Finite Group Databasebeta

← C2×#24918613 #36276452

Subgroups — C2×#24918613 #36276452

Order 21248.

2006 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
1C2×#19128255C211yesmaximal, characteristic
2C2×C4⋊Q8×D83C211yesmaximal, characteristic
3#24918613C2116yesmaximal, direct product factor
4C2×#19128105C212yesmaximal, complemented
5C2×#19128050C212yesmaximal, complemented
6C2×#19128038C212yesmaximal
7C2×#19128035C212yesmaximal
8C2×#19127997C211yesmaximal, characteristic, complemented
9C2×#19127996C211yesmaximal, characteristic, complemented
10C2×#19127993C211yesmaximal, characteristic, complemented
11C2×#19127753C211yesmaximal, characteristic
12C2×#477×C83C211yesmaximal, characteristic, complemented
13C22×Q8×D83—21no—
14#19128255—24no—
15#19128255C2218yes—
16C2×#14581792C2211yescharacteristic
17C2×#14581791—21no—
18C2×#14581791C2212yes—
19C2×#14581791C2211yescharacteristic
20C2×#14581790—21no—
21C2×#14581790C2211yescharacteristic
22C4⋊Q8×D83—24no—
23C4⋊Q8×D83C2218yes—
24C2×#14581782—21no—
25C2×#14581772C2211yescharacteristic
26C2×#14581770—21no—
27C2×#14581768C2211yescharacteristic
28C2×C4×Dic166—21no—
29C2×C4×Dic166C2211yescharacteristic
30C2×#14581749—21no—
31C2×C4⋊C4×D83—21no—
32C2×C4⋊C4×D83C2212yes—
33C2×C42×D83C2211yescharacteristic
34#19128105C22116yescomplemented
35#19128050C22116yescomplemented
36#19128038C22116yes—
37#19128035C22116yes—
38#19127997C2218yescomplemented
39#19127996C2218yescomplemented
40#19127993C2218yescomplemented
41#19127753C2218yes—
42C2×#14581726C2212yes—
43C2×#14581725C2212yes—
44C2×#14581725C2212yes—
45C2×C4⋊(C83⋊C8)C2211yescharacteristic
46C2×#14581717C2212yes—
47C2×#14581716C2212yes—
48C2×#14581715C2212yes—
49C2×(C2×D83)⋊C8C2212yes—
50C2×C8×Dic83C2212yes—
51C2×C4⋊Q8×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53#477×C83C2218yescomplemented
54C2×#107×C83C2212yes—
55C2×C4⋊C8×C83C2211yescharacteristic
56C2×#103×C83C2212yes—
57C22×Dic166—21no—
58C22×Dic166—22no—
59C22×Dic166—21no—
60C22×Dic166C2311yescharacteristic
61C2×Q8×D83—28no—
62C2×Q8×D83—48no—
63C22×C4×D83—22no—
64C22×C4×D83D411yescharacteristic
65C22×C4×D83D411yescharacteristic
66C22×C4×D83C2311yescharacteristic
67C22×C83⋊C8C2312yes—
68#14581792—22no—
69#14581792C2314yes—
70#14581791—24no—
71#14581791—28no—
72#14581791—22no—
73#14581791C2314yes—
74#14581791C2318yes—
75#14581790—28no—
76#14581790—22no—
77#14581790C2314yes—
78#14581782—28no—
79#14581772—22no—
80#14581772C2314yes—
81#14581770—28no—
82#14581768—22no—
83#14581768C2314yes—
84C4×Dic166—28no—
85C4×Dic166—22no—
86C4×Dic166C2314yes—
87#14581749—28no—
88C2×C4⋊Dic83—21no—
