Finite Group Databasebeta

← #29681240

Subgroups — #29681240

Order 15616.

2258 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840274C211yesmaximal, characteristic
2C22×C4⋊C4×D61C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4C2×#16839761C211yesmaximal, characteristic, complemented
5C2×#16839725C211yesmaximal, characteristic, complemented
6#21446135C214yesmaximal, complemented
7#21446125C214yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×C4×D61—21no—
12C23×C4×D61C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16840274—24no—
15#16840274C418yescomplemented
16C2×#14519761—21no—
17C2×#14519757C2212yes—
18C2×#14519752C2212yes—
19C2×#14519747—21no—
20C2×C4×D244—22no—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×#11758196—21no—
23C2×C4⋊C4×D61—24no—
24C2×C4⋊C4×D61—24no—
25C2×C4⋊C4×D61—22no—
26C2×C4⋊C4×D61C2214yes—
27#16839764—24no—
28#16839761—24no—
29C2×#14519702C2212yescomplemented
30C2×#14519700C2212yescomplemented
31C2×#14519699C2212yescomplemented
32C2×#14519697C2212yescomplemented
33#16839725—24no—
34C2×#14519689C2214yes—
35#16839524C2214yescomplemented
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C22×C4⋊C4×C61C2211yescharacteristic
39C61×#340C2214yescomplemented
40C24×D61D411yescharacteristic
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222C2×C414yescomplemented
47C22×Dic122C2311yescharacteristic
48C22×D244—22no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C23×Dic61—21no—
52C23×Dic61D411yescharacteristic
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61—22no—
56C22×C4×D61—22no—
57C22×C4×D61—28no—
58C22×C4×D61—24no—
59C22×C4×D61—21no—
60C22×C4×D61—48no—
61C22×C4×D61C2×C414yes—
62C22×C4×D61D412yes—
63C22×C4×D61D414yes—
64C22×C4×D61C2312yes—
65C2×#11758194—22no—
66C2×#11758194—22no—
67C2×#11758194—22no—
68C22×#92418C2312yes—
69#14519761—28no—
70#14519757—24no—
71#14519757C2×C418yescomplemented
72#14519752—24no—
73#14519752C2×C418yescomplemented
74#14519747—28no—
75C4×D244—216no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61D414yescomplemented
80C2×C4⋊Dic61C2312yes—
81C2×#11758196—28no—
82C2×#11758196—24no—
83C4⋊C4×D61—216no—
84C4⋊C4×D61—28no—
85C4⋊C4×D61—28no—
86C2×#11758198—22no—
87C2×#11758198—22no—
88#14519702—24no—
89#14519700—24no—
90#14519699—24no—
91#14519697—24no—
92#14519689—28no—
93C23×C4×C61—21no—
94C23×C4×C61C2311yescharacteristic
95C2×M16×C61—22no—
96C2×M16×C61—22no—
97C2×M16×C61—22no—
98C22×C8×C61C2312yes—
99#117×C61—22no—
100#117×C61C2×C414yescomplemented
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61D414yescomplemented
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2312yes—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C24×C61C2×D411yescharacteristic
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—24no—
112C23×D61—24no—
113C23×D61—21no—
114C23×D61C22⋊C414yes—
115C23×D61C2×D412yes—
116C22×C4×C61—24no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—44no—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61#4112yes—
126C22×C4×C61#4112yes—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—24no—
129C2×C22⋊D61—42no—
130C2×C22⋊D61—42no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—24no—
134C22×Dic61—22no—
135C22×Dic61—21no—
136C22×Dic61—44no—
137C22×Dic61C22⋊C414yes—
138C22×Dic61C2×D412yes—
139#11758222—28no—
140#11758222—28no—
141#11758222—44no—
