Finite Group Databasebeta

← C31×#18214 #21474973

Subgroups — C31×#18214 #21474973

Order 7936.

982 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×#3101C211yesmaximal, characteristic, complemented
3C31×#3100C211yesmaximal, characteristic
4C31×#2536C211yesmaximal, characteristic, complemented
5C31×#1609C214yesmaximal, complemented
6C31×#1597C214yesmaximal, complemented
7C31×#1518C211yesmaximal, characteristic, complemented
8C31×#1517C211yesmaximal, characteristic, complemented
9C31×#1461C211yesmaximal, characteristic, complemented
10C31×#559—21no—
11C31×#557—21no—
12C31×#556—21no—
13C31×#556—22no—
14C31×#554—21no—
15C31×#552—22no—
16C31×#539C2214yes—
17C31×#536—21no—
18C31×#533C2211yescharacteristic
19C31×#532C2211yescharacteristic
20C31×#529—22no—
21C2×C31×C22⋊Q8—21no—
22C31×#520—21no—
23C2×C4×Q8×C31C2211yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C31×#483C2212yes—
26C31×#480C2212yes—
27C31×#476—21no—
28C31×#474—21no—
29C31×#461—21no—
30C31×#459—21no—
31C31×#450C2212yes—
32C31×#447C2212yes—
33C4×SD16×C31—22no—
34C4×D8×C31—22no—
35C31×#431C2211yescharacteristic
36C31×#431C2211yescharacteristic
37C31×#398C2212yes—
38C31×#394C2212yes—
39C31×#388—22no—
40C31×#382C2211yescharacteristic
41C31×#380—21no—
42C31×#328C2214yes—
43C31×#327C2214yes—
44C31×#324C2214yes—
45C2×#41×C31—21no—
46C22×Q8×C31D411yescharacteristic
47C22×D4×C31C2311yescharacteristic
48C23×C4×C31—21no—
49C2×SD16×C31—22no—
50C2×D8×C31—22no—
51C31×C4⋊Q8—22no—
52C31×C4⋊Q8D411yescharacteristic
53C31×C4⋊Q8D411yescharacteristic
54C31×C4⋊Q8D411yescharacteristic
55C31×#127D411yescharacteristic
56C31×#126—24no—
57C31×#126—24no—
58C31×#125—22no—
59C31×#125—24no—
60C31×#125—21no—
61C31×#124—22no—
62C31×#124—22no—
63C31×#124—21no—
64C31×#123—24no—
65C31×#123—24no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—24no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—44no—
71C31×C22⋊Q8C2312yes—
72C31×#121—21no—
73C31×#121C2312yes—
74C4×Q8×C31—22no—
75C4×Q8×C31—22no—
76C4×Q8×C31—21no—
77C4×Q8×C31—21no—
78C4×Q8×C31—21no—
79C4×Q8×C31—21no—
80C4×Q8×C31—21no—
81C4×Q8×C31D412yes—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—24no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31—21no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31D412yes—
91C31×#117—22no—
92C31×#117—24no—
93C31×#117—21no—
94C2×C4⋊C4×C31—22no—
95C2×C4⋊C4×C31—22no—
96C2×C4⋊C4×C31—21no—
97C2×C4⋊C4×C31D411yescharacteristic
98C2×C4⋊C4×C31C2311yescharacteristic
99C2×C4⋊C4×C31C2312yes—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—22no—
103C2×C42×C31D412yes—
104C2×C42×C31C2311yescharacteristic
105C31×#107—22no—
106C31×#106—22no—
107C31×C4⋊C8—21no—
108C31×C4⋊C8—21no—
109C31×#103—22no—
110C31×#102—22no—
111C31×#102—22no—
112C31×#102—22no—
113C31×C22⋊C8C2312yes—
114C31×C22⋊C8C2312yes—
115C4×C8×C31—21no—
116C4×C8×C31—21no—
117C31×#95—22no—
118C31×#95—21no—
119C31×#95C2312yes—
120C24×C31—21no—
121#41×C31—42no—
122#41×C31—42no—
123C2×Q8×C31—21no—
124C2×Q8×C31—21no—
125C2×Q8×C31—21no—
126C2×Q8×C31—21no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31C2×D412yes—
130C2×D4×C31—22no—
131C2×D4×C31—21no—
132C2×D4×C31—21no—
133C2×D4×C31—21no—
134C2×D4×C31—21no—
135C2×D4×C31—42no—
136C2×D4×C31C2×D412yes—
137C22×C4×C31—24no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—41no—
142C22×C4×C31—41no—
143C22×C4×C31—42no—
144C22×C4×C31—42no—
145C22×C4×C31C2×D412yes—
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2411yescharacteristic
150SD16×C31—44no—
