Finite Group Databasebeta

← C31×#32997 #21489756

Subgroups — C31×#32997 #21489756

Order 7936.

1326 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#520C211yesmaximal, characteristic, complemented
2C31×#3094C211yesmaximal, characteristic, complemented
3C31×#3006C211yesmaximal, characteristic
4C31×#2504C211yesmaximal, characteristic, complemented
5C31×#1883C214yesmaximal, complemented
6C31×#1860C214yesmaximal, complemented
7C31×#1826C211yesmaximal, characteristic, complemented
8C31×#1821C211yesmaximal, characteristic, complemented
9C31×#1789C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C2×C31×AGL(1,ℤ/8)C2212yes—
12C2×C31×#121C2211yescharacteristic
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×#482—22no—
25C31×#481—21no—
26C31×#480—21no—
27C31×#449—22no—
28C31×#448—21no—
29C31×#447—21no—
30C31×#428C2212yescomplemented
31C2×C31×#104C2214yes—
32C31×#418C2212yescomplemented
33C31×#417C2212yes—
34C31×#407C2211yescharacteristic
35C31×#407C2211yescharacteristic
36C31×#397—22no—
37C31×#390C2212yescomplemented
38C31×#386—22no—
39C31×#382C2212yescomplemented
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×D4×C31—21no—
50C22×D4×C31—21no—
51C22×D4×C31D411yescharacteristic
52C22×D4×C31C2311yescharacteristic
53C23×C4×C31—21no—
54C23×C4×C31—22no—
55C23×C4×C31D412yes—
56C23×C4×C31C2311yescharacteristic
57C31×AGL(1,ℤ/8)—28no—
58C2×SD16×C31—22no—
59C2×D8×C31—22no—
60C2×M16×C31C2312yes—
61C2×M16×C31C2312yes—
62C31×#121—21no—
63C31×#121—21no—
64C31×#121—21no—
65C31×#121—21no—
66C4×Q8×C31—24no—
67C4×Q8×C31—24no—
68C4×D4×C31—28no—
69C4×D4×C31—24no—
70C4×D4×C31—24no—
71C4×D4×C31—28no—
72C31×#117—24no—
73C31×#117—21no—
74C31×#117—21no—
75C31×#117—24no—
76C31×#117D412yes—
77C31×#117C2312yes—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—24no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—24no—
85C2×C4⋊C4×C31D412yes—
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2312yes—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—24no—
95C2×C42×C31—22no—
96C2×C42×C31—21no—
97C2×C42×C31C2312yes—
98C31×#107—22no—
99C31×#106—22no—
100C31×#104—28no—
101C31×#103—22no—
102C31×#102—22no—
103C31×#102—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×C31—21no—
113C24×C31C2×D411yescharacteristic
114#41×C31—28no—
115#41×C31—28no—
116#41×C31—24no—
117#41×C31—24no—
118#41×C31—24no—
119#41×C31#4118yes—
120C2×Q8×C31—24no—
121C2×Q8×C31—21no—
122C2×Q8×C31C2×D412yes—
123C2×D4×C31—28no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—24no—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—24no—
131C2×D4×C31—44no—
132C2×D4×C31—41no—
133C2×D4×C31—41no—
134C2×D4×C31C2×D412yes—
135C22×C4×C31—28no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—24no—
140C22×C4×C31—21no—
141C22×C4×C31—22no—
142C22×C4×C31—21no—
143C22×C4×C31—28no—
144C22×C4×C31—22no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—82no—
148C22×C4×C31—84no—
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D414yes—
155C22×C4×C31C2411yescharacteristic
156SD16×C31—44no—
157D8×C31—44no—
158M16×C31—24no—
159M16×C31—24no—
160C2×C8×C31—22no—
161C2×C8×C31—22no—
162C4⋊C4×C31—28no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31C2×D412yes—
172C4⋊C4×C31C2×D412yes—
173C22⋊C4×C31—28no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—42no—
177C22⋊C4×C31—41no—
178C22⋊C4×C31—41no—
179C22⋊C4×C31—84no—
180C42×C31—28no—
181C42×C31—22no—
182C42×C31—22no—
183C42×C31—24no—
184#32997C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—24no—
186C23×C31—22no—
187C23×C31—22no—
188C23×C31—22no—
189C23×C31—41no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—41no—
