Finite Group Databasebeta

← #29828810

Subgroups — #29828810

Order 15616.

1576 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16839959C211yesmaximal, characteristic, complemented
3C2×#16839899C212yesmaximal, complemented
4#21454161C211yesmaximal, characteristic
5#21446176C212yesmaximal, complemented
6#21446171C212yesmaximal, complemented
7#21446133C212yesmaximal, complemented
8#21445825C211yesmaximal, characteristic, complemented
9#21445824C211yesmaximal, characteristic, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C2×C61×#460C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×#11758200C2212yes—
14C2×#14519748—21no—
15C2×#14519748—21no—
16C2×#14519748C2212yes—
17C2×Q8×Dic61C2211yescharacteristic
18C2×C4×Dic122—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742—21no—
21C2×#14519742C2211yescharacteristic
22C4×Q8×D61—24no—
23C4×Q8×D61—24no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C2211yescharacteristic
26C2×C42×D61—21no—
27C2×C42×D61C2211yescharacteristic
28#16839959—24no—
29#16839899—28no—
30C2×C4⋊#92418C2211yescharacteristic
31C2×#14519702C2212yes—
32C2×#14519701C2212yes—
33C2×#14519698C2212yes—
34C2×#14519689C2212yes—
35#16839665—21no—
36#16839665—21no—
37#16839665C2212yes—
38#16839632C2211yescharacteristic
39#16839525C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C22×SD16×C61C2212yes—
43C2×#127×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C61×#460—24no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×#102×C61C2212yes—
48C22×Dic122—21no—
49C22×Dic122C2311yescharacteristic
50C2×Q8×D61—22no—
51C2×Q8×D61—22no—
52C2×Q8×D61—22no—
53C2×Q8×D61—22no—
54C2×Q8×D61—22no—
55C2×Q8×D61D414yescomplemented
56C22×C22⋊D61—22no—
57C22×C4×D61—21no—
58C22×C4×D61—21no—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×#11758200—24no—
63C2×#11758200—24no—
64C2×#11758200—24no—
65C22×#92418C2312yes—
66#14519748—24no—
67#14519748—24no—
68#14519748—44no—
69#14519748—44no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72C4×Dic122—28no—
73C4×Dic122—22no—
74C4×Dic122D414yescomplemented
75#14519742—28no—
76#14519742—22no—
77#14519742—22no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C4⋊C4×D61—28no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C2×C22⋊Dic61—22no—
90C2×C22⋊Dic61—22no—
91C2×C22⋊Dic61—22no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61D411yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C42×D61—28no—
99C42×D61—22no—
100C42×D61—22no—
101C4⋊#92418—22no—
102#14519702—24no—
103#14519701—24no—
104#14519698—24no—
105#14519689—24no—
106C22×Q8×C61C2311yescharacteristic
107C22×D4×C61—21no—
108C22×D4×C61—21no—
109C22×D4×C61C2312yes—
110C2×SD16×C61—24no—
111C2×SD16×C61—24no—
112C2×SD16×C61—24no—
113C22×C8×C61C2312yes—
114#127×C61—22no—
115#127×C61—22no—
116C4×Q8×C61—22no—
117C4×Q8×C61D414yescomplemented
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C4⋊C8×C61—22no—
123#102×C61—24no—
124C24×C61—22no—
125C23×D61C2×D411yescharacteristic
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61C2×D412yes—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61#4112yes—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—42no—
139C2×D4×C61—42no—
140C2×D4×C61—42no—
141C2×D4×C61—42no—
142C2×D4×C61—42no—
143C2×D4×C61—42no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—44no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—44no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158Q8×D61—28no—
159Q8×D61—28no—
160Q8×D61—28no—
161Q8×D61—28no—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—24no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—24no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—41no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61SD1614yescomplemented
178C2×C4×D61C2×D412yes—
179C2×C4×D61#4112yes—
180C2×Dic122—24no—
181C2×Dic122—24no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—24no—
186C2×Dic122C2×D412yes—
187C2×Dic122C2×D412yes—
188C2×Dic122#4112yes—
189SD16×C61—48no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C2×C8×C61—22no—
193C4⋊C4×C61—24no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61C2×D412yes—
196C42×C61—24no—
197C42×C61C2×D412yes—
198C42×C61C2×D412yes—
199C22⋊Dic61—44no—
200C22⋊Dic61—44no—
201C22⋊Dic61—44no—
202#11758200—48no—
203C4⋊Dic61—24no—
204C4⋊Dic61—21no—
205C4⋊Dic61—22no—
206C4⋊Dic61—22no—
207C4⋊Dic61—42no—
208C4⋊Dic61—42no—
209C4⋊Dic61C2×D412yes—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213C4×Dic61—24no—