89C2×C4⋊Dic83—21no—
90C2×C4⋊Dic83C2312yes—
91C2×C4⋊Dic83C2311yescharacteristic
92C2×Dic83⋊C4—22no—
93C2×Dic83⋊C4—22no—
94C2×Dic83⋊C4—21no—
95C2×Dic83⋊C4—21no—
96C2×Dic83⋊C4—21no—
97C2×Dic83⋊C4—21no—
98C2×Dic83⋊C4—21no—
99C4⋊C4×D83—28no—
100C4⋊C4×D83—24no—
101C4⋊C4×D83C2318yes—
102C2×(C2×D83)⋊C4—21no—
103C2×(C2×D83)⋊C4—21no—
104C2×C4×Dic83—21no—
105C2×C4×Dic83D411yescharacteristic
106C2×C4×Dic83D411yescharacteristic
107C2×C4×Dic83C2311yescharacteristic
108C42×D83D414yes—
109C42×D83C2314yes—
110#14581726C2318yes—
111#14581725C2318yes—
112#14581725C2318yes—
113C4⋊(C83⋊C8)C2314yes—
114#14581717C2318yes—
115#14581716C2318yes—
116#14581715C2318yes—
117(C2×D83)⋊C8C2318yes—
118C8×Dic83C2318yes—
119C22×Q8×C83—21no—
120C22×Q8×C83C2311yescharacteristic
121C22×C8×C83C2312yes—
122C4⋊Q8×C83—22no—
123C4⋊Q8×C83C2314yes—
124C4×Q8×C83—22no—
125C4×Q8×C83C2314yes—
126C2×C4⋊C4×C83—21no—
127C2×C4⋊C4×C83—21no—
128C2×C4⋊C4×C83C2311yescharacteristic
129C2×C4⋊C4×C83C2312yes—
130C2×C42×C83—21no—
131C2×C42×C83C2311yescharacteristic
132#107×C83C2318yes—
133C4⋊C8×C83C2314yes—
134#103×C83C2318yes—
135C23×D83C2×D411yescharacteristic
136C2×Dic166—22no—
137C2×Dic166—22no—
138C2×Dic166—21no—
139C2×Dic166—24no—
140C2×Dic166—24no—
141C2×Dic166—28no—
142C2×Dic166—44no—
143C2×Dic166—48no—
144C2×Dic166—44no—
145C2×Dic166C2412yes—
146Q8×D83—432no—
147C22×Dic83—22no—
148C22×Dic83—21no—
149C22×Dic83—21no—
150C22×Dic83C2×D411yescharacteristic
151C22×Dic83C2×D411yescharacteristic
152C22×Dic83C2×D411yescharacteristic
153C2×C4×D83—22no—
154C2×C4×D83—22no—
155C2×C4×D83—24no—
156C2×C4×D83—28no—
157C2×C4×D83—48no—
158C2×C4×D83Q1618yes—
159C2×C4×D83C2×D412yes—
160C2×C4×D83C2×D414yes—
161C2×C4×D83C2×D414yes—
162C2×C4×D83C2412yes—
163C2×C83⋊C8—24no—
164C2×C83⋊C8C2414yes—
165C4⋊Dic83—21no—
166C4⋊Dic83—22no—
167C4⋊Dic83—24no—
168C4⋊Dic83—24no—
169C4⋊Dic83C2412yes—
170C4⋊Dic83C2414yes—
171Dic83⋊C4—24no—
172Dic83⋊C4—24no—
173Dic83⋊C4—24no—
174Dic83⋊C4—24no—
175Dic83⋊C4—24no—
176Dic83⋊C4—28no—
177Dic83⋊C4—28no—
178(C2×D83)⋊C4—24no—
179(C2×D83)⋊C4—24no—
180C4×Dic83—24no—
181C4×Dic83C2×D414yes—
182C4×Dic83C2×D412yes—
183C4×Dic83C2×D414yes—
184C4×Dic83C2412yes—
185C2×Q8×C83—21no—
186C2×Q8×C83—24no—
187C2×Q8×C83—22no—
188C2×Q8×C83—22no—
189C2×Q8×C83—44no—
190C2×Q8×C83C2412yes—
191C22×C4×C83—22no—
192C22×C4×C83—21no—
193C22×C4×C83—21no—
194C22×C4×C83C2×D411yescharacteristic
195C22×C4×C83C2×D411yescharacteristic
196C22×C4×C83C2411yescharacteristic
197C2×C8×C83—24no—
198C2×C8×C83C2414yes—
199C4⋊C4×C83—24no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—24no—
202C4⋊C4×C83—22no—
203C4⋊C4×C83C2412yes—
