Finite Group Databasebeta

← C4×#2122 #492992

Subgroups — C4×#2122 #492992

Order 512.

1427 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2C2×#2122C212yesmaximal
3C4×#413C211yesmaximal, characteristic, complemented
4C4×#410C211yesmaximal, characteristic, complemented
5#12622C214yesmaximal, complemented
6C4×AΣL(1,𝔽16)C214yesmaximal, complemented
7#12575C212yesmaximal, complemented
8C2×#535C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C411yescharacteristic, complemented
11C2×#413C2212yes—
12C2×#410C2212yes—
13C2×C4×M16—21no—
14#2328—21no—
15#2326—21no—
16#2324—21no—
17#2324—21no—
18#2321C412yescomplemented
19#2321C2212yes—
20#2317C411yescharacteristic, complemented
21#2311C2211yescharacteristic
22#2309C411yescharacteristic, complemented
23C4×#127—21no—
24C4×#124—21no—
25C4×#121—21no—
26C4×#121—21no—
27C4×#120C412yescomplemented
28C4×#120C2212yes—
29C4×#117—21no—
30C4×#117C411yescharacteristic, complemented
31#2122C4116yesdirect product factor
32C2×AΣL(1,𝔽16)C2218yes—
33C4×#101—21no—
34C4×#101—21no—
35C4×#100C2212yes—
36C4×#100C2212yes—
37C4×#99—21no—
38C4×#99—21no—
39C4×#99C2212yes—
40C4×#99C2212yes—
41#1742—21no—
42#1740—21no—
43#1390C2218yes—
44C22×#41C2311yescharacteristic
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C22×M16—22no—
48#535—24no—
49#535C2×C418yes—
50C2×#127—21no—
51C2×#124—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#120C2×C412yes—
55C2×#120C2312yes—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—41no—
69C2×C4×D4—41no—
70C2×C4×D4—41no—
71C2×C4×D4—41no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—22no—
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C22⋊C4C2×C412yes—
86C22×C42C2×C411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89#413—24no—
90#413C2×C418yescomplemented
91C2×#101—22no—
92C2×#100C2314yes—
93#410—24no—
94#410C2×C418yescomplemented
95C2×#99—22no—
96C2×#99C2314yes—
97C4×M16—22no—
98C4×M16—22no—
99C4×M16—41no—
100C4×M16—41no—
101C2×#97—41no—
102C2×C4×C8—41no—
103#391—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—22no—
108#390—22no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386C2×C412yes—
122#386C2312yes—
123#385—22no—
124#385—22no—
125#385C2×C412yes—
126#385C2×C412yes—
127#384—21no—
128#382—21no—
129#380—21no—
130#380—21no—
131#379C2×C412yes—
132#379C2×C412yes—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—41no—
136C4×C4⋊C4—41no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4C2×C412yes—
142C4×C22⋊C4C2×C412yes—
143#376—21no—
144#376—41no—
145C43—21no—
146C43—41no—
147AΣL(1,𝔽16)C2×C4132yescomplemented
148#343—22no—
149#343—22no—
150#342—22no—
151#342—22no—
152C25C22×C412yes—
153C2×#41—41no—
154C2×#41—41no—
155C2×#41—42no—
156C2×#41C22×C414yes—
157C2×#41#4114yes—
158C22×Q8C2×D411yescharacteristic
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—22no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—42no—
167C22×D4C4214yescomplemented
168C22×D4C22×C411yescharacteristic
169C22×D4C2×D411yescharacteristic
170C22×D4C2411yescharacteristic
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—84no—
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×M16—28no—
187C2×M16—24no—
188C2×M16—44no—
189C2×M16—42no—
190C22×C8—42no—
191#127—28no—
192#124—28no—
193#121—28no—
194#121—28no—
195#120—24no—
196#120—24no—
197#120C4218yescomplemented
198#120C22×C418yes—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—84no—
226C4×D4—84no—
227#117—22no—
228#117—28no—
229#117—41no—
230#117—41no—
231#117—41no—
232#117—41no—
233#117—41no—
234#117—41no—
235#117—44no—
236#117—42no—
237#117C4214yescomplemented
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22×C412yes—
269C2×C22⋊C4C22×C412yes—
270C2×C22⋊C4C22×C411yescharacteristic
271C2×C22⋊C4C22×C411yescharacteristic
272C2×C22⋊C4C2×D411yescharacteristic
