Finite Group Databasebeta

← C31×#33000 #21489759

Subgroups — C31×#33000 #21489759

Order 7936.

1300 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
1C2×C31×#520C211yesmaximal, characteristic
2C31×#3095C211yesmaximal, characteristic, complemented
3C31×#3007C211yesmaximal, characteristic
4C31×#2502C211yesmaximal, characteristic, complemented
5C31×#1882C214yesmaximal, complemented
6C31×#1859C214yesmaximal, complemented
7C31×#1826C211yesmaximal, characteristic, complemented
8C31×#1821C211yesmaximal, characteristic, complemented
9C31×#1789C211yesmaximal, characteristic
10C22×#41×C31C2211yescharacteristic
11C2×C31×#137C2212yes—
12C2×C31×C22⋊Q8C2211yescharacteristic
13C31×#520—22no—
14C31×#520—22no—
15C31×#520—22no—
16C31×#520C2214yes—
17C2×C4×Q8×C31—21no—
18C2×C4×D4×C31—21no—
19C2×C4×D4×C31—22no—
20C2×C31×#117—21no—
21C2×C31×#117C2211yescharacteristic
22C22×C4⋊C4×C31—21no—
23C22×C4⋊C4×C31C2211yescharacteristic
24C31×#485—22no—
25C31×#484—21no—
26C31×#483—21no—
27C31×#452—22no—
28C31×#451—21no—
29C31×#450—21no—
30C31×#428C2212yescomplemented
31C2×C31×#104C2214yes—
32C31×#419C2212yescomplemented
33C31×#417C2212yes—
34C31×#407C2211yescharacteristic
35C31×#407C2211yescharacteristic
36C31×#398—22no—
37C31×#391C2212yescomplemented
38C31×#385—22no—
39C31×#384C2212yescomplemented
40C4×C22⋊C4×C31C2211yescharacteristic
41C31×#343C2214yescomplemented
42C31×#339C2214yescomplemented
43C2×#41×C31—28no—
44C2×#41×C31—21no—
45C2×#41×C31—21no—
46C2×#41×C31D412yes—
47C2×#41×C31C2312yes—
48C22×Q8×C31D411yescharacteristic
49C22×Q8×C31C2311yescharacteristic
50C22×D4×C31—21no—
51C22×D4×C31D411yescharacteristic
52C23×C4×C31—21no—
53C23×C4×C31—22no—
54C23×C4×C31D412yes—
55C23×C4×C31C2311yescharacteristic
56C31×#137—28no—
57C2×Q16×C31—22no—
58C2×SD16×C31—22no—
59C2×M16×C31C2312yes—
60C2×M16×C31C2312yes—
61C31×C22⋊Q8—21no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8—21no—
65C4×Q8×C31—24no—
66C4×Q8×C31—24no—
67C4×D4×C31—28no—
68C4×D4×C31—24no—
69C4×D4×C31—24no—
70C4×D4×C31—28no—
71C31×#117—24no—
72C31×#117—21no—
73C31×#117—21no—
74C31×#117—24no—
75C31×#117D412yes—
76C31×#117C2312yes—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—24no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—24no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—24no—
86C2×C4⋊C4×C31D412yes—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2312yes—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C42×C31—22no—
96C2×C42×C31—21no—
97C2×C42×C31C2312yes—
98C31×#107—22no—
99C31×#106—22no—
100C31×#104—28no—
101C31×#103—22no—
102C31×#103—22no—
103C31×#103—22no—
104C31×#102—22no—
105C31×C22⋊C8—21no—
106C31×C22⋊C8—21no—
107C31×C22⋊C8—21no—
108C31×C22⋊C8—21no—
109C31×#95—21no—
110C31×#95—24no—
111C24×C31—21no—
112C24×C31C2×D411yescharacteristic
113#41×C31—28no—
114#41×C31—28no—
115#41×C31—24no—
116#41×C31—24no—
117#41×C31—24no—
118#41×C31#4118yes—
119C2×Q8×C31—24no—
120C2×Q8×C31—21no—
121C2×Q8×C31—21no—
122C2×Q8×C31—21no—
123C2×Q8×C31—24no—
124C2×Q8×C31C2×D412yes—
125C2×D4×C31—28no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—24no—
129C2×D4×C31—21no—
130C2×D4×C31C2×D412yes—
131C22×C4×C31—28no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—24no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—22no—
140C22×C4×C31—28no—
141C22×C4×C31—22no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—82no—
145C22×C4×C31—84no—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D414yes—
152C22×C4×C31C2411yescharacteristic
153Q16×C31—44no—
154SD16×C31—44no—
155M16×C31—24no—
156M16×C31—24no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C4⋊C4×C31—28no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—41no—