204C4⋊C4×C83C2414yes—
205C42×C83—24no—
206C42×C83C2×D412yes—
207C42×C83C2412yes—
208C23×C83C22×D411yescharacteristic
209C22×D83—22no—
210C22×D83—22no—
211C22×D83C2×Q1614yes—
212C22×D83C22×D412yes—
213Q8×C83—22no—
214Q8×C83—22no—
215Q8×C83—48no—
216Q8×C83—42no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—24no—
221C2×C4×C83—42no—
222C2×C4×C83—44no—
223C2×C4×C83—42no—
224C2×C4×C83C2×Q1614yes—
225C2×C4×C83C22×D412yes—
226C2×C4×C83C22×D411yescharacteristic
227C2×C4×C83C22×D412yes—
228C2×C4×C832−1+412yes—
229C2×C4×C832−1+412yes—
230C2×C4×C83C2511yescharacteristic
231C2×Dic83—22no—
232C2×Dic83—22no—
233C2×Dic83—22no—
234C2×Dic83—24no—
235C2×Dic83—42no—
236C2×Dic83—44no—
237C2×Dic83—42no—
238C2×Dic83C2×Q1614yes—
239C2×Dic83AGL(1,ℤ/8)14yes—
240C2×Dic83C22×D412yes—
241C2×Dic83C22×D412yes—
242C2×Dic83C22×D412yes—
243C4×D83—24no—
244C4×D83—22no—
245C4×D83—22no—
246C4×D83—28no—
247C4×D83—416no—
248C4×D83C2×Q16116yes—
249Dic166—22no—
250Dic166—22no—
251Dic166—48no—
252Dic166—42no—
253Dic166—416no—
254Dic166—48no—
255C8×C83—24no—
256C83⋊C8—24no—
257C22×C83#53612yes—
258C22×C83C22×Q1612yes—
259C22×C83C2×AGL(1,ℤ/8)12yes—
260C22×C83C23×D411yescharacteristic
261C2×D83—22no—
262C2×D83—28no—
263C2×D83—22no—
264C2×D83C22×Q1614yes—
265C4×C83—22no—
266C4×C83—42no—
267C4×C83—44no—
268C4×C83—42no—
269C4×C83#53612yes—
270C4×C83C22×Q1614yes—
271C4×C83C2×2−1+411yescharacteristic
272C4×C83C2×2−1+411yescharacteristic
273Dic83—22no—
274Dic83—42no—
275Dic83—44no—
276Dic83—42no—
277Dic83C22×Q1614yes—
278Dic83C2×AGL(1,ℤ/8)14yes—
279C2×#3230—831nomaximal, Sylow
280C2×C83#323014yescomplemented
281C2×C83C2×#53611yescharacteristic
282C2×C83C23×Q1611yescharacteristic
283C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
284D83—24no—
285C2×#533—831no—
286C22×C4⋊Q8—831no—
287#3230—8316no—
288C2×#500—832no—
289C2×#487—832no—
290C2×#484—832no—
291C2×#484—832no—
292C2×#477—831no—
293C2×#477—831no—
294C2×#477—831no—
295C2×#477D8311yescharacteristic, complemented
296C2×#431—831no—
297C83C2×#323011yescharacteristic, complemented, Sylow
298C23×Q8—1661no—
299#533—838no—
300#533—1664no—
301C2×C4⋊Q8—838no—
302C2×C4⋊Q8—832no—
303C2×C4⋊Q8—831no—
304C2×C4⋊Q8—831no—
305C2×C4⋊Q8—831no—
306C2×C4⋊Q8—1661no—
307C2×C4⋊Q8—1664no—
308C2×C4⋊Q8—1661no—
309C2×C4⋊Q8C2×D8311yescharacteristic
310C2×#125—1661no—
311C2×C22⋊Q8—831no—
312C2×C22⋊Q8—831no—
313C2×C22⋊Q8—1661no—
314C2×C4×Q8—831no—
315C2×C4×Q8—1661no—
316C2×C4×Q8C2×D8311yescharacteristic
317C2×#117—1661no—
318C22×C4⋊C4—832no—
319C22×C4⋊C4—1661no—
320C22×C42—831no—
321#500—8316no—
322#487—8316no—
323#484—8316no—
324#484—8316no—
325#477—838no—
