Finite Group Databasebeta

← C2×#21455765 #30547817

Subgroups — C2×#21455765 #30547817

Order 15616.

2326 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519753C211yesmaximal, characteristic
2C2×#117×D61C211yesmaximal, characteristic
3#21455765C2116yesmaximal, direct product factor
4C2×#16840096C212yesmaximal, complemented
5C2×#16840102C212yesmaximal, complemented
6C2×#16840101C212yesmaximal, complemented
7C2×#16840091C212yesmaximal, complemented
8C2×#16839805C211yesmaximal, characteristic, complemented
9C2×#16839804C211yesmaximal, characteristic, complemented
10C2×#16839771C211yesmaximal, characteristic, complemented
11C2×#16839725C211yesmaximal, characteristic
12C2×C61×#419C211yesmaximal, characteristic, complemented
13C23×C4×D61C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15C2×#14519753—21no—
16C2×#14519753—24no—
17C2×#14519753—21no—
18C2×#14519753C2212yes—
19C2×#14519753C2212yes—
20C2×#14519753C2218yes—
21C2×#14519736C2211yescharacteristic
22C2×#14519735C2211yescharacteristic
23C2×#14519735C2211yescharacteristic
24C2×#14519734—21no—
25C2×#14519733—21no—
26#117×D61—24no—
27#117×D61C2218yes—
28C22×C4⋊Dic61C2211yescharacteristic
29C22×#11758196—21no—
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C4⋊C4×D61C2211yescharacteristic
32C22×#11758198—21no—
33C2×C22⋊C4×D61—21no—
34C2×C42×D61—21no—
35#16840102C22116yescomplemented
36#16840101C22116yescomplemented
37#16840096C22116yescomplemented
38#16840091C22116yescomplemented
39#16839805C2218yescomplemented
40#16839804C2218yescomplemented
41C2×#14519711C2212yescomplemented
42C2×#14519710C2212yescomplemented
43C2×#14519709C2212yescomplemented
44C2×#14519708C2212yescomplemented
45#16839771C2218yescomplemented
46C2×#14519701C2212yescomplemented
47C2×#14519701C2212yescomplemented
48#16839725C2218yes—
49C2×#14519689C2212yes—
50C2×#14519689C2212yes—
51C23×Q8×C61C2211yescharacteristic
52C22×M16×C61C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C61×#419C2218yescomplemented
55C2×#103×C61C2212yescomplemented
56C2×#103×C61C2212yescomplemented
57C24×D61D411yescharacteristic
58C23×Dic61—21no—
59C23×Dic61D411yescharacteristic
60C22×C4×D61—24no—
61C22×C4×D61—24no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D414yes—
66C22×C4×D61D411yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C22×C4×D61C2314yes—
69C22×C4×D61C2311yescharacteristic
70C2×#11758194—24no—
71C2×#11758194C2314yes—
72C22×#92418C2312yes—
73#14519753—28no—
74#14519753—24no—
75#14519753—28no—
76#14519753—24no—
77#14519753C2318yes—
78#14519753C2318yes—
79#14519736—22no—
80#14519736C2314yes—
81#14519735—22no—
82#14519735—22no—
83#14519735C2314yes—
84#14519735C2314yes—
85#14519734—28no—
86#14519733—28no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—22no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2314yes—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—22no—
96C2×#11758196—22no—
97C2×#11758196—28no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61—22no—
100C4⋊C4×D61C2314yes—
101C4⋊C4×D61C2314yes—
102C2×C22⋊Dic61—21no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×#11758198—22no—
106C2×#11758198—28no—
107C2×#11758198—22no—
108C22⋊C4×D61—28no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61—21no—
112C42×D61—28no—
113#14519711C2318yescomplemented
114#14519710C2318yescomplemented
115#14519709C2318yescomplemented
116#14519708C2318yescomplemented
117#14519701C2318yescomplemented
118#14519701C2318yescomplemented
119#14519689C2318yes—
120#14519689C2318yes—
121C22×Q8×C61—21no—
122C22×Q8×C61—22no—
123C22×Q8×C61—24no—
124C22×Q8×C61—24no—
125C22×Q8×C61C2312yes—
126C22×Q8×C61C2314yes—
127C23×C4×C61—21no—
128C23×C4×C61C2311yescharacteristic
129C2×M16×C61—24no—
130C2×M16×C61C2314yes—
131C22×C8×C61C2312yes—
132#117×C61—22no—
133#117×C61C2314yes—