142#11758222—44no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—28no—
148C2×D244—21no—
149C2×D244—44no—
150C2×D244—44no—
151C2×D244C22×C412yescomplemented
152C2×C4×D61—28no—
153C2×C4×D61—28no—
154C2×C4×D61—24no—
155C2×C4×D61—28no—
156C2×C4×D61—28no—
157C2×C4×D61—24no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—216no—
163C2×C4×D61—24no—
164C2×C4×D61—432no—
165C2×C4×D61—48no—
166C2×C4×D61—48no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61C22⋊C418yes—
170C2×C4×D61C22×C414yes—
171C2×C4×D61C2×D414yes—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122C22×C412yescomplemented
178M16×C61—44no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C22×C414yescomplemented
186C4⋊C4×C61C2×D414yes—
187C22⋊C4×C61—24no—
188C42×C61—24no—
189#11758198—28no—
190#11758198—28no—
191C4⋊Dic61—24no—
192C4⋊Dic61—22no—
193C4⋊Dic61—28no—
194C4⋊Dic61C2×D414yes—
195#11758196—216no—
196#11758194—44no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61C2×C22⋊C412yes—
203C23×C61#12012yes—
204C23×C61#12312yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—28no—
207C22×D61—216no—
208C22×D61—28no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—28no—
214C22×D61—216no—
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61C2×C22⋊C414yes—
221C22×D61AGL(1,ℤ/8)14yes—
222C22×D61#13714yes—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—48no—
226C2×C4×C61—44no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C2×C22⋊C412yes—
230C2×C4×C61C4×D414yes—
231C2×C4×C61C4×D414yes—
232C2×C4×C61#12012yes—
233C2×C4×C61#12114yes—
234C2×C4×C61#12114yes—
235C2×C4×C61#12312yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—48no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—28no—
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—48no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—44no—
252C2×Dic61C2×C22⋊C414yes—
253D244—22no—
254D244—22no—
255D244—416no—
256D244—41no—
257D244—41no—
258C4×D61—24no—
259C4×D61—24no—
260C4×D61—24no—
261C4×D61—24no—
262C4×D61—216no—
263C4×D61—432no—
264C4×D61—48no—
265Dic122—22no—
266Dic122—22no—
267Dic122—41no—
268Dic122—41no—
269C8×C61—42no—
270#92418—42no—
271#20909—611nomaximal, Sylow
272C22×C61—24no—
273C22×C61—24no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61#38614yes—
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61C2×#12011yescharacteristic
279C22×C61C2×#12311yescharacteristic
280C22×C61C2×AGL(1,ℤ/8)12yes—
281C22×C61C2×#13712yes—
282C2×D61—232no—
283C2×D61—24no—
284C2×D61—24no—
285C2×D61—24no—
286C2×D61—24no—
287C2×D61—24no—
288C2×D61—24no—
289C2×D61—44no—
290C2×D61—42no—
291C2×D61—82no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61#41718yescomplemented
295C4×C61—44no—
296C4×C61—44no—
297C4×C61#38614yes—
298C4×C61C2×C4×D411yescharacteristic
299C4×C61C2×C4×D411yescharacteristic
300C4×C61C2×#12111yescharacteristic
301C4×C61C2×#12111yescharacteristic
302Dic61—24no—
303Dic61—44no—
304Dic61—42no—
305C2×#525—611no—
306C23×C4⋊C4—611no—
307C2×#417—611no—
308C2×#417—611no—
309C2×#407—611no—
310#1874—614no—
311#1872—614no—
312#1736—611no—
313#1736D6111yescharacteristic, complemented
314C2×C61—24no—
315C2×C61C2×#38611yescharacteristic
316C2×C61C2×#41712yes—
317C2×C61#300311yescharacteristic
318C2×C61#300411yescharacteristic
319C2×C61#309212yes—
320D61—28no—
321D61—82no—
322C22×#41—611no—
323C24×C4—611no—