169C4⋊C4×C31—41no—
170C4⋊C4×C31—84no—
171C4⋊C4×C31C2×D412yes—
172C4⋊C4×C31C2×D412yes—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—28no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—41no—
178C22⋊C4×C31—41no—
179C42×C31—28no—
180C42×C31—22no—
181C42×C31—22no—
182C42×C31—24no—
183#33000C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—24no—
185C23×C31—22no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—41no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31#12011yescharacteristic
192C23×C31#12011yescharacteristic
193C23×C31C22×D411yescharacteristic
194Q8×C31—24no—
195Q8×C31—22no—
196Q8×C31—42no—
197Q8×C31—42no—
198Q8×C31#12314yes—
199D4×C31—28no—
200D4×C31—24no—
201D4×C31—28no—
202D4×C31—22no—
203D4×C31#12314yes—
204C2×C4×C31—24no—
205C2×C4×C31—28no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—28no—
212C2×C4×C31—22no—
213C2×C4×C31—24no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—42no—
218C2×C4×C31—44no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—44no—
223C2×C4×C31—42no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—82no—
228C2×C4×C31—88no—
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C2×C4×C31C22×D411yescharacteristic
233C2×C4×C31C2×#4114yes—
234C8×C31—42no—
235C8×C31—42no—
236C2×#520C2×C3111yescharacteristic
237#3095C2×C3111yescharacteristic, complemented
238#3007C2×C3111yescharacteristic
239#2502C2×C3111yescharacteristic, complemented
240#1882C2×C3114yescomplemented
241#1859C2×C3114yescomplemented
242#1826C2×C3111yescharacteristic, complemented
243#1821C2×C3111yescharacteristic, complemented
244#1789C2×C3111yescharacteristic
245C22×C31—21no—
246C22×C31—21no—
247C22×C31—21no—
248C22×C31—21no—
249C22×C31—21no—
250C22×C31—24no—
251C22×C31—21no—
252C22×C31—21no—
253C22×C31—44no—
254C22×C31—48no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31#45612yes—
258C22×C31C2×#12011yescharacteristic
259C22×C31#52512yes—
260C4×C31—21no—
261C4×C31—24no—
262C4×C31—21no—
263C4×C31—44no—
264C4×C31—42no—
265C4×C31—44no—
266C4×C31—42no—
267C4×C31—81no—
268C4×C31—81no—
269C4×C31C2×#12312yes—
270C4×C31#52512yes—
271C22×#41C22×C3111yescharacteristic
272C2×#137C22×C3112yes—
273C2×C22⋊Q8C22×C3111yescharacteristic
274#520—22no—
275#520—22no—
276#520—22no—
277#520C22×C3114yes—
278C2×C4×Q8—21no—
279C2×C4×D4—21no—
280C2×C4×D4—22no—
281C2×#117—21no—
282C2×#117C22×C3111yescharacteristic
283C22×C4⋊C4—21no—
284C22×C4⋊C4C22×C3111yescharacteristic
285#485—22no—
286#484—21no—
287#483—21no—
288#452—22no—
289#451—21no—
290#450—21no—
291#428C22×C3112yescomplemented
292C2×#104C22×C3114yes—
293#419C22×C3112yescomplemented
294#417C22×C3112yes—
295#407C22×C3111yescharacteristic
296#407C22×C3111yescharacteristic
297#398—22no—
298#391C22×C3112yescomplemented
299#385—22no—
300#384C22×C3112yescomplemented
301C4×C22⋊C4C22×C3111yescharacteristic
302#343C22×C3114yescomplemented
303#339C22×C3114yescomplemented
304C2×C31—22no—
305C2×C31—22no—
306C2×C31—41no—
307C2×C31—44no—
308C2×C31#284911yescharacteristic
309C2×C31C2×#45611yescharacteristic
310C2×C31#301911yescharacteristic
311C2×#41—28no—
312C2×#41—21no—
313C2×#41—21no—
314C2×#41D4×C3112yes—
315C2×#41C23×C3112yes—
316C22×Q8D4×C3111yescharacteristic
317C22×Q8C23×C3111yescharacteristic
318C22×D4—21no—
319C22×D4D4×C3111yescharacteristic
320C23×C4—21no—
321C23×C4—22no—
322C23×C4D4×C3112yes—
323C23×C4C23×C3111yescharacteristic
324#137—28no—
325C2×Q16—22no—
326C2×SD16—22no—
327C2×M16C23×C3112yes—
328C2×M16C23×C3112yes—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C4×Q8—24no—
334C4×Q8—24no—
335C4×D4—28no—