326#477—838no—
327#477—838no—
328#477C2×D8318yescomplemented
329#431—838no—
330C2×#107—832no—
331C2×#107—832no—
332C2×#107—832no—
333C2×#107C2×D8312yes—
334C2×C4⋊C8—831no—
335C2×C4⋊C8C2×D8311yescharacteristic
336C2×#103—832no—
337C2×#103—832no—
338C2×#103—832no—
339C2×#103C2×D8312yes—
340C2×C22⋊C8—832no—
341C2×C4×C8—832no—
342C22×Q8—21no—
343C22×Q8—3328no—
344C22×Q8—831no—
345C22×Q8—1661no—
346C22×Q8—1662no—
347C22×Q8—1668no—
348C22×Q8—1661no—
349C22×Q8C22×D8311yescharacteristic
350C23×C4—831no—
351C23×C4—831no—
352C23×C4—831no—
353C23×C4—1662no—
354C22×C8—832no—
355C22×C8C22×D8312yes—
356C4⋊Q8—22no—
357C4⋊Q8—838no—
358C4⋊Q8—834no—
359C4⋊Q8—834no—
360C4⋊Q8—834no—
361C4⋊Q8—1668no—
362C4⋊Q8—1664no—
363C4⋊Q8—1662no—
364C4⋊Q8—1668no—
365C4⋊Q8—1662no—
366C4⋊Q8—1662no—
367C4⋊Q8C22×D8314yes—
368#125—1668no—
369C22⋊Q8—834no—
370C22⋊Q8—834no—
371C22⋊Q8—1668no—
372C22⋊Q8—1662no—
373C22⋊Q8—1662no—
374C4×Q8—22no—
375C4×Q8—834no—
376C4×Q8—1668no—
377C4×Q8—1662no—
378C4×Q8C22×D8314yes—
379#117—1668no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—832no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—838no—
385C2×C4⋊C4—1664no—
386C2×C4⋊C4—1668no—
387C2×C4⋊C4—1662no—
388C2×C4⋊C4—1662no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4C22×D8312yes—
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C42—21no—
401C2×C42—831no—
402C2×C42—831no—
403C2×C42—831no—
404C2×C42—834no—
405C2×C42—834no—
406C2×C42—1661no—
407C2×C42C22×D8311yescharacteristic
408#107—838no—
409#107—838no—
410#107—838no—
411#107C22×D8318yes—
412C4⋊C8—834no—
413C4⋊C8C22×D8314yes—
414#103—838no—
415#103—838no—
416#103—838no—
417#103C22×D8318yes—
418C22⋊C8—838no—
419C4×C8—838no—
420C24—831no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—3324no—
427C2×Q8—3328no—
428C2×Q8—3324no—
429C2×Q8—33232no—
430C2×Q8—832no—
431C2×Q8—1664no—
432C2×Q8—1668no—
433C2×Q8—1662no—
434C2×Q8—1662no—
435C2×Q8—1661no—
436C2×Q8—1664no—
437C2×Q8C23×D8312yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—3328no—
442C22×C4—831no—
443C22×C4—831no—
444C22×C4—838no—
445C22×C4—834no—
446C22×C4—831no—
447C22×C4—832no—
448C22×C4—834no—
449C22×C4—832no—
450C22×C4—1668no—
451C22×C4—1664no—
452C22×C4—1661no—
453C22×C4—1661no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4D4×D8311yescharacteristic
458C22×C4D4×D8311yescharacteristic
459C22×C4C23×D8311yescharacteristic
460C2×C8—24no—
461C2×C8—834no—
462C2×C8—1664no—
463C2×C8C23×D8314yes—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—834no—
469C4⋊C4—832no—
470C4⋊C4—1668no—
471C4⋊C4—1668no—
472C4⋊C4—1664no—
473C4⋊C4—1664no—
474C4⋊C4—1664no—
