Finite Group Databasebeta

← C2×#24918664 #36276513

Subgroups — C2×#24918664 #36276513

Order 21248.

1770 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128256C211yesmaximal, characteristic
2C2×#125×D83C211yesmaximal, characteristic
3#24918664C2116yesmaximal, direct product factor
4C2×#19128111C211yesmaximal, characteristic, complemented
5C2×#19128110C211yesmaximal, characteristic, complemented
6C2×#19128069C211yesmaximal, characteristic, complemented
7C2×#19128035C212yesmaximal
8C2×#19128030C212yesmaximal
9C2×#19128010C212yesmaximal, complemented
10C2×#19128006C212yesmaximal, complemented
11C2×#19127756C211yesmaximal, characteristic
12C2×#491×C83C211yesmaximal, characteristic, complemented
13#19128256—24no—
14#19128256C2218yes—
15C2×#14581792C2211yescharacteristic
16C2×#14581784C2211yescharacteristic
17C2×#14581783—21no—
18C2×#14581783C2211yescharacteristic
19C2×#14581783C2211yescharacteristic
20C2×#14581782—21no—
21C2×#14581782—21no—
22#125×D83—24no—
23#125×D83C2218yes—
24C2×#14581772—21no—
25C2×#14581768C2212yes—
26C2×Q8×Dic83C2212yes—
27C2×#14581757—21no—
28C2×#14581750—21no—
29C2×C4⋊C4×D83—21no—
30C2×C4⋊C4×D83—21no—
31C2×C4⋊C4×D83C2211yescharacteristic
32C2×C4⋊C4×D83C2211yescharacteristic
33C2×C42×D83C2211yescharacteristic
34#19128111C2218yescomplemented
35#19128110C2218yescomplemented
36#19128069C2218yescomplemented
37#19128035C22116yes—
38#19128030C22116yes—
39#19128010C22116yescomplemented
40#19128006C22116yescomplemented
41#19127756C2218yes—
42C2×#14581726C2212yes—
43C2×#14581725C2212yes—
44C2×#14581724C2212yes—
45C2×#14581723C2212yes—
46C2×Dic83⋊C8C2212yes—
47C2×#14581716C2212yes—
48C2×#14581716C2212yes—
49C2×(C2×D83)⋊C8C2212yes—
50C2×#14581703C2211yescharacteristic
51C2×C4⋊Q8×C83C2211yescharacteristic
52C2×#125×C83C2211yescharacteristic
53#491×C83C2218yescomplemented
54C2×#103×C83C2212yes—
55C2×#103×C83C2212yes—
56C2×#97×C83C2211yescharacteristic
57C22×Dic166—21no—
58C22×C4×D83—21no—
59C22×C4×D83—21no—
60C22×C4×D83D411yescharacteristic
61C22×C4×D83D411yescharacteristic
62C22×C4×D83C2311yescharacteristic
63C22×C83⋊C8C2312yes—
64#14581792—22no—
65#14581792C2314yes—
66#14581784—22no—
67#14581784C2314yes—
68#14581783—22no—
69#14581783—22no—
70#14581783—28no—
71#14581783C2314yes—
72#14581783C2314yes—
73#14581782—28no—
74#14581782—28no—
75#14581772—28no—
76#14581768—24no—
77#14581768C2318yes—
78Q8×Dic83—24no—
79Q8×Dic83C2318yes—
80#14581757—28no—
81#14581750—28no—
82C2×C4⋊Dic83—21no—
83C2×C4⋊Dic83—21no—
84C2×C4⋊Dic83—22no—
85C2×C4⋊Dic83—21no—
86C2×C4⋊Dic83C2311yescharacteristic
87C2×C4⋊Dic83C2311yescharacteristic
88C2×C4⋊Dic83C2312yes—
89C2×Dic83⋊C4—21no—
90C2×Dic83⋊C4—21no—
91C2×Dic83⋊C4—21no—
92C2×Dic83⋊C4—21no—
93C2×Dic83⋊C4—22no—
94C2×Dic83⋊C4—21no—
95C2×Dic83⋊C4—21no—
96C4⋊C4×D83—28no—
97C4⋊C4×D83—28no—
98C4⋊C4×D83—22no—
99C4⋊C4×D83—22no—
100C4⋊C4×D83C2314yes—
101C4⋊C4×D83C2314yes—
102C2×(C2×D83)⋊C4—22no—
103C2×C4×Dic83—22no—
104C2×C4×Dic83D411yescharacteristic
105C2×C4×Dic83D411yescharacteristic
106C2×C4×Dic83C2311yescharacteristic
