Finite Group Databasebeta

← C2×#24918778 #36277185

Subgroups — C2×#24918778 #36277185

Order 21248.

2178 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128255C211yesmaximal, characteristic, complemented
2C2×#19128252C211yesmaximal, characteristic
3#24918778C2116yesmaximal, direct product factor
4C2×#19127983C211yesmaximal, characteristic, complemented
5C2×#19127990C212yesmaximal, complemented
6C2×#19127933C212yesmaximal, complemented
7C2×#19127931C212yesmaximal, complemented
8C2×#19127902C211yesmaximal, characteristic, complemented
9C2×#19127901C211yesmaximal, characteristic, complemented
10C2×#19127905C212yesmaximal, complemented
11C2×C4⋊C8×D83C211yesmaximal, characteristic, complemented
12C2×#475×C83C211yesmaximal, characteristic, complemented
13C22×#13050591—21no—
14C22×#13050579C2212yes—
15C22×#13050577C2212yes—
16C22×C8×D83C2212yes—
17#19128255—24no—
18#19128255C2218yescomplemented
19#19128252—24no—
20#19128252C2218yes—
21C2×#14581791C2211yescharacteristic
22C2×#14581790C2211yescharacteristic
23C2×#14581789C2211yescharacteristic
24C2×#14581782—21no—
25C2×#14581781—21no—
26C2×#14581772C2211yescharacteristic
27C2×#14581770—21no—
28C2×#14581768C2211yescharacteristic
29C2×#14581767—21no—
30C2×#14581767C2212yes—
31C2×C4×Dic166—21no—
32C2×C4×Dic166C2211yescharacteristic
33C2×#14581754—21no—
34C2×#14581754C2212yes—
35C2×#14581749—21no—
36C2×C42×D83C2211yescharacteristic
37#19127990C22116yescomplemented
38#19127983C2218yescomplemented
39#19127933C22116yescomplemented
40#19127931C22116yescomplemented
41#19127905C22116yescomplemented
42#19127902C2218yescomplemented
43#19127901C2218yescomplemented
44C2×#14581723C2212yes—
45C2×C4⋊(C83⋊C8)C2211yescharacteristic
46C2×Dic83⋊C8C2212yes—
47C4⋊C8×D83C2218yescomplemented
48C2×#14581715C2212yes—
49C2×#14581714C2212yes—
50C2×#14581712C2212yes—
51C2×C4⋊Q8×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53#475×C83C2218yescomplemented
54C2×#106×C83C2212yes—
55C2×C4⋊C8×C83C2211yescharacteristic
56C2×#103×C83C2212yes—
57C2×#13050591—28no—
58C2×#13050591—48no—
59C22×Dic166—21no—
60C22×Dic166C2311yescharacteristic
61C22×D332—22no—
62C22×D332—21no—
63C22×D332—21no—
64C22×D332C2312yes—
65C22×C4×D83—22no—
66C22×C4×D83D411yescharacteristic
67C22×C4×D83D411yescharacteristic
68C22×C4×D83C2311yescharacteristic
69C2×#13050579—44no—
70C2×#13050579C2318yes—
71C2×#13050577—44no—
72C2×#13050577C2318yes—
73C22×C83⋊C8C2312yes—
74C2×C8×D83—44no—
75C2×C8×D83C2318yes—
76#14581791—22no—
77#14581791C2314yes—
78#14581790—22no—
79#14581790C2314yes—
80#14581789—22no—
81#14581789C2314yes—
82#14581782—28no—
83#14581781—28no—
84#14581772—22no—
85#14581772C2314yes—
86#14581770—28no—
87#14581768—22no—
88#14581768C2314yes—
89#14581767—28no—
90#14581767—24no—
91#14581767C2318yes—
92C4×Dic166—28no—
93C4×Dic166—22no—
94C4×Dic166C2314yes—
