Finite Group Databasebeta

← C2×C31×#1639 #21475240

Subgroups — C2×C31×#1639 #21475240

Order 7936.

1690 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D42×C31C211yesmaximal, characteristic
2C2×C31×#486C211yesmaximal, characteristic, complemented
3C2×C31×#474C211yesmaximal, characteristic, complemented
4C2×C31×#461C211yesmaximal, characteristic, complemented
5C31×#1639C218yesmaximal, direct product factor
6C2×C31×#331C211yesmaximal, characteristic, complemented
7C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
9C23×D4×C31—21no—
10C23×D4×C31—21no—
11C22×SD16×C31—21no—
12D42×C31—22no—
13D42×C31—22no—
14D42×C31—22no—
15D42×C31C2214yes—
16C2×C31×C4⋊Q8C2211yescharacteristic, complemented
17C2×C31×#127C2211yescharacteristic
18C2×C31×#121—21no—
19C2×C31×#121—21no—
20C2×C31×#120—22no—
21C2×C4×D4×C31C2211yescharacteristic
22C2×C4×D4×C31C2211yescharacteristic
23C31×#486—22no—
24C31×#486C2214yescomplemented
25C31×#474—22no—
26C31×#474C2214yescomplemented
27C31×#461—22no—
28C31×#461C2214yescomplemented
29C2×C31×#107—21no—
30C2×C31×C4⋊C8C2211yescharacteristic, complemented
31C2×C31×C4⋊C8C2211yescharacteristic, complemented
32C2×C31×#103—21no—
33C2×C31×#102—22no—
34C2×C31×#102—21no—
35C2×C4×C8×C31C2211yescharacteristic, complemented
36C31×#331C2214yescomplemented
37C31×D4⋊C8C2214yescomplemented
38C31×D4⋊C8C2214yescomplemented
39C25×C31—22no—
40C22×Q8×C31—21no—
41C22×D4×C31—22no—
42C22×D4×C31—22no—
43C22×D4×C31—24no—
44C22×D4×C31—24no—
45C22×D4×C31—24no—
46C22×D4×C31—24no—
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31—44no—
50C22×D4×C31—44no—
51C22×D4×C31—44no—
52C22×D4×C31—44no—
53C22×D4×C31D411yescharacteristic
54C22×D4×C31D411yescharacteristic
55C22×D4×C31D412yes—
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C2×SD16×C31—24no—
59C2×SD16×C31—44no—
60C22×C8×C31—21no—
61C22×C8×C31—21no—
62C22×C8×C31—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8C2312yescomplemented
67C31×#127—21no—
68C31×#127—21no—
69C31×#127D412yes—
70C31×#127C2312yes—
71C31×#121—24no—
72C31×#121—24no—
73C31×#121—24no—
74C31×#121—24no—
75C31×#120—28no—
76C31×#120—28no—
77C4×D4×C31—21no—
78C4×D4×C31—21no—
79C4×D4×C31—21no—
80C4×D4×C31—21no—
81C4×D4×C31D412yes—
82C4×D4×C31D412yes—
83C4×D4×C31C2312yes—
84C4×D4×C31C2312yes—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C42×C31C2311yescharacteristic
93C31×#107—24no—
94C31×C4⋊C8—21no—
95C31×C4⋊C8—21no—
96C31×C4⋊C8C2312yescomplemented
97C31×C4⋊C8C2312yescomplemented
98C31×#103—24no—
99C31×#102—24no—
100C31×#102—28no—
101C4×C8×C31—21no—
102C4×C8×C31C2312yescomplemented
103C24×C31—21no—
104C24×C31—21no—
105C24×C31—28no—
106C24×C31—22no—
107C24×C31—48no—
108C24×C31—48no—
109C24×C31—84no—
110C2×Q8×C31—22no—
111C2×Q8×C31—41no—
112C2×Q8×C31—42no—
113C2×Q8×C31—42no—
114C2×D4×C31—28no—
115C2×D4×C31—28no—
116C2×D4×C31—21no—
117C2×D4×C31—21no—
118C2×D4×C31—22no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—44no—
126C2×D4×C31—48no—
127C2×D4×C31—48no—
128C2×D4×C31—416no—
129C2×D4×C31—416no—
130C2×D4×C31—48no—
131C2×D4×C31—48no—
132C2×D4×C31—48no—
133C2×D4×C31—48no—
134C2×D4×C31—42no—
