Finite Group Databasebeta

← #29712197

Subgroups — #29712197

Order 15616.

1500 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
1C2×#16839895C211yesmaximal, characteristic, complemented
2C22×#14519697C211yesmaximal, characteristic, complemented
3#21449175C211yesmaximal, characteristic, complemented
4#21446962C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446248C211yesmaximal, characteristic, complemented
7#21446167C212yesmaximal, complemented
8#21446141C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445909C211yesmaximal, characteristic, complemented
11C2×C61×#553C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758199C2212yes—
14C23×#92418C2211yescharacteristic
15C2×#14519751C2212yes—
16C2×#14519745—21no—
17#16839895—24no—
18C2×#14519698C2212yescomplemented
19C2×#14519697—24no—
20C2×#14519697C2212yes—
21C2×#14519697C2212yes—
22C2×#14519697C2212yes—
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2212yescomplemented
25#16839679—21no—
26#16839671—21no—
27#16839664C2212yes—
28#16839622C2211yescharacteristic
29#16839617—21no—
30#16839524C2212yescomplemented
31#16839524C2212yescomplemented
32#16839524C2212yescomplemented
33#16839517C2212yescomplemented
34C61×#553—24no—
35C61×#553—24no—
36C2×#126×C61—21no—
37C2×#125×C61C2211yescharacteristic, complemented
38C2×#123×C61—21no—
39C2×#121×C61—21no—
40C2×#121×C61C2211yescharacteristic
41C2×#121×C61C2211yescharacteristic, complemented
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2211yescharacteristic
44C2×C4×D4×C61C2211yescharacteristic, complemented
45C2×#117×C61C2211yescharacteristic, complemented
46C22×C4⋊C4×C61C2211yescharacteristic
47C24×D61—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—42no—
53C22×D244C2312yes—
54C22×C22⋊D61—22no—
55C2×#11758199—24no—
56C2×#11758199—24no—
57C2×#11758199—42no—
58C22×#92418—22no—
59C22×#92418—42no—
60C22×#92418C2312yes—
61C22×#92418C2312yes—
62#14519751—24no—
63#14519751—42no—
64#14519745—44no—
65C2×C4⋊Dic61C2312yes—
66C2×C22⋊Dic61—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70#14519698—42no—
71#14519697—24no—
72#14519697—24no—
73#14519697—24no—
74C22⋊#92418—22no—
75C22⋊#92418—42no—
76#14519684—21no—
77C22×D4×C61—21no—
78C22×D4×C61—21no—
79C22×D4×C61D411yescharacteristic
80C22×D4×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61—21no—
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2311yescharacteristic
85#126×C61—28no—
86#125×C61—22no—
87#125×C61—22no—
88#123×C61—28no—
89#121×C61—28no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61D414yescomplemented
94C4×D4×C61—28no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99#117×C61—22no—
100#117×C61—22no—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D414yescomplemented
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C22⋊C4×C61D411yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C24×C61—21no—
119C24×C61C2×D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C23×D61—22no—
122C23×D61—44no—
123C23×D61—42no—
124C23×D61—42no—
125C23×D61—42no—
126C23×D61—82no—
127C2×D4×C61—22no—
128C2×D4×C61—24no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—24no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61—82no—
138C2×D4×C61C2×D412yes—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—24no—
145C22×C4×C61—24no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—82no—
151C22×C4×C61—82no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D412yes—
158C22×C4×C61C2411yescharacteristic
159C2×C22⋊D61—44no—
160C2×C22⋊D61—44no—
161C2×C22⋊D61—82no—
162C22×Dic61—22no—
163C2×D244—28no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—22no—
169C2×D244—48no—
170C2×D244—44no—
171C2×D244—44no—
172C2×D244—42no—
173C2×D244#4114yes—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D412yes—
182C4⋊C4×C61C2×D412yes—
183C4⋊C4×C61C2×D412yes—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—24no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—24no—
189C22⋊C4×C61—22no—
190C42×C61—24no—
191C42×C61—21no—
192C42×C61—21no—
193C22⋊Dic61—42no—
194#11758199—48no—
195#11758198—44no—
