Finite Group Databasebeta

← C2×C31×#1880 #21477682

Subgroups — C2×C31×#1880 #21477682

Order 7936.

2038 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×SD16×C31C211yesmaximal, characteristic
2C22×C4×Q8×C31C211yesmaximal, characteristic, complemented
3C22×C31×#103C211yesmaximal, characteristic, complemented
4C22×C31×#102C211yesmaximal, characteristic
5C2×C31×#390C211yesmaximal, characteristic
6C31×#1880C218yesmaximal, direct product factor
7C2×C31×#340C211yesmaximal, characteristic, complemented
8C2×C31×#336C211yesmaximal, characteristic, complemented
9C23×Q8×C31C2211yescharacteristic
10C23×D4×C31C2211yescharacteristic
11C22×SD16×C31—24no—
12C22×SD16×C31—24no—
13C22×SD16×C31C418yescomplemented
14C22×SD16×C31C2214yes—
15C23×C8×C31C2211yescharacteristic
16C23×C8×C31C2211yescharacteristic
17C2×C4×Q8×C31—24no—
18C2×C4×Q8×C31—24no—
19C2×C4×Q8×C31—22no—
20C2×C4×Q8×C31C2214yescomplemented
21C22×C4⋊C4×C31—21no—
22C22×C4⋊C4×C31C2211yescharacteristic
23C22×C4⋊C4×C31C2211yescharacteristic
24C22×C22⋊C4×C31—21no—
25C22×C22⋊C4×C31—21no—
26C22×C42×C31—21no—
27C22×C42×C31C2211yescharacteristic
28C2×C31×#103—24no—
29C2×C31×#103C2214yescomplemented
30C2×C31×#102—24no—
31C2×C31×#102C2214yes—
32C31×#390—22no—
33C31×#390C2214yes—
34C31×#340C2214yescomplemented
35C31×#336C2214yescomplemented
36C25×C31—21no—
37C22×Q8×C31—22no—
38C22×Q8×C31—21no—
39C22×Q8×C31—22no—
40C22×Q8×C31—22no—
41C22×Q8×C31C2×C414yes—
42C22×Q8×C31D414yes—
43C22×Q8×C31D414yes—
44C22×Q8×C31C2312yes—
45C22×D4×C31—22no—
46C22×D4×C31—22no—
47C22×D4×C31—22no—
48C22×D4×C31—21no—
49C22×D4×C31—22no—
50C22×D4×C31—22no—
51C22×D4×C31C2×C414yes—
52C22×D4×C31C2312yes—
53C23×C4×C31—21no—
54C23×C4×C31—21no—
55C23×C4×C31—21no—
56C23×C4×C31D411yescharacteristic
57C23×C4×C31D411yescharacteristic
58C23×C4×C31C2311yescharacteristic
59C2×SD16×C31—232no—
60C2×SD16×C31—28no—
61C2×SD16×C31—28no—
62C2×SD16×C31C2×C4116yescomplemented
63C22×C8×C31—22no—
64C22×C8×C31—22no—
65C22×C8×C31—22no—
66C22×C8×C31—22no—
67C22×C8×C31—22no—
68C22×C8×C31C2×C414yes—
69C22×C8×C31C2312yes—
70C22×C8×C31C2312yes—
71C4×Q8×C31—216no—
72C4×Q8×C31—28no—
73C4×Q8×C31—28no—
74C2×C4⋊C4×C31—28no—
75C2×C4⋊C4×C31—24no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31D414yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C4⋊C4×C31C2312yes—
84C2×C22⋊C4×C31—24no—
85C2×C22⋊C4×C31—24no—
86C2×C22⋊C4×C31—44no—
87C2×C22⋊C4×C31—44no—
88C2×C42×C31—28no—
89C2×C42×C31—24no—
90C2×C42×C31—22no—
91C2×C42×C31—22no—
92C2×C42×C31D412yes—
93C2×C42×C31C2312yes—
94C31×#103—28no—
95C31×#102—28no—
96C24×C31—24no—
97C24×C31—22no—
98C24×C31—42no—
99C24×C31—42no—
100C24×C31—42no—
101C24×C31—42no—
102C24×C31—42no—
103C24×C31—42no—
104C24×C31C2×D411yescharacteristic
105C2×Q8×C31—24no—
106C2×Q8×C31—28no—
107C2×Q8×C31—24no—
108C2×Q8×C31—24no—
109C2×Q8×C31—28no—
110C2×Q8×C31—28no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—22no—
114C2×Q8×C31C22⋊C4116yescomplemented
115C2×Q8×C31C22×C414yes—
116C2×Q8×C31C2×D414yes—
117C2×Q8×C31C2×D414yes—
118C2×D4×C31—28no—
119C2×D4×C31—28no—
120C2×D4×C31—24no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—42no—
127C2×D4×C31—44no—
128C2×D4×C31—42no—
129C2×D4×C31—44no—
130C2×D4×C31C22×C414yes—
131C22×C4×C31—24no—
132C22×C4×C31—22no—
133C22×C4×C31—24no—
