Finite Group Databasebeta

← C2×#24907654 #35405113

Subgroups — C2×#24907654 #35405113

Order 21248.

1966 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D83C211yesmaximal, characteristic
2C22×#14581723C211yesmaximal, characteristic, complemented
3C22×(C2×D83)⋊C8C211yesmaximal, characteristic
4C2×#19127558C211yesmaximal, characteristic
5#24907654C218yesmaximal, direct product factor
6C2×#19127464C211yesmaximal, characteristic, complemented
7C2×#19127463C211yesmaximal, characteristic, complemented
8C22×#106×C83C211yesmaximal, characteristic, complemented
9C23×C4×D83—21no—
10C23×C4×D83C2211yescharacteristic
11C23×C83⋊C8C2211yescharacteristic
12C22×C4⋊Dic83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×Dic83⋊C4—21no—
15C2×C4⋊C4×D83—24no—
16C2×C4⋊C4×D83—24no—
17C2×C4⋊C4×D83—22no—
18C2×C4⋊C4×D83C2214yes—
19C22×(C2×D83)⋊C4—21no—
20C2×#14581723—24no—
21C2×#14581723C2214yescomplemented
22C2×(C2×D83)⋊C8C418yescomplemented
23C2×(C2×D83)⋊C8C2214yes—
24#19127558—22no—
25#19127558C2214yes—
26C2×C83⋊#95—21no—
27#19127464C2214yescomplemented
28#19127463C2214yescomplemented
29C23×C8×C83C2211yescharacteristic
30C22×C4⋊C4×C83C2211yescharacteristic
31C22×C4⋊C4×C83C2211yescharacteristic
32C2×#106×C83—24no—
33C2×#106×C83C2214yescomplemented
34C24×D83D411yescharacteristic
35C23×Dic83—21no—
36C23×Dic83—21no—
37C23×Dic83D411yescharacteristic
38C22×C4×D83—28no—
39C22×C4×D83—24no—
40C22×C4×D83—22no—
41C22×C4×D83—22no—
42C22×C4×D83—22no—
43C22×C4×D83—22no—
44C22×C4×D83—44no—
45C22×C4×D83—44no—
46C22×C4×D83C2×C414yes—
47C22×C4×D83D412yes—
48C22×C4×D83Q814yes—
49C22×C4×D83C2312yes—
50C22×C83⋊C8—22no—
51C22×C83⋊C8—22no—
52C22×C83⋊C8C2×C414yescomplemented
53C22×C83⋊C8C2312yes—
54C2×C4⋊Dic83—21no—
55C2×C4⋊Dic83—22no—
56C2×C4⋊Dic83—21no—
57C2×C4⋊Dic83—22no—
58C2×C4⋊Dic83—22no—
59C2×C4⋊Dic83D414yes—
60C2×C4⋊Dic83C2312yes—
61C2×C4⋊Dic83C2312yes—
62C2×Dic83⋊C4—28no—
63C2×Dic83⋊C4—24no—
64C4⋊C4×D83—216no—
65C4⋊C4×D83—28no—
66C4⋊C4×D83—28no—
67C2×(C2×D83)⋊C4—24no—
68C2×(C2×D83)⋊C4—44no—
69#14581723—28no—
70(C2×D83)⋊C8C2×C4116yescomplemented
71C83⋊#95—24no—
72C23×C4×C83—21no—
73C23×C4×C83—21no—
74C23×C4×C83C2311yescharacteristic
75C22×C8×C83—22no—
76C22×C8×C83—22no—
77C22×C8×C83C2×C414yescomplemented
78C22×C8×C83C2312yes—
79C2×C4⋊C4×C83—21no—
80C2×C4⋊C4×C83—22no—
81C2×C4⋊C4×C83—22no—
82C2×C4⋊C4×C83—22no—
83C2×C4⋊C4×C83—21no—
84C2×C4⋊C4×C83D414yes—
85C2×C4⋊C4×C83C2312yes—
86C2×C4⋊C4×C83C2312yes—
87#106×C83—28no—
88C23×D83—22no—
89C23×D83—22no—
90C23×D83—22no—
91C23×D83—22no—
92C23×D83C4⋊C414yes—
93C23×D83SD1614yes—
94C23×D83SD1614yes—
