Finite Group Databasebeta

← C31×#18584 #21475343

Subgroups — C31×#18584 #21475343

Order 7936.

882 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3509C211yesmaximal, characteristic, complemented
2C31×#3254C211yesmaximal, characteristic, complemented
3C31×#3141C211yesmaximal, characteristic, complemented
4C31×#3106C211yesmaximal, characteristic
5C31×#1638C212yesmaximal, complemented
6C31×#1636C212yesmaximal, complemented
7C31×#1628C212yesmaximal, complemented
8C31×#1626C212yesmaximal, complemented
9C31×#1549C211yesmaximal, characteristic, complemented
10C31×#1489C211yesmaximal, characteristic, complemented
11C31×#1485C211yesmaximal, characteristic, complemented
12C31×#557—21no—
13C31×#557C2212yes—
14C31×#556—21no—
15C31×#556C2212yes—
16C31×#552—22no—
17C31×#552—22no—
18C31×#541—22no—
19C31×#536—21no—
20C31×#533—21no—
21C31×#533—21no—
22C31×#533C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#519—21no—
26C4×#41×C31C2211yescharacteristic
27C31×#489—22no—
28C31×#487C2212yes—
29C31×#486C2212yes—
30C31×#482—21no—
31C31×#477C2211yescharacteristic
32C31×#477C2212yes—
33C31×#476C2211yescharacteristic
34C31×#474—21no—
35C31×#468—22no—
36C31×#461—21no—
37C31×#460C2211yescharacteristic
38C31×#460C2212yes—
39C31×#459C2211yescharacteristic
40C31×#449—21no—
41C4×SD16×C31—21no—
42C4×D8×C31—21no—
43C31×#431C2211yescharacteristic
44C31×#423C2211yescharacteristic
45C31×#417—22no—
46C4×M16×C31C2211yescharacteristic
47C31×#332C2212yes—
48C31×#332C2212yes—
49C31×Q8⋊C8C2212yes—
50C31×Q8⋊C8C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×D4⋊C8C2212yes—
53C2×#41×C31—21no—
54C2×#41×C31D411yescharacteristic
55C22×Q8×C31—21no—
56C2×SD16×C31—21no—
57C2×SD16×C31—22no—
58C2×D8×C31—21no—
59C2×M16×C31—21no—
60C2×M16×C31—21no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8D411yescharacteristic
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8D412yes—
67C31×C4⋊Q8C2311yescharacteristic
68C31×C4⋊Q8C2312yes—
69C31×#127C2311yescharacteristic
70C31×#126—24no—
71C31×#126—24no—
72C31×#125—22no—
73C31×#125—24no—
74C31×#125—22no—
75C31×#125—24no—
76C31×#125—21no—
77C31×#124—22no—
78C31×#124—21no—
79C31×#123—24no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8D412yes—
86C31×#121—21no—
87C31×#121—21no—
88C4×Q8×C31—22no—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31—22no—
92C4×Q8×C31—22no—
93C4×Q8×C31—22no—
94C4×Q8×C31—21no—
95C4×Q8×C31D411yescharacteristic
96C4×Q8×C31C2311yescharacteristic
97C4×Q8×C31C2312yes—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31D411yescharacteristic
103C4×D4×C31C2311yescharacteristic
104C31×#117—24no—
105C31×#117—22no—
106C31×#117—21no—
107C31×#117—22no—
108C31×#117—21no—
109C31×#117D411yescharacteristic
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31D412yes—
112C2×C42×C31—21no—
113C2×C42×C31C2311yescharacteristic
114C31×#107—21no—
115C31×#107—22no—
116C31×#106—21no—
117C31×C4⋊C8—21no—
118C31×C4⋊C8C2312yes—
119C31×C4⋊C8C2311yescharacteristic
120C31×C4⋊C8C2311yescharacteristic
121C31×#103—21no—
122C31×#103—22no—
123C31×#103—24no—
124C31×#102—21no—
125C31×#102—24no—
126C31×#102—21no—
127C31×#102—21no—
128C31×#102—22no—
129C31×C22⋊C8—21no—
130C31×#97—21no—
131C4×C8×C31—21no—
132C4×C8×C31C2312yes—
133#41×C31—41no—
134#41×C31—41no—
135#41×C31—44no—
136C2×Q8×C31—21no—
137C2×Q8×C31—22no—
138C2×Q8×C31—22no—
139C2×Q8×C31—21no—
140C2×Q8×C31—42no—
141C2×Q8×C31—42no—
142C2×Q8×C31C2×D412yes—
143C2×Q8×C31C2×D411yescharacteristic
144C2×D4×C31—21no—
145C2×D4×C31—21no—
146C2×D4×C31—21no—
147C2×D4×C31—21no—
148C2×D4×C31—41no—
149C2×D4×C31C2×D411yescharacteristic
150C22×C4×C31—22no—
151C22×C4×C31—21no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—41no—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158SD16×C31—42no—
159SD16×C31—82no—
160D8×C31—42no—
161M16×C31—42no—
162M16×C31—81no—
163C2×C8×C31—21no—
164C2×C8×C31—21no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C4⋊C4×C31—24no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31C2×D414yes—