134C22×C4×C31—24no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—24no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—42no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31—42no—
147C22×C4×C31—42no—
148C22×C4×C31SD1614yescomplemented
149C22×C4×C31C22×C412yes—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D412yes—
156C22×C4×C31#4112yes—
157C22×C4×C31#4112yes—
158C22×C4×C31C2411yescharacteristic
159SD16×C31—232no—
160C2×C8×C31—28no—
161C2×C8×C31—22no—
162C2×C8×C31—22no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—44no—
167C2×C8×C31C22×C414yes—
168C4⋊C4×C31—216no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—42no—
174C4⋊C4×C31C2×D414yes—
175C22⋊C4×C31—48no—
176C22⋊C4×C31—48no—
177C42×C31—216no—
178C42×C31—24no—
179C42×C31—22no—
180C42×C31—22no—
181C2×#1880C3111yesmaximal, characteristic, direct product factor, Sylow
182C23×C31—24no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—48no—
190C23×C31—48no—
191C23×C31—48no—
192C23×C31—84no—
193C23×C31—84no—
194C23×C31—84no—
195C23×C31—84no—
196C23×C31C4×D412yes—
197C23×C31#12012yes—
198C23×C31#12112yes—
199C23×C31C2×SD1612yes—
200C23×C31#13512yes—
201C23×C31AGL(1,ℤ/8)12yes—
202C23×C31#13712yes—
203C23×C31C22×D411yescharacteristic
204Q8×C31—24no—
205Q8×C31—24no—
206Q8×C31—24no—
207Q8×C31—216no—
208Q8×C31C2×C22⋊C418yescomplemented
209D4×C31—24no—
210D4×C31—24no—
211D4×C31—42no—
212D4×C31—42no—
213D4×C31—48no—
214C2×C4×C31—24no—
215C2×C4×C31—22no—
216C2×C4×C31—24no—
217C2×C4×C31—24no—
218C2×C4×C31—22no—
219C2×C4×C31—216no—
220C2×C4×C31—28no—
221C2×C4×C31—22no—
222C2×C4×C31—24no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—44no—
228C2×C4×C31—48no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—84no—
233C2×C4×C31C2×C22⋊C414yes—
234C2×C4×C31C4×D414yes—
235C2×C4×C31C4×D412yes—
236C2×C4×C31#12012yes—
237C2×C4×C31#12114yes—
238C2×C4×C31#12112yes—
239C2×C4×C31#12314yes—
240C2×C4×C31C2×SD1614yescomplemented
241C2×C4×C31C23×C411yescharacteristic
242C2×C4×C31C22×D411yescharacteristic
243C2×C4×C31C22×D411yescharacteristic
244C2×C4×C31C22×D411yescharacteristic
245C2×C4×C31C2×#4111yescharacteristic
246C2×C4×C31C2×#4111yescharacteristic
247C8×C31—24no—
248C8×C31—42no—
249C23×SD16C2×C3111yescharacteristic
250C22×C4×Q8C2×C3111yescharacteristic, complemented
251C22×#103C2×C3111yescharacteristic, complemented
252C22×#102C2×C3111yescharacteristic
253C2×#390C2×C3111yescharacteristic
254#1880C2×C3118yesdirect product factor
255C2×#340C2×C3111yescharacteristic, complemented
256C2×#336C2×C3111yescharacteristic, complemented
257C22×C31—44no—
258C22×C31—44no—
259C22×C31—44no—
260C22×C31—816no—
261C22×C31—82no—
262C22×C31—82no—
263C22×C31—82no—
264C22×C31—82no—
265C22×C31#38614yes—
266C22×C31C4×SD1614yes—
267C22×C31#44014yes—
268C22×C31#44814yes—
269C22×C31#44914yes—
270C22×C31#46014yes—
271C22×C31#46414yes—
272C22×C31C2×C4×D411yescharacteristic
273C22×C31C2×#12011yescharacteristic
274C22×C31C2×#12111yescharacteristic
275C22×C31C22×SD1611yescharacteristic
276C22×C31C2×#13511yescharacteristic
277C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C31C2×#13711yescharacteristic
279C4×C31—24no—
280C4×C31—28no—
281C4×C31—42no—
282C4×C31—44no—
283C4×C31—82no—
284C4×C31#38614yes—
285C4×C31C22×C22⋊C411yescharacteristic
286C4×C31C2×C4×D411yescharacteristic
287C4×C31C2×#12111yescharacteristic
288C4×C31C2×#12311yescharacteristic
