Finite Group Databasebeta

← C2×#21446128 #29681216

Subgroups — C2×#21446128 #29681216

Order 15616.

1966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519753C211yesmaximal, characteristic
2C22×#14519735C211yesmaximal, characteristic
3C22×#14519701C211yesmaximal, characteristic, complemented
4C22×#14519689C211yesmaximal, characteristic
5#21446128C218yesmaximal, direct product factor
6C2×#16839525C211yesmaximal, characteristic, complemented
7C2×#16839524C211yesmaximal, characteristic, complemented
8C22×#103×C61C211yesmaximal, characteristic, complemented
9C23×C4×D61C2211yescharacteristic
10C23×#92418C2211yescharacteristic
11C2×#14519753—22no—
12C2×#14519753—24no—
13C2×#14519753C418yescomplemented
14C2×#14519753C2214yes—
15C2×#14519735—24no—
16C2×#14519735—24no—
17C2×#14519735—22no—
18C2×#14519735C2214yes—
19C22×C4⋊Dic61C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758196—21no—
22C22×#11758198—21no—
23C22×#11758198—21no—
24C22×C4×Dic61—21no—
25C2×#14519701—24no—
26C2×#14519701C2214yescomplemented
27C2×#14519689—24no—
28C2×#14519689C2214yes—
29#16839525C2214yescomplemented
30#16839524C2214yescomplemented
31C23×Q8×C61C2211yescharacteristic
32C23×C8×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C2×#103×C61—24no—
35C2×#103×C61C2214yescomplemented
36C24×D61D411yescharacteristic
37C23×Dic61—21no—
38C23×Dic61—21no—
39C23×Dic61D411yescharacteristic
40C22×C4×D61—22no—
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61C2×C414yes—
45C22×C4×D61D412yes—
46C22×C4×D61D414yes—
47C22×C4×D61C2312yes—
48C22×#92418—22no—
49C22×#92418—22no—
50C22×#92418—22no—
51C22×#92418C2312yes—
52#14519753—216no—
53#14519753—28no—
54#14519753C2×C4116yescomplemented
55#14519735—216no—
56#14519735—28no—
57#14519735—28no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61C2×C414yes—
63C2×C4⋊Dic61D414yes—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—28no—
67C2×#11758196—24no—
68C2×#11758198—28no—
69C2×#11758198—28no—
70C2×#11758198—24no—
71C2×#11758198—24no—
72C2×C4×Dic61—28no—
73C2×C4×Dic61—24no—
74#14519701—28no—
75#14519689—28no—
76C22×Q8×C61—22no—
77C22×Q8×C61—21no—
78C22×Q8×C61—22no—
79C22×Q8×C61—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61—22no—
82C22×Q8×C61C2×C414yes—
83C22×Q8×C61C2312yes—
84C23×C4×C61—21no—
85C23×C4×C61—21no—
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61—22no—
88C22×C8×C61—22no—
89C22×C8×C61—22no—
90C22×C8×C61C2312yes—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61D414yes—
94C2×C4⋊C4×C61C2312yes—
95#103×C61—28no—
96C24×C61C2×D411yescharacteristic
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61—22no—
103C23×D61C22⋊C414yes—
104C23×D61C2×D412yes—
105C2×Q8×C61—24no—
106C2×Q8×C61—22no—
107C2×Q8×C61—28no—
108C2×Q8×C61—28no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—42no—
114C2×Q8×C61—44no—
115C2×Q8×C61—44no—
116C2×Q8×C61—42no—
117C2×Q8×C61C22×C414yes—
118C22×C4×C61—24no—
119C22×C4×C61—24no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—24no—
137C22×Dic61—24no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61C22⋊C414yes—
145C22×Dic61C2×D412yes—
146C2×C4×D61—28no—
147C2×C4×D61—28no—
148C2×C4×D61—24no—
149C2×C4×D61—28no—
150C2×C4×D61—28no—
151C2×C4×D61—24no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61C22⋊C418yes—
157C2×C4×D61C22×C414yes—
158C2×C4×D61C2×D414yes—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—44no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61C2×D414yes—
166#11758198—216no—
167#11758198—216no—
168C4⋊Dic61—24no—
169C4⋊Dic61—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61C22×C414yes—
173C4⋊Dic61C2×D414yes—
174#11758196—216no—
175C4×Dic61—216no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—44no—
180C23×C61C2×C22⋊C412yes—
181C23×C61#12012yes—
182C23×C61#12312yes—
183C23×C61#13512yes—
184C23×C61#13512yes—
185C23×C61#13712yes—
186C23×C61#13712yes—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—28no—