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C22⋊C4×C61—21no—
137C2×C42×C61—21no—
138#103×C61C2318yescomplemented
139#103×C61C2318yescomplemented
140C24×C61C2×D411yescharacteristic
141C23×D61—24no—
142C23×D61—24no—
143C23×D61—42no—
144C23×D61C2×D411yescharacteristic
145C23×D61C2×D414yes—
146C23×D61C2×D411yescharacteristic
147C2×Q8×C61—24no—
148C2×Q8×C61—22no—
149C2×Q8×C61—24no—
150C2×Q8×C61—28no—
151C2×Q8×C61—24no—
152C2×Q8×C61—28no—
153C2×Q8×C61—48no—
154C2×Q8×C61—44no—
155C2×Q8×C61C2414yes—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—24no—
159C22×C4×C61—22no—
160C22×C4×C61—44no—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D412yes—
163C22×C4×C61#4114yes—
164C22×C4×C61#4114yes—
165C22×C4×C61C2411yescharacteristic
166C22×C4×C61C2412yes—
167C22×Dic61—24no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—24no—
171C22×Dic61—22no—
172C22×Dic61—44no—
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175C22×Dic61C2×D414yes—
176C2×C4×D61—24no—
177C2×C4×D61—216no—
178C2×C4×D61—24no—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—28no—
182C2×C4×D61—22no—
183C2×C4×D61—22no—
184C2×C4×D61—28no—
185C2×C4×D61—24no—
186C2×C4×D61—24no—
187C2×C4×D61—44no—
188C2×C4×D61—44no—
189C2×C4×D61C2×D414yes—
190C2×C4×D61C2×D412yes—
191C2×C4×D61C2×D414yes—
192C2×C4×D61C2×D412yes—
193C2×C4×D61C2412yes—
194C2×C4×D61C2412yes—
195M16×C61—28no—
196C2×C8×C61—24no—
197C2×C8×C61C2414yes—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61C2412yes—
201C4⋊C4×C61C2412yes—
202C22⋊C4×C61—24no—
203C42×C61—24no—
204C22⋊Dic61—24no—
205#11758198—24no—
206#11758198—24no—
207#11758198—28no—
208#11758198—28no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—24no—
212C4⋊Dic61—22no—
213C4⋊Dic61C2412yes—
214C4⋊Dic61C2412yes—
215C4⋊Dic61C2414yes—
216#11758196—24no—
217#11758196—24no—
218#11758196—28no—
219#11758196—28no—
220C4×Dic61—24no—
221C4×Dic61—24no—
222C4×Dic61—24no—
223#11758194—28no—
224C2×#92418—24no—
225C2×#92418C2414yes—
226C23×C61—22no—
227C23×C61—22no—
228C23×C61#12314yes—
229C23×C61C22×D412yes—
230C23×C61C22×D411yescharacteristic
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—28no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—28no—
238C22×D61—44no—
239C22×D61—416no—
240C22×D61#13714yes—
241C22×D61C22×D412yes—
242C22×D61C22×D412yes—
243Q8×C61—24no—
244Q8×C61—24no—
245Q8×C61—48no—
246Q8×C61—44no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—24no—
250C2×C4×C61—48no—
251C2×C4×C61—44no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61#12318yes—
255C2×C4×C61#12314yes—
256C2×C4×C61C22×D412yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4112yes—
259C2×C4×C61C2×#4114yes—
260C2×C4×C61C2×#4112yes—
261C2×C4×C61C2×#4114yes—
262C2×C4×C61C2511yescharacteristic
263C2×Dic61—28no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—24no—
269C2×Dic61—48no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—44no—
273C2×Dic61C22×D412yes—
274C2×Dic61C22×D412yes—
275C4×D61—28no—
276C4×D61—24no—
277C4×D61—28no—
278C4×D61—24no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—48no—
284C4×D61—48no—
285C8×C61—24no—
286#92418—24no—
287C2×#3095—611nomaximal, Sylow
288C22×C61—28no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61C2×#12314yes—
292C22×C61C2×#12312yes—
293C22×C61C2×#13712yes—
294C22×C61#57512yes—
295C22×C61C23×D411yescharacteristic
296C2×D61—28no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—416no—
303C2×D61—44no—
304C2×D61C2×#13714yes—
305C4×C61—44no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61C2×#12314yes—
309C4×C61C2×#12312yes—
310C4×C61C22×#4111yescharacteristic
311C4×C61C22×#4111yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—42no—
315Dic61—42no—
316Dic61—44no—
317C22×C22⋊Q8—611no—
318C22×#117—611no—
319#3095—6116no—
320C2×#484—612no—