95#14581754—28no—
96#14581754—24no—
97#14581754C2318yes—
98#14581749—28no—
99C2×C4⋊Dic83—21no—
100C2×C4⋊Dic83C2311yescharacteristic
101C2×Dic83⋊C4—21no—
102C2×Dic83⋊C4—21no—
103C2×Dic83⋊C4—21no—
104C2×Dic83⋊C4—21no—
105C2×Dic83⋊C4—21no—
106C2×(C2×D83)⋊C4—21no—
107C2×(C2×D83)⋊C4—21no—
108C2×(C2×D83)⋊C4—22no—
109C2×(C2×D83)⋊C4—22no—
110C2×C4×Dic83—21no—
111C2×C4×Dic83D411yescharacteristic
112C2×C4×Dic83D411yescharacteristic
113C2×C4×Dic83C2311yescharacteristic
114C42×D83D414yes—
115C42×D83C2314yes—
116#14581723C2318yes—
117C4⋊(C83⋊C8)C2314yes—
118Dic83⋊C8C2318yes—
119#14581715C2318yes—
120#14581714C2318yes—
121#14581712C2318yes—
122C22×Q8×C83—21no—
123C22×Q8×C83C2311yescharacteristic
124C22×C8×C83C2312yes—
125C4⋊Q8×C83—22no—
126C4⋊Q8×C83C2314yes—
127C4×Q8×C83—22no—
128C4×Q8×C83C2314yes—
129C2×C4⋊C4×C83—21no—
130C2×C4⋊C4×C83—21no—
131C2×C4⋊C4×C83C2311yescharacteristic
132C2×C4⋊C4×C83C2312yes—
133C2×C42×C83—21no—
134C2×C42×C83C2311yescharacteristic
135#106×C83C2318yes—
136C4⋊C8×C83C2314yes—
137#103×C83C2318yes—
138C23×D83—22no—
139C23×D83C2×D411yescharacteristic
140#13050591—432no—
141C2×Dic166—22no—
142C2×Dic166—22no—
143C2×Dic166—21no—
144C2×Dic166—24no—
145C2×Dic166—44no—
146C2×Dic166C2412yes—
147C2×D332—24no—
148C2×D332—24no—
149C2×D332—28no—
150C2×D332—22no—
151C2×D332—24no—
152C2×D332—24no—
153C2×D332—44no—
154C2×D332—48no—
155C2×D332—44no—
156C2×D332C2414yes—
157C22×Dic83—21no—
158C22×Dic83—21no—
159C22×Dic83C2×D411yescharacteristic
160C22×Dic83C2×D411yescharacteristic
161C22×Dic83C2×D411yescharacteristic
162C2×C4×D83—22no—
163C2×C4×D83—22no—
164C2×C4×D83—28no—
165C2×C4×D83—42no—
166C2×C4×D83—48no—
167C2×C4×D83SD1618yescomplemented
168C2×C4×D83C2×D412yes—
169C2×C4×D83C2×D414yes—
170C2×C4×D83C2×D414yes—
171C2×C4×D83C2412yes—
172#13050579—48no—
173#13050577—48no—
174C2×C83⋊C8—24no—
175C2×C83⋊C8C2414yes—
176C8×D83—48no—
177C4⋊Dic83—21no—
178C4⋊Dic83—24no—
179C4⋊Dic83C2412yes—
180Dic83⋊C4—24no—
181Dic83⋊C4—24no—
182Dic83⋊C4—24no—
183Dic83⋊C4—24no—
184Dic83⋊C4—24no—
185(C2×D83)⋊C4—24no—
186(C2×D83)⋊C4—24no—
187(C2×D83)⋊C4—28no—
188(C2×D83)⋊C4—28no—
189C4×Dic83—24no—
190C4×Dic83C2×D414yes—
191C4×Dic83C2×D414yes—
192C4×Dic83C2×D412yes—
193C4×Dic83C2412yes—
194C2×Q8×C83—21no—
195C2×Q8×C83—24no—
196C2×Q8×C83—22no—
197C2×Q8×C83—22no—
198C2×Q8×C83—44no—
199C2×Q8×C83C2412yes—
200C22×C4×C83—22no—
201C22×C4×C83—21no—
202C22×C4×C83—21no—
203C22×C4×C83C2×D411yescharacteristic
204C22×C4×C83C2×D411yescharacteristic
205C22×C4×C83C2411yescharacteristic
206C2×C8×C83—24no—
207C2×C8×C83C2414yes—
208C4⋊C4×C83—24no—
209C4⋊C4×C83—22no—
210C4⋊C4×C83—24no—
211C4⋊C4×C83—21no—
212C4⋊C4×C83C2414yes—
213C4⋊C4×C83C2412yes—
214C42×C83—24no—
215C42×C83C2×D412yes—
216C42×C83C2412yes—
217C23×C83C22×D411yescharacteristic
218C22×D83—22no—
219C22×D83—22no—
220C22×D83—24no—
