Finite Group Databasebeta

← #481×#6923 #30617189

Subgroups — #481×#6923 #30617189

Order 15616.

1060 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×#6923C211yesmaximal, characteristic
2C2×C4⋊C4×#6923C211yesmaximal, characteristic, complemented
3#21461897C212yesmaximal, complemented
4#21461881C212yesmaximal, complemented
5#21461806C211yesmaximal, characteristic
6#21461769C211yesmaximal, characteristic, complemented
7#21461723C211yesmaximal, characteristic, complemented
8#106×#6923C212yesmaximal, complemented
9#102×#6923C212yesmaximal, complemented
10C22⋊C8×#6923C211yesmaximal, characteristic, complemented
11#481×D61C211yesmaximal, characteristic
12C2×D4×#6923—21no—
13C2×D4×#6923C2211yescharacteristic
14C22×C4⋊#6923C2211yescharacteristic
15C22×C4×#6923—21no—
16C22×C4×#6923C2211yescharacteristic
17C2×#14519845C2212yes—
18C2×#14519839C2212yes—
19D61⋊(C22⋊C8)C2211yescharacteristic
20C2×C8×#6923C2212yes—
21#16840714C2211yescharacteristic
22#16840713C2211yescharacteristic
23#16840712—21no—
24#16840710—21no—
25#16840704C2211yescharacteristic
26C4⋊C4×#6923—21no—
27C4⋊C4×#6923—21no—
28C4⋊C4×#6923C2211yescharacteristic
29C4⋊C4×#6923C2212yes—
30C4⋊C4×#6923C2212yes—
31C22⋊C4×#6923—21no—
32C2×(C2×#6923)⋊C4—21no—
33#16840636C2212yes—
34#16840635C2212yes—
35#16840635C2212yes—
36#16840635C2212yes—
37(C2×#6923)⋊C8C2212yes—
38#121×D61C2211yescharacteristic
39C2×C4⋊C4×D61C2211yescharacteristic
40#16840082C412yescomplemented
41#16840081C411yescharacteristic, complemented
42#16840080C411yescharacteristic, complemented
43#16840079C412yescomplemented
44#16840078C411yescharacteristic, complemented
45#106×D61C2212yes—
46#102×D61C2212yes—
47C22⋊C8×D61C2211yescharacteristic
48C61×#481C411yescharacteristic, complemented
49C23×#6923—21no—
50C23×#6923—21no—
51C23×#6923D411yescharacteristic
52C23×#6923D411yescharacteristic
53C2×D61⋊C8C2312yes—
54D4×#6923—24no—
55D4×#6923—44no—
56C2×C4⋊#6923—21no—
57C2×C4⋊#6923—24no—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923C2311yescharacteristic
60C2×C4⋊#6923C2311yescharacteristic
61C2×C4⋊#6923C2312yes—
62C2×C22⋊#6923—21no—
63C2×C22⋊#6923—21no—
64C2×C22⋊#6923—21no—
65C2×C22⋊#6923—21no—
66C2×C4×#6923—24no—
67C2×C4×#6923—22no—
68C2×C4×#6923—22no—
69C2×C4×#6923—21no—
70C2×C4×#6923—44no—
71C2×C4×#6923D411yescharacteristic
72C2×C4×#6923D411yescharacteristic
73C2×C4×#6923C2311yescharacteristic
74C2×C4×#6923C2311yescharacteristic
75#14519845—24no—
76#14519839—24no—
77#14519834—21no—
78C22⋊#92420—21no—
79C8×#6923—24no—
80(C2×#6923)⋊C4—21no—
81(C2×#6923)⋊C4—21no—
82(C2×#6923)⋊C4—22no—
83(C2×#6923)⋊C4—22no—
84(C2×#6923)⋊C4—21no—
85C2×D4×D61—21no—
86C2×D4×D61C2311yescharacteristic
87C22×C4×D61—21no—
88C22×C4×D61C2311yescharacteristic
89C2×C8×D61C2312yes—
90#14519752C2×C411yescharacteristic
91#14519751C2×C411yescharacteristic
92#14519750—21no—
93#14519749—21no—
94#14519748C2×C411yescharacteristic
95C2×C4⋊Dic61C2×C411yescharacteristic, complemented
96C2×#11758196—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61C2×C412yescomplemented
100C4⋊C4×D61C2311yescharacteristic
101C4⋊C4×D61C2312yes—
102C22⋊C4×D61—21no—
103#14519709C2×C412yescomplemented
104#14519708C2×C412yescomplemented
105#14519708C2×C412yescomplemented
106#14519699C2×C412yescomplemented
107#14519698C2×C412yescomplemented
108#14519697C2×C412yescomplemented
109C22⋊#92418C2×C411yescharacteristic, complemented
110#14519689C2×C412yescomplemented
111#121×C61C2×C411yescharacteristic
112C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
