Finite Group Databasebeta

← #30616255

Subgroups — #30616255

Order 15616.

1658 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840743C212yesmaximal, complemented
2C2×#16840742C212yesmaximal, complemented
3#21462268C211yesmaximal, characteristic, complemented
4C4×#14519856C211yesmaximal, characteristic, complemented
5#21461960C218yesmaximal, complemented
6C2×#16840311C211yesmaximal, characteristic
7C2×#14519856C2212yes—
8C2×#14519856C2212yes—
9#16840743—28no—
10#16840742—28no—
11C2×#14519836C2218yes—
12C2×#14519835C2218yes—
13#16840708—21no—
14#16840706—21no—
15#16840704C2214yes—
16C4×C4⋊#6923—22no—
17C4×C4⋊#6923—22no—
18C4×C22⋊#6923C2214yes—
19#16840698—21no—
20C42×#6923—21no—
21C22×#11758226C2211yescharacteristic
22C22×#11758224C2211yescharacteristic
23#16840311—21no—
24#16840311—21no—
25#16840311—21no—
26#16840311—21no—
27#16840311—22no—
28#16840311—22no—
29C2×#14519775C411yescharacteristic, complemented
30C2×#14519773C411yescharacteristic, complemented
31C2×#14519765C411yescharacteristic, complemented
32C2×#14519750C2214yes—
33C2×#14519737C414yescomplemented
34C2×C42×D61C2211yescharacteristic
35C2×#124×C61C411yescharacteristic, complemented
36#14519856—24no—
37#14519856—24no—
38#14519856—24no—
39#14519856D418yescomplemented
40C2×C4⋊#6923—22no—
41C2×C4⋊#6923—24no—
42C2×C4⋊#6923—22no—
43C2×C22⋊#6923C2314yes—
44C2×C22⋊#6923C2314yes—
45C2×C4×#6923—22no—
46C2×C4×#6923—22no—
47C2×C4×#6923—24no—
48#14519836—216no—
49#14519835—216no—
50(C2×#6923)⋊C4—24no—
51(C2×#6923)⋊C4—24no—
52C2×#11758226—24no—
53C2×#11758226—21no—
54C2×#11758226—21no—
55C2×#11758226—21no—
56C2×#11758226—21no—
57C2×#11758226—24no—
58C2×#11758224—24no—
59C2×#11758224—21no—
60C2×#11758224—21no—
61C2×#11758224—21no—
62C2×#11758224—21no—
63C2×#11758224—24no—
64C22×Dic122C2×C411yescharacteristic
65C22×D244C2×C411yescharacteristic
66C22×D244C2312yes—
67C22×C22⋊D61C2×C414yes—
68C22×C22⋊D61C2312yes—
69C23×Dic61C2×C412yes—
70C22×C4×D61C2×C412yes—
71C22×C4×D61C2312yes—
72C22×C4×D61C2311yescharacteristic
73#14519775—21no—
74#14519775—22no—
75#14519775—21no—
76#14519773—21no—
77#14519773—22no—
78#14519773—21no—
79#14519765—21no—
80#14519765—21no—
81#14519765—22no—
82#14519750—24no—
83#14519750—24no—
84#14519750—24no—
85#14519750—24no—
86#14519737—24no—
87#14519737—24no—
88#14519737—24no—
89#14519737—24no—
90C2×#11758196C2×C414yes—
91C2×#11758198C2×C414yes—
92C2×C4×Dic61C2×C412yes—
93C2×C4×Dic61C2×C411yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D412yes—
99C42×D61D412yes—
100C22×Q8×C61C2×C411yescharacteristic
101C22×D4×C61C2×C411yescharacteristic
102#124×C61—21no—
103#124×C61—21no—
104#124×C61—22no—
105C2×C22⋊C4×C61C2×C414yes—
106C2×C42×C61C2×C411yescharacteristic
107C24×C61C22×C412yes—
108C23×D61C22×C412yes—
109C23×D61C2411yescharacteristic
110C22×#6923—24no—
111C22×#6923—24no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—24no—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—24no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—48no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—28no—
131C22×Dic61C22×C412yes—
132C22×Dic61C22×C412yes—
133C22×Dic61C22×C411yescharacteristic
134#11758226—44no—
135#11758226—44no—
136#11758226—42no—
137#11758226—42no—
138#11758226—42no—
139#11758226—42no—
140#11758224—44no—
141#11758224—44no—
142#11758224—42no—
143#11758224—42no—
144#11758224—42no—
145#11758224—42no—
146C2×D244—24no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—24no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—24no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—28no—
156C2×C4×D61—44no—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D414yes—
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61#4114yes—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—24no—
170C22⋊#6923—24no—
171C22⋊#6923—24no—
172C22⋊#6923—24no—
173C22⋊#6923C2×D418yes—
174C4⋊#6923—28no—
175C4⋊#6923—48no—
176C4⋊#6923—44no—
177C4×#6923—28no—
178C4×#6923—48no—
