Finite Group Databasebeta

← #29681247

Subgroups — #29681247

Order 15616.

1954 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840274C211yesmaximal, characteristic
2C22×#14519735C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4C2×#16839761C211yesmaximal, characteristic, complemented
5C2×#16839725C211yesmaximal, characteristic, complemented
6#21446140C214yesmaximal, complemented
7#21446130C214yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13#16840274—24no—
14#16840274C418yescomplemented
15C2×#14519761—21no—
16C2×#14519757C2212yes—
17C2×#14519752C2212yes—
18C2×#14519747—21no—
19C2×C4×D244—22no—
20C2×#14519735—24no—
21C2×#14519735—24no—
22C2×#14519735—22no—
23C2×#14519735C2214yes—
24C22×C4⋊Dic61C2211yescharacteristic
25C22×#11758198—21no—
26C22×C4×Dic61—21no—
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×C414yes—
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—21no—
58C22×C4×D61C2×C414yes—
59C22×C4×D61D412yes—
60C22×C4×D61D414yes—
61C22×C4×D61C2312yes—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194—22no—
65C22×#92418C2312yes—
66#14519761—28no—
67#14519757—24no—
68#14519757C2×C418yescomplemented
69#14519752—24no—
70#14519752C2×C418yescomplemented
71#14519747—28no—
72C4×D244—216no—
73#14519735—216no—
74#14519735—28no—
75#14519735—28no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61D414yescomplemented
80C2×C4⋊Dic61C2312yes—
81C2×#11758198—28no—
82C2×#11758198—24no—
83C2×#11758198—22no—
84C2×#11758198—22no—
85C2×C4×Dic61—28no—
86C2×C4×Dic61—24no—
87#14519702—24no—
88#14519700—24no—
89#14519699—24no—
90#14519697—24no—
91#14519689—28no—
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61—22no—
97C22×C8×C61C2312yes—
98#117×C61—22no—
99#117×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61D414yescomplemented
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C24×C61C2×D411yescharacteristic
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—21no—
111C23×D61—42no—
112C23×D61—42no—
113C23×D61C22⋊C414yes—
114C23×D61C2×D412yes—
115C22×C4×C61—24no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—44no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—42no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—24no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—44no—
136C22×Dic61C22⋊C414yes—
137C22×Dic61C2×D412yes—
138#11758222—28no—
139#11758222—28no—
140#11758222—44no—
141#11758222—44no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—28no—
147C2×D244—21no—
148C2×D244—44no—
149C2×D244—44no—
150C2×D244C22×C412yes—
151C2×C4×D61—28no—
152C2×C4×D61—28no—
153C2×C4×D61—24no—
154C2×C4×D61—28no—
155C2×C4×D61—28no—
156C2×C4×D61—24no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—24no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61C22⋊C418yes—
165C2×C4×D61C22×C414yes—
166C2×C4×D61C2×D414yes—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173M16×C61—44no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61C22×C414yes—
181C4⋊C4×C61C2×D414yes—
182C22⋊C4×C61—24no—
183C42×C61—24no—
184#11758198—216no—
185#11758198—28no—
186#11758198—28no—
187C4⋊Dic61—24no—
188C4⋊Dic61—22no—
189C4⋊Dic61—28no—
190C4⋊Dic61C2×D414yes—
191C4×Dic61—216no—
192#11758194—44no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61C2×C22⋊C412yes—
199C23×C61#12012yes—
200C23×C61#12312yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—28no—
209C22×D61—22no—
210C22×D61—48no—
211C22×D61—48no—
212C22×D61—44no—
213C22×D61—44no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61C2×C22⋊C414yes—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—48no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—44no—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61C4×D414yes—
227C2×C4×C61C4×D414yes—
228C2×C4×C61#12012yes—
229C2×C4×C61#12114yes—
230C2×C4×C61#12114yes—
