Finite Group Databasebeta

← C2×#21455769 #30547821

Subgroups — C2×#21455769 #30547821

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×#14519757C211yesmaximal, characteristic
2C2×#117×D61C211yesmaximal, characteristic
3#21455769C2116yesmaximal, direct product factor
4C2×#16840099C212yesmaximal, complemented
5C2×#16840106C212yesmaximal, complemented
6C2×#16840105C212yesmaximal, complemented
7C2×#16840094C212yesmaximal, complemented
8C2×#16839802C211yesmaximal, characteristic, complemented
9C2×#16839772C211yesmaximal, characteristic, complemented
10C2×#16839770C211yesmaximal, characteristic, complemented
11C2×#16839725C211yesmaximal, characteristic
12C2×C61×#428C211yesmaximal, characteristic, complemented
13C23×Dic122C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C22×#11758194C2211yescharacteristic
16C2×#14519757—21no—
17C2×#14519757—24no—
18C2×#14519757—21no—
19C2×#14519757C2212yes—
20C2×#14519757C2212yes—
21C2×#14519757C2218yes—
22C2×#14519736C2211yescharacteristic
23C2×#14519735C2211yescharacteristic
24C2×#14519735C2211yescharacteristic
25C2×#14519734—21no—
26C2×#14519733—21no—
27#117×D61—24no—
28#117×D61C2218yes—
29C22×C4⋊Dic61—21no—
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C4⋊C4×D61C2211yescharacteristic
32C22×#11758198—21no—
33C2×C22⋊C4×D61—21no—
34C2×C42×D61—21no—
35#16840106C22116yescomplemented
36#16840105C22116yescomplemented
37#16840099C22116yescomplemented
38#16840094C22116yescomplemented
39#16839802C2218yescomplemented
40C2×#14519710C2212yescomplemented
41C2×#14519708C2212yescomplemented
42#16839772C2218yescomplemented
43#16839770C2218yescomplemented
44C2×#14519702C2212yescomplemented
45C2×#14519702C2212yescomplemented
46C2×#14519700C2212yescomplemented
47C2×#14519700C2212yescomplemented
48#16839725C2218yes—
49C2×#14519689C2212yes—
50C2×#14519689C2212yes—
51C22×M16×C61C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C61×#428C2218yescomplemented
55C2×#107×C61C2212yescomplemented
56C2×#106×C61C2212yescomplemented
57C24×D61D411yescharacteristic
58C22×Dic122—24no—
59C22×Dic122—24no—
60C22×Dic122—21no—
61C22×Dic122—22no—
62C22×Dic122C2312yes—
63C22×Dic122C2314yes—
64C23×Dic61—21no—
65C23×Dic61D411yescharacteristic
66C22×C4×D61—24no—
67C22×C4×D61—24no—
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61D411yescharacteristic
71C22×C4×D61D414yes—
72C22×C4×D61D411yescharacteristic
73C22×C4×D61C2311yescharacteristic
74C22×C4×D61C2314yes—
75C22×C4×D61C2311yescharacteristic
76C2×#11758194—24no—
77C2×#11758194C2314yes—
78C22×#92418C2312yes—
79#14519757—28no—
80#14519757—24no—
81#14519757—28no—
82#14519757—24no—
83#14519757C2318yes—
84#14519757C2318yes—
85#14519736—22no—
86#14519736C2314yes—
87#14519735—22no—
88#14519735—22no—
89#14519735C2314yes—
90#14519735C2314yes—
91#14519734—28no—
92#14519733—28no—
93C2×C4⋊Dic61—22no—
94C2×C4⋊Dic61—22no—
95C2×C4⋊Dic61—28no—
96C2×C4⋊Dic61C2311yescharacteristic
97C2×C4⋊Dic61C2311yescharacteristic
98C2×#11758196—21no—
99C2×#11758196—21no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61—22no—
102C4⋊C4×D61C2314yes—
103C4⋊C4×D61C2314yes—
104C2×C22⋊Dic61—21no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×#11758198—22no—
108C2×#11758198—28no—
109C2×#11758198—22no—
110C22⋊C4×D61—28no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61—21no—
114C42×D61—28no—
115#14519710C2318yescomplemented
116#14519708C2318yescomplemented
117#14519702C2318yescomplemented
118#14519702C2318yescomplemented
119#14519700C2318yescomplemented
120#14519700C2318yescomplemented
121#14519689C2318yes—
122#14519689C2318yes—
123C23×C4×C61—21no—
124C23×C4×C61C2311yescharacteristic
125C2×M16×C61—24no—
126C2×M16×C61C2314yes—
127C22×C8×C61C2312yes—
128#117×C61—22no—
129#117×C61C2314yes—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—22no—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2312yes—
