Finite Group Databasebeta

← #29685661

Subgroups — #29685661

Order 15616.

1952 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519739C211yesmaximal, characteristic, complemented
2C2×#16839895C211yesmaximal, characteristic
3C2×#16839871C212yesmaximal, complemented
4#121×Dic61C211yesmaximal, characteristic, complemented
5#21446164C212yesmaximal, complemented
6#21446081C212yesmaximal, complemented
7#21446012C211yesmaximal, characteristic, complemented
8#21445751C211yesmaximal, characteristic, complemented
9#21445643C212yesmaximal, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1874C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×C4×D61—21no—
14C22×#11758199C2212yes—
15C2×D4×Dic61—21no—
16C2×D4×Dic61C2211yescharacteristic, complemented
17C2×#14519739—28no—
18C2×#14519739—21no—
19C2×#14519739—21no—
20C2×#14519739C2212yes—
21C2×#14519739C2212yes—
22C22×#11758198—21no—
23C22×C4×Dic61C2211yescharacteristic
24#16839895C418yescomplemented
25#16839871—28no—
26C2×#14519708C2212yescomplemented
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2212yescomplemented
32#16839665C2211yescharacteristic, complemented
33#16839661C2211yescharacteristic, complemented
34#16839655—21no—
35#16839642—21no—
36#16839622C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic, complemented
38C22⋊C4×Dic61—21no—
39#16839524C2212yescomplemented
40#16839516C2212yescomplemented
41C2×#121×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#102×C61C2212yescomplemented
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61—21no—
47C22×D244—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—24no—
51C22×D244C2×C414yescomplemented
52C22×D244C2312yes—
53C23×Dic61—21no—
54C23×Dic61D411yescharacteristic
55C23×Dic61D411yescharacteristic
56C22×C4×D61—28no—
57C22×C4×D61—22no—
58C22×C4×D61—22no—
59C22×C4×D61—44no—
60C22×C4×D61—44no—
61C2×#11758199—24no—
62C2×#11758199—24no—
63C2×#11758199C2×C418yescomplemented
64C22×#92418C2312yes—
65D4×Dic61—24no—
66D4×Dic61—44no—
67#14519739—28no—
68#14519739—28no—
69#14519739—24no—
70#14519739—24no—
71#14519739—24no—
72#14519739D418yescomplemented
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—28no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×C4×Dic61—24no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519708—24no—
90#14519699—24no—
91#14519699—24no—
92#14519697C2×C418yescomplemented
93C22⋊#92418C2×C414yescomplemented
94#14519687—24no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98C22×D4×C61—21no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103#121×C61—22no—
104#121×C61C2×C414yescomplemented
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110#102×C61—24no—
111C22⋊C8×C61—22no—
112C24×C61—21no—
113C24×C61C2×D411yescharacteristic
114C23×D61—24no—
115C23×D61—24no—
116C23×D61—24no—
117C23×D61—22no—
118C23×D61—42no—
119C23×D61—42no—
120C23×D61—42no—
121C23×D61—42no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61C22×C412yescomplemented
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61#4112yes—
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—41no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×D244—24no—
156C2×D244—216no—
157C2×D244—24no—
158C2×D244—24no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—216no—
163C2×D244C22×C414yes—
164C2×D244C2×D414yes—
165C2×D244#4114yes—
166C2×C4×D61—28no—
167C2×C4×D61—28no—
168C2×C4×D61—432no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61—44no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C412yescomplemented
180C4⋊C4×C61C2×D414yes—
181C22⋊C4×C61—24no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186#11758199—216no—
187#11758198—28no—
188#11758198—28no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—42no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195C4×Dic61—42no—
196C4×Dic61C2×D414yes—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418C22×C414yescomplemented
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C4×D412yes—
212C23×C61AGL(1,ℤ/8)12yes—
213C23×C61AGL(1,ℤ/8)12yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—216no—
216C22×D61—24no—
217C22×D61—24no—
