Finite Group Databasebeta

← C2×#21446204 #29681562

Subgroups — C2×#21446204 #29681562

Order 15616.

1966 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
1C22×#14519766C211yesmaximal, characteristic
2C2×#16840236C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4C22×#14519693C211yesmaximal, characteristic
5#21446204C218yesmaximal, direct product factor
6C2×#16839523C211yesmaximal, characteristic, complemented
7C2×#16839521C211yesmaximal, characteristic, complemented
8C2×C61×#419C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C2×#14519766—22no—
12C2×#14519766—24no—
13C2×#14519766C418yescomplemented
14C2×#14519766C2214yes—
15#16840236—22no—
16#16840236—22no—
17#16840236—22no—
18#16840236C2214yes—
19C2×C4×D244—22no—
20C2×#14519739—22no—
21C2×#14519736C2211yescharacteristic
22C22×#11758198—22no—
23C2×C22⋊C4×D61—22no—
24C22×C4×Dic61C2211yescharacteristic
25#16839771—22no—
26#16839771C2214yescomplemented
27C2×#14519701—22no—
28C2×#14519693—24no—
29C2×#14519693C2214yes—
30#16839523C2214yescomplemented
31#16839521C2214yescomplemented
32C23×Q8×C61C2211yescharacteristic
33C22×M16×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C61×#419—22no—
36C61×#419C2214yescomplemented
37C2×#103×C61—22no—
38C24×D61D412yes—
39C22×D244—21no—
40C22×D244—21no—
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244C2×C414yes—
46C22×D244D412yes—
47C22×D244D414yes—
48C22×D244C2312yes—
49C23×Dic61—21no—
50C22×C4×D61—24no—
51C2×#11758194—21no—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C2×#11758194C2312yes—
55C22×#92418—21no—
56#14519766—216no—
57#14519766—28no—
58#14519766C2×C4116yescomplemented
59C4×D244—28no—
60C4×D244—28no—
61#14519739—28no—
62#14519739—28no—
63#14519736—21no—
64#14519736—21no—
65#14519736D412yes—
66#14519736C2312yes—
67C2×C4⋊Dic61D412yes—
68C2×C22⋊Dic61—21no—
69C2×#11758198—24no—
70C2×#11758198—216no—
71C2×#11758198—28no—
72C22⋊C4×D61—28no—
73C22⋊C4×D61—28no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61C2×C414yescomplemented
78C2×C4×Dic61C2312yes—
79#14519701—28no—
80#14519693—28no—
81C22×Q8×C61—22no—
82C22×Q8×C61—21no—
83C22×Q8×C61—21no—
84C22×Q8×C61—21no—
85C22×Q8×C61—21no—
86C22×Q8×C61—21no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61C2×C414yescomplemented
90C22×Q8×C61C2312yes—
91C23×C4×C61—21no—
92C23×C4×C61C2311yescharacteristic
93C2×M16×C61—21no—
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61C2312yes—
97C22×C8×C61—21no—
98#117×C61—21no—
99#117×C61—21no—
100#117×C61D412yes—
101#117×C61C2312yes—
102C2×C4⋊C4×C61D412yes—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61—21no—
105#103×C61—28no—
106C24×C61C2×D411yescharacteristic
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—24no—
110C23×D61—48no—
111C23×D61C22⋊C418yes—
112C23×D61C2×D414yes—
113C2×Q8×C61—24no—
114C2×Q8×C61—22no—
115C2×Q8×C61—24no—
116C2×Q8×C61—24no—
117C2×Q8×C61—24no—
118C2×Q8×C61—24no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—41no—
124C2×Q8×C61—41no—
125C2×Q8×C61—42no—
126C2×Q8×C61—42no—
127C2×Q8×C61—42no—
128C2×Q8×C61—42no—
129C2×Q8×C61—42no—