113#106×C61C2×C412yescomplemented
114#102×C61C2×C412yescomplemented
115C22⋊C8×C61C2×C411yescharacteristic, complemented
116C23×D61—21no—
117C23×D61C2×D411yescharacteristic
118C22×#6923—22no—
119C22×#6923—22no—
120C22×#6923—24no—
121C22×#6923—22no—
122C22×#6923—22no—
123C22×#6923—22no—
124C22×#6923—21no—
125C22×#6923—21no—
126C22×#6923—41no—
127C22×#6923—42no—
128C22×#6923—42no—
129C22×#6923—42no—
130C22×#6923—42no—
131C22×#6923—42no—
132C22×#6923SD1614yes—
133C22×#6923C2×D411yescharacteristic
134C22×#6923C2×D411yescharacteristic
135C22×#6923C2×D411yescharacteristic
136C22×#6923C2×D411yescharacteristic
137C2×D4×C61—21no—
138C2×D4×C61C22×C411yescharacteristic
139C22×C4×C61—21no—
140C22×C4×C61C22×C411yescharacteristic
141C2×C22⋊D61—21no—
142C2×C22⋊D61—21no—
143C2×C22⋊D61—21no—
144C22×Dic61—21no—
145C22×Dic61#4111yescharacteristic
146D4×D61—22no—
147D4×D61—22no—
148D4×D61—42no—
149D4×D61—42no—
150C2×D244—21no—
151C2×D244C22×C411yescharacteristic
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—21no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61C22×C411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61#4111yescharacteristic
162C2×C4×D61C2411yescharacteristic
163C22⋊#6923—42no—
164C22⋊#6923—42no—
165C22⋊#6923—42no—
166C22⋊#6923—42no—
167C2×#92420—22no—
168C4⋊#6923—24no—
169C4⋊#6923—22no—
170C4⋊#6923—22no—
171C4⋊#6923—22no—
172C4⋊#6923—42no—
173C4⋊#6923#4114yes—
174C4×#6923—22no—
175C4×#6923—22no—
176C4×#6923—22no—
177C4×#6923—44no—
178C4×#6923—42no—
179C4×#6923—44no—
180C4×#6923#4114yes—
181D61⋊C8—22no—
182D61⋊C8—22no—
183C2×C8×C61C22×C412yes—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C22×C412yes—
186C4⋊C4×C61C22×C411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊Dic61—21no—
189#11758198—21no—
190#11758198—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C22×C412yes—
193C4⋊Dic61C22×C411yescharacteristic
194#11758196—22no—
195#11758196—22no—
196#11758196—21no—
197C2×#92418C22×C412yes—
198C8×D61—22no—
199C8×D61—22no—
200C23×C61—21no—
201C23×C61C4×D411yescharacteristic
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61C4×D411yescharacteristic
214C22×D61C2×SD1612yes—
215C22×D61#13512yes—
216C22×D61C22×D411yescharacteristic
217C2×#6923—21no—
218C2×#6923—21no—
219C2×#6923—21no—
220C2×#6923—21no—
221C2×#6923—28no—
222C2×#6923—21no—
223C2×#6923—24no—
224C2×#6923—21no—
225C2×#6923—21no—
226C2×#6923—44no—
227C2×#6923—48no—
228C2×#6923—42no—
229C2×#6923—42no—
230C2×#6923—41no—
231C2×#6923—41no—
232C2×#6923—41no—
233C2×#6923—41no—
234C2×#6923—41no—
235C2×#6923—41no—
236C2×#6923—84no—
237C2×#6923#12312yes—
238C2×#6923#12312yes—
239C2×#6923C2×SD1612yes—
240C2×#6923C2×SD1612yes—
241C2×#6923AGL(1,ℤ/8)12yes—
242D4×C61—22no—
243D4×C61—42no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—42no—
248C2×C4×C61C4×D411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C22⋊D61—42no—
251C22⋊D61—42no—
252C22⋊D61—42no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—42no—
257C2×Dic61C4×D411yescharacteristic
258C2×Dic61C2×#4111yescharacteristic
259D244—22no—
260D244—42no—
261C4×D61—21no—
262C4×D61—21no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—44no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61C2×#4112yes—
271C4×D61C2×#4112yes—
272#92420—42no—
273C8×C61—22no—
274#92418—22no—
275C4×#481—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61C4×SD1612yes—
282C22×C61C2×C4×D411yescharacteristic
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61C4×SD1612yes—
297C2×D61C2×#12311yescharacteristic