336C4×D4—24no—
337C4×D4—24no—
338C4×D4—28no—
339#117—24no—
340#117—21no—
341#117—21no—
342#117—24no—
343#117D4×C3112yes—
344#117C23×C3112yes—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4D4×C3112yes—
355C2×C4⋊C4D4×C3111yescharacteristic
356C2×C4⋊C4D4×C3111yescharacteristic
357C2×C4⋊C4C23×C3111yescharacteristic
358C2×C4⋊C4C23×C3112yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42C23×C3112yes—
366#107—22no—
367#106—22no—
368#104—28no—
369#103—22no—
370#103—22no—
371#103—22no—
372#102—22no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377#95—21no—
378#95—24no—
379C31#3300011yescharacteristic, direct product factor, Sylow
380C24—21no—
381C24C2×D4×C3111yescharacteristic
382#41—28no—
383#41—28no—
384#41—24no—
385#41—24no—
386#41—24no—
387#41#41×C3118yes—
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—24no—
393C2×Q8C2×D4×C3112yes—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4C2×D4×C3112yes—
400C22×C4—28no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—82no—
414C22×C4—84no—
415C22×C4C2×D4×C3111yescharacteristic
416C22×C4C2×D4×C3111yescharacteristic
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C2×D4×C3111yescharacteristic
419C22×C4C2×D4×C3111yescharacteristic
420C22×C4C2×D4×C3114yes—
421C22×C4C24×C3111yescharacteristic
422Q16—44no—
423SD16—44no—
424M16—24no—
425M16—24no—
426C2×C8—22no—
427C2×C8—22no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—84no—
440C4⋊C4C2×D4×C3112yes—
441C4⋊C4C2×D4×C3112yes—
442C22⋊C4—24no—
443C22⋊C4—28no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C42—28no—
449C42—22no—
450C42—22no—
451C42—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23C31×#12011yescharacteristic
460C23C31×#12011yescharacteristic
461C23C22×D4×C3111yescharacteristic
462Q8—24no—
463Q8—22no—
464Q8—42no—
465Q8—42no—
466Q8C31×#12314yes—
467D4—28no—
468D4—24no—
469D4—28no—
470D4—22no—
471D4C31×#12314yes—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—28no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—82no—
496C2×C4—88no—
497C2×C4C31×#12011yescharacteristic
498C2×C4C31×#12011yescharacteristic
499C2×C4C22×D4×C3111yescharacteristic
500C2×C4C22×D4×C3111yescharacteristic
501C2×C4C2×#41×C3114yes—
502C8—42no—
503C8—42no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—24no—
510C22—21no—
511C22—21no—
512C22—44no—
513C22—48no—
514C22—42no—
515C22—42no—
516C22C31×#45612yes—
517C22C2×C31×#12011yescharacteristic
518C22C31×#52512yes—
519C4—21no—
520C4—24no—
521C4—21no—
522C4—44no—
523C4—42no—
524C4—44no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4C2×C31×#12312yes—
529C4C31×#52512yes—
530C2—22no—
531C2—22no—
532C2—41no—
533C2—44no—
534C2C31×#284911yescharacteristic
535C2C2×C31×#45611yescharacteristic
536C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3095C2semidirect
4. C31×#2502C2semidirect
5. C31×#1882C2semidirect
6. C31×#1859C2semidirect
7. C31×#1826C2semidirect
8. C31×#1821C2semidirect
30. C31×#428C22semidirect
32. C31×#419C22semidirect
37. C31×#391C22semidirect
39. C31×#384C22semidirect
41. C31×#343C22semidirect
42. C31×#339C22semidirect
183. #33000C31direct (N × H)
237. #3095C2×C31semidirect
239. #2502C2×C31semidirect
240. #1882C2×C31semidirect
241. #1859C2×C31semidirect
242. #1826C2×C31semidirect
243. #1821C2×C31semidirect
291. #428C22×C31semidirect
293. #419C22×C31semidirect
298. #391C22×C31semidirect
300. #384C22×C31semidirect
302. #343C22×C31semidirect
303. #339C22×C31semidirect
379. C31#33000direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5202
C31×#30952
C31×#30072
C31×#25022
C31×#18822
C31×#18592
C31×#18262
C31×#18212
C31×#17892
#3300031