Finite Group Databasebeta

← C2×#24918696 #36276539

Subgroups — C2×#24918696 #36276539

Order 21248.

2018 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581791C211yesmaximal, characteristic, complemented
2C2×#19128237C211yesmaximal, characteristic
3#24918696C2116yesmaximal, direct product factor
4C2×#19128109C212yesmaximal, complemented
5C2×#19128062C212yesmaximal, complemented
6C2×#19128034C211yesmaximal, characteristic, complemented
7C2×#19128020C211yesmaximal, characteristic
8C2×#19128019C211yesmaximal, characteristic
9C2×#19127984C212yesmaximal, complemented
10C2×#19127980C212yesmaximal, complemented
11C2×#19127755C211yesmaximal, characteristic, complemented
12C2×#484×C83C211yesmaximal, characteristic, complemented
13C23×Dic166—21no—
14C2×#14581791—24no—
15C2×#14581791—21no—
16C2×#14581791—21no—
17C2×#14581791C2212yes—
18C2×#14581791C2212yes—
19C2×#14581791C2218yescomplemented
20#19128237—24no—
21#19128237C2218yes—
22C2×#14581787—21no—
23C2×#14581783C2211yescharacteristic
24C2×#14581781C2211yescharacteristic
25C2×#14581776—21no—
26C2×#14581766C2211yescharacteristic
27C2×Q8×Dic83C2211yescharacteristic
28C2×#14581754C2211yescharacteristic
29C2×(C22⋊D83)⋊C4—21no—
30C2×#14581750—21no—
31C22×C4⋊Dic83C2211yescharacteristic
32C22×Dic83⋊C4—21no—
33C22×C4×Dic83C2211yescharacteristic
34#19128109C22116yescomplemented
35#19128062C22116yescomplemented
36#19128034C2218yescomplemented
37#19128020C2218yes—
38#19128019C2218yes—
39#19127984C22116yescomplemented
40#19127980C22116yescomplemented
41#19127755C2218yescomplemented
42C2×#14581725C2212yes—
43C2×#14581724C2212yes—
44C2×#14581723C2212yes—
45C2×Dic83⋊C8C2212yes—
46C2×#14581716C2212yes—
47C2×#14581714C2212yes—
48C2×#14581712C2212yes—
49C2×C22⋊(C83⋊C8)C2211yescharacteristic
50C2×#14581702C2212yes—
51C2×C22⋊Q8×C83C2211yescharacteristic
52C22×C4⋊C4×C83C2211yescharacteristic
53#484×C83C2218yescomplemented
54C2×#107×C83C2212yes—
55C2×#103×C83C2212yes—
56C2×C22⋊C8×C83C2211yescharacteristic
57C22×Dic166—22no—
58C22×Dic166—22no—
59C22×Dic166—28no—
60C22×Dic166—48no—
61C22×D332C2311yescharacteristic
62C22×C22⋊D83—21no—
63C23×Dic83—21no—
64C23×Dic83D411yescharacteristic
65C23×Dic83D411yescharacteristic
66C22×C4×D83—21no—
67C22×C83⋊C8C2312yes—
68#14581791—24no—
69#14581791—24no—
70#14581791—28no—
71#14581791—28no—
72#14581791C2318yes—
73#14581791C2318yes—
74#14581787—28no—
75#14581783—22no—
76#14581783C2314yes—
77#14581781—22no—
78#14581781C2314yes—
79#14581776—28no—
80#14581766—22no—
81#14581766C2314yes—
82Q8×Dic83—22no—
83Q8×Dic83C2314yes—
84#14581754—22no—
85#14581754C2314yes—
86(C22⋊D83)⋊C4—28no—
87#14581750—28no—
88C2×C4⋊Dic83—21no—
