Finite Group Databasebeta

← C2×#24918683 #36276525

Subgroups — C2×#24918683 #36276525

Order 21248.

2018 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D83C211yesmaximal, characteristic, complemented
2C2×#19128233C211yesmaximal, characteristic
3#24918683C2116yesmaximal, direct product factor
4C2×#19128108C212yesmaximal, complemented
5C2×#19128059C212yesmaximal, complemented
6C2×#19128033C212yesmaximal, complemented
7C2×#19128013C212yesmaximal
8C2×#19127997C211yesmaximal, characteristic, complemented
9C2×#19127993C211yesmaximal, characteristic, complemented
10C2×#19127991C211yesmaximal, characteristic, complemented
11C2×#19127757C211yesmaximal, characteristic, complemented
12C2×#476×C83C211yesmaximal, characteristic, complemented
13C22×Q8×D83—21no—
14C2×#14581792C2211yescharacteristic
15C2×#14581791—21no—
16C2×#14581791C2212yes—
17C2×#14581790—21no—
18C4⋊Q8×D83—24no—
19C4⋊Q8×D83C2218yescomplemented
20#19128233—24no—
21#19128233C2218yes—
22C2×#14581785—21no—
23C2×#14581783C2211yescharacteristic
24C2×#14581779C2211yescharacteristic
25C2×#14581774—21no—
26C2×#14581772C2211yescharacteristic
27C2×#14581763C2211yescharacteristic
28C2×C4×Dic166C2211yescharacteristic
29C2×C4×C22⋊D83—21no—
30C2×#14581751—21no—
31C2×C4⋊C4×D83—21no—
32C2×C4⋊C4×D83C2212yes—
33C2×C42×D83C2211yescharacteristic
34#19128108C22116yescomplemented
35#19128059C22116yescomplemented
36#19128033C22116yescomplemented
37#19128013C22116yes—
38#19127997C2218yescomplemented
39#19127993C2218yescomplemented
40#19127991C2218yescomplemented
41#19127757C2218yescomplemented
42C2×#14581726C2212yes—
43C2×#14581725C2212yes—
44C2×#14581723C2212yes—
45C2×C4⋊(C83⋊C8)C2211yescharacteristic
46C2×#14581717C2212yes—
47C2×#14581715C2212yes—
48C2×#14581713C2212yes—
49C2×(C2×D83)⋊C8C2212yes—
50C2×#14581702C2212yes—
51C2×C4⋊Q8×C83C2211yescharacteristic
52C2×C4×D4×C83C2211yescharacteristic
53#476×C83C2218yescomplemented
54C2×#106×C83C2212yes—
55C2×C4⋊C8×C83C2211yescharacteristic
56C2×#102×C83C2212yes—
57C22×Dic166—22no—
58C22×Dic166—21no—
59C22×Dic166C2311yescharacteristic
60C2×Q8×D83—28no—
61C2×Q8×D83—48no—
62C22×C22⋊D83—21no—
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#14581791C2318yes—
73#14581790—28no—
74#14581785—28no—
75#14581783—22no—
76#14581783C2314yes—
77#14581779—22no—
78#14581779C2314yes—
79#14581774—28no—
80#14581772—22no—
81#14581772C2314yes—
82#14581763—22no—
83#14581763C2314yes—
84C4×Dic166—22no—
85C4×Dic166C2314yes—
86C4×C22⋊D83—28no—
87#14581751—28no—
88C2×C4⋊Dic83—21no—
89C2×C4⋊Dic83—21no—
90C2×C4⋊Dic83—21no—
91C2×C4⋊Dic83C2312yes—
92C2×C4⋊Dic83C2311yescharacteristic
93C2×Dic83⋊C4—22no—