218C22×D61—24no—
219C22×D61—48no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61—48no—
223C22×D61—48no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—48no—
229D4×C61—24no—
230D4×C61—48no—
231D4×C61—42no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—48no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—41no—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D412yes—
246C2×C4×C61C2×#4112yes—
247C2×C4×C61C2×#4111yescharacteristic
248C2×Dic61—28no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—44no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—84no—
266C2×Dic61#12012yes—
267C2×Dic61#12012yes—
268C2×Dic61AGL(1,ℤ/8)12yes—
269D244—216no—
270D244—24no—
271D244—24no—
272D244—24no—
273D244C4×D418yescomplemented
274C4×D61—416no—
275C4×D61—416no—
276C8×C61—42no—
277#92418—24no—
278#21452—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61#44214yescomplemented
293C22×C61C2×C4×D411yescharacteristic
294C22×C61C2×#12011yescharacteristic
295C22×C61#52511yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C61#57511yescharacteristic
300C2×D61—28no—
301C2×D61—432no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—44no—
305C2×D61—44no—
306C2×D61—44no—
307C2×D61—44no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—44no—
311C4×C61C2×C4×D412yes—
312C4×C61#52512yes—
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—41no—
317Dic61—81no—
318Dic61—81no—
319C22×C4×D4—611no—
320C2×#447—611no—
321C2×#442—612no—
322C4×#121—611no—
323#1879—612no—
324#1874D6111yescharacteristic, complemented
325#1864—612no—
326#1853—611no—
327#1786—611no—
328#1762—612no—
329#1749—611no—
330C2×C61—24no—
331C2×C61—82no—
332C2×C61C4×#12011yescharacteristic
333C2×C61C2×#44211yescharacteristic
334C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
335C2×C61#300311yescharacteristic
336C2×C61#300611yescharacteristic
337C2×C61#310911yescharacteristic
338C2×C61#311411yescharacteristic
339D61—48no—
340C23×D4—611no—
341C24×C4—1221no—
342C22×D8—612no—
343C2×#121C2×D6111yescharacteristic
344C2×C4×D4—611no—
345C2×C4×D4—612no—
346C2×C4×D4—612no—
347C2×C4×D4—1221no—
348C2×C4×D4—1221no—
349C2×C4×D4—1228no—
350C2×C4×D4—1221no—
351C22×C4⋊C4C2×D6111yescharacteristic
352C22×C22⋊C4—1221no—
353C22×C42—611no—
354#447—618no—
355#442—1228no—
356C2×#106—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×#102—612no—
360C2×#102C2×D6112yescomplemented
361C2×C22⋊C8—611no—
362C2×C22⋊C8C2×D6111yescharacteristic
363C2×#97—612no—
364#390—611no—
365#389—611no—
366#388—1221no—
367#386—1221no—
368#380—611no—
369C4×C4⋊C4—611no—
370C4×C22⋊C4—1221no—
371#340—612no—
372#340C2×D6112yescomplemented
373#336—612no—
374C61#2145211yescharacteristic, complemented, Sylow
375C25—1221no—
376C22×D4—21no—
377C22×D4—614no—
378C22×D4—612no—
379C22×D4—1224no—
380C22×D4—1224no—
381C22×D4—1221no—
382C22×D4—1222no—
383C22×D4C22×D6111yescharacteristic
384C23×C4—21no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—1222no—
388C23×C4—1222no—
389C23×C4—1228no—
390C23×C4—1221no—
391C23×C4—2444no—
392C23×C4—2444no—
393C23×C4C22×D6111yescharacteristic
394C2×D8—618no—
395C2×D8—1224no—
396C2×D8—1224no—
397C22×C8—612no—
398C22×C8C22×D6112yes—
399#121—22no—
400#121C4×D6114yescomplemented
401C4×D4—618no—
402C4×D4—1224no—
403C4×D4—1224no—
404C4×D4—1224no—
405C4×D4—1228no—
406C4×D4—1228no—
407C4×D4—1224no—
408C4×D4—2444no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6112yes—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1228no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—1224no—
429C2×C42—1221no—
430#106—1224no—
431#102—24no—
432#102—618no—
433#102—1224no—
434#102—1224no—
435C22⋊C8—22no—
436C22⋊C8—614no—
437#97—1224no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441C24—21no—
442C24—1224no—
443C24—1224no—
444C24—1222no—
445C24—1224no—
446C24—2442no—
447C24—2442no—
448C24—2442no—
449C24—2442no—
450C24D4×D6111yescharacteristic
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—614no—
458C2×D4—614no—
459C2×D4—614no—
460C2×D4—12216no—
461C2×D4—1224no—
462C2×D4—1224no—
463C2×D4—12216no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1224no—
