Finite Group Databasebeta

← #29681261

Subgroups — #29681261

Order 15616.

2278 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519752C211yesmaximal, characteristic
2C2×#117×D61C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C2×#16839766C211yesmaximal, characteristic, complemented
5C2×#16839725C211yesmaximal, characteristic, complemented
6#21446140C214yesmaximal, complemented
7#21446135C214yesmaximal, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C23×D244—21no—
11C23×D244C2211yescharacteristic
12C23×C4×D61C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C2×#14519752—21no—
15C2×#14519752—21no—
16C2×#14519752—24no—
17C2×#14519752C418yescomplemented
18C2×#14519752C2212yes—
19C2×#14519752C2212yes—
20C2×#14519736C2211yescharacteristic
21C2×#14519735C2212yescomplemented
22C2×#14519734—21no—
23C2×#14519733—21no—
24#117×D61—24no—
25#117×D61—24no—
26C2×C4⋊C4×D61C2212yescomplemented
27C22×#11758198—21no—
28C2×C22⋊C4×D61—21no—
29C2×C42×D61—21no—
30#16839768—24no—
31#16839766—24no—
32C2×#14519699C2214yescomplemented
33C2×#14519697C2214yescomplemented
34#16839725—24no—
35C2×#14519689C2212yescomplemented
36C2×#14519689C2212yescomplemented
37#16839524C2214yescomplemented
38C22×M16×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#340C2214yescomplemented
42C24×D61—21no—
43C24×D61D411yescharacteristic
44C22×D244—22no—
45C22×D244—22no—
46C22×D244—22no—
47C22×D244—22no—
48C22×D244—28no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—22no—
53C22×D244—48no—
54C22×D244C2×C414yescomplemented
55C22×D244C2312yes—
56C23×Dic61D411yescharacteristic
57C22×C4×D61—22no—
58C22×C4×D61—22no—
59C22×C4×D61—22no—
60C22×C4×D61—22no—
61C22×C4×D61—22no—
62C22×C4×D61C2×C414yes—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D414yes—
66C22×C4×D61C2311yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C2×#11758194—22no—
69C2×#11758194—22no—
70C2×#11758194—22no—
71C22×#92418C2312yes—
72#14519752—28no—
73#14519752—28no—
74#14519752—24no—
75#14519752—24no—
76#14519752C2×C418yes—
77#14519752C2×C418yescomplemented
78#14519736—22no—
79#14519736D414yescomplemented
80#14519735—24no—
81#14519735—24no—
82#14519734—28no—
83#14519733—28no—
84C2×C4⋊Dic61C2312yes—
85C2×#11758196—22no—
86C4⋊C4×D61—24no—
87C4⋊C4×D61—24no—
88C2×C22⋊Dic61—21no—
89C2×#11758198—22no—
90C2×#11758198—28no—
91C2×#11758198—22no—
92C2×#11758198—22no—
93C22⋊C4×D61—28no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—22no—
96C42×D61—28no—
97#14519699—28no—
98#14519697—28no—
99#14519689—24no—
100#14519689—24no—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C2×M16×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61—22no—
106C22×C8×C61C2312yes—
107#117×C61—22no—
108#117×C61D414yescomplemented
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61C2×C414yescomplemented
112C2×C4⋊C4×C61C2312yes—
113C2×C4⋊C4×C61C2312yes—
114C2×C22⋊C4×C61—21no—
115C2×C42×C61—21no—
116C24×C61C2×D411yescharacteristic
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—24no—
120C23×D61—24no—
121C23×D61—22no—
122C23×D61—42no—
123C23×D61—44no—
124C23×D61—44no—
125C23×D61—42no—
126C23×D61—42no—
127C23×D61C22⋊C414yes—
128C23×D61C2×D411yescharacteristic
129C23×D61C2×D411yescharacteristic
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—44no—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61C22⋊C414yes—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×D244—28no—
149C2×D244—28no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—28no—
155C2×D244—28no—
156C2×D244—24no—
157C2×D244—22no—
158C2×D244—48no—
