Finite Group Databasebeta

← #29652667

Subgroups — #29652667

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455052C212yesmaximal, complemented
2C2×#16839682C211yesmaximal, characteristic, complemented
3#21444793C212yesmaximal, complemented
4#21444788C212yesmaximal, complemented
5#21444782C214yesmaximal, complemented
6#21444323C212yesmaximal, complemented
7#21444149C211yesmaximal, characteristic, complemented
8C61×#3492C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C2×#14519807—22no—
11C22×#11758196—21no—
12#16839917—22no—
13#16839917C2214yes—
14#16839900—22no—
15#16839900C2214yes—
16#16839770—22no—
17#16839726C2212yes—
18#16839682—21no—
19#16839682—21no—
20#16839682C2212yes—
21#16839682C2212yes—
22C2×#14519684—21no—
23C2×#14519684C2211yescharacteristic
24#16839499C2214yes—
25#16839496C2214yes—
26#16839488C2212yes—
27#16839488C2212yes—
28C61×#554—22no—
29C61×#553—22no—
30C61×#552—22no—
31C61×#547—22no—
32C61×#544C2211yescharacteristic
33C61×#539—21no—
34C61×#539C2212yes—
35C61×#537C2211yescharacteristic
36C61×#525—21no—
37C61×#525—21no—
38C61×#525C2212yes—
39C61×#520—21no—
40C22×C4⋊C4×C61C2211yescharacteristic
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—21no—
44C22×Dic122—44no—
45C22×Dic122C2312yes—
46C23×Dic61—21no—
47C23×Dic61—21no—
48#14519807—48no—
49C2×#11758202—24no—
50C2×#11758200—24no—
51C2×#11758194—22no—
52C2×#11758196—22no—
53C2×#11758196—22no—
54C2×#11758196—44no—
55#14519700—24no—
56#14519700—24no—
57C22⋊#92418—22no—
58C22⋊#92418C2314yes—
59#14519684—21no—
60#14519684—22no—
61#14519684—22no—
62#14519684C2312yes—
63C2×#41×C61—21no—
64C2×#41×C61—21no—
65C2×#41×C61D412yes—
66C23×C4×C61D411yescharacteristic
67C23×C4×C61D411yescharacteristic
68C23×C4×C61C2311yescharacteristic
69C4⋊Q8×C61—22no—
70#126×C61—22no—
71#126×C61—22no—
72#125×C61—22no—
73#125×C61—22no—
74#125×C61—22no—
75#123×C61—22no—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79#123×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—21no—
82C22⋊Q8×C61—21no—
83C22⋊Q8×C61—21no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61D412yes—
88C22⋊Q8×C61C2312yes—
89#121×C61—21no—
90#121×C61—21no—
91#121×C61—21no—
92#121×C61—22no—
93#121×C61—21no—
94#121×C61D412yes—
95#121×C61C2312yes—
96#120×C61—22no—
97C4×Q8×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105#117×C61—21no—
106#117×C61—22no—
107#117×C61—21no—
108#117×C61D412yes—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61D411yescharacteristic
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C24×C61C2×D411yescharacteristic
117#41×C61—42no—
118#41×C61—44no—
119#41×C61—42no—
120#41×C61—82no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61C2×D412yes—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—21no—
129C2×D4×C61C2×D412yes—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—42no—
139C22×C4×C61—41no—
140C22×C4×C61—42no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—24no—
150C22×Dic61—24no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61—41no—
157C22×Dic61—41no—
158C2×Dic122—28no—
159C2×Dic122—24no—
160C2×Dic122—44no—
161C2×Dic122—48no—
162C2×Dic122—44no—
163C2×Dic122—44no—
164C2×Dic122—42no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61C2×D412yes—
180C4⋊C4×C61C2×D412yes—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61—22no—
193#11758202—48no—
194#11758200—48no—
195#11758196—44no—
196#11758196—44no—
197#11758194—44no—
198C2×#92418—24no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61#12011yescharacteristic
205C23×C61#12011yescharacteristic
206C23×C61C22×D411yescharacteristic
207Q8×C61—42no—
208Q8×C61—42no—
209Q8×C61—81no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—44no—
215D4×C61—44no—
216D4×C61—81no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—44no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61—81no—
239C2×C4×C61—82no—
240C2×C4×C61#12012yes—
241C2×C4×C61#13712yescomplemented
242C2×C4×C61C22×D412yes—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—44no—
248C2×Dic61—48no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—44no—
252Dic122—28no—
253Dic122—44no—
254Dic122—48no—
255#92418—44no—
256#18165—611nomaximal, Sylow
257C22×C61—24no—
258C22×C61—24no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61#45612yes—
269C22×C61B(4,𝔽2)12yes—
270C22×C61C2×#12011yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C22×C61#57511yescharacteristic
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—22no—
276C4×C61—41no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—42no—
282Dic61—44no—
283Dic61—44no—
284#3492D6111yescharacteristic