231C2×C4×C61#12312yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C22⋊D61—48no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—28no—
242C2×Dic61—22no—
243C2×Dic61—24no—
244C2×Dic61—48no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—44no—
248C2×Dic61C2×C22⋊C414yes—
249D244—22no—
250D244—22no—
251D244—416no—
252D244—41no—
253D244—41no—
254C4×D61—24no—
255C4×D61—24no—
256C4×D61—24no—
257C4×D61—24no—
258C4×D61—216no—
259C4×D61—48no—
260Dic122—22no—
261Dic122—22no—
262Dic122—41no—
263Dic122—41no—
264C8×C61—42no—
265#92418—42no—
266#20910—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61#38614yes—
272C22×C61C22×C22⋊C411yescharacteristic
273C22×C61C2×#12011yescharacteristic
274C22×C61C2×#12311yescharacteristic
275C22×C61#57512yes—
276C22×C61#57512yes—
277C2×D61—24no—
278C2×D61—24no—
279C2×D61—24no—
280C2×D61—416no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—44no—
286C2×D61—42no—
287C2×D61—82no—
288C2×D61—81no—
289C2×D61—81no—
290C4×C61—44no—
291C4×C61—44no—
292C4×C61#38614yes—
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×C4×D411yescharacteristic
295C4×C61C2×#12111yescharacteristic
296C4×C61C2×#12111yescharacteristic
297Dic61—24no—
298Dic61—44no—
299Dic61—42no—
300C2×#525—611no—
301C22×#117—611no—
302C2×#417—611no—
303C2×#417—611no—
304C2×#407—611no—
305#1875—614no—
306#1873—614no—
307#1736—611no—
308#1736D6111yescharacteristic, complemented
309C2×C61—24no—
310C2×C61C2×#38611yescharacteristic
311C2×C61#289612yes—
312C2×C61#300511yescharacteristic
313C2×C61#300511yescharacteristic
314C2×C61#309312yes—
315D61—44no—
316D61—82no—
317C22×#41—611no—
318C24×C4—611no—
319C22×M16—611no—
320C22×M16C2×D6111yescharacteristic
321#525—618no—
322#525—1224no—
323C2×#123—1221no—
324C2×C22⋊Q8—612no—
325C2×#121—612no—
326C2×#120—1221no—
327C2×C4×D4—1222no—
328C2×#117—614no—
329C2×#117—1224no—
330C2×#117—1222no—
331C2×#117—1224no—
332C2×#117C2×D6111yescharacteristic
333C22×C4⋊C4—611no—
334C22×C4⋊C4C2×D6111yescharacteristic
335C22×C22⋊C4—1221no—
336C22×C42—1221no—
337#417—1224no—
338#417—1224no—
339C2×#103—612no—
340C2×#103—612no—
341C2×#102—612no—
342C2×#102—612no—
343#407—1224no—
344C2×C22⋊C8—614no—
345#340—614no—
346#340C2×D6114yescomplemented
347C61#2091011yescharacteristic, complemented, Sylow
348C25—611no—
349C2×#41—614no—
350C2×#41—1222no—
351C2×#41—1222no—
352C2×#41—1222no—
353C2×#41—1222no—
354C2×#41—1222no—
355C22×Q8—611no—
356C22×D4—611no—
357C22×D4—1222no—
358C22×D4—1221no—
359C23×C4—21no—
360C23×C4—611no—
361C23×C4—614no—
362C23×C4—612no—
363C23×C4—614no—
364C23×C4—612no—
365C23×C4—1222no—
366C23×C4—1222no—
367C23×C4—1222no—
368C23×C4—1222no—
369C23×C4—1221no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372C2×M16—22no—
373C2×M16—22no—
374C2×M16—22no—
375C2×M16—1222no—
376C2×M16—1222no—
377C2×M16—1222no—
378C22×C8—612no—
379C22×C8C22×D6112yes—
380#123—1228no—
381C22⋊Q8—618no—
382C22⋊Q8—1224no—
383#121—618no—
384#121—1224no—
385#120—1228no—
386C4×D4—12216no—
387#117—22no—
388#117—1228no—
389#117—12216no—
390#117—1228no—
391#117C4×D6114yes—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—614no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4D24414yescomplemented
400C2×C4⋊C4C22×D6112yes—
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1228no—
407C2×C42—21no—
408C2×C42—1224no—
409C2×C42—1228no—
410#103—1224no—
411#103—1224no—
412#102—1224no—
413#102—1224no—
414C22⋊C8—1228no—
415C24—612no—
416C24—614no—
417C24—1222no—
418C24—1222no—
419C24—1221no—
420C24—2442no—
421C24—2442no—
422C24D4×D6111yescharacteristic
423#41—1228no—
424#41—1228no—
425#41—2444no—
426#41—2444no—
427C2×Q8—612no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×D4—612no—
434C2×D4—1228no—
435C2×D4—1224no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—2444no—
442C2×D4—2444no—
443C2×D4—2442no—
444C2×D4—2442no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—44no—
