Finite Group Databasebeta

← C2×C31×#1625 #21475245

Subgroups — C2×C31×#1625 #21475245

Order 7936.

1322 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic, complemented
2C2×C31×#486C211yesmaximal, characteristic, complemented
3C2×C31×#459C211yesmaximal, characteristic, complemented
4C2×C31×#459C211yesmaximal, characteristic, complemented
5C31×#1625C218yesmaximal, direct product factor
6C2×C31×#332C211yesmaximal, characteristic, complemented
7C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×#41×C31—21no—
10C22×D8×C31—21no—
11C22×D8×C31—21no—
12C31×#547—22no—
13C31×#547—22no—
14C31×#547—22no—
15C31×#547C2214yescomplemented
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#127C2211yescharacteristic
18C2×C31×#123—22no—
19C2×C31×C22⋊Q8—21no—
20C2×C31×#121—21no—
21C2×C4×D4×C31C2211yescharacteristic
22C2×C4×D4×C31C2211yescharacteristic
23C22×C4⋊C4×C31—21no—
24C31×#486—22no—
25C31×#486C2214yescomplemented
26C31×#459—22no—
27C31×#459—22no—
28C31×#459C2214yescomplemented
29C31×#459C2214yescomplemented
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×C4⋊C8C2211yescharacteristic
32C2×C31×#102—22no—
33C2×C31×#102—21no—
34C2×C31×#102—21no—
35C2×C4×C8×C31C2211yescharacteristic
36C31×#332C2214yescomplemented
37C31×D4⋊C8C2214yescomplemented
38C31×D4⋊C8C2214yescomplemented
39C2×#41×C31—24no—
40C2×#41×C31—24no—
41C2×#41×C31—44no—
42C2×#41×C31—44no—
43C22×Q8×C31—21no—
44C22×D4×C31—22no—
45C22×D4×C31—21no—
46C22×D4×C31—21no—
47C22×D4×C31—21no—
48C22×D4×C31D411yescharacteristic
49C22×D4×C31D411yescharacteristic
50C23×C4×C31—22no—
51C23×C4×C31—21no—
52C23×C4×C31—21no—
53C2×D8×C31—24no—
54C2×D8×C31—24no—
55C2×D8×C31—44no—
56C2×D8×C31—44no—
57C22×C8×C31—21no—
58C22×C8×C31—21no—
59C22×C8×C31—21no—
60C31×C4⋊Q8—21no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8D412yes—
63C31×C4⋊Q8C2312yes—
64C31×#127—21no—
65C31×#127—21no—
66C31×#127—21no—
67C31×#127C2312yes—
68C31×#123—28no—
69C31×#123—28no—
70C31×C22⋊Q8—24no—
71C31×C22⋊Q8—24no—
72C31×#121—24no—
73C31×#121—24no—
74C4×D4×C31—21no—
75C4×D4×C31—21no—
76C4×D4×C31—21no—
77C4×D4×C31—21no—
78C4×D4×C31D412yes—
79C4×D4×C31D412yes—
80C4×D4×C31C2312yes—
81C4×D4×C31C2312yes—
82C2×C4⋊C4×C31—22no—
83C2×C4⋊C4×C31—24no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—24no—
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D412yes—
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C42×C31C2311yescharacteristic
93C31×C4⋊C8—21no—
94C31×C4⋊C8—21no—
95C31×C4⋊C8C2312yes—
96C31×C4⋊C8C2312yes—
97C31×#102—24no—
98C31×#102—24no—
99C31×#102—28no—
100C4×C8×C31—21no—
101C4×C8×C31C2312yes—
102C24×C31—21no—
103C24×C31—21no—
104C24×C31—41no—
105#41×C31—48no—
106#41×C31—416no—
107#41×C31—48no—
108C2×Q8×C31—22no—
109C2×Q8×C31—22no—
110C2×Q8×C31—42no—
111C2×Q8×C31—42no—
112C2×D4×C31—28no—
113C2×D4×C31—21no—
114C2×D4×C31—21no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31—22no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—44no—
122C2×D4×C31—48no—
123C2×D4×C31—41no—
124C2×D4×C31—41no—
125C2×D4×C31—41no—
126C2×D4×C31—42no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31—42no—
130C2×D4×C31D814yescomplemented
131C2×D4×C31C2×D412yes—
132C2×D4×C31C2×D412yes—
133C22×C4×C31—28no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—48no—
142C22×C4×C31—42no—
143C22×C4×C31—42no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149D8×C31—48no—
150D8×C31—48no—
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C2×C8×C31—22no—
154C2×C8×C31—42no—
155C2×C8×C31—42no—
156C2×C8×C31—42no—
157C4⋊C4×C31—28no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31C2×D412yes—
164C4⋊C4×C31C2×D412yes—
165C4⋊C4×C31C2×D414yes—
166C22⋊C4×C31—28no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C42×C31C2×D411yescharacteristic
172C42×C31C2×D411yescharacteristic
173C42×C31C2×D411yescharacteristic
174C42×C31C2411yescharacteristic
175C2×#1625C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31—22no—
