Finite Group Databasebeta

← C4×#1683 #211133

Subgroups — C4×#1683 #211133

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#487C211yesmaximal, characteristic, complemented
2C4×#476C211yesmaximal, characteristic
3C4×#474C211yesmaximal, characteristic
4C4×C8×D4C211yesmaximal, characteristic
5C2×#1683C211yesmaximal, characteristic
6#9737C211yesmaximal, characteristic, complemented
7#9616C211yesmaximal, characteristic, complemented
8#9568C211yesmaximal, characteristic, complemented
9#9562C211yesmaximal, characteristic, complemented
10#9219C211yesmaximal, characteristic
11#9215C211yesmaximal, characteristic
12#8648C211yesmaximal, characteristic
13C4×#335C211yesmaximal, characteristic, complemented
14C4×#332C211yesmaximal, characteristic, complemented
15C4×#333C211yesmaximal, characteristic, complemented
16C2×#487C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C22×C4×C8—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1988C2211yescharacteristic
27#1987C2211yescharacteristic
28#1956C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1933—21no—
32#1933—21no—
33#1920—21no—
34#1918—21no—
35#1864—21no—
36#1862—21no—
37#1834C2211yescharacteristic
38#1802C2211yescharacteristic
39#1802C2211yescharacteristic
40C4×#107—21no—
41C4×#106—21no—
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103—21no—
48C4×#103—21no—
49C4×#102—21no—
50C4×#102—21no—
51C8×C22⋊C4—21no—
52C8×C22⋊C4—21no—
53C4×C22⋊C8—21no—
54C42×C8C2211yescharacteristic
55#1683C418yesdirect product factor
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×C8—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C22×C42—21no—
78#487—22no—
79#487C2×C414yescomplemented
80#476—22no—
81#476C2×C414yes—
82#474—22no—
83#474C2×C414yes—
84C8×D4—22no—
85C8×D4—22no—
86C8×D4—22no—
87C8×D4C2×C414yes—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—44no—
114C2×C4×C8D411yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—21no—
121#391—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C43C2311yescharacteristic
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#335—22no—
157#335C2×C414yescomplemented
158#333—22no—
159#333C2×C414yescomplemented
160#332—22no—
161#332C2×C414yescomplemented
162C22×Q8—21no—
163C22×Q8—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C22×C8—24no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—24no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8C22×C412yes—
191C4⋊Q8C22×C412yes—
192C4×Q8—42no—
193C4×Q8—42no—
194C4×Q8—82no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4Q1614yes—
201C4×D4C22×C412yes—
202C4×D4#4112yes—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—24no—
236#107—42no—
237#106—24no—
238#106—42no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C412yes—
249C4⋊C8C22×C412yes—
250C4⋊C8#4112yes—
251#103—24no—
252#103—24no—
253#103—42no—
254#103—42no—
255#102—24no—
256#102—24no—
257#102—42no—
258#102—42no—
259C22⋊C8—24no—
260C22⋊C8—24no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—41no—
267C4×C8—42no—
268C4×C8—42no—
269C4×C8—42no—
270C4×C8—42no—
271C4×C8—42no—
272C4×C8C22×C412yes—
273C4×C8#4112yes—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281C24—21no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—81no—
291C2×Q8—81no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4C4×D412yes—
296C2×D4C2×Q1612yes—
297C2×D4#13512yes—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—48no—
338C2×C8—48no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8C4×D412yes—
354C2×C8C4×D412yes—
355C2×C8#13512yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4C4×D412yes—
380C4⋊C4C2×Q1612yes—
381C4⋊C4#13512yes—
382C22⋊C4—24no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—24no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—22no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—42no—
395C42—42no—
396C42—82no—
397C42—82no—
398C42C2×Q1612yes—
399C42#13512yes—
400C42#13512yes—
401C42AGL(1,ℤ/8)12yes—
402C42C23×C411yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C23—22no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23C2×#12111yescharacteristic
414Q8—42no—
415Q8—42no—
416Q8—82no—
417Q8—82no—
418Q8—81no—
419Q8—81no—
420D4—22no—
421D4—22no—
422D4—22no—
423D4C4×Q1614yes—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—48no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4#46312yes—
461C2×C4#46312yes—
462C2×C4#46712yes—
463C2×C4#46712yes—
464C2×C4#47112yes—
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C4×#4111yescharacteristic
468C2×C4#52511yescharacteristic
469C2×C4#52911yescharacteristic
470C2×C4#53211yescharacteristic
471C2×C4C22×Q1611yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4C2×#13511yescharacteristic
475C2×C4C2×#13511yescharacteristic
476C2×C4C2×#13511yescharacteristic
477C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C2×C4#57511yescharacteristic
479C8—22no—
480C8—22no—
481C8—22no—
482C8—41no—
483C8—42no—
484C8—42no—
485C8—42no—
486C8—42no—
487C8—81no—
488C8—81no—
489C22—24no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22C4×#12111yescharacteristic
497C22C2×#46311yescharacteristic
498C22#303511yescharacteristic
499C22C2×#46711yescharacteristic
500C22#305011yescharacteristic
501C22C2×#47111yescharacteristic
502C22#306711yescharacteristic
503C4—21no—
504C4—21no—
505C4—22no—
506C4—22no—
507C4—42no—
508C4—42no—
509C4—41no—
510C4—42no—
511C4—82no—
512C4—82no—
513C4—161no—
514C4—161no—
515C4#168314yesdirect product factor
516C4C2×C4×Q1611yescharacteristic
517C4C4×#13511yescharacteristic
518C4C4×#13511yescharacteristic
519C4C4×AGL(1,ℤ/8)11yescharacteristic
520C4#310211yescharacteristic
521C4#310511yescharacteristic
522C4#316811yescharacteristic
523C4#317011yescharacteristic
524C2—44no—
525C2C2×#168311yescharacteristic
526C2#1911011yescharacteristic
527C2#1912611yescharacteristic
528C2#1912711yescharacteristic
529C2C4×#46311yescharacteristic
530C2C4×#46711yescharacteristic
531C2C4×#47111yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×#121
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#487C2semidirect
6. #9737C2semidirect
7. #9616C2semidirect
8. #9568C2semidirect
9. #9562C2semidirect
13. C4×#335C2semidirect
14. C4×#332C2semidirect
15. C4×#333C2semidirect
55. #1683C4direct (N × H)
79. #487C2×C4semidirect
157. #335C2×C4semidirect
159. #333C2×C4semidirect
161. #332C2×C4semidirect
515. C4#1683direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#4872
C4×#4762
C4×#4742
C4×C8×D42
C2×#16832
#97372
#96162
#95682
#95622
#92192
#92152
#86482
C4×#3352
C4×#3322
C4×#3332