Finite Group Databasebeta

← #29685656

Subgroups — #29685656

Order 15616.

1952 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
1C2×#16839896C211yesmaximal, characteristic
2C2×#16839872C212yesmaximal, complemented
3C4×#14519751C211yesmaximal, characteristic, complemented
4#21446167C212yesmaximal, complemented
5#21446136C211yesmaximal, characteristic, complemented
6#21446080C212yesmaximal, complemented
7#21446012C211yesmaximal, characteristic, complemented
8#21445750C211yesmaximal, characteristic, complemented
9#21445644C212yesmaximal, complemented
10#21445581C211yesmaximal, characteristic, complemented
11C22×C4×D4×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519751C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic, complemented
15C2×C4×C22⋊D61—21no—
16#16839896C418yescomplemented
17#16839872—28no—
18C2×#14519708C2212yescomplemented
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×#14519697C2212yescomplemented
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25C2×#14519688C2212yescomplemented
26#16839664C2211yescharacteristic, complemented
27#16839660C2211yescharacteristic, complemented
28#16839653—21no—
29#16839641—21no—
30#16839622C2211yescharacteristic
31C4×C4⋊Dic61C2211yescharacteristic, complemented
32C4×#11758198—21no—
33#16839524C2212yescomplemented
34#16839524C2212yescomplemented
35#16839517C2212yescomplemented
36C23×D4×C61C2211yescharacteristic
37C24×C4×C61—21no—
38C2×C4×D4×C61—28no—
39C2×C4×D4×C61—21no—
40C2×C4×D4×C61—21no—
41C2×C4×D4×C61C2212yes—
42C2×C4×D4×C61C2212yes—
43C22×C4⋊C4×C61C2211yescharacteristic
44C22×C22⋊C4×C61—21no—
45C22×C42×C61C2211yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61—21no—
49C2×#11758199—24no—
50C2×#11758199—24no—
51C2×#11758199C2×C418yescomplemented
52C22×#92418C2312yes—
53C22×#92418C2312yes—
54#14519751—22no—
55#14519751C2×C414yescomplemented
56C4×D244—24no—
57C4×C22⋊D61—44no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68#14519708—24no—
69#14519698—24no—
70#14519698—24no—
71#14519698C2×C418yescomplemented
72#14519697—24no—
73C22⋊#92418—22no—
74C22⋊#92418C2×C414yescomplemented
75#14519688—24no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79C25×C61—21no—
80C22×D4×C61—24no—
81C22×D4×C61—24no—
82C22×D4×C61—21no—
83C22×D4×C61—22no—
84C22×D4×C61C2×C414yescomplemented
85C22×D4×C61C2312yes—
86C23×C4×C61—28no—
87C23×C4×C61—21no—
88C23×C4×C61—22no—
89C23×C4×C61—22no—
90C23×C4×C61—44no—
91C23×C4×C61—44no—
92C23×C4×C61D411yescharacteristic
93C23×C4×C61D411yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C4×D4×C61—28no—
96C4×D4×C61—28no—
97C4×D4×C61—24no—
98C4×D4×C61—24no—
99C4×D4×C61—24no—
100C4×D4×C61D418yescomplemented
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—24no—
103C2×C4⋊C4×C61C2312yes—
104C2×C22⋊C4×C61—28no—
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—22no—
107C2×C42×C61—24no—
108C2×C42×C61D411yescharacteristic
109C2×C42×C61D411yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112C24×C61—24no—
113C24×C61—24no—
114C24×C61—24no—
115C24×C61—22no—
116C24×C61—42no—
117C24×C61—42no—
118C24×C61—42no—
119C24×C61—42no—
120C24×C61C2×D411yescharacteristic
121C23×D61—21no—
122C2×D4×C61—24no—
123C2×D4×C61—24no—
124C2×D4×C61—216no—
125C2×D4×C61—216no—
126C2×D4×C61—24no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61C22×C414yes—
131C2×D4×C61C2×D414yes—
132C2×D4×C61#4114yes—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—24no—
139C22×C4×C61—28no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—28no—
144C22×C4×C61—41no—
145C22×C4×C61—432no—
146C22×C4×C61—44no—
147C22×C4×C61—44no—
148C22×C4×C61—44no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61—44no—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61#4112yes—
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—24no—
161C2×C22⋊D61—42no—
162C2×C22⋊D61—42no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—21no—
169C2×D244C22×C412yescomplemented
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C2×D414yes—
176C22⋊C4×C61—28no—
177C22⋊C4×C61—28no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61—22no—
181C42×C61C2×D414yes—
182C22⋊Dic61—42no—
183#11758199—216no—
184#11758198—24no—
185#11758198—42no—
186#11758198—42no—
187#11758198—42no—