107C42×D83D414yes—
108C42×D83C2314yes—
109#14581726C2318yes—
110#14581725C2318yes—
111#14581724C2318yes—
112#14581723C2318yes—
113Dic83⋊C8C2318yes—
114#14581716C2318yes—
115#14581716C2318yes—
116(C2×D83)⋊C8C2318yes—
117#14581703C2314yes—
118C22×Q8×C83C2312yes—
119C22×C8×C83C2312yes—
120C4⋊Q8×C83—22no—
121C4⋊Q8×C83C2314yes—
122#125×C83—22no—
123#125×C83C2314yes—
124C2×C4⋊C4×C83—21no—
125C2×C4⋊C4×C83—21no—
126C2×C4⋊C4×C83—21no—
127C2×C4⋊C4×C83—21no—
128C2×C4⋊C4×C83C2311yescharacteristic
129C2×C4⋊C4×C83C2311yescharacteristic
130C2×C42×C83C2311yescharacteristic
131#103×C83C2318yes—
132#103×C83C2318yes—
133#97×C83C2314yes—
134C23×D83C2×D411yescharacteristic
135C2×Dic166—24no—
136C2×Dic166—44no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—22no—
140C22×Dic83C2×D411yescharacteristic
141C22×Dic83C2×D411yescharacteristic
142C22×Dic83C2×D411yescharacteristic
143C2×C4×D83—24no—
144C2×C4×D83—24no—
145C2×C4×D83—24no—
146C2×C4×D83—24no—
147C2×C4×D83—42no—
148C2×C4×D83—44no—
149C2×C4×D83—44no—
150C2×C4×D83C2×D412yes—
151C2×C4×D83C2×D414yes—
152C2×C4×D83C2×D414yes—
153C2×C4×D83C2412yes—
154C2×C83⋊C8—24no—
155C2×C83⋊C8C2414yes—
156C4⋊Dic83—22no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—24no—
160C4⋊Dic83—28no—
161C4⋊Dic83—24no—
162C4⋊Dic83—24no—
163C4⋊Dic83C2414yes—
164C4⋊Dic83C2412yes—
165C4⋊Dic83C2412yes—
166Dic83⋊C4—24no—
167Dic83⋊C4—24no—
168Dic83⋊C4—28no—
169Dic83⋊C4—24no—
170Dic83⋊C4—24no—
171Dic83⋊C4—24no—
172Dic83⋊C4—24no—
173(C2×D83)⋊C4—28no—
174C4×Dic83—28no—
175C4×Dic83C2×D414yes—
176C4×Dic83C2×D414yes—
177C4×Dic83C2×D412yes—
178C4×Dic83C2412yes—
179C2×Q8×C83—22no—
180C2×Q8×C83—28no—
181C2×Q8×C83C2414yes—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—22no—
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2411yescharacteristic
188C2×C8×C83—24no—
189C2×C8×C83C2414yes—
190C4⋊C4×C83—24no—
191C4⋊C4×C83—24no—
192C4⋊C4×C83—24no—
193C4⋊C4×C83—24no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83C2412yes—
197C4⋊C4×C83C2412yes—
198C42×C83C2×D412yes—
199C42×C83C2412yes—
200C23×C83C22×D411yescharacteristic
201C22×D83—22no—
202C22×D83—22no—
203C22×D83#13714yes—
204C22×D83C22×D412yes—
205Q8×C83—28no—
206Q8×C83—44no—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83—24no—
212C2×C4×C83—44no—
213C2×C4×C83—42no—
214C2×C4×C83—42no—
215C2×C4×C83C22×D412yes—
216C2×C4×C83C22×D412yes—
217C2×C4×C83C22×D411yescharacteristic
218C2×C4×C832−1+414yes—
219C2×C4×C83C2511yescharacteristic
220C2×Dic83—22no—
221C2×Dic83—22no—
222C2×Dic83—22no—
223C2×Dic83—24no—
224C2×Dic83—22no—
225C2×Dic83—22no—
226C2×Dic83—42no—
227C2×Dic83—42no—
228C2×Dic83—44no—
229C2×Dic83C22×D412yes—
230C2×Dic83C22×D412yes—
231C2×Dic83C22×D412yes—
232C4×D83—28no—
233C4×D83—24no—
234C4×D83—24no—
235C4×D83—44no—
236C4×D83—48no—
237C4×D83—48no—
238Dic166—48no—
239C8×C83—24no—
240C83⋊C8—24no—
241C22×C83#53612yes—
242C22×C83C2×#13712yes—
243C22×C83#57512yes—
244C22×C83C23×D411yescharacteristic
245C2×D83—22no—
246C2×D83—28no—
247C2×D83—22no—
248C2×D83C2×#13714yes—
249C4×C83—22no—
250C4×C83—22no—