130C2×Q8×C61C22×C414yescomplemented
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—41no—
135C22×C4×C61—41no—
136C22×C4×C61—42no—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61#4112yes—
145C22×C4×C61#4112yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—24no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—44no—
151C2×D244—24no—
152C2×D244—24no—
153C2×D244—24no—
154C2×D244—24no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—24no—
158C2×D244—24no—
159C2×D244—24no—
160C2×D244—24no—
161C2×D244—24no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244C22⋊C418yes—
167C2×D244C22×C414yes—
168C2×D244C2×D414yes—
169C2×C4×D61—216no—
170C2×C4×D61—88no—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—42no—
174C2×C8×C61—22no—
175C2×C8×C61—42no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61C2×D414yes—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—22no—
181C42×C61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184#11758198—216no—
185#11758198—232no—
186C4⋊Dic61—22no—
187C4⋊Dic61C2×D414yes—
188C4×Dic61—24no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61C22×C414yescomplemented
193#11758194—22no—
194#11758194—22no—
195#11758194—42no—
196C2×#92418—22no—
197C2×#92418—42no—
198C23×C61—24no—
199C23×C61C2×C22⋊C412yes—
200C23×C61#12011yescharacteristic
201C23×C61#12011yescharacteristic
202C23×C61#12312yes—
203C23×C61C22×D411yescharacteristic
204C22×D61—24no—
205C22×D61—28no—
206C22×D61—24no—
207C22×D61—28no—
208C22×D61—48no—
209C22×D61—416no—
210C22×D61—48no—
211C22×D61—88no—
212C22×D61—84no—
213C22×D61C2×C22⋊C418yes—
214Q8×C61—24no—
215Q8×C61—24no—
216Q8×C61—42no—
217Q8×C61—42no—
218Q8×C61—44no—
219Q8×C61—44no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—44no—
225C2×C4×C61—44no—
226C2×C4×C61—44no—
227C2×C4×C61C2×C22⋊C412yes—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61C4×D412yes—
231C2×C4×C61C4×D412yes—
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12112yes—
235C2×C4×C61#12112yes—
236C2×C4×C61#12112yes—
237C2×C4×C61#12112yes—
238C2×C4×C61#12312yes—
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—48no—
248C2×Dic61—44no—
249C2×Dic61—44no—
250D244—24no—
251D244—24no—
252D244—24no—
253D244—24no—
254D244—28no—
255D244—28no—
256C4×D61—816no—
257C8×C61—42no—
258#92418—42no—
259C2×#1885—611nomaximal, Sylow
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61#38612yes—
265C22×C61#38612yes—
266C22×C61#45514yes—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61C2×#12011yescharacteristic
269C22×C61C2×#12311yescharacteristic
270C2×D61—28no—
271C2×D61—416no—
272C2×D61—48no—
273C2×D61—48no—
274C2×D61—84no—
275C2×D61—88no—
276C4×C61—44no—
277C4×C61—44no—
278C4×C61#38612yes—
279C4×C61#38612yes—
280C4×C61C2×C4×D411yescharacteristic
281C4×C61C2×C4×D411yescharacteristic
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×#12111yescharacteristic
284Dic61—44no—
285Dic61—44no—
286C22×#124—611no—
287C2×#518—611no—
288C2×#419—611no—
289C2×#419D6111yescharacteristic, complemented
290C22×#100—611no—
291#1885—618no—
292C2×#339—611no—