95C23×D83C2×D412yes—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—24no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—42no—
102C22×Dic83—42no—
103C22×Dic83—42no—
104C22×Dic83—42no—
105C22×Dic83—42no—
106C22×Dic83C4⋊C414yes—
107C22×Dic83C2×D412yes—
108C2×C4×D83—28no—
109C2×C4×D83—28no—
110C2×C4×D83—24no—
111C2×C4×D83—28no—
112C2×C4×D83—28no—
113C2×C4×D83—24no—
114C2×C4×D83—22no—
115C2×C4×D83—22no—
116C2×C4×D83—22no—
117C2×C4×D83—22no—
118C2×C4×D83—216no—
119C2×C4×D83—48no—
120C2×C4×D83—432no—
121C2×C4×D83—48no—
122C2×C4×D83C4⋊C418yes—
123C2×C4×D83C22×C414yes—
124C2×C4×D83C2×Q814yes—
125C2×C83⋊C8—28no—
126C2×C83⋊C8—22no—
127C2×C83⋊C8—22no—
128C2×C83⋊C8C22×C414yescomplemented
129C4⋊Dic83—24no—
130C4⋊Dic83—22no—
131C4⋊Dic83—22no—
132C4⋊Dic83—22no—
133C4⋊Dic83—42no—
134C4⋊Dic83C2×D414yes—
135Dic83⋊C4—216no—
136(C2×D83)⋊C4—48no—
137C24×C83C2×D411yescharacteristic
138C22×C4×C83—24no—
139C22×C4×C83—22no—
140C22×C4×C83—22no—
141C22×C4×C83—42no—
142C22×C4×C83—42no—
143C22×C4×C83—42no—
144C22×C4×C83—42no—
145C22×C4×C83—42no—
146C22×C4×C83C22×C412yes—
147C22×C4×C83C2×D411yescharacteristic
148C22×C4×C83C2×D412yes—
149C22×C4×C83C2×D412yes—
150C22×C4×C83C2×Q812yes—
151C22×C4×C83#4112yes—
152C22×C4×C83#4112yes—
153C22×C4×C83C2411yescharacteristic
154C2×C8×C83—28no—
155C2×C8×C83—22no—
156C2×C8×C83—22no—
157C2×C8×C83C22×C414yescomplemented
158C4⋊C4×C83—24no—
159C4⋊C4×C83—22no—
160C4⋊C4×C83—22no—
161C4⋊C4×C83—22no—
162C4⋊C4×C83—42no—
163C4⋊C4×C83C2×D414yes—
164C23×C83C2×C4⋊C412yes—
165C23×C83#12012yes—
166C23×C83#12312yes—
167C23×C83C2×SD1612yes—
168C23×C83C2×SD1612yes—
169C23×C83AGL(1,ℤ/8)12yes—
170C23×C83#13712yes—
171C23×C83C22×D411yescharacteristic
172C22×D83—22no—
173C22×D83—22no—
174C22×D83—22no—
175C22×D83—22no—
176C22×D83—28no—
177C22×D83—28no—
178C22×D83—28no—
179C22×D83—28no—
180C22×D83—28no—
181C22×D83—28no—
182C22×D83#106116yescomplemented
183C22×D83C2×C4⋊C414yes—
184C22×D83C2×SD1614yes—
185C22×D83C2×SD1614yes—
186C2×C4×C83—24no—
187C2×C4×C83—42no—
188C2×C4×C83—42no—
189C2×C4×C83—48no—
190C2×C4×C83—42no—
191C2×C4×C83—42no—
192C2×C4×C83—42no—
193C2×C4×C83—84no—
194C2×C4×C83C2×C4⋊C412yes—
195C2×C4×C83C4×D414yes—
196C2×C4×C83C4×D414yes—
197C2×C4×C83#12012yes—
198C2×C4×C83C22⋊Q814yes—
199C2×C4×C83C22⋊Q814yes—
200C2×C4×C83#12312yes—
201C2×C4×C83C23×C411yescharacteristic
202C2×C4×C83C22×D411yescharacteristic
203C2×C4×C83C22×D411yescharacteristic
204C2×C4×C83C22×Q811yescharacteristic
205C2×C4×C83C2×#4111yescharacteristic
206C2×C4×C83C2×#4111yescharacteristic
207C2×Dic83—22no—
208C2×Dic83—22no—
209C2×Dic83—28no—
210C2×Dic83—24no—
211C2×Dic83—42no—
212C2×Dic83—42no—
213C2×Dic83—48no—
