81no—
234C22×C31—81no—
235C22×C31—81no—
236C22×C31—81no—
237C22×C31—81no—
238C22×C31—81no—
239C22×C31—81no—
240C22×C31—81no—
241C22×C31#39211yescharacteristic
242C22×C31#44711yescharacteristic
243C22×C31#44911yescharacteristic
244C4×C31—21no—
245C4×C31—21no—
246C4×C31—21no—
247C4×C31—21no—
248C4×C31—82no—
249C4×C31—81no—
250C4×C31—81no—
251C4×C31—81no—
252C4×C31#44711yescharacteristic
253C4×C31#44911yescharacteristic
254C4×C31#46611yescharacteristic
255C4×C31#46911yescharacteristic
256C2×AGL(1,ℤ/8)—21no—
257#532C22×C3111yescharacteristic
258C2×#121—21no—
259C2×C4×D4C22×C3111yescharacteristic
260C2×C4×D4C22×C3111yescharacteristic
261#480—21no—
262#474—21no—
263#463—22no—
264#461—21no—
265#459—21no—
266#459—21no—
267#447—21no—
268C4×D8—22no—
269#431C22×C3111yescharacteristic
270#431C22×C3111yescharacteristic
271#423C22×C3111yescharacteristic
272#417—21no—
273#397—21no—
274#392—21no—
275#386—22no—
276#382C22×C3111yescharacteristic
277#380—21no—
278#327—22no—
279D4⋊C8—22no—
280D4⋊C8—22no—
281C2×C31—21no—
282C2×C31—21no—
283C2×C31—82no—
284C2×C31—82no—
285C2×C31—161no—
286C2×C31#199911yescharacteristic
287C2×C31#200811yescharacteristic
288C2×C31#201011yescharacteristic
289C2×#41—21no—
290C22×D4—22no—
291C22×D4D4×C3111yescharacteristic
292C22×D4D4×C3111yescharacteristic
293C23×C4—21no—
294C23×C4—21no—
295AGL(1,ℤ/8)—42no—
296AGL(1,ℤ/8)—42no—
297C2×SD16—22no—
298C2×D8—22no—
299C2×D8—22no—
300C2×M16—21no—
301C4⋊Q8D4×C3111yescharacteristic, complemented
302#127D4×C3111yescharacteristic, complemented
303#124—21no—
304#121—21no—
305#121—21no—
306#121—42no—
307#121—42no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4D4×C3111yescharacteristic
320C2×C4⋊C4D4×C3111yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C42C23×C3111yescharacteristic
326#107—22no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8D4×C3112yescomplemented
331#103—22no—
332#102—22no—
333#102—22no—
334#102—22no—
335C22⋊C8—21no—
336C22⋊C8—21no—
337C4×C8—21no—
338C4×C8D4×C3112yescomplemented
339#95—22no—
340#95—21no—
341#95D4×C3112yes—
342C31#999611yescharacteristic, direct product factor, Sylow
343C24—21no—
344C24—21no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349C2×Q8—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—41no—
364C2×D4—82no—
365C2×D4D8×C3112yescomplemented
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—82no—
379C22×C4C2×D4×C3111yescharacteristic
380C22×C4C2×D4×C3111yescharacteristic
381C22×C4C2×D4×C3111yescharacteristic
382SD16—44no—
383D8—44no—
384D8—44no—
385M16—41no—
386M16—41no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—41no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C22⋊C4—44no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—82no—
406C42C2×D4×C3111yescharacteristic
407C42C2×D4×C3111yescharacteristic
408C42C2×D4×C3111yescharacteristic
409C42#41×C3111yescharacteristic
410C23—22no—
411C23—22no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23C31×#12011yescharacteristic
424Q8—41no—
425Q8—41no—
426D4—22no—
427D4—22no—
428D4—22no—
429D4—22no—
430D4—41no—
431D4—41no—
432D4—42no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—88no—
439D4—81no—
440D4—81no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4C31×#12011yescharacteristic
461C2×C4C31×#12011yescharacteristic
462C2×C4C31×#12111yescharacteristic
463C2×C4C31×#12111yescharacteristic
464C2×C4C31×#12111yescharacteristic
465C2×C4C31×#12711yescharacteristic
466C2×C4C2×D8×C3111yescharacteristic
467C2×C4C31×#13511yescharacteristic
468C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
469C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
470C8—42no—
471C8—42no—
472C8—81no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22C31×#39211yescharacteristic
491C22C31×#44711yescharacteristic
492C22C31×#44911yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—82no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4C31×#44711yescharacteristic
502C4C31×#44911yescharacteristic
503C4C31×#46611yescharacteristic
504C4C31×#46911yescharacteristic
505C2—21no—
506C2—21no—
507C2—82no—
508C2—82no—
509C2—161no—
510C2C31×#199911yescharacteristic
511C2C31×#200811yescharacteristic
512C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3100C2semidirect
2. C31×#3036C2semidirect
3. C31×#2535C2semidirect
4. C31×#1517C2semidirect
5. C31×#1486C2semidirect
6. C31×#1478C2semidirect
7. C31×#1286C2semidirect
45. C31×C4⋊Q8D4semidirect
46. C31×#127D4semidirect
74. C31×C4⋊C8D4semidirect
82. C4×C8×C31D4semidirect
108. C2×D4×C31D8semidirect
153. #9996C31direct (N × H)
217. #3100C2×C31semidirect
218. #3036C2×C31semidirect
219. #2535C2×C31semidirect
220. #1517C2×C31semidirect
221. #1486C2×C31semidirect
222. #1478C2×C31semidirect
223. #1286C2×C31semidirect
301. C4⋊Q8D4×C31semidirect
302. #127D4×C31semidirect
330. C4⋊C8D4×C31semidirect
338. C4×C8D4×C31semidirect
342. C31#9996direct (N × H)
365. C2×D4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31002
C31×#30362
C31×#25352
C31×#15172
C31×#14862
C31×#14782
C31×#12862
#999631