289C23×Q8C22×C3111yescharacteristic
290C23×D4C22×C3111yescharacteristic
291C22×SD16—24no—
292C22×SD16—24no—
293C22×SD16C4×C3118yescomplemented
294C22×SD16C22×C3114yes—
295C23×C8C22×C3111yescharacteristic
296C23×C8C22×C3111yescharacteristic
297C2×C4×Q8—24no—
298C2×C4×Q8—24no—
299C2×C4×Q8—22no—
300C2×C4×Q8C22×C3114yescomplemented
301C22×C4⋊C4—21no—
302C22×C4⋊C4C22×C3111yescharacteristic
303C22×C4⋊C4C22×C3111yescharacteristic
304C22×C22⋊C4—21no—
305C22×C22⋊C4—21no—
306C22×C42—21no—
307C22×C42C22×C3111yescharacteristic
308C2×#103—24no—
309C2×#103C22×C3114yescomplemented
310C2×#102—24no—
311C2×#102C22×C3114yes—
312#390—22no—
313#390C22×C3114yes—
314#340C22×C3114yescomplemented
315#336C22×C3114yescomplemented
316C2×C31—84no—
317C2×C31#188018yesdirect product factor
318C2×C31C2×#38611yescharacteristic
319C2×C31C2×C4×SD1611yescharacteristic
320C2×C31C2×#44011yescharacteristic
321C2×C31C2×#44811yescharacteristic
322C2×C31C2×#44911yescharacteristic
323C2×C31C2×#46011yescharacteristic
324C2×C31C2×#46411yescharacteristic
325C25—21no—
326C22×Q8—22no—
327C22×Q8—21no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8C2×C4×C3114yes—
331C22×Q8D4×C3114yes—
332C22×Q8D4×C3114yes—
333C22×Q8C23×C3112yes—
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4—21no—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4C2×C4×C3114yes—
341C22×D4C23×C3112yes—
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4D4×C3111yescharacteristic
346C23×C4D4×C3111yescharacteristic
347C23×C4C23×C3111yescharacteristic
348C2×SD16—232no—
349C2×SD16—28no—
350C2×SD16—28no—
351C2×SD16C2×C4×C31116yescomplemented
352C22×C8—22no—
353C22×C8—22no—
354C22×C8—22no—
355C22×C8—22no—
356C22×C8—22no—
357C22×C8C2×C4×C3114yes—
358C22×C8C23×C3112yes—
359C22×C8C23×C3112yes—
360C4×Q8—216no—
361C4×Q8—28no—
362C4×Q8—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4D4×C3114yes—
371C2×C4⋊C4C23×C3112yes—
372C2×C4⋊C4C23×C3112yes—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4—44no—
377C2×C42—28no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—24no—
381C2×C42D4×C3112yes—
382C2×C42C23×C3112yes—
383#103—28no—
384#102—28no—
385C31C2×#188011yescharacteristic, direct product factor, Sylow
386C24—24no—
387C24—22no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C2×D4×C3111yescharacteristic
395C2×Q8—24no—
396C2×Q8—28no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8C22⋊C4×C31116yescomplemented
405C2×Q8C22×C4×C3114yes—
406C2×Q8C2×D4×C3114yes—
407C2×Q8C2×D4×C3114yes—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4C22×C4×C3114yes—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4SD16×C3114yescomplemented
439C22×C4C22×C4×C3112yes—
440C22×C4C2×D4×C3112yes—
441C22×C4C2×D4×C3112yes—
442C22×C4C2×D4×C3112yes—
443C22×C4C2×D4×C3112yes—
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3112yes—
446C22×C4#41×C3112yes—
447C22×C4#41×C3112yes—
448C22×C4C24×C3111yescharacteristic
449SD16—232no—
450C2×C8—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—44no—
457C2×C8C22×C4×C3114yes—
458C4⋊C4—216no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4C2×D4×C3114yes—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C42—216no—
468C42—24no—
469C42—22no—
470C42—22no—
471C23—24no—
472C23—48no—
473C23—48no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—48no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23C4×D4×C3112yes—
