Finite Group Databasebeta

← #29689059

Subgroups — #29689059

Order 15616.

1882 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#121×D61C211yesmaximal, characteristic
2C2×#16839725C211yesmaximal, characteristic, complemented
3#380×D61C211yesmaximal, characteristic, complemented
4#21446258C211yesmaximal, characteristic, complemented
5#21446257C211yesmaximal, characteristic, complemented
6#21446255C211yesmaximal, characteristic, complemented
7#21446135C212yesmaximal, complemented
8#21446133C212yesmaximal, complemented
9#21445908C212yesmaximal, complemented
10#21445748C212yesmaximal, complemented
11C61×#1895C211yesmaximal, characteristic, complemented
12C22×D4×D61—21no—
13C22×D4×D61C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C22×#11758194C2211yescharacteristic, complemented
16C2×#14519752C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×#14519750—21no—
19C2×#14519749—21no—
20C2×#14519748C2211yescharacteristic
21#121×D61—24no—
22#121×D61C418yescomplemented
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C4⋊C4×D61C2212yescomplemented
25C2×C22⋊C4×D61—21no—
26C2×C22⋊C4×D61—21no—
27C2×#14519699C2212yescomplemented
28C2×#14519698C2212yescomplemented
29C2×#14519697C2212yescomplemented
30#16839725—24no—
31C2×#14519689C2212yes—
32C2×#14519689C2212yescomplemented
33#16839622C2211yescharacteristic, complemented
34#16839621—21no—
35#16839620—21no—
36#16839619C2212yes—
37#95×D61—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×M16×C61C2211yescharacteristic, complemented
42C2×#121×C61C2211yescharacteristic
43C2×#102×C61C2212yescomplemented
44C61×#380C2211yescharacteristic, complemented
45C61×#340C2212yescomplemented
46C24×D61—21no—
47C24×D61D411yescharacteristic
48C22×D244—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C2×D4×D61—28no—
54C2×D4×D61—22no—
55C2×D4×D61—42no—
56C2×D4×D61—42no—
57C2×D4×D61—48no—
58C2×D4×D61C2×C414yescomplemented
59C23×Dic61D411yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61—22no—
62C22×C4×D61—42no—
63C22×C4×D61—42no—
64C22×C4×D61C2×C414yes—
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61D414yes—
68C22×C4×D61C2311yescharacteristic
69C22×C4×D61C2311yescharacteristic
70C2×#11758194—22no—
71C2×#11758194—42no—
72C22×#92418C2312yes—
73#14519752—22no—
74#14519752C2×C414yes—
75#14519751—22no—
76#14519751C2×C414yescomplemented
77#14519750—28no—
78#14519749—28no—
79#14519748—22no—
80#14519748C2×C414yes—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—22no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61—42no—
87C4⋊C4×D61—42no—
88C4⋊C4×D61C2×C414yescomplemented
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—22no—
94C22⋊C4×D61—28no—
95C22⋊C4×D61—44no—
96#14519699—42no—
97#14519698—42no—
98#14519697—42no—
99#14519689—24no—
100#14519689—24no—
101#14519684—21no—
102#14519684—22no—
103C22×D4×C61—21no—
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C2×M16×C61—22no—
107C2×M16×C61—42no—
108C22×C8×C61C2312yes—
109#121×C61—22no—
110#121×C61C2×C414yescomplemented
111C2×C4⋊C4×C61C2312yes—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115#102×C61—42no—
116#95×C61—21no—
117C24×C61—21no—
118C24×C61C2×D411yescharacteristic
119C23×D61—24no—
120C23×D61—22no—
121C23×D61—22no—
122C23×D61—21no—
123C23×D61—24no—
124C23×D61—42no—
125C23×D61—44no—
126C23×D61—44no—
127C23×D61—42no—
128C23×D61—42no—
129C23×D61C22⋊C414yes—
130C23×D61C2×D411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C2×D4×C61—24no—
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—44no—
137C2×D4×C61C22×C412yes—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—24no—
147C2×C22⋊D61—24no—