214C2×Dic83—42no—
215C2×Dic83—42no—
216C2×Dic83—42no—
217C2×Dic83—84no—
218C2×Dic83C2×C4⋊C414yes—
219C4×D83—24no—
220C4×D83—24no—
221C4×D83—24no—
222C4×D83—24no—
223C4×D83—216no—
224C4×D83—432no—
225C8×C83—24no—
226C83⋊C8—24no—
227C22×C83#38514yes—
228C22×C83C2×#10614yes—
229C22×C83#42814yes—
230C22×C83#44814yes—
231C22×C83#45014yes—
232C22×C83#48114yes—
233C22×C83#48314yes—
234C22×C83C22×C4⋊C411yescharacteristic
235C22×C83C2×#12011yescharacteristic
236C22×C83C2×#12311yescharacteristic
237C22×C83C22×SD1611yescharacteristic
238C22×C83C22×SD1611yescharacteristic
239C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
240C22×C83C2×#13711yescharacteristic
241C2×D83—24no—
242C2×D83—24no—
243C2×D83—232no—
244C2×D83—24no—
245C2×D83—24no—
246C2×D83—24no—
247C2×D83—24no—
248C2×D83C2×#10618yescomplemented
249C4×C83—44no—
250C4×C83—82no—
251C4×C83#38514yes—
252C4×C83C2×C4×D411yescharacteristic
253C4×C83C2×C4×D411yescharacteristic
254C4×C83C2×C22⋊Q811yescharacteristic
255C4×C83C2×C22⋊Q811yescharacteristic
256Dic83—24no—
257Dic83—44no—
258Dic83—82no—
259C2×#1853—831nomaximal, Sylow
260C2×C83#185318yescomplemented
261C2×C83C2×#38511yescharacteristic
262C2×C83C22×#10611yescharacteristic
263C2×C83C2×#42811yescharacteristic
264C2×C83C2×#45011yescharacteristic
265C2×C83C2×#44811yescharacteristic
266C2×C83C2×#48311yescharacteristic
267C2×C83C2×#48111yescharacteristic
268D83—28no—
269C23×C4⋊C4—831no—
270C22×#106—831no—
271C22×#106D8311yescharacteristic, complemented
272C22×C22⋊C8—831no—
273C2×#380—831no—
274#1853—838no—
275C2×#340—831no—
276C2×#340—831no—
277C83C2×#185311yescharacteristic, complemented, Sylow
278C24×C4—831no—
279C24×C4—1661no—
280C23×C8—831no—
281C23×C8C2×D8311yescharacteristic
282C22×C4⋊C4—831no—
283C22×C4⋊C4—831no—
284C22×C4⋊C4—834no—
285C22×C4⋊C4—1664no—
286C22×C4⋊C4—1662no—
287C22×C4⋊C4—1661no—
288C22×C4⋊C4—1664no—
289C22×C4⋊C4C2×D8311yescharacteristic
290C22×C4⋊C4C2×D8311yescharacteristic
291C22×C22⋊C4—1661no—
292C2×#106—24no—
293C2×#106—834no—
294C2×#106—1664no—
295C2×#106C2×D8314yescomplemented
296C2×C22⋊C8—838no—
297C2×C22⋊C8—834no—
298#380—834no—
299#380—1662no—
300C2×#95—1661no—
301#340—834no—
302#340—834no—
303C25—831no—
304C23×C4—21no—
305C23×C4—21no—
306C23×C4—3324no—
307C23×C4—3324no—
308C23×C4—831no—
309C23×C4—834no—
310C23×C4—832no—
311C23×C4—834no—
312C23×C4—832no—
313C23×C4—1664no—
314C23×C4—1668no—
315C23×C4—1661no—
316C23×C4—1661no—
317C23×C4—1662no—
318C23×C4—1662no—
319C23×C4—1662no—
320C23×C4—1662no—
321C23×C4C22×D8311yescharacteristic
322C22×C8—22no—
323C22×C8—22no—
324C22×C8—834no—
325C22×C8—832no—
326C22×C8—1662no—
327C22×C8—1662no—
