Finite Group Databasebeta

← #29681547

Subgroups — #29681547

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459022C211yesmaximal, characteristic, complemented
2C2×#16839779C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic, complemented
4#21448401C211yesmaximal, characteristic, complemented
5#21446197C218yesmaximal, complemented
6#21445557C211yesmaximal, characteristic, complemented
7#21445530C211yesmaximal, characteristic, complemented
8C22×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic
9C22×#11758194C2211yescharacteristic
10#16840245—22no—
11#16840245C2214yescomplemented
12#16840240—21no—
13#16840237—22no—
14#16840237—21no—
15#41×Dic61—22no—
16#41×Dic61—22no—
17C2×#14519736—22no—
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839779—24no—
22C2×#14519704C2214yescomplemented
23#16839767—22no—
24#16839767C2214yescomplemented
25C2×#14519698—22no—
26#16839665—22no—
27#16839665C2214yescomplemented
28#16839618—22no—
29#16839530C2214yescomplemented
30#16839523C2214yescomplemented
31C22×#41×C61C2211yescharacteristic
32C23×D4×C61C2211yescharacteristic
33C2×AGL(1,ℤ/8)×C61—24no—
34C2×AGL(1,ℤ/8)×C61—22no—
35C2×AGL(1,ℤ/8)×C61C418yescomplemented
36C2×AGL(1,ℤ/8)×C61C2214yes—
37C22×SD16×C61—21no—
38C22×D8×C61—21no—
39C22×M16×C61C2211yescharacteristic
40C2×#11758194—21no—
41C2×#11758194—22no—
42C2×#11758194—22no—
43C2×#11758194C2312yes—
44C22×#92418—21no—
45Q8×Dic61—24no—
46Q8×Dic61—24no—
47D4×Dic61—28no—
48D4×Dic61—24no—
49D4×Dic61—24no—
50D4×Dic61—28no—
51#14519736—24no—
52#14519736—24no—
53#14519736—21no—
54#14519736—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—24no—
60#14519736—22no—
61#14519736—22no—
62#14519736D412yes—
63#14519736C2312yes—
64C2×C4⋊Dic61—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D412yes—
67C2×C4⋊Dic61C2312yes—
68C2×C22⋊Dic61—24no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61—21no—
71C2×C22⋊Dic61—28no—
72C2×C22⋊Dic61D412yes—
73C2×C4×Dic61—24no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61D412yes—
76C2×C4×Dic61C2312yes—
77#14519704—28no—
78#14519698—28no—
79C25×C61—21no—
80C2×#41×C61—21no—
81C2×#41×C61—21no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61C2×C414yes—
85C2×#41×C61D414yes—
86C2×#41×C61D412yes—
87C2×#41×C61C2312yes—
88C22×Q8×C61D411yescharacteristic
89C22×D4×C61—21no—
90C22×D4×C61—21no—
91C22×D4×C61—21no—
92C22×D4×C61—22no—
93C22×D4×C61—21no—
94C22×D4×C61—21no—
95C22×D4×C61—21no—
96C22×D4×C61—22no—
97C22×D4×C61—22no—
98C22×D4×C61C2×C414yescomplemented
99C22×D4×C61D411yescharacteristic
100C22×D4×C61C2312yes—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103AGL(1,ℤ/8)×C61—216no—
104AGL(1,ℤ/8)×C61—28no—
105AGL(1,ℤ/8)×C61C2×C4116yescomplemented
106C2×SD16×C61—28no—
107C2×SD16×C61—24no—
108C2×D8×C61—28no—
109C2×D8×C61—24no—
110C2×M16×C61—22no—
111C2×M16×C61—21no—
112C2×M16×C61C2×C414yescomplemented
113C2×M16×C61C2312yes—
114C22×C8×C61—21no—
115C24×C61—21no—
116C24×C61—22no—
117C24×C61—24no—
118C24×C61—41no—
119C24×C61—41no—
120C24×C61—42no—
121C24×C61—84no—
122C24×C61C2×D411yescharacteristic
123#41×C61—24no—
124#41×C61—24no—
125#41×C61—24no—
126#41×C61—24no—
127#41×C61—24no—
128#41×C61—22no—
129#41×C61—22no—
130#41×C61C22⋊C418yescomplemented
131#41×C61C22×C414yes—
132#41×C61C2×D414yes—
133C2×Q8×C61—21no—
134C2×Q8×C61—22no—
135C2×Q8×C61C22⋊C414yescomplemented
136C2×Q8×C61C2×D412yes—
137C2×D4×C61—24no—
138C2×D4×C61—22no—
