Finite Group Databasebeta

← C31×#18839 #21475598

Subgroups — C31×#18839 #21475598

Order 7936.

936 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic, complemented
2C31×#3254C211yesmaximal, characteristic, complemented
3C31×#3210C211yesmaximal, characteristic, complemented
4C31×#3109C211yesmaximal, characteristic
5C31×#1637C212yesmaximal, complemented
6C31×#1632C212yesmaximal, complemented
7C31×#1627C212yesmaximal, complemented
8C31×#1622C212yesmaximal, complemented
9C31×#1559C211yesmaximal, characteristic, complemented
10C31×#1497C211yesmaximal, characteristic, complemented
11C31×#1486C211yesmaximal, characteristic, complemented
12C31×#554—21no—
13C31×#554—22no—
14C31×#554—22no—
15C31×#554C2212yes—
16D4×Q8×C31—21no—
17D4×Q8×C31C2212yes—
18C31×#544—21no—
19C31×#543—22no—
20C31×#533C2211yescharacteristic
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C2×C31×C22⋊Q8—22no—
24C31×#518—21no—
25C2×C4×D4×C31C2211yescharacteristic
26C31×#494C2211yescharacteristic, complemented
27C31×#492C2211yescharacteristic, complemented
28C31×#487C2211yescharacteristic, complemented
29C31×#486C2211yescharacteristic, complemented
30C31×#481—21no—
31C31×#477C2212yescomplemented
32C31×#476C2212yes—
33C31×#474—21no—
34C31×#468—22no—
35C31×#464—22no—
36C31×#461—21no—
37C31×#460C2212yescomplemented
38C31×#459C2212yes—
39C31×#452—22no—
40C31×#447—21no—
41C31×#442—22no—
42C31×#432C2211yescharacteristic
43C31×#431C2211yescharacteristic
44C31×#423C2211yescharacteristic
45C31×#333C2212yescomplemented
46C31×#332C2212yescomplemented
47C31×Q8⋊C8C2212yescomplemented
48C31×Q8⋊C8C2212yescomplemented
49C31×D4⋊C8C2212yes—
50C31×D4⋊C8C2212yes—
51C2×#41×C31—21no—
52C22×Q8×C31—22no—
53C22×D4×C31D411yescharacteristic
54C23×C4×C31—21no—
55C2×Q16×C31—22no—
56C2×SD16×C31—22no—
57C2×SD16×C31—22no—
58C2×D8×C31—22no—
59C2×M16×C31—21no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8D411yescharacteristic
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8C2311yescharacteristic
65C31×C4⋊Q8C2311yescharacteristic
66C31×C4⋊Q8C2311yescharacteristic
67C31×#127C2311yescharacteristic
68C31×#125—22no—
69C31×#125—22no—
70C31×#125—21no—
71C31×#125—21no—
72C31×#124—21no—
73C31×C22⋊Q8—24no—
74C31×C22⋊Q8—24no—
75C31×C22⋊Q8—24no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—21no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—24no—
83C31×C22⋊Q8D412yes—
84C31×C22⋊Q8D412yes—
85C31×#121—21no—
86C31×#121—21no—
87C4×Q8×C31—21no—
88C4×Q8×C31—21no—
89C4×Q8×C31D412yes—
90C4×Q8×C31C2312yes—
91C4×D4×C31—22no—
92C4×D4×C31—22no—
93C4×D4×C31—22no—
94C4×D4×C31—22no—
95C4×D4×C31—21no—
96C4×D4×C31—21no—
97C4×D4×C31—21no—
98C4×D4×C31—21no—
99C4×D4×C31D412yes—
100C4×D4×C31C2312yes—
101C31×#117—21no—
102C31×#117—21no—
103C2×C4⋊C4×C31—24no—
104C2×C4⋊C4×C31—22no—
105C2×C4⋊C4×C31D411yescharacteristic
106C2×C22⋊C4×C31—24no—
107C2×C22⋊C4×C31—21no—
108C2×C42×C31C2311yescharacteristic
109C31×#107—22no—
110C31×#106—22no—
111C31×C4⋊C8—21no—