135C2×C4⋊C4×C61C2314yes—
136C2×C22⋊C4×C61—21no—
137C2×C42×C61—21no—
138#107×C61C2318yescomplemented
139#106×C61C2318yescomplemented
140C24×C61C2×D411yescharacteristic
141C23×D61—24no—
142C23×D61—24no—
143C23×D61—42no—
144C23×D61C2×D411yescharacteristic
145C23×D61C2×D414yes—
146C23×D61C2×D411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—24no—
150C22×C4×C61—22no—
151C22×C4×C61—44no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61#4114yes—
155C22×C4×C61#4114yes—
156C22×C4×C61C2411yescharacteristic
157C22×C4×C61C2412yes—
158C22×Dic61—24no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—24no—
162C22×Dic61—22no—
163C22×Dic61—44no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D414yes—
167C2×C4×D61—24no—
168C2×C4×D61—216no—
169C2×C4×D61—24no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—28no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—28no—
176C2×C4×D61—24no—
177C2×C4×D61—24no—
178C2×C4×D61—44no—
179C2×C4×D61—44no—
180C2×C4×D61C2×D414yes—
181C2×C4×D61C2×D412yes—
182C2×C4×D61C2×D414yes—
183C2×C4×D61C2×D412yes—
184C2×C4×D61C2412yes—
185C2×C4×D61C2412yes—
186C2×Dic122—24no—
187C2×Dic122—28no—
188C2×Dic122—24no—
189C2×Dic122—28no—
190C2×Dic122—24no—
191C2×Dic122—22no—
192C2×Dic122—48no—
193C2×Dic122—44no—
194C2×Dic122C2414yes—
195M16×C61—28no—
196C2×C8×C61—24no—
197C2×C8×C61C2414yes—
198C4⋊C4×C61—24no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61C2412yes—
203C4⋊C4×C61C2414yes—
204C4⋊C4×C61C2412yes—
205C22⋊C4×C61—24no—
206C42×C61—24no—
207C22⋊Dic61—24no—
208#11758198—24no—
209#11758198—24no—
210#11758198—28no—
211#11758198—28no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—28no—
215C4⋊Dic61—28no—
216C4⋊Dic61C2412yes—
217C4⋊Dic61C2412yes—
218#11758196—24no—
219#11758196—24no—
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—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—48no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—44no—
250C2×C4×C61#12318yes—
251C2×C4×C61#12314yes—
252C2×C4×C61C22×D412yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4112yes—
255C2×C4×C61C2×#4114yes—
256C2×C4×C61C2×#4112yes—
257C2×C4×C61C2×#4114yes—
258C2×C4×C61C2511yescharacteristic
259C2×Dic61—28no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—24no—
265C2×Dic61—48no—
266C2×Dic61—44no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61C22×D412yes—
270C2×Dic61C22×D412yes—
271C4×D61—28no—
272C4×D61—24no—
273C4×D61—28no—
274C4×D61—24no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—48no—
280C4×D61—48no—
281Dic122—24no—
282Dic122—24no—
283Dic122—48no—
284Dic122—44no—
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—42no—
306C4×C61—42no—
307C4×C61—44no—
308C4×C61C2×#12314yes—
309C4×C61C2×#12312yes—
310C4×C61C22×#4111yescharacteristic
311C4×C61C22×#4111yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—44no—
315Dic61—42no—
316Dic61—42no—
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×#428D6111yescharacteristic, complemented
326C2×#419—611no—
327C2×#419—611no—
328C2×#407—611no—
329C2×C61—24no—
330C2×C61#309514yescomplemented
331C2×C61C22×#12311yescharacteristic
332C2×C61C22×#13711yescharacteristic
333C2×C61C2×#57511yescharacteristic
334D61—24no—
335D61—42no—
336C23×Q8—611no—
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—1221no—
357C22×C4⋊C4C2×D6111yescharacteristic
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#428C2×D6118yescomplemented
367C2×#107—612no—
368C2×#107C2×D6112yescomplemented
369C2×#106—612no—
370C2×#106C2×D6112yescomplemented
371#419—618no—
372#419—618no—
373C2×#103—612no—
374C2×#103—612no—
375C2×#103—612no—
376C2×#103—612no—
377#407—618no—
378C2×C22⋊C8—612no—
379C2×C22⋊C8—612no—
380C61C2×#309511yescharacteristic, complemented, Sylow
381C25—611no—
382C22×Q8—612no—
383C22×Q8—614no—
384C22×Q8—1222no—
385C22×Q8—1221no—
386C22×Q8—1224no—
387C22×Q8—1224no—