179C4×#6923—44no—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22⋊C411yescharacteristic
184C42×C61C22⋊C412yes—
185#11758198—24no—
186#11758198—24no—
187#11758196—24no—
188#11758196—24no—
189C4×Dic61C22⋊C412yes—
190C4×Dic61C22⋊C412yes—
191C4×Dic61C22⋊C412yes—
192C4×Dic61C22⋊C412yes—
193C4×Dic61C22⋊C411yescharacteristic
194C4×Dic61C22⋊C412yes—
195C4×Dic61C22⋊C411yescharacteristic
196C23×C61—22no—
197C23×C61—24no—
198C23×C61—44no—
199C23×C61C23×C411yescharacteristic
200C22×D61—22no—
201C22×D61—24no—
202C22×D61—42no—
203C22×D61—44no—
204C22×D61C22×D411yescharacteristic
205C22×D61C22×D411yescharacteristic
206C22×D61C22×D412yes—
207C22×D61C2×#4112yes—
208C2×#6923—28no—
209C2×#6923—48no—
210C2×#6923—48no—
211C2×#6923—44no—
212Q8×C61—42no—
213Q8×C61—42no—
214D4×C61—42no—
215D4×C61—42no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—42no—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C4×D414yes—
224C22⋊D61—48no—
225C22⋊D61—44no—
226C22⋊D61—42no—
227C22⋊D61—42no—
228C22⋊D61—48no—
229C2×Dic61—24no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—44no—
233C2×Dic61—44no—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×C22⋊C411yescharacteristic
236C2×Dic61C2×C22⋊C411yescharacteristic
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61C4×D414yes—
239D244—44no—
240D244—42no—
241D244—42no—
242D244—42no—
243D244—42no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—44no—
249C4×D61—44no—
250C4×D61#12114yes—
251C4×D61#12114yes—
252C4×D61#12412yes—
253C4×D61#12412yes—
254C4×D61#12712yes—
255C4×D61#12712yes—
256Dic122—42no—
257Dic122—42no—
258#21212—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—44no—
261C22×C61—42no—
262C22×C61—84no—
263C22×C61C22×C22⋊C411yescharacteristic
264C22×C61C2×C4×D412yes—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—44no—
270C2×D61—42no—
271C2×D61—84no—
272C2×D61C2×#12112yes—
273C2×D61C2×#12411yescharacteristic
274C2×D61C2×#12711yescharacteristic
275#6923—48no—
276#6923—84no—
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61#39012yes—
281C4×C61#39012yes—
282Dic61—22no—
283Dic61—42no—
284Dic61#39012yes—
285Dic61#39012yes—
286Dic61#41014yescomplemented
287C2×#532—611no—
288C2×#410—612no—
289C2×#410—612no—
290#2285—611no—
291C4×#117—611no—
292#1964—618no—
293C2×C61—22no—
294C2×C61—82no—
295C2×C61C2×#39011yescharacteristic
296C2×C61C2×#41012yes—
297D61—22no—
298D61—21no—
299D61—21no—
300D61—82no—
301C22×#41—611no—
302C22×#41—611no—
303#532—1221no—
304#532—1221no—
305#532—1222no—
306#532—1222no—
307#532—1221no—
308#532—1221no—
309C2×C4⋊Q8—611no—
310C2×#127—611no—
311C2×#124—611no—
312C2×#124#692311yescharacteristic, complemented
313C2×#121—614no—
314C2×C4×D4—614no—
315C2×#117—612no—
316C2×#117—612no—
317C22×C42—611no—
318#410—1228no—
319#410—1228no—
320C2×#99—618no—
321C2×#99—618no—
322#384—1221no—
323#382—1221no—
324#380—614no—
325C4×C4⋊C4—1222no—
326C4×C4⋊C4—1222no—
327C4×C22⋊C4—614no—
328#376—1221no—
329C43—1221no—
330C61#2121211yescharacteristic, complemented, Sylow
331C2×#41—1221no—
332C2×#41—1221no—
333C2×#41—1221no—
334C2×#41—1221no—
335C2×#41—1224no—
336C2×#41—1224no—
337C2×#41—1221no—
338C2×#41—1221no—
339C2×#41—1221no—
340C2×#41—1221no—
341C2×#41—1224no—
342C2×#41—1224no—
343C22×Q8—611no—
344C22×Q8C2×#692311yescharacteristic
345C22×D4—612no—
346C22×D4—614no—
347C22×D4—611no—
348C22×D4—612no—
349C22×D4C2×#692311yescharacteristic
350C23×C4—611no—
351C23×C4—612no—
352C23×C4—612no—
353C23×C4—612no—
354C4⋊Q8—1221no—
355C4⋊Q8—1222no—
356C4⋊Q8—1221no—
357#127—1221no—
358#127—1222no—
359#127—1221no—
360#124—21no—
361#124—21no—
362#124—22no—
363#124—1221no—
364#124—1221no—
365#124—1222no—
366#121—1224no—
367#121—1224no—
368#121—1224no—
369#121—1224no—
370C4×D4—1224no—
371C4×D4—1224no—
372C4×D4—1224no—
373C4×D4—1224no—
374#117—618no—
375#117—1224no—
376#117—1224no—
377#117—1224no—
378C2×C4⋊C4—614no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1224no—
382C2×C22⋊C4—614no—
383C2×C22⋊C4—614no—
384C2×C22⋊C4—614no—
385C2×C22⋊C4C2×#692314yes—
