Finite Group Databasebeta

← C31×#9946 #21466705

Subgroups — C31×#9946 #21466705

Order 7936.

1060 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3186C211yesmaximal, characteristic, complemented
2C2×C31×#447C211yesmaximal, characteristic, complemented
3C31×#2433C211yesmaximal, characteristic, complemented
4C31×#1964C211yesmaximal, characteristic, complemented
5C31×#1786C211yesmaximal, characteristic, complemented
6C31×#1519C211yesmaximal, characteristic, complemented
7C31×#1282C211yesmaximal, characteristic, complemented
8C23×D4×C31C2211yescharacteristic
9C22×D8×C31—21no—
10C2×C31×#127—21no—
11C2×C31×#121C2211yescharacteristic
12C2×C31×#121C2211yescharacteristic
13C2×C31×#120—22no—
14C31×#480—21no—
15C31×#480—21no—
16C31×#466—22no—
17C31×#447—21no—
18C31×#447—22no—
19C31×#447—21no—
20C31×#447—22no—
21C2×C31×#102—21no—
22C2×C31×#102—22no—
23C2×C31×#99—22no—
24C2×C31×C22⋊C8C2211yescharacteristic
25C2×C31×C22⋊C8C2211yescharacteristic
26C31×#392—22no—
27C31×#390—21no—
28C31×#380C2211yescharacteristic
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#327—22no—
31C25×C31—21no—
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31—24no—
35C22×D4×C31—21no—
36C22×D4×C31—21no—
37C22×D4×C31—21no—
38C22×D4×C31—21no—
39C22×D4×C31—21no—
40C22×D4×C31—44no—
41C22×D4×C31D411yescharacteristic
42C22×D4×C31D412yes—
43C22×D4×C31D411yescharacteristic, complemented
44C23×C4×C31C2311yescharacteristic
45C2×D8×C31—24no—
46C2×D8×C31—22no—
47C2×D8×C31—44no—
48C22×C8×C31—21no—
49C22×C8×C31D412yescomplemented
50C31×#127—44no—
51C31×#121—21no—
52C31×#121—21no—
53C31×#121—21no—
54C31×#121—21no—
55C31×#121—21no—
56C31×#121—21no—
57C31×#121—21no—
58C31×#121—21no—
59C31×#120—44no—
60C31×#120—44no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31D411yescharacteristic
63C2×C4⋊C4×C31D411yescharacteristic, complemented
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—21no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31D412yes—
69C2×C42×C31—21no—
70C31×#107—22no—
71C31×#102—24no—
72C31×#102—22no—
73C31×#102—22no—
74C31×#102—44no—
75C31×#99—44no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8—21no—
79C31×C22⋊C8D412yescomplemented
80C31×#95—21no—
81C31×#95—21no—
82C24×C31—22no—
83C24×C31—21no—
84C24×C31—21no—
85C24×C31—44no—
86C24×C31—41no—
87C24×C31—41no—
88C24×C31—42no—
89C24×C31—42no—
90C24×C31—82no—
91C24×C31C2×D411yescharacteristic
92C2×D4×C31—24no—
93C2×D4×C31—24no—
94C2×D4×C31—24no—
95C2×D4×C31—22no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—22no—
100C2×D4×C31—21no—
101C2×D4×C31—42no—
102C2×D4×C31—42no—
103C2×D4×C31—42no—
104C2×D4×C31—42no—
105C2×D4×C31—42no—
106C2×D4×C31—48no—
107C2×D4×C31—44no—
108C2×D4×C31—44no—
109C2×D4×C31—44no—
110C2×D4×C31—48no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—41no—
114C2×D4×C31—41no—
115C2×D4×C31—42no—
116C2×D4×C31—42no—
117C2×D4×C31—42no—
118C2×D4×C31—42no—
119C2×D4×C31—41no—
120C2×D4×C31—41no—
121C2×D4×C31—84no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—42no—
126C22×C4×C31—41no—
127C22×C4×C31C2×D411yescharacteristic
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31#4111yescharacteristic
133D8×C31—48no—
134D8×C31—82no—
135C2×C8×C31—22no—
136C2×C8×C31—22no—
137C2×C8×C31—42no—
138C2×C8×C31—42no—
139C4⋊C4×C31—21no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—21no—
142C4⋊C4×C31—21no—
143C4⋊C4×C31—42no—
144C22⋊C4×C31—22no—
145C22⋊C4×C31—48no—
146C22⋊C4×C31—41no—
147C22⋊C4×C31—41no—
148C22⋊C4×C31—41no—
149C22⋊C4×C31—41no—
150C22⋊C4×C31—42no—
151C22⋊C4×C31D814yescomplemented
152C42×C31—42no—
