Finite Group Databasebeta

← #29732214

Subgroups — #29732214

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16839988C212yesmaximal
3#21454149C211yesmaximal, characteristic
4#21447153C211yesmaximal, characteristic, complemented
5#21447062C211yesmaximal, characteristic, complemented
6#21446955C211yesmaximal, characteristic, complemented
7#21446239C212yesmaximal, complemented
8#21446176C212yesmaximal, complemented
9#21446089C212yesmaximal, complemented
10#336×D61C211yesmaximal, characteristic, complemented
11C61×#2013C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840317—28no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768—21no—
16C2×#14519759—21no—
17C2×#14519757—21no—
18C2×#14519753C2212yes—
19C2×#14519748C2211yescharacteristic
20C2×Q8×Dic61C2212yes—
21C2×#14519742—21no—
22C2×#14519735—21no—
23C2×C42×D61C2211yescharacteristic
24#16839988—28no—
25C2×#14519709C2212yes—
26C2×#14519701C2212yes—
27C2×#14519701C2212yes—
28C2×#14519698C2212yes—
29#16839665C2211yescharacteristic
30#16839661C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839656—21no—
33#16839655—21no—
34#16839638—21no—
35#16839619C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C22×SD16×C61C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#390C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×C22⋊D61—21no—
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C22×#92418C2312yes—
51C2×C8×D61—24no—
52C2×C8×D61—24no—
53C2×C8×D61—42no—
54#14519777—24no—
55#14519768—28no—
56#14519759—44no—
57#14519757—28no—
58#14519753—24no—
59#14519753D418yes—
60#14519748—24no—
61Q8×Dic61—24no—
62Q8×Dic61—24no—
63#14519742—28no—
64#14519735—28no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×#11758198—22no—
77C2×#11758198—21no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C42×D61—24no—
83#14519709—24no—
84#14519701—24no—
85#14519701—42no—
86#14519698—24no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90C22×Q8×C61C2312yes—
91C22×D4×C61C2311yescharacteristic
92C2×SD16×C61—24no—
93C2×SD16×C61—24no—
94C2×SD16×C61—42no—
95C22×C8×C61C2312yes—
96C4⋊Q8×C61—24no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#103×C61—24no—
104C24×C61—21no—
105C23×D61C2×D411yescharacteristic
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61C2×D414yes—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—41no—
138C2×C4×D61SD1614yescomplemented
139C2×C4×D61C2×D414yes—
140C2×Dic122—24no—
141C2×Dic122—42no—
142C2×Dic122—42no—
143SD16×C61—48no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61#4114yes—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C22⋊C4×C61—42no—
151C22⋊C4×C61—42no—
152C42×C61—22no—
153C22⋊Dic61—42no—
154C22⋊Dic61—42no—
155C22⋊Dic61—42no—
156#11758198—28no—
157#11758198—42no—
158C4⋊Dic61—24no—
159C4⋊Dic61—28no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—42no—
165C4⋊Dic61C2×D414yes—
166#11758196—28no—
167#11758196—42no—
168C4×Dic61—28no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C8×D61—48no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61C22×D411yescharacteristic
184C22×D61—21no—
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61#12012yes—
189C22×D61C2×SD1612yes—
190C22×D61#13712yes—
191Q8×C61—24no—
192Q8×C61—24no—
193Q8×C61—42no—
194Q8×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—24no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61C2×SD1612yes—
208C2×C4×C61#13512yes—
209C2×C4×C61C22×D412yes—
210C2×C4×C61C2×#4112yes—
211C2×C4×C61H(5,𝔽2)11yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C22⋊D61—84no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—24no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61#12012yes—
231C2×Dic61C2×SD1612yes—
232C4×D61—28no—
233C4×D61—24no—
234C4×D61—24no—
235C4×D61—44no—
236Dic122—48no—
237C8×C61—24no—
238#92418—42no—
239#25497—611nomaximal, Sylow
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61C2×#12011yescharacteristic
248C22×C61#53211yescharacteristic
249C22×C61#53711yescharacteristic
250C22×C61C22×SD1611yescharacteristic
251C22×C61C2×#13511yescharacteristic
252C22×C61C2×#13711yescharacteristic
253C22×C61#57511yescharacteristic
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61#44814yes—
261C4×C61—22no—
262C4×C61—22no—
263C4×C61—44no—
264C4×C61—81no—
265C4×C61#54612yes—
266C4×C61#54712yes—
267Dic61—22no—
268Dic61—22no—
269Dic61—44no—
270Dic61—41no—
271Dic61—81no—
272Dic61—81no—
273C2×#533—611no—
274C2×#465—612no—
275#2854—611no—
276#2058—611no—
277#2043—611no—
278#2023—611no—
279#2013D6111yescharacteristic, complemented
280#1892—612no—
281#1880—612no—
282#1865—612no—
283C2×#336—611no—
284C2×C61—161no—
285C2×C61#243011yescharacteristic
286C2×C61C2×#44811yescharacteristic
287C2×C61#300211yescharacteristic
288C2×C61#316711yescharacteristic
289C2×C61#318111yescharacteristic
290C2×C61#324611yescharacteristic