293C2×#339—611no—
294C2×C61—21no—
295C2×C61—21no—
296C2×C61—22no—
297C2×C61#188514yescomplemented
298C2×C61C2×#38611yescharacteristic
299C2×C61C2×#45512yes—
300D61—48no—
301C23×Q8C2×D6111yescharacteristic
302C23×D4—611no—
303C22×M16—611no—
304C22×M16C2×D6111yescharacteristic
305C2×#124—618no—
306C2×#124—614no—
307C2×#124—1222no—
308C2×#124—1224no—
309#518—614no—
310#518—1222no—
311#518—1222no—
312#518—1222no—
313C2×C4×D4—1222no—
314C2×C4×D4—1222no—
315C2×#117—611no—
316C2×#117C2×D6111yescharacteristic
317C22×C22⋊C4—1222no—
318C22×C22⋊C4—1222no—
319C22×C42—611no—
320#419—22no—
321#419—614no—
322#419—1222no—
323#419C2×D6114yescomplemented
324C2×#103—22no—
325C2×#103—1222no—
326C2×#100—614no—
327C2×#100—1224no—
328#339—614no—
329#339—614no—
330C61C2×#188511yescharacteristic, complemented, Sylow
331C25—612no—
332C22×Q8—21no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—21no—
336C22×Q8—21no—
337C22×Q8—22no—
338C22×Q8—21no—
339C22×Q8—21no—
340C22×Q8C4×D6114yescomplemented
341C22×Q8C22×D6112yes—
342C22×D4—614no—
343C22×D4—612no—
344C22×D4—614no—
345C22×D4—612no—
346C22×D4—1222no—
347C22×D4—1221no—
348C22×D4—1221no—
349C22×D4—1222no—
350C22×D4—1221no—
351C22×D4—1221no—
352C23×C4—21no—
353C23×C4—1224no—
354C23×C4—1221no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—21no—
357C2×M16—22no—
358C2×M16—22no—
359C2×M16—612no—
360C2×M16—1221no—
361C2×M16—1222no—
362C2×M16—1222no—
363C2×M16C22×D6112yes—
364C22×C8—21no—
365C22×C8—1221no—
366#124—6116no—
367#124—1228no—
368#124—12216no—
369C4×D4—1228no—
370C4×D4—1228no—
371C4×D4—1228no—
372C4×D4—1228no—
373#117—21no—
374#117—21no—
375#117—612no—
376#117—612no—
377#117—1221no—
378#117—1221no—
379#117C22⋊D6112yes—
380#117C22×D6112yes—
381C2×C4⋊C4—612no—
382C2×C4⋊C4C22⋊D6112yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1228no—
385C2×C22⋊C4—1228no—
386C2×C22⋊C4—1228no—
387C2×C22⋊C4—12216no—
388C2×C22⋊C4—1224no—
389C2×C22⋊C4—1221no—
390C2×C42—21no—
391C2×C42—614no—
392C2×C42—612no—
393C2×C42—1222no—
394C2×C42—1221no—
395C2×C42—1221no—
396#103—28no—
397#103—1228no—
398#100—1228no—
399C24—614no—
400C24—618no—
401C24—1222no—
402C24—1224no—
403C24—1222no—
404C24—2448no—
405C24D4×D6111yescharacteristic
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8C2×C4×D6114yescomplemented
424C2×D4—614no—
425C2×D4—614no—
426C2×D4—618no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1224no—
432C2×D4—1224no—
433C2×D4—1224no—
434C2×D4—1224no—
435C2×D4—1222no—
436C2×D4—1222no—
437C2×D4—1224no—
438C2×D4—1224no—
439C2×D4—1224no—
440C2×D4—1224no—
441C2×D4—1224no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—12216no—
449C22×C4—1222no—
450C22×C4—1224no—
451C22×C4—1222no—
452C22×C4—4888no—
453C22×C4—2444no—
454C22×C4C2×C4×D6112yes—
455C22×C4#1175822212yes—
456C22×C4D4×D6112yes—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460C22×C4#1175822412yes—
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464M16—22no—
465M16—22no—
466M16—42no—
467M16—1222no—
468M16—1222no—
469M16—2442no—
470C2×C8—22no—
471C2×C8—42no—
472C2×C8—1222no—
473C2×C8—2442no—
