Finite Group Databasebeta

← #29683043

Subgroups — #29683043

Order 15616.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839871C211yesmaximal, characteristic
2#21448281C211yesmaximal, characteristic, complemented
3C4×#14519739C211yesmaximal, characteristic, complemented
4#21446448C211yesmaximal, characteristic, complemented
5#21446404C211yesmaximal, characteristic, complemented
6#21446395C211yesmaximal, characteristic, complemented
7#21446170C211yesmaximal, characteristic
8#21446164C211yesmaximal, characteristic
9#21446083C211yesmaximal, characteristic, complemented
10#21446063C211yesmaximal, characteristic, complemented
11#21445718C211yesmaximal, characteristic, complemented
12#21445646C211yesmaximal, characteristic, complemented
13#21445643C211yesmaximal, characteristic, complemented
14#21445566C211yesmaximal, characteristic, complemented
15C61×#1925C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519733—21no—
22C2×C42×D61—21no—
23#16839871C418yescomplemented
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839656—21no—
36#16839650—21no—
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C22⋊C4×Dic61—21no—
41C4×#11758198—21no—
42#16839609—21no—
43#16839609—21no—
44C42×Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C23×Dic61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758199—22no—
66C2×#11758199—22no—
67C2×#11758199C2×C414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C4×D244—22no—
71C4×D244—22no—
72D4×Dic61—22no—
73D4×Dic61C2×C414yescomplemented
74#14519739—22no—
75#14519739—22no—
76#14519739—22no—
77#14519739C2×C414yes—
78#14519733—44no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61D411yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99C42×D61—44no—
100#14519709—22no—
101#14519708C2×C414yescomplemented
102#14519699—22no—
103#14519699C2×C414yes—
104#14519698—22no—
105#14519697—22no—
106#14519687—22no—
107#14519687C2×C414yescomplemented
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C22×D4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#107×C61—22no—
124#102×C61—22no—
125#102×C61C2×C414yescomplemented
126#95×C61—21no—
127C24×C61—21no—
128C23×D61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C22×C412yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C22×C412yes—
155C2×D244#4112yes—
156C2×D244#4112yes—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C22×C412yes—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C412yes—
177C4⋊C4×C61#4112yes—
178C22⋊C4×C61—42no—
179C22⋊C4×C61—42no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—42no—
183C22⋊Dic61—24no—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186#11758199—28no—
187#11758198—24no—
188#11758198—24no—
189#11758198—24no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C22×C412yes—
195C4⋊Dic61#4112yes—
196#11758196—42no—
197#11758196—42no—
198C4×Dic61—22no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—22no—
202C4×Dic61—41no—
203C4×Dic61—41no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61C22×C412yes—
208C4×Dic61#4112yes—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C22×C412yes—
215C23×C61—22no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61AGL(1,ℤ/8)12yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—48no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61C4×D412yes—
275C2×Dic61C4×D412yes—
276C2×Dic61AGL(1,ℤ/8)12yes—
277D244—22no—
278D244—22no—
279D244—22no—
280D244#11714yescomplemented
281C4×D61—48no—
282C4×D61—48no—
283C4×D61—48no—
284C8×C61—22no—
285C8×C61—41no—
286#92418—22no—
287#92418—41no—
288#21122—611nomaximal, Sylow
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61C2×C4×D411yescharacteristic
297C22×C61#52511yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57511yescharacteristic
303C2×D61—24no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—42no—
314C4×C61—81no—
315C4×C61C2×#11711yescharacteristic
316C4×C61C4×#4111yescharacteristic
317C4×C61#52911yescharacteristic
318C4×C61#52911yescharacteristic
319Dic61—22no—
320Dic61—22no—
321Dic61—22no—
322Dic61—41no—
323Dic61—42no—
324Dic61—42no—
325Dic61—81no—
326C2×#442—611no—
327#2293—611no—
328C42×D4—611no—
329#1926—611no—
330#1925D6111yescharacteristic, complemented
331#1920—611no—
332#1919—611no—
333#1879—611no—
334#1879—611no—
335#1864—611no—
336#1862—611no—
337#1776—611no—
338#1762—611no—
339#1762—611no—
340#1745—611no—
341C2×C61—82no—
342C2×C61#229011yescharacteristic
343C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
344C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
345C2×C61#310911yescharacteristic
346C2×C61#311111yescharacteristic
347C2×C61#311411yescharacteristic
348C2×C61#311511yescharacteristic
349D61—44no—
350C22×D8—611no—
351C2×C4×D4—611no—
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×#117—1221no—
356C22×C42—1221no—
357#442—618no—
358C2×#107C2×D6111yescharacteristic
359C2×#106—611no—