291C2×C61#324811yescharacteristic
292D61—24no—
293C22×SD16C2×D6112yes—
294C23×C8—612no—
295#533—1228no—
296C2×C4⋊Q8—611no—
297C2×C4⋊Q8C2×D6111yescharacteristic
298C2×#125—1221no—
299C2×#123—1221no—
300C2×C22⋊Q8—612no—
301C2×C22⋊Q8—1221no—
302C2×#121—611no—
303C2×C4×Q8—612no—
304C2×C4×Q8—1221no—
305C2×#117—1221no—
306C22×C42—611no—
307#465—1228no—
308C2×#106—612no—
309C2×#103—612no—
310C2×#103—612no—
311C2×#103C2×D6112yes—
312C2×#102—612no—
313#390—611no—
314#390C2×D6111yescharacteristic
315#389—611no—
316#389—611no—
317#388—1221no—
318#388—1221no—
319#386—1221no—
320#380—611no—
321#340—612no—
322#340—612no—
323#336—612no—
324#336—611no—
325#336C2×D6111yescharacteristic
326C61#2549711yescharacteristic, complemented, Sylow
327C22×Q8—1221no—
328C22×Q8C22×D6112yes—
329C22×D4—1221no—
330C22×D4C22×D6111yescharacteristic
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C2×SD16—24no—
335C2×SD16—24no—
336C2×SD16—42no—
337C22×C8—612no—
338C22×C8—1224no—
339C22×C8—1224no—
340C22×C8—2442no—
341C22×C8C22×D6112yes—
342C4⋊Q8—24no—
343C4⋊Q8—1224no—
344#125—1228no—
345#123—2444no—
346C22⋊Q8—618no—
347C22⋊Q8—1228no—
348C22⋊Q8—1224no—
349#121—1224no—
350C4×Q8—1228no—
351C4×Q8—1224no—
352C4×Q8—1224no—
353#117—1228no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—1224no—
376C2×C42—1222no—
377C2×C42C22×D6111yescharacteristic
378#106—1224no—
379#103—24no—
380#103—1224no—
381#103—2442no—
382#102—1224no—
383#95—1221no—
384#95—1221no—
385#95—1221no—
386C24—21no—
387C24—611no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—1224no—
394C2×Q8—2442no—
395C2×Q8—2442no—
396C2×Q8D4×D6114yes—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—2442no—
402C2×D4—2442no—
403C2×D4—2442no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—614no—
409C22×C4—614no—
410C22×C4—611no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—2441no—
422C22×C4D4×D6111yescharacteristic
423C22×C4D4×D6111yescharacteristic
424C22×C4C23×D6111yescharacteristic
425SD16—48no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—1222no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—2448no—
432C2×C8#1175822414yes—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—614no—
437C4⋊C4—1228no—
438C4⋊C4—1224no—
439C4⋊C4—1228no—
440C4⋊C4—1222no—
441C4⋊C4—1222no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—2442no—
445C4⋊C4—2442no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—1228no—
449C22⋊C4—2442no—
450C22⋊C4—2442no—
451C22⋊C4—2442no—
452C22⋊C4—2442no—
453C42—22no—
454C42—1222no—
455C42—1222no—
456C42—1228no—
457C42—1222no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—612no—
466C23—612no—
467C23—612no—
468C23—1221no—
469C23—1221no—
470C23—1221no—
471C23—1221no—
472C23C2×D4×D6111yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—42no—
476Q8—42no—
477Q8—2448no—
478D4—42no—
479D4—42no—
480D4—4884no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—612no—
492C2×C4—612no—
493C2×C4—1228no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1224no—
499C2×C4—1224no—
500C2×C4—1221no—
501C2×C4—1224no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—2444no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4SD16×D6112yes—
514C2×C4#1451979812yes—
515C2×C4C2×D4×D6112yes—
516C2×C4C2×#1175822412yes—
517C2×C4#1451982211yescharacteristic
518C2×C4#1451982411yescharacteristic
519C8—24no—
520C8—2442no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—614no—
529C22—1222no—
530C22—1222no—
531C22—1222no—
532C22—1222no—
533C22—1222no—
534C22—1222no—
535C22#120×D6111yescharacteristic
536C22#1684031011yescharacteristic
537C22#1684034111yescharacteristic
538C22C2×SD16×D6111yescharacteristic
539C22C2×#1451979811yescharacteristic
540C22#137×D6111yescharacteristic
541C22#1684056111yescharacteristic
542C4—22no—
543C4—22no—
544C4—44no—
545C4—81no—
546C4—1222no—
547C4—1222no—
548C4—4881no—
549C4—4881no—
550C4—2441no—
551C4—2444no—
552C4#1684039912yes—
553C4#1684041212yes—
554C2—161no—
555C2—1224no—
556C2#2144942211yescharacteristic
557C2#448×D6111yescharacteristic
558C2#2145503611yescharacteristic
559C2#2145637611yescharacteristic
560C2#2145652511yescharacteristic
561C2#2145713211yescharacteristic
562C2#2145715211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2013C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenter#2013D61
hypocenterC61#25497
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
4. #21447153C2semidirect
5. #21447062C2semidirect
6. #21446955C2semidirect
7. #21446239C2semidirect
8. #21446176C2semidirect
9. #21446089C2semidirect
10. #336×D61C2semidirect
11. C61×#2013C2semidirect
138. C2×C4×D61SD16semidirect
279. #2013D61semidirect
326. C61#25497semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168399882
#214541492
#214471532
#214470622
#214469552
#214462392
#214461762
#214460892
#336×D612
C61×#20132
#2549761