148C2×C22⋊D61—24no—
149C2×C22⋊D61—44no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—44no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61—42no—
155C22×Dic61C22⋊C414yes—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158D4×D61—48no—
159D4×D61—44no—
160D4×D61—44no—
161D4×D61—432no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—24no—
165C2×D244—21no—
166C2×D244—44no—
167C2×D244C22×C412yes—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—24no—
173C2×C4×D61—48no—
174C2×C4×D61—44no—
175C2×C4×D61—44no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C4×D61—44no—
179C2×C4×D61—44no—
180C2×C4×D61—44no—
181C2×C4×D61—44no—
182C2×C4×D61C22⋊C414yes—
183C2×C4×D61C22⋊C414yes—
184C2×C4×D61C22×C412yes—
185C2×C4×D61C22×C412yes—
186C2×C4×D61C2×D412yes—
187C2×C4×D61C2×D412yes—
188M16×C61—44no—
189C2×C8×C61—22no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61C22×C412yes—
196C22⋊C4×C61—24no—
197C22⋊C4×C61—42no—
198C22⋊Dic61—24no—
199C22⋊Dic61—42no—
200#11758198—24no—
201#11758198—24no—
202#11758198—44no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61C22×C412yes—
207#11758196—24no—
208#11758196—44no—
209#11758194—44no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213C23×C61—22no—
214C23×C61—21no—
215C23×C61—22no—
216C23×C61—21no—
217C23×C61—22no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C2×C22⋊C412yes—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—21no—
227C22×D61—22no—
228C22×D61—21no—
229C22×D61—28no—
230C22×D61—28no—
231C22×D61—22no—
232C22×D61—44no—
233C22×D61—48no—
234C22×D61—48no—
235C22×D61—48no—
236C22×D61—48no—
237C22×D61—42no—
238C22×D61—42no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—48no—
242C22×D61—88no—
243C22×D61—88no—
244C22×D61C2×C22⋊C412yes—
245C22×D61C2×C22⋊C412yes—
246C22×D61AGL(1,ℤ/8)12yes—
247C22×D61AGL(1,ℤ/8)14yes—
248C22×D61#13712yes—
249D4×C61—24no—
250D4×C61—48no—
251D4×C61—41no—
252D4×C61—41no—
253C2×C4×C61—22no—
254C2×C4×C61—24no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61C2×C22⋊C412yes—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61H(5,𝔽2)12yes—
266C2×C4×C61H(5,𝔽2)12yes—
267C22⋊D61—48no—
268C22⋊D61—48no—
269C22⋊D61—48no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—44no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280D244—24no—
281D244—48no—
282D244—41no—
283D244—41no—
284C4×D61—24no—
285C4×D61—24no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—44no—
291C4×D61—44no—
292C4×D61—48no—
293C4×D61—88no—
294C8×C61—42no—
295#92418—42no—
296#21764—611nomaximal, Sylow
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—82no—
306C22×C61—82no—
307C22×C61C22×C22⋊C411yescharacteristic
308C22×C61#53412yes—
309C22×C61C2×AGL(1,ℤ/8)12yes—
310C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C61C2×#13711yescharacteristic
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—24no—
317C2×D61—24no—
318C2×D61—24no—
319C2×D61—24no—
320C2×D61—416no—
321C2×D61—42no—
322C2×D61—42no—
323C2×D61—42no—
324C2×D61—816no—
325C2×D61—82no—
326C2×D61—82no—
327C2×D61#41714yes—
328C2×D61#41814yes—
329C4×C61—22no—
330C4×C61—42no—
331C4×C61—82no—
332C4×C61#51812yes—
333C4×C61#53411yescharacteristic
334C4×C61#53411yescharacteristic
335Dic61—22no—
336Dic61—22no—
337Dic61—42no—
338Dic61—82no—
339C22×#121—611no—
340C2×#407—611no—
341C2×#380—611no—
342#1895—611no—
343#1895—611no—
344#1895—611no—
345#1895D6111yescharacteristic, complemented
346#1874—612no—
347#1874—612no—
348#1824—612no—
349#1786—612no—
350C2×C61—42no—
351C2×C61—82no—
352C2×C61#230811yescharacteristic
353C2×C61C2×#41811yescharacteristic
354C2×C61C2×#41711yescharacteristic