298C2×D61#52511yescharacteristic
299C2×D61C22×SD1611yescharacteristic
300C2×D61C2×#13511yescharacteristic
301C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
302C2×D61#57511yescharacteristic
303#6923—22no—
304#6923—22no—
305#6923—41no—
306#6923—44no—
307#6923—81no—
308#6923—81no—
309#6923#48114yesdirect product factor
310C4×C61—21no—
311C4×C61—41no—
312C4×C61—42no—
313C4×C61C4×#4112yes—
314Dic61—21no—
315Dic61—41no—
316Dic61—42no—
317Dic61#52912yes—
318C2×#481—611no—
319C4×#121—611no—
320C2×C4×C4⋊C4—611no—
321#1926—612no—
322#1919—612no—
323#1874—611no—
324#1853—611no—
325#1824—611no—
326C4×#106—612no—
327C4×#102—612no—
328C4×C22⋊C8—611no—
329C2×C61—22no—
330C2×C61—81no—
331C2×C61C4×#12311yescharacteristic
332C2×C61C2×C4×SD1611yescharacteristic
333C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
334D61—22no—
335D61—81no—
336D61C2×#48111yescharacteristic
337D61#309111yescharacteristic
338D61#310311yescharacteristic
339D61#310511yescharacteristic
340C2×#121—611no—
341C2×C4×D4—611no—
342C2×C4×D4—1221no—
343C22×C4⋊C4—611no—
344C22×C4⋊C4—611no—
345C22×C42—611no—
346C22×C42—1221no—
347#481—611no—
348#481—611no—
349#481—612no—
350#481—612no—
351#481—611no—
352#481#692311yescharacteristic, direct product factor
353C2×#106—612no—
354C2×#106—612no—
355C2×#102—612no—
356C2×#102—612no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8—611no—
359C2×C4×C8—612no—
360#390—611no—
361#389—611no—
362#388—1221no—
363#386—1221no—
364#380—611no—
365C4×C4⋊C4—612no—
366C4×C4⋊C4—612no—
367C4×C4⋊C4—611no—
368C4×C4⋊C4—1221no—
369C4×C4⋊C4—1221no—
370C4×C22⋊C4—1221no—
371C2×#95—1221no—
372#340—612no—
373#340—612no—
374#340—612no—
375#340—612no—
376#336—612no—
377C61C4×#48111yescharacteristic, complemented, Sylow
378C22×D4—611no—
379C22×D4—1221no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—1221no—
384C23×C4—1221no—
385C23×C4—1221no—
386C22×C8—612no—
387C22×C8—612no—
388#121—611no—
389#121—611no—
390#121—611no—
391#121—1221no—
392#121—1221no—
393#121C2×#692311yescharacteristic
394C4×D4—1224no—
395C4×D4—2444no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1224no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C2×#692311yescharacteristic, complemented
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—1224no—
420C2×C42—1221no—
421C2×C42—1222no—
422C2×C42—1222no—
423C2×C42—2444no—
424#106—612no—
425#106—612no—
426#106—612no—
427#106—1224no—
428#106C2×#692312yescomplemented
429#102—612no—
430#102—612no—
431#102—612no—
432#102—1224no—
433#102C2×#692312yescomplemented
434C22⋊C8—611no—
435C22⋊C8—612no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8C2×#692311yescharacteristic, complemented
439C4×C8—1224no—
440#95—1222no—
441#95—1222no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445C24—611no—
446C24—1221no—
447C2×D4—21no—
448C2×D4—611no—
449C2×D4—1221no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—2442no—
456C2×D4—2442no—
457C2×D4C22×#692311yescharacteristic
458C22×C4—21no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—614no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1224no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4—2442no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2441no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4C22×#692311yescharacteristic
492C2×C8—612no—
493C2×C8—1222no—
494C2×C8—1222no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8C22×#692312yes—
499C4⋊C4—21no—
500C4⋊C4—614no—
501C4⋊C4—612no—
502C4⋊C4—611no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1221no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—2442no—
