Finite Group Databasebeta

← C4×#1691 #211405

Subgroups — C4×#1691 #211405

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#498C211yesmaximal, characteristic, complemented
2C4×#476C211yesmaximal, characteristic
3C4×#461C211yesmaximal, characteristic
4C4×#436C211yesmaximal, characteristic
5C2×#1691C211yesmaximal, characteristic
6#9741C211yesmaximal, characteristic, complemented
7#9618C211yesmaximal, characteristic, complemented
8#9575C211yesmaximal, characteristic, complemented
9#9573C211yesmaximal, characteristic, complemented
10#9225C211yesmaximal, characteristic
11#9223C211yesmaximal, characteristic
12#8650C211yesmaximal, characteristic
13C4×#335C211yesmaximal, characteristic, complemented
14C4×#332C211yesmaximal, characteristic, complemented
15C4×#333C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×C4×SD16—21no—
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C4×M16—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1987C2211yescharacteristic
29#1981C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1941—21no—
33#1941—21no—
34#1927—21no—
35#1920—21no—
36#1877—21no—
37#1864—21no—
38#1844C2211yescharacteristic
39#1802C2211yescharacteristic
40#1802C2211yescharacteristic
41C4×#106—21no—
42C4×#106—21no—
43C4×#106—21no—
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×#103—21no—
50C4×#102—21no—
51#1753—21no—
52#1753—21no—
53C4×C22⋊C8—21no—
54C42×C8C2211yescharacteristic
55#1691C418yesdirect 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
68C22×SD16—21no—
69C22×M16—21no—
70C22×M16—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C42—21no—
79#498—22no—
80#498C2×C414yescomplemented
81#476—22no—
82#476C2×C414yes—
83#461—22no—
84#461C2×C414yes—
85C4×SD16—44no—
86#436—22no—
87#436—22no—
88#436—22no—
89#436C2×C414yes—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C4×M16—44no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#391—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C43C2311yescharacteristic
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#340—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#335—22no—
158#335C2×C414yescomplemented
159#333—22no—
160#333C2×C414yescomplemented
161#332—22no—
162#332C2×C414yescomplemented
163C22×Q8—21no—
164C22×Q8—21no—
165C22×D4C2×D411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C2×SD16—24no—
170C2×SD16—42no—
171C2×SD16—42no—
172C2×M16—24no—
173C2×M16—24no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—42no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—82no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4C22×C412yes—
206C4×D4#4112yes—
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—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42C2411yescharacteristic
239#106—24no—
240#106—24no—
241#106—24no—
242#106—42no—
243#106—42no—
244#106—42no—
245#106—42no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8C22×C412yes—
255C4⋊C8C22×C412yes—
256C4⋊C8C22×C412yes—
257C4⋊C8#4112yes—
258#103—24no—
259#103—42no—
260#103—42no—
261#102—24no—
262#102—42no—
263#102—42no—
264C22⋊C8—24no—
265C22⋊C8—24no—
266#97—42no—
267#97—42no—
268C4×C8—22no—
269C4×C8—22no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—41no—
273C4×C8—42no—
274C4×C8—42no—
275C4×C8—42no—
276C4×C8SD1614yescomplemented
277C4×C8C22×C412yes—
278C4×C8#4112yes—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—41no—
283#95—41no—
284#95—41no—
285#95—41no—
286C24—21no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—81no—
296C2×Q8—81no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4C4×D412yes—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329SD16—48no—
330M16—48no—
331M16—48no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
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×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8C4×D412yes—
358C2×C8C4×D412yes—
359C2×C8C2×SD1612yes—
360C2×C8C2×SD1612yes—
361C2×C8#13512yes—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—81no—
384C4⋊C4—81no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4C4×D412yes—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—42no—
402C42—82no—
403C42—82no—
404C42C2×SD1612yes—
405C42AGL(1,ℤ/8)12yes—
406C42#13712yes—
407C42#13712yes—
408C42C23×C411yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C23—22no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23C2×#12111yescharacteristic
420Q8—42no—
421Q8—42no—
422Q8—82no—
423Q8—82no—
424Q8—81no—
425Q8—81no—
426D4—22no—
427D4—22no—
428D4—41no—
429D4—41no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—84no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—84no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4#46112yes—
467C2×C4#46112yes—
468C2×C4#46812yes—
469C2×C4#46812yes—
470C2×C4#47112yes—
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C4×#4111yescharacteristic
474C2×C4#52511yescharacteristic
475C2×C4#52911yescharacteristic
476C2×C4#53211yescharacteristic
477C2×C4C22×SD1611yescharacteristic
478C2×C4C2×#13511yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×#13711yescharacteristic
481C2×C4C2×#13711yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57511yescharacteristic
485C8—22no—
486C8—22no—
487C8—42no—
488C8—41no—
489C8—42no—
490C8—42no—
491C8—42no—
492C8—81no—
493C8—81no—
494C8C4×SD1614yescomplemented
495C22—42no—
496C22—42no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22C4×#12111yescharacteristic
503C22C2×#46111yescharacteristic
504C22#303511yescharacteristic
505C22C2×#46811yescharacteristic
506C22#305411yescharacteristic
507C22C2×#47111yescharacteristic
508C22#306711yescharacteristic
509C4—21no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—42no—
515C4—82no—
516C4—82no—
517C4—81no—
518C4—81no—
519C4—161no—
520C4—161no—
521C4#169114yesdirect product factor
522C4C2×C4×SD1611yescharacteristic
523C4C4×AGL(1,ℤ/8)11yescharacteristic
524C4C4×#13711yescharacteristic
525C4C4×#13711yescharacteristic
526C4#310111yescharacteristic
527C4#310711yescharacteristic
528C4#317111yescharacteristic
529C4#317211yescharacteristic
530C2—82no—
531C2C2×#169111yescharacteristic
532C2#1913711yescharacteristic
533C2#1915611yescharacteristic
534C2#1915911yescharacteristic
535C2C4×#46111yescharacteristic
536C2C4×#46811yescharacteristic
537C2C4×#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×#498C2semidirect
6. #9741C2semidirect
7. #9618C2semidirect
8. #9575C2semidirect
9. #9573C2semidirect
13. C4×#335C2semidirect
14. C4×#332C2semidirect
15. C4×#333C2semidirect
55. #1691C4direct (N × H)
80. #498C2×C4semidirect
158. #335C2×C4semidirect
160. #333C2×C4semidirect
162. #332C2×C4semidirect
276. C4×C8SD16semidirect
494. C8C4×SD16semidirect
521. C4#1691direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#4982
C4×#4762
C4×#4612
C4×#4362
C2×#16912
#97412
#96182
#95752
#95732
#92252
#92232
#86502
C4×#3352
C4×#3322
C4×#3332