Finite Group Databasebeta

← C2×#21444360 #29647266

Subgroups — C2×#21444360 #29647266

Order 15616.

1510 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2C2×#16839726C211yesmaximal, characteristic, complemented
3#21444360C218yesmaximal, direct product factor
4C2×#16839498C212yesmaximal, complemented
5C2×#16839495C212yesmaximal, complemented
6C2×C61×#380C211yesmaximal, characteristic, complemented
7C22×#11758222C411yescharacteristic, complemented
8C22×#11758194—21no—
9#16840273—22no—
10#16840273—22no—
11#16840273C414yescomplemented
12#16840273C2214yes—
13C2×#14519760—21no—
14C2×#14519756C411yescharacteristic, complemented
15C2×#14519756C2211yescharacteristic
16C2×#14519751C411yescharacteristic, complemented
17C2×#14519751C2211yescharacteristic
18C2×#14519746—21no—
19C2×C4×C22⋊D61—21no—
20C2×C4×C22⋊D61—21no—
21C2×#14519736C411yescharacteristic, complemented
22#16839726—22no—
23#16839726C2214yescomplemented
24C2×C22⋊#92418—21no—
25C2×C22⋊#92418C2212yes—
26#16839498C2218yescomplemented
27#16839495C2218yescomplemented
28C24×C4×C61C2211yescharacteristic
29C22×C4⋊C4×C61—21no—
30C22×C22⋊C4×C61—21no—
31C61×#380—22no—
32C61×#380C2214yescomplemented
33C2×#95×C61C2212yes—
34C2×#11758222—22no—
35C2×#11758222—44no—
36C2×#11758222C2×C414yescomplemented
37C22×Dic122C2×C411yescharacteristic
38C22×D244C2×C411yescharacteristic
39C22×C22⋊D61—21no—
40C22×C22⋊D61—21no—
41C22×C22⋊D61—21no—
42C22×C4×D61—21no—
43C2×#11758194—24no—
44C2×#11758194—44no—
45C22×#92418—22no—
46#14519760—24no—
47#14519760—24no—
48#14519756—22no—
49#14519756—21no—
50#14519756—21no—
51#14519756C2×C412yes—
52#14519756C2×C414yescomplemented
53#14519756C2312yes—
54#14519751—22no—
55#14519751—21no—
56#14519751—21no—
57#14519751C2×C412yes—
58#14519751C2×C414yescomplemented
59#14519751C2312yes—
60#14519746—24no—
61#14519746—24no—
62C4×C22⋊D61—24no—
63C4×C22⋊D61—24no—
64C4×C22⋊D61—24no—
65C4×C22⋊D61—24no—
66#14519736—22no—
67#14519736C2×C414yescomplemented
68C2×C4⋊Dic61C2×C411yescharacteristic
69C2×C4⋊Dic61C2×C411yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—21no—
78C22⋊#92418—24no—
79C22⋊#92418—22no—
80C22⋊#92418C2314yes—
81C25×C61D411yescharacteristic
82C23×C4×C61—22no—
83C23×C4×C61—44no—
84C23×C4×C61C2×C411yescharacteristic
85C23×C4×C61C2×C411yescharacteristic
86C23×C4×C61C2×C412yes—
87C23×C4×C61D411yescharacteristic
88C23×C4×C61D411yescharacteristic
89C23×C4×C61D412yes—
90C23×C4×C61D411yescharacteristic
91C23×C4×C61D412yes—
92C23×C4×C61C2311yescharacteristic
93C23×C4×C61C2312yes—
94C2×C4⋊C4×C61—24no—
95C2×C4⋊C4×C61—44no—
96C2×C22⋊C4×C61—24no—
97C2×C22⋊C4×C61—44no—
98#95×C61—22no—
99#95×C61C2314yes—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—42no—
103C24×C61—42no—
104C24×C61C22⋊C411yescharacteristic
105C24×C61C22⋊C411yescharacteristic
106C24×C61C22⋊C412yes—
107C24×C61C2×D411yescharacteristic
108C24×C61C2×D412yes—
109C23×D61—21no—
110C22×C4×C61—24no—
111C22×C4×C61—24no—
112C22×C4×C61—24no—
113C22×C4×C61—24no—
114C22×C4×C61—44no—
115C22×C4×C61—44no—
116C22×C4×C61—44no—
117C22×C4×C61—44no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—44no—
122C22×C4×C61—44no—
123C22×C4×C61—44no—
124C22×C4×C61C22⋊C412yes—
125C22×C4×C61C22⋊C411yescharacteristic
126C22×C4×C61C22⋊C412yes—
127C22×C4×C61C22⋊C412yes—
128C22×C4×C61C22⋊C412yes—
129C22×C4×C61C22⋊C412yes—
130C22×C4×C61C22⋊C411yescharacteristic
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C412yes—
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—24no—
143C2×C22⋊D61—22no—
144C2×C22⋊D61—24no—
145C2×C22⋊D61—22no—
146C2×C22⋊D61—44no—
147C2×C22⋊D61—44no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153#11758222—48no—
154#11758222—48no—
155C2×D244—21no—
156C2×D244—42no—
157C2×D244C22×C412yes—
158C2×C4×D61—24no—
159C2×C4×D61—44no—
160C2×Dic122—21no—
161C2×Dic122—42no—
162C2×Dic122C22×C412yes—
163C4⋊C4×C61—48no—
164C22⋊C4×C61—48no—
165C22⋊Dic61—24no—