195C23×C31—42no—
196C23×C31#12011yescharacteristic
197C23×C31#12011yescharacteristic
198C23×C31C22×D411yescharacteristic
199Q8×C31—24no—
200Q8×C31—22no—
201Q8×C31#12314yes—
202D4×C31—28no—
203D4×C31—24no—
204D4×C31—28no—
205D4×C31—22no—
206D4×C31—42no—
207D4×C31—42no—
208D4×C31—82no—
209D4×C31—82no—
210D4×C31#12314yes—
211C2×C4×C31—24no—
212C2×C4×C31—28no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—28no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—24no—
222C2×C4×C31—22no—
223C2×C4×C31—44no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—44no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—82no—
233C2×C4×C31—88no—
234C2×C4×C31#12011yescharacteristic
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31C22×D411yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C2×C4×C31C2×#4114yes—
239C8×C31—42no—
240C8×C31—42no—
241C2×#520C2×C3111yescharacteristic, complemented
242#3094C2×C3111yescharacteristic, complemented
243#3006C2×C3111yescharacteristic
244#2504C2×C3111yescharacteristic, complemented
245#1883C2×C3114yescomplemented
246#1860C2×C3114yescomplemented
247#1826C2×C3111yescharacteristic, complemented
248#1821C2×C3111yescharacteristic, complemented
249#1789C2×C3111yescharacteristic, complemented
250C22×C31—21no—
251C22×C31—21no—
252C22×C31—21no—
253C22×C31—21no—
254C22×C31—21no—
255C22×C31—24no—
256C22×C31—21no—
257C22×C31—21no—
258C22×C31—44no—
259C22×C31—48no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—84no—
263C22×C31—81no—
264C22×C31—81no—
265C22×C31—81no—
266C22×C31#45412yes—
267C22×C31C2×#12011yescharacteristic
268C22×C31#52512yes—
269C4×C31—21no—
270C4×C31—24no—
271C4×C31—21no—
272C4×C31—42no—
273C4×C31—44no—
274C4×C31—42no—
275C4×C31—82no—
276C4×C31—81no—
277C4×C31C2×#12312yes—
278C4×C31#52512yes—
279C22×#41C22×C3111yescharacteristic
280C2×AGL(1,ℤ/8)C22×C3112yes—
281C2×#121C22×C3111yescharacteristic
282#520—22no—
283#520—22no—
284#520—22no—
285#520C22×C3114yes—
286C2×C4×Q8—21no—
287C2×C4×D4—21no—
288C2×C4×D4—22no—
289C2×#117—21no—
290C2×#117C22×C3111yescharacteristic
291C22×C4⋊C4—21no—
292C22×C4⋊C4C22×C3111yescharacteristic
293#482—22no—
294#481—21no—
295#480—21no—
296#449—22no—
297#448—21no—
298#447—21no—
299#428C22×C3112yescomplemented
300C2×#104C22×C3114yes—
301#418C22×C3112yescomplemented
302#417C22×C3112yes—
303#407C22×C3111yescharacteristic
304#407C22×C3111yescharacteristic
305#397—22no—
306#390C22×C3112yescomplemented
307#386—22no—
308#382C22×C3112yescomplemented
309C4×C22⋊C4C22×C3111yescharacteristic
310#343C22×C3114yescomplemented
311#339C22×C3114yescomplemented
312C2×C31—22no—
313C2×C31—22no—
314C2×C31—41no—
315C2×C31—44no—
316C2×C31—161no—
317C2×C31#284911yescharacteristic
318C2×C31C2×#45411yescharacteristic
319C2×C31#301911yescharacteristic
320C2×#41—28no—
321C2×#41—21no—
322C2×#41—21no—
323C2×#41D4×C3112yes—
324C2×#41C23×C3112yes—
325C22×Q8D4×C3111yescharacteristic
326C22×D4—21no—
327C22×D4—21no—
328C22×D4D4×C3111yescharacteristic
329C22×D4C23×C3111yescharacteristic
330C23×C4—21no—
331C23×C4—22no—
332C23×C4D4×C3112yes—
333C23×C4C23×C3111yescharacteristic
334AGL(1,ℤ/8)—28no—
335C2×SD16—22no—
336C2×D8—22no—
337C2×M16C23×C3112yes—
338C2×M16C23×C3112yes—
339#121—21no—
340#121—21no—
341#121—21no—
342#121—21no—
343C4×Q8—24no—
344C4×Q8—24no—
345C4×D4—28no—
346C4×D4—24no—
347C4×D4—24no—
348C4×D4—28no—
349#117—24no—
350#117—21no—
351#117—21no—
352#117—24no—
353#117D4×C3112yes—
354#117C23×C3112yes—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4D4×C3112yes—
363C2×C4⋊C4D4×C3111yescharacteristic
364C2×C4⋊C4D4×C3111yescharacteristic