251C4×C83—44no—
252C4×C83—42no—
253C4×C83—42no—
254C4×C83#53612yes—
255C4×C83C2×2−1+412yes—
256Dic83—22no—
257Dic83—22no—
258Dic83—22no—
259Dic83—42no—
260Dic83—44no—
261Dic83—42no—
262C2×#3236—831nomaximal, Sylow
263C2×C83#323614yescomplemented
264C2×C83C2×#53611yescharacteristic
265C2×C83C22×#13711yescharacteristic
266C2×C83C2×#57511yescharacteristic
267D83—24no—
268C2×#533—831no—
269C22×#125—831no—
270#3236—8316no—
271C2×#501—831no—
272C2×#501—831no—
273C2×#491—831no—
274C2×#491D8311yescharacteristic, complemented
275C2×#484—832no—
276C2×#483—832no—
277C2×#479—832no—
278C2×#479—832no—
279C2×#432—831no—
280C83C2×#323611yescharacteristic, complemented, Sylow
281#533—838no—
282#533—1664no—
283C2×C4⋊Q8—831no—
284C2×C4⋊Q8C2×D8311yescharacteristic
285C2×#125—838no—
286C2×#125—831no—
287C2×#125—831no—
288C2×#125—831no—
289C2×#125—1661no—
290C2×#125—1661no—
291C2×#125—1661no—
292C2×#125—1664no—
293C2×#125C2×D8311yescharacteristic
294C2×C22⋊Q8—832no—
295C2×C22⋊Q8—1661no—
296C2×C4×Q8—832no—
297C2×C4×Q8—1661no—
298C2×#117—1661no—
299C22×C4⋊C4—831no—
300C22×C4⋊C4—831no—
301C22×C4⋊C4—1661no—
302C22×C4⋊C4—1661no—
303C22×C42—831no—
304#501—838no—
305#501—838no—
306#491—838no—
307#491C2×D8318yescomplemented
308#484—8316no—
309#483—8316no—
310#479—8316no—
311#479—8316no—
312#432—838no—
313C2×#107—832no—
314C2×#107—832no—
315C2×#106—832no—
316C2×#106—832no—
317C2×C4⋊C8—832no—
318C2×#103—832no—
319C2×#103—832no—
320C2×#103C2×D8312yes—
321C2×#103C2×D8312yes—
322C2×C22⋊C8—832no—
323C2×#97—831no—
324C2×#97C2×D8311yescharacteristic
325C22×Q8—1661no—
326C22×Q8C22×D8312yes—
327C23×C4—831no—
328C23×C4—831no—
329C23×C4—831no—
330C23×C4—1661no—
331C23×C4—1661no—
332C22×C8—832no—
333C22×C8C22×D8312yes—
334C4⋊Q8—22no—
335C4⋊Q8—834no—
336C4⋊Q8—1662no—
337C4⋊Q8C22×D8314yes—
338#125—22no—
339#125—834no—
340#125—834no—
341#125—834no—
342#125—1668no—
343#125—1668no—
344#125—1662no—
345#125—1662no—
346#125—1668no—
347#125—1662no—
348#125C22×D8314yes—
349C22⋊Q8—838no—
350C22⋊Q8—1668no—
351C22⋊Q8—1664no—
352C4×Q8—838no—
353C4×Q8—1668no—
354C4×Q8—1664no—
355#117—1668no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—831no—
361C2×C4⋊C4—831no—
362C2×C4⋊C4—832no—
363C2×C4⋊C4—834no—
364C2×C4⋊C4—834no—
365C2×C4⋊C4—1662no—
366C2×C4⋊C4—1662no—
367C2×C4⋊C4—1668no—
368C2×C4⋊C4—1668no—
369C2×C4⋊C4—1661no—
370C2×C4⋊C4—1661no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1662no—
376C2×C4⋊C4—1662no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C4⋊C4C22×D8311yescharacteristic
382C2×C22⋊C4—1662no—
383C2×C42—831no—
384C2×C42—831no—
385C2×C42—831no—
386C2×C42—834no—
387C2×C42—834no—
388C2×C42—1662no—
389C2×C42C22×D8311yescharacteristic
390#107—838no—
391#107—838no—
392#106—838no—
393#106—838no—
394C4⋊C8—838no—
395#103—838no—
396#103—838no—
397#103C22×D8318yes—
398#103C22×D8318yes—
399C22⋊C8—838no—
400#97—834no—
401#97C22×D8314yes—
402C24—831no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—3324no—
406C2×Q8—1664no—
407C2×Q8C23×D8314yes—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—3322no—
