Finite Group Databasebeta

← C31×#18993 #21475752

Subgroups — C31×#18993 #21475752

Order 7936.

1012 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#562C211yesmaximal, characteristic, complemented
2C31×#3234C211yesmaximal, characteristic, complemented
3C31×#3213C211yesmaximal, characteristic, complemented
4C31×#3153C211yesmaximal, characteristic, complemented
5C31×#1661C214yesmaximal, complemented
6C31×#1656C214yesmaximal, complemented
7C31×#1533C211yesmaximal, characteristic, complemented
8C31×#1528C211yesmaximal, characteristic, complemented
9C31×#1527C211yesmaximal, characteristic, complemented
10C2×C31×#137—21no—
11C2×C31×AGL(1,ℤ/8)—21no—
12C31×#562—22no—
13C31×#562—22no—
14C31×#562—22no—
15C31×#562C2214yes—
16C31×#532C2211yescharacteristic
17C2×C31×#125C2211yescharacteristic
18C2×C31×#124C2211yescharacteristic
19C2×C31×#124C2211yescharacteristic
20C2×C31×#123—21no—
21C2×C31×#123—21no—
22C2×C31×C22⋊Q8—21no—
23C2×C31×C22⋊Q8—21no—
24C2×C31×#121—21no—
25C2×C31×#121—21no—
26C31×#501C2212yescomplemented
27C31×#497C2212yescomplemented
28C31×#496C2212yescomplemented
29C31×#496C2212yescomplemented
30C31×#495—22no—
31C31×#490C2212yescomplemented
32C31×#489C2212yescomplemented
33C31×#481—21no—
34C31×#480—21no—
35C31×#478—22no—
36C31×#463—22no—
37C31×#448—21no—
38C31×#447—21no—
39C31×#432C2211yescharacteristic
40C31×#432C2211yescharacteristic
41C4×M16×C31C2211yescharacteristic
42C31×#330C2214yescomplemented
43C31×#329C2214yescomplemented
44C31×#329C2214yescomplemented
45C2×#41×C31—21no—
46C22×Q8×C31D411yescharacteristic
47C22×Q8×C31D411yescharacteristic
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31D411yescharacteristic
51C22×D4×C31D411yescharacteristic
52C23×C4×C31—21no—
53C23×C4×C31—21no—
54C31×#137—44no—
55C31×AGL(1,ℤ/8)—44no—
56C2×Q16×C31—22no—
57C2×SD16×C31—22no—
58C2×SD16×C31—22no—
59C2×SD16×C31—22no—
60C2×D8×C31—22no—
61C2×D8×C31—22no—
62C2×M16×C31—21no—
63C31×C4⋊Q8C2311yescharacteristic
64C31×#127C2311yescharacteristic
65C31×#125—21no—
66C31×#125—21no—
67C31×#125D412yes—
68C31×#125C2312yes—
69C31×#124—21no—
70C31×#124—21no—
71C31×#124—21no—
72C31×#124—21no—
73C31×#124—21no—
74C31×#124D412yes—
75C31×#124D412yes—
76C31×#124C2312yes—
77C31×#124C2312yes—
78C31×#123—24no—
79C31×#123—24no—
80C31×#123—24no—
81C31×#123—24no—
82C31×C22⋊Q8—24no—
83C31×C22⋊Q8—24no—
84C31×C22⋊Q8—24no—
85C31×C22⋊Q8—24no—
86C31×#121—24no—
87C31×#121—24no—
88C31×#121—24no—
89C31×#121—24no—
90C31×#121—21no—
91C31×#121—21no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C2×C4⋊C4×C31—21no—
95C2×C4⋊C4×C31—21no—
96C2×C4⋊C4×C31—21no—
97C2×C4⋊C4×C31—21no—
98C2×C4⋊C4×C31D411yescharacteristic
99C2×C4⋊C4×C31D411yescharacteristic
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—21no—
104C2×C22⋊C4×C31—21no—
105C2×C22⋊C4×C31—21no—
106C2×C42×C31C2311yescharacteristic
107C31×#107—22no—
108C31×#106—22no—
109C31×C4⋊C8—21no—
110C31×C4⋊C8—21no—
111C31×#103—22no—
112C31×#102—22no—
113C31×#102—22no—
114C31×#102—22no—
115C31×C22⋊C8—21no—
116C31×C22⋊C8—21no—
117C31×#97C2312yes—
118C31×#97C2312yes—
119C31×#97C2312yes—
120C4×C8×C31—21no—
121C24×C31—21no—
122C24×C31—21no—
123#41×C31—42no—
124#41×C31—42no—
125C2×Q8×C31—21no—
126C2×Q8×C31—21no—
127C2×Q8×C31—21no—
128C2×Q8×C31—42no—
129C2×Q8×C31—81no—
130C2×Q8×C31C2×D412yes—
131C2×Q8×C31C2×D412yes—
132C2×D4×C31—24no—
133C2×D4×C31—24no—
134C2×D4×C31—21no—
135C2×D4×C31—21no—
136C2×D4×C31—21no—
137C2×D4×C31—21no—
138C2×D4×C31—21no—
139C2×D4×C31—42no—
140C2×D4×C31—41no—
141C2×D4×C31—81no—
142C2×D4×C31—82no—
143C2×D4×C31—82no—
144C2×D4×C31C2×D412yes—
145C2×D4×C31C2×D412yes—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—24no—
151C22×C4×C31—24no—
152C22×C4×C31—41no—
153C22×C4×C31—82no—