468C2×D4C2×C4×D6112yescomplemented
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—1228no—
481C22×C4—1228no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—2444no—
490C22×C4—24432no—
491C22×C4—2444no—
492C22×C4—2444no—
493C22×C4—2444no—
494C22×C4—2442no—
495C22×C4—2441no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4C2×C4×D6112yes—
499C22×C4D4×D6111yescharacteristic
500C22×C4#1175822412yes—
501C22×C4C23×D6111yescharacteristic
502D8—12216no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—614no—
507C2×C8—1222no—
508C2×C8—1222no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—2442no—
515C4⋊C4C2×C4×D6112yescomplemented
516C4⋊C4D4×D6114yes—
517C22⋊C4—24no—
518C22⋊C4—1228no—
519C22⋊C4—1228no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—614no—
525C42—1222no—
526C42—1222no—
527C42—1222no—
528C42—2442no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—12216no—
541C23—1224no—
542C23—1224no—
543C23—1224no—
544C23—2448no—
545C23—2448no—
546C23—2448no—
547C23—2448no—
548C23—2448no—
549C23—2442no—
550C23—2442no—
551C23—2442no—
552C23—2442no—
553C23—2448no—
554C23#1451973712yes—
555C23#1451980112yes—
556C23#1451980412yes—
557C23C2×D4×D6111yescharacteristic
558D4—24no—
559D4—48no—
560D4—42no—
561D4—618no—
562D4—12216no—
563D4—1224no—
564D4—1224no—
565D4—1224no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—1228no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—4884no—
591C2×C4—24416no—
592C2×C4—24416no—
593C2×C4—2444no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4#1451973712yes—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C2×D4×D6112yes—
603C2×C4#41×D6112yes—
604C2×C4C2×#1175822411yescharacteristic
605C8—42no—
606C8—1224no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—1228no—
621C22—24432no—
622C22—2444no—
623C22—2444no—
624C22—2444no—
625C22—2444no—
626C22—2444no—
627C22—2444no—
628C22#1683987114yescomplemented
629C22C2×#1451973711yescharacteristic
630C22#120×D6111yescharacteristic
631C22#1684027011yescharacteristic
632C22AGL(1,ℤ/8)×D6111yescharacteristic
633C22C2×#1451980111yescharacteristic
634C22C2×#1451980411yescharacteristic
635C22#1684056111yescharacteristic
636C4—22no—
637C4—42no—
638C4—44no—
639C4—1222no—
640C4—1222no—
641C4—1222no—
642C4—4881no—
643C4—4881no—
644C4—2441no—
645C4C4×D4×D6112yes—
646C4#1684027212yes—
647C2—24no—
648C2—82no—
649C2—2448no—
650C2#2144832411yescharacteristic
651C2C2×#1683987111yescharacteristic
652C2#2145463211yescharacteristic
653C2#2145504311yescharacteristic
654C2#2145507011yescharacteristic
655C2#2145590711yescharacteristic
656C2#2145595411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radicalC2×#16839895C2
cosocleC22×C4×C61C24
hypercenter#1874D61
hypocenterC61#21452
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. C22×#14519739C2semidirect
3. C2×#16839871C2semidirect
4. #121×Dic61C2semidirect
5. #21446164C2semidirect
6. #21446081C2semidirect
7. #21446012C2semidirect
8. #21445751C2semidirect
9. #21445643C2semidirect
10. #21445583C2semidirect
11. C61×#1874C2semidirect
16. C2×D4×Dic61C22semidirect
24. #16839895C4semidirect
26. C2×#14519708C22semidirect
31. C2×#14519687C22semidirect
32. #16839665C22semidirect
33. #16839661C22semidirect
37. C4⋊C4×Dic61C22semidirect
39. #16839524C22semidirect
40. #16839516C22semidirect
43. C2×#102×C61C22semidirect
45. C61×#340C22semidirect
51. C22×D244C2×C4semidirect
63. C2×#11758199C2×C4semidirect
72. #14519739D4semidirect
92. #14519697C2×C4semidirect
93. C22⋊#92418C2×C4semidirect
104. #121×C61C2×C4semidirect
128. C2×D4×C61C22×C4semidirect
179. C4⋊C4×C61C22×C4semidirect
199. C2×#92418C22×C4semidirect
273. D244C4×D4semidirect
292. C22×C61#442semidirect
324. #1874D61semidirect
360. C2×#102C2×D61semidirect
372. #340C2×D61semidirect
374. C61#21452semidirect
400. #121C4×D61semidirect
468. C2×D4C2×C4×D61semidirect
515. C4⋊C4C2×C4×D61semidirect
628. C22#16839871semidirect

Maximal subgroups

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

subgroupindex
C22×#145197392
C2×#168398952
C2×#168398712
#121×Dic612
#214461642
#214460812
#214460122
#214457512
#214456432
#214455832
C61×#18742
#2145261