159C2×D244—432no—
160C2×D244—48no—
161C2×D244—42no—
162C2×D244—44no—
163C2×D244—44no—
164C2×D244—42no—
165C2×D244C22×C414yes—
166C2×C4×D61—28no—
167C2×C4×D61—28no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—28no—
171C2×C4×D61—28no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×C4×D61—21no—
181C2×C4×D61—21no—
182C2×C4×D61—28no—
183C2×C4×D61—44no—
184C2×C4×D61—44no—
185C2×C4×D61C22⋊C414yes—
186C2×C4×D61C22⋊C414yes—
187C2×C4×D61C22×C412yes—
188C2×C4×D61C22×C412yes—
189C2×C4×D61C2×D412yes—
190C2×C4×D61C2×D412yes—
191M16×C61—44no—
192C2×C8×C61—22no—
193C2×C8×C61—22no—
194C2×C8×C61—22no—
195C4⋊C4×C61—24no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61C22×C414yes—
199C4⋊C4×C61C2×D414yescomplemented
200C22⋊C4×C61—24no—
201C42×C61—24no—
202C22⋊Dic61—24no—
203#11758198—28no—
204#11758198—28no—
205#11758198—28no—
206C4⋊Dic61—22no—
207C4⋊Dic61C2×D414yescomplemented
208#11758196—28no—
209C4×Dic61—28no—
210C4×Dic61—24no—
211#11758194—44no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—22no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61C2×C22⋊C412yes—
218C23×C61#12012yes—
219C23×C61#12312yes—
220C23×C61C22×D411yescharacteristic
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—22no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—28no—
229C22×D61—28no—
230C22×D61—24no—
231C22×D61—44no—
232C22×D61—48no—
233C22×D61—48no—
234C22×D61—48no—
235C22×D61—48no—
236C22×D61—42no—
237C22×D61—42no—
238C22×D61—44no—
239C22×D61—44no—
240C22×D61—48no—
241C22×D61—88no—
242C22×D61—88no—
243C22×D61C2×C22⋊C412yes—
244C22×D61C2×C22⋊C412yes—
245C22×D61AGL(1,ℤ/8)14yes—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—48no—
249C2×C4×C61—42no—
250C2×C4×C61—44no—
251C2×C4×C61—42no—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C4×D414yes—
254C2×C4×C61C4×D414yes—
255C2×C4×C61#12012yes—
256C2×C4×C61#12114yes—
257C2×C4×C61#12114yes—
258C2×C4×C61#12312yes—
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—28no—
270C2×Dic61—24no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C412yes—
275D244—24no—
276D244—24no—
277D244—416no—
278D244—416no—
279D244—42no—
280D244—42no—
281D244—48no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—28no—
291C4×D61—28no—
292C4×D61—416no—
293C8×C61—42no—
294#92418—42no—
295#20912—611nomaximal, Sylow
296C22×C61—24no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—24no—
300C22×C61#38614yes—
301C22×C61C22×C22⋊C411yescharacteristic
302C22×C61C2×#12011yescharacteristic
303C22×C61C2×#12311yescharacteristic
304C22×C61C2×AGL(1,ℤ/8)12yes—
305C22×C61#57512yes—
306C2×D61—24no—
307C2×D61—22no—
308C2×D61—24no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—416no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—44no—
317C2×D61—44no—
318C2×D61—44no—
319C2×D61—816no—
320C2×D61—84no—
321C2×D61—84no—
322C2×D61#41814yescomplemented
323C4×C61—44no—
324C4×C61—44no—
325C4×C61#38614yes—
326C4×C61C2×C4×D411yescharacteristic
327C4×C61C2×C4×D411yescharacteristic
328C4×C61C2×#12111yescharacteristic
329C4×C61C2×#12111yescharacteristic
330Dic61—22no—
331Dic61—22no—
332Dic61—44no—
333C22×#121—611no—
334C22×#117—611no—
335C2×#418—611no—
336C2×#418—611no—
337C2×#407—611no—
338#1875—614no—
339#1874—614no—
340#1736—611no—
341#1736D6111yescharacteristic, complemented
342C2×C61—24no—
343C2×C61C2×#38611yescharacteristic
344C2×C61C2×#41811yescharacteristic
345C2×C61#289611yescharacteristic
346C2×C61#300612yes—
347C2×C61#309412yes—
348D61—24no—
349D61—42no—
350D61—84no—
351C23×D4—611no—
352C23×D4—1221no—
353C24×C4—611no—
354C22×M16—611no—