355C2×C61#319212yes—
356C2×C61#319312yes—
357D61—24no—
358D61—24no—
359D61—82no—
360C23×D4—611no—
361C23×D4—1221no—
362C24×C4—611no—
363C22×M16—611no—
364C22×M16C2×D6111yescharacteristic, complemented
365C2×#121—611no—
366C2×#121—611no—
367C2×#121—618no—
368C2×#121—611no—
369C2×#121—1221no—
370C2×#121—1224no—
371C2×#121—1221no—
372C2×#121C2×D6111yescharacteristic
373C22×C4⋊C4—611no—
374C22×C4⋊C4—612no—
375C22×C22⋊C4—1221no—
376C22×C22⋊C4—1221no—
377C2×#102—612no—
378C2×#102—612no—
379C2×#102—612no—
380C2×#102C2×D6112yescomplemented
381#407—1224no—
382C2×C22⋊C8—612no—
383C2×C22⋊C8—612no—
384#380—611no—
385#380—612no—
386#380—1221no—
387#380—1221no—
388#380C2×D6111yescharacteristic, complemented
389C2×#95—1221no—
390#340—612no—
391#340—612no—
392#340—612no—
393#340C2×D6112yescomplemented
394C61#2176411yescharacteristic, complemented, Sylow
395C25—611no—
396C25—1221no—
397C22×D4—21no—
398C22×D4—614no—
399C22×D4—611no—
400C22×D4—1221no—
401C22×D4—1222no—
402C22×D4—1221no—
403C22×D4—1228no—
404C22×D4—1221no—
405C22×D4—1221no—
406C22×D4—2442no—
407C22×D4—2442no—
408C22×D4—2448no—
409C22×D4C22×D6111yescharacteristic
410C23×C4—611no—
411C23×C4—611no—
412C23×C4—611no—
413C23×C4—614no—
414C23×C4—611no—
415C23×C4—611no—
416C23×C4—614no—
417C23×C4—1221no—
418C23×C4—1222no—
419C23×C4—2442no—
420C23×C4—2442no—
421C23×C4C22×D6111yescharacteristic
422C2×M16—22no—
423C2×M16—42no—
424C2×M16—1222no—
425C2×M16—2442no—
426C22×C8—612no—
427C22×C8C22×D6112yes—
428#121—22no—
429#121—614no—
430#121—614no—
431#121—614no—
432#121—1228no—
433#121—1222no—
434#121—1222no—
435#121—1228no—
436#121—1222no—
437#121C4×D6114yescomplemented
438C2×C4⋊C4—611no—
439C2×C4⋊C4—614no—
440C2×C4⋊C4—612no—
441C2×C4⋊C4—1222no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1222no—
444C2×C4⋊C4—2442no—
445C2×C4⋊C4—2442no—
446C2×C4⋊C4C22×D6112yes—
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—1228no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1222no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—2444no—
457#102—42no—
458#102—2442no—
459#102—2442no—
460#102—2442no—
461C22⋊C8—1224no—
462C22⋊C8—1224no—
463#95—21no—
464#95—1222no—
465#95—1221no—
466C24—21no—
467C24—611no—
468C24—611no—
469C24—614no—
470C24—1222no—
471C24—1224no—
472C24—1222no—
473C24—1224no—
474C24—1221no—
475C24—2442no—
476C24—2442no—
477C24—2442no—
478C24—2444no—
479C24—2444no—
480C24D4×D6111yescharacteristic
481C2×D4—24no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—44no—
486C2×D4—612no—
487C2×D4—1224no—
488C2×D4—1224no—
489C2×D4—1224no—
490C2×D4—1224no—
491C2×D4—1221no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4—2444no—
495C2×D4—2448no—
496C2×D4—2444no—
497C2×D4—2444no—
498C2×D4—2444no—
499C2×D4—24432no—
500C2×D4—2444no—
501C2×D4—2444no—
502C2×D4C2×C4×D6112yes—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—614no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—612no—
513C22×C4—612no—
514C22×C4—614no—
515C22×C4—614no—
516C22×C4—1224no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1221no—
522C22×C4—1222no—
523C22×C4—2448no—
524C22×C4—2444no—
525C22×C4—2444no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4—2444no—
529C22×C4—2444no—
530C22×C4—2444no—
531C22×C4—2444no—
532C22×C4—2442no—
533C22×C4C2×C4×D6112yes—
534C22×C4D4×D6112yes—
535C22×C4D4×D6111yescharacteristic
536C22×C4C23×D6111yescharacteristic
537M16—44no—
538M16—2444no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—1222no—
543C2×C8—1222no—
544C2×C8—1222no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—612no—
549C4⋊C4—1224no—
550C4⋊C4—1221no—
551C4⋊C4—1222no—
552C4⋊C4—1222no—