196#11758198—44no—
197#11758198—82no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—44no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—22no—
215C23×C61—42no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—42no—
219C23×C61#12012yes—
220C23×C61#13512yes—
221C23×C61AGL(1,ℤ/8)12yes—
222C23×C61C22×D411yescharacteristic
223C22×D61—24no—
224C22×D61—48no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61—42no—
231C22×D61—88no—
232C22×D61—88no—
233C22×D61—84no—
234C22×D61—84no—
235D4×C61—24no—
236D4×C61—44no—
237D4×C61—48no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—84no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—24no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—42no—
262C2×C4×C61—84no—
263C2×C4×C61—84no—
264C2×C4×C61#12012yes—
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4112yes—
268C2×C4×C61H(5,𝔽2)12yes—
269C22⋊D61—88no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273D244—24no—
274D244—24no—
275D244—48no—
276D244—42no—
277D244—42no—
278#92418—42no—
279#92418—42no—
280#23963—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—24no—
284C22×C61—24no—
285C22×C61—41no—
286C22×C61—42no—
287C22×C61—41no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—41no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—82no—
294C22×C61—82no—
295C22×C61#44914yes—
296C22×C61C2×#12011yescharacteristic
297C22×C61#54612yes—
298C22×C61C2×#13511yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61#57612yes—
301C2×D61—44no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—816no—
305C2×D61—82no—
306C2×D61—82no—
307C2×D61—82no—
308C2×D61—82no—
309C4×C61—22no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—42no—
313C4×C61—41no—
314C4×C61—42no—
315C4×C61—42no—
316C4×C61#52511yescharacteristic
317C4×C61#53511yescharacteristic
318C4×C61#54612yes—
319Dic61—82no—
320C2×#553D6111yescharacteristic, complemented
321C2×#447—611no—
322C22×#102—611no—
323#2408—611no—
324#2025—612no—
325#1999—612no—
326#1893—611no—
327#1879—612no—
328#1875—612no—
329#1869—611no—
330#1824—611no—
331C2×C61—24no—
332C2×C61—81no—
333C2×C61—82no—
334C2×C61#240411yescharacteristic
335C2×C61C2×#44911yescharacteristic
336C2×C61#300811yescharacteristic
337C2×C61#329712yes—
338C2×C61#331812yes—
339D61—84no—
340C23×D4—611no—
341C22×D8—612no—
342C23×C8—611no—
343#553—24no—
344#553—24no—
345C2×#126—21no—
346C2×#125C2×D6111yescharacteristic, complemented
347C2×#123—21no—
348C2×#121—21no—
349C2×#121—612no—
350C2×#121C2×D6111yescharacteristic
351C2×#121C2×D6111yescharacteristic, complemented
352C2×#120—1221no—
353C2×C4×D4—21no—
354C2×C4×D4C2×D6111yescharacteristic
355C2×C4×D4C2×D6111yescharacteristic, complemented
356C2×#117C2×D6111yescharacteristic, complemented
357C22×C4⋊C4C2×D6111yescharacteristic
358#447—1224no—
359C2×#102—612no—
360C2×#102—612no—
361C2×#102—612no—
362C2×#102—612no—
363C2×#102—1224no—
364C2×C22⋊C8—611no—
365C2×C22⋊C8—612no—
366#394—1221no—
367#392—1221no—
368#390—612no—
369#380—611no—
370#379—1221no—
371#340—612no—
372#340—612no—
373#340—612no—
374#336—612no—
375C61#2396311yescharacteristic, complemented, Sylow
376C25—1221no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—612no—
380C22×D4—1224no—
381C22×D4—1222no—
382C22×D4—1222no—
383C22×D4—1222no—
384C22×D4—1221no—
385C22×D4—2442no—
386C22×D4C22⋊D6111yescharacteristic
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4C22×D6111yescharacteristic
391C23×C4C22×D6111yescharacteristic
392C2×D8—1224no—
393C2×D8—1224no—
394C2×D8—2442no—
395C22×C8—612no—
396C22×C8—612no—
397C22×C8—1222no—
398C22×C8—2442no—
399#126—28no—
400#125—22no—
401#125—22no—
402#123—28no—
403#121—28no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121—1224no—
408#121—2442no—
409#121C22⋊D6114yescomplemented
410#120—2444no—
411C4×D4—28no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416#117—22no—
417#117—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—612no—
423C2×C4⋊C4C22⋊D6111yescharacteristic
424C2×C4⋊C4C22⋊D6114yescomplemented
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1222no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1222no—