475C4⋊C4—1664no—
476C4⋊C4—1664no—
477C4⋊C4—1662no—
478C4⋊C4—1661no—
479C4⋊C4—1664no—
480C4⋊C4—1664no—
481C4⋊C4C23×D8314yes—
482C4⋊C4C23×D8312yes—
483C22⋊C4—1664no—
484C22⋊C4—1664no—
485C42—24no—
486C42—834no—
487C42—834no—
488C42—832no—
489C42—832no—
490C42—1664no—
491C42D4×D8312yes—
492C42C23×D8312yes—
493C23—834no—
494C23—832no—
495C23—1662no—
496C23—1662no—
497C23C2×D4×D8311yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—48no—
501Q8—42no—
502Q8—3328no—
503Q8—33216no—
504Q8—3328no—
505Q8—3322no—
506Q8—1662no—
507Q8—1662no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—33216no—
516C2×C4—3324no—
517C2×C4—3322no—
518C2×C4—3322no—
519C2×C4—834no—
520C2×C4—834no—
521C2×C4—832no—
522C2×C4—832no—
523C2×C4—8316no—
524C2×C4—832no—
525C2×C4—1668no—
526C2×C4—1662no—
527C2×C4—1662no—
528C2×C4—1664no—
529C2×C4—1662no—
530C2×C4—1664no—
531C2×C4—1662no—
532C2×C4—1662no—
533C2×C4Q16×D8314yes—
534C2×C4C2×D4×D8312yes—
535C2×C4C2×D4×D8312yes—
536C2×C4C2×D4×D8311yescharacteristic
537C2×C4#1458183912yes—
538C2×C4#1458184012yes—
539C2×C4C24×D8311yescharacteristic
540C8—24no—
541C8—1664no—
542C22—834no—
543C22—1662no—
544C22—1668no—
545C22—1662no—
546C22#1912827412yes—
547C22C2×Q16×D8312yes—
548C22AGL(1,ℤ/8)×D8312yes—
549C22C22×D4×D8311yescharacteristic
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—3324no—
555C4—3322no—
556C4—3322no—
557C4—834no—
558C4—834no—
559C4—1662no—
560C4#1912827412yes—
561C4C2×Q16×D8314yes—
562C4C2×#1458183911yescharacteristic
563C4C2×#1458184011yescharacteristic
564C2—1664no—
565C2#2491861314yesdirect product factor
566C2C2×#1912827411yescharacteristic
567C2C22×Q16×D8311yescharacteristic
568C2C2×AGL(1,ℤ/8)×D8311yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#477×C83C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
hypercenterC2×#477D83
hypocenterC83C2×#3230
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #24918613C2direct (N × H)
4. C2×#19128105C2semidirect
5. C2×#19128050C2semidirect
8. C2×#19127997C2semidirect
9. C2×#19127996C2semidirect
10. C2×#19127993C2semidirect
12. C2×#477×C83C2semidirect
34. #19128105C22semidirect
35. #19128050C22semidirect
38. #19127997C22semidirect
39. #19127996C22semidirect
40. #19127993C22semidirect
53. #477×C83C22semidirect
280. C2×C83#3230semidirect
295. C2×#477D83semidirect
297. C83C2×#3230semidirect
328. #477C2×D83semidirect
565. C2#24918613direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282552
C2×C4⋊Q8×D832
#249186132
C2×#191281052
C2×#191280502
C2×#191280382
C2×#191280352
C2×#191279972
C2×#191279962
C2×#191279932
C2×#191277532
C2×#477×C832
C2×#323083