94C2×Dic83⋊C4—22no—
95C2×Dic83⋊C4—21no—
96C2×Dic83⋊C4—21no—
97C4⋊C4×D83—28no—
98C4⋊C4×D83—24no—
99C4⋊C4×D83C2318yes—
100C2×C22⋊Dic83—21no—
101C2×C22⋊Dic83—21no—
102C2×C22⋊Dic83—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#14581723C2318yes—
113C4⋊(C83⋊C8)C2314yes—
114#14581717C2318yes—
115#14581715C2318yes—
116#14581713C2318yes—
117(C2×D83)⋊C8C2318yes—
118#14581702C2318yes—
119C22×Q8×C83—21no—
120C22×D4×C83C2311yescharacteristic
121C23×C4×C83—21no—
122C22×C8×C83C2312yes—
123C4⋊Q8×C83—22no—
124C4⋊Q8×C83C2314yes—
125C4×D4×C83—22no—
126C4×D4×C83C2314yes—
127C2×C4⋊C4×C83—21no—
128C2×C4⋊C4×C83C2311yescharacteristic
129C2×C4⋊C4×C83C2312yes—
130C2×C22⋊C4×C83—21no—
131C2×C42×C83C2311yescharacteristic
132#106×C83C2318yes—
133C4⋊C8×C83C2314yes—
134#102×C83C2318yes—
135C23×D83C2×D411yescharacteristic
136C2×Dic166—22no—
137C2×Dic166—22no—
138C2×Dic166—21no—
139C2×Dic166—24no—
140C2×Dic166—28no—
141C2×Dic166—48no—
142C2×Dic166—44no—
143C2×Dic166C2412yes—
144Q8×D83—432no—
145C2×C22⋊D83—24no—
146C2×C22⋊D83—44no—
147C22×Dic83—22no—
148C22×Dic83—21no—
149C22×Dic83—21no—
150C22×Dic83C2×D411yescharacteristic
151C22×Dic83C2×D411yescharacteristic
152C22×Dic83C2×D411yescharacteristic
153C2×C4×D83—24no—
154C2×C4×D83—24no—
155C2×C4×D83—22no—
156C2×C4×D83—22no—
157C2×C4×D83—28no—
158C2×C4×D83—48no—
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⋊Dic83—24no—
170C4⋊Dic83C2412yes—
171C4⋊Dic83C2414yes—
172Dic83⋊C4—24no—
173Dic83⋊C4—24no—
174Dic83⋊C4—28no—
175Dic83⋊C4—28no—
176C22⋊Dic83—24no—
177C22⋊Dic83—24no—
178C22⋊Dic83—24no—
179(C2×D83)⋊C4—24no—
180C4×Dic83—24no—
181C4×Dic83C2×D412yes—
182C4×Dic83C2×D414yes—
183C4×Dic83C2×D414yes—
184C4×Dic83C2412yes—
185C24×C83—21no—
186C2×Q8×C83—24no—
187C2×Q8×C83—44no—
188C2×D4×C83—21no—
189C2×D4×C83—22no—
190C2×D4×C83—22no—
191C2×D4×C83C2412yes—
192C22×C4×C83—22no—
193C22×C4×C83—21no—
194C22×C4×C83—24no—
195C22×C4×C83—44no—
196C22×C4×C83C2×D411yescharacteristic
197C22×C4×C83C2×D411yescharacteristic
198C22×C4×C83C2411yescharacteristic
199C2×C8×C83—24no—
200C2×C8×C83C2414yes—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—24no—
203C4⋊C4×C83—22no—
204C4⋊C4×C83C2412yes—
205C4⋊C4×C83C2414yes—
206C22⋊C4×C83—24no—
207C42×C83C2×D412yes—
208C42×C83C2412yes—
209C23×C83—22no—
210C23×C83—42no—
211C23×C83—42no—
212C23×C83—42no—
213C23×C83C22×D411yescharacteristic
214C22×D83—22no—
215C22×D83—22no—
216C22×D83AGL(1,ℤ/8)14yes—
217C22×D83C22×D412yes—
218Q8×C83—48no—
219D4×C83—22no—
220D4×C83—22no—
221D4×C83—42no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—24no—
225C2×C4×C83—44no—
226C2×C4×C83—42no—
227C2×C4×C83—48no—