355C22×M16C2×D6111yescharacteristic
356C2×#121—618no—
357C2×#121—612no—
358C2×#121—612no—
359C2×#121—1221no—
360C2×#121—1224no—
361C2×#121—1221no—
362C2×#117—611no—
363C2×#117—612no—
364C2×#117—1221no—
365C2×#117—1221no—
366C2×#117—1224no—
367C2×#117—1224no—
368C2×#117C2×D6111yescharacteristic
369C22×C4⋊C4—612no—
370C22×C4⋊C4C2×D6111yescharacteristic
371C22×C22⋊C4—1221no—
372C22×C22⋊C4—1221no—
373C22×C42—1221no—
374#418—1224no—
375#418—1224no—
376C2×#102—614no—
377C2×#102—614no—
378#407—1224no—
379C2×C22⋊C8—612no—
380C2×C22⋊C8—612no—
381#340—614no—
382#340C2×D6114yescomplemented
383C61#2091211yescharacteristic, complemented, Sylow
384C25—611no—
385C25—1221no—
386C22×D4—614no—
387C22×D4—612no—
388C22×D4—1222no—
389C22×D4—1222no—
390C22×D4—1228no—
391C22×D4—1222no—
392C22×D4—1222no—
393C22×D4—1222no—
394C22×D4—1222no—
395C22×D4—1221no—
396C22×D4—1222no—
397C22×D4—2448no—
398C23×C4—21no—
399C23×C4—611no—
400C23×C4—614no—
401C23×C4—611no—
402C23×C4—614no—
403C23×C4—611no—
404C23×C4—611no—
405C23×C4—611no—
406C23×C4—1222no—
407C23×C4—1222no—
408C23×C4—1222no—
409C23×C4—1222no—
410C23×C4—1222no—
411C23×C4C22×D6111yescharacteristic
412C2×M16—22no—
413C2×M16—22no—
414C2×M16—22no—
415C2×M16—1222no—
416C2×M16—1222no—
417C2×M16—1222no—
418C22×C8—612no—
419C22×C8C22×D6112yes—
420#121—618no—
421#121—618no—
422#121—1224no—
423#121—1224no—
424#121—1228no—
425#121—1228no—
426#117—22no—
427#117—614no—
428#117—1228no—
429#117—1228no—
430#117—1224no—
431#117—1222no—
432#117—1224no—
433#117D24414yescomplemented
434C2×C4⋊C4—22no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—612no—
437C2×C4⋊C4—1224no—
438C2×C4⋊C4—1222no—
439C2×C4⋊C4—1224no—
440C2×C4⋊C4C4×D6114yescomplemented
441C2×C4⋊C4C22×D6112yes—
442C2×C4⋊C4C22×D6112yes—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—1228no—
445C2×C22⋊C4—1222no—
446C2×C22⋊C4—1222no—
447C2×C22⋊C4—1228no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1221no—
450C2×C42—21no—
451C2×C42—1228no—
452C2×C42—1222no—
453C2×C42—1221no—
454#102—1228no—
455#102—1228no—
456C22⋊C8—1224no—
457C22⋊C8—1224no—
458C24—611no—
459C24—611no—
460C24—614no—
461C24—1224no—
462C24—1222no—
463C24—1222no—
464C24—1222no—
465C24—1224no—
466C24—2442no—
467C24—2444no—
468C24—2444no—
469C24—2442no—
470C24—2442no—
471C24D4×D6111yescharacteristic
472C2×D4—614no—
473C2×D4—1228no—
474C2×D4—1228no—
475C2×D4—1222no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1224no—
481C2×D4—1228no—
482C2×D4—1228no—
483C2×D4—2448no—
484C2×D4—24432no—
485C2×D4—2448no—
486C2×D4—2442no—
487C2×D4—2444no—
488C2×D4—2444no—
489C2×D4—2442no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—44no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—614no—
497C22×C4—612no—
498C22×C4—612no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—614no—
502C22×C4—614no—
503C22×C4—1228no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1228no—
515C22×C4—1228no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1228no—
519C22×C4—1228no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—2444no—
524C22×C4—2444no—
525C22×C4C2×C4×D6112yes—
526C22×C4C2×D24412yes—
527C22×C4#1175822212yes—
528C22×C4D4×D6112yes—
529C22×C4D4×D6112yes—
530C22×C4D4×D6111yescharacteristic
531C22×C4#1175822612yes—
532C22×C4C23×D6111yescharacteristic
533M16—44no—
534M16—2444no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—1222no—
539C2×C8—1222no—
540C2×C8—1222no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—24no—
544C4⋊C4—614no—
545C4⋊C4—1228no—
546C4⋊C4—1222no—
547C4⋊C4C2×C4×D6114yes—
548C4⋊C4C2×D24414yescomplemented
