Finite Group Databasebeta

← C4×#16840366 #29839120

Subgroups — C4×#16840366 #29839120

Order 15616.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840366C211yesmaximal, characteristic
2#21448606C214yesmaximal, complemented
3#21448578C214yesmaximal, complemented
4#21448558C211yesmaximal, characteristic, complemented
5#21448545C211yesmaximal, characteristic, complemented
6#21448515C212yesmaximal, complemented
7#21448456C212yesmaximal, complemented
8#21448427C211yesmaximal, characteristic, complemented
9C4×#14519769C211yesmaximal, characteristic, complemented
10C4×#14519758C214yesmaximal, complemented
11C4×#14519749C214yesmaximal, complemented
12C4×#14519748C212yesmaximal, complemented
13C4×D4×Dic61C212yesmaximal, complemented
14C4×#14519734C211yesmaximal, characteristic, complemented
15C4×#127×C61C211yesmaximal, characteristic, complemented
16#16840366C4116yesdirect product factor
17C2×#14519769C2211yescharacteristic
18C2×#14519758C2214yes—
19C2×#14519749C2214yes—
20C2×#14519748C2212yes—
21C2×D4×Dic61C2212yes—
22C2×D4×Dic61C2212yes—
23C2×C4×C22⋊D61C2214yes—
24C2×#14519734C2211yescharacteristic
25C2×#14519734C2211yescharacteristic
26C22×C4×Dic61C2214yes—
27C2×C42×D61C2211yescharacteristic
28#16839665C2212yes—
29#16839662C2214yes—
30#16839660C2212yes—
31#16839658C2212yes—
32#16839657C2214yes—
33#16839656C2214yes—
34#16839652C2214yes—
35#16839650C2214yes—
36#16839650C2214yes—
37#16839650C2214yes—
38#16839646C2214yes—
39#16839643C2214yes—
40#16839643C2214yes—
41#16839642C2214yes—
42#16839642C2214yes—
43#16839639C2214yes—
44#16839638C2214yes—
45#16839634C2214yes—
46#16839628C2211yescharacteristic
47#16839623C2214yes—
48#16839620C2214yes—
49#16839619C2212yes—
50C4⋊C4×Dic61C2212yes—
51C4⋊C4×Dic61C2212yes—
52C4×#11758196C2214yes—
53C4×#11758196C2212yes—
54C4×#11758196C2212yes—
55C4×C4⋊Dic61C2212yes—
56C22⋊C4×Dic61C2214yes—
57C22⋊C4×Dic61C2214yes—
58C4×#11758198C2212yes—
59C4×#11758198C2212yes—
60C4×C22⋊Dic61C2214yes—
61C4×C22⋊Dic61C2214yes—
62#16839608C2211yescharacteristic
63#16839608C2211yescharacteristic
64C42×Dic61C2211yescharacteristic
65C2×#127×C61C2211yescharacteristic
66C2×C4×D4×C61C2212yes—
67C2×C4×D4×C61C2214yes—
68C61×#390C2212yes—
69C61×#390C2214yes—
70C61×#384C2211yescharacteristic
71C43×C61C2211yescharacteristic
72C22×C22⋊D61C2314yes—
73C23×Dic61C2314yes—
74C23×Dic61C2314yes—
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2311yescharacteristic
77C22×C4×D61C2311yescharacteristic
78#14519769C2×C418yescomplemented
79#14519758C2×C4132yescomplemented
80#14519749C2×C4132yescomplemented
81#14519748C2×C4116yescomplemented
82D4×Dic61—28no—
83D4×Dic61C2×C4116yescomplemented
84C4×C22⋊D61—48no—
85#14519734—24no—
86#14519734C2×C418yescomplemented
87C2×C4⋊Dic61C2312yes—
88C2×C4⋊Dic61C2312yes—
89C2×#11758196C2312yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C2×#11758196C2312yes—
93C2×#11758196C2314yes—
94C2×C22⋊Dic61C2314yes—
95C2×C22⋊Dic61C2314yes—
96C2×C22⋊Dic61C2314yes—
97C2×C22⋊Dic61C2314yes—
98C2×C22⋊Dic61C2314yes—
99C2×#11758198C2312yes—
100C2×#11758198C2314yes—
101C2×#11758198C2312yes—
102C2×C4×Dic61—48no—
103C2×C4×Dic61C2311yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C2×C4×Dic61C2312yes—
107C2×C4×Dic61C2314yes—
108C2×C4×Dic61C2314yes—
109C2×C4×Dic61C2314yes—
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C42×D61—24no—
114#14519684C2312yes—
115#14519684C2312yes—
116#14519684C2312yes—
117#14519684C2312yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#14519684C2314yes—