285#3004—612no—
286C2×#395—611no—
287#1605—612no—
288#1604—612no—
289#1603—614no—
290#1504—612no—
291#1463—611no—
292C2×C61—21no—
293C2×C61—21no—
294C2×C61—22no—
295C2×C61—22no—
296C2×C61—82no—
297C2×C61#300711yescharacteristic
298C2×C61C2×#45611yescharacteristic
299C2×C61C2×B(4,𝔽2)11yescharacteristic
300C23×Q8—611no—
301C2×#137—1222no—
302#554—22no—
303#553—22no—
304#552—22no—
305#547—22no—
306#544C2×D6111yescharacteristic
307#539—21no—
308#539C2×D6112yes—
309#537C2×D6111yescharacteristic
310#525—21no—
311#525—21no—
312#525C2×D6112yes—
313#520—21no—
314C22×C4⋊C4—1221no—
315C22×C4⋊C4C2×D6111yescharacteristic
316#451—614no—
317#451—1222no—
318#448—614no—
319#448—1222no—
320#419—1222no—
321#407—612no—
322#395—612no—
323#395—612no—
324#395—1221no—
325#395—1221no—
326C2×#95—611no—
327C2×#95—1221no—
328#328—614no—
329#327—614no—
330#324—612no—
331#324—612no—
332C61#1816511yescharacteristic, complemented, Sylow
333C2×#41—21no—
334C2×#41—21no—
335C2×#41C22⋊D6112yes—
336C22×Q8—612no—
337C22×Q8—1221no—
338C22×Q8—1222no—
339C22×Q8—1222no—
340C22×Q8—2444no—
341C23×C4—1221no—
342C23×C4—1221no—
343C23×C4C22⋊D6111yescharacteristic
344C23×C4C22⋊D6111yescharacteristic
345C23×C4C22×D6111yescharacteristic
346#137—2448no—
347C2×Q16—1224no—
348C2×SD16—1224no—
349C2×M16—1222no—
350C4⋊Q8—22no—
351#126—22no—
352#126—22no—
353#125—22no—
354#125—22no—
355#125—22no—
356#123—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—21no—
367C22⋊Q8—22no—
368C22⋊Q8C22⋊D6112yes—
369C22⋊Q8C22×D6112yes—
370#121—21no—
371#121—21no—
372#121—21no—
373#121—22no—
374#121—21no—
375#121C22⋊D6112yes—
376#121C22×D6112yes—
377#120—22no—
378C4×Q8—22no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—22no—
386#117—22no—
387#117—21no—
388#117—21no—
389#117C22⋊D6112yes—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—2444no—
396C2×C4⋊C4C22⋊D6111yescharacteristic
397C2×C4⋊C4C22⋊D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400#103—1224no—
401#103—1224no—
402C22⋊C8—614no—
403C22⋊C8—1222no—
404#95—612no—
405#95—1221no—
406#95—1222no—
407#95—1222no—
408C24C2×C22⋊D6111yescharacteristic
409#41—42no—
410#41—44no—
411#41—42no—
412#41—82no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—1228no—
416C2×Q8—1224no—
417C2×Q8—2444no—
418C2×Q8—2444no—
419C2×Q8—2444no—
420C2×Q8—2448no—
421C2×Q8—2442no—
422C2×Q8C2×C22⋊D6112yes—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4C2×C22⋊D6112yes—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4—2441no—
448C22×C4—2441no—
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6112yes—
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6112yes—
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457Q16—2448no—
458SD16—2448no—
459M16—2444no—
460C2×C8—1224no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—2444no—
476C4⋊C4—2444no—
477C4⋊C4C2×C22⋊D6112yes—
478C4⋊C4C2×C22⋊D6112yes—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C23—22no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23#1451974611yescharacteristic
497C23#1451974611yescharacteristic
498C23C22×C22⋊D6111yescharacteristic
499Q8—42no—
500Q8—42no—
501Q8—81no—
502Q8—1228no—
503Q8—2444no—
504Q8—2448no—
505D4—42no—
506D4—42no—
507D4—44no—
508D4—44no—
509D4—42no—
510D4—42no—
511D4—81no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—2442no—
538C2×C4—2448no—
539C2×C4—2442no—
540C2×C4—2444no—
541C2×C4—2444no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4#1451974612yes—
545C2×C4#1451980712yescomplemented
546C2×C4C22×C22⋊D6112yes—
547C8—2444no—
548C22—24no—
549C22—24no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22#1683994512yes—
560C22#1683994912yes—
561C22C2×#1451974611yescharacteristic
562C22C2×#1451980711yescharacteristic
563C22#1684056011yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—42no—
573C4—2444no—
574C4—2444no—
575C2—21no—
576C2—22no—
577C2—22no—
578C2—21no—
579C2—82no—
580C2#2145507911yescharacteristic
581C2C2×#1683994511yescharacteristic
582C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3492C2
socleC22×C61C2×#120
hypercenter#3492D61
hypocenterC61#18165
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
1. #21455052C2semidirect
2. C2×#16839682C2semidirect
3. #21444793C2semidirect
4. #21444788C2semidirect
5. #21444782C2semidirect
6. #21444323C2semidirect
7. #21444149C2semidirect
241. C2×C4×C61#137semidirect
332. C61#18165semidirect
545. C2×C4#14519807semidirect

Maximal subgroups

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

subgroupindex
#214550522
C2×#168396822
#214447932
#214447882
#214447822
#214443232
#214441492
C61×#34922
#1816561