184C4⋊C4×C31C2×D411yescharacteristic
185C4⋊C4×C31C2×D412yes—
186C4⋊C4×C31C2×D411yescharacteristic
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—21no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—41no—
194C22⋊C4×C31—41no—
195C42×C31—24no—
196C42×C31—24no—
197C42×C31—21no—
198C42×C31—21no—
199C42×C31—21no—
200C42×C31—22no—
201C42×C31C2×D411yescharacteristic
202C42×C31C2×D411yescharacteristic
203C42×C31C2×D411yescharacteristic
204C42×C31C2411yescharacteristic
205#18584C3111yesmaximal, characteristic, direct product factor, Sylow
206C23×C31—21no—
207C23×C31—41no—
208C23×C31#12011yescharacteristic
209Q8×C31—21no—
210Q8×C31—21no—
211Q8×C31—22no—
212Q8×C31—22no—
213Q8×C31—44no—
214Q8×C31—42no—
215Q8×C31—42no—
216Q8×C31—42no—
217Q8×C31—42no—
218D4×C31—21no—
219D4×C31—21no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—41no—
223D4×C31—41no—
224D4×C31—41no—
225D4×C31—41no—
226D4×C31—82no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—22no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—24no—
237C2×C4×C31—21no—
238C2×C4×C31—42no—
239C2×C4×C31—44no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—42no—
243C2×C4×C31—41no—
244C2×C4×C31—82no—
245C2×C4×C31#12011yescharacteristic
246C2×C4×C31#12011yescharacteristic
247C2×C4×C31#12011yescharacteristic
248C2×C4×C31C22×D411yescharacteristic
249C2×C4×C31C22×D411yescharacteristic
250C2×C4×C31C22×D411yescharacteristic
251C8×C31—42no—
252C8×C31—41no—
253C8×C31—41no—
254C8×C31—42no—
255#3509C2×C3111yescharacteristic, complemented
256#3254C2×C3111yescharacteristic, complemented
257#3141C2×C3111yescharacteristic, complemented
258#3106C2×C3111yescharacteristic
259#1638C2×C3112yescomplemented
260#1636C2×C3112yescomplemented
261#1628C2×C3112yescomplemented
262#1626C2×C3112yescomplemented
263#1549C2×C3111yescharacteristic, complemented
264#1489C2×C3111yescharacteristic, complemented
265#1485C2×C3111yescharacteristic, complemented
266C22×C31—21no—
267C22×C31—21no—
268C22×C31—21no—
269C22×C31—41no—
270C22×C31—41no—
271C22×C31—41no—
272C22×C31—81no—
273C22×C31—81no—
274C22×C31—81no—
275C22×C31C2×#12011yescharacteristic
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—21no—
279C4×C31—41no—
280C4×C31—44no—
281C4×C31—41no—
282C4×C31—42no—
283C4×C31—41no—
284C4×C31—42no—
285C4×C31—81no—
286C4×C31#45412yes—
287C4×C31#57511yescharacteristic
288C4×C31#57511yescharacteristic
289C4×C31#57711yescharacteristic
290C4×C31#57811yescharacteristic
291#557—21no—
292#557C22×C3112yes—
293#556—21no—
294#556C22×C3112yes—
295#552—22no—
296#552—22no—
297#541—22no—
298#536—21no—
299#533—21no—
300#533—21no—
301#533C22×C3111yescharacteristic
302#532C22×C3111yescharacteristic
303C2×C4⋊Q8C22×C3111yescharacteristic
304#519—21no—
305C4×#41C22×C3111yescharacteristic
306#489—22no—
307#487C22×C3112yes—
308#486C22×C3112yes—
309#482—21no—
310#477C22×C3111yescharacteristic
311#477C22×C3112yes—
312#476C22×C3111yescharacteristic
313#474—21no—
314#468—22no—
315#461—21no—
316#460C22×C3111yescharacteristic
317#460C22×C3112yes—
318#459C22×C3111yescharacteristic
319#449—21no—
320C4×SD16—21no—
321C4×D8—21no—
322#431C22×C3111yescharacteristic
323#423C22×C3111yescharacteristic
324#417—22no—
325C4×M16C22×C3111yescharacteristic
326#332C22×C3112yes—
327#332C22×C3112yes—
328Q8⋊C8C22×C3112yes—
329Q8⋊C8C22×C3112yes—
330Q8⋊C8C22×C3112yes—
331D4⋊C8C22×C3112yes—
332C2×C31—41no—
333C2×C31—81no—
334C2×C31—161no—
335C2×C31#300511yescharacteristic
336C2×C31#301311yescharacteristic
337C2×C31C2×#45411yescharacteristic
338C2×#41—21no—
339C2×#41D4×C3111yescharacteristic
340C22×Q8—21no—
341C2×SD16—21no—
342C2×SD16—22no—
343C2×D8—21no—
344C2×M16—21no—
345C2×M16—21no—
346C4⋊Q8—22no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8D4×C3111yescharacteristic
350C4⋊Q8D4×C3112yes—
351C4⋊Q8D4×C3111yescharacteristic
352C4⋊Q8C23×C3111yescharacteristic
353C4⋊Q8C23×C3112yes—
354#127C23×C3111yescharacteristic
355#126—24no—
356#126—24no—
357#125—22no—
358#125—24no—
359#125—22no—
360#125—24no—