89C2×C4⋊Dic83—21no—
90C2×C4⋊Dic83—22no—
91C2×C4⋊Dic83—21no—
92C2×C4⋊Dic83C2312yes—
93C2×C4⋊Dic83C2314yes—
94C2×C4⋊Dic83C2311yescharacteristic
95C2×Dic83⋊C4—22no—
96C2×Dic83⋊C4—22no—
97C2×Dic83⋊C4—28no—
98C2×Dic83⋊C4—21no—
99C2×(C2×D83)⋊C4—21no—
100C2×(C2×D83)⋊C4—21no—
101C2×(C2×D83)⋊C4—21no—
102C2×(C2×D83)⋊C4—21no—
103C2×C4×Dic83—21no—
104C2×C4×Dic83D411yescharacteristic
105C2×C4×Dic83D411yescharacteristic
106C2×C4×Dic83D414yes—
107C2×C4×Dic83C2311yescharacteristic
108C2×C4×Dic83C2311yescharacteristic
109C2×C4×Dic83C2314yes—
110#14581725C2318yes—
111#14581724C2318yes—
112#14581723C2318yes—
113Dic83⋊C8C2318yes—
114#14581716C2318yes—
115#14581714C2318yes—
116#14581712C2318yes—
117C22⋊(C83⋊C8)C2314yes—
118#14581702C2318yes—
119C22×Q8×C83C2311yescharacteristic
120C23×C4×C83—21no—
121C23×C4×C83C2311yescharacteristic
122C22×C8×C83C2312yes—
123C22⋊Q8×C83—22no—
124C22⋊Q8×C83C2314yes—
125C2×C4⋊C4×C83—22no—
126C2×C4⋊C4×C83—21no—
127C2×C4⋊C4×C83—21no—
128C2×C4⋊C4×C83C2314yes—
129C2×C4⋊C4×C83C2312yes—
130C2×C4⋊C4×C83C2311yescharacteristic
131C2×C22⋊C4×C83—21no—
132#107×C83C2318yes—
133#103×C83C2318yes—
134C22⋊C8×C83C2314yes—
135C23×D83—21no—
136C2×Dic166—28no—
137C2×Dic166—28no—
138C2×Dic166—48no—
139C2×Dic166—432no—
140C2×Dic166—48no—
141C2×D332—24no—
142C2×D332—21no—
143C2×D332C2412yes—
144C2×C22⋊D83—24no—
145C2×C22⋊D83—44no—
146C22×Dic83—24no—
147C22×Dic83—24no—
148C22×Dic83—24no—
149C22×Dic83—22no—
150C22×Dic83—21no—
151C22×Dic83—44no—
152C22×Dic83C2×D414yes—
153C22×Dic83C2×D411yescharacteristic
154C22×Dic83C2×D414yes—
155C22×Dic83C2×D411yescharacteristic
156C22×Dic83C2×D411yescharacteristic
157C22×Dic83C2×D411yescharacteristic
158C2×C4×D83—24no—
159C2×C4×D83—44no—
160C2×C83⋊C8—24no—
161C2×C83⋊C8C2414yes—
162C4⋊Dic83—21no—
163C4⋊Dic83—24no—
164C4⋊Dic83—22no—
165C4⋊Dic83—24no—
166C4⋊Dic83—24no—
167C4⋊Dic83C2412yes—
168C4⋊Dic83C2414yes—
169Dic83⋊C4—24no—
170Dic83⋊C4—28no—
171Dic83⋊C4—28no—
172(C2×D83)⋊C4—24no—
173(C2×D83)⋊C4—24no—
174(C2×D83)⋊C4—24no—
175(C2×D83)⋊C4—24no—
176C4×Dic83—24no—
177C4×Dic83C2×D412yes—
178C4×Dic83C2×D414yes—
179C4×Dic83C2×D414yes—
180C4×Dic83C2×D412yes—
181C4×Dic83C2412yes—
182C4×Dic83C2412yes—
183C24×C83C2×D411yescharacteristic
184C2×Q8×C83—21no—
185C2×Q8×C83—24no—
186C2×Q8×C83C2412yes—
187C22×C4×C83—24no—
188C22×C4×C83—21no—
189C22×C4×C83—24no—
190C22×C4×C83—22no—
191C22×C4×C83—21no—
192C22×C4×C83—44no—
193C22×C4×C83C2×D412yes—
194C22×C4×C83C2×D411yescharacteristic
195C22×C4×C83C2412yes—
196C22×C4×C83C2411yescharacteristic
197C2×C8×C83—24no—
198C2×C8×C83C2414yes—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—24no—
201C4⋊C4×C83—24no—
202C4⋊C4×C83—22no—