435C2×C22⋊C4—1222no—
436C2×C22⋊C4C22⋊D6111yescharacteristic
437C2×C22⋊C4C22⋊D6111yescharacteristic
438C2×C42—21no—
439C2×C42C22×D6111yescharacteristic
440#102—1224no—
441#102—1224no—
442#102—1224no—
443#102—2442no—
444C22⋊C8—1222no—
445C22⋊C8—2442no—
446#95—1221no—
447C24—21no—
448C24—1222no—
449C24—4882no—
450C24—2444no—
451C24—2442no—
452C24—2442no—
453C24—2442no—
454C24C2×C22⋊D6111yescharacteristic
455C24C2×C22⋊D6111yescharacteristic
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—82no—
467C2×D4—614no—
468C2×D4—1228no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1222no—
474C2×D4—4882no—
475C2×D4—2448no—
476C2×D4—2444no—
477C2×D4—2444no—
478C2×D4—2444no—
479C2×D4—2444no—
480C2×D4—2442no—
481C2×D4C2×C22⋊D6112yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—1222no—
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6112yes—
500C22×C4C2×C22⋊D6112yes—
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503D8—2448no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2444no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4C2×C22⋊D6112yes—
521C4⋊C4C2×C22⋊D6112yes—
522C4⋊C4C2×C22⋊D6112yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—22no—
527C22⋊C4—24no—
528C22⋊C4—22no—
529C22⋊C4—4882no—
530C22⋊C4—2444no—
531C22⋊C4—2444no—
532C22⋊C4—2442no—
533C42—24no—
534C42—21no—
535C42—21no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—1224no—
549C23—4884no—
550C23—4888no—
551C23—4888no—
552C23—4884no—
553C23—2448no—
554C23—2442no—
555C23—2442no—
556C23—2442no—
557C23—2442no—
558C23—2442no—
559C23—2442no—
560C23#1451974612yes—
561C23#1451979712yes—
562C23#1451980212yes—
563C23C22×C22⋊D6111yescharacteristic
564D4—24no—
565D4—44no—
566D4—48no—
567D4—41no—
568D4—41no—
569D4—84no—
570D4—1224no—
571D4—1224no—
572D4—4888no—
573D4—2448no—
574D4—2442no—
575D4—2442no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4#1451974612yes—
603C2×C4C22×C22⋊D6111yescharacteristic
604C2×C4C22×C22⋊D6111yescharacteristic
605C2×C4#41×D6112yes—
606C2×C4#1451982312yes—
607C8—2442no—
608C8—2442no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—42no—
614C22—41no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22—42no—
621C22—82no—
622C22—82no—
623C22—48816no—
624C22—4882no—
625C22—4882no—
626C22—4882no—
627C22—4882no—
628C22—2444no—
629C22—2444no—
630C22—2444no—
631C22#1683990714yes—
632C22C2×#1451974611yescharacteristic
633C22#1684040012yes—
634C22C2×#1451979711yescharacteristic
635C22C2×#1451980211yescharacteristic
636C22#1684056512yes—
637C4—22no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—4882no—
645C4#1684026911yescharacteristic
646C4#1684032611yescharacteristic
647C4#1684040012yes—
648C2—24no—
649C2—81no—
650C2—82no—
651C2—4884no—
652C2#2144913411yescharacteristic
653C2C2×#1683990711yescharacteristic
654C2#2145508711yescharacteristic
655C2#2145766312yes—
656C2#2145795412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#553C2
socleC23×C61C22×D4
hypercenterC2×#553D61
hypocenterC61#23963
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×#16839895C2semidirect
2. C22×#14519697C2semidirect
3. #21449175C2semidirect
4. #21446962C2semidirect
5. #21446815C2semidirect
6. #21446248C2semidirect
7. #21446167C2semidirect
8. #21446141C2semidirect
9. #21446112C2semidirect
10. #21445909C2semidirect
11. C2×C61×#553C2semidirect
18. C2×#14519698C22semidirect
24. C2×C22⋊#92418C22semidirect
30. #16839524C22semidirect
31. #16839524C22semidirect
32. #16839524C22semidirect
33. #16839517C22semidirect
37. C2×#125×C61C22semidirect
41. C2×#121×C61C22semidirect
44. C2×C4×D4×C61C22semidirect
45. C2×#117×C61C22semidirect
93. #121×C61D4semidirect
106. C2×C4⋊C4×C61D4semidirect
320. C2×#553D61semidirect
346. C2×#125C2×D61semidirect
351. C2×#121C2×D61semidirect
355. C2×C4×D4C2×D61semidirect
356. C2×#117C2×D61semidirect
375. C61#23963semidirect
409. #121C22⋊D61semidirect
424. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398952
C22×#145196972
#214491752
#214469622
#214468152
#214462482
#214461672
#214461412
#214461122
#214459092
C2×C61×#5532
#2396361