112C31×C4⋊C8—21no—
113C31×C4⋊C8C2312yes—
114C31×C4⋊C8C2312yes—
115C31×#103—22no—
116C31×#103—22no—
117C31×#102—22no—
118C31×#102—22no—
119C31×#102—22no—
120C31×#102—22no—
121C31×C22⋊C8—21no—
122C31×C22⋊C8—21no—
123C31×#97—21no—
124C4×C8×C31C2311yescharacteristic
125C4×C8×C31C2311yescharacteristic
126C24×C31—21no—
127#41×C31—82no—
128C2×Q8×C31—24no—
129C2×Q8×C31—21no—
130C2×Q8×C31—21no—
131C2×Q8×C31—21no—
132C2×Q8×C31—82no—
133C2×Q8×C31—82no—
134C2×Q8×C31C2×D412yes—
135C2×Q8×C31C2×D412yes—
136C2×D4×C31—24no—
137C2×D4×C31—21no—
138C2×D4×C31—21no—
139C2×D4×C31—21no—
140C2×D4×C31—21no—
141C2×D4×C31—41no—
142C2×D4×C31C2×D412yes—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—41no—
149C22×C4×C31—81no—
150C22×C4×C31—81no—
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154Q16×C31—82no—
155SD16×C31—82no—
156SD16×C31—82no—
157D8×C31—44no—
158M16×C31—81no—
159C2×C8×C31—21no—
160C2×C8×C31—21no—
161C2×C8×C31—22no—
162C2×C8×C31—22no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—24no—
165C4⋊C4×C31—24no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31C2×D412yes—
180C4⋊C4×C31C2×D412yes—
181C4⋊C4×C31C2×D412yes—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—41no—
189C22⋊C4×C31—41no—
190C42×C31—22no—
191C42×C31—22no—
192C42×C31C2×D411yescharacteristic
193C42×C31C2×D411yescharacteristic
194C42×C31C2×D411yescharacteristic
195C42×C31C2411yescharacteristic
196#18839C3111yesmaximal, characteristic, direct product factor, Sylow
197C23×C31—24no—
198C23×C31—22no—
199C23×C31—41no—
200C23×C31—81no—
201C23×C31—81no—
202C23×C31#12011yescharacteristic
203Q8×C31—22no—
204Q8×C31—22no—
205Q8×C31—42no—
206Q8×C31—42no—
207Q8×C31—41no—
208Q8×C31—41no—
209Q8×C31—81no—
210Q8×C31—84no—
211D4×C31—24no—
212D4×C31—22no—
213D4×C31—42no—
214D4×C31—42no—
215D4×C31—41no—
216D4×C31—41no—
217D4×C31—82no—
218D4×C31#13714yes—
219C2×C4×C31—22no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—41no—
231C2×C4×C31—82no—
232C2×C4×C31—82no—
233C2×C4×C31—81no—
234C2×C4×C31—84no—
235C2×C4×C31—84no—
236C2×C4×C31#12011yescharacteristic
237C2×C4×C31#12011yescharacteristic
238C2×C4×C31#12011yescharacteristic
239C2×C4×C31C22×D411yescharacteristic
240C2×C4×C31C22×D411yescharacteristic
241C2×C4×C31C22×D411yescharacteristic
242C8×C31—41no—
243C8×C31—41no—
244C8×C31—42no—
245C8×C31—42no—
246#3501C2×C3111yescharacteristic, complemented
247#3254C2×C3111yescharacteristic, complemented
248#3210C2×C3111yescharacteristic, complemented
249#3109C2×C3111yescharacteristic
250#1637C2×C3112yescomplemented
251#1632C2×C3112yescomplemented
252#1627C2×C3112yescomplemented
253#1622C2×C3112yescomplemented
254#1559C2×C3111yescharacteristic, complemented
255#1497C2×C3111yescharacteristic, complemented
256#1486C2×C3111yescharacteristic, complemented
257C22×C31—21no—
258C22×C31—24no—
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—84no—
264C22×C31—81no—
265C22×C31—81no—
266C22×C31—81no—
267C22×C31C2×#12011yescharacteristic
268C4×C31—21no—
