Finite Group Databasebeta

← C2×C31×#1609 #21474864

Subgroups — C2×C31×#1609 #21474864

Order 7936.

1060 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#539C211yesmaximal, characteristic
2C2×C31×#483C211yesmaximal, characteristic, complemented
3C2×C31×#480C211yesmaximal, characteristic
4C2×C31×#398C211yesmaximal, characteristic, complemented
5C31×#1609C218yesmaximal, direct product factor
6C2×C31×#328C211yesmaximal, characteristic, complemented
7C2×C31×#327C211yesmaximal, characteristic
8C2×C31×#324C211yesmaximal, characteristic, complemented
9C31×#539—22no—
10C31×#539—22no—
11C31×#539—22no—
12C31×#539C2214yes—
13C2×C31×#126—21no—
14C2×C31×#125—21no—
15C2×C31×#123—21no—
16C2×C31×#123—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8C2211yescharacteristic
19C2×C31×#121C2211yescharacteristic
20C2×C4×D4×C31—21no—
21C22×C4⋊C4×C31—21no—
22C22×C4⋊C4×C31C2211yescharacteristic
23C22×C4⋊C4×C31C2211yescharacteristic
24C31×#483—22no—
25C31×#483C2214yescomplemented
26C31×#480—22no—
27C31×#480C2214yes—
28C2×C31×#107—21no—
29C2×C31×#106—21no—
30C2×C31×#103—21no—
31C2×C31×#102—21no—
32C2×C31×C22⋊C8C2211yescharacteristic
33C2×C31×C22⋊C8C2211yescharacteristic
34C31×#398—22no—
35C31×#398C2214yescomplemented
36C2×C31×#95—21no—
37C2×C31×#95C2211yescharacteristic
38C31×#328C2214yescomplemented
39C31×#327C2214yes—
40C31×#324C2214yescomplemented
41C22×Q8×C31D411yescharacteristic
42C22×D4×C31—21no—
43C22×D4×C31D411yescharacteristic
44C23×C4×C31—21no—
45C23×C4×C31—21no—
46C23×C4×C31—21no—
47C23×C4×C31D411yescharacteristic
48C23×C4×C31D411yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C22×C8×C31—21no—
51C22×C8×C31—21no—
52C31×#126—24no—
53C31×#126—24no—
54C31×#125—24no—
55C31×#125—24no—
56C31×#123—24no—
57C31×#123—24no—
58C31×#123—24no—
59C31×#123—24no—
60C31×C22⋊Q8—24no—
61C31×C22⋊Q8—24no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8D412yes—
65C31×C22⋊Q8C2312yes—
66C31×#121—21no—
67C31×#121—21no—
68C31×#121D412yes—
69C31×#121C2312yes—
70C4×D4×C31—24no—
71C4×D4×C31—24no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—24no—
86C2×C4⋊C4×C31—44no—
87C2×C4⋊C4×C31D412yes—
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31C2312yes—
91C2×C4⋊C4×C31C2312yes—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C42×C31—21no—
98C31×#107—24no—
99C31×#106—24no—
100C31×#103—24no—
101C31×#102—24no—
102C31×C22⋊C8—21no—
103C31×C22⋊C8—21no—
104C31×C22⋊C8C2312yes—
105C31×C22⋊C8C2312yes—
106C31×#95—21no—
107C31×#95—24no—
108C31×#95C2312yes—
109C24×C31—21no—
110C24×C31C2×D411yescharacteristic
111C2×Q8×C31—21no—
112C2×Q8×C31—42no—
113C2×Q8×C31C2×D412yes—
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—21no—
117C2×D4×C31—42no—
118C2×D4×C31—42no—
119C2×D4×C31—42no—
120C2×D4×C31C2×D412yes—
121C22×C4×C31—24no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—24no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—42no—
135C22×C4×C31—42no—
136C22×C4×C31—42no—
137C22×C4×C31—42no—
138C22×C4×C31—41no—
139C22×C4×C31—41no—
140C22×C4×C31—42no—
141C22×C4×C31—42no—
142C22×C4×C31—82no—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2411yescharacteristic
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C2×C8×C31—42no—
154C2×C8×C31—42no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—48no—
169C4⋊C4×C31—41no—
