Finite Group Databasebeta

← C31×#18668 #21475427

Subgroups — C31×#18668 #21475427

Order 7936.

936 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3507C211yesmaximal, characteristic, complemented
2C31×#3245C211yesmaximal, characteristic
3C31×#3167C211yesmaximal, characteristic, complemented
4C31×#3037C211yesmaximal, characteristic, complemented
5C31×#1652C212yesmaximal, complemented
6C31×#1644C212yesmaximal, complemented
7C31×#1628C212yesmaximal, complemented
8C31×#1622C212yesmaximal, complemented
9C31×#1550C211yesmaximal, characteristic, complemented
10C31×#1492C211yesmaximal, characteristic, complemented
11C2×C31×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×C31×#137—21no—
13C31×#559—21no—
14C31×#557—21no—
15C31×#556—22no—
16C31×#556C2212yes—
17C31×#554C2212yes—
18C31×#552—22no—
19C31×#539—22no—
20C31×#536—21no—
21C31×#533C2211yescharacteristic
22C31×#532C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C31×#529—22no—
25C31×#520—21no—
26C2×C4×Q8×C31C2211yescharacteristic
27C31×#498C2211yescharacteristic
28C31×#492C2211yescharacteristic
29C31×#487C2211yescharacteristic
30C31×#486C2211yescharacteristic
31C31×#474—21no—
32C31×#474C2212yes—
33C31×#466—22no—
34C31×#465—22no—
35C31×#464—22no—
36C31×#462C2212yes—
37C31×#460C2212yes—
38C31×#459—21no—
39C31×#459C2212yes—
40C4×SD16×C31—22no—
41C31×#423C2211yescharacteristic
42C2×C31×C4⋊C8C2211yescharacteristic
43C31×#417—21no—
44C2×C4×C8×C31C2211yescharacteristic
45C31×#333C2212yes—
46C31×#332C2212yes—
47C31×Q8⋊C8C2212yes—
48C31×Q8⋊C8C2212yes—
49C31×D4⋊C8C2212yes—
50C31×D4⋊C8C2212yes—
51C2×#41×C31—21no—
52C2×#41×C31—21no—
53C22×Q8×C31D411yescharacteristic
54C31×#137—44no—
55C2×Q16×C31—22no—
56C2×SD16×C31—22no—
57C2×SD16×C31—22no—
58C2×D8×C31—22no—
59C2×M16×C31—21no—
60C22×C8×C31—21no—
61C22×C8×C31—21no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8D411yescharacteristic
64C31×C4⋊Q8C2311yescharacteristic
65C31×C4⋊Q8C2311yescharacteristic
66C31×C4⋊Q8C2311yescharacteristic
67C31×#127D411yescharacteristic
68C31×#127C2311yescharacteristic
69C31×#126—24no—
70C31×#126—24no—
71C31×#125—24no—
72C31×#125—22no—
73C31×#125—21no—
74C31×#124—22no—
75C31×#124—22no—
76C31×#124—21no—
77C31×#124—21no—
78C31×#123—24no—
79C31×#123—24no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—24no—
83C31×C22⋊Q8—21no—
84C31×C22⋊Q8D412yes—
85C31×#121—21no—
86C31×#121—21no—
87C31×#121—21no—
88C31×#121D412yes—
89C4×Q8×C31—22no—
90C4×Q8×C31—22no—
91C4×Q8×C31—21no—
92C4×Q8×C31—21no—
93C4×Q8×C31—21no—
94C4×Q8×C31D412yes—
95C4×Q8×C31D412yes—
96C4×Q8×C31C2312yes—
97C4×D4×C31—24no—
98C4×D4×C31—22no—
99C4×D4×C31—22no—
100C4×D4×C31—21no—
101C4×D4×C31—21no—
102C4×D4×C31—21no—
103C4×D4×C31C2312yes—
104C31×#117—24no—
105C31×#117—21no—
106C31×#117—22no—
107C2×C4⋊C4×C31—21no—
108C2×C4⋊C4×C31—22no—
109C2×C4⋊C4×C31D411yescharacteristic
110C2×C42×C31—21no—
111C2×C42×C31C2311yescharacteristic
112C31×#107—22no—
113C31×#106—22no—
114C31×C4⋊C8—21no—
115C31×C4⋊C8—21no—
116C31×C4⋊C8C2312yes—
117C31×C4⋊C8C2312yes—
118C31×#103—22no—
119C31×#103—22no—
120C31×#102—22no—
121C31×#102—22no—
122C31×#102—22no—
123C31×#102—22no—
124C4×C8×C31—21no—
125C4×C8×C31C2311yescharacteristic
126C4×C8×C31C2311yescharacteristic
127#41×C31—42no—
128#41×C31—42no—
129#41×C31—42no—
130#41×C31—42no—
131C2×Q8×C31—21no—
132C2×Q8×C31—21no—
133C2×Q8×C31—21no—
134C2×Q8×C31—22no—
135C2×Q8×C31—22no—
136C2×Q8×C31—21no—
137C2×Q8×C31C2×D412yes—
138C2×Q8×C31C2×D412yes—
139C2×D4×C31—22no—
140C2×D4×C31—21no—
141C2×D4×C31—21no—
142C2×D4×C31—21no—
143C2×D4×C31—21no—
144C2×D4×C31—21no—
145C2×D4×C31—41no—
146C2×D4×C31C2×D412yes—
147C22×C4×C31—21no—
148C22×C4×C31—24no—
149C22×C4×C31—21no—
150C22×C4×C31—22no—
151C22×C4×C31—41no—
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2×D411yescharacteristic
155Q16×C31—44no—
156SD16×C31—44no—
157SD16×C31—44no—
158D8×C31—82no—
159M16×C31—42no—
160C2×C8×C31—21no—