221C22×D83—44no—
222C22×D83—44no—
223C22×D83—44no—
224C22×D83C2×SD1614yes—
225C22×D83C22×D412yes—
226Q8×C83—22no—
227Q8×C83—22no—
228Q8×C83—42no—
229Q8×C83—48no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—24no—
234C2×C4×C83—42no—
235C2×C4×C83—44no—
236C2×C4×C83—42no—
237C2×C4×C83C2×SD1614yes—
238C2×C4×C83C22×D412yes—
239C2×C4×C83C22×D411yescharacteristic
240C2×C4×C83C22×D412yes—
241C2×C4×C83H(5,𝔽2)12yes—
242C2×C4×C832−1+412yes—
243C2×C4×C83C2511yescharacteristic
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—42no—
249C2×Dic83—42no—
250C2×Dic83C2×SD1614yes—
251C2×Dic83C22×D412yes—
252C2×Dic83C22×D412yes—
253C2×Dic83C22×D412yes—
254D332—24no—
255D332—24no—
256D332—48no—
257D332—416no—
258D332—44no—
259D332—48no—
260C4×D83—24no—
261C4×D83—22no—
262C4×D83—22no—
263C4×D83—44no—
264C4×D83—416no—
265C4×D83C2×SD16116yescomplemented
266Dic166—22no—
267Dic166—22no—
268Dic166—48no—
269Dic166—42no—
270C8×C83—24no—
271C83⋊C8—24no—
272C22×C83#53712yes—
273C22×C83C22×SD1612yes—
274C22×C83#57512yes—
275C22×C83C23×D411yescharacteristic
276C2×D83—22no—
277C2×D83—28no—
278C2×D83—22no—
279C2×D83—44no—
280C2×D83—44no—
281C2×D83—44no—
282C2×D83—88no—
283C2×D83C22×SD1614yes—
284C4×C83—22no—
285C4×C83—42no—
286C4×C83—44no—
287C4×C83—42no—
288C4×C83#53712yes—
289C4×C83C22×SD1614yes—
290C4×C83C2×H(5,𝔽2)11yescharacteristic
291C4×C83C2×2−1+411yescharacteristic
292Dic83—22no—
293Dic83—22no—
294Dic83—42no—
295Dic83—42no—
296Dic83C22×SD1614yes—
297C2×#3246—831nomaximal, Sylow
298C2×C83#324614yescomplemented
299C2×C83C2×#53711yescharacteristic
300C2×C83C23×SD1611yescharacteristic
301C2×C83C2×#57511yescharacteristic
302D83—24no—
303D83—84no—
304C2×#533—831no—
305C2×#532—831no—
306#3246—8316no—
307C2×#475—831no—
308C2×#475D8311yescharacteristic, complemented
309C2×#476—832no—
310C2×#465—832no—
311C2×#465—832no—
312C2×#460—831no—
313C2×#460—831no—
314C2×#461—832no—
315C22×C4⋊C8—831no—
316C83C2×#324611yescharacteristic, complemented, Sylow
317C22×#41—1661no—
318C22×SD16—832no—
319C22×SD16—832no—
320C23×C8—832no—
321#533—838no—
322#533—1664no—
323#532—838no—
324#532—1664no—
325C2×C4⋊Q8—831no—
326C2×C4⋊Q8—831no—
327C2×C4⋊Q8C2×D8311yescharacteristic
328C2×#127—831no—
329C2×#125—1661no—
330C2×#124—1661no—
331C2×C22⋊Q8—831no—
332C2×C22⋊Q8—831no—
333C2×C22⋊Q8—1661no—
334C2×#121—832no—
335C2×#121—1661no—
336C2×C4×Q8—831no—
337C2×C4×Q8—1661no—
338C2×C4×Q8C2×D8311yescharacteristic
339C2×C4×D4—832no—
340C2×C4×D4—1661no—
341C2×#117—1661no—
342C22×C42—831no—
343#476—8316no—
344#475—838no—
345#475C2×D8318yescomplemented
346#465—8316no—
347#465—8316no—
348#461—8316no—
349#460—838no—
350#460—838no—
351C2×#106—832no—
352C2×#106C2×D8312yes—
353C2×C4⋊C8—832no—
354C2×C4⋊C8—831no—
355C2×C4⋊C8—838no—
356C2×C4⋊C8C2×D8311yescharacteristic
357C2×#103—832no—
358C2×#103C2×D8312yes—
359C2×#102—832no—