553C4⋊C4—2444no—
554C4⋊C4C2×C4×D6112yes—
555C22⋊C4—24no—
556C22⋊C4—42no—
557C22⋊C4—1224no—
558C22⋊C4—1224no—
559C22⋊C4—1224no—
560C22⋊C4—2444no—
561C22⋊C4—2442no—
562C23—21no—
563C23—22no—
564C23—21no—
565C23—22no—
566C23—22no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—612no—
572C23—612no—
573C23—612no—
574C23—614no—
575C23—612no—
576C23—1228no—
577C23—1228no—
578C23—1221no—
579C23—1222no—
580C23—1221no—
581C23—1222no—
582C23—1222no—
583C23—1222no—
584C23—4888no—
585C23—4888no—
586C23—2444no—
587C23—2448no—
588C23—2448no—
589C23—2448no—
590C23—2448no—
591C23—2448no—
592C23—2442no—
593C23—2441no—
594C23—2441no—
595C23—2442no—
596C23C22⋊C4×D6112yes—
597C23C2×D4×D6111yescharacteristic
598D4—24no—
599D4—48no—
600D4—41no—
601D4—41no—
602D4—1224no—
603D4—2448no—
604D4—2448no—
605D4—2448no—
606D4—2448no—
607D4—2441no—
608D4—2441no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—612no—
619C2×C4—612no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1224no—
623C2×C4—1224no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—4888no—
630C2×C4—2444no—
631C2×C4—2444no—
632C2×C4—2448no—
633C2×C4—2444no—
634C2×C4—2442no—
635C2×C4—2442no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4C22⋊C4×D6112yes—
639C2×C4C22×C4×D6111yescharacteristic
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4#1451982212yes—
642C2×C4#1451982312yes—
643C8—42no—
644C8—2442no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—82no—
654C22—82no—
655C22—614no—
656C22—614no—
657C22—1222no—
658C22—1222no—
659C22—1224no—
660C22—1224no—
661C22—1224no—
662C22—1224no—
663C22—1222no—
664C22—1222no—
665C22—48816no—
666C22—4882no—
667C22—4882no—
668C22—24416no—
669C22—2442no—
670C22—2442no—
671C22—2442no—
672C22C2×C22⋊C4×D6111yescharacteristic
673C22#1684032012yes—
674C22AGL(1,ℤ/8)×D6111yescharacteristic
675C22AGL(1,ℤ/8)×D6112yes—
676C22#137×D6111yescharacteristic
677C4—22no—
678C4—42no—
679C4—82no—
680C4—1222no—
681C4—1222no—
682C4—4882no—
683C4—2442no—
684C4#1684023412yes—
685C4#1684032211yescharacteristic
686C4#1684032311yescharacteristic
687C2—42no—
688C2—82no—
689C2—1224no—
690C2—1224no—
691C2—4882no—
692C2#2144839311yescharacteristic
693C2#418×D6111yescharacteristic
694C2#417×D6111yescharacteristic
695C2#2145661512yes—
696C2#2145662012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1895C2
socleC23×C61C22×D4
prime radicalC2×#121×D61C2
cosocleC22×C4×C61C24
hypercenter#1895D61
hypocenterC61#21764
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
2. C2×#16839725C2semidirect
3. #380×D61C2semidirect
4. #21446258C2semidirect
5. #21446257C2semidirect
6. #21446255C2semidirect
7. #21446135C2semidirect
8. #21446133C2semidirect
9. #21445908C2semidirect
10. #21445748C2semidirect
11. C61×#1895C2semidirect
15. C22×#11758194C22semidirect
22. #121×D61C4semidirect
24. C2×C4⋊C4×D61C22semidirect
27. C2×#14519699C22semidirect
28. C2×#14519698C22semidirect
29. C2×#14519697C22semidirect
32. C2×#14519689C22semidirect
33. #16839622C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
41. C22×M16×C61C22semidirect
43. C2×#102×C61C22semidirect
44. C61×#380C22semidirect
45. C61×#340C22semidirect
58. C2×D4×D61C2×C4semidirect
76. #14519751C2×C4semidirect
88. C4⋊C4×D61C2×C4semidirect
110. #121×C61C2×C4semidirect
345. #1895D61semidirect
364. C22×M16C2×D61semidirect
380. C2×#102C2×D61semidirect
388. #380C2×D61semidirect
393. #340C2×D61semidirect
394. C61#21764semidirect
437. #121C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#121×D612
C2×#168397252
#380×D612
#214462582
#214462572
#214462552
#214461352
#214461332
#214459082
#214457482
C61×#18952
#2176461