Finite Group Databasebeta

← C31×#18816 #21475575

Subgroups — C31×#18816 #21475575

Order 7936.

918 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic, complemented
2C31×#3216C211yesmaximal, characteristic, complemented
3C31×#3206C211yesmaximal, characteristic, complemented
4C31×#3115C211yesmaximal, characteristic
5C31×#1645C212yesmaximal, complemented
6C31×#1644C212yesmaximal, complemented
7C31×#1623C212yesmaximal, complemented
8C31×#1622C212yesmaximal, complemented
9C31×#1558C211yesmaximal, characteristic, complemented
10C31×#1497C211yesmaximal, characteristic, complemented
11C31×#1488C211yesmaximal, characteristic, complemented
12Q82×C31C2212yes—
13C31×#557—21no—
14C31×#554—22no—
15C31×#554C2212yes—
16D4×Q8×C31—22no—
17D4×Q8×C31—21no—
18C31×#544—22no—
19C31×#544—21no—
20C31×#533—21no—
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C2×C31×C4⋊Q8—21no—
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#520—21no—
26C4×#41×C31C2211yescharacteristic
27C31×#497—21no—
28C31×#492C2212yes—
29C31×#489—21no—
30C31×#487C2212yes—
31C31×#482—21no—
32C31×#475C2211yescharacteristic
33C31×#474C2211yescharacteristic
34C31×#462C2212yes—
35C31×#461—22no—
36C31×#460C2211yescharacteristic
37C31×#460C2212yes—
38C31×#459C2211yescharacteristic
39C31×#452—22no—
40C31×#449—21no—
41C31×#448—22no—
42C31×#442—21no—
43C31×#440—21no—
44C31×#432C2211yescharacteristic
45C31×#432C2211yescharacteristic
46C31×#431C2211yescharacteristic
47C31×#333C2212yes—
48C31×#333C2212yes—
49C31×Q8⋊C8C2212yes—
50C31×Q8⋊C8C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×D4⋊C8C2212yes—
53C2×#41×C31—21no—
54C2×#41×C31D411yescharacteristic
55C22×Q8×C31—22no—
56C22×Q8×C31D412yes—
57C2×Q16×C31—22no—
58C2×SD16×C31—21no—
59C2×SD16×C31—24no—
60C2×SD16×C31—22no—
61C2×D8×C31—21no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8—24no—
65C31×C4⋊Q8—22no—
66C31×C4⋊Q8—22no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8D412yes—
70C31×C4⋊Q8D411yescharacteristic
71C31×C4⋊Q8D411yescharacteristic
72C31×C4⋊Q8C2311yescharacteristic
73C31×C4⋊Q8C2312yes—
74C31×#127C2311yescharacteristic
75C31×#125—22no—
76C31×#125—24no—
77C31×#125—22no—
78C31×#125—21no—
79C31×#124—21no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—24no—
82C31×C22⋊Q8—24no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—24no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8D412yes—
87C31×#121—21no—
88C31×#121—21no—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31—22no—
92C4×Q8×C31—21no—
93C4×Q8×C31D411yescharacteristic
94C4×Q8×C31C2312yes—
95C4×Q8×C31C2311yescharacteristic
96C4×D4×C31—24no—
97C4×D4×C31—22no—
98C4×D4×C31—21no—
99C4×D4×C31—21no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31D411yescharacteristic
103C4×D4×C31C2311yescharacteristic
104C31×#117—22no—
105C31×#117—21no—
106C31×#117—21no—
107C31×#117D411yescharacteristic
108C2×C4⋊C4×C31—22no—
109C2×C4⋊C4×C31—21no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31—22no—
112C2×C42×C31—21no—
113C2×C42×C31C2311yescharacteristic
114C31×#107—21no—
115C31×#106—21no—
116C31×C4⋊C8—21no—
117C31×C4⋊C8C2311yescharacteristic
118C31×C4⋊C8C2312yes—
119C31×C4⋊C8C2311yescharacteristic
120C31×#103—21no—
121C31×#103—24no—
122C31×#103—22no—
123C31×#102—21no—
124C31×#102—21no—
125C31×#102—21no—
126C31×#102—22no—
127C31×C22⋊C8—21no—
128C31×C22⋊C8—21no—
129C31×C22⋊C8—21no—
130C31×#97—21no—
131C31×#97—21no—
132C4×C8×C31C2312yes—
133#41×C31—22no—
134#41×C31—22no—
135#41×C31—82no—
136C2×Q8×C31—24no—
137C2×Q8×C31—22no—
138C2×Q8×C31—21no—
139C2×Q8×C31—21no—
140C2×Q8×C31—84no—
141C2×Q8×C31—82no—
142C2×Q8×C31C2×D412yes—
143C2×Q8×C31C2×D411yescharacteristic
144C2×Q8×C31C2×D414yes—
145C2×D4×C31—21no—
146C2×D4×C31—21no—
147C2×D4×C31—21no—
148C2×D4×C31—21no—
149C2×D4×C31—41no—
150C2×D4×C31C2×D411yescharacteristic
151C22×C4×C31—21no—
152C22×C4×C31—24no—
153C22×C4×C31—22no—
154C22×C4×C31—21no—
155C22×C4×C31—21no—
156C22×C4×C31—41no—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160Q16×C31—82no—
