Finite Group Databasebeta

← C4×#2123 #492993

Subgroups — C4×#2123 #492993

Order 512.

1267 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2C2×#2123C212yesmaximal
3C4×#413C211yesmaximal, characteristic
4C4×#410C211yesmaximal, characteristic
5#12621C214yesmaximal, complemented
6C4×#352C214yesmaximal, complemented
7#12575C212yesmaximal
8C2×#536C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic, complemented
11C2×#413C2212yes—
12C2×#410C2212yes—
13C2×C4×M16—21no—
14#2327—21no—
15#2326—21no—
16#2323—21no—
17#2323—21no—
18#2322C412yescomplemented
19#2322C2212yes—
20#2317C411yescharacteristic, complemented
21#2311C2211yescharacteristic
22#2310C411yescharacteristic, complemented
23C4×C4⋊Q8—21no—
24C4×#124—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27C4×#123C412yescomplemented
28C4×#123C2212yes—
29C4×#117—21no—
30C4×#117C411yescharacteristic, complemented
31#2123C4116yesdirect product factor
32C2×#352C2218yes—
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×Q8C2×C411yescharacteristic
46C22×M16—22no—
47#536—24no—
48#536C2×C418yes—
49C2×C4⋊Q8—21no—
50C2×#124—21no—
51C2×#123C2×C412yes—
52C2×#123C2312yes—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C4×#41—41no—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—41no—
64C2×C4×Q8—41no—
65C2×C4×Q8—41no—
66C2×C4×Q8—41no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—22no—
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C412yes—
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C412yes—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87#413—24no—
88#413C2×C418yes—
89C2×#101—22no—
90C2×#100C2314yes—
91#410—24no—
92#410C2×C418yes—
93C2×#99—22no—
94C2×#99C2314yes—
95C4×M16—22no—
96C4×M16—22no—
97C4×M16—41no—
98C4×M16—41no—
99C2×#97—41no—
100C2×C4×C8—41no—
101#391—21no—
102#391—21no—
103#391—21no—
104#391—21no—
105#391—21no—
106#390—21no—
107#389—22no—
108#389—22no—
109#389—21no—
110#389—21no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#386C2×C412yes—
120#386C2312yes—
121#385—22no—
122#385—22no—
123#385C2×C412yes—
124#385C2×C412yes—
125#384—21no—
126#382—21no—
127#381C2×C412yes—
128#381C2×C412yes—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—22no—
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—
147#352C2×C4132yescomplemented
148#343—22no—
149#343—22no—
150#342—22no—
151#342—22no—
152C2×#41—41no—
153C2×#41—41no—
154C2×#41—42no—
155C2×#41C22×C414yes—
156C2×#41#4114yes—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—42no—
163C22×Q8C4214yescomplemented
164C22×Q8C2×D411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C22×D4C2×D411yescharacteristic
167C22×D4C2411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×M16—28no—
180C2×M16—24no—
181C2×M16—44no—
182C2×M16—42no—
183C22×C8—42no—
184C4⋊Q8—28no—
185#124—28no—
186#123—24no—
187#123—24no—
188#123C4218yescomplemented
189#123C22×C418yes—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—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—41no—
216C4×D4—41no—
217C4×D4—41no—
218#117—22no—
219#117—28no—
220#117—41no—
221#117—41no—
222#117—41no—
223#117—41no—
224#117—41no—
225#117—41no—
226#117—44no—
227#117—42no—
228#117C4214yescomplemented
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22×C412yes—
258C2×C22⋊C4C22×C412yes—
259C2×C22⋊C4C22×C411yescharacteristic
260C2×C22⋊C4C22×C411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296#101—28no—
297#100—24no—
298#100C22×C418yes—
299#99—28no—
300#99—24no—
301#99C22×C418yes—
302#97—42no—
303#97—42no—
304C4×C8—42no—
305C4×C8—42no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—42no—
317#95—42no—
318C24C2×C22⋊C411yescharacteristic
319C24C23×C411yescharacteristic
320C24C22×D411yescharacteristic
321#41—44no—
322#41—44no—
323#41—44no—
324#41—44no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—42no—
338C2×Q8C4×D414yes—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4C2×C4212yes—
346C2×D4#11712yes—
347C2×D4C4×D414yes—
348C2×D4C23×C412yes—
349C2×D4C2×#4112yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—82no—
387C22×C4C2×C4212yes—
388C22×C4C2×C4214yes—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4#11712yes—
401C22×C4C4×D414yes—
402C22×C4C4×D414yes—
403C22×C4C22×D411yescharacteristic
404M16—28no—
405M16—44no—
406M16—48no—
407C2×C8—44no—
408C2×C8—48no—
409C4⋊C4—28no—
410C4⋊C4—24no—
411C4⋊C4—28no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C22⋊C4—22no—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4C2×C4214yes—
443C42—24no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—44no—
463C42—44no—
464C42—42no—
465C42—48no—
466C42—42no—
467C42—42no—
468C42—82no—
469C42—84no—
470C42—84no—
471C42#9911yescharacteristic
472C42#9911yescharacteristic
473C42#9911yescharacteristic
474C42#9911yescharacteristic
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C4×C22⋊C412yes—
488C23C22×C4211yescharacteristic
489C23C22×C22⋊C411yescharacteristic
490C23C2×#11711yescharacteristic
491C23C2×C4×D412yes—
492Q8—44no—
493Q8—48no—
494Q8—44no—
495Q8—44no—
496D4—42no—
497D4—42no—
498D4—44no—
499D4—42no—
500D4—42no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4C4×C22⋊C412yes—
560C2×C4C4×C22⋊C412yes—
561C2×C4C4×C22⋊C412yes—
562C2×C4C2×#9911yescharacteristic
563C2×C4C2×#9911yescharacteristic
564C2×C4C2×#9911yescharacteristic
565C2×C4C2×#9911yescharacteristic
566C2×C4C2×#9911yescharacteristic
567C2×C4C2×#9911yescharacteristic
568C2×C4#41012yes—
569C2×C4C2×C4×D412yes—
570C8—48no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—21no—
576C22—22no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—41no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22C2×C4×C22⋊C411yescharacteristic
591C22C22×#9911yescharacteristic
592C22C2×#41011yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—44no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—44no—
607C4—82no—
608C4—88no—
609C4—82no—
610C4C4×#9912yes—
611C4C4×#9912yes—
612C4#212312yesdirect product factor
613C2—21no—
614C2—21no—
615C2—42no—
616C2—41no—
617C2—41no—
618C2—42no—
619C2C2×C4×#9911yescharacteristic
620C2C2×#212311yescharacteristic
621C2#3334711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
socleC22C22×#99
prime radicalC22×#41C23
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
5. #12621C2semidirect
6. C4×#352C2semidirect
10. C22×C4×Q8C4semidirect
18. #2322C4semidirect
20. #2317C4semidirect
22. #2310C4semidirect
27. C4×#123C4semidirect
30. C4×#117C4semidirect
31. #2123C4direct (N × H)
147. #352C2×C4semidirect
163. C22×Q8C42semidirect
188. #123C42semidirect
228. #117C42semidirect
612. C4#2123direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#5362
C2×#21232
C4×#4132
C4×#4102
#126212
C4×#3522
#125752