Finite Group Databasebeta

← C4×#2120 #492987

Subgroups — C4×#2120 #492987

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C2×#2120C212yesmaximal
3C4×#413C211yesmaximal, characteristic, complemented
4C4×#410C211yesmaximal, characteristic, complemented
5C4×#352C214yesmaximal, complemented
6C4×AΣL(1,𝔽16)C214yesmaximal, complemented
7#12573C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C23×C42C411yescharacteristic, complemented
11C2×#413C2212yes—
12C2×#410C2212yes—
13C2×C4×M16—21no—
14C4×C22⋊Q8—22no—
15C4×#123C412yescomplemented
16C4×#123C2212yes—
17C4×#121—22no—
18C4×#120C412yescomplemented
19C4×#120C2212yes—
20#2297C2211yescharacteristic
21#2296C411yescharacteristic, complemented
22#2290—21no—
23#2290—21no—
24#2280C411yescharacteristic, complemented
25C42×D4—21no—
26C42×D4—21no—
27C4×#117—21no—
28C4×#117C411yescharacteristic, complemented
29#2120C4116yesdirect product factor
30C2×#352C2214yes—
31C2×AΣL(1,𝔽16)C2214yes—
32C4×#101—22no—
33C4×#100C2214yes—
34C4×#99—22no—
35C4×#99C2214yes—
36#1742—21no—
37#1740—21no—
38#1390C2218yes—
39C22×#41C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C22×M16—22no—
43#525—24no—
44#525C2×C418yes—
45C2×#123C2×C411yescharacteristic
46C2×#123C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×#121—21no—
49C2×#120C2×C411yescharacteristic
50C2×#120C2311yescharacteristic
51C4×#41—41no—
52C4×#41—42no—
53C4×#41—41no—
54C2×C4×Q8D412yes—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4C2312yes—
62C2×#117—22no—
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C42—21no—
69C22×C42—22no—
70C22×C42—21no—
71C22×C42—41no—
72C22×C42—42no—
73C22×C42—41no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C412yes—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#413—24no—
80#413C2×C418yescomplemented
81C2×#101—22no—
82C2×#100C2314yes—
83#410—24no—
84#410C2×C418yescomplemented
85C2×#99—22no—
86C2×#99C2314yes—
87C4×M16—24no—
88C4×M16—42no—
89C2×#97—41no—
90C2×C4×C8—41no—
91#391—21no—
92#390—21no—
93#389—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#386C2×C412yes—
101#386C2312yes—
102#385—22no—
103#385—22no—
104#385C2×C412yes—
105#385C2×C412yes—
106#381C2×C412yes—
107#380—22no—
108#379C2×C412yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—42no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—24no—
124C4×C22⋊C4C2×C412yes—
125C4×C22⋊C4C2×C412yes—
126#376—21no—
127#376—41no—
128C43—21no—
129C43—41no—
130#352C2×C4116yescomplemented
131AΣL(1,𝔽16)C2×C4116yescomplemented
132#343—24no—
133#342—24no—
134C25C22×C411yescharacteristic
135C2×#41—41no—
136C2×#41—41no—
137C2×#41—42no—
138C2×#41C22×C414yes—
139C2×#41#4114yes—
140C22×Q8C2×D411yescharacteristic
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C22×C411yescharacteristic
144C22×D4C2×D411yescharacteristic
145C22×D4C2411yescharacteristic
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—48no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4C4214yescomplemented
158C23×C4C22⋊C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C2×M16—28no—
167C2×M16—24no—
168C2×M16—44no—
169C2×M16—42no—
170C22×C8—42no—
171#123—22no—
172#123—22no—
173#123C4214yescomplemented
174#123C22×C414yes—
175C22⋊Q8—28no—
176#121—28no—
177#120—22no—
178#120—22no—
179#120C4214yescomplemented
180#120C22×C414yes—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—48no—
192C4×D4—48no—
193#117—22no—
194#117—28no—
195#117—41no—
196#117—41no—
197#117—41no—
198#117—41no—
199#117—41no—
200#117—41no—
201#117—44no—
202#117—42no—
203#117C4214yescomplemented
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22×C411yescharacteristic
225C2×C22⋊C4C22×C411yescharacteristic
226C2×C22⋊C4C22×C412yes—
227C2×C22⋊C4C22×C412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—42no—
251C2×C42—41no—
252C2×C42—42no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C412yes—