228C2×C4×C83#13714yes—
229C2×C4×C83C22×D412yes—
230C2×C4×C83C22×D411yescharacteristic
231C2×C4×C83C22×D412yes—
232C2×C4×C832−1+412yes—
233C2×C4×C832−1+412yes—
234C2×C4×C83C2511yescharacteristic
235C22⋊D83—48no—
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—24no—
242C2×Dic83—42no—
243C2×Dic83—44no—
244C2×Dic83—42no—
245C2×Dic83C22×D412yes—
246C2×Dic83C22×D412yes—
247C2×Dic83C22×D412yes—
248C4×D83—28no—
249C4×D83—24no—
250C4×D83—28no—
251C4×D83—22no—
252C4×D83—22no—
253C4×D83—416no—
254Dic166—22no—
255Dic166—22no—
256Dic166—416no—
257Dic166—48no—
258Dic166—42no—
259C8×C83—24no—
260C83⋊C8—24no—
261C22×C83—42no—
262C22×C83—42no—
263C22×C83—42no—
264C22×C83—84no—
265C22×C83#53612yes—
266C22×C83C2×AGL(1,ℤ/8)12yes—
267C22×C83C2×#13712yes—
268C22×C83C23×D411yescharacteristic
269C2×D83—22no—
270C2×D83—28no—
271C2×D83—22no—
272C2×D83C2×AGL(1,ℤ/8)14yes—
273C4×C83—22no—
274C4×C83—44no—
275C4×C83—42no—
276C4×C83#53612yes—
277C4×C83C2×#13714yes—
278C4×C83C2×2−1+411yescharacteristic
279C4×C83C2×2−1+411yescharacteristic
280Dic83—22no—
281Dic83—22no—
282Dic83—22no—
283Dic83—42no—
284Dic83—44no—
285Dic83—42no—
286C2×#3238—831nomaximal, Sylow
287C2×C83—82no—
288C2×C83#323814yescomplemented
289C2×C83C2×#53611yescharacteristic
290C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
291C2×C83C22×#13711yescharacteristic
292D83—24no—
293C22×C4⋊Q8—831no—
294C2×#529—831no—
295#3238—8316no—
296C2×#501—832no—
297C2×#489—832no—
298C2×#483—832no—
299C2×#480—832no—
300C2×#477—831no—
301C2×#477—831no—
302C2×#476—831no—
303C2×#476D8311yescharacteristic, complemented
304C2×#432—831no—
305C83C2×#323811yescharacteristic, complemented, Sylow
306C23×Q8—1661no—
307C2×C4⋊Q8—838no—
308C2×C4⋊Q8—832no—
309C2×C4⋊Q8—831no—
310C2×C4⋊Q8—1661no—
311C2×C4⋊Q8—1664no—
312C2×C4⋊Q8—1661no—
313C2×C4⋊Q8C2×D8311yescharacteristic
314#529—838no—
315#529—1664no—
316C2×#126—1661no—
317C2×#125—831no—
318C2×#124—831no—
319C2×#123—1661no—
320C2×C22⋊Q8—831no—
321C2×#121—831no—
322C2×C4×Q8—831no—
323C2×C4×D4—1661no—
324C2×C4×D4C2×D8311yescharacteristic
325C2×#117—1661no—
326C22×C4⋊C4—832no—
327C22×C4⋊C4—1661no—
328C22×C42—831no—
329#501—8316no—
330#489—8316no—
331#483—8316no—
332#480—8316no—
333#477—838no—
334#477—838no—
335#476—838no—
336#476C2×D8318yescomplemented
337#432—838no—
338C2×#107—832no—
339C2×#107—832no—
340C2×#106—832no—
341C2×#106C2×D8312yes—
342C2×C4⋊C8—831no—
343C2×C4⋊C8C2×D8311yescharacteristic
344C2×#103—832no—
345C2×#103—832no—
346C2×#102—832no—
347C2×#102C2×D8312yes—
348C2×C22⋊C8—832no—
349C2×#97—832no—
350C22×Q8—21no—
351C22×Q8—3328no—
352C22×Q8—831no—
353C22×Q8—1662no—
354C22×Q8—1668no—