269C4×C31—21no—
270C4×C31—21no—
271C4×C31—21no—
272C4×C31—42no—
273C4×C31—42no—
274C4×C31—42no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—81no—
278C4×C31C2×#13712yes—
279C4×C31#57611yescharacteristic
280C4×C31#57711yescharacteristic
281#554—21no—
282#554—22no—
283#554—22no—
284#554C22×C3112yes—
285D4×Q8—21no—
286D4×Q8C22×C3112yes—
287#544—21no—
288#543—22no—
289#533C22×C3111yescharacteristic
290#533C22×C3111yescharacteristic
291#532C22×C3111yescharacteristic
292C2×C22⋊Q8—22no—
293#518—21no—
294C2×C4×D4C22×C3111yescharacteristic
295#494C22×C3111yescharacteristic, complemented
296#492C22×C3111yescharacteristic, complemented
297#487C22×C3111yescharacteristic, complemented
298#486C22×C3111yescharacteristic, complemented
299#481—21no—
300#477C22×C3112yescomplemented
301#476C22×C3112yes—
302#474—21no—
303#468—22no—
304#464—22no—
305#461—21no—
306#460C22×C3112yescomplemented
307#459C22×C3112yes—
308#452—22no—
309#447—21no—
310#442—22no—
311#432C22×C3111yescharacteristic
312#431C22×C3111yescharacteristic
313#423C22×C3111yescharacteristic
314#333C22×C3112yescomplemented
315#332C22×C3112yescomplemented
316Q8⋊C8C22×C3112yescomplemented
317Q8⋊C8C22×C3112yescomplemented
318D4⋊C8C22×C3112yes—
319D4⋊C8C22×C3112yes—
320C2×C31—41no—
321C2×C31—44no—
322C2×C31—161no—
323C2×C31#300411yescharacteristic
324C2×C31#301211yescharacteristic
325C2×C31#302111yescharacteristic
326C2×#41—21no—
327C22×Q8—22no—
328C22×D4D4×C3111yescharacteristic
329C23×C4—21no—
330C2×Q16—22no—
331C2×SD16—22no—
332C2×SD16—22no—
333C2×D8—22no—
334C2×M16—21no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337C4⋊Q8D4×C3111yescharacteristic
338C4⋊Q8D4×C3111yescharacteristic
339C4⋊Q8C23×C3111yescharacteristic
340C4⋊Q8C23×C3111yescharacteristic
341C4⋊Q8C23×C3111yescharacteristic
342#127C23×C3111yescharacteristic
343#125—22no—
344#125—22no—
345#125—21no—
346#125—21no—
347#124—21no—
348C22⋊Q8—24no—
349C22⋊Q8—24no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—24no—
358C22⋊Q8D4×C3112yes—
359C22⋊Q8D4×C3112yes—
360#121—21no—
361#121—21no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8D4×C3112yes—
365C4×Q8C23×C3112yes—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4D4×C3112yes—
375C4×D4C23×C3112yes—
376#117—21no—
377#117—21no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4D4×C3111yescharacteristic
381C2×C22⋊C4—24no—
382C2×C22⋊C4—21no—
383C2×C42C23×C3111yescharacteristic
384#107—22no—
385#106—22no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8C23×C3112yes—
389C4⋊C8C23×C3112yes—
390#103—22no—
391#103—22no—
392#102—22no—
393#102—22no—
394#102—22no—
395#102—22no—
396C22⋊C8—21no—
397C22⋊C8—21no—
398#97—21no—
399C4×C8C23×C3111yescharacteristic
400C4×C8C23×C3111yescharacteristic
401C31#1883911yescharacteristic, direct product factor, Sylow
402C24—21no—
403#41—82no—
404C2×Q8—24no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8C2×D4×C3112yes—
411C2×Q8C2×D4×C3112yes—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—41no—