321C2×#485—612no—
322C2×#485—612no—
323C2×#483—612no—
324C2×#428—611no—
325C2×#428—611no—
326C2×#419—611no—
327C2×#419D6111yescharacteristic, complemented
328C2×#407—611no—
329C2×C61—24no—
330C2×C61#309514yescomplemented
331C2×C61C22×#12311yescharacteristic
332C2×C61C22×#13711yescharacteristic
333C2×C61C2×#57511yescharacteristic
334D61—24no—
335D61—42no—
336C23×Q8C2×D6111yescharacteristic
337C24×C4—611no—
338C22×M16—611no—
339C22×M16C2×D6111yescharacteristic
340C2×C22⋊Q8—612no—
341C2×C22⋊Q8—612no—
342C2×C22⋊Q8—618no—
343C2×C22⋊Q8—1224no—
344C2×C22⋊Q8—1221no—
345C2×C22⋊Q8—1221no—
346C2×#117—611no—
347C2×#117—618no—
348C2×#117—611no—
349C2×#117—611no—
350C2×#117—1221no—
351C2×#117—1221no—
352C2×#117—1224no—
353C2×#117C2×D6111yescharacteristic
354C22×C4⋊C4—611no—
355C22×C4⋊C4—611no—
356C22×C4⋊C4—611no—
357C22×C4⋊C4—1221no—
358C22×C22⋊C4—1221no—
359C22×C22⋊C4—1221no—
360C22×C42—1221no—
361#485—6116no—
362#485—6116no—
363#484—6116no—
364#483—6116no—
365#428—618no—
366#428—618no—
367C2×#107—612no—
368C2×#107—612no—
369C2×#106—612no—
370C2×#106—612no—
371#419—618no—
372#419C2×D6118yescomplemented
373C2×#103—612no—
374C2×#103—612no—
375C2×#103C2×D6112yescomplemented
376C2×#103C2×D6112yescomplemented
377#407—618no—
378C2×C22⋊C8—612no—
379C2×C22⋊C8—612no—
380C61C2×#309511yescharacteristic, complemented, Sylow
381C25—611no—
382C22×Q8—22no—
383C22×Q8—21no—
384C22×Q8—24no—
385C22×Q8—24no—
386C22×Q8C22×D6112yes—
387C22×Q8C22×D6114yes—
388C23×C4—21no—
389C23×C4—611no—
390C23×C4—614no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—614no—
396C23×C4—1221no—
397C23×C4—1221no—
398C23×C4—1224no—
399C23×C4—1224no—
400C23×C4—1221no—
401C23×C4C22×D6111yescharacteristic
402C2×M16—24no—
403C2×M16—614no—
404C2×M16—1224no—
405C2×M16C22×D6114yes—
406C22×C8—612no—
407C22×C8C22×D6112yes—
408C22⋊Q8—618no—
409C22⋊Q8—618no—
410C22⋊Q8—1228no—
411C22⋊Q8—1224no—
412C22⋊Q8—1224no—
413C22⋊Q8—1228no—
414#117—22no—
415#117—614no—
416#117—614no—
417#117—614no—
418#117—1228no—
419#117—1228no—
420#117—1222no—
421#117—1222no—
422#117—1222no—
423#117C22×D6114yes—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—612no—
427C2×C4⋊C4—614no—
428C2×C4⋊C4—614no—
429C2×C4⋊C4—614no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1228no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4—1222no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1228no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1228no—
446C2×C22⋊C4—1222no—
447C2×C22⋊C4—1222no—
448C2×C22⋊C4—1221no—
449C2×C42—21no—
450C2×C42—1228no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42—1221no—
454#107—618no—
455#107—618no—
456#106—618no—
457#106—618no—
458#103—618no—
459#103—618no—
460#103C22×D6118yescomplemented
461#103C22×D6118yescomplemented
462C22⋊C8—618no—
463C22⋊C8—618no—
464C24—611no—
465C24—611no—
466C24—614no—
467C24—1224no—
468C24—1224no—
469C24—2442no—
470C24D4×D6111yescharacteristic
471C2×Q8—24no—
472C2×Q8—22no—
473C2×Q8—24no—
474C2×Q8—28no—
475C2×Q8—24no—
476C2×Q8—28no—
477C2×Q8—48no—
478C2×Q8—44no—
479C2×Q8C23×D6114yes—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—614no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—614no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—614no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—1224no—
495C22×C4—1224no—
496C22×C4—1224no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1228no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1228no—
503C22×C4—1224no—
504C22×C4—12216no—
505C22×C4—1224no—
506C22×C4—1224no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1222no—
510C22×C4—2444no—
511C22×C4—2444no—
512C22×C4—2444no—
513C22×C4#1175822214yes—
514C22×C4D4×D6112yes—
515C22×C4D4×D6111yescharacteristic
516C22×C4#1175822414yes—
517C22×C4C23×D6112yes—
518C22×C4C23×D6111yescharacteristic