328C22×C8C4×D8314yescomplemented
329C22×C8C22×D8312yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—834no—
336C2×C4⋊C4—832no—
337C2×C4⋊C4—832no—
338C2×C4⋊C4—1668no—
339C2×C4⋊C4—16616no—
340C2×C4⋊C4—1661no—
341C2×C4⋊C4—1662no—
342C2×C4⋊C4—1662no—
343C2×C4⋊C4—1664no—
344C2×C4⋊C4—1668no—
345C2×C4⋊C4—1661no—
346C2×C4⋊C4—1662no—
347C2×C4⋊C4—1668no—
348C2×C4⋊C4C22⋊D8314yes—
349C2×C4⋊C4C22×D8312yes—
350C2×C4⋊C4C22×D8312yes—
351C2×C22⋊C4—3324no—
352C2×C22⋊C4—1664no—
353#106—28no—
354#106—1668no—
355C22⋊C8—8316no—
356#95—1664no—
357C24—834no—
358C24—834no—
359C24—832no—
360C24—834no—
361C24—1662no—
362C24—1662no—
363C24—1662no—
364C24—1662no—
365C24D4×D8311yescharacteristic
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—3328no—
375C22×C4—33232no—
376C22×C4—3328no—
377C22×C4—3322no—
378C22×C4—3322no—
379C22×C4—3322no—
380C22×C4—3322no—
381C22×C4—3322no—
382C22×C4—832no—
383C22×C4—834no—
384C22×C4—834no—
385C22×C4—834no—
386C22×C4—838no—
387C22×C4—16616no—
388C22×C4—1662no—
389C22×C4—1664no—
390C22×C4—1662no—
391C22×C4—1662no—
392C22×C4—1662no—
393C22×C4—1664no—
394C22×C4—1668no—
395C22×C4—1668no—
396C22×C4—1664no—
397C22×C4—1662no—
398C22×C4—1662no—
399C22×C4—1662no—
400C22×C4—1662no—
401C22×C4—1668no—
402C22×C4—1668no—
403C22×C4C2×C4×D8312yes—
404C22×C4D4×D8311yescharacteristic
405C22×C4D4×D8312yes—
406C22×C4C2×C22⋊D8312yes—
407C22×C4Q8×D8312yes—
408C22×C4#1305059012yes—
409C22×C4#1305059112yes—
410C22×C4C23×D8311yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—28no—
414C2×C8—834no—
415C2×C8—1662no—
416C2×C8—1662no—
417C2×C8—1668no—
418C2×C8C2×C4×D8314yescomplemented
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—3322no—
425C4⋊C4—834no—
426C4⋊C4—16616no—
427C4⋊C4—1662no—
428C4⋊C4—1662no—
429C4⋊C4—1664no—
430C4⋊C4—1662no—
431C4⋊C4C2×C22⋊D8314yes—
432C22⋊C4—3328no—
433C23—834no—
434C23—834no—
435C23—834no—
436C23—8316no—
437C23—1668no—
438C23—1668no—
439C23—1668no—
440C23—1662no—
441C23—1662no—
442C23—1662no—
443C23—1662no—
444C23—1668no—
445C23—1668no—
446C23—1668no—
447C23C4⋊C4×D8312yes—
448C23#1458176012yes—
449C23#1458177712yes—
450C23SD16×D8312yes—
451C23SD16×D8312yes—
452C23#1458181912yes—
453C23#1458182112yes—
454C23C2×D4×D8311yescharacteristic
455C2×C4—24no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—33232no—
464C2×C4—3328no—
465C2×C4—3322no—
466C2×C4—3322no—
467C2×C4—3322no—
468C2×C4—3322no—
469C2×C4—3322no—
470C2×C4—834no—
471C2×C4—6644no—
472C2×C4—1664no—
473C2×C4—1668no—
474C2×C4—1662no—
475C2×C4—1662no—
476C2×C4—16616no—
477C2×C4—1664no—
478C2×C4—1664no—
479C2×C4—1664no—
480C2×C4—1664no—
481C2×C4C4⋊C4×D8312yes—