195C22×D61—28no—
196C22×D61—28no—
197C22×D61—44no—
198C22×D61—44no—
199C22×D61—44no—
200C22×D61—44no—
201C22×D61C2×C22⋊C414yes—
202Q8×C61—24no—
203Q8×C61—24no—
204Q8×C61—42no—
205Q8×C61—42no—
206Q8×C61—48no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—48no—
214C2×C4×C61—48no—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61C4×D414yes—
217C2×C4×C61C4×D414yes—
218C2×C4×C61#12012yes—
219C2×C4×C61#12114yes—
220C2×C4×C61#12114yes—
221C2×C4×C61#12312yes—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—28no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—48no—
238C2×Dic61—48no—
239C2×Dic61C2×C22⋊C414yes—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—24no—
243C4×D61—24no—
244C4×D61—216no—
245C8×C61—42no—
246#92418—42no—
247C2×#1873—611nomaximal, Sylow
248C22×C61#38614yes—
249C22×C61#41614yes—
250C22×C61#41914yes—
251C22×C61#45214yes—
252C22×C61#45214yes—
253C22×C61#48514yes—
254C22×C61#48514yes—
255C22×C61C22×C22⋊C411yescharacteristic
256C22×C61C2×#12011yescharacteristic
257C22×C61C2×#12311yescharacteristic
258C22×C61C2×#13511yescharacteristic
259C22×C61C2×#13511yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C22×C61C2×#13711yescharacteristic
262C2×D61—24no—
263C2×D61—24no—
264C2×D61—24no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—416no—
268C2×D61—42no—
269C2×D61—42no—
270C4×C61—44no—
271C4×C61—44no—
272C4×C61#38614yes—
273C4×C61C2×C4×D411yescharacteristic
274C4×C61C2×C4×D411yescharacteristic
275C4×C61C2×#12111yescharacteristic
276C4×C61C2×#12111yescharacteristic
277Dic61—24no—
278Dic61—44no—
279Dic61—44no—
280C22×C22⋊Q8—611no—
281C22×#117—611no—
282C22×#103—611no—
283C22×#103D6111yescharacteristic, complemented
284C22×C22⋊C8—611no—
285#1873—618no—
286C2×#340—611no—
287C2×#340—611no—
288C2×C61#187318yescomplemented
289C2×C61C2×#38611yescharacteristic
290C2×C61C2×#41611yescharacteristic
291C2×C61C2×#41911yescharacteristic
292C2×C61C2×#45211yescharacteristic
293C2×C61C2×#45211yescharacteristic
294C2×C61C2×#48511yescharacteristic
295C2×C61C2×#48511yescharacteristic
296D61—44no—
297C23×Q8C2×D6111yescharacteristic
298C24×C4—611no—
299C23×C8—611no—
300C23×C8C2×D6111yescharacteristic
301C2×C22⋊Q8—618no—
302C2×C22⋊Q8—614no—
303C2×C22⋊Q8—1222no—
304C2×C22⋊Q8—1224no—
305C2×#117—614no—
306C2×#117—1224no—
307C2×#117—1222no—
308C2×#117—1224no—
309C22×C4⋊C4—611no—
310C22×C4⋊C4—611no—
311C22×C4⋊C4—1221no—
312C22×C4⋊C4C2×D6111yescharacteristic
313C22×C22⋊C4—1221no—
314C22×C22⋊C4—1221no—
315C22×C42—1221no—
316C2×#103—24no—
317C2×#103—614no—
318C2×#103—1224no—
319C2×#103C2×D6114yescomplemented
320C2×C22⋊C8—614no—
321C2×C22⋊C8—1224no—
322#340—614no—
323#340—614no—
324C61C2×#187311yescharacteristic, complemented, Sylow
325C25—611no—
326C22×Q8—21no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8C4×D6114yes—
333C22×Q8C22×D6112yes—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—611no—
337C23×C4—614no—
338C23×C4—612no—
339C23×C4—614no—
340C23×C4—612no—
341C23×C4—1222no—
342C23×C4—1222no—
343C23×C4—1222no—
344C23×C4—1222no—
345C23×C4—1221no—
346C23×C4—1221no—
347C23×C4C22×D6111yescharacteristic
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—612no—
352C22×C8—1222no—
353C22×C8—1222no—
354C22×C8—1222no—
355C22×C8C22×D6112yes—
356C22⋊Q8—6116no—
357C22⋊Q8—1228no—
358C22⋊Q8—12216no—
359#117—1228no—
360#117—12216no—
361#117—1228no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—614no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4C22⋊D6114yes—
375C2×C4⋊C4C22×D6112yes—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—1228no—
379C2×C22⋊C4—1228no—
380C2×C42—1224no—
381C2×C42—1228no—
382#103—28no—
383#103—1228no—
384C22⋊C8—1228no—
385C24—612no—
386C24—614no—
387C24—1222no—
388C24—1222no—
389C24—1222no—
390C24—1222no—
391C24—1222no—
392C24—1222no—
393C24D4×D6111yescharacteristic
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8C2×C4×D6114yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—612no—