153#9946C3111yesmaximal, characteristic, direct product factor, Sylow
154C23×C31—24no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—44no—
158C23×C31—44no—
159C23×C31—42no—
160C23×C31—44no—
161C23×C31—42no—
162C23×C31—42no—
163C23×C31—42no—
164C23×C31—44no—
165C23×C31—41no—
166C23×C31—41no—
167C23×C31—42no—
168C23×C31—42no—
169C23×C31—82no—
170C23×C31—82no—
171C23×C31—81no—
172C23×C31—81no—
173C23×C31—88no—
174C23×C31—88no—
175C23×C31—82no—
176C23×C31—81no—
177C23×C31—81no—
178C23×C31#12011yescharacteristic
179C23×C31#12111yescharacteristic
180C23×C31#12711yescharacteristic
181C23×C31C2×D812yes—
182C23×C31AGL(1,ℤ/8)12yes—
183D4×C31—24no—
184D4×C31—24no—
185D4×C31—42no—
186D4×C31—42no—
187D4×C31—44no—
188D4×C31—42no—
189D4×C31—44no—
190D4×C31—42no—
191D4×C31—42no—
192D4×C31—81no—
193D4×C31—816no—
194D4×C31—82no—
195D4×C31—82no—
196D4×C31—81no—
197D4×C31—81no—
198D4×C31—81no—
199D4×C31—81no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—24no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—81no—
214C2×C4×C31—81no—
215C2×C4×C31—81no—
216C2×C4×C31#12011yescharacteristic
217C2×C4×C31#12011yescharacteristic
218C2×C4×C31#12111yescharacteristic
219C2×C4×C31#12111yescharacteristic
220C8×C31—42no—
221C8×C31—42no—
222#3186C2×C3111yescharacteristic, complemented
223C2×#447C2×C3111yescharacteristic, complemented
224#2433C2×C3111yescharacteristic, complemented
225#1964C2×C3111yescharacteristic, complemented
226#1786C2×C3111yescharacteristic, complemented
227#1519C2×C3111yescharacteristic, complemented
228#1282C2×C3111yescharacteristic, complemented
229C22×C31—21no—
230C22×C31—21no—
231C22×C31—21no—
232C22×C31—21no—
233C22×C31—21no—
234C22×C31—21no—
235C22×C31—42no—
236C22×C31—42no—
237C22×C31—48no—
238C22×C31—44no—
239C22×C31—84no—
240C22×C31—84no—
241C22×C31—88no—
242C22×C31—84no—
243C22×C31—84no—
244C22×C31—82no—
245C22×C31—82no—
246C22×C31—81no—
247C22×C31—81no—
248C22×C31—81no—
249C22×C31—81no—
250C22×C31—81no—
251C22×C31—81no—
252C22×C31—162no—
253C22×C31#39211yescharacteristic
254C22×C31#44712yes—
255C22×C31#45912yes—
256C22×C31#49311yescharacteristic
257C22×C31#49611yescharacteristic
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—44no—
262C4×C31—81no—
263C4×C31—81no—
264C4×C31—161no—
265C4×C31B(4,𝔽2)12yes—
266C23×D4C22×C3111yescharacteristic
267C22×D8—21no—
268C2×#127—21no—
269C2×#121C22×C3111yescharacteristic
270C2×#121C22×C3111yescharacteristic
271C2×#120—22no—
272#480—21no—
273#480—21no—
274#466—22no—
275#447—21no—
276#447—22no—
277#447—21no—
278#447—22no—
279C2×#102—21no—
280C2×#102—22no—
281C2×#99—22no—
282C2×C22⋊C8C22×C3111yescharacteristic
283C2×C22⋊C8C22×C3111yescharacteristic
284#392—22no—
285#390—21no—
286#380C22×C3111yescharacteristic
287C4×C22⋊C4C22×C3111yescharacteristic
288#327—22no—
289C2×C31—22no—
290C2×C31—21no—
291C2×C31—22no—
292C2×C31—21no—
293C2×C31—84no—
294C2×C31—161no—
295C2×C31—161no—
296C2×C31#200811yescharacteristic
297C2×C31#201111yescharacteristic
298C2×C31#202111yescharacteristic
299C25—21no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—24no—
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—21no—
306C22×D4—21no—
307C22×D4—21no—
308C22×D4—44no—
309C22×D4D4×C3111yescharacteristic
310C22×D4D4×C3112yes—
311C22×D4D4×C3111yescharacteristic, complemented
312C23×C4C23×C3111yescharacteristic
313C2×D8—24no—
314C2×D8—22no—
315C2×D8—44no—
316C22×C8—21no—
317C22×C8D4×C3112yescomplemented
318#127—44no—
319#121—21no—
320#121—21no—
321#121—21no—
322#121—21no—
323#121—21no—
324#121—21no—
325#121—21no—
326#121—21no—
327#120—44no—