449C22×C4—612no—
450C22×C4—614no—
451C22×C4—614no—
452C22×C4—614no—
453C22×C4—618no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1228no—
461C22×C4—1228no—
462C22×C4—1224no—
463C22×C4—1228no—
464C22×C4—1228no—
465C22×C4—1224no—
466C22×C4—1222no—
467C22×C4—1221no—
468C22×C4—1224no—
469C22×C4—1224no—
470C22×C4—2444no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4C2×C4×D6112yes—
474C22×C4C2×D24412yes—
475C22×C4#1175822212yes—
476C22×C4D4×D6112yes—
477C22×C4D4×D6112yes—
478C22×C4D4×D6111yescharacteristic
479C22×C4#1175822612yes—
480C22×C4C23×D6111yescharacteristic
481M16—44no—
482M16—2444no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—614no—
493C4⋊C4—1228no—
494C4⋊C4—1222no—
495C4⋊C4—1224no—
496C4⋊C4C2×C4×D6114yes—
497C4⋊C4C2×D24414yes—
498C22⋊C4—24no—
499C22⋊C4—1228no—
500C22⋊C4—1228no—
501C22⋊C4—12216no—
502C42—24no—
503C42—12216no—
504C23—22no—
505C23—22no—
506C23—614no—
507C23—1222no—
508C23—1222no—
509C23—1222no—
510C23—1222no—
511C23—1222no—
512C23—1222no—
513C23—1228no—
514C23—1222no—
515C23—2448no—
516C23—2448no—
517C23—2444no—
518C23—2444no—
519C23—2441no—
520C23—2441no—
521C23—2442no—
522C23—2442no—
523C23C22⋊C4×D6112yes—
524C23#1451974712yes—
525C23#1451976212yes—
526C23C2×D4×D6111yescharacteristic
527Q8—1222no—
528Q8—1222no—
529Q8—2441no—
530Q8—2441no—
531D4—1222no—
532D4—1222no—
533D4—24416no—
534D4—2448no—
535D4—2441no—
536D4—2441no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—614no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1228no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—12216no—
552C2×C4—1222no—
553C2×C4—1224no—
554C2×C4—2448no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2444no—
558C2×C4—2448no—
559C2×C4C22⋊C4×D6112yes—
560C2×C4#1451973914yes—
561C2×C4C4×D24414yes—
562C2×C4#1451974712yes—
563C2×C4#1451975014yes—
564C2×C4#1451975214yes—
565C2×C4#1451976212yes—
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C22×D24411yescharacteristic
570C2×C4C2×#1175822211yescharacteristic
571C2×C4C2×#1175822611yescharacteristic
572C8—42no—
573C8—2442no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—24no—
578C22—1224no—
579C22—1224no—
580C22—1224no—
581C22—4882no—
582C22—4881no—
583C22—4881no—
584C22—24416no—
585C22—2442no—
586C22—2444no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22#1683964014yes—
592C22C2×C22⋊C4×D6111yescharacteristic
593C22C2×#1451974711yescharacteristic
594C22C2×#1451976211yescharacteristic
595C22#1684055812yes—
596C22#1684056112yes—
597C4—44no—
598C4—44no—
599C4—1224no—
600C4—2442no—
601C4—2444no—
602C4#1683964014yes—
603C4C2×#1451973911yescharacteristic
604C4C2×C4×D24411yescharacteristic
605C4C2×#1451975011yescharacteristic
606C4C2×#1451975211yescharacteristic
607C2—24no—
608C2—4882no—
609C2—2444no—
610C2C2×#1683964011yescharacteristic
611C2#2145433512yes—
612C2#2145506311yescharacteristic
613C2#2145506511yescharacteristic
614C2#2145575312yes—
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#20910
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×#14519735C2semidirect
3. C2×#16839764C2semidirect
4. C2×#16839761C2semidirect
5. C2×#16839725C2semidirect
6. #21446140C2semidirect
7. #21446130C2semidirect
8. #21445540C2semidirect
9. C61×#1736C2semidirect
14. #16840274C4semidirect
29. C2×#14519702C22semidirect
30. C2×#14519700C22semidirect
31. C2×#14519699C22semidirect
32. C2×#14519697C22semidirect
35. #16839524C22semidirect
39. C61×#340C22semidirect
68. #14519757C2×C4semidirect
70. #14519752C2×C4semidirect
79. C2×C4⋊Dic61D4semidirect
102. C2×C4⋊C4×C61D4semidirect
308. #1736D61semidirect
346. #340C2×D61semidirect
347. C61#20910semidirect
399. C2×C4⋊C4D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168402742
C22×#145197352
C2×#168397642
C2×#168397612
C2×#168397252
#214461402
#214461302
#214455402
C61×#17362
#2091061