386C2×C42—612no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—612no—
391C2×C42—612no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—1222no—
395C2×C42—1222no—
396C2×C42—1224no—
397C2×C42C2×#692311yescharacteristic
398#99—12216no—
399#99—12216no—
400#95—1224no—
401#95—1224no—
402C24—611no—
403C24—612no—
404C24C22×#692312yes—
405#41—2444no—
406#41—2444no—
407#41—2442no—
408#41—2442no—
409#41—2442no—
410#41—2442no—
411#41—2444no—
412#41—2444no—
413#41—2442no—
414#41—2442no—
415#41—2442no—
416#41—2442no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—24no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1224no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—24no—
426C2×D4—1224no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1224no—
430C2×D4—1224no—
431C2×D4—1224no—
432C2×D4—1224no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1224no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1224no—
439C2×D4—2448no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1228no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1228no—
459C22×C4—1224no—
460C22×C4—1224no—
461C22×C4—2444no—
462C22×C4C22×#692312yes—
463C22×C4C22×#692312yes—
464C22×C4C22×#692311yescharacteristic
465C4⋊C4—1224no—
466C4⋊C4—1224no—
467C4⋊C4—1228no—
468C4⋊C4—2444no—
469C4⋊C4—2448no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—618no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C42—612no—
479C42—612no—
480C42—612no—
481C42—612no—
482C42—611no—
483C42—612no—
484C42—611no—
485C42—1228no—
486C42—2444no—
487C42—2448no—
488C42C22⋊#692311yescharacteristic
489C42C22⋊#692311yescharacteristic
490C42C22⋊#692312yes—
491C23—22no—
492C23—24no—
493C23—44no—
494C23—611no—
495C23—611no—
496C23—612no—
497C23—612no—
498C23—1224no—
499C23—1222no—
500C23—2442no—
501C23—2444no—
502C23C23×#692311yescharacteristic
503Q8—42no—
504Q8—42no—
505Q8—2442no—
506Q8—2442no—
507D4—42no—
508D4—42no—
509D4—2444no—
510D4—2442no—
511D4—2442no—
512D4—2448no—
513D4—2448no—
514D4—2444no—
515D4—2442no—
516D4—2442no—
517D4—2442no—
518D4—2442no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—612no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—614no—
526C2×C4—614no—
527C2×C4—614no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1228no—
537C2×C4—1222no—
538C2×C4—1224no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2444no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2444no—
545C2×C4—2444no—
546C2×C4—2448no—
547C2×C4—2448no—
548C2×C4C2×C22⋊#692312yes—
549C2×C4C2×C22⋊#692312yes—
550C2×C4C2×C22⋊#692311yescharacteristic
551C2×C4C2×C22⋊#692311yescharacteristic
552C2×C4D4×#692314yes—
553C22—22no—
554C22—42no—
555C22—44no—
556C22—84no—
557C22—612no—
558C22—611no—
559C22—611no—
560C22—1222no—
561C22—1222no—
562C22—4884no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22—2444no—
567C22C22×C22⋊#692311yescharacteristic
568C22C2×D4×#692312yes—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—612no—
573C4—612no—
574C4—614no—
575C4—1222no—
576C4—4884no—
577C4—2448no—
578C4—2442no—
579C4#1684071412yes—
580C4#1684071412yes—
581C2—22no—
582C2—82no—
583C2—1221no—
584C2—1222no—
585C2—1221no—
586C2—4882no—
587C2C2×#1684071411yescharacteristic
588C2D61⋊#41012yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#124×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840311C2
cosocleC23×D61C24
hypercenterC2×#124#6923
hypocenterC61#21212
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16840743C2semidirect
2. C2×#16840742C2semidirect
3. #21462268C2semidirect
4. C4×#14519856C2semidirect
5. #21461960C2semidirect
29. C2×#14519775C4semidirect
30. C2×#14519773C4semidirect
31. C2×#14519765C4semidirect
33. C2×#14519737C4semidirect
35. C2×#124×C61C4semidirect
39. #14519856D4semidirect
286. Dic61#410semidirect
312. C2×#124#6923semidirect
330. C61#21212semidirect

Maximal subgroups

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

subgroupindex
C2×#168407432
C2×#168407422
#214622682
C4×#145198562
#214619602
C2×#168403112
#2121261