124#14519684C2314yes—
125C22×D4×C61C2312yes—
126C22×D4×C61C2314yes—
127C23×C4×C61C2314yes—
128#127×C61C2×C418yescomplemented
129C4×D4×C61—28no—
130C4×D4×C61—48no—
131C2×C4⋊C4×C61C2312yes—
132C2×C4⋊C4×C61C2314yes—
133C2×C22⋊C4×C61C2314yes—
134C2×C22⋊C4×C61C2314yes—
135C2×C22⋊C4×C61C2314yes—
136C2×C42×C61C2311yescharacteristic
137C2×C42×C61C2312yes—
138C2×C42×C61C2312yes—
139C2×C42×C61C2311yescharacteristic
140C2×C42×C61C2311yescharacteristic
141C24×C61C2414yes—
142C23×D61C2411yescharacteristic
143C2×D4×C61—24no—
144C2×D4×C61—24no—
145C2×D4×C61—44no—
146C2×D4×C61—44no—
147C2×D4×C61C22×C418yes—
148C2×D4×C61C22×C4116yes—
149C22×C4×C61—28no—
150C22×C4×C61—28no—
151C22×C4×C61—28no—
152C22×C4×C61C2412yes—
153C22×C4×C61C2414yes—
154C22×C4×C61C2412yes—
155C22×C4×C61C2411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—44no—
159C2×C22⋊D61—44no—
160C2×C22⋊D61C22×C4116yes—
161C22×Dic61—28no—
162C22×Dic61—44no—
163C22×Dic61—44no—
164C22×Dic61—44no—
165C22×Dic61—44no—
166C22×Dic61C22×C4116yes—
167C22×Dic61C2411yescharacteristic
168C22×Dic61C2411yescharacteristic
169C22×Dic61C2414yes—
170C22×Dic61C2411yescharacteristic
171C22×Dic61C2412yes—
172C22×Dic61C2414yes—
173C22×Dic61C2412yes—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—24no—
179C2×C4×D61—42no—
180C2×C4×D61C22×C414yes—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—44no—
183C22⋊C4×C61—28no—
184C22⋊C4×C61—44no—
185C22⋊C4×C61—44no—
186C42×C61—22no—
187C42×C61—24no—
188C42×C61—42no—
189C42×C61C22×C414yes—
190C42×C61#4114yes—
191C22⋊Dic61—28no—
192C22⋊Dic61—44no—
193C22⋊Dic61—44no—
194C22⋊Dic61C22×C4116yes—
195C22⋊Dic61C22×C4116yes—
196#11758198—24no—
197#11758198—44no—
198#11758198C22×C418yes—
199C4⋊Dic61—24no—
200C4⋊Dic61C22×C418yes—
201#11758196—24no—
202#11758196—44no—
203#11758196—44no—
204#11758196C22×C418yes—
205#11758196C22×C4116yes—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—24no—
210C4×Dic61—28no—
211C4×Dic61—44no—
212C4×Dic61—44no—
213C4×Dic61—42no—
214C4×Dic61C22×C414yes—
215C4×Dic61#4114yes—
216C23×C61—24no—
217C23×C61—24no—
218C23×C61—24no—
219C23×C61—24no—
220C23×C61—24no—
221C23×C61—24no—
222C23×C61C23×C418yes—
223C23×C61C2511yescharacteristic
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61C23×C412yes—
229D4×C61—216no—
230D4×C61—416no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—416no—
241C2×C4×C61C23×C414yes—
242C2×C4×C61C23×C412yes—
243C2×C4×C61C2×#4112yes—
244C2×C4×C61C2×#4112yes—
245C2×C4×C61C2×#4112yes—
246C2×C4×C61C2×#4112yes—
247C2×C4×C61H(5,𝔽2)14yes—
248C22⋊D61—416no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—24no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—24no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—22no—
264C2×Dic61—416no—
265C2×Dic61—416no—
266C2×Dic61C23×C414yes—
267C2×Dic61C23×C418yes—
268C2×Dic61C23×C412yes—
269C2×Dic61C2×#4114yes—
270C4×D61—28no—
271C4×D61—44no—
272C4×D61—44no—
273C4×#540—611nomaximal, Sylow
274C22×C61—28no—
275C22×C61—28no—
276C22×C61—28no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61—44no—
281C22×C61C24×C411yescharacteristic
282C22×C61C22×#4111yescharacteristic
283C22×C61C22×#4111yescharacteristic
284C22×C61C2×H(5,𝔽2)12yes—
285C22×C61#58512yes—
286C2×D61—22no—
287C2×D61—24no—
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—42no—
291C4×C61—24no—
292C4×C61—22no—
293C4×C61—44no—
294C4×C61—42no—
295C4×C61C4×#4114yes—
296C4×C61#54014yescomplemented
297Dic61—22no—
298Dic61—24no—
299Dic61—28no—
300Dic61—24no—