135C2×D4×C31—42no—
136C2×D4×C31—42no—
137C2×D4×C31—42no—
138C2×D4×C31D814yes—
139C2×D4×C31SD1614yescomplemented
140C2×D4×C31C2×D412yes—
141C2×D4×C31C2×D412yes—
142C2×D4×C31C2×D414yes—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—42no—
150C22×C4×C31—42no—
151C22×C4×C31—42no—
152C22×C4×C31—42no—
153C22×C4×C31—41no—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157SD16×C31—48no—
158C2×C8×C31—22no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C2×C8×C31—42no—
162C2×C8×C31—42no—
163C2×C8×C31—42no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—41no—
169C4⋊C4×C31—41no—
170C4⋊C4×C31C2×D412yes—
171C4⋊C4×C31C2×D412yes—
172C22⋊C4×C31—28no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C42×C31C2×D411yescharacteristic
178C42×C31C2×D411yescharacteristic
179C42×C31C2×D411yescharacteristic
180C42×C31C2411yescharacteristic
181C2×#1639C3111yesmaximal, characteristic, direct product factor, Sylow
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—24no—
185C23×C31—24no—
186C23×C31—24no—
187C23×C31—416no—
188C23×C31—416no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—44no—
194C23×C31—44no—
195C23×C31—44no—
196C23×C31—88no—
197C23×C31—832no—
198C23×C31#12011yescharacteristic
199Q8×C31—42no—
200Q8×C31—42no—
201Q8×C31—82no—
202D4×C31—22no—
203D4×C31—22no—
204D4×C31—24no—
205D4×C31—24no—
206D4×C31—44no—
207D4×C31—416no—
208D4×C31—416no—
209D4×C31—42no—
210D4×C31—42no—
211D4×C31—42no—
212D4×C31—42no—
213D4×C31—44no—
214D4×C31—44no—
215D4×C31—82no—
216D4×C31C2×D814yes—
217D4×C31C2×SD1614yescomplemented
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
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—44no—
228C2×C4×C31—44no—
229C2×C4×C31—42no—
230C2×C4×C31—82no—
231C2×C4×C31#12011yescharacteristic
232C2×C4×C31#12011yescharacteristic
233C2×C4×C31#12011yescharacteristic
234C2×C4×C31C2×D812yes—
235C2×C4×C31C2×SD1612yes—
236C2×C4×C31AGL(1,ℤ/8)12yes—
237C2×C4×C31AGL(1,ℤ/8)12yes—
238C2×C4×C31C22×D411yescharacteristic
239C2×C4×C31C22×D411yescharacteristic
240C2×C4×C31C22×D411yescharacteristic
241C8×C31—42no—
242C8×C31—42no—
243C8×C31—42no—
244C2×D42C2×C3111yescharacteristic
245C2×#486C2×C3111yescharacteristic, complemented
246C2×#474C2×C3111yescharacteristic, complemented
247C2×#461C2×C3111yescharacteristic, complemented
248#1639C2×C3118yesdirect product factor
249C2×#331C2×C3111yescharacteristic, complemented
250C2×D4⋊C8C2×C3111yescharacteristic, complemented
251C2×D4⋊C8C2×C3111yescharacteristic, complemented
252C22×C31—24no—
253C22×C31—24no—
254C22×C31—48no—
255C22×C31—48no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—44no—
261C22×C31—44no—
262C22×C31—44no—
263C22×C31—832no—
264C22×C31—88no—
265C22×C31#44712yes—
266C22×C31#45012yes—
267C22×C31#45512yes—
268C22×C31C2×#12011yescharacteristic
269C4×C31—21no—
270C4×C31—21no—
271C4×C31—42no—
272C4×C31—42no—
273C4×C31—82no—
274C4×C31#44712yes—
275C4×C31#45012yes—
276C4×C31C22×D811yescharacteristic
277C4×C31C22×SD1611yescharacteristic
278C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
279C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
280C23×D4—21no—