203C4⋊C4×C83C2412yes—
204C4⋊C4×C83C2414yes—
205C22⋊C4×C83—24no—
206C23×C83—22no—
207C23×C83—22no—
208C23×C83#13714yes—
209C23×C83C22×D412yes—
210C23×C83C22×D411yescharacteristic
211C22×D83—22no—
212C22×D83—42no—
213C22×D83—42no—
214C22×D83—42no—
215Q8×C83—24no—
216Q8×C83—42no—
217C2×C4×C83—22no—
218C2×C4×C83—24no—
219C2×C4×C83—24no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
222C2×C4×C83—24no—
223C2×C4×C83—48no—
224C2×C4×C83—42no—
225C2×C4×C83—44no—
226C2×C4×C83—42no—
227C2×C4×C83C22×D412yes—
228C2×C4×C83C22×D411yescharacteristic
229C2×C4×C832−1+414yes—
230C2×C4×C83C2511yescharacteristic
231C22⋊D83—48no—
232C2×Dic83—22no—
233C2×Dic83—28no—
234C2×Dic83—22no—
235C2×Dic83—22no—
236C2×Dic83—28no—
237C2×Dic83—22no—
238C2×Dic83—24no—
239C2×Dic83—48no—
240C2×Dic83—44no—
241C2×Dic83—42no—
242C2×Dic83AGL(1,ℤ/8)14yes—
243C2×Dic83C22×D412yes—
244C2×Dic83C22×D412yes—
245C2×Dic83C22×D412yes—
246C2×Dic83C22×D412yes—
247D332—24no—
248D332—42no—
249C4×D83—48no—
250Dic166—416no—
251Dic166—416no—
252C8×C83—24no—
253C83⋊C8—24no—
254C22×C83—22no—
255C22×C83—28no—
256C22×C83—22no—
257C22×C83#53612yes—
258C22×C83C2×AGL(1,ℤ/8)12yes—
259C22×C83C2×#13714yes—
260C22×C83C2×#13712yes—
261C22×C83C23×D411yescharacteristic
262C2×D83—42no—
263C2×D83—42no—
264C2×D83—42no—
265C2×D83—84no—
266C4×C83—22no—
267C4×C83—42no—
268C4×C83—44no—
269C4×C83—42no—
270C4×C83#53612yes—
271C4×C83C2×2−1+412yes—
272Dic83—22no—
273Dic83—22no—
274Dic83—22no—
275Dic83—44no—
276Dic83—42no—
277Dic83C2×AGL(1,ℤ/8)14yes—
278C2×#3239—831nomaximal, Sylow
279C2×C83—24no—
280C2×C83#323914yescomplemented
281C2×C83C2×#53611yescharacteristic
282C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
283C2×C83C22×#13711yescharacteristic
284D83—82no—
285C22×C4⋊Q8—831no—
286C2×#529—831no—
287#3239—8316no—
288C2×#501—832no—
289C2×#489—832no—
290C2×#484—831no—
291C2×#484D8311yescharacteristic, complemented
292C2×#481—831no—
293C2×#481—831no—
294C2×#475—832no—
295C2×#474—832no—
296C2×#432—831no—
297C83C2×#323911yescharacteristic, complemented, Sylow
298C23×Q8—1661no—
299C2×C4⋊Q8—832no—
300C2×C4⋊Q8—832no—
301C2×C4⋊Q8—838no—
302C2×C4⋊Q8—1661no—
303C2×C4⋊Q8—1661no—
304C2×C4⋊Q8—1664no—
305#529—838no—
306#529—1664no—
307C2×#126—1661no—
308C2×#125—831no—
309C2×#124—831no—
310C2×#123—1661no—
311C2×C22⋊Q8C2×D8311yescharacteristic
312C2×#121—831no—
313C2×C4×Q8—831no—
314C2×C4×D4—831no—
315C2×C4×D4—1661no—
316C2×#117—1661no—
317C22×C4⋊C4—831no—
318C22×C4⋊C4—1661no—
319C22×C4⋊C4C2×D8311yescharacteristic
320C22×C42—831no—
321#501—8316no—
322#489—8316no—
323#484—838no—
324#484C2×D8318yescomplemented
325#481—838no—
326#481—838no—
327#475—8316no—
328#474—8316no—