161C2×C8×C31—21no—
162C2×C8×C31—22no—
163C2×C8×C31—22no—
164C2×C8×C31—42no—
165C2×C8×C31—81no—
166C4⋊C4×C31—24no—
167C4⋊C4×C31—24no—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—21no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—21no—
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—21no—
180C4⋊C4×C31C2×D412yes—
181C4⋊C4×C31C2×D412yes—
182C4⋊C4×C31C2×D412yes—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—24no—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—41no—
190C22⋊C4×C31—41no—
191C42×C31—24no—
192C42×C31—22no—
193C42×C31—22no—
194C42×C31—22no—
195C42×C31C2×D411yescharacteristic
196C42×C31C2×D411yescharacteristic
197C42×C31C2×D411yescharacteristic
198C42×C31C2411yescharacteristic
199#18668C3111yesmaximal, characteristic, direct product factor, Sylow
200C23×C31—22no—
201C23×C31—41no—
202C23×C31#12011yescharacteristic
203Q8×C31—22no—
204Q8×C31—22no—
205Q8×C31—42no—
206Q8×C31—41no—
207Q8×C31—41no—
208Q8×C31—42no—
209Q8×C31—42no—
210Q8×C31—41no—
211Q8×C31—41no—
212D4×C31—22no—
213D4×C31—22no—
214D4×C31—42no—
215D4×C31—44no—
216D4×C31—41no—
217D4×C31—41no—
218D4×C31—41no—
219D4×C31—41no—
220D4×C31—42no—
221D4×C31—81no—
222D4×C31—82no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—41no—
235C2×C4×C31—81no—
236C2×C4×C31—84no—
237C2×C4×C31—82no—
238C2×C4×C31#12011yescharacteristic
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#13512yes—
242C2×C4×C31AGL(1,ℤ/8)12yes—
243C2×C4×C31C22×D411yescharacteristic
244C2×C4×C31C22×D411yescharacteristic
245C2×C4×C31C22×D411yescharacteristic
246C8×C31—22no—
247C8×C31—41no—
248C8×C31—42no—
249C8×C31—42no—
250#3507C2×C3111yescharacteristic, complemented
251#3245C2×C3111yescharacteristic
252#3167C2×C3111yescharacteristic, complemented
253#3037C2×C3111yescharacteristic, complemented
254#1652C2×C3112yescomplemented
255#1644C2×C3112yescomplemented
256#1628C2×C3112yescomplemented
257#1622C2×C3112yescomplemented
258#1550C2×C3111yescharacteristic, complemented
259#1492C2×C3111yescharacteristic, complemented
260C2×Q8⋊C8C2×C3111yescharacteristic, complemented
261C22×C31—21no—
262C22×C31—21no—
263C22×C31—42no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—81no—
267C22×C31—81no—
268C22×C31—81no—
269C22×C31#44912yes—
270C22×C31C2×#12011yescharacteristic
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—42no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—42no—
278C4×C31—42no—
279C4×C31—81no—
280C4×C31#44912yes—
281C4×C31C2×#13511yescharacteristic
282C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
283C4×C31#57711yescharacteristic
284C4×C31#57811yescharacteristic
285C2×#137—21no—
286#559—21no—
287#557—21no—
288#556—22no—
289#556C22×C3112yes—
290#554C22×C3112yes—
291#552—22no—
292#539—22no—
293#536—21no—
294#533C22×C3111yescharacteristic
295#532C22×C3111yescharacteristic
296#532C22×C3111yescharacteristic
297#529—22no—
298#520—21no—
299C2×C4×Q8C22×C3111yescharacteristic
300#498C22×C3111yescharacteristic
301#492C22×C3111yescharacteristic
302#487C22×C3111yescharacteristic
303#486C22×C3111yescharacteristic
304#474—21no—
305#474C22×C3112yes—
306#466—22no—
307#465—22no—
308#464—22no—
309#462C22×C3112yes—
310#460C22×C3112yes—
311#459—21no—
312#459C22×C3112yes—
313C4×SD16—22no—
314#423C22×C3111yescharacteristic
315C2×C4⋊C8C22×C3111yescharacteristic
316#417—21no—
317C2×C4×C8C22×C3111yescharacteristic
318#333C22×C3112yes—
319#332C22×C3112yes—
320Q8⋊C8C22×C3112yes—
321Q8⋊C8C22×C3112yes—
322D4⋊C8C22×C3112yes—
323D4⋊C8C22×C3112yes—
324C2×C31—22no—
325C2×C31—82no—
326C2×C31—161no—
327C2×C31C2×#44911yescharacteristic
328C2×C31#301311yescharacteristic
329C2×C31#301811yescharacteristic
330C2×#41—21no—
331C2×#41—21no—
332C22×Q8D4×C3111yescharacteristic
333#137—44no—
334C2×Q16—22no—
335C2×SD16—22no—
336C2×SD16—22no—
337C2×D8—22no—
338C2×M16—21no—
339C22×C8—21no—
340C22×C8—21no—
341C4⋊Q8—22no—
342C4⋊Q8D4×C3111yescharacteristic