324C24×C4—1221no—
325C22×M16—611no—
326C22×M16C2×D6111yescharacteristic
327#525—618no—
328#525—1224no—
329C2×#123—1221no—
330C2×C22⋊Q8—612no—
331C2×#121—612no—
332C2×#120—1221no—
333C2×C4×D4—1222no—
334C2×#117C2×D6111yescharacteristic
335C22×C4⋊C4—611no—
336C22×C4⋊C4—614no—
337C22×C4⋊C4—1224no—
338C22×C4⋊C4—1222no—
339C22×C4⋊C4—1221no—
340C22×C4⋊C4—1224no—
341C22×C4⋊C4C2×D6111yescharacteristic
342#417—1224no—
343#417—1224no—
344C2×#103—612no—
345C2×#103—612no—
346C2×#102—612no—
347C2×#102—612no—
348#407—1224no—
349C2×C22⋊C8—614no—
350#340—614no—
351#340C2×D6114yescomplemented
352C61#2090911yescharacteristic, complemented, Sylow
353C25—611no—
354C2×#41—614no—
355C2×#41—1222no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1222no—
359C2×#41—1222no—
360C22×Q8—611no—
361C22×D4—611no—
362C22×D4—1222no—
363C22×D4—1221no—
364C23×C4—21no—
365C23×C4—611no—
366C23×C4—614no—
367C23×C4—612no—
368C23×C4—614no—
369C23×C4—612no—
370C23×C4—1224no—
371C23×C4—1228no—
372C23×C4—1222no—
373C23×C4—1222no—
374C23×C4—1222no—
375C23×C4—1222no—
376C23×C4—1221no—
377C23×C4—1221no—
378C23×C4—2448no—
379C23×C4C22×D6111yescharacteristic
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—1222no—
384C2×M16—1222no—
385C2×M16—1222no—
386C22×C8—612no—
387C22×C8C22×D6112yes—
388#123—1228no—
389C22⋊Q8—618no—
390C22⋊Q8—1224no—
391#121—618no—
392#121—1224no—
393#120—1228no—
394C4×D4—12216no—
395#117—22no—
396#117C4×D6114yescomplemented
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—614no—
400C2×C4⋊C4—612no—
401C2×C4⋊C4—1228no—
402C2×C4⋊C4—12216no—
403C2×C4⋊C4—1224no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1228no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1228no—
409C2×C4⋊C4D24414yescomplemented
410C2×C4⋊C4C22×D6112yes—
411C2×C4⋊C4C22×D6112yes—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1222no—
415C2×C42—21no—
416#103—1224no—
417#103—1224no—
418#102—1224no—
419#102—1224no—
420C22⋊C8—1228no—
421C24—612no—
422C24—614no—
423C24—1224no—
424C24—1224no—
425C24—1222no—
426C24—1222no—
427C24—1221no—
428C24D4×D6111yescharacteristic
429#41—1228no—
430#41—1228no—
431#41—2444no—
432#41—2444no—
433C2×Q8—612no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×D4—612no—
440C2×D4—1228no—
441C2×D4—1224no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—2444no—
448C2×D4—2444no—
449C2×D4—2442no—
450C2×D4—2442no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—612no—
456C22×C4—614no—
457C22×C4—614no—
458C22×C4—614no—
459C22×C4—618no—
460C22×C4—12216no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1228no—
468C22×C4—1228no—
469C22×C4—1224no—
470C22×C4—1228no—
471C22×C4—1228no—
472C22×C4—1224no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—1224no—
476C22×C4—1224no—
477C22×C4—2448no—
478C22×C4—24432no—
479C22×C4—2448no—
480C22×C4—2444no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4C2×C4×D6112yes—
484C22×C4C2×D24412yes—
485C22×C4#1175822212yes—
486C22×C4D4×D6112yes—
487C22×C4D4×D6112yes—
488C22×C4D4×D6111yescharacteristic
489C22×C4#1175822612yes—
490C22×C4C23×D6111yescharacteristic
491M16—44no—
492M16—2444no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—1222no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—614no—
503C4⋊C4—1228no—
504C4⋊C4—12216no—
505C4⋊C4—1222no—
506C4⋊C4—1224no—
507C4⋊C4C2×C4×D6114yescomplemented
508C4⋊C4C2×D24414yes—