214C4×Dic61—24no—
215C4×Dic61—24no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C2×#92418—22no—
221C2×#92418—22no—
222C2×#92418—22no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61—42no—
226C23×C61—42no—
227C23×C61—42no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61C22×D411yescharacteristic
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61#12112yes—
236C22×D61C2×SD1612yes—
237C22×D61AGL(1,ℤ/8)12yes—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61#12114yescomplemented
242D4×C61—42no—
243D4×C61—42no—
244D4×C61—42no—
245D4×C61—42no—
246D4×C61—84no—
247D4×C61—84no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61#12012yes—
257C2×C4×C61#12012yes—
258C2×C4×C61C2×SD1612yes—
259C2×C4×C61#13712yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C22⋊D61—88no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—22no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283C2×Dic61#12112yes—
284C2×Dic61C2×SD1612yes—
285C4×D61—216no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—28no—
291C4×D61—44no—
292C4×D61—48no—
293Dic122—216no—
294Dic122—22no—
295Dic122—22no—
296Dic122—22no—
297Dic122#12114yescomplemented
298C8×C61—42no—
299#92418—42no—
300#32903—611nomaximal, Sylow
301C22×C61—82no—
302C22×C61—82no—
303C22×C61—82no—
304C22×C61—82no—
305C22×C61—82no—
306C22×C61—82no—
307C22×C61—82no—
308C22×C61C2×#12011yescharacteristic
309C22×C61C2×#12111yescharacteristic
310C22×C61#52511yescharacteristic
311C22×C61C22×SD1611yescharacteristic
312C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
313C22×C61C2×#13711yescharacteristic
314C22×C61#57511yescharacteristic
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—22no—
321C2×D61#46014yescomplemented
322C4×C61—24no—
323C4×C61—22no—
324C4×C61—42no—
325C4×C61#44814yescomplemented
326C4×C61C2×#12111yescharacteristic
327C4×C61C2×#12111yescharacteristic
328C4×C61#52511yescharacteristic
329C4×C61#52511yescharacteristic
330Dic61—22no—
331Dic61—24no—
332Dic61—22no—
333Dic61—41no—
334Dic61—42no—
335Dic61—82no—
336C22×C4×Q8—611no—
337C2×#460—611no—
338C2×#460D6111yescharacteristic, complemented
339C2×#448—612no—
340#2856—611no—
341#1880—612no—
342#1880—612no—
343#1874—612no—
344#1803—611no—
345#1803—611no—
346#1797—611no—
347C2×C61—162no—
348C2×C61#284711yescharacteristic
349C2×C61C2×#44811yescharacteristic
350C2×C61#300611yescharacteristic
351C2×C61C2×#46011yescharacteristic
352C2×C61#303711yescharacteristic
353C2×C61#310311yescharacteristic
354C2×C61#311211yescharacteristic
355D61—24no—
356C23×Q8—611no—
357C22×SD16—612no—
358C22×SD16C2×D6112yes—
359C2×#127C2×D6111yescharacteristic
360C2×#121—612no—
361C2×#121—1221no—
362C2×#121—1221no—
363C2×C4×Q8—611no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×Q8—1221no—
367C2×C4×Q8—1221no—
368C2×C4×Q8—1224no—
369C2×C4×Q8—1224no—
370C2×C4×Q8C2×D6111yescharacteristic
371C22×C4⋊C4—611no—
372C22×C4⋊C4—1221no—
373C22×C42—611no—
374C22×C42—1221no—
375#460—24no—
376#460—1224no—
377#448—1228no—
378C2×C4⋊C8—611no—
379C2×C4⋊C8C2×D6111yescharacteristic
380C2×#103—612no—
381C2×#103—612no—
382C2×#102—612no—
383C2×#102C2×D6112yes—
384C2×C22⋊C8—612no—
385#390—612no—
386#390—1221no—
387#390—1221no—
388#384—611no—
389#340—612no—
390#340—612no—
391#336—612no—
392C61#3290311yescharacteristic, complemented, Sylow
393C22×Q8—614no—
394C22×Q8—611no—
395C22×Q8—1221no—
396C22×Q8—1222no—
397C22×Q8—1222no—
398C22×Q8—1222no—
399C22×Q8—1222no—
400C22×Q8—1222no—
401C22×Q8C22×D6111yescharacteristic
402C22×D4—21no—
403C22×D4—21no—
404C22×D4—1222no—
405C22×D4C22×D6112yes—
406C23×C4—611no—
407C23×C4—611no—
408C23×C4—611no—
409C23×C4—1221no—
410C23×C4—1221no—
411C2×SD16—24no—
412C2×SD16—24no—
413C2×SD16—24no—
414C2×SD16—1224no—
415C2×SD16—1224no—
416C2×SD16—1224no—
417C22×C8—612no—
418C22×C8C22×D6112yes—
419#127—22no—
420#127—22no—
421#121—1224no—
422#121—1224no—
423#121—2444no—
424#121—2444no—
425C4×Q8—22no—
426C4×Q8—614no—
427C4×Q8—1228no—
428C4×Q8—1222no—
429C4×Q8—1228no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×Q8—1222no—
434C4×Q8C22⋊D6114yescomplemented
435C2×C4⋊C4—21no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—612no—
438C2×C4⋊C4—1222no—
439C2×C4⋊C4—1228no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1222no—
450C2×C22⋊C4—1222no—
451C2×C42—21no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—1228no—
456C2×C42—1221no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42—1222no—
460C2×C42—1222no—
461C2×C42C22×D6111yescharacteristic
462C4⋊C8—22no—