188C4⋊Dic61—21no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C22×C412yescomplemented
192#11758196—42no—
193C4×Dic61—42no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418C22×C414yescomplemented
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—216no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—24no—
208C23×C61—48no—
209C23×C61—48no—
210C23×C61—48no—
211C23×C61—48no—
212C23×C61—48no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—48no—
218C23×C61C4×D412yes—
219C23×C61AGL(1,ℤ/8)12yes—
220C23×C61AGL(1,ℤ/8)12yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229D4×C61—216no—
230D4×C61—24no—
231D4×C61—24no—
232D4×C61—24no—
233D4×C61C4×D418yescomplemented
234C2×C4×C61—28no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—21no—
246C2×C4×C61—24no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—44no—
250C2×C4×C61—416no—
251C2×C4×C61—416no—
252C2×C4×C61—48no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61C4×D412yes—
256C2×C4×C61#12012yes—
257C2×C4×C61#12012yes—
258C2×C4×C61AGL(1,ℤ/8)12yes—
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C22×D412yes—
261C2×C4×C61C2×#4112yes—
262C2×C4×C61C2×#4111yescharacteristic
263C22⋊D61—48no—
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273D244—24no—
274D244—42no—
275C4×D61—84no—
276#92418—24no—
277#92418—42no—
278#21452—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—28no—
286C22×C61—432no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61—44no—
290C22×C61—44no—
291C22×C61—44no—
292C22×C61—44no—
293C22×C61#44214yescomplemented
294C22×C61C2×C4×D411yescharacteristic
295C22×C61C2×#12011yescharacteristic
296C22×C61#52511yescharacteristic
297C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61#57511yescharacteristic
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—44no—
313C4×C61—41no—
314C4×C61C2×C4×D412yes—
315C4×C61#52512yes—
316Dic61—42no—
317Dic61—81no—
318Dic61—81no—
319C22×C4×D4D6111yescharacteristic, complemented
320C2×#447—611no—
321C2×#442—612no—
322C4×#121—611no—
323#1879—612no—
324#1874—611no—
325#1864—612no—
326#1853—611no—
327#1786—611no—
328#1762—612no—
329#1749—611no—
330C2×C61—24no—
331C2×C61—48no—
332C2×C61C4×#12011yescharacteristic
333C2×C61C2×#44211yescharacteristic
334C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
335C2×C61#300311yescharacteristic
336C2×C61#300611yescharacteristic
337C2×C61#310911yescharacteristic
338C2×C61#311411yescharacteristic
339D61—82no—
340C23×D4C2×D6111yescharacteristic
341C24×C4—21no—
342C22×D8—612no—
343C2×#121—611no—
344C2×C4×D4—28no—
345C2×C4×D4—21no—
346C2×C4×D4—21no—
347C2×C4×D4—611no—
348C2×C4×D4—1221no—
349C2×C4×D4C2×D6112yes—
350C2×C4×D4C2×D6112yes—
351C22×C4⋊C4C2×D6111yescharacteristic
352C22×C22⋊C4—21no—
353C22×C42C2×D6111yescharacteristic
354#447—618no—
355#442—1228no—
356C2×#106—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×#102—612no—
360C2×#102—612no—
361C2×C22⋊C8—611no—
362C2×C22⋊C8—611no—
363C2×#97—612no—
364#390—611no—
365#389—611no—
366#388—1221no—
367#386—1221no—
368#380—611no—
369C4×C4⋊C4—611no—
370C4×C22⋊C4—1221no—
371#340—612no—
372#340—612no—
373#336—612no—
374C61#2145211yescharacteristic, complemented, Sylow
375C25—21no—
376C22×D4—24no—
377C22×D4—24no—
378C22×D4—22no—
379C22×D4—21no—
380C22×D4—611no—
381C22×D4—1221no—
382C22×D4C4×D6114yescomplemented
383C22×D4C22×D6112yes—
384C23×C4—28no—
385C23×C4—21no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—44no—
389C23×C4—44no—
390C23×C4—1221no—
391C23×C4C22⋊D6111yescharacteristic
392C23×C4C22⋊D6111yescharacteristic
393C23×C4C22×D6111yescharacteristic
394C2×D8—618no—
395C2×D8—1224no—
396C2×D8—1224no—
397C22×C8—612no—
398C22×C8—612no—
399#121—614no—
400#121—1222no—
401C4×D4—28no—
402C4×D4—28no—
403C4×D4—24no—
404C4×D4—24no—
405C4×D4—24no—
406C4×D4—1224no—
407C4×D4—2444no—
408C4×D4C22⋊D6118yescomplemented
409C2×C4⋊C4—21no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22×D6112yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—24no—
425C2×C42—1221no—
426C2×C42C22⋊D6111yescharacteristic
427C2×C42C22⋊D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430#106—1224no—
431#102—618no—
432#102—1224no—
433#102—1224no—
434#102—1224no—
435C22⋊C8—614no—
436C22⋊C8—1222no—
437#97—1224no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441C24—24no—
442C24—24no—
443C24—22no—