139C2×D4×C61—24no—
140C2×D4×C61—22no—
141C2×D4×C61—24no—
142C2×D4×C61—24no—
143C2×D4×C61—24no—
144C2×D4×C61—24no—
145C2×D4×C61—24no—
146C2×D4×C61—21no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—22no—
151C2×D4×C61—22no—
152C2×D4×C61—22no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C2×D4×C61—42no—
156C2×D4×C61—42no—
157C2×D4×C61—42no—
158C2×D4×C61—42no—
159C2×D4×C61—42no—
160C2×D4×C61C22⋊C414yescomplemented
161C2×D4×C61C22×C414yes—
162C2×D4×C61C2×D412yes—
163C22×C4×C61—21no—
164C22×C4×C61—24no—
165C22×C4×C61—22no—
166C22×C4×C61—24no—
167C22×C4×C61—22no—
168C22×C4×C61C22×C412yes—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D412yes—
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61#4112yes—
176C22×C4×C61#4112yes—
177C22×C4×C61C2411yescharacteristic
178C22×Dic61—28no—
179C22×Dic61—22no—
180C22×Dic61—22no—
181C22×Dic61C2×D414yes—
182SD16×C61—216no—
183D8×C61—216no—
184M16×C61—24no—
185M16×C61—22no—
186M16×C61C22×C414yes—
187C2×C8×C61—24no—
188C2×C8×C61—22no—
189C22⋊Dic61—28no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—42no—
195C22⋊Dic61—88no—
196C4⋊Dic61—28no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—42no—
201C4⋊Dic61C2×D414yes—
202C4×Dic61—28no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209#11758194—22no—
210#11758194—22no—
211#11758194—42no—
212C2×#92418—22no—
213C2×#92418—42no—
214C23×C61—22no—
215C23×C61—24no—
216C23×C61—24no—
217C23×C61—42no—
218C23×C61—48no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—44no—
222C23×C61—44no—
223C23×C61—82no—
224C23×C61—84no—
225C23×C61—84no—
226C23×C61—88no—
227C23×C61—88no—
228C23×C61C4×D412yes—
229C23×C61#12011yescharacteristic
230C23×C61#12011yescharacteristic
231C23×C61#12112yes—
232C23×C61C22×D411yescharacteristic
233Q8×C61—22no—
234Q8×C61—24no—
235Q8×C61C2×C22⋊C414yes—
236D4×C61—24no—
237D4×C61—24no—
238D4×C61—28no—
239D4×C61—24no—
240D4×C61—24no—
241D4×C61—22no—
242D4×C61—24no—
243D4×C61—42no—
244D4×C61—42no—
245D4×C61—44no—
246D4×C61—44no—
247D4×C61C2×C22⋊C414yes—
248C2×C4×C61—22no—
249C2×C4×C61—24no—
250C2×C4×C61—24no—
251C2×C4×C61—24no—
252C2×C4×C61—28no—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C4×D412yes—
256C2×C4×C61C4×D412yes—
257C2×C4×C61C4×D412yes—
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61#12011yescharacteristic
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12112yes—
263C2×C4×C61#12312yes—
264C2×C4×C61#12312yes—
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×Dic61—24no—
272C2×Dic61—24no—
273C2×Dic61—44no—
274C2×Dic61—44no—
275C2×Dic61—42no—
276C2×Dic61—82no—
277C2×Dic61—88no—
278C8×C61—24no—
279#92418—42no—
280#20945—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—24no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—48no—
288C22×C61—44no—
289C22×C61—44no—
290C22×C61—84no—
291C22×C61—84no—
292C22×C61—88no—
293C22×C61—88no—
294C22×C61—84no—
295C22×C61#38612yes—
296C22×C61#38612yes—
297C22×C61C2×C4×D411yescharacteristic
298C22×C61C2×#12011yescharacteristic
299C22×C61C2×#12111yescharacteristic
300C4×C61—24no—
301C4×C61#38612yes—
302C4×C61#38612yes—
303C4×C61C22×C22⋊C411yescharacteristic
304C4×C61C2×C4×D411yescharacteristic
305C4×C61C2×#12111yescharacteristic
306C4×C61C2×#12311yescharacteristic
307Dic61—44no—
308Dic61—44no—
309Dic61—82no—
310C22×AGL(1,ℤ/8)D6111yescharacteristic
311#3428—611no—
312C2×#421—611no—
313C2×#418—611no—
314#2309—611no—
315#1883—618no—