360C2×#102—832no—
361C2×#41—3328no—
362C2×#41—1668no—
363C22×Q8—21no—
364C22×Q8—831no—
365C22×Q8—1661no—
366C22×Q8C22×D8311yescharacteristic
367C22×D4—832no—
368C22×D4—1662no—
369C22×D4—1661no—
370C22×D4—1661no—
371C23×C4—831no—
372C23×C4—831no—
373C23×C4—831no—
374C23×C4—1662no—
375C2×SD16—3324no—
376C2×SD16—3324no—
377C2×SD16—838no—
378C2×SD16—838no—
379C22×C8—3324no—
380C22×C8—838no—
381C22×C8—832no—
382C22×C8C22×D8312yes—
383C4⋊Q8—22no—
384C4⋊Q8—834no—
385C4⋊Q8—834no—
386C4⋊Q8—1662no—
387C4⋊Q8—1662no—
388C4⋊Q8C22×D8314yes—
389#127—834no—
390#127—1662no—
391#125—1668no—
392#124—1668no—
393C22⋊Q8—834no—
394C22⋊Q8—834no—
395C22⋊Q8—1668no—
396C22⋊Q8—1662no—
397C22⋊Q8—1662no—
398#121—838no—
399#121—1664no—
400#121—1668no—
401C4×Q8—22no—
402C4×Q8—834no—
403C4×Q8—1668no—
404C4×Q8—1662no—
405C4×Q8C22×D8314yes—
406C4×D4—838no—
407C4×D4—1664no—
408C4×D4—1668no—
409#117—1668no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—831no—
413C2×C4⋊C4—1661no—
414C2×C4⋊C4—1661no—
415C2×C4⋊C4—1661no—
416C2×C4⋊C4—1661no—
417C2×C4⋊C4—1661no—
418C2×C4⋊C4—1661no—
419C2×C4⋊C4C22×D8312yes—
420C2×C4⋊C4C22×D8311yescharacteristic
421C2×C22⋊C4—1662no—
422C2×C22⋊C4—1662no—
423C2×C22⋊C4—1661no—
424C2×C22⋊C4—1661no—
425C2×C42—21no—
426C2×C42—831no—
427C2×C42—831no—
428C2×C42—831no—
429C2×C42—834no—
430C2×C42—834no—
431C2×C42—1661no—
432C2×C42C22×D8311yescharacteristic
433#106—838no—
434#106C22×D8318yes—
435C4⋊C8—838no—
436C4⋊C8—834no—
437C4⋊C8C22×D8314yes—
438#103—838no—
439#103C22×D8318yes—
440#102—838no—
441#102—838no—
442C24—831no—
443C24—1662no—
444#41—33232no—
445C2×Q8—21no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—44no—
450C2×Q8—3324no—
451C2×Q8—832no—
452C2×Q8—1664no—
453C2×Q8—1662no—
454C2×Q8—1662no—
455C2×Q8—1661no—
456C2×Q8C23×D8312yes—
457C2×D4—3324no—
458C2×D4—3328no—
459C2×D4—3324no—
460C2×D4—834no—
461C2×D4—1668no—
462C2×D4—1664no—
463C2×D4—1662no—
464C2×D4—1664no—
465C2×D4—1664no—
466C2×D4—1664no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—3322no—
471C22×C4—3328no—
472C22×C4—834no—
473C22×C4—831no—
474C22×C4—838no—
475C22×C4—834no—
476C22×C4—831no—
477C22×C4—831no—
478C22×C4—832no—
479C22×C4—832no—
480C22×C4—1668no—
481C22×C4—1661no—
482C22×C4—1661no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4D4×D8311yescharacteristic
486C22×C4D4×D8311yescharacteristic
487C22×C4C23×D8311yescharacteristic
488SD16—3328no—
489SD16—3328no—
490C2×C8—24no—
491C2×C8—3328no—
492C2×C8—834no—
493C2×C8—1664no—
494C2×C8C23×D8314yes—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—832no—
500C4⋊C4—1664no—
501C4⋊C4—1664no—
502C4⋊C4—1664no—
503C4⋊C4—1664no—
504C4⋊C4—1664no—
505C4⋊C4—1661no—
506C4⋊C4—1664no—
507C4⋊C4C23×D8312yes—
508C4⋊C4C23×D8314yes—
509C22⋊C4—1668no—
510C22⋊C4—1668no—
511C22⋊C4—1664no—
512C22⋊C4—1664no—
513C42—24no—
514C42—834no—