301Dic61—41no—
302Dic61—41no—
303Dic61—44no—
304C2×#540—611no—
305#2335—614no—
306#2333—614no—
307#2330—611no—
308#2328—611no—
309#2324—612no—
310#2319—612no—
311#2314—611no—
312C4×#127D6111yescharacteristic, complemented
313C4×#125—611no—
314C4×#123—614no—
315C4×#121—612no—
316C4×#121—614no—
317C42×D4—612no—
318C4×#117—611no—
319C2×C61—48no—
320C2×C61C2×C4×#4111yescharacteristic
321C2×C61C4×H(5,𝔽2)12yes—
322C2×C61C2×#54011yescharacteristic
323C2×C61#345511yescharacteristic
324C2×C61#346012yes—
325D61—42no—
326#540—6116no—
327C2×#127C2×D6111yescharacteristic
328C2×#125—611no—
329C2×#123—614no—
330C2×#121—612no—
331C2×#121—614no—
332C2×C4×D4—612no—
333C2×C4×D4—612no—
334C2×C4×D4—614no—
335C2×C4×D4C2×D6112yes—
336C2×C4×D4C2×D6114yes—
337C2×#117—611no—
338C2×#117—611no—
339C22×C42—614no—
340C22×C42—611no—
341#390—612no—
342#390—614no—
343#390C2×D6112yes—
344#390C2×D6114yes—
345#389—612no—
346#389—614no—
347#389—612no—
348#388—614no—
349#388—614no—
350#388—614no—
351#388—614no—
352#388—614no—
353#387—614no—
354#387—614no—
355#387—614no—
356#386—614no—
357#386—614no—
358#386—614no—
359#386—614no—
360#385—614no—
361#384C2×D6111yescharacteristic
362#382—611no—
363#381—614no—
364#380—612no—
365#380—614no—
366C4×C4⋊C4—612no—
367C4×C4⋊C4—612no—
368C4×C4⋊C4—612no—
369C4×C4⋊C4—614no—
370C4×C4⋊C4—612no—
371C4×C4⋊C4—612no—
372C4×C22⋊C4—614no—
373C4×C22⋊C4—614no—
374C4×C22⋊C4—614no—
375C4×C22⋊C4—614no—
376C4×C22⋊C4—612no—
377C4×C22⋊C4—612no—
378#376—611no—
379#376—611no—
380C43—611no—
381C43C2×D6111yescharacteristic
382C61C4×#54011yescharacteristic, complemented, Sylow
383C22×D4—614no—
384C22×D4C22×D6112yes—
385C22×D4C22×D6114yes—
386C23×C4—614no—
387C23×C4—611no—
388C23×C4—614no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4C22×D6114yes—
392#127C4×D6118yescomplemented
393#125—618no—
394#123—6132no—
395#121—6116no—
396#121—6132no—
397C4×D4—28no—
398C4×D4—48no—
399C4×D4—6116no—
400C4×D4—1228no—
401C4×D4—2448no—
402#117—618no—
403#117—1224no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—614no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—612no—
409C2×C4⋊C4—614no—
410C2×C4⋊C4—614no—
411C2×C4⋊C4C22×D6112yes—
412C2×C4⋊C4C22×D6114yes—
413C2×C22⋊C4—614no—
414C2×C22⋊C4—614no—
415C2×C22⋊C4—612no—
416C2×C22⋊C4—614no—
417C2×C22⋊C4—612no—
418C2×C22⋊C4—614no—
419C2×C22⋊C4—614no—
420C2×C22⋊C4—614no—
421C2×C22⋊C4C22×D6114yes—
422C2×C22⋊C4C22×D6114yes—
423C2×C22⋊C4C22×D6114yes—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—614no—
429C2×C42—612no—
430C2×C42—611no—
431C2×C42—612no—
432C2×C42—614no—
433C2×C42—614no—
434C2×C42—1224no—
435C2×C42—2448no—
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6112yes—
438C2×C42C22×D6112yes—
439C2×C42C22×D6111yescharacteristic
440C2×C42C22×D6111yescharacteristic
441#95—612no—
442#95—612no—
443#95—612no—
444#95—612no—
445#95—614no—
446#95—614no—
447#95—614no—
448#95—614no—
449#95—614no—
450#95—614no—
451#95—614no—
452C24—611no—
453C24C23×D6114yes—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—6116no—
459C2×D4—2444no—
460C2×D4—2444no—
461C2×D4C2×C4×D6118yes—
462C2×D4C2×C4×D61116yes—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—611no—
469C22×C4—614no—
470C22×C4—6116no—
471C22×C4—614no—
472C22×C4—611no—
473C22×C4—614no—
474C22×C4—611no—
475C22×C4—1228no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1224no—
481C22×C4—2442no—
482C22×C4—2444no—
483C22×C4—2444no—
484C22×C4—2444no—
485C22×C4—2444no—
486C22×C4C23×D6114yes—
487C22×C4C23×D6112yes—