355C22×Q8—1661no—
356C22×D4—1661no—
357C22×D4C22×D8311yescharacteristic
358C23×C4—21no—
359C23×C4—831no—
360C23×C4—831no—
361C23×C4—831no—
362C23×C4—1662no—
363C22×C8—832no—
364C22×C8C22×D8312yes—
365C4⋊Q8—22no—
366C4⋊Q8—838no—
367C4⋊Q8—834no—
368C4⋊Q8—1668no—
369C4⋊Q8—1664no—
370C4⋊Q8—1668no—
371C4⋊Q8—1662no—
372C4⋊Q8C22×D8314yes—
373#126—1668no—
374#125—834no—
375#125—1662no—
376#124—834no—
377#124—1662no—
378#123—1668no—
379C22⋊Q8—834no—
380C22⋊Q8—1662no—
381#121—834no—
382#121—1662no—
383C4×Q8—834no—
384C4×Q8—1662no—
385C4×D4—22no—
386C4×D4—1668no—
387C4×D4C22×D8314yes—
388#117—1668no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—832no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—838no—
393C2×C4⋊C4—1664no—
394C2×C4⋊C4—1668no—
395C2×C4⋊C4—1662no—
396C2×C4⋊C4—1662no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4C22×D8312yes—
403C2×C4⋊C4C22×D8311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1661no—
406C2×C22⋊C4—1661no—
407C2×C22⋊C4—1661no—
408C2×C22⋊C4—1661no—
409C2×C42—831no—
410C2×C42—831no—
411C2×C42—831no—
412C2×C42—834no—
413C2×C42—834no—
414C2×C42—1661no—
415C2×C42C22×D8311yescharacteristic
416#107—838no—
417#107—838no—
418#106—838no—
419#106C22×D8318yes—
420C4⋊C8—834no—
421C4⋊C8C22×D8314yes—
422#103—838no—
423#103—838no—
424#102—838no—
425#102C22×D8318yes—
426C22⋊C8—838no—
427#97—838no—
428C24—21no—
429C24—831no—
430C2×Q8—24no—
431C2×Q8—44no—
432C2×Q8—3324no—
433C2×Q8—3328no—
434C2×Q8—33232no—
435C2×Q8—832no—
436C2×Q8—1668no—
437C2×Q8—1662no—
438C2×Q8—1662no—
439C2×Q8—1661no—
440C2×Q8—1664no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—3324no—
445C2×D4—1664no—
446C2×D4C23×D8312yes—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—3328no—
452C22×C4—831no—
453C22×C4—831no—
454C22×C4—831no—
455C22×C4—834no—
456C22×C4—832no—
457C22×C4—834no—
458C22×C4—832no—
459C22×C4—1668no—
460C22×C4—1664no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1662no—
464C22×C4—1664no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4D4×D8311yescharacteristic
468C22×C4D4×D8311yescharacteristic
469C22×C4C23×D8311yescharacteristic
470C2×C8—24no—
471C2×C8—834no—
472C2×C8—1664no—
473C2×C8C23×D8314yes—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—834no—
478C4⋊C4—832no—
479C4⋊C4—1668no—
480C4⋊C4—1668no—
481C4⋊C4—1664no—
482C4⋊C4—1664no—
483C4⋊C4—1664no—
484C4⋊C4—1662no—
485C4⋊C4—1661no—
486C4⋊C4—1664no—
487C4⋊C4—1664no—
488C4⋊C4C23×D8314yes—
489C4⋊C4C23×D8312yes—
490C22⋊C4—24no—
491C22⋊C4—1664no—
492C22⋊C4—1664no—
493C22⋊C4—1664no—
494C22⋊C4—1664no—
495C42—834no—
496C42—832no—
497C42—834no—
498C42—832no—
499C42—1664no—