549C22⋊C4—24no—
550C22⋊C4—1228no—
551C22⋊C4—1228no—
552C22⋊C4—1228no—
553C22⋊C4—1224no—
554C42—24no—
555C42—1228no—
556C42—1224no—
557C23—22no—
558C23—22no—
559C23—612no—
560C23—614no—
561C23—612no—
562C23—1228no—
563C23—1221no—
564C23—1221no—
565C23—1222no—
566C23—1221no—
567C23—1221no—
568C23—1222no—
569C23—1222no—
570C23—1228no—
571C23—1224no—
572C23—4888no—
573C23—4888no—
574C23—2444no—
575C23—2448no—
576C23—2448no—
577C23—2448no—
578C23—2448no—
579C23—2442no—
580C23—2442no—
581C23—2444no—
582C23—2444no—
583C23—2448no—
584C23C22⋊C4×D6112yes—
585C23#1451974712yes—
586C23#1451976212yes—
587C23C2×D4×D6111yescharacteristic
588D4—1224no—
589D4—1224no—
590D4—24416no—
591D4—24416no—
592D4—2442no—
593D4—2442no—
594D4—2448no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1228no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1228no—
617C2×C4—1228no—
618C2×C4—1224no—
619C2×C4—24416no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4C22⋊C4×D6112yes—
623C2×C4#1451973914yes—
624C2×C4C4×D24414yes—
625C2×C4#1451974712yes—
626C2×C4#1451975014yes—
627C2×C4#1451975214yes—
628C2×C4#1451976212yes—
629C2×C4C22×C4×D6111yescharacteristic
630C2×C4C2×D4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C22×D24411yescharacteristic
633C2×C4C2×#1175822211yescharacteristic
634C2×C4C2×#1175822611yescharacteristic
635C8—42no—
636C8—2442no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—22no—
641C22—614no—
642C22—1222no—
643C22—1222no—
644C22—1224no—
645C22—1224no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—48816no—
650C22—4884no—
651C22—4884no—
652C22—24416no—
653C22—2444no—
654C22—2444no—
655C22—2444no—
656C22—2442no—
657C22—2442no—
658C22#1683964014yes—
659C22C2×C22⋊C4×D6111yescharacteristic
660C22C2×#1451974711yescharacteristic
661C22C2×#1451976211yescharacteristic
662C22AGL(1,ℤ/8)×D6112yes—
663C22#1684056112yes—
664C4—44no—
665C4—44no—
666C4—1222no—
667C4—1222no—
668C4—2444no—
669C4#1683964014yes—
670C4C2×#1451973911yescharacteristic
671C4C2×C4×D24411yescharacteristic
672C4C2×#1451975011yescharacteristic
673C4C2×#1451975211yescharacteristic
674C2—24no—
675C2—1224no—
676C2—4884no—
677C2—2442no—
678C2C2×#1683964011yescharacteristic
679C2#418×D6111yescharacteristic
680C2#2145433711yescharacteristic
681C2#2145507212yes—
682C2#2145576112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#14519752C2
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20912
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#117×D61C2semidirect
3. C2×#16839768C2semidirect
4. C2×#16839766C2semidirect
5. C2×#16839725C2semidirect
6. #21446140C2semidirect
7. #21446135C2semidirect
8. #21445538C2semidirect
9. C61×#1736C2semidirect
17. C2×#14519752C4semidirect
21. C2×#14519735C22semidirect
26. C2×C4⋊C4×D61C22semidirect
32. C2×#14519699C22semidirect
33. C2×#14519697C22semidirect
35. C2×#14519689C22semidirect
36. C2×#14519689C22semidirect
37. #16839524C22semidirect
41. C61×#340C22semidirect
54. C22×D244C2×C4semidirect
77. #14519752C2×C4semidirect
79. #14519736D4semidirect
108. #117×C61D4semidirect
111. C2×C4⋊C4×C61C2×C4semidirect
199. C4⋊C4×C61C2×D4semidirect
207. C4⋊Dic61C2×D4semidirect
322. C2×D61#418semidirect
341. #1736D61semidirect
382. #340C2×D61semidirect
383. C61#20912semidirect
433. #117D244semidirect
440. C2×C4⋊C4C4×D61semidirect
548. C4⋊C4C2×D244semidirect

Maximal subgroups

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

subgroupindex
C22×#145197522
C2×#117×D612
C2×#168397682
C2×#168397662
C2×#168397252
#214461402
#214461352
#214455382
C61×#17362
#2091261