343C4⋊Q8C23×C3111yescharacteristic
344C4⋊Q8C23×C3111yescharacteristic
345C4⋊Q8C23×C3111yescharacteristic
346#127D4×C3111yescharacteristic
347#127C23×C3111yescharacteristic
348#126—24no—
349#126—24no—
350#125—24no—
351#125—22no—
352#125—21no—
353#124—22no—
354#124—22no—
355#124—21no—
356#124—21no—
357#123—24no—
358#123—24no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—24no—
362C22⋊Q8—21no—
363C22⋊Q8D4×C3112yes—
364#121—21no—
365#121—21no—
366#121—21no—
367#121D4×C3112yes—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8D4×C3112yes—
374C4×Q8D4×C3112yes—
375C4×Q8C23×C3112yes—
376C4×D4—24no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C23×C3112yes—
383#117—24no—
384#117—21no—
385#117—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4D4×C3111yescharacteristic
389C2×C42—21no—
390C2×C42C23×C3111yescharacteristic
391#107—22no—
392#106—22no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8C23×C3112yes—
396C4⋊C8C23×C3112yes—
397#103—22no—
398#103—22no—
399#102—22no—
400#102—22no—
401#102—22no—
402#102—22no—
403C4×C8—21no—
404C4×C8C23×C3111yescharacteristic
405C4×C8C23×C3111yescharacteristic
406C31#1866811yescharacteristic, direct product factor, Sylow
407#41—42no—
408#41—42no—
409#41—42no—
410#41—42no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8C2×D4×C3112yes—
418C2×Q8C2×D4×C3112yes—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—41no—
426C2×D4C2×D4×C3112yes—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435Q16—44no—
436SD16—44no—
437SD16—44no—
438D8—82no—
439M16—42no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—42no—
445C2×C8—81no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4C2×D4×C3112yes—
461C4⋊C4C2×D4×C3112yes—
462C4⋊C4C2×D4×C3112yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42C2×D4×C3111yescharacteristic
476C42C2×D4×C3111yescharacteristic
477C42C2×D4×C3111yescharacteristic
478C42C24×C3111yescharacteristic
479C23—22no—
480C23—41no—
481C23C31×#12011yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491D4—22no—
492D4—22no—
493D4—42no—
494D4—44no—
495D4—42no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—81no—
501D4—82no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—81no—
515C2×C4—84no—
516C2×C4—82no—
517C2×C4C31×#12011yescharacteristic
518C2×C4C31×#12011yescharacteristic
519C2×C4C31×#12011yescharacteristic
520C2×C4C31×#13512yes—
521C2×C4C31×AGL(1,ℤ/8)12yes—
522C2×C4C22×D4×C3111yescharacteristic
523C2×C4C22×D4×C3111yescharacteristic
524C2×C4C22×D4×C3111yescharacteristic
525C8—22no—
526C8—41no—
527C8—42no—
528C8—42no—
529C22—21no—
530C22—21no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22C31×#44912yes—
538C22C2×C31×#12011yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—81no—
548C4C31×#44912yes—
549C4C2×C31×#13511yescharacteristic
550C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
551C4C31×#57711yescharacteristic
552C4C31×#57811yescharacteristic
553C2—22no—
554C2—82no—
555C2—161no—
556C2C2×C31×#44911yescharacteristic
557C2C31×#301311yescharacteristic
558C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3245C2
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×#3507C2semidirect
3. C31×#3167C2semidirect
4. C31×#3037C2semidirect
5. C31×#1652C2semidirect
6. C31×#1644C2semidirect
7. C31×#1628C2semidirect
8. C31×#1622C2semidirect
9. C31×#1550C2semidirect
10. C31×#1492C2semidirect
11. C2×C31×Q8⋊C8C2semidirect
199. #18668C31direct (N × H)
250. #3507C2×C31semidirect
252. #3167C2×C31semidirect
253. #3037C2×C31semidirect
254. #1652C2×C31semidirect
255. #1644C2×C31semidirect
256. #1628C2×C31semidirect
257. #1622C2×C31semidirect
258. #1550C2×C31semidirect
259. #1492C2×C31semidirect
260. C2×Q8⋊C8C2×C31semidirect
406. C31#18668direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35072
C31×#32452
C31×#31672
C31×#30372
C31×#16522
C31×#16442
C31×#16282
C31×#16222
C31×#15502
C31×#14922
C2×C31×Q8⋊C82
#1866831