166C22⋊Dic61—22no—
167C22⋊Dic61—24no—
168C22⋊Dic61—22no—
169#11758198—22no—
170#11758198—24no—
171#11758198—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C22×C412yes—
175C4⋊Dic61C22×C412yes—
176#11758196—22no—
177#11758196—24no—
178#11758196—22no—
179C4×Dic61—24no—
180#11758194—48no—
181C2×#92418—24no—
182C2×#92418—44no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—24no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—48no—
199C23×C61—48no—
200C23×C61C2×C22⋊C412yes—
201C23×C61C2×C22⋊C412yes—
202C23×C61C2×C22⋊C411yescharacteristic
203C23×C61C2×C22⋊C411yescharacteristic
204C23×C61AGL(1,ℤ/8)12yes—
205C23×C61#13712yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—44no—
229C2×C4×C61—44no—
230C2×C4×C61—44no—
231C2×C4×C61—88no—
232C2×C4×C61#9914yes—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C412yes—
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—48no—
240C22⋊D61—48no—
241C22⋊D61—42no—
242C22⋊D61—42no—
243C22⋊D61—44no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250D244—42no—
251D244—42no—
252C4×D61—48no—
253Dic122—42no—
254Dic122—42no—
255#92418—44no—
256C2×#1513—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—24no—
262C22×C61—24no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61—48no—
272C22×C61—44no—
273C22×C61—44no—
274C22×C61C2×#9912yes—
275C22×C61#41712yes—
276C22×C61#41712yes—
277C22×C61#42112yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61C2×#13711yescharacteristic
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—84no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—84no—
291C4×C61C2×#9914yes—
292Dic61—42no—
293Dic61—42no—
294Dic61—42no—
295C2×#525—611no—
296C2×#407—611no—
297C2×#380D6111yescharacteristic, complemented
298#1513—618no—
299C2×#328—612no—
300C2×#327—612no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—22no—
304C2×C61—42no—
305C2×C61—42no—
306C2×C61#151314yescomplemented
307C2×C61C22×#9911yescharacteristic
308C2×C61C2×#41711yescharacteristic
309C2×C61C2×#42111yescharacteristic
310D61—82no—
311C22×#41—611no—
312C24×C4C2×D6111yescharacteristic
313C22×M16—1221no—
314#525—614no—
315#525—614no—
316#525—1222no—
317#525—1222no—
318C2×#123—1221no—
319C2×C22⋊Q8—611no—
320C2×C22⋊Q8—611no—
321C2×#121—611no—
322C2×#121—611no—
323C2×#120—1221no—
324C2×C4×D4—1221no—
325C2×C4×D4—1221no—
326C2×#117—611no—
327C22×C4⋊C4—21no—
328C22×C22⋊C4—21no—
329#407—614no—
330#407—1222no—
331C2×C22⋊C8—612no—
332C2×C22⋊C8—1221no—
333#380—22no—
334#380C2×D6114yescomplemented
335C2×#95C2×D6112yes—
336#328—618no—
337#327—618no—
338C61C2×#151311yescharacteristic, complemented, Sylow
339C25C22⋊D6111yescharacteristic
340C2×#41—614no—
341C2×#41—1222no—
342C2×#41—2444no—
343C22×Q8—611no—
344C22×D4—611no—
345C22×D4—1221no—
346C22×D4—1221no—
347C22×D4—1221no—
348C23×C4—22no—
349C23×C4—44no—
350C23×C4—1221no—
351C23×C4C4×D6111yescharacteristic
352C23×C4C4×D6111yescharacteristic
353C23×C4C4×D6112yes—
354C23×C4D24411yescharacteristic
355C23×C4D24411yescharacteristic
356C23×C4D24412yes—
357C23×C4C22⋊D6111yescharacteristic
358C23×C4C22⋊D6112yes—
359C23×C4C22×D6111yescharacteristic
360C23×C4C22×D6112yes—
361C2×M16—1224no—
362C2×M16—2444no—
363C22×C8—1222no—
364#123—1224no—
365#123—1224no—
366C22⋊Q8—612no—
367C22⋊Q8—614no—
368C22⋊Q8—612no—
369C22⋊Q8—1222no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372#121—612no—
373#121—614no—
374#121—612no—
375#121—1222no—
376#121—1221no—
377#121—1221no—
378#120—1224no—
379#120—1224no—
380C4×D4—1224no—
381C4×D4—1224no—
382C4×D4—1224no—
383C4×D4—1224no—
384#117—614no—
385#117—1222no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—44no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—44no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C42—1221no—
400C22⋊C8—614no—
401C22⋊C8—1222no—
402C22⋊C8—1224no—
403#95—22no—
404#95C22×D6114yes—
405C24—22no—
406C24—22no—
407C24—42no—
408C24—42no—