412C22×C4—3324no—
413C22×C4—3324no—
414C22×C4—834no—
415C22×C4—831no—
416C22×C4—831no—
417C22×C4—831no—
418C22×C4—834no—
419C22×C4—832no—
420C22×C4—832no—
421C22×C4—1664no—
422C22×C4—1664no—
423C22×C4—1664no—
424C22×C4—1662no—
425C22×C4—1661no—
426C22×C4—1661no—
427C22×C4—1664no—
428C22×C4D4×D8311yescharacteristic
429C22×C4D4×D8311yescharacteristic
430C22×C4C23×D8311yescharacteristic
431C2×C8—24no—
432C2×C8—834no—
433C2×C8—1664no—
434C2×C8C23×D8314yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—832no—
442C4⋊C4—832no—
443C4⋊C4—834no—
444C4⋊C4—1664no—
445C4⋊C4—1664no—
446C4⋊C4—1664no—
447C4⋊C4—1664no—
448C4⋊C4—1664no—
449C4⋊C4—1664no—
450C4⋊C4—1668no—
451C4⋊C4—1668no—
452C4⋊C4—1661no—
453C4⋊C4—1661no—
454C4⋊C4—1662no—
455C4⋊C4—1664no—
456C4⋊C4—1664no—
457C4⋊C4—1664no—
458C4⋊C4C23×D8312yes—
459C4⋊C4C23×D8312yes—
460C22⋊C4—1668no—
461C42—834no—
462C42—832no—
463C42—834no—
464C42—832no—
465C42—1668no—
466C42D4×D8312yes—
467C42C23×D8312yes—
468C23—834no—
469C23—832no—
470C23—1662no—
471C23—1662no—
472C23C2×D4×D8311yescharacteristic
473Q8—28no—
474Q8—44no—
475Q8—3328no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—3324no—
485C2×C4—3328no—
486C2×C4—3328no—
487C2×C4—3322no—
488C2×C4—3322no—
489C2×C4—3324no—
490C2×C4—832no—
491C2×C4—832no—
492C2×C4—832no—
493C2×C4—1668no—
494C2×C4—1664no—
495C2×C4—1662no—
496C2×C4—1662no—
497C2×C4—1662no—
498C2×C4—1662no—
499C2×C4—1662no—
500C2×C4—1664no—
501C2×C4—1664no—
502C2×C4C2×D4×D8312yes—
503C2×C4C2×D4×D8312yes—
504C2×C4C2×D4×D8311yescharacteristic
505C2×C4#1458183914yes—
506C2×C4C24×D8311yescharacteristic
507C8—24no—
508C8—1664no—
509C22—834no—
510C22—1662no—
511C22—1668no—
512C22—1662no—
513C22#1912827112yes—
514C22#137×D8312yes—
515C22#1912850012yes—
516C22C22×D4×D8311yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—44no—
522C4—3322no—
523C4—3322no—
524C4—3324no—
525C4—1662no—
526C4—1662no—
527C4—1662no—
528C4#1912827112yes—
529C4C2×#1458183912yes—
530C2—1664no—
531C2#2491866414yesdirect product factor
532C2C2×#1912827111yescharacteristic
533C2C2×#137×D8311yescharacteristic
534C2C2×#1912850011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#491×C83C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
hypercenterC2×#491D83
hypocenterC83C2×#3236
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
3. #24918664C2direct (N × H)
4. C2×#19128111C2semidirect
5. C2×#19128110C2semidirect
6. C2×#19128069C2semidirect
9. C2×#19128010C2semidirect
10. C2×#19128006C2semidirect
12. C2×#491×C83C2semidirect
34. #19128111C22semidirect
35. #19128110C22semidirect
36. #19128069C22semidirect
39. #19128010C22semidirect
40. #19128006C22semidirect
53. #491×C83C22semidirect
263. C2×C83#3236semidirect
274. C2×#491D83semidirect
280. C83C2×#3236semidirect
307. #491C2×D83semidirect
531. C2#24918664direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282562
C2×#125×D832
#249186642
C2×#191281112
C2×#191281102
C2×#191280692
C2×#191280352
C2×#191280302
C2×#191280102
C2×#191280062
C2×#191277562
C2×#491×C832
C2×#323683