151D8×C31—44no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—44no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31C2×D412yes—
172C4⋊C4×C31C2×D412yes—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—24no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—44no—
180C22⋊C4×C31—44no—
181C22⋊C4×C31—41no—
182C22⋊C4×C31—41no—
183C42×C31—24no—
184C42×C31—22no—
185C42×C31—22no—
186C42×C31—22no—
187C42×C31C2×D411yescharacteristic
188C42×C31C2×D411yescharacteristic
189C42×C31C2×D411yescharacteristic
190C42×C31C2×D411yescharacteristic
191#18214C3111yesmaximal, characteristic, direct product factor, Sylow
192C23×C31—22no—
193C23×C31—22no—
194C23×C31—41no—
195C23×C31—41no—
196C23×C31—41no—
197C23×C31—41no—
198C23×C31—41no—
199C23×C31—41no—
200C23×C31C22×D411yescharacteristic
201Q8×C31—42no—
202Q8×C31—42no—
203Q8×C31—42no—
204Q8×C31—42no—
205Q8×C31—42no—
206D4×C31—22no—
207D4×C31—22no—
208D4×C31—42no—
209D4×C31—42no—
210D4×C31—41no—
211D4×C31—41no—
212D4×C31—42no—
213D4×C31—42no—
214D4×C31—42no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—44no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—84no—
233C2×C4×C31—82no—
234C2×C4×C31—82no—
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31#12011yescharacteristic
237C2×C4×C31#12011yescharacteristic
238C2×C4×C31#12011yescharacteristic
239C2×C4×C31#13512yes—
240C2×C4×C31AGL(1,ℤ/8)12yes—
241C2×C4×C31C22×D411yescharacteristic
242C2×C4×C31C22×D411yescharacteristic
243C8×C31—42no—
244C8×C31—42no—
245#3507C2×C3111yescharacteristic, complemented
246#3101C2×C3111yescharacteristic, complemented
247#3100C2×C3111yescharacteristic
248#2536C2×C3111yescharacteristic, complemented
249#1609C2×C3114yescomplemented
250#1597C2×C3114yescomplemented
251#1518C2×C3111yescharacteristic, complemented
252#1517C2×C3111yescharacteristic, complemented
253#1461C2×C3111yescharacteristic, complemented
254C22×C31—21no—
255C22×C31—21no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—81no—
263C22×C31—81no—
264C22×C31—81no—
265C22×C31—81no—
266C22×C31C2×#12011yescharacteristic
267C22×C31C2×#13511yescharacteristic
268C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
269C4×C31—21no—
270C4×C31—21no—
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—42no—
274C4×C31—42no—
275C4×C31—42no—
276C4×C31—42no—
277C4×C31—42no—
278C4×C31—82no—
279C4×C31—81no—
280C4×C31#44912yes—
281C4×C31#44912yes—
282#559—21no—
283#557—21no—
284#556—21no—
285#556—22no—
286#554—21no—
287#552—22no—
288#539C22×C3114yes—
289#536—21no—
290#533C22×C3111yescharacteristic
291#532C22×C3111yescharacteristic
292#529—22no—
293C2×C22⋊Q8—21no—
294#520—21no—
295C2×C4×Q8C22×C3111yescharacteristic
296C2×C4×D4C22×C3111yescharacteristic
297#483C22×C3112yes—
298#480C22×C3112yes—
299#476—21no—
300#474—21no—
301#461—21no—
302#459—21no—
303#450C22×C3112yes—
304#447C22×C3112yes—
305C4×SD16—22no—
306C4×D8—22no—
307#431C22×C3111yescharacteristic
308#431C22×C3111yescharacteristic
309#398C22×C3112yes—
310#394C22×C3112yes—
311#388—22no—
312#382C22×C3111yescharacteristic
313#380—21no—
314#328C22×C3114yes—
315#327C22×C3114yes—
316#324C22×C3114yes—
317C2×C31—21no—
318C2×C31—21no—
319C2×C31—82no—
320C2×C31—82no—
321C2×C31C2×#44911yescharacteristic
322C2×C31#302011yescharacteristic
323C2×C31#302511yescharacteristic
324C2×#41—21no—
325C22×Q8D4×C3111yescharacteristic
326C22×D4C23×C3111yescharacteristic
327C23×C4—21no—
328C2×SD16—22no—
329C2×D8—22no—
330C4⋊Q8—22no—
331C4⋊Q8D4×C3111yescharacteristic
332C4⋊Q8D4×C3111yescharacteristic
333C4⋊Q8D4×C3111yescharacteristic