329#432—838no—
330C2×#107—832no—
331C2×#107C2×D8312yes—
332C2×#106—832no—
333C2×#106—832no—
334C2×C4⋊C8—832no—
335C2×#103—832no—
336C2×#103C2×D8312yes—
337C2×#102—832no—
338C2×#102—832no—
339C2×C22⋊C8—831no—
340C2×C22⋊C8C2×D8311yescharacteristic
341C2×#97—832no—
342C22×Q8—3328no—
343C22×Q8—1668no—
344C22×Q8—1662no—
345C22×Q8—1662no—
346C22×Q8C22×D8311yescharacteristic
347C22×D4—831no—
348C22×D4—1661no—
349C23×C4—21no—
350C23×C4—831no—
351C23×C4—831no—
352C23×C4—1661no—
353C23×C4—1661no—
354C23×C4C22×D8311yescharacteristic
355C22×C8—832no—
356C22×C8C22×D8312yes—
357C4⋊Q8—838no—
358C4⋊Q8—838no—
359C4⋊Q8—1668no—
360C4⋊Q8—1664no—
361C4⋊Q8—1664no—
362C4⋊Q8—1668no—
363#126—1668no—
364#125—834no—
365#125—1662no—
366#124—834no—
367#124—1662no—
368#123—1668no—
369C22⋊Q8—22no—
370C22⋊Q8C22×D8314yes—
371#121—834no—
372#121—1662no—
373C4×Q8—834no—
374C4×Q8—1662no—
375C4×D4—834no—
376C4×D4—1662no—
377C4×D4—1668no—
378#117—1668no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—832no—
383C2×C4⋊C4—834no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—1668no—
386C2×C4⋊C4—1662no—
387C2×C4⋊C4—1662no—
388C2×C4⋊C4—1661no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1662no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4C22×D8312yes—
394C2×C4⋊C4C22×D8314yes—
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1661no—
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C42—831no—
402C2×C42—831no—
403C2×C42—834no—
404C2×C42—831no—
405C2×C42—831no—
406C2×C42—834no—
407C2×C42—1661no—
408#107—838no—
409#107C22×D8318yes—
410#106—838no—
411#106—838no—
412C4⋊C8—838no—
413#103—838no—
414#103C22×D8318yes—
415#102—838no—
416#102—838no—
417C22⋊C8—834no—
418C22⋊C8C22×D8314yes—
419#97—838no—
420C24—1661no—
421C24D4×D8311yescharacteristic
422C2×Q8—21no—
423C2×Q8—24no—
424C2×Q8—3328no—
425C2×Q8—33232no—
426C2×Q8—3328no—
427C2×Q8—1668no—
428C2×Q8—1668no—
429C2×Q8C23×D8312yes—
430C2×D4—3324no—
431C2×D4—832no—
432C2×D4—1664no—
433C2×D4—1661no—
434C2×D4—1664no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—44no—
441C22×C4—3324no—
442C22×C4—3324no—
443C22×C4—831no—
444C22×C4—834no—
445C22×C4—834no—
446C22×C4—831no—
447C22×C4—831no—
448C22×C4—831no—
449C22×C4—1664no—
450C22×C4—1664no—
451C22×C4—1661no—
452C22×C4—1662no—
453C22×C4—1664no—
454C22×C4—1664no—
455C22×C4D4×D8312yes—
456C22×C4D4×D8311yescharacteristic
457C22×C4C23×D8312yes—
458C22×C4C23×D8311yescharacteristic
459C2×C8—24no—
460C2×C8—834no—
461C2×C8—1664no—
462C2×C8C23×D8314yes—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—834no—
468C4⋊C4—832no—
469C4⋊C4—1668no—
470C4⋊C4—1668no—
471C4⋊C4—1664no—
472C4⋊C4—1664no—