361#125—21no—
362#124—22no—
363#124—21no—
364#123—24no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—21no—
370C22⋊Q8D4×C3112yes—
371#121—21no—
372#121—21no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×Q8—21no—
380C4×Q8D4×C3111yescharacteristic
381C4×Q8C23×C3111yescharacteristic
382C4×Q8C23×C3112yes—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4D4×C3111yescharacteristic
388C4×D4C23×C3111yescharacteristic
389#117—24no—
390#117—22no—
391#117—21no—
392#117—22no—
393#117—21no—
394#117D4×C3111yescharacteristic
395C2×C4⋊C4—21no—
396C2×C4⋊C4D4×C3112yes—
397C2×C42—21no—
398C2×C42C23×C3111yescharacteristic
399#107—21no—
400#107—22no—
401#106—21no—
402C4⋊C8—21no—
403C4⋊C8C23×C3112yes—
404C4⋊C8C23×C3111yescharacteristic
405C4⋊C8C23×C3111yescharacteristic
406#103—21no—
407#103—22no—
408#103—24no—
409#102—21no—
410#102—24no—
411#102—21no—
412#102—21no—
413#102—22no—
414C22⋊C8—21no—
415#97—21no—
416C4×C8—21no—
417C4×C8C23×C3112yes—
418C31#1858411yescharacteristic, direct product factor, Sylow
419#41—41no—
420#41—41no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8C2×D4×C3112yes—
429C2×Q8C2×D4×C3111yescharacteristic
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—41no—
435C2×D4C2×D4×C3111yescharacteristic
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—41no—
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443C22×C4C2×D4×C3111yescharacteristic
444SD16—42no—
445SD16—82no—
446D8—42no—
447M16—42no—
448M16—81no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—22no—
452C2×C8—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4C2×D4×C3114yes—
470C4⋊C4C2×D4×C3111yescharacteristic
471C4⋊C4C2×D4×C3112yes—
472C4⋊C4C2×D4×C3111yescharacteristic
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—21no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C42—24no—
482C42—24no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—22no—
487C42C2×D4×C3111yescharacteristic
488C42C2×D4×C3111yescharacteristic
489C42C2×D4×C3111yescharacteristic
490C42C24×C3111yescharacteristic
491C23—21no—
492C23—41no—
493C23C31×#12011yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—22no—
497Q8—22no—
498Q8—44no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—42no—
503D4—21no—
504D4—21no—
505D4—42no—
506D4—42no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—82no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—82no—
530C2×C4C31×#12011yescharacteristic
531C2×C4C31×#12011yescharacteristic
532C2×C4C31×#12011yescharacteristic
533C2×C4C22×D4×C3111yescharacteristic
534C2×C4C22×D4×C3111yescharacteristic
535C2×C4C22×D4×C3111yescharacteristic
536C8—42no—
537C8—41no—
538C8—41no—
539C8—42no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22C2×C31×#12011yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—44no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—42no—
559C4—81no—
560C4C31×#45412yes—
561C4C31×#57511yescharacteristic
562C4C31×#57511yescharacteristic
563C4C31×#57711yescharacteristic
564C4C31×#57811yescharacteristic
565C2—41no—
566C2—81no—
567C2—161no—
568C2C31×#300511yescharacteristic
569C2C31×#301311yescharacteristic
570C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3106C2
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×#3509C2semidirect
2. C31×#3254C2semidirect
3. C31×#3141C2semidirect
5. C31×#1638C2semidirect
6. C31×#1636C2semidirect
7. C31×#1628C2semidirect
8. C31×#1626C2semidirect
9. C31×#1549C2semidirect
10. C31×#1489C2semidirect
11. C31×#1485C2semidirect
205. #18584C31direct (N × H)
255. #3509C2×C31semidirect
256. #3254C2×C31semidirect
257. #3141C2×C31semidirect
259. #1638C2×C31semidirect
260. #1636C2×C31semidirect
261. #1628C2×C31semidirect
262. #1626C2×C31semidirect
263. #1549C2×C31semidirect
264. #1489C2×C31semidirect
265. #1485C2×C31semidirect
418. C31#18584direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35092
C31×#32542
C31×#31412
C31×#31062
C31×#16382
C31×#16362
C31×#16282
C31×#16262
C31×#15492
C31×#14892
C31×#14852
#1858431