486C23C31×#12012yes—
487C23C31×#12112yes—
488C23C2×SD16×C3112yes—
489C23C31×#13512yes—
490C23C31×AGL(1,ℤ/8)12yes—
491C23C31×#13712yes—
492C23C22×D4×C3111yescharacteristic
493Q8—24no—
494Q8—24no—
495Q8—24no—
496Q8—216no—
497Q8C2×C22⋊C4×C3118yescomplemented
498D4—24no—
499D4—24no—
500D4—42no—
501D4—42no—
502D4—48no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—216no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4C2×C22⋊C4×C3114yes—
523C2×C4C4×D4×C3114yes—
524C2×C4C4×D4×C3112yes—
525C2×C4C31×#12012yes—
526C2×C4C31×#12114yes—
527C2×C4C31×#12112yes—
528C2×C4C31×#12314yes—
529C2×C4C2×SD16×C3114yescomplemented
530C2×C4C23×C4×C3111yescharacteristic
531C2×C4C22×D4×C3111yescharacteristic
532C2×C4C22×D4×C3111yescharacteristic
533C2×C4C22×D4×C3111yescharacteristic
534C2×C4C2×#41×C3111yescharacteristic
535C2×C4C2×#41×C3111yescharacteristic
536C8—24no—
537C8—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—816no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22C31×#38614yes—
547C22C4×SD16×C3114yes—
548C22C31×#44014yes—
549C22C31×#44814yes—
550C22C31×#44914yes—
551C22C31×#46014yes—
552C22C31×#46414yes—
553C22C2×C4×D4×C3111yescharacteristic
554C22C2×C31×#12011yescharacteristic
555C22C2×C31×#12111yescharacteristic
556C22C22×SD16×C3111yescharacteristic
557C22C2×C31×#13511yescharacteristic
558C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
559C22C2×C31×#13711yescharacteristic
560C4—24no—
561C4—28no—
562C4—42no—
563C4—44no—
564C4—82no—
565C4C31×#38614yes—
566C4C22×C22⋊C4×C3111yescharacteristic
567C4C2×C4×D4×C3111yescharacteristic
568C4C2×C31×#12111yescharacteristic
569C4C2×C31×#12311yescharacteristic
570C2—84no—
571C2C31×#188018yesdirect product factor
572C2C2×C31×#38611yescharacteristic
573C2C2×C4×SD16×C3111yescharacteristic
574C2C2×C31×#44011yescharacteristic
575C2C2×C31×#44811yescharacteristic
576C2C2×C31×#44911yescharacteristic
577C2C2×C31×#46011yescharacteristic
578C2C2×C31×#46411yescharacteristic
subgroupG/H
center, socleC24×C31C2×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C4×Q8×C31C2semidirect
3. C22×C31×#103C2semidirect
6. C31×#1880C2direct (N × H)
7. C2×C31×#340C2semidirect
8. C2×C31×#336C2semidirect
13. C22×SD16×C31C4semidirect
20. C2×C4×Q8×C31C22semidirect
29. C2×C31×#103C22semidirect
34. C31×#340C22semidirect
35. C31×#336C22semidirect
62. C2×SD16×C31C2×C4semidirect
114. C2×Q8×C31C22⋊C4semidirect
148. C22×C4×C31SD16semidirect
181. C2×#1880C31direct (N × H)
208. Q8×C31C2×C22⋊C4semidirect
240. C2×C4×C31C2×SD16semidirect
250. C22×C4×Q8C2×C31semidirect
251. C22×#103C2×C31semidirect
254. #1880C2×C31direct (N × H)
255. C2×#340C2×C31semidirect
256. C2×#336C2×C31semidirect
293. C22×SD16C4×C31semidirect
300. C2×C4×Q8C22×C31semidirect
309. C2×#103C22×C31semidirect
314. #340C22×C31semidirect
315. #336C22×C31semidirect
317. C2×C31#1880direct (N × H)
351. C2×SD16C2×C4×C31semidirect
385. C31C2×#1880direct (N × H)
404. C2×Q8C22⋊C4×C31semidirect
438. C22×C4SD16×C31semidirect
497. Q8C2×C22⋊C4×C31semidirect
529. C2×C4C2×SD16×C31semidirect
571. C2C31×#1880direct (N × H)

Maximal subgroups

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

subgroupindex
C23×SD16×C312
C22×C4×Q8×C312
C22×C31×#1032
C22×C31×#1022
C2×C31×#3902
C31×#18802
C2×C31×#3402
C2×C31×#3362
C2×#188031