360C2×#106—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#102C2×D6111yescharacteristic
366C2×#102C2×D6111yescharacteristic
367C2×#97—611no—
368C2×#97—611no—
369#390—611no—
370#390C2×D6111yescharacteristic
371#389—611no—
372#389—611no—
373#388—1221no—
374#388—1221no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4C2×D6111yescharacteristic
379C4×C22⋊C4—1221no—
380C4×C22⋊C4—1221no—
381C4×C22⋊C4—1221no—
382C4×C22⋊C4—1221no—
383C43—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340C2×D6111yescharacteristic
390#336—611no—
391#336—611no—
392#336—611no—
393#336C2×D6111yescharacteristic
394C61#2112211yescharacteristic, complemented, Sylow
395C22×D4—611no—
396C22×D4C22×D6111yescharacteristic
397C23×C4—1221no—
398C23×C4—1221no—
399C23×C4—1221no—
400C23×C4—1221no—
401C2×D8—614no—
402C2×D8—1222no—
403C2×D8—1222no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C4×D4—614no—
409C4×D4—614no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416#117—2444no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—21no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—2444no—
444C2×C42C22×D6111yescharacteristic
445#107—22no—
446#106—614no—
447#106—1222no—
448#102—22no—
449#102—614no—
450#102—1222no—
451#102—1222no—
452#102—1222no—
453#102C4×D6114yescomplemented
454#97—614no—
455#97—1222no—
456#95—21no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—21no—
463C24—1221no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—612no—
470C2×D4—612no—
471C2×D4—612no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4C2×C4×D6112yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1224no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4C23×D6111yescharacteristic
501D8—1228no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—612no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8C2×C4×D6112yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—612no—
519C4⋊C4—612no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4C2×C4×D6112yes—
527C4⋊C4#1175822412yes—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—1224no—
531C22⋊C4—1224no—
532C22⋊C4—1224no—
533C22⋊C4—1224no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C42—21no—
537C42—21no—
538C42—42no—
539C42—612no—
540C42—612no—
541C42—1222no—
542C42—1221no—
543C42—1221no—
544C42—1222no—
545C42—2441no—
546C42—2441no—
547C42—2442no—
548C42—2442no—
549C42—2442no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—1222no—
558C23—1222no—
559C23—1222no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23C2×D4×D6111yescharacteristic
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569D4—614no—
570D4—1222no—
571D4—1222no—
572D4—1222no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—2448no—
602C2×C4—2448no—
603C2×C4—2448no—
604C2×C4—2448no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4#1451980112yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4#41×D6111yescharacteristic
617C2×C4#41×D6111yescharacteristic
618C2×C4C2×#1175822411yescharacteristic
619C8—22no—
620C8—41no—
621C8—1222no—
622C8—2441no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—1224no—
631C22—2442no—
632C22—2442no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22C4×D4×D6111yescharacteristic
638C22#1684027011yescharacteristic
639C22#1684027211yescharacteristic
640C22AGL(1,ℤ/8)×D6111yescharacteristic
641C22C2×#1451980111yescharacteristic
642C22#1684055911yescharacteristic
643C22#1684056111yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—81no—
649C4—1222no—
650C4—1222no—
651C4—1222no—
652C4—4881no—
653C4—2442no—
654C4—2441no—
655C4—2442no—
656C4#117×D6111yescharacteristic
657C4C4×#1175822411yescharacteristic
658C4#1684029111yescharacteristic
659C4#1684029511yescharacteristic
660C2—82no—
661C2—2444no—
662C2#2144826111yescharacteristic
663C2C4×#1451980111yescharacteristic
664C2#2145462411yescharacteristic
665C2#2145590911yescharacteristic
666C2#2145592711yescharacteristic
667C2#2145595411yescharacteristic
668C2#2145596511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1925C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenter#1925D61
hypocenterC61#21122
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
2. #21448281C2semidirect
3. C4×#14519739C2semidirect
4. #21446448C2semidirect
5. #21446404C2semidirect
6. #21446395C2semidirect
9. #21446083C2semidirect
10. #21446063C2semidirect
11. #21445718C2semidirect
12. #21445646C2semidirect
13. #21445643C2semidirect
14. #21445566C2semidirect
15. C61×#1925C2semidirect
23. #16839871C4semidirect
73. D4×Dic61C2×C4semidirect
101. #14519708C2×C4semidirect
107. #14519687C2×C4semidirect
125. #102×C61C2×C4semidirect
280. D244#117semidirect
330. #1925D61semidirect
394. C61#21122semidirect
453. #102C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398712
#214482812
C4×#145197392
#214464482
#214464042
#214463952
#214461702
#214461642
#214460832
#214460632
#214457182
#214456462
#214456432
#214455662
C61×#19252
#2112261