409C24—1221no—
410C24#1175819811yescharacteristic
411C24#1175819811yescharacteristic
412C24#1175819812yes—
413C24C2×C22⋊D6111yescharacteristic
414C24C2×C22⋊D6112yes—
415#41—2448no—
416#41—2448no—
417C2×Q8—612no—
418C2×Q8—1221no—
419C2×Q8—2442no—
420C2×D4—612no—
421C2×D4—1222no—
422C2×D4—1224no—
423C2×D4—1222no—
424C2×D4—1224no—
425C2×D4—1221no—
426C2×D4—2442no—
427C2×D4—2444no—
428C2×D4—2444no—
429C2×D4—2442no—
430C2×D4—2442no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1224no—
449C22×C4—2444no—
450C22×C4#1175819812yes—
451C22×C4#1175819812yes—
452C22×C4#1175819812yes—
453C22×C4#1175819811yescharacteristic
454C22×C4#1175819812yes—
455C22×C4#1175819812yes—
456C22×C4#1175819811yescharacteristic
457C22×C4C2×C4×D6112yes—
458C22×C4C2×C4×D6111yescharacteristic
459C22×C4C2×C4×D6112yes—
460C22×C4C2×C4×D6111yescharacteristic
461C22×C4C2×D24412yes—
462C22×C4C2×D24411yescharacteristic
463C22×C4C2×D24411yescharacteristic
464C22×C4C2×D24412yes—
465C22×C4C2×C22⋊D6112yes—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468M16—2448no—
469C2×C8—1224no—
470C2×C8—2444no—
471C4⋊C4—48no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—1222no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1224no—
478C4⋊C4—1222no—
479C22⋊C4—48no—
480C22⋊C4—1222no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—1224no—
486C22⋊C4—1222no—
487C42—1224no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—48no—
504C23—48no—
505C23—1222no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23C2×#1175819812yes—
510C23C2×#1175819812yes—
511C23C2×#1175819811yescharacteristic
512C23C2×#1175819811yescharacteristic
513C23#1451980312yes—
514C23#1451980812yes—
515C23C22×C22⋊D6111yescharacteristic
516Q8—2442no—
517Q8—2442no—
518D4—2442no—
519D4—2442no—
520D4—2448no—
521D4—2448no—
522D4—2444no—
523D4—2442no—
524D4—2442no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—88no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2448no—
553C2×C4#1451969114yes—
554C2×C4C2×#1175819811yescharacteristic
555C2×C4C2×#1175819812yes—
556C2×C4C2×#1175819811yescharacteristic
557C2×C4C2×#1175819812yes—
558C2×C4C22×C4×D6111yescharacteristic
559C2×C4C22×D24411yescharacteristic
560C8—2444no—
561C22—24no—
562C22—24no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—4884no—
579C22—2442no—
580C22—2442no—
581C22—2442no—
582C22C2×#1451969112yes—
583C22#1683976112yes—
584C22#1683976112yes—
585C22#1683978012yes—
586C22C22×#1175819811yescharacteristic
587C22C2×#1451980311yescharacteristic
588C22C2×#1451980811yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—84no—
595C4—2442no—
596C4—2442no—
597C4—2442no—
598C4C2×#1451969114yes—
599C2—21no—
600C2—22no—
601C2—21no—
602C2—42no—
603C2—42no—
604C2—4882no—
605C2#2144436014yesdirect product factor
606C2C22×#1451969111yescharacteristic
607C2C2×#1683976111yescharacteristic
608C2C2×#1683978011yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#380C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#16840273C2
cosocleC22×C4×C61C24
hypercenterC2×#380D61
hypocenterC61C2×#1513
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16839726C2semidirect
3. #21444360C2direct (N × H)
4. C2×#16839498C2semidirect
5. C2×#16839495C2semidirect
6. C2×C61×#380C2semidirect
7. C22×#11758222C4semidirect
11. #16840273C4semidirect
14. C2×#14519756C4semidirect
16. C2×#14519751C4semidirect
21. C2×#14519736C4semidirect
23. #16839726C22semidirect
26. #16839498C22semidirect
27. #16839495C22semidirect
32. C61×#380C22semidirect
36. C2×#11758222C2×C4semidirect
52. #14519756C2×C4semidirect
58. #14519751C2×C4semidirect
67. #14519736C2×C4semidirect
297. C2×#380D61semidirect
306. C2×C61#1513semidirect
334. #380C2×D61semidirect
338. C61C2×#1513semidirect
605. C2#21444360direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402732
C2×#168397262
#214443602
C2×#168394982
C2×#168394952
C2×C61×#3802
C2×#151361