334#127D4×C3111yescharacteristic
335#126—24no—
336#126—24no—
337#125—22no—
338#125—24no—
339#125—21no—
340#124—22no—
341#124—22no—
342#124—21no—
343#123—24no—
344#123—24no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347C22⋊Q8—24no—
348C22⋊Q8—21no—
349C22⋊Q8—44no—
350C22⋊Q8C23×C3112yes—
351#121—21no—
352#121C23×C3112yes—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8D4×C3112yes—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—24no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4D4×C3112yes—
370#117—22no—
371#117—24no—
372#117—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4D4×C3111yescharacteristic
377C2×C4⋊C4C23×C3111yescharacteristic
378C2×C4⋊C4C23×C3112yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C42D4×C3112yes—
383C2×C42C23×C3111yescharacteristic
384#107—22no—
385#106—22no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388#103—22no—
389#102—22no—
390#102—22no—
391#102—22no—
392C22⋊C8C23×C3112yes—
393C22⋊C8C23×C3112yes—
394C4×C8—21no—
395C4×C8—21no—
396#95—22no—
397#95—21no—
398#95C23×C3112yes—
399C31#1821411yescharacteristic, direct product factor, Sylow
400C24—21no—
401#41—42no—
402#41—42no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8C2×D4×C3112yes—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4C2×D4×C3112yes—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C2×D4×C3112yes—
426C22×C4C2×D4×C3112yes—
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C24×C3111yescharacteristic
430SD16—44no—
431D8—44no—
432C2×C8—22no—
433C2×C8—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4C2×D4×C3112yes—
452C4⋊C4C2×D4×C3112yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42C2×D4×C3111yescharacteristic
468C42C2×D4×C3111yescharacteristic
469C42C2×D4×C3111yescharacteristic
470C42C2×D4×C3111yescharacteristic
471C23—22no—
472C23—22no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23C22×D4×C3111yescharacteristic
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485D4—22no—
486D4—22no—
487D4—42no—
488D4—42no—
489D4—41no—
490D4—41no—
491D4—42no—
492D4—42no—
493D4—42no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4C31×#12011yescharacteristic
515C2×C4C31×#12011yescharacteristic
516C2×C4C31×#12011yescharacteristic
517C2×C4C31×#12011yescharacteristic
518C2×C4C31×#13512yes—
519C2×C4C31×AGL(1,ℤ/8)12yes—
520C2×C4C22×D4×C3111yescharacteristic
521C2×C4C22×D4×C3111yescharacteristic
522C8—42no—
523C8—42no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22C2×C31×#12011yescharacteristic
537C22C2×C31×#13511yescharacteristic
538C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—82no—
549C4—81no—
550C4C31×#44912yes—
551C4C31×#44912yes—
552C2—21no—
553C2—21no—
554C2—82no—
555C2—82no—
556C2C2×C31×#44911yescharacteristic
557C2C31×#302011yescharacteristic
558C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3100C2
LCSG ▷ C22×C4 ▷ 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
2. C31×#3101C2semidirect
4. C31×#2536C2semidirect
5. C31×#1609C2semidirect
6. C31×#1597C2semidirect
7. C31×#1518C2semidirect
8. C31×#1517C2semidirect
9. C31×#1461C2semidirect
191. #18214C31direct (N × H)
245. #3507C2×C31semidirect
246. #3101C2×C31semidirect
248. #2536C2×C31semidirect
249. #1609C2×C31semidirect
250. #1597C2×C31semidirect
251. #1518C2×C31semidirect
252. #1517C2×C31semidirect
253. #1461C2×C31semidirect
399. C31#18214direct (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×#31012
C31×#31002
C31×#25362
C31×#16092
C31×#15972
C31×#15182
C31×#15172
C31×#14612
#1821431