488C22×C4C23×D6112yes—
489C22×C4C23×D6111yescharacteristic
490C22×C4C23×D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—44no—
494C4⋊C4—618no—
495C4⋊C4—6116no—
496C4⋊C4—618no—
497C4⋊C4—1224no—
498C4⋊C4—1224no—
499C4⋊C4—2444no—
500C4⋊C4—2444no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—6116no—
505C22⋊C4—6116no—
506C22⋊C4—618no—
507C22⋊C4—1224no—
508C22⋊C4—1228no—
509C22⋊C4—2444no—
510C22⋊C4—2444no—
511C22⋊C4—2444no—
512C42—22no—
513C42—24no—
514C42—42no—
515C42—614no—
516C42—614no—
517C42—1222no—
518C42—1222no—
519C42—1222no—
520C42—1224no—
521C42—1228no—
522C42—2444no—
523C42—2444no—
524C42—2442no—
525C42C2×C4×D6114yes—
526C42#1175822414yes—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—612no—
534C23—1221no—
535C23—1221no—
536C23—1222no—
537C23—1222no—
538C23C22×C4×D6118yes—
539C23C24×D6111yescharacteristic
540D4—216no—
541D4—416no—
542D4—24416no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—416no—
553C2×C4—614no—
554C2×C4—612no—
555C2×C4—618no—
556C2×C4—614no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1224no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1224no—
565C2×C4—1228no—
566C2×C4—1224no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1224no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1224no—
573C2×C4—2444no—
574C2×C4—24416no—
575C2×C4—24416no—
576C2×C4—2444no—
577C2×C4C22×C4×D6114yes—
578C2×C4C22×C4×D6112yes—
579C2×C4#41×D6112yes—
580C2×C4C2×#1175822412yes—
581C2×C4C2×#1175822412yes—
582C2×C4C2×#1175822412yes—
583C2×C4#1451982214yes—
584C22—28no—
585C22—28no—
586C22—28no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—1222no—
592C22—1224no—
593C22—2441no—
594C22—2442no—
595C22—2441no—
596C22C23×C4×D6111yescharacteristic
597C22C2×#41×D6111yescharacteristic
598C22C22×#1175822411yescharacteristic
599C22C2×#1451982212yes—
600C22#1684059812yes—
601C4—22no—
602C4—24no—
603C4—42no—
604C4—44no—
605C4—1224no—
606C4—1222no—
607C4—1224no—
608C4—1228no—
609C4—2441no—
610C4—2444no—
611C4—2441no—
612C4C4×#1175822414yes—
613C4#1684036614yesdirect product factor
614C2—48no—
615C2—2442no—
616C2C2×C4×#1175822411yescharacteristic
617C2C4×#1451982212yes—
618C2C2×#1684036611yescharacteristic
619C2#2145920811yescharacteristic
620C2#2145923312yes—
subgroupG/H
centerC22×C4C23×D61
commutatorC22×C61C24×C4
FrattiniC23C24×D61
FittingC4×#127×C61C2
socle, cosocleC23×C61C25
prime radicalC2×#16840366C2
hypercenterC4×#127D61
hypocenterC61C4×#540
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. #21448606C2semidirect
3. #21448578C2semidirect
4. #21448558C2semidirect
5. #21448545C2semidirect
6. #21448515C2semidirect
7. #21448456C2semidirect
8. #21448427C2semidirect
9. C4×#14519769C2semidirect
10. C4×#14519758C2semidirect
11. C4×#14519749C2semidirect
12. C4×#14519748C2semidirect
13. C4×D4×Dic61C2semidirect
14. C4×#14519734C2semidirect
15. C4×#127×C61C2semidirect
16. #16840366C4direct (N × H)
78. #14519769C2×C4semidirect
79. #14519758C2×C4semidirect
80. #14519749C2×C4semidirect
81. #14519748C2×C4semidirect
83. D4×Dic61C2×C4semidirect
86. #14519734C2×C4semidirect
128. #127×C61C2×C4semidirect
296. C4×C61#540semidirect
312. C4×#127D61semidirect
382. C61C4×#540semidirect
392. #127C4×D61semidirect
613. C4#16840366direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403662
#214486062
#214485782
#214485582
#214485452
#214485152
#214484562
#214484272
C4×#145197692
C4×#145197582
C4×#145197492
C4×#145197482
C4×D4×Dic612
C4×#145197342
C4×#127×C612
C4×#54061