416C22×C4—614no—
417C22×C4—614no—
418C22×C4—614no—
419C22×C4—618no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1228no—
427C22×C4—1228no—
428C22×C4—1224no—
429C22×C4—1228no—
430C22×C4—1228no—
431C22×C4—1224no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1224no—
435C22×C4—1224no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4C2×C4×D6112yes—
441C22×C4#1175822212yes—
442C22×C4D4×D6112yes—
443C22×C4D4×D6112yes—
444C22×C4D4×D6111yescharacteristic
445C22×C4#1175822412yes—
446C22×C4C2×C22⋊D6112yes—
447C22×C4C23×D6111yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—2444no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—614no—
459C4⋊C4—614no—
460C4⋊C4—12216no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4C2×C22⋊D6114yes—
466C22⋊C4—12216no—
467C22⋊C4—12216no—
468C42—12216no—
469C23—614no—
470C23—1222no—
471C23—1222no—
472C23—1228no—
473C23—1228no—
474C23—1222no—
475C23—1222no—
476C23—1222no—
477C23—1222no—
478C23—1228no—
479C23—2444no—
480C23—2444no—
481C23—2444no—
482C23—2444no—
483C23C22⋊C4×D6112yes—
484C23#1451974512yes—
485C23#1451975912yes—
486C23#1451979812yes—
487C23#1451980012yes—
488C23#1451980612yes—
489C23#1451980912yes—
490C23C2×D4×D6111yescharacteristic
491Q8—24no—
492Q8—24no—
493Q8—42no—
494Q8—42no—
495Q8—48no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—614no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1228no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—12216no—
513C2×C4—1224no—
514C2×C4—1224no—
515C2×C4—2448no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2448no—
521C2×C4C22⋊C4×D6112yes—
522C2×C4#1451973714yes—
523C2×C4C4×C22⋊D6114yes—
524C2×C4#1451974512yes—
525C2×C4#1451974814yes—
526C2×C4#1451975014yes—
527C2×C4#1451975912yes—
528C2×C4C22×C4×D6111yescharacteristic
529C2×C4C2×D4×D6111yescharacteristic
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4C2×#1175822411yescharacteristic
533C2×C4C2×#1175822211yescharacteristic
534C8—42no—
535C8—2442no—
536C22—1224no—
537C22—1224no—
538C22—1224no—
539C22—2442no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—24416no—
544C22#1683963814yes—
545C22#1683975714yes—
546C22#1683976914yes—
547C22#1683992114yes—
548C22#1683992314yes—
549C22#1684010114yes—
550C22#1684010214yes—
551C22C2×C22⋊C4×D6111yescharacteristic
552C22C2×#1451974511yescharacteristic
553C22C2×#1451975911yescharacteristic
554C22C2×#1451979811yescharacteristic
555C22C2×#1451980011yescharacteristic
556C22C2×#1451980611yescharacteristic
557C22C2×#1451980911yescharacteristic
558C4—44no—
559C4—44no—
560C4—1224no—
561C4—2444no—
562C4—2444no—
563C4#1683963814yes—
564C4C2×#1451973711yescharacteristic
565C4C2×C4×C22⋊D6111yescharacteristic
566C4C2×#1451974811yescharacteristic
567C4C2×#1451975011yescharacteristic
568C2—2444no—
569C2#2144612818yesdirect product factor
570C2C2×#1683963811yescharacteristic
571C2C2×#1683975711yescharacteristic
572C2C2×#1683976911yescharacteristic
573C2C2×#1683992111yescharacteristic
574C2C2×#1683992311yescharacteristic
575C2C2×#1684010111yescharacteristic
576C2C2×#1684010211yescharacteristic
subgroupG/H
centerC24D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC24×C61C2×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC22×#103D61
hypocenterC61C2×#1873
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#14519701C2semidirect
5. #21446128C2direct (N × H)
6. C2×#16839525C2semidirect
7. C2×#16839524C2semidirect
8. C22×#103×C61C2semidirect
13. C2×#14519753C4semidirect
26. C2×#14519701C22semidirect
29. #16839525C22semidirect
30. #16839524C22semidirect
35. C2×#103×C61C22semidirect
54. #14519753C2×C4semidirect
283. C22×#103D61semidirect
288. C2×C61#1873semidirect
319. C2×#103C2×D61semidirect
324. C61C2×#1873semidirect
569. C2#21446128direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197532
C22×#145197352
C22×#145197012
C22×#145196892
#214461282
C2×#168395252
C2×#168395242
C22×#103×C612
C2×#187361