170C4⋊C4×C31—41no—
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—22no—
180C42×C31—24no—
181C2×#1609C3111yesmaximal, characteristic, direct product factor, Sylow
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31#12011yescharacteristic
189C23×C31#12011yescharacteristic
190C23×C31#13512yes—
191C23×C31AGL(1,ℤ/8)12yes—
192C23×C31C22×D411yescharacteristic
193Q8×C31—42no—
194Q8×C31—81no—
195D4×C31—42no—
196D4×C31—42no—
197D4×C31—42no—
198D4×C31—44no—
199D4×C31—81no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—24no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—48no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—81no—
228C2×C4×C31—84no—
229C2×C4×C31—82no—
230C2×C4×C31—82no—
231C2×C4×C31#12011yescharacteristic
232C2×C4×C31#12011yescharacteristic
233C2×C4×C31C22×D411yescharacteristic
234C2×C4×C31C22×D411yescharacteristic
235C8×C31—42no—
236C8×C31—42no—
237C2×#539C2×C3111yescharacteristic
238C2×#483C2×C3111yescharacteristic, complemented
239C2×#480C2×C3111yescharacteristic
240C2×#398C2×C3111yescharacteristic, complemented
241#1609C2×C3118yesdirect product factor
242C2×#328C2×C3111yescharacteristic, complemented
243C2×#327C2×C3111yescharacteristic
244C2×#324C2×C3111yescharacteristic, complemented
245C22×C31—24no—
246C22×C31—24no—
247C22×C31—21no—
248C22×C31—21no—
249C22×C31—21no—
250C22×C31—21no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—84no—
255C22×C31#44912yes—
256C22×C31#44912yes—
257C22×C31#45612yes—
258C22×C31#45812yes—
259C22×C31C2×#12011yescharacteristic
260C22×C31C2×#13511yescharacteristic
261C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
262C4×C31—21no—
263C4×C31—21no—
264C4×C31—21no—
265C4×C31—21no—
266C4×C31—44no—
267C4×C31—42no—
268C4×C31—42no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—42no—
272C4×C31—82no—
273#539—22no—
274#539—22no—
275#539—22no—
276#539C22×C3114yes—
277C2×#126—21no—
278C2×#125—21no—
279C2×#123—21no—
280C2×#123—21no—
281C2×C22⋊Q8—21no—
282C2×C22⋊Q8C22×C3111yescharacteristic
283C2×#121C22×C3111yescharacteristic
284C2×C4×D4—21no—
285C22×C4⋊C4—21no—
286C22×C4⋊C4C22×C3111yescharacteristic
287C22×C4⋊C4C22×C3111yescharacteristic
288#483—22no—
289#483C22×C3114yescomplemented
290#480—22no—
291#480C22×C3114yes—
292C2×#107—21no—
293C2×#106—21no—
294C2×#103—21no—
295C2×#102—21no—
296C2×C22⋊C8C22×C3111yescharacteristic
297C2×C22⋊C8C22×C3111yescharacteristic
298#398—22no—
299#398C22×C3114yescomplemented
300C2×#95—21no—
301C2×#95C22×C3111yescharacteristic
302#328C22×C3114yescomplemented
303#327C22×C3114yes—
304#324C22×C3114yescomplemented
305C2×C31—21no—
306C2×C31—21no—
307C2×C31—22no—
308C2×C31—82no—
309C2×C31#160914yesdirect product factor
310C2×C31C2×#44911yescharacteristic
311C2×C31C2×#45611yescharacteristic
312C2×C31C2×#45811yescharacteristic
313C22×Q8D4×C3111yescharacteristic
314C22×D4—21no—
315C22×D4D4×C3111yescharacteristic
316C23×C4—21no—
317C23×C4—21no—
318C23×C4—21no—
319C23×C4D4×C3111yescharacteristic
320C23×C4D4×C3111yescharacteristic
321C23×C4C23×C3111yescharacteristic
322C22×C8—21no—
323C22×C8—21no—
324#126—24no—
325#126—24no—
326#125—24no—
327#125—24no—
328#123—24no—
329#123—24no—
330#123—24no—
331#123—24no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8D4×C3112yes—
337C22⋊Q8C23×C3112yes—
338#121—21no—