281C23×D4—21no—
282C22×SD16—21no—
283D42—22no—
284D42—22no—
285D42—22no—
286D42C22×C3114yes—
287C2×C4⋊Q8C22×C3111yescharacteristic, complemented
288C2×#127C22×C3111yescharacteristic
289C2×#121—21no—
290C2×#121—21no—
291C2×#120—22no—
292C2×C4×D4C22×C3111yescharacteristic
293C2×C4×D4C22×C3111yescharacteristic
294#486—22no—
295#486C22×C3114yescomplemented
296#474—22no—
297#474C22×C3114yescomplemented
298#461—22no—
299#461C22×C3114yescomplemented
300C2×#107—21no—
301C2×C4⋊C8C22×C3111yescharacteristic, complemented
302C2×C4⋊C8C22×C3111yescharacteristic, complemented
303C2×#103—21no—
304C2×#102—22no—
305C2×#102—21no—
306C2×C4×C8C22×C3111yescharacteristic, complemented
307#331C22×C3114yescomplemented
308D4⋊C8C22×C3114yescomplemented
309D4⋊C8C22×C3114yescomplemented
310C2×C31—44no—
311C2×C31—44no—
312C2×C31—84no—
313C2×C31#163914yesdirect product factor
314C2×C31C2×#44711yescharacteristic
315C2×C31C2×#45011yescharacteristic
316C2×C31C2×#45511yescharacteristic
317C25—22no—
318C22×Q8—21no—
319C22×D4—22no—
320C22×D4—22no—
321C22×D4—24no—
322C22×D4—24no—
323C22×D4—24no—
324C22×D4—24no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—44no—
328C22×D4—44no—
329C22×D4—44no—
330C22×D4—44no—
331C22×D4D4×C3111yescharacteristic
332C22×D4D4×C3111yescharacteristic
333C22×D4D4×C3112yes—
334C23×C4—21no—
335C23×C4—21no—
336C2×SD16—24no—
337C2×SD16—44no—
338C22×C8—21no—
339C22×C8—21no—
340C22×C8—21no—
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8C23×C3112yescomplemented
345#127—21no—
346#127—21no—
347#127D4×C3112yes—
348#127C23×C3112yes—
349#121—24no—
350#121—24no—
351#121—24no—
352#121—24no—
353#120—28no—
354#120—28no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4D4×C3112yes—
360C4×D4D4×C3112yes—
361C4×D4C23×C3112yes—
362C4×D4C23×C3112yes—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4D4×C3111yescharacteristic
366C2×C4⋊C4D4×C3111yescharacteristic
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C42C23×C3111yescharacteristic
371#107—24no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8C23×C3112yescomplemented
375C4⋊C8C23×C3112yescomplemented
376#103—24no—
377#102—24no—
378#102—28no—
379C4×C8—21no—
380C4×C8C23×C3112yescomplemented
381C31C2×#163911yescharacteristic, direct product factor, Sylow
382C24—21no—
383C24—21no—
384C24—28no—
385C24—22no—
386C24—48no—
387C24—48no—
388C24—84no—
389C2×Q8—22no—
390C2×Q8—41no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—416no—
406C2×D4—416no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4D8×C3114yes—
418C2×D4SD16×C3114yescomplemented
419C2×D4C2×D4×C3112yes—
420C2×D4C2×D4×C3112yes—
421C2×D4C2×D4×C3114yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436SD16—48no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4C2×D4×C3112yes—
450C4⋊C4C2×D4×C3112yes—
451C22⋊C4—28no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C42C2×D4×C3111yescharacteristic
457C42C2×D4×C3111yescharacteristic
458C42C2×D4×C3111yescharacteristic
459C42C24×C3111yescharacteristic
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—416no—
466C23—416no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—88no—