177C23×C31—24no—
178C23×C31—22no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—82no—
186C23×C31—82no—
187C23×C31—82no—
188C23×C31#12011yescharacteristic
189Q8×C31—42no—
190Q8×C31—42no—
191Q8×C31—44no—
192D4×C31—22no—
193D4×C31—22no—
194D4×C31—22no—
195D4×C31—24no—
196D4×C31—44no—
197D4×C31—416no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—42no—
201D4×C31—42no—
202D4×C31—42no—
203D4×C31—82no—
204D4×C31—82no—
205D4×C31—82no—
206D4×C31C2×D814yescomplemented
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—24no—
211C2×C4×C31—416no—
212C2×C4×C31—42no—
213C2×C4×C31—44no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—44no—
220C2×C4×C31—44no—
221C2×C4×C31#12011yescharacteristic
222C2×C4×C31#12011yescharacteristic
223C2×C4×C31#12011yescharacteristic
224C2×C4×C31C2×D812yes—
225C2×C4×C31#13512yes—
226C2×C4×C31AGL(1,ℤ/8)12yes—
227C2×C4×C31AGL(1,ℤ/8)12yes—
228C2×C4×C31C22×D411yescharacteristic
229C2×C4×C31C22×D411yescharacteristic
230C2×C4×C31C22×D411yescharacteristic
231C8×C31—42no—
232C8×C31—42no—
233C8×C31—42no—
234C2×#547C2×C3111yescharacteristic, complemented
235C2×#486C2×C3111yescharacteristic, complemented
236C2×#459C2×C3111yescharacteristic, complemented
237C2×#459C2×C3111yescharacteristic, complemented
238#1625C2×C3118yesdirect product factor
239C2×#332C2×C3111yescharacteristic, complemented
240C2×D4⋊C8C2×C3111yescharacteristic, complemented
241C2×D4⋊C8C2×C3111yescharacteristic, complemented
242C22×C31—24no—
243C22×C31—48no—
244C22×C31—42no—
245C22×C31—42no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—84no—
250C22×C31—82no—
251C22×C31—82no—
252C22×C31—82no—
253C22×C31—164no—
254C22×C31#44712yes—
255C22×C31#44912yes—
256C22×C31#45412yes—
257C22×C31C2×#12011yescharacteristic
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—42no—
261C4×C31—44no—
262C4×C31—42no—
263C4×C31#44712yes—
264C4×C31#44912yes—
265C4×C31C22×D811yescharacteristic
266C4×C31C2×#13511yescharacteristic
267C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
268C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
269C22×#41—21no—
270C22×D8—21no—
271C22×D8—21no—
272#547—22no—
273#547—22no—
274#547—22no—
275#547C22×C3114yescomplemented
276C2×C4⋊Q8C22×C3111yescharacteristic
277C2×#127C22×C3111yescharacteristic
278C2×#123—22no—
279C2×C22⋊Q8—21no—
280C2×#121—21no—
281C2×C4×D4C22×C3111yescharacteristic
282C2×C4×D4C22×C3111yescharacteristic
283C22×C4⋊C4—21no—
284#486—22no—
285#486C22×C3114yescomplemented
286#459—22no—
287#459—22no—
288#459C22×C3114yescomplemented
289#459C22×C3114yescomplemented
290C2×C4⋊C8C22×C3111yescharacteristic
291C2×C4⋊C8C22×C3111yescharacteristic
292C2×#102—22no—
293C2×#102—21no—
294C2×#102—21no—
295C2×C4×C8C22×C3111yescharacteristic
296#332C22×C3114yescomplemented
297D4⋊C8C22×C3114yescomplemented
298D4⋊C8C22×C3114yescomplemented
299C2×C31—44no—
300C2×C31—82no—
301C2×C31—162no—
302C2×C31#162514yesdirect product factor
303C2×C31C2×#44711yescharacteristic
304C2×C31C2×#44911yescharacteristic
305C2×C31C2×#45411yescharacteristic
306C2×#41—24no—
307C2×#41—24no—
308C2×#41—44no—
309C2×#41—44no—
310C22×Q8—21no—
311C22×D4—22no—
312C22×D4—21no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4D4×C3111yescharacteristic
316C22×D4D4×C3111yescharacteristic
317C23×C4—22no—
318C23×C4—21no—
319C23×C4—21no—
320C2×D8—24no—
321C2×D8—24no—
322C2×D8—44no—
323C2×D8—44no—
324C22×C8—21no—
325C22×C8—21no—
326C22×C8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8D4×C3112yes—
330C4⋊Q8C23×C3112yes—
331#127—21no—
332#127—21no—
333#127—21no—
334#127C23×C3112yes—
335#123—28no—
336#123—28no—
337C22⋊Q8—24no—
338C22⋊Q8—24no—
339#121—24no—
340#121—24no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4D4×C3112yes—
346C4×D4D4×C3112yes—
347C4×D4C23×C3112yes—
348C4×D4C23×C3112yes—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4D4×C3111yescharacteristic
354C2×C4⋊C4D4×C3111yescharacteristic