463C4⋊C8—1222no—
464#103—1224no—
465#103—1224no—
466#102—24no—
467#102—1224no—
468C22⋊C8—1224no—
469C24—22no—
470C24—611no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—612no—
475C2×Q8—612no—
476C2×Q8—612no—
477C2×Q8—1224no—
478C2×Q8—1224no—
479C2×Q8—1224no—
480C2×Q8—1228no—
481C2×Q8—1228no—
482C2×Q8—1228no—
483C2×Q8—1228no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8D4×D6112yes—
488C2×Q8#1175822412yes—
489C2×Q8C2×C22⋊D6112yes—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—2444no—
503C2×D4—2444no—
504C2×D4—2444no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—612no—
511C22×C4—614no—
512C22×C4—611no—
513C22×C4—1224no—
514C22×C4—1224no—
515C22×C4—1224no—
516C22×C4—1222no—
517C22×C4—1224no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1221no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—2441no—
529C22×C4—2442no—
530C22×C4—2442no—
531C22×C4D4×D6111yescharacteristic
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534SD16—48no—
535SD16—2448no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—1222no—
540C2×C8—1222no—
541C2×C8—1222no—
542C4⋊C4—21no—
543C4⋊C4—24no—
544C4⋊C4—612no—
545C4⋊C4—1224no—
546C4⋊C4—1224no—
547C4⋊C4—1224no—
548C4⋊C4—1221no—
549C4⋊C4—1224no—
550C4⋊C4—1222no—
551C4⋊C4—1222no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4C2×C22⋊D6112yes—
555C22⋊C4—2444no—
556C22⋊C4—2444no—
557C22⋊C4—2444no—
558C42—24no—
559C42—1224no—
560C42—1224no—
561C42—1221no—
562C42—1221no—
563C42—1222no—
564C42—1222no—
565C42—1224no—
566C42D4×D6112yes—
567C42C2×C22⋊D6112yes—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—612no—
576C23—612no—
577C23—612no—
578C23—1221no—
579C23—1221no—
580C23—1221no—
581C23—1221no—
582C23C2×D4×D6111yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—22no—
586Q8—614no—
587Q8—12216no—
588Q8—1222no—
589Q8—1222no—
590Q8—1222no—
591Q8#1451974814yescomplemented
592D4—42no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—84no—
597D4—84no—
598D4—4888no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—12216no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1228no—
622C2×C4—1222no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—2444no—
627C2×C4—2448no—
628C2×C4—2442no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4#1451974512yes—
634C2×C4#1451974512yes—
635C2×C4SD16×D6112yes—
636C2×C4#1451980612yes—
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4C22×C22⋊D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C2×C4C2×#1175822411yescharacteristic
643C8—42no—
644C8—2442no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—614no—
653C22—1222no—
654C22—1222no—
655C22—1222no—
656C22—1222no—
657C22—1222no—
658C22—1222no—
659C22C2×#1451974511yescharacteristic
660C22#121×D6111yescharacteristic
661C22#1684027211yescharacteristic
662C22C2×SD16×D6111yescharacteristic
663C22AGL(1,ℤ/8)×D6111yescharacteristic
664C22C2×#1451980611yescharacteristic
665C22#1684055811yescharacteristic
666C4—24no—
667C4—22no—
668C4—42no—
669C4—1222no—
670C4—1224no—
671C4—1222no—
672C4—4882no—
673C4—2441no—
674C4—2442no—
675C4#1683989914yescomplemented
676C4#121×D6111yescharacteristic
677C4C2×#1451974811yescharacteristic
678C4#1684026911yescharacteristic
679C4#1684027211yescharacteristic
680C2—162no—
681C2—1224no—
682C2#2145409811yescharacteristic
683C2C2×#1683989911yescharacteristic
684C2#2145506711yescharacteristic
685C2#460×D6111yescharacteristic
686C2#2145528611yescharacteristic
687C2#2145584911yescharacteristic
688C2#2145594011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#460C2
socleC23×C61C22×D4
prime radical#21454161C2
hypercenterC2×#460D61
hypocenterC61#32903
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×C4×Q8×D61C2semidirect
2. C2×#16839959C2semidirect
3. C2×#16839899C2semidirect
5. #21446176C2semidirect
6. #21446171C2semidirect
7. #21446133C2semidirect
8. #21445825C2semidirect
9. #21445824C2semidirect
10. #21445796C2semidirect
11. C2×C61×#460C2semidirect
55. C2×Q8×D61D4semidirect
74. C4×Dic122D4semidirect
117. C4×Q8×C61D4semidirect
177. C2×C4×D61SD16semidirect
241. Q8×C61#121semidirect
297. Dic122#121semidirect
321. C2×D61#460semidirect
325. C4×C61#448semidirect
338. C2×#460D61semidirect
392. C61#32903semidirect
434. C4×Q8C22⋊D61semidirect
591. Q8#14519748semidirect
675. C4#16839899semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
C2×#168399592
C2×#168398992
#214541612
#214461762
#214461712
#214461332
#214458252
#214458242
#214457962
C2×C61×#4602
#3290361