500C42D4×D8312yes—
501C42C23×D8312yes—
502C23—22no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—834no—
507C23—832no—
508C23—1662no—
509C23—1662no—
510C23C2×D4×D8311yescharacteristic
511Q8—48no—
512Q8—3328no—
513Q8—33216no—
514Q8—3322no—
515Q8—1662no—
516Q8—1662no—
517D4—22no—
518D4—22no—
519D4—42no—
520D4—3328no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—48no—
527C2×C4—33216no—
528C2×C4—3324no—
529C2×C4—3322no—
530C2×C4—3322no—
531C2×C4—832no—
532C2×C4—832no—
533C2×C4—832no—
534C2×C4—1668no—
535C2×C4—1662no—
536C2×C4—1662no—
537C2×C4—1664no—
538C2×C4—1662no—
539C2×C4—1662no—
540C2×C4—1662no—
541C2×C4—1668no—
542C2×C4—1664no—
543C2×C4—1662no—
544C2×C4—1662no—
545C2×C4#1458182114yes—
546C2×C4C2×D4×D8312yes—
547C2×C4C2×D4×D8312yes—
548C2×C4C2×D4×D8311yescharacteristic
549C2×C4#1458183912yes—
550C2×C4#1458184012yes—
551C2×C4C24×D8311yescharacteristic
552C8—24no—
553C8—1664no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—84no—
558C22—834no—
559C22—1662no—
560C22—1668no—
561C22—1662no—
562C22#1912827412yes—
563C22AGL(1,ℤ/8)×D8312yes—
564C22C2×#1458182112yes—
565C22C22×D4×D8311yescharacteristic
566C4—22no—
567C4—44no—
568C4—42no—
569C4—3324no—
570C4—3322no—
571C4—3322no—
572C4—1662no—
573C4—1662no—
574C4—1662no—
575C4#1912827412yes—
576C4C2×#1458182114yes—
577C4C2×#1458183911yescharacteristic
578C4C2×#1458184011yescharacteristic
579C2—82no—
580C2—1664no—
581C2#2491868314yesdirect product factor
582C2C2×#1912827411yescharacteristic
583C2C2×AGL(1,ℤ/8)×D8311yescharacteristic
584C2C22×#1458182111yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#476×C83C2
socleC23×C83C22×D4
prime radicalC2×#14581763C22
hypercenterC2×#476D83
hypocenterC83C2×#3238
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D83C2semidirect
3. #24918683C2direct (N × H)
4. C2×#19128108C2semidirect
5. C2×#19128059C2semidirect
6. C2×#19128033C2semidirect
8. C2×#19127997C2semidirect
9. C2×#19127993C2semidirect
10. C2×#19127991C2semidirect
11. C2×#19127757C2semidirect
12. C2×#476×C83C2semidirect
19. C4⋊Q8×D83C22semidirect
34. #19128108C22semidirect
35. #19128059C22semidirect
36. #19128033C22semidirect
38. #19127997C22semidirect
39. #19127993C22semidirect
40. #19127991C22semidirect
41. #19127757C22semidirect
53. #476×C83C22semidirect
288. C2×C83#3238semidirect
303. C2×#476D83semidirect
305. C83C2×#3238semidirect
336. #476C2×D83semidirect
581. C2#24918683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D832
C2×#191282332
#249186832
C2×#191281082
C2×#191280592
C2×#191280332
C2×#191280132
C2×#191279972
C2×#191279932
C2×#191279912
C2×#191277572
C2×#476×C832
C2×#323883