161SD16×C31—42no—
162SD16×C31—82no—
163SD16×C31—84no—
164D8×C31—42no—
165C2×C8×C31—22no—
166C2×C8×C31—21no—
167C2×C8×C31—21no—
168C2×C8×C31—22no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—24no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31—21no—
185C4⋊C4×C31—21no—
186C4⋊C4×C31C2×D412yes—
187C4⋊C4×C31C2×D411yescharacteristic
188C4⋊C4×C31C2×D411yescharacteristic
189C22⋊C4×C31—24no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—21no—
194C22⋊C4×C31—41no—
195C22⋊C4×C31—41no—
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#18816C3111yesmaximal, characteristic, direct product factor, Sylow
206C23×C31—21no—
207C23×C31—41no—
208C23×C31#12011yescharacteristic
209Q8×C31—21no—
210Q8×C31—22no—
211Q8×C31—22no—
212Q8×C31—44no—
213Q8×C31—44no—
214Q8×C31—42no—
215Q8×C31—42no—
216Q8×C31—84no—
217Q8×C31#13712yes—
218D4×C31—24no—
219D4×C31—21no—
220D4×C31—42no—
221D4×C31—41no—
222D4×C31—41no—
223D4×C31—42no—
224D4×C31—82no—
225D4×C31#13712yes—
226C2×C4×C31—24no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—24no—
235C2×C4×C31—22no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—41no—
240C2×C4×C31—82no—
241C2×C4×C31—82no—
242C2×C4×C31—84no—
243C2×C4×C31#12011yescharacteristic
244C2×C4×C31#12011yescharacteristic
245C2×C4×C31#12011yescharacteristic
246C2×C4×C31C22×D411yescharacteristic
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C22×D411yescharacteristic
249C8×C31—42no—
250C8×C31—41no—
251C8×C31—41no—
252C8×C31—42no—
253#3503C2×C3111yescharacteristic, complemented
254#3216C2×C3111yescharacteristic, complemented
255#3206C2×C3111yescharacteristic, complemented
256#3115C2×C3111yescharacteristic
257#1645C2×C3112yescomplemented
258#1644C2×C3112yescomplemented
259#1623C2×C3112yescomplemented
260#1622C2×C3112yescomplemented
261#1558C2×C3111yescharacteristic, complemented
262#1497C2×C3111yescharacteristic, complemented
263#1488C2×C3111yescharacteristic, complemented
264C22×C31—21no—
265C22×C31—22no—
266C22×C31—21no—
267C22×C31—21no—
268C22×C31—41no—
269C22×C31—41no—
270C22×C31—81no—
271C22×C31—81no—
272C22×C31—81no—
273C22×C31C2×#12011yescharacteristic
274C4×C31—21no—
275C4×C31—22no—
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—21no—
279C4×C31—41no—
280C4×C31—42no—
281C4×C31—41no—
282C4×C31—44no—
283C4×C31—42no—
284C4×C31—81no—
285C4×C31C2×#13711yescharacteristic
286C4×C31C2×#13711yescharacteristic
287C4×C31#57712yes—
288Q82C22×C3112yes—
289#557—21no—
290#554—22no—
291#554C22×C3112yes—
292D4×Q8—22no—
293D4×Q8—21no—
294#544—22no—
295#544—21no—
296#533—21no—
297#533C22×C3111yescharacteristic
298#532C22×C3111yescharacteristic
299C2×C4⋊Q8—21no—
300C2×C4⋊Q8C22×C3111yescharacteristic
301#520—21no—
302C4×#41C22×C3111yescharacteristic
303#497—21no—
304#492C22×C3112yes—
305#489—21no—
306#487C22×C3112yes—
307#482—21no—
308#475C22×C3111yescharacteristic
309#474C22×C3111yescharacteristic
310#462C22×C3112yes—
311#461—22no—
312#460C22×C3111yescharacteristic
313#460C22×C3112yes—
314#459C22×C3111yescharacteristic
315#452—22no—
316#449—21no—
317#448—22no—
318#442—21no—
319#440—21no—
320#432C22×C3111yescharacteristic
321#432C22×C3111yescharacteristic
322#431C22×C3111yescharacteristic
323#333C22×C3112yes—
324#333C22×C3112yes—
325Q8⋊C8C22×C3112yes—
326Q8⋊C8C22×C3112yes—
327Q8⋊C8C22×C3112yes—
328D4⋊C8C22×C3112yes—
329C2×C31—41no—
330C2×C31—42no—
331C2×C31—161no—
332C2×C31#300411yescharacteristic
333C2×C31#301011yescharacteristic
334C2×C31#301911yescharacteristic
335C2×#41—21no—
336C2×#41D4×C3111yescharacteristic
337C22×Q8—22no—
338C22×Q8D4×C3112yes—
339C2×Q16—22no—
340C2×SD16—21no—
341C2×SD16—24no—
342C2×SD16—22no—
343C2×D8—21no—
344C4⋊Q8—22no—
345C4⋊Q8—22no—
346C4⋊Q8—24no—
347C4⋊Q8—22no—
348C4⋊Q8—22no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8D4×C3112yes—
352C4⋊Q8D4×C3111yescharacteristic
353C4⋊Q8D4×C3111yescharacteristic
354C4⋊Q8C23×C3111yescharacteristic
355C4⋊Q8C23×C3112yes—