482C2×C4C4×C22⋊D8314yes—
483C2×C4#1458175414yes—
484C2×C4#1458176012yes—
485C2×C4#1458176814yes—
486C2×C4#1458177014yes—
487C2×C4#1458177712yes—
488C2×C4C22×C4×D8311yescharacteristic
489C2×C4C2×D4×D8311yescharacteristic
490C2×C4C22×C22⋊D8311yescharacteristic
491C2×C4C2×Q8×D8311yescharacteristic
492C2×C4C2×#1305059011yescharacteristic
493C2×C4C2×#1305059111yescharacteristic
494C8—24no—
495C8—1664no—
496C22—838no—
497C22—1664no—
498C22—1664no—
499C22—1664no—
500C22—1664no—
501C22—16632no—
502C22—1664no—
503C22—1664no—
504C22#1912757214yes—
505C22#106×D8314yes—
506C22#1912774014yes—
507C22#1912783814yes—
508C22#1912785014yes—
509C22#1912802114yes—
510C22#1912803314yes—
511C22C2×C4⋊C4×D8311yescharacteristic
512C22C2×#1458176011yescharacteristic
513C22C2×#1458177711yescharacteristic
514C22C2×SD16×D8311yescharacteristic
515C22C2×SD16×D8311yescharacteristic
516C22C2×#1458181911yescharacteristic
517C22C2×#1458182111yescharacteristic
518C4—44no—
519C4—82no—
520C4—3324no—
521C4—6642no—
522C4—1664no—
523C4#1912757214yes—
524C4C2×C4×C22⋊D8311yescharacteristic
525C4C2×#1458175411yescharacteristic
526C4C2×#1458176811yescharacteristic
527C4C2×#1458177011yescharacteristic
528C2—1668no—
529C2#2490765418yesdirect product factor
530C2C2×#1912757211yescharacteristic
531C2C2×#106×D8311yescharacteristic
532C2C2×#1912774011yescharacteristic
533C2C2×#1912785011yescharacteristic
534C2C2×#1912783811yescharacteristic
535C2C2×#1912803311yescharacteristic
536C2C2×#1912802111yescharacteristic
subgroupG/H
centerC24D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC22×#106×C83C2
socleC24×C83C2×D4
prime radicalC24×D83D4
cosocleC22×C4×C83C24
hypercenterC22×#106D83
hypocenterC83C2×#1853
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
2. C22×#14581723C2semidirect
5. #24907654C2direct (N × H)
6. C2×#19127464C2semidirect
7. C2×#19127463C2semidirect
8. C22×#106×C83C2semidirect
21. C2×#14581723C22semidirect
22. C2×(C2×D83)⋊C8C4semidirect
27. #19127464C22semidirect
28. #19127463C22semidirect
33. C2×#106×C83C22semidirect
52. C22×C83⋊C8C2×C4semidirect
70. (C2×D83)⋊C8C2×C4semidirect
77. C22×C8×C83C2×C4semidirect
128. C2×C83⋊C8C22×C4semidirect
157. C2×C8×C83C22×C4semidirect
182. C22×D83#106semidirect
248. C2×D83C2×#106semidirect
260. C2×C83#1853semidirect
271. C22×#106D83semidirect
277. C83C2×#1853semidirect
295. C2×#106C2×D83semidirect
328. C22×C8C4×D83semidirect
418. C2×C8C2×C4×D83semidirect
529. C2#24907654direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D832
C22×#145817232
C22×(C2×D83)⋊C82
C2×#191275582
#249076542
C2×#191274642
C2×#191274632
C22×#106×C832
C2×#185383