365C2×C4⋊C4C23×C3111yescharacteristic
366C2×C4⋊C4C23×C3112yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—24no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42C23×C3112yes—
375#107—22no—
376#106—22no—
377#104—28no—
378#103—22no—
379#102—22no—
380#102—22no—
381#102—22no—
382C22⋊C8—21no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385C22⋊C8—21no—
386#95—21no—
387#95—24no—
388C31#3299711yescharacteristic, direct product factor, Sylow
389C24—21no—
390C24—21no—
391C24C2×D4×C3111yescharacteristic
392#41—28no—
393#41—28no—
394#41—24no—
395#41—24no—
396#41—24no—
397#41#41×C3118yes—
398C2×Q8—24no—
399C2×Q8—21no—
400C2×Q8C2×D4×C3112yes—
401C2×D4—28no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—24no—
409C2×D4—44no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4C2×D4×C3112yes—
413C22×C4—28no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—82no—
426C22×C4—84no—
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C2×D4×C3111yescharacteristic
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3114yes—
433C22×C4C24×C3111yescharacteristic
434SD16—44no—
435D8—44no—
436M16—24no—
437M16—24no—
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—28no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4C2×D4×C3112yes—
450C4⋊C4C2×D4×C3112yes—
451C22⋊C4—28no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—84no—
458C42—28no—
459C42—22no—
460C42—22no—
461C42—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23C31×#12011yescharacteristic
474C23C31×#12011yescharacteristic
475C23C22×D4×C3111yescharacteristic
476Q8—24no—
477Q8—22no—
478Q8C31×#12314yes—
479D4—28no—
480D4—24no—
481D4—28no—
482D4—22no—
483D4—42no—
484D4—42no—
485D4—82no—
486D4—82no—
487D4C31×#12314yes—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—82no—
510C2×C4—88no—
511C2×C4C31×#12011yescharacteristic
512C2×C4C31×#12011yescharacteristic
513C2×C4C22×D4×C3111yescharacteristic
514C2×C4C22×D4×C3111yescharacteristic
515C2×C4C2×#41×C3114yes—
516C8—42no—
517C8—42no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—24no—
524C22—21no—
525C22—21no—
526C22—44no—
527C22—48no—
528C22—42no—
529C22—42no—
530C22—84no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22C31×#45412yes—
535C22C2×C31×#12011yescharacteristic
536C22C31×#52512yes—
537C4—21no—
538C4—24no—
539C4—21no—
540C4—42no—
541C4—44no—
542C4—42no—
543C4—82no—
544C4—81no—
545C4C2×C31×#12312yes—
546C4C31×#52512yes—
547C2—22no—
548C2—22no—
549C2—41no—
550C2—44no—
551C2—161no—
552C2C31×#284911yescharacteristic
553C2C2×C31×#45411yescharacteristic
554C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3006C2
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
1. C2×C31×#520C2semidirect
2. C31×#3094C2semidirect
4. C31×#2504C2semidirect
5. C31×#1883C2semidirect
6. C31×#1860C2semidirect
7. C31×#1826C2semidirect
8. C31×#1821C2semidirect
9. C31×#1789C2semidirect
30. C31×#428C22semidirect
32. C31×#418C22semidirect
37. C31×#390C22semidirect
39. C31×#382C22semidirect
41. C31×#343C22semidirect
42. C31×#339C22semidirect
184. #32997C31direct (N × H)
241. C2×#520C2×C31semidirect
242. #3094C2×C31semidirect
244. #2504C2×C31semidirect
245. #1883C2×C31semidirect
246. #1860C2×C31semidirect
247. #1826C2×C31semidirect
248. #1821C2×C31semidirect
249. #1789C2×C31semidirect
299. #428C22×C31semidirect
301. #418C22×C31semidirect
306. #390C22×C31semidirect
308. #382C22×C31semidirect
310. #343C22×C31semidirect
311. #339C22×C31semidirect
388. C31#32997direct (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×#30942
C31×#30062
C31×#25042
C31×#18832
C31×#18602
C31×#18262
C31×#18212
C31×#17892
#3299731