356#127C23×C3111yescharacteristic
357#125—22no—
358#125—24no—
359#125—22no—
360#125—21no—
361#124—21no—
362C22⋊Q8—22no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—22no—
366C22⋊Q8—24no—
367C22⋊Q8—21no—
368C22⋊Q8D4×C3112yes—
369#121—21no—
370#121—21no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×Q8—21no—
375C4×Q8D4×C3111yescharacteristic
376C4×Q8C23×C3111yescharacteristic
377C4×Q8C23×C3112yes—
378C4×D4—24no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4D4×C3111yescharacteristic
385C4×D4C23×C3111yescharacteristic
386#117—22no—
387#117—21no—
388#117—21no—
389#117D4×C3111yescharacteristic
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C42—21no—
395C2×C42C23×C3111yescharacteristic
396#107—21no—
397#106—21no—
398C4⋊C8—21no—
399C4⋊C8C23×C3112yes—
400C4⋊C8C23×C3111yescharacteristic
401C4⋊C8C23×C3111yescharacteristic
402#103—21no—
403#103—24no—
404#103—22no—
405#102—21no—
406#102—21no—
407#102—21no—
408#102—22no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412#97—21no—
413#97—21no—
414C4×C8C23×C3112yes—
415C31#1881611yescharacteristic, direct product factor, Sylow
416#41—22no—
417#41—22no—
418#41—82no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—84no—
424C2×Q8—82no—
425C2×Q8C2×D4×C3112yes—
426C2×Q8C2×D4×C3111yescharacteristic
427C2×Q8C2×D4×C3114yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—41no—
433C2×D4C2×D4×C3111yescharacteristic
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—41no—
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4C2×D4×C3111yescharacteristic
442C22×C4C2×D4×C3111yescharacteristic
443Q16—82no—
444SD16—42no—
445SD16—82no—
446SD16—84no—
447D8—42no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4C2×D4×C3112yes—
470C4⋊C4C2×D4×C3111yescharacteristic
471C4⋊C4C2×D4×C3111yescharacteristic
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C42—24no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—22no—
484C42C2×D4×C3111yescharacteristic
485C42C2×D4×C3111yescharacteristic
486C42C2×D4×C3111yescharacteristic
487C42C24×C3111yescharacteristic
488C23—21no—
489C23—41no—
490C23C31×#12011yescharacteristic
491Q8—21no—
492Q8—22no—
493Q8—22no—
494Q8—44no—
495Q8—44no—
496Q8—42no—
497Q8—42no—
498Q8—84no—
499Q8C31×#13712yes—
500D4—24no—
501D4—21no—
502D4—42no—
503D4—41no—
504D4—41no—
505D4—42no—
506D4—82no—
507D4C31×#13712yes—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4C31×#12011yescharacteristic
526C2×C4C31×#12011yescharacteristic
527C2×C4C31×#12011yescharacteristic
528C2×C4C22×D4×C3111yescharacteristic
529C2×C4C22×D4×C3111yescharacteristic
530C2×C4C22×D4×C3111yescharacteristic
531C8—42no—
532C8—41no—
533C8—41no—
534C8—42no—
535C22—21no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22C2×C31×#12011yescharacteristic
545C4—21no—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—41no—
551C4—42no—
552C4—41no—
553C4—44no—
554C4—42no—
555C4—81no—
556C4C2×C31×#13711yescharacteristic
557C4C2×C31×#13711yescharacteristic
558C4C31×#57712yes—
559C2—41no—
560C2—42no—
561C2—161no—
562C2C31×#300411yescharacteristic
563C2C31×#301011yescharacteristic
564C2C31×#301911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3115C2
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×#3503C2semidirect
2. C31×#3216C2semidirect
3. C31×#3206C2semidirect
5. C31×#1645C2semidirect
6. C31×#1644C2semidirect
7. C31×#1623C2semidirect
8. C31×#1622C2semidirect
9. C31×#1558C2semidirect
10. C31×#1497C2semidirect
11. C31×#1488C2semidirect
205. #18816C31direct (N × H)
253. #3503C2×C31semidirect
254. #3216C2×C31semidirect
255. #3206C2×C31semidirect
257. #1645C2×C31semidirect
258. #1644C2×C31semidirect
259. #1623C2×C31semidirect
260. #1622C2×C31semidirect
261. #1558C2×C31semidirect
262. #1497C2×C31semidirect
263. #1488C2×C31semidirect
415. C31#18816direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#32162
C31×#32062
C31×#31152
C31×#16452
C31×#16442
C31×#16232
C31×#16222
C31×#15582
C31×#14972
C31×#14882
#1881631