475C23—832no—
476C23C31×#12011yescharacteristic
477Q8—42no—
478Q8—42no—
479Q8—82no—
480D4—24no—
481D4—24no—
482D4—22no—
483D4—22no—
484D4—44no—
485D4—416no—
486D4—416no—
487D4—44no—
488D4—44no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—82no—
494D4C2×D8×C3114yes—
495D4C2×SD16×C3114yescomplemented
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4C31×#12011yescharacteristic
510C2×C4C31×#12011yescharacteristic
511C2×C4C31×#12011yescharacteristic
512C2×C4C2×D8×C3112yes—
513C2×C4C2×SD16×C3112yes—
514C2×C4C31×AGL(1,ℤ/8)12yes—
515C2×C4C31×AGL(1,ℤ/8)12yes—
516C2×C4C22×D4×C3111yescharacteristic
517C2×C4C22×D4×C3111yescharacteristic
518C2×C4C22×D4×C3111yescharacteristic
519C8—42no—
520C8—42no—
521C8—42no—
522C22—24no—
523C22—24no—
524C22—48no—
525C22—48no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—832no—
534C22—88no—
535C22C31×#44712yes—
536C22C31×#45012yes—
537C22C31×#45512yes—
538C22C2×C31×#12011yescharacteristic
539C4—21no—
540C4—21no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4C31×#44712yes—
545C4C31×#45012yes—
546C4C22×D8×C3111yescharacteristic
547C4C22×SD16×C3111yescharacteristic
548C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
549C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
550C2—44no—
551C2—44no—
552C2—84no—
553C2C31×#163914yesdirect product factor
554C2C2×C31×#44711yescharacteristic
555C2C2×C31×#45011yescharacteristic
556C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×D42×C31C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#486C2semidirect
3. C2×C31×#474C2semidirect
4. C2×C31×#461C2semidirect
5. C31×#1639C2direct (N × H)
6. C2×C31×#331C2semidirect
7. C2×C31×D4⋊C8C2semidirect
8. C2×C31×D4⋊C8C2semidirect
16. C2×C31×C4⋊Q8C22semidirect
24. C31×#486C22semidirect
26. C31×#474C22semidirect
28. C31×#461C22semidirect
30. C2×C31×C4⋊C8C22semidirect
31. C2×C31×C4⋊C8C22semidirect
35. C2×C4×C8×C31C22semidirect
36. C31×#331C22semidirect
37. C31×D4⋊C8C22semidirect
38. C31×D4⋊C8C22semidirect
66. C31×C4⋊Q8C23semidirect
96. C31×C4⋊C8C23semidirect
97. C31×C4⋊C8C23semidirect
102. C4×C8×C31C23semidirect
139. C2×D4×C31SD16semidirect
181. C2×#1639C31direct (N × H)
217. D4×C31C2×SD16semidirect
245. C2×#486C2×C31semidirect
246. C2×#474C2×C31semidirect
247. C2×#461C2×C31semidirect
248. #1639C2×C31direct (N × H)
249. C2×#331C2×C31semidirect
250. C2×D4⋊C8C2×C31semidirect
251. C2×D4⋊C8C2×C31semidirect
287. C2×C4⋊Q8C22×C31semidirect
295. #486C22×C31semidirect
297. #474C22×C31semidirect
299. #461C22×C31semidirect
301. C2×C4⋊C8C22×C31semidirect
302. C2×C4⋊C8C22×C31semidirect
306. C2×C4×C8C22×C31semidirect
307. #331C22×C31semidirect
308. D4⋊C8C22×C31semidirect
309. D4⋊C8C22×C31semidirect
313. C2×C31#1639direct (N × H)
344. C4⋊Q8C23×C31semidirect
374. C4⋊C8C23×C31semidirect
375. C4⋊C8C23×C31semidirect
380. C4×C8C23×C31semidirect
381. C31C2×#1639direct (N × H)
418. C2×D4SD16×C31semidirect
495. D4C2×SD16×C31semidirect
553. C2C31×#1639direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D42×C312
C2×C31×#4862
C2×C31×#4742
C2×C31×#4612
C31×#16392
C2×C31×#3312
C2×C31×D4⋊C82
C2×#163931