519M16—28no—
520M16—1228no—
521C2×C8—24no—
522C2×C8—614no—
523C2×C8—1224no—
524C2×C8C23×D6114yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—612no—
528C4⋊C4—612no—
529C4⋊C4—614no—
530C4⋊C4—1224no—
531C4⋊C4—1224no—
532C4⋊C4—1228no—
533C4⋊C4—1228no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1224no—
537C4⋊C4—1222no—
538C4⋊C4C23×D6112yes—
539C4⋊C4C23×D6112yes—
540C22⋊C4—24no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C22⋊C4—1228no—
544C22⋊C4—1228no—
545C22⋊C4—1224no—
546C42—24no—
547C42—1224no—
548C42—1224no—
549C42—1224no—
550C23—22no—
551C23—22no—
552C23—614no—
553C23—612no—
554C23—612no—
555C23—1228no—
556C23—1222no—
557C23—1222no—
558C23—1228no—
559C23—1222no—
560C23—1222no—
561C23—1222no—
562C23—2444no—
563C23—24416no—
564C23#1451975914yes—
565C23C2×D4×D6112yes—
566C23C2×D4×D6111yescharacteristic
567Q8—24no—
568Q8—24no—
569Q8—48no—
570Q8—44no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—1228no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1228no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1228no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1224no—
594C2×C4—2448no—
595C2×C4—2448no—
596C2×C4—2448no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2444no—
600C2×C4#1451975918yes—
601C2×C4#1451975914yes—
602C2×C4C2×D4×D6112yes—
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4C2×#1175822412yes—
605C2×C4C2×#1175822414yes—
606C2×C4C2×#1175822212yes—
607C2×C4C2×#1175822214yes—
608C2×C4C24×D6111yescharacteristic
609C8—24no—
610C8—1224no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—614no—
615C22—1222no—
616C22—1222no—
617C22—1228no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—24416no—
622C22—2444no—
623C22C2×#1451975914yes—
624C22C2×#1451975912yes—
625C22#137×D6112yes—
626C22#1684056112yes—
627C22C22×D4×D6111yescharacteristic
628C4—42no—
629C4—42no—
630C4—44no—
631C4—1222no—
632C4—1222no—
633C4—2442no—
634C4—2442no—
635C4—2444no—
636C4C2×#1451975914yes—
637C4C2×#1451975912yes—
638C4C22×#1175822411yescharacteristic
639C4C22×#1175822211yescharacteristic
640C2—24no—
641C2—1224no—
642C2—2442no—
643C2#2145576514yesdirect product factor
644C2C22×#1451975911yescharacteristic
645C2C2×#137×D6111yescharacteristic
646C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC24×D61D4
hypercenterC2×#419D61
hypocenterC61C2×#3095
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. #21455765C2direct (N × H)
4. C2×#16840096C2semidirect
5. C2×#16840102C2semidirect
6. C2×#16840101C2semidirect
7. C2×#16840091C2semidirect
8. C2×#16839805C2semidirect
9. C2×#16839804C2semidirect
10. C2×#16839771C2semidirect
12. C2×C61×#419C2semidirect
35. #16840102C22semidirect
36. #16840101C22semidirect
37. #16840096C22semidirect
38. #16840091C22semidirect
39. #16839805C22semidirect
40. #16839804C22semidirect
41. C2×#14519711C22semidirect
42. C2×#14519710C22semidirect
43. C2×#14519709C22semidirect
44. C2×#14519708C22semidirect
45. #16839771C22semidirect
46. C2×#14519701C22semidirect
47. C2×#14519701C22semidirect
54. C61×#419C22semidirect
55. C2×#103×C61C22semidirect
56. C2×#103×C61C22semidirect
113. #14519711C23semidirect
114. #14519710C23semidirect
115. #14519709C23semidirect
116. #14519708C23semidirect
117. #14519701C23semidirect
118. #14519701C23semidirect
138. #103×C61C23semidirect
139. #103×C61C23semidirect
327. C2×#419D61semidirect
330. C2×C61#3095semidirect
372. #419C2×D61semidirect
375. C2×#103C2×D61semidirect
376. C2×#103C2×D61semidirect
380. C61C2×#3095semidirect
460. #103C22×D61semidirect
461. #103C22×D61semidirect
643. C2#21455765direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197532
C2×#117×D612
#214557652
C2×#168400962
C2×#168401022
C2×#168401012
C2×#168400912
C2×#168398052
C2×#168398042
C2×#168397712
C2×#168397252
C2×C61×#4192
C2×#309561