328#120—44no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4D4×C3111yescharacteristic
331C2×C4⋊C4D4×C3111yescharacteristic, complemented
332C2×C22⋊C4—24no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4D4×C3112yes—
337C2×C42—21no—
338#107—22no—
339#102—24no—
340#102—22no—
341#102—22no—
342#102—44no—
343#99—44no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8D4×C3112yescomplemented
348#95—21no—
349#95—21no—
350C31#994611yescharacteristic, direct product factor, Sylow
351C24—22no—
352C24—21no—
353C24—21no—
354C24—44no—
355C24—41no—
356C24—41no—
357C24—42no—
358C24—42no—
359C24—82no—
360C24C2×D4×C3111yescharacteristic
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—48no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—84no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4C2×D4×C3111yescharacteristic
397C22×C4C2×D4×C3111yescharacteristic
398C22×C4C2×D4×C3111yescharacteristic
399C22×C4C2×D4×C3111yescharacteristic
400C22×C4C2×D4×C3111yescharacteristic
401C22×C4#41×C3111yescharacteristic
402D8—48no—
403D8—82no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—42no—
407C2×C8—42no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—42no—
413C22⋊C4—22no—
414C22⋊C4—48no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—42no—
420C22⋊C4D8×C3114yescomplemented
421C42—42no—
422C23—24no—
423C23—41no—
424C23—41no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—88no—
442C23—88no—
443C23—82no—
444C23—81no—
445C23—81no—
446C23C31×#12011yescharacteristic
447C23C31×#12111yescharacteristic
448C23C31×#12711yescharacteristic
449C23C2×D8×C3112yes—
450C23C31×AGL(1,ℤ/8)12yes—
451D4—24no—
452D4—24no—
453D4—42no—
454D4—42no—
455D4—44no—
456D4—42no—
457D4—44no—
458D4—42no—
459D4—42no—
460D4—81no—
461D4—816no—
462D4—82no—
463D4—82no—
464D4—81no—
465D4—81no—
466D4—81no—
467D4—81no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4C31×#12011yescharacteristic
485C2×C4C31×#12011yescharacteristic
486C2×C4C31×#12111yescharacteristic
487C2×C4C31×#12111yescharacteristic
488C8—42no—
489C8—42no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—42no—
497C22—42no—
498C22—48no—
499C22—44no—
500C22—84no—
501C22—84no—
502C22—88no—
503C22—84no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—162no—
514C22C31×#39211yescharacteristic
515C22C31×#44712yes—
516C22C31×#45912yes—
517C22C31×#49311yescharacteristic
518C22C31×#49611yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—44no—
523C4—81no—
524C4—81no—
525C4—161no—
526C4C31×B(4,𝔽2)12yes—
527C2—22no—
528C2—21no—
529C2—22no—
530C2—21no—
531C2—84no—
532C2—161no—
533C2—161no—
534C2C31×#200811yescharacteristic
535C2C31×#201111yescharacteristic
536C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
LCSG ▷ C23×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. C31×#3186C2semidirect
2. C2×C31×#447C2semidirect
3. C31×#2433C2semidirect
4. C31×#1964C2semidirect
5. C31×#1786C2semidirect
6. C31×#1519C2semidirect
7. C31×#1282C2semidirect
43. C22×D4×C31D4semidirect
49. C22×C8×C31D4semidirect
63. C2×C4⋊C4×C31D4semidirect
79. C31×C22⋊C8D4semidirect
151. C22⋊C4×C31D8semidirect
153. #9946C31direct (N × H)
222. #3186C2×C31semidirect
223. C2×#447C2×C31semidirect
224. #2433C2×C31semidirect
225. #1964C2×C31semidirect
226. #1786C2×C31semidirect
227. #1519C2×C31semidirect
228. #1282C2×C31semidirect
311. C22×D4D4×C31semidirect
317. C22×C8D4×C31semidirect
331. C2×C4⋊C4D4×C31semidirect
347. C22⋊C8D4×C31semidirect
350. C31#9946direct (N × H)
420. C22⋊C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31862
C2×C31×#4472
C31×#24332
C31×#19642
C31×#17862
C31×#15192
C31×#12822
#994631