339#121—21no—
340#121D4×C3112yes—
341#121C23×C3112yes—
342C4×D4—24no—
343C4×D4—24no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4D4×C3112yes—
360C2×C4⋊C4D4×C3111yescharacteristic
361C2×C4⋊C4D4×C3111yescharacteristic
362C2×C4⋊C4C23×C3112yes—
363C2×C4⋊C4C23×C3112yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C42—21no—
370#107—24no—
371#106—24no—
372#103—24no—
373#102—24no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8C23×C3112yes—
377C22⋊C8C23×C3112yes—
378#95—24no—
379#95—21no—
380#95C23×C3112yes—
381C31C2×#160911yescharacteristic, direct product factor, Sylow
382C24—21no—
383C24C2×D4×C3111yescharacteristic
384C2×Q8—21no—
385C2×Q8—42no—
386C2×Q8C2×D4×C3112yes—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C2×D4×C3112yes—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—82no—
416C22×C4C2×D4×C3112yes—
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C2×D4×C3111yescharacteristic
419C22×C4C2×D4×C3112yes—
420C22×C4C2×D4×C3111yescharacteristic
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C2×D4×C3111yescharacteristic
423C22×C4C24×C3111yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—42no—
427C2×C8—42no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—48no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4C2×D4×C3112yes—
445C4⋊C4C2×D4×C3112yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C42—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23C31×#12011yescharacteristic
461C23C31×#12011yescharacteristic
462C23C31×#13512yes—
463C23C31×AGL(1,ℤ/8)12yes—
464C23C22×D4×C3111yescharacteristic
465Q8—42no—
466Q8—81no—
467D4—42no—
468D4—44no—
469D4—42no—
470D4—42no—
471D4—81no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—84no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4C31×#12011yescharacteristic
504C2×C4C31×#12011yescharacteristic
505C2×C4C22×D4×C3111yescharacteristic
506C2×C4C22×D4×C3111yescharacteristic
507C8—42no—
508C8—42no—
509C22—24no—
510C22—24no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—84no—
519C22C31×#44912yes—
520C22C31×#44912yes—
521C22C31×#45612yes—
522C22C31×#45812yes—
523C22C2×C31×#12011yescharacteristic
524C22C2×C31×#13511yescharacteristic
525C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—44no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—82no—
537C2—21no—
538C2—22no—
539C2—21no—
540C2—82no—
541C2C31×#160914yesdirect product factor
542C2C2×C31×#44911yescharacteristic
543C2C2×C31×#45611yescharacteristic
544C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C31×#483C2semidirect
4. C2×C31×#398C2semidirect
5. C31×#1609C2direct (N × H)
6. C2×C31×#328C2semidirect
8. C2×C31×#324C2semidirect
25. C31×#483C22semidirect
35. C31×#398C22semidirect
38. C31×#328C22semidirect
40. C31×#324C22semidirect
181. C2×#1609C31direct (N × H)
238. C2×#483C2×C31semidirect
240. C2×#398C2×C31semidirect
241. #1609C2×C31direct (N × H)
242. C2×#328C2×C31semidirect
244. C2×#324C2×C31semidirect
289. #483C22×C31semidirect
299. #398C22×C31semidirect
302. #328C22×C31semidirect
304. #324C22×C31semidirect
309. C2×C31#1609direct (N × H)
381. C31C2×#1609direct (N × H)
541. C2C31×#1609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5392
C2×C31×#4832
C2×C31×#4802
C2×C31×#3982
C31×#16092
C2×C31×#3282
C2×C31×#3272
C2×C31×#3242
C2×#160931