154C22×C4×C31—82no—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158Q16×C31—44no—
159SD16×C31—44no—
160SD16×C31—44no—
161SD16×C31—82no—
162D8×C31—44no—
163D8×C31—82no—
164M16×C31—24no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31C2×D412yes—
179C4⋊C4×C31C2×D412yes—
180C22⋊C4×C31—24no—
181C22⋊C4×C31—24no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—41no—
191C22⋊C4×C31—41no—
192C42×C31C2×D411yescharacteristic
193C42×C31C2×D411yescharacteristic
194C42×C31C2×D411yescharacteristic
195C42×C31C2411yescharacteristic
196#18993C3111yesmaximal, characteristic, direct product factor, Sylow
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—41no—
206C23×C31#12011yescharacteristic
207Q8×C31—42no—
208Q8×C31—42no—
209Q8×C31—41no—
210Q8×C31—41no—
211Q8×C31—81no—
212Q8×C31—81no—
213D4×C31—42no—
214D4×C31—42no—
215D4×C31—41no—
216D4×C31—42no—
217D4×C31—41no—
218D4×C31—42no—
219D4×C31—81no—
220D4×C31—81no—
221D4×C31—84no—
222D4×C31—84no—
223D4×C31—82no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—82no—
237C2×C4×C31—84no—
238C2×C4×C31—84no—
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31C22×D411yescharacteristic
243C2×C4×C31C22×D411yescharacteristic
244C2×C4×C31C22×D411yescharacteristic
245C8×C31—24no—
246C8×C31—42no—
247C8×C31—42no—
248C2×#562C2×C3111yescharacteristic, complemented
249#3234C2×C3111yescharacteristic, complemented
250#3213C2×C3111yescharacteristic, complemented
251#3153C2×C3111yescharacteristic, complemented
252#1661C2×C3114yescomplemented
253#1656C2×C3114yescomplemented
254#1533C2×C3111yescharacteristic, complemented
255#1528C2×C3111yescharacteristic, complemented
256#1527C2×C3111yescharacteristic, complemented
257C22×C31—21no—
258C22×C31—21no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—84no—
266C22×C31—84no—
267C22×C31—81no—
268C22×C31—81no—
269C22×C31—81no—
270C22×C31#45312yes—
271C22×C31C2×#12011yescharacteristic
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—42no—
279C4×C31—42no—
280C4×C31—42no—
281C4×C31—42no—
282C4×C31—81no—
283C2×#137—21no—
284C2×AGL(1,ℤ/8)—21no—
285#562—22no—
286#562—22no—
287#562—22no—
288#562C22×C3114yes—
289#532C22×C3111yescharacteristic
290C2×#125C22×C3111yescharacteristic
291C2×#124C22×C3111yescharacteristic
292C2×#124C22×C3111yescharacteristic
293C2×#123—21no—
294C2×#123—21no—
295C2×C22⋊Q8—21no—
296C2×C22⋊Q8—21no—
297C2×#121—21no—
298C2×#121—21no—
299#501C22×C3112yescomplemented
300#497C22×C3112yescomplemented
301#496C22×C3112yescomplemented
302#496C22×C3112yescomplemented
303#495—22no—
304#490C22×C3112yescomplemented
305#489C22×C3112yescomplemented
306#481—21no—
307#480—21no—
308#478—22no—
309#463—22no—
310#448—21no—
311#447—21no—
312#432C22×C3111yescharacteristic
313#432C22×C3111yescharacteristic
314C4×M16C22×C3111yescharacteristic
315#330C22×C3114yescomplemented
316#329C22×C3114yescomplemented
317#329C22×C3114yescomplemented
318C2×C31—22no—
319C2×C31—82no—
320C2×C31—82no—
321C2×C31—161no—
322C2×C31C2×#45311yescharacteristic
323C2×C31#301911yescharacteristic
324C2×C31#302011yescharacteristic
325C2×#41—21no—
326C22×Q8D4×C3111yescharacteristic
327C22×Q8D4×C3111yescharacteristic
328C22×D4—21no—
329C22×D4—21no—
330C22×D4D4×C3111yescharacteristic
331C22×D4D4×C3111yescharacteristic
332C23×C4—21no—
333C23×C4—21no—
334#137—44no—
335AGL(1,ℤ/8)—44no—
336C2×Q16—22no—
337C2×SD16—22no—
338C2×SD16—22no—
339C2×SD16—22no—
340C2×D8—22no—
341C2×D8—22no—
342C2×M16—21no—
343C4⋊Q8C23×C3111yescharacteristic
344#127C23×C3111yescharacteristic
345#125—21no—
346#125—21no—
347#125D4×C3112yes—
348#125C23×C3112yes—
349#124—21no—
350#124—21no—
351#124—21no—
352#124—21no—
353#124—21no—
354#124D4×C3112yes—
355#124D4×C3112yes—
356#124C23×C3112yes—
357#124C23×C3112yes—
358#123—24no—
359#123—24no—
360#123—24no—