263C2×C42C22⋊C412yes—
264C2×C42C22⋊C411yescharacteristic
265C2×C42C2×D412yes—
266#101—28no—
267#100—24no—
268#100C22×C418yes—
269#99—28no—
270#99—24no—
271#99C22×C418yes—
272#97—44no—
273C4×C8—44no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—44no—
283C24—21no—
284C24—21no—
285C24—41no—
286C24—44no—
287C24—41no—
288C24C2×C4212yes—
289C24C2×C22⋊C411yescharacteristic
290C24C23×C411yescharacteristic
291C24C22×D411yescharacteristic
292#41—44no—
293#41—44no—
294#41—44no—
295#41—44no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8C4×D414yes—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4C2×C4212yes—
310C2×D4#11712yes—
311C2×D4C4×D414yes—
312C2×D4C23×C412yes—
313C2×D4C2×#4112yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4C2×C4212yes—
366C22×C4C2×C4212yes—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4#11712yes—
375C22×C4C4×D414yes—
376C22×C4C4×D414yes—
377C22×C4C22×D411yescharacteristic
378M16—28no—
379M16—44no—
380M16—48no—
381C2×C8—44no—
382C2×C8—48no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C22⋊C4—21no—
394C22⋊C4—21no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—28no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C22⋊C4C2×C4212yes—
410C22⋊C4C2×C4212yes—
411C42—24no—
412C42—22no—
413C42—22no—
414C42—21no—
415C42—21no—
416C42—22no—
417C42—22no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—22no—
423C42—22no—
424C42—42no—
425C42—44no—
426C42—44no—
427C42—44no—
428C42—48no—
429C42—42no—
430C42—42no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42—88no—
435C42#9911yescharacteristic
436C42#9912yes—
437C42#9911yescharacteristic
438C23—24no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—48no—
457C23—41no—
458C23—41no—
459C23C4×C22⋊C412yes—
460C23C22×C4211yescharacteristic
461C23C22×C22⋊C411yescharacteristic
462C23C2×#11711yescharacteristic
463C23C2×C4×D412yes—
464Q8—44no—
465D4—42no—
466D4—42no—
467D4—44no—
468D4—42no—
469D4—42no—
470D4—48no—
471D4—48no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—48no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—416no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—88no—
524C2×C4C4×C22⋊C412yes—
525C2×C4C4×C22⋊C412yes—
526C2×C4C4×C22⋊C412yes—
527C2×C4C2×#9912yes—
528C2×C4C2×#9912yes—
529C2×C4C2×#9911yescharacteristic
530C2×C4C2×#9911yescharacteristic
531C2×C4#41012yes—
532C2×C4C2×C4×D412yes—
533C8—48no—
534C22—22no—
535C22—24no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—42no—
545C22—44no—
546C22—42no—
547C22—44no—
548C22—44no—
549C22—48no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22C2×C4×C22⋊C411yescharacteristic
558C22C22×#9911yescharacteristic
559C22C2×#41011yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—82no—
572C4—88no—
573C4C4×#9912yes—
574C4C4×#9912yes—
575C4#212014yesdirect product factor
576C2—21no—
577C2—21no—
578C2—42no—
579C2—44no—
580C2—41no—
581C2—41no—
582C2—42no—
583C2C2×C4×#9911yescharacteristic
584C2C2×#212012yes—
subgroupG/H
centerC42#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
socleC22C22×#99
prime radicalC2×#525C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C42 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#413C2semidirect
4. C4×#410C2semidirect
5. C4×#352C2semidirect
6. C4×AΣL(1,𝔽16)C2semidirect
7. #12573C2semidirect
10. C23×C42C4semidirect
15. C4×#123C4semidirect
18. C4×#120C4semidirect
21. #2296C4semidirect
24. #2280C4semidirect
28. C4×#117C4semidirect
29. #2120C4direct (N × H)
80. #413C2×C4semidirect
84. #410C2×C4semidirect
130. #352C2×C4semidirect
131. AΣL(1,𝔽16)C2×C4semidirect
157. C23×C4C42semidirect
173. #123C42semidirect
179. #120C42semidirect
203. #117C42semidirect
575. C4#2120direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#5252
C2×#21202
C4×#4132
C4×#4102
C4×#3522
C4×AΣL(1,𝔽16)2
#125732