273C2×C22⋊C4C2×D411yescharacteristic
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303#101—28no—
304#100—24no—
305#100C22×C418yes—
306#99—28no—
307#99—24no—
308#99C22×C418yes—
309#97—42no—
310#97—42no—
311C4×C8—42no—
312C4×C8—42no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—42no—
320#95—42no—
321C24—22no—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—82no—
326C24—82no—
327C24C2×C4214yes—
328C24C2×C22⋊C411yescharacteristic
329C24C23×C411yescharacteristic
330C24C22×D411yescharacteristic
331#41—44no—
332#41—44no—
333#41—44no—
334#41—44no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8C4×D414yes—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4C2×C4212yes—
362C2×D4#11712yes—
363C2×D4C4×D414yes—
364C2×D4C23×C412yes—
365C2×D4C2×#4112yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—81no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4C2×C4212yes—
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4#11712yes—
421C22×C4C4×D414yes—
422C22×C4C4×D414yes—
423C22×C4C22×D411yescharacteristic
424M16—28no—
425M16—44no—
426M16—48no—
427C2×C8—44no—
428C2×C8—48no—
429C4⋊C4—24no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—42no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C22⋊C4—22no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4C2×C4214yes—
465C42—24no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—42no—
485C42—44no—
486C42—48no—
487C42—82no—
488C42—84no—
489C42—84no—
490C42#9911yescharacteristic
491C42#9911yescharacteristic
492C42#9911yescharacteristic
493C42#9911yescharacteristic
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—24no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—84no—
516C23—84no—
517C23—84no—
518C23—84no—
519C23—88no—
520C23C4×C22⋊C412yes—
521C23C22×C4211yescharacteristic
522C23C22×C22⋊C411yescharacteristic
523C23C2×#11711yescharacteristic
524C23C2×C4×D412yes—
525Q8—44no—
526D4—42no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—42no—
531D4—48no—
532D4—44no—
533D4—44no—
534D4—88no—
535D4—88no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—84no—
587C2×C4—88no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4C4×C22⋊C412yes—
593C2×C4C4×C22⋊C412yes—
594C2×C4C4×C22⋊C412yes—
595C2×C4C2×#9911yescharacteristic
596C2×C4C2×#9911yescharacteristic
597C2×C4C2×#9911yescharacteristic
598C2×C4C2×#9911yescharacteristic
599C2×C4C2×#9911yescharacteristic
600C2×C4C2×#9911yescharacteristic
601C2×C4#41012yes—
602C2×C4C2×C4×D412yes—
603C8—48no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—44no—
624C22—81no—
625C22—81no—
626C22—82no—
627C22—82no—
628C22—88no—
629C22—88no—
630C22—82no—
631C22—82no—
632C22C2×C4×C22⋊C411yescharacteristic
633C22C22×#9911yescharacteristic
634C22C2×#41011yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—44no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—82no—
649C4—88no—
650C4—82no—
651C4C4×#9912yes—
652C4C4×#9912yes—
653C4#212212yesdirect product factor
654C2—21no—
655C2—21no—
656C2—42no—
657C2—41no—
658C2—41no—
659C2—42no—
660C2—84no—
661C2C2×C4×#9911yescharacteristic
662C2C2×#212211yescharacteristic
663C2#3334711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
socleC22C22×#99
prime radicalC2×#535C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #12622C2semidirect
6. C4×AΣL(1,𝔽16)C2semidirect
7. #12575C2semidirect
10. C22×C4×D4C4semidirect
18. #2321C4semidirect
20. #2317C4semidirect
22. #2309C4semidirect
27. C4×#120C4semidirect
30. C4×#117C4semidirect
31. #2122C4direct (N × H)
90. #413C2×C4semidirect
94. #410C2×C4semidirect
147. AΣL(1,𝔽16)C2×C4semidirect
167. C22×D4C42semidirect
197. #120C42semidirect
237. #117C42semidirect
653. C4#2122direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#5352
C2×#21222
C4×#4132
C4×#4102
#126222
C4×AΣL(1,𝔽16)2
#125752