361#123—24no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—24no—
366#121—24no—
367#121—24no—
368#121—24no—
369#121—24no—
370#121—21no—
371#121—21no—
372C4×D4—21no—
373C4×D4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4D4×C3111yescharacteristic
379C2×C4⋊C4D4×C3111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42C23×C3111yescharacteristic
387#107—22no—
388#106—22no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391#103—22no—
392#102—22no—
393#102—22no—
394#102—22no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397#97C23×C3112yes—
398#97C23×C3112yes—
399#97C23×C3112yes—
400C4×C8—21no—
401C31#1899311yescharacteristic, direct product factor, Sylow
402C24—21no—
403C24—21no—
404#41—42no—
405#41—42no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8C2×D4×C3112yes—
412C2×Q8C2×D4×C3112yes—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—41no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×D4×C3112yes—
426C2×D4C2×D4×C3112yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—41no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C2×D4×C3111yescharacteristic
439Q16—44no—
440SD16—44no—
441SD16—44no—
442SD16—82no—
443D8—44no—
444D8—82no—
445M16—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4C2×D4×C3112yes—
460C4⋊C4C2×D4×C3112yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C42C2×D4×C3111yescharacteristic
474C42C2×D4×C3111yescharacteristic
475C42C2×D4×C3111yescharacteristic
476C42C24×C3111yescharacteristic
477C23—22no—
478C23—22no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23C31×#12011yescharacteristic
487Q8—42no—
488Q8—42no—
489Q8—41no—
490Q8—41no—
491Q8—81no—
492Q8—81no—
493D4—42no—
494D4—42no—
495D4—41no—
496D4—42no—
497D4—41no—
498D4—42no—
499D4—81no—
500D4—81no—
501D4—84no—
502D4—84no—
503D4—82no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4C31×#12011yescharacteristic
520C2×C4C31×#12011yescharacteristic
521C2×C4C31×#12011yescharacteristic
522C2×C4C22×D4×C3111yescharacteristic
523C2×C4C22×D4×C3111yescharacteristic
524C2×C4C22×D4×C3111yescharacteristic
525C8—24no—
526C8—42no—
527C8—42no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—84no—
537C22—84no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C31×#45312yes—
542C22C2×C31×#12011yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—81no—
554C2—22no—
555C2—82no—
556C2—82no—
557C2—161no—
558C2C2×C31×#45311yescharacteristic
559C2C31×#301911yescharacteristic
560C2C31×#302011yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
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. C2×C31×#562C2semidirect
2. C31×#3234C2semidirect
3. C31×#3213C2semidirect
4. C31×#3153C2semidirect
5. C31×#1661C2semidirect
6. C31×#1656C2semidirect
7. C31×#1533C2semidirect
8. C31×#1528C2semidirect
9. C31×#1527C2semidirect
26. C31×#501C22semidirect
27. C31×#497C22semidirect
28. C31×#496C22semidirect
29. C31×#496C22semidirect
31. C31×#490C22semidirect
32. C31×#489C22semidirect
42. C31×#330C22semidirect
43. C31×#329C22semidirect
44. C31×#329C22semidirect
196. #18993C31direct (N × H)
248. C2×#562C2×C31semidirect
249. #3234C2×C31semidirect
250. #3213C2×C31semidirect
251. #3153C2×C31semidirect
252. #1661C2×C31semidirect
253. #1656C2×C31semidirect
254. #1533C2×C31semidirect
255. #1528C2×C31semidirect
256. #1527C2×C31semidirect
299. #501C22×C31semidirect
300. #497C22×C31semidirect
301. #496C22×C31semidirect
302. #496C22×C31semidirect
304. #490C22×C31semidirect
305. #489C22×C31semidirect
315. #330C22×C31semidirect
316. #329C22×C31semidirect
317. #329C22×C31semidirect
401. C31#18993direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5622
C31×#32342
C31×#32132
C31×#31532
C31×#16612
C31×#16562
C31×#15332
C31×#15282
C31×#15272
#1899331