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—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
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⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6112yes—
440C2×C4⋊C4C22×D6114yes—
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#107C22×D6118yescomplemented
456#106—618no—
457#106C22×D6118yescomplemented
458#103—618no—
459#103—618no—
460#103—618no—
461#103—618no—
462C22⋊C8—618no—
463C22⋊C8—618no—
464C24—611no—
465C24—611no—
466C24—614no—
467C24—1224no—
468C24—1224no—
469C24—2442no—
470C24D4×D6111yescharacteristic
471C2×Q8—614no—
472C2×Q8—1224no—
473C2×Q8—1222no—
474C2×Q8—1224no—
475C2×Q8—1228no—
476C2×Q8—1224no—
477C2×Q8—1228no—
478C2×Q8—2448no—
479C2×Q8—2444no—
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#1175822614yes—
517C22×C4C23×D6112yes—
518C22×C4C23×D6111yescharacteristic
519M16—28no—
520M16—1228no—
521C2×C8—24no—
522C2×C8—614no—
523C2×C8—1224no—
524C2×C8C23×D6114yes—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1228no—
534C4⋊C4—1228no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4C23×D6112yes—
538C4⋊C4C23×D6112yes—
539C4⋊C4C23×D6114yes—
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#1451976214yes—
565C23C2×D4×D6112yes—
566C23C2×D4×D6111yescharacteristic
567Q8—1224no—
568Q8—1224no—
569Q8—2448no—
570Q8—2444no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
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—2444no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4#1451976218yes—
601C2×C4#1451976214yes—
602C2×C4C2×D4×D6112yes—
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4C2×#1175822212yes—
605C2×C4C2×#1175822214yes—
606C2×C4C2×#1175822612yes—
607C2×C4C2×#1175822614yes—
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×#1451976214yes—
624C22C2×#1451976212yes—
625C22#137×D6112yes—
626C22#1684055812yes—
627C22C22×D4×D6111yescharacteristic
628C4—42no—
629C4—44no—
630C4—42no—
631C4—1222no—
632C4—1222no—
633C4—2442no—
634C4—2442no—
635C4—2444no—
636C4C2×#1451976214yes—
637C4C2×#1451976212yes—
638C4C22×#1175822211yescharacteristic
639C4C22×#1175822611yescharacteristic
640C2—24no—
641C2—1224no—
642C2—2442no—
643C2#2145576914yesdirect product factor
644C2C22×#1451976211yescharacteristic
645C2C2×#137×D6111yescharacteristic
646C2C2×#1684055811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC24×D61D4
hypercenterC2×#428D61
hypocenterC61C2×#3095
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #21455769C2direct (N × H)
4. C2×#16840099C2semidirect
5. C2×#16840106C2semidirect
6. C2×#16840105C2semidirect
7. C2×#16840094C2semidirect
8. C2×#16839802C2semidirect
9. C2×#16839772C2semidirect
10. C2×#16839770C2semidirect
12. C2×C61×#428C2semidirect
35. #16840106C22semidirect
36. #16840105C22semidirect
37. #16840099C22semidirect
38. #16840094C22semidirect
39. #16839802C22semidirect
40. C2×#14519710C22semidirect
41. C2×#14519708C22semidirect
42. #16839772C22semidirect
43. #16839770C22semidirect
44. C2×#14519702C22semidirect
45. C2×#14519702C22semidirect
46. C2×#14519700C22semidirect
47. C2×#14519700C22semidirect
54. C61×#428C22semidirect
55. C2×#107×C61C22semidirect
56. C2×#106×C61C22semidirect
115. #14519710C23semidirect
116. #14519708C23semidirect
117. #14519702C23semidirect
118. #14519702C23semidirect
119. #14519700C23semidirect
120. #14519700C23semidirect
138. #107×C61C23semidirect
139. #106×C61C23semidirect
325. C2×#428D61semidirect
330. C2×C61#3095semidirect
366. #428C2×D61semidirect
368. C2×#107C2×D61semidirect
370. C2×#106C2×D61semidirect
380. C61C2×#3095semidirect
455. #107C22×D61semidirect
457. #106C22×D61semidirect
643. C2#21455769direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197572
C2×#117×D612
#214557692
C2×#168400992
C2×#168401062
C2×#168401052
C2×#168400942
C2×#168398022
C2×#168397722
C2×#168397702
C2×#168397252
C2×C61×#4282
C2×#309561