355C2×C4⋊C4D4×C3112yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C42C23×C3111yescharacteristic
360C4⋊C8—21no—
361C4⋊C8—21no—
362C4⋊C8C23×C3112yes—
363C4⋊C8C23×C3112yes—
364#102—24no—
365#102—24no—
366#102—28no—
367C4×C8—21no—
368C4×C8C23×C3112yes—
369C31C2×#162511yescharacteristic, direct product factor, Sylow
370C24—21no—
371C24—21no—
372C24—41no—
373#41—416no—
374#41—48no—
375#41—48no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×D4—28no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4D8×C3114yescomplemented
399C2×D4C2×D4×C3112yes—
400C2×D4C2×D4×C3112yes—
401C22×C4—28no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3111yescharacteristic
416C22×C4C2×D4×C3111yescharacteristic
417D8—48no—
418D8—48no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C4⋊C4—28no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4C2×D4×C3112yes—
432C4⋊C4C2×D4×C3112yes—
433C4⋊C4C2×D4×C3114yes—
434C22⋊C4—28no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C42C2×D4×C3111yescharacteristic
440C42C2×D4×C3111yescharacteristic
441C42C2×D4×C3111yescharacteristic
442C42C24×C3111yescharacteristic
443C23—24no—
444C23—22no—
445C23—22no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23C31×#12011yescharacteristic
456Q8—44no—
457Q8—42no—
458Q8—42no—
459D4—24no—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—44no—
464D4—416no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—82no—
471D4—82no—
472D4—82no—
473D4C2×D8×C3114yescomplemented
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—416no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4C31×#12011yescharacteristic
489C2×C4C31×#12011yescharacteristic
490C2×C4C31×#12011yescharacteristic
491C2×C4C2×D8×C3112yes—
492C2×C4C31×#13512yes—
493C2×C4C31×AGL(1,ℤ/8)12yes—
494C2×C4C31×AGL(1,ℤ/8)12yes—
495C2×C4C22×D4×C3111yescharacteristic
496C2×C4C22×D4×C3111yescharacteristic
497C2×C4C22×D4×C3111yescharacteristic
498C8—42no—
499C8—42no—
500C8—42no—
501C22—24no—
502C22—48no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—84no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—164no—
513C22C31×#44712yes—
514C22C31×#44912yes—
515C22C31×#45412yes—
516C22C2×C31×#12011yescharacteristic
517C4—21no—
518C4—21no—
519C4—42no—
520C4—44no—
521C4—42no—
522C4C31×#44712yes—
523C4C31×#44912yes—
524C4C22×D8×C3111yescharacteristic
525C4C2×C31×#13511yescharacteristic
526C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
527C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
528C2—44no—
529C2—82no—
530C2—162no—
531C2C31×#162514yesdirect product factor
532C2C2×C31×#44711yescharacteristic
533C2C2×C31×#44911yescharacteristic
534C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC42C24×C31
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
1. C2×C31×#547C2semidirect
2. C2×C31×#486C2semidirect
3. C2×C31×#459C2semidirect
4. C2×C31×#459C2semidirect
5. C31×#1625C2direct (N × H)
6. C2×C31×#332C2semidirect
7. C2×C31×D4⋊C8C2semidirect
8. C2×C31×D4⋊C8C2semidirect
15. C31×#547C22semidirect
25. C31×#486C22semidirect
28. C31×#459C22semidirect
29. C31×#459C22semidirect
36. C31×#332C22semidirect
37. C31×D4⋊C8C22semidirect
38. C31×D4⋊C8C22semidirect
130. C2×D4×C31D8semidirect
175. C2×#1625C31direct (N × H)
206. D4×C31C2×D8semidirect
234. C2×#547C2×C31semidirect
235. C2×#486C2×C31semidirect
236. C2×#459C2×C31semidirect
237. C2×#459C2×C31semidirect
238. #1625C2×C31direct (N × H)
239. C2×#332C2×C31semidirect
240. C2×D4⋊C8C2×C31semidirect
241. C2×D4⋊C8C2×C31semidirect
275. #547C22×C31semidirect
285. #486C22×C31semidirect
288. #459C22×C31semidirect
289. #459C22×C31semidirect
296. #332C22×C31semidirect
297. D4⋊C8C22×C31semidirect
298. D4⋊C8C22×C31semidirect
302. C2×C31#1625direct (N × H)
369. C31C2×#1625direct (N × H)
398. C2×D4D8×C31semidirect
473. D4C2×D8×C31semidirect
531. C2C31×#1625direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5472
C2×C31×#4862
C2×C31×#4592
C31×#16252
C2×C31×#3322
C2×C31×D4⋊C82
C2×#162531