509C22⋊C4—24no—
510C22⋊C4—1228no—
511C22⋊C4—1228no—
512C42—24no—
513C23—22no—
514C23—22no—
515C23—614no—
516C23—614no—
517C23—614no—
518C23—12216no—
519C23—1228no—
520C23—12216no—
521C23—1228no—
522C23—1222no—
523C23—1222no—
524C23—1222no—
525C23—1222no—
526C23—1228no—
527C23—1222no—
528C23—2441no—
529C23—2441no—
530C23—2442no—
531C23—2442no—
532C23C22⋊C4×D6112yes—
533C23#1451974712yes—
534C23#1451976212yes—
535C23C2×D4×D6111yescharacteristic
536Q8—1222no—
537Q8—1222no—
538Q8—2441no—
539Q8—2441no—
540D4—1222no—
541D4—1222no—
542D4—24416no—
543D4—2448no—
544D4—2441no—
545D4—2441no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—614no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1228no—
556C2×C4—1224no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—12216no—
561C2×C4—1222no—
562C2×C4—1224no—
563C2×C4—24432no—
564C2×C4—2448no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2444no—
568C2×C4—2448no—
569C2×C4C22⋊C4×D6112yes—
570C2×C4#1451973914yes—
571C2×C4C4×D24414yes—
572C2×C4#1451974712yes—
573C2×C4#1451975014yes—
574C2×C4#1451975214yes—
575C2×C4#1451976212yes—
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C22×D24411yescharacteristic
580C2×C4C2×#1175822211yescharacteristic
581C2×C4C2×#1175822611yescharacteristic
582C8—42no—
583C8—2442no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—24no—
588C22—618no—
589C22—12232no—
590C22—1224no—
591C22—1224no—
592C22—1224no—
593C22—1224no—
594C22—1224no—
595C22—1224no—
596C22—4882no—
597C22—4881no—
598C22—4881no—
599C22—2442no—
600C22—2444no—
601C22#1683964014yes—
602C22C2×C22⋊C4×D6111yescharacteristic
603C22C2×#1451974711yescharacteristic
604C22C2×#1451976211yescharacteristic
605C22AGL(1,ℤ/8)×D6112yes—
606C22#137×D6112yes—
607C4—44no—
608C4—44no—
609C4—1224no—
610C4—2442no—
611C4—2444no—
612C4#1683964014yes—
613C4C2×#1451973911yescharacteristic
614C4C2×C4×D24411yescharacteristic
615C4C2×#1451975011yescharacteristic
616C4C2×#1451975211yescharacteristic
617C2—24no—
618C2—1228no—
619C2—4882no—
620C2C2×#1683964011yescharacteristic
621C2#417×D6112yes—
622C2#2145504811yescharacteristic
623C2#2145505511yescharacteristic
624C2#2145574512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC2×#16840274C2
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20909
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D61C2semidirect
3. C2×#16839764C2semidirect
4. C2×#16839761C2semidirect
5. C2×#16839725C2semidirect
6. #21446135C2semidirect
7. #21446125C2semidirect
8. #21445540C2semidirect
9. C61×#1736C2semidirect
15. #16840274C4semidirect
29. C2×#14519702C22semidirect
30. C2×#14519700C22semidirect
31. C2×#14519699C22semidirect
32. C2×#14519697C22semidirect
35. #16839524C22semidirect
39. C61×#340C22semidirect
46. C2×#11758222C2×C4semidirect
71. #14519757C2×C4semidirect
73. #14519752C2×C4semidirect
79. C2×C4⋊Dic61D4semidirect
100. #117×C61C2×C4semidirect
103. C2×C4⋊C4×C61D4semidirect
151. C2×D244C22×C4semidirect
177. C2×Dic122C22×C4semidirect
185. C4⋊C4×C61C22×C4semidirect
294. C2×D61#417semidirect
313. #1736D61semidirect
351. #340C2×D61semidirect
352. C61#20909semidirect
396. #117C4×D61semidirect
409. C2×C4⋊C4D244semidirect
507. C4⋊C4C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402742
C22×C4⋊C4×D612
C2×#168397642
C2×#168397612
C2×#168397252
#214461352
#214461252
#214455402
C61×#17362
#2090961