418C2×D4C2×D4×C3112yes—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430Q16—82no—
431SD16—82no—
432SD16—82no—
433D8—44no—
434M16—81no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4C2×D4×C3112yes—
456C4⋊C4C2×D4×C3112yes—
457C4⋊C4C2×D4×C3112yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C42—22no—
467C42—22no—
468C42C2×D4×C3111yescharacteristic
469C42C2×D4×C3111yescharacteristic
470C42C2×D4×C3111yescharacteristic
471C42C24×C3111yescharacteristic
472C23—24no—
473C23—22no—
474C23—41no—
475C23—81no—
476C23—81no—
477C23C31×#12011yescharacteristic
478Q8—22no—
479Q8—22no—
480Q8—42no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8—81no—
485Q8—84no—
486D4—24no—
487D4—22no—
488D4—42no—
489D4—42no—
490D4—41no—
491D4—41no—
492D4—82no—
493D4C31×#13714yes—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4C31×#12011yescharacteristic
512C2×C4C31×#12011yescharacteristic
513C2×C4C31×#12011yescharacteristic
514C2×C4C22×D4×C3111yescharacteristic
515C2×C4C22×D4×C3111yescharacteristic
516C2×C4C22×D4×C3111yescharacteristic
517C8—41no—
518C8—41no—
519C8—42no—
520C8—42no—
521C22—21no—
522C22—24no—
523C22—21no—
524C22—21no—
525C22—42no—
526C22—42no—
527C22—84no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22C2×C31×#12011yescharacteristic
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—81no—
542C4C2×C31×#13712yes—
543C4C31×#57611yescharacteristic
544C4C31×#57711yescharacteristic
545C2—41no—
546C2—44no—
547C2—161no—
548C2C31×#300411yescharacteristic
549C2C31×#301211yescharacteristic
550C2C31×#302111yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3109C2
LCSG ▷ 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×#3501C2semidirect
2. C31×#3254C2semidirect
3. C31×#3210C2semidirect
5. C31×#1637C2semidirect
6. C31×#1632C2semidirect
7. C31×#1627C2semidirect
8. C31×#1622C2semidirect
9. C31×#1559C2semidirect
10. C31×#1497C2semidirect
11. C31×#1486C2semidirect
26. C31×#494C22semidirect
27. C31×#492C22semidirect
28. C31×#487C22semidirect
29. C31×#486C22semidirect
31. C31×#477C22semidirect
37. C31×#460C22semidirect
45. C31×#333C22semidirect
46. C31×#332C22semidirect
47. C31×Q8⋊C8C22semidirect
48. C31×Q8⋊C8C22semidirect
196. #18839C31direct (N × H)
246. #3501C2×C31semidirect
247. #3254C2×C31semidirect
248. #3210C2×C31semidirect
250. #1637C2×C31semidirect
251. #1632C2×C31semidirect
252. #1627C2×C31semidirect
253. #1622C2×C31semidirect
254. #1559C2×C31semidirect
255. #1497C2×C31semidirect
256. #1486C2×C31semidirect
295. #494C22×C31semidirect
296. #492C22×C31semidirect
297. #487C22×C31semidirect
298. #486C22×C31semidirect
300. #477C22×C31semidirect
306. #460C22×C31semidirect
314. #333C22×C31semidirect
315. #332C22×C31semidirect
316. Q8⋊C8C22×C31semidirect
317. Q8⋊C8C22×C31semidirect
401. C31#18839direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#32542
C31×#32102
C31×#31092
C31×#16372
C31×#16322
C31×#16272
C31×#16222
C31×#15592
C31×#14972
C31×#14862
#1883931