444C24—24no—
445C24—42no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—1221no—
450C24C2×C22⋊D6111yescharacteristic
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—216no—
454C2×D4—216no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—612no—
460C2×D4—1224no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4—2442no—
465C2×D4—2442no—
466C2×D4C2×C4×D6114yes—
467C2×D4#1175822214yes—
468C2×D4C2×C22⋊D6114yes—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—28no—
480C22×C4—41no—
481C22×C4—432no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4C2×C4×D6112yes—
495C22×C4#1175822212yes—
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502D8—12216no—
503C2×C8—614no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—612no—
512C4⋊C4—1221no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—2442no—
516C4⋊C4C2×C22⋊D6114yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—1224no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—2442no—
528C42C2×C22⋊D6114yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—216no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—48no—
547C23—1222no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23C4×C22⋊D6112yes—
555C23#1451980212yes—
556C23#1451980312yes—
557C23C22×C22⋊D6111yescharacteristic
558D4—216no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4—1224no—
563D4—2448no—
564D4—2442no—
565D4C4×C22⋊D6118yescomplemented
566C2×C4—28no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—1222no—
588C2×C4—4884no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4C4×C22⋊D6112yes—
598C2×C4#1451974612yes—
599C2×C4#1451974612yes—
600C2×C4#1451980312yes—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C22×C22⋊D6112yes—
603C2×C4C2×#1175822212yes—
604C2×C4C2×#1175822211yescharacteristic
605C8—1224no—
606C8—2442no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—432no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—4881no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—2442no—
626C22—2442no—
627C22—2442no—
628C22#1683987214yescomplemented
629C22C2×C4×C22⋊D6111yescharacteristic
630C22C2×#1451974611yescharacteristic
631C22#1684027311yescharacteristic
632C22C2×#1451980211yescharacteristic
633C22C2×#1451980311yescharacteristic
634C22C2×#1451980311yescharacteristic
635C22#1684056011yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—44no—
641C4—41no—
642C4—4881no—
643C4—4881no—
644C4—2442no—
645C4C2×C4×C22⋊D6112yes—
646C4#1684027312yes—
647C2—24no—
648C2—48no—
649C2—4882no—
650C2C4×#1451974611yescharacteristic
651C2C2×#1683987211yescharacteristic
652C2C4×#1451980311yescharacteristic
653C2#2145504611yescharacteristic
654C2#2145507111yescharacteristic
655C2#2145591011yescharacteristic
656C2#2145595711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×C4×D4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839896C2
cosocleC22×C4×C61C24
hypercenterC22×C4×D4D61
hypocenterC61#21452
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#16839872C2semidirect
3. C4×#14519751C2semidirect
4. #21446167C2semidirect
5. #21446136C2semidirect
6. #21446080C2semidirect
7. #21446012C2semidirect
8. #21445750C2semidirect
9. #21445644C2semidirect
10. #21445581C2semidirect
11. C22×C4×D4×C61C2semidirect
14. C2×C4×D244C22semidirect
16. #16839896C4semidirect
18. C2×#14519708C22semidirect
22. C2×#14519697C22semidirect
25. C2×#14519688C22semidirect
26. #16839664C22semidirect
27. #16839660C22semidirect
31. C4×C4⋊Dic61C22semidirect
33. #16839524C22semidirect
34. #16839524C22semidirect
35. #16839517C22semidirect
51. C2×#11758199C2×C4semidirect
55. #14519751C2×C4semidirect
71. #14519698C2×C4semidirect
74. C22⋊#92418C2×C4semidirect
84. C22×D4×C61C2×C4semidirect
100. C4×D4×C61D4semidirect
169. C2×D244C22×C4semidirect
191. C4⋊Dic61C22×C4semidirect
199. C2×#92418C22×C4semidirect
233. D4×C61C4×D4semidirect
293. C22×C61#442semidirect
319. C22×C4×D4D61semidirect
374. C61#21452semidirect
382. C22×D4C4×D61semidirect
408. C4×D4C22⋊D61semidirect
565. D4C4×C22⋊D61semidirect
628. C22#16839872semidirect

Maximal subgroups

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

subgroupindex
C2×#168398962
C2×#168398722
C4×#145197512
#214461672
#214461362
#214460802
#214460122
#214457502
#214456442
#214455812
C22×C4×D4×C612
#2145261