515C42—832no—
516C42—834no—
517C42—832no—
518C42—1664no—
519C42D4×D8312yes—
520C42C23×D8312yes—
521C23—3324no—
522C23—3324no—
523C23—3324no—
524C23—834no—
525C23—832no—
526C23—1664no—
527C23—1662no—
528C23—1662no—
529C23C2×D4×D8311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—48no—
533Q8—42no—
534Q8—3328no—
535Q8—3322no—
536Q8—1662no—
537Q8—1662no—
538D4—3328no—
539D4—33216no—
540D4—3328no—
541D4—3324no—
542D4—1664no—
543D4—1664no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—3324no—
552C2×C4—33216no—
553C2×C4—3322no—
554C2×C4—3322no—
555C2×C4—834no—
556C2×C4—832no—
557C2×C4—8316no—
558C2×C4—832no—
559C2×C4—832no—
560C2×C4—1662no—
561C2×C4—1662no—
562C2×C4—1662no—
563C2×C4—1662no—
564C2×C4—1662no—
565C2×C4—1662no—
566C2×C4—1664no—
567C2×C4SD16×D8314yes—
568C2×C4C2×D4×D8312yes—
569C2×C4C2×D4×D8312yes—
570C2×C4C2×D4×D8311yescharacteristic
571C2×C4#1458183812yes—
572C2×C4#1458184012yes—
573C2×C4C24×D8311yescharacteristic
574C8—24no—
575C8—1664no—
576C22—3324no—
577C22—3324no—
578C22—3324no—
579C22—834no—
580C22—6648no—
581C22—1662no—
582C22—1662no—
583C22—1668no—
584C22#1912828112yes—
585C22C2×SD16×D8312yes—
586C22#1912850012yes—
587C22C22×D4×D8311yescharacteristic
588C4—22no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—3322no—
593C4—3322no—
594C4—834no—
595C4—1662no—
596C4—1662no—
597C4#1912828112yes—
598C4C2×SD16×D8314yes—
599C4C2×#1458183811yescharacteristic
600C4C2×#1458184011yescharacteristic
601C2—6644no—
602C2—1664no—
603C2#2491877814yesdirect product factor
604C2C2×#1912828111yescharacteristic
605C2C22×SD16×D8311yescharacteristic
606C2C2×#1912850011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#475×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128252C2
hypercenterC2×#475D83
hypocenterC83C2×#3246
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
1. C2×#19128255C2semidirect
3. #24918778C2direct (N × H)
4. C2×#19127983C2semidirect
5. C2×#19127990C2semidirect
6. C2×#19127933C2semidirect
7. C2×#19127931C2semidirect
8. C2×#19127902C2semidirect
9. C2×#19127901C2semidirect
10. C2×#19127905C2semidirect
11. C2×C4⋊C8×D83C2semidirect
12. C2×#475×C83C2semidirect
18. #19128255C22semidirect
37. #19127990C22semidirect
38. #19127983C22semidirect
39. #19127933C22semidirect
40. #19127931C22semidirect
41. #19127905C22semidirect
42. #19127902C22semidirect
43. #19127901C22semidirect
47. C4⋊C8×D83C22semidirect
53. #475×C83C22semidirect
167. C2×C4×D83SD16semidirect
265. C4×D83C2×SD16semidirect
298. C2×C83#3246semidirect
308. C2×#475D83semidirect
316. C83C2×#3246semidirect
345. #475C2×D83semidirect
603. C2#24918778direct (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×#191282522
#249187782
C2×#191279832
C2×#191279902
C2×#191279332
C2×#191279312
C2×#191279022
C2×#191279012
C2×#191279052
C2×C4⋊C8×D832
C2×#475×C832
C2×#324683