473C4⋊C4—1664no—
474C4⋊C4—1664no—
475C4⋊C4—1662no—
476C4⋊C4—1661no—
477C4⋊C4C23×D8314yes—
478C4⋊C4C23×D8312yes—
479C22⋊C4—24no—
480C22⋊C4—1664no—
481C22⋊C4—1664no—
482C22⋊C4—1664no—
483C22⋊C4—1664no—
484C42—834no—
485C42—832no—
486C42—832no—
487C42—834no—
488C42—832no—
489C42—832no—
490C42—1664no—
491C23—22no—
492C23—22no—
493C23—3322no—
494C23—3322no—
495C23—3322no—
496C23—1662no—
497C23#1458182414yes—
498C23C2×D4×D8312yes—
499C23C2×D4×D8311yescharacteristic
500Q8—24no—
501Q8—42no—
502Q8—33216no—
503Q8—33216no—
504D4—3328no—
505D4—3322no—
506D4—1664no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—3328no—
518C2×C4—3328no—
519C2×C4—3324no—
520C2×C4—3322no—
521C2×C4—834no—
522C2×C4—832no—
523C2×C4—832no—
524C2×C4—832no—
525C2×C4—832no—
526C2×C4—1662no—
527C2×C4—1664no—
528C2×C4—1668no—
529C2×C4—1662no—
530C2×C4—1668no—
531C2×C4—1662no—
532C2×C4—1662no—
533C2×C4C2×D4×D8312yes—
534C2×C4C2×D4×D8311yescharacteristic
535C2×C4#1458183914yes—
536C2×C4C24×D8311yescharacteristic
537C8—24no—
538C8—1664no—
539C22—22no—
540C22—28no—
541C22—22no—
542C22—3322no—
543C22—3322no—
544C22—3322no—
545C22—6644no—
546C22#1912827312yes—
547C22AGL(1,ℤ/8)×D8312yes—
548C22C2×#1458182414yes—
549C22C2×#1458182412yes—
550C22C22×D4×D8311yescharacteristic
551C4—22no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—3324no—
556C4—3322no—
557C4—834no—
558C4—1662no—
559C4—1662no—
560C4—1662no—
561C4#1912827312yes—
562C4C2×#1458183912yes—
563C2—24no—
564C2—6642no—
565C2#2491869614yesdirect product factor
566C2C2×#1912827311yescharacteristic
567C2C2×AGL(1,ℤ/8)×D8311yescharacteristic
568C2C22×#1458182411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#484×C83C2
socleC23×C83C22×D4
prime radicalC2×#14581766C22
hypercenterC2×#484D83
hypocenterC83C2×#3239
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
1. C22×#14581791C2semidirect
3. #24918696C2direct (N × H)
4. C2×#19128109C2semidirect
5. C2×#19128062C2semidirect
6. C2×#19128034C2semidirect
9. C2×#19127984C2semidirect
10. C2×#19127980C2semidirect
11. C2×#19127755C2semidirect
12. C2×#484×C83C2semidirect
19. C2×#14581791C22semidirect
34. #19128109C22semidirect
35. #19128062C22semidirect
36. #19128034C22semidirect
39. #19127984C22semidirect
40. #19127980C22semidirect
41. #19127755C22semidirect
53. #484×C83C22semidirect
280. C2×C83#3239semidirect
291. C2×#484D83semidirect
297. C83C2×#3239semidirect
324. #484C2×D83semidirect
565. C2#24918696direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817912
C2×#191282372
#249186962
C2×#191281092
C2×#191280622
C2×#191280342
C2×#191280202
C2×#191280192
C2×#191279842
C2×#191279802
C2×#191277552
C2×#484×C832
C2×#323983