474C4⋊C4—22no—
475C4⋊C4—614no—
476C4⋊C4—1222no—
477C4⋊C4C2×C22⋊D6114yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—12232no—
481C22⋊C4—12216no—
482C22⋊C4—1222no—
483C22⋊C4—1222no—
484C42—22no—
485C42—22no—
486C42—614no—
487C42—1222no—
488C42—1222no—
489C42—1224no—
490C42—1222no—
491C23—24no—
492C23—618no—
493C23—1224no—
494C23—1228no—
495C23—1224no—
496C23—1228no—
497C23—4884no—
498C23—4888no—
499C23—2448no—
500C23—24416no—
501C23—2448no—
502C23C22⋊C4×D6112yes—
503C23#1451974511yescharacteristic
504C23#1451974511yescharacteristic
505C23#1451975912yes—
506C23C2×D4×D6111yescharacteristic
507Q8—24no—
508Q8—24no—
509Q8—42no—
510Q8—42no—
511Q8—44no—
512Q8—44no—
513D4—1224no—
514D4—1224no—
515D4—1224no—
516D4—1224no—
517D4—1228no—
518D4—1228no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—48816no—
529C2×C4—2444no—
530C2×C4—2448no—
531C2×C4—2444no—
532C2×C4C22⋊C4×D6112yes—
533C2×C4#1451973712yes—
534C2×C4#1451973712yes—
535C2×C4C4×C22⋊D6112yes—
536C2×C4C4×C22⋊D6112yes—
537C2×C4#1451974511yescharacteristic
538C2×C4#1451974511yescharacteristic
539C2×C4#1451974812yes—
540C2×C4#1451974812yes—
541C2×C4#1451975012yes—
542C2×C4#1451975012yes—
543C2×C4#1451975912yes—
544C2×C4C22×C4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4C2×#1175822411yescharacteristic
549C2×C4C2×#1175822211yescharacteristic
550C8—42no—
551C8—2442no—
552C22—22no—
553C22—22no—
554C22—24no—
555C22—24no—
556C22—1228no—
557C22—4884no—
558C22—4888no—
559C22—2448no—
560C22—2448no—
561C22—24416no—
562C22#1683963812yes—
563C22#1683963812yes—
564C22#1683993814yes—
565C22C2×C22⋊C4×D6111yescharacteristic
566C22C2×#1451974511yescharacteristic
567C22C2×#1451975911yescharacteristic
568C4—44no—
569C4—44no—
570C4—2444no—
571C4—2444no—
572C4#1683963812yes—
573C4#1683963812yes—
574C4C2×#1451973711yescharacteristic
575C4C2×C4×C22⋊D6111yescharacteristic
576C4C2×#1451974811yescharacteristic
577C4C2×#1451975011yescharacteristic
578C2—21no—
579C2—21no—
580C2—22no—
581C2—2448no—
582C2#2144620414yesdirect product factor
583C2C2×#1683963811yescharacteristic
584C2C2×#1683993812yes—
subgroupG/H
centerC23#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61#120
prime radicalC23×D244C22
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61C2×#1885
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
5. #21446204C2direct (N × H)
6. C2×#16839523C2semidirect
7. C2×#16839521C2semidirect
8. C2×C61×#419C2semidirect
13. C2×#14519766C4semidirect
26. #16839771C22semidirect
30. #16839523C22semidirect
31. #16839521C22semidirect
36. C61×#419C22semidirect
58. #14519766C2×C4semidirect
77. C2×C4×Dic61C2×C4semidirect
89. C22×Q8×C61C2×C4semidirect
130. C2×Q8×C61C22×C4semidirect
192. C4×Dic61C22×C4semidirect
289. C2×#419D61semidirect
297. C2×C61#1885semidirect
323. #419C2×D61semidirect
330. C61C2×#1885semidirect
340. C22×Q8C4×D61semidirect
423. C2×Q8C2×C4×D61semidirect
582. C2#21446204direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197662
C2×#168402362
C2×#168397712
C22×#145196932
#214462042
C2×#168395232
C2×#168395212
C2×C61×#4192
C2×#188561