316#1742—611no—
317#1733—611no—
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—22no—
321C2×C61—22no—
322C2×C61—44no—
323C2×C61—84no—
324C2×C61C2×#38611yescharacteristic
325C2×C61#302112yes—
326C22×#41C2×D6111yescharacteristic
327C23×D4C2×D6111yescharacteristic
328C2×AGL(1,ℤ/8)—24no—
329C2×AGL(1,ℤ/8)—22no—
330C2×AGL(1,ℤ/8)Dic6118yescomplemented
331C2×AGL(1,ℤ/8)C2×D6114yes—
332C22×SD16—21no—
333C22×D8—21no—
334C22×M16—611no—
335C22×M16C2×D6111yescharacteristic
336#520—614no—
337#520—1222no—
338#519—1221no—
339#518—1222no—
340#518—1221no—
341C4×#41—1222no—
342C4×#41—1222no—
343C2×#117—611no—
344C2×#117—611no—
345C2×#117—611no—
346C2×#117—1222no—
347#421—1224no—
348C2×#104—614no—
349#418—614no—
350#418—1222no—
351C2×#102—1222no—
352#390—614no—
353#390—1222no—
354#379—1222no—
355#343—614no—
356#339—614no—
357C61#2094511yescharacteristic, complemented, Sylow
358C25—21no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41C2×Dic6114yes—
364C2×#41C22⋊D6114yes—
365C2×#41C22⋊D6112yes—
366C2×#41C22×D6112yes—
367C22×Q8C22⋊D6111yescharacteristic
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4—21no—
372C22×D4—22no—
373C22×D4—22no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4—22no—
377C22×D4C2×Dic6114yescomplemented
378C22×D4C22⋊D6111yescharacteristic
379C22×D4C22×D6112yes—
380C23×C4—21no—
381C23×C4C22×D6111yescharacteristic
382AGL(1,ℤ/8)—216no—
383AGL(1,ℤ/8)—28no—
384AGL(1,ℤ/8)C2×Dic61116yescomplemented
385C2×SD16—28no—
386C2×SD16—24no—
387C2×D8—28no—
388C2×D8—24no—
389C2×M16—21no—
390C2×M16—22no—
391C2×M16—612no—
392C2×M16—1221no—
393C2×M16—1222no—
394C2×M16—1222no—
395C2×M16C2×Dic6114yescomplemented
396C2×M16C22×D6112yes—
397C22×C8—21no—
398C22×C8—1221no—
399C4×Q8—1224no—
400C4×Q8—1224no—
401C4×D4—1228no—
402C4×D4—1228no—
403C4×D4—1224no—
404C4×D4—1224no—
405#117—612no—
406#117—612no—
407#117—1224no—
408#117—1224no—
409#117—1224no—
410#117—1221no—
411#117—1221no—
412#117—1221no—
413#117—1221no—
414#117—1221no—
415#117—1221no—
416#117—1222no—
417#117—1222no—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—1224no—
421C2×C4⋊C4—1221no—
422C2×C22⋊C4—612no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1228no—
426C2×C22⋊C4—1221no—
427C2×C42—612no—
428C2×C42—612no—
429C2×C42—1224no—
430C2×C42—1221no—
431#104—1228no—
432#102—1228no—
433C24—21no—
434C24—22no—
435C24—24no—
436C24—41no—
437C24—41no—
438C24—42no—
439C24—84no—
440C24D4×D6111yescharacteristic
441#41—22no—
442#41—22no—
443#41—24no—
444#41—24no—
445#41—24no—
446#41—24no—
447#41—24no—
448#41C22⋊Dic6118yescomplemented
449#41C22×Dic6114yes—
450#41C2×C22⋊D6114yes—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8C22⋊Dic6114yescomplemented
454C2×Q8C2×C22⋊D6112yes—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4C22⋊Dic6114yescomplemented
479C2×D4C22×Dic6114yes—
480C2×D4C2×C22⋊D6112yes—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—614no—
487C22×C4—1228no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4D4×D6111yescharacteristic
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4#1175822412yes—
494C22×C4#1175822412yes—
495C22×C4C22×Dic6112yes—
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6112yes—
499C22×C4C23×D6111yescharacteristic
500SD16—216no—
501D8—216no—
502M16—22no—
503M16—24no—
504M16—1222no—
505M16—1222no—
506M16—2442no—
507M16C22×Dic6114yes—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—1222no—