512C4⋊C4C22×#692312yes—
513C4⋊C4C22×#692311yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—1221no—
516C22⋊C4—1221no—
517C22⋊C4—1221no—
518C22⋊C4—2442no—
519C22⋊C4—2442no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C42—614no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C42—2444no—
527C42—2442no—
528C42—2444no—
529C23—21no—
530C23—611no—
531C23—612no—
532C23—612no—
533C23—611no—
534C23—1221no—
535C23—1221no—
536C23—1221no—
537C23—1221no—
538C23—1221no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23D4×#692311yescharacteristic
546D4—22no—
547D4—42no—
548D4—1222no—
549D4—2442no—
550D4—2442no—
551D4—2442no—
552D4—2442no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—611no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—611no—
566C2×C4—1222no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1222no—
570C2×C4—1228no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1224no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1222no—
582C2×C4—4884no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2442no—
586C2×C4—2441no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2448no—
599C2×C4D4×#692311yescharacteristic
600C2×C4C23×#692311yescharacteristic
601C8—22no—
602C8—1222no—
603C8—2442no—
604C22—21no—
605C22—21no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—611no—
610C22—611no—
611C22—611no—
612C22—611no—
613C22—611no—
614C22—611no—
615C22—612no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—1221no—
621C22—1221no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—2441no—
627C22—2441no—
628C22—2441no—
629C22SD16×#692312yes—
630C22C2×D4×#692311yescharacteristic
631C4—21no—
632C4—41no—
633C4—42no—
634C4—614no—
635C4—612no—
636C4—1222no—
637C4—1222no—
638C4—1221no—
639C4—4881no—
640C4—4881no—
641C4—2441no—
642C4—2444no—
643C4—2441no—
644C4—2442no—
645C4#41×#692312yes—
646C2—22no—
647C2—81no—
648C2—611no—
649C2—611no—
650C2—611no—
651C2—611no—
652C2—1222no—
653C2—4881no—
654C2#123×#692311yescharacteristic
655C2C2×SD16×#692311yescharacteristic
656C2AGL(1,ℤ/8)×#692311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#481C4
socleC22×C61C2×C4×D4
prime radical#121×D61C22
cosocleC2×C4×D61C24
hypercenter#481#6923
hypocenterC61C4×#481
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×#6923C2semidirect
3. #21461897C2semidirect
4. #21461881C2semidirect
6. #21461769C2semidirect
7. #21461723C2semidirect
8. #106×#6923C2semidirect
9. #102×#6923C2semidirect
10. C22⋊C8×#6923C2semidirect
40. #16840082C4semidirect
41. #16840081C4semidirect
42. #16840080C4semidirect
43. #16840079C4semidirect
44. #16840078C4semidirect
48. C61×#481C4semidirect
95. C2×C4⋊Dic61C2×C4semidirect
99. C4⋊C4×D61C2×C4semidirect
103. #14519709C2×C4semidirect
104. #14519708C2×C4semidirect
105. #14519708C2×C4semidirect
106. #14519699C2×C4semidirect
107. #14519698C2×C4semidirect
108. #14519697C2×C4semidirect
109. C22⋊#92418C2×C4semidirect
110. #14519689C2×C4semidirect
112. C2×C4⋊C4×C61C2×C4semidirect
113. #106×C61C2×C4semidirect
114. #102×C61C2×C4semidirect
115. C22⋊C8×C61C2×C4semidirect
309. #6923#481direct (N × H)
352. #481#6923direct (N × H)
377. C61C4×#481semidirect
409. C2×C4⋊C4C2×#6923semidirect
428. #106C2×#6923semidirect
433. #102C2×#6923semidirect
438. C22⋊C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#121×#69232
C2×C4⋊C4×#69232
#214618972
#214618812
#214618062
#214617692
#214617232
#106×#69232
#102×#69232
C22⋊C8×#69232
#481×D612
C4×#48161