511C2×C8—2442no—
512C4⋊C4—614no—
513C4⋊C4—1228no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—2442no—
518C22⋊C4—1228no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—4888no—
524C22⋊C4—2442no—
525C42—1228no—
526C42—1222no—
527C42—1222no—
528C42—1222no—
529C42—1222no—
530C42—1222no—
531C42—1222no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—42no—
536C23—48no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—82no—
542C23—84no—
543C23—84no—
544C23—88no—
545C23—88no—
546C23D4×Dic6112yes—
547C23#1451974511yescharacteristic
548C23#1451974511yescharacteristic
549C23#1451974912yes—
550C23C2×D4×D6111yescharacteristic
551Q8—22no—
552Q8—24no—
553Q8C2×C22⋊Dic6114yes—
554D4—24no—
555D4—24no—
556D4—28no—
557D4—24no—
558D4—24no—
559D4—22no—
560D4—24no—
561D4—42no—
562D4—42no—
563D4—44no—
564D4—44no—
565D4C2×C22⋊Dic6114yes—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—4882no—
574C2×C4—4888no—
575C2×C4—2444no—
576C2×C4—2442no—
577C2×C4—2444no—
578C2×C4C2×C22⋊Dic6112yes—
579C2×C4C2×C22⋊Dic6112yes—
580C2×C4D4×Dic6112yes—
581C2×C4D4×Dic6112yes—
582C2×C4D4×Dic6112yes—
583C2×C4#1451974511yescharacteristic
584C2×C4#1451974511yescharacteristic
585C2×C4#1451974912yes—
586C2×C4#1451974912yes—
587C2×C4#1451974912yes—
588C2×C4#1451975812yes—
589C2×C4#1451975812yes—
590C2×C4C23×Dic6111yescharacteristic
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C2×C4C22×C22⋊D6111yescharacteristic
594C2×C4C2×#1175822411yescharacteristic
595C2×C4C2×#1175822411yescharacteristic
596C8—24no—
597C8—2442no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—48no—
605C22—44no—
606C22—44no—
607C22—84no—
608C22—84no—
609C22—88no—
610C22—88no—
611C22—84no—
612C22#1683964212yes—
613C22#1683964212yes—
614C22C2×D4×Dic6111yescharacteristic
615C22C2×#1451974511yescharacteristic
616C22C2×#1451974911yescharacteristic
617C4—24no—
618C4—4882no—
619C4—2444no—
620C4—2444no—
621C4#1683964212yes—
622C4#1683964212yes—
623C4C22×C22⋊Dic6111yescharacteristic
624C4C2×D4×Dic6111yescharacteristic
625C4C2×#1451974911yescharacteristic
626C4C2×#1451975811yescharacteristic
627C2—21no—
628C2—21no—
629C2—22no—
630C2—22no—
631C2—44no—
632C2—84no—
633C2C2×#1683964211yescharacteristic
634C2#2145518812yes—
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC22×AGL(1,ℤ/8)×C61C2
socleC22×C61C2×#120
cosocleC22×C4×C61C24
hypercenterC22×AGL(1,ℤ/8)D61
hypocenterC61#20945
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #21459022C2semidirect
2. C2×#16839779C2semidirect
3. C2×#16839767C2semidirect
4. #21448401C2semidirect
5. #21446197C2semidirect
6. #21445557C2semidirect
7. #21445530C2semidirect
11. #16840245C22semidirect
22. C2×#14519704C22semidirect
24. #16839767C22semidirect
27. #16839665C22semidirect
29. #16839530C22semidirect
30. #16839523C22semidirect
35. C2×AGL(1,ℤ/8)×C61C4semidirect
98. C22×D4×C61C2×C4semidirect
105. AGL(1,ℤ/8)×C61C2×C4semidirect
112. C2×M16×C61C2×C4semidirect
130. #41×C61C22⋊C4semidirect
135. C2×Q8×C61C22⋊C4semidirect
160. C2×D4×C61C22⋊C4semidirect
330. C2×AGL(1,ℤ/8)Dic61semidirect
357. C61#20945semidirect
377. C22×D4C2×Dic61semidirect
384. AGL(1,ℤ/8)C2×Dic61semidirect
395. C2×M16C2×Dic61semidirect
448. #41C22⋊Dic61semidirect
453. C2×Q8C22⋊Dic61semidirect
478. C2×D4C22⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
#214590222
C2×#168397792
C2×#168397672
#214484012
#214461972
#214455572
#214455302
C22×AGL(1,ℤ/8)×C612
#2094561