Finite Group Databasebeta

← #30609646

Subgroups — #30609646

Order 15616.

2842 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
1C23×C4×#6923C211yesmaximal, characteristic, complemented
2C2×#16840710C212yesmaximal
3C2×#16840705C211yesmaximal, characteristic, complemented
4#21461000C214yesmaximal, complemented
5#21460998C214yesmaximal, complemented
6#21460992C212yesmaximal, complemented
7C2×#123×D61C211yesmaximal, characteristic
8C24×#6923C2212yes—
9C22×C22⋊#6923C2212yes—
10C22×C22⋊#6923C2214yes—
11C22×C4×#6923—28no—
12C22×C4×#6923—24no—
13C22×C4×#6923C2214yes—
14#16840710—28no—
15#16840705—24no—
16C2×(C2×#6923)⋊C4C2218yes—
17C2×(C2×#6923)⋊C4C2214yes—
18C2×(C2×#6923)⋊C4C2214yes—
19C22×D4×D61C2211yescharacteristic
20C23×C4×D61C2211yescharacteristic
21C2×#14519762C412yescomplemented
22C2×#14519761C411yescharacteristic, complemented
23C2×#14519760C411yescharacteristic, complemented
24C2×#14519759C412yescomplemented
25C2×#14519758C411yescharacteristic, complemented
26#123×D61—24no—
27#123×D61—24no—
28C2×C4⋊C4×D61C2212yes—
29C2×C22⋊C4×D61C2211yescharacteristic
30C2×C22⋊C4×D61C2212yes—
31C2×#123×C61C411yescharacteristic, complemented
32C23×#6923—24no—
33C23×#6923—24no—
34C23×#6923—216no—
35C23×#6923D418yes—
36C23×#6923C2314yes—
37C23×#6923C2314yes—
38C2×C22⋊#6923—24no—
39C2×C22⋊#6923—24no—
40C2×C22⋊#6923—24no—
41C2×C22⋊#6923—28no—
42C2×C22⋊#6923—216no—
43C2×C4×#6923—232no—
44C2×C4×#6923—216no—
45C2×C4×#6923—28no—
46(C2×#6923)⋊C4—28no—
47(C2×#6923)⋊C4—216no—
48(C2×#6923)⋊C4—28no—
49C24×D61C2311yescharacteristic
50C24×D61C2311yescharacteristic
51C22×D244C2×C411yescharacteristic
52C22×C22⋊D61C2×C411yescharacteristic
53C22×C22⋊D61C2×C411yescharacteristic
54C2×D4×D61—24no—
55C2×D4×D61—22no—
56C2×D4×D61—22no—
57C2×D4×D61—24no—
58C23×Dic61C2×C411yescharacteristic
59C22×C4×D61—24no—
60C22×C4×D61—24no—
61C22×C4×D61C2×C412yes—
62C22×C4×D61D414yes—
63C22×C4×D61D414yes—
64C22×C4×D61C2312yes—
65C22×C4×D61C2311yescharacteristic
66C22×C4×D61C2312yes—
67#14519762—28no—
68#14519761—24no—
69#14519760—24no—
70#14519759—28no—
71#14519758—24no—
72C2×C4⋊Dic61C2×C412yescomplemented
73C2×#11758196C2×C412yescomplemented
74C2×#11758196C2×C412yescomplemented
75C4⋊C4×D61—24no—
76C4⋊C4×D61—24no—
77C2×C22⋊Dic61C2×C411yescharacteristic
78C2×C22⋊Dic61C2×C412yescomplemented
79C2×#11758198C2×C412yes—
80C2×#11758198C2×C412yescomplemented
81C2×#11758198C2×C412yescomplemented
82C22⋊C4×D61—22no—
83C22⋊C4×D61—22no—
84C22⋊C4×D61—24no—
85C22⋊C4×D61—24no—
86C22×D4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C411yescharacteristic
88#123×C61—24no—
89C2×C4⋊C4×C61C2×C412yescomplemented
90C2×C22⋊C4×C61C2×C411yescharacteristic
91C2×C22⋊C4×C61C2×C412yescomplemented
92C24×C61C22×C411yescharacteristic
93C24×C61C22×C411yescharacteristic
94C23×D61—28no—
95C23×D61—22no—
96C23×D61—21no—
97C23×D61—21no—
98C23×D61—22no—
99C23×D61—28no—
100C23×D61C22×C411yescharacteristic
101C23×D61C22×C411yescharacteristic
102C23×D61C2×D412yes—
103C23×D61C2×D412yes—
104C23×D61C2×D414yes—
105C23×D61#4114yes—
106C23×D61C2411yescharacteristic
107C22×#6923—232no—
108C22×#6923—232no—
109C22×#6923—216no—
110C22×#6923—24no—
111C22×#6923—24no—
112C22×#6923—28no—
113C22×#6923—232no—
114C22×#6923—24no—
115C22×#6923—24no—
116C22×#6923—28no—
117C22×#6923—24no—
118C22×#6923—28no—
119C22×#6923C2×D418yes—
120C22×#6923#41116yes—
121C2×D4×C61—22no—
122C2×D4×C61—24no—
123C22×C4×C61—24no—
124C22×C4×C61C22⋊C414yescomplemented
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
128C2×C22⋊D61—22no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—24no—
132C22×Dic61—24no—
133C22×Dic61C22⋊C414yescomplemented
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C22×C412yes—
136C22×Dic61C22×C412yes—
137D4×D61—44no—
138D4×D61—48no—
139D4×D61—44no—
140C2×D244—22no—
141C2×D244—24no—
142C2×C4×D61—28no—
143C2×C4×D61—28no—
144C2×C4×D61—216no—
145C2×C4×D61—28no—
146C2×C4×D61—24no—
147C2×C4×D61—24no—
148C2×C4×D61—22no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—28no—
155C2×C4×D61C22⋊C418yescomplemented
156C2×C4×D61C2×D414yes—
157C2×C4×D61C2×D414yes—
158C22⋊#6923—48no—
159C22⋊#6923—416no—
160C4×#6923—232no—
161C4⋊C4×C61—24no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—24no—
164C22⋊Dic61—22no—
165C22⋊Dic61—24no—
166#11758198—24no—
167#11758198—24no—
168#11758198—24no—
169C4⋊Dic61—24no—
170#11758196—24no—
171#11758196—24no—
172C23×C61—24no—
173C23×C61—21no—
174C23×C61—22no—
175C23×C61—24no—
176C23×C61C2×C22⋊C412yes—
177C23×C61C4×D414yes—
178C23×C61C23×C411yescharacteristic
179C22×D61—24no—
180C22×D61—28no—
181C22×D61—28no—
182C22×D61—28no—
183C22×D61—216no—
184C22×D61—28no—
185C22×D61—21no—
186C22×D61—22no—
187C22×D61—24no—
188C22×D61—41no—
189C22×D61—42no—
190C22×D61—44no—
191C22×D61—41no—
192C22×D61—44no—
193C22×D61—44no—
194C22×D61—48no—
195C22×D61—44no—
196C22×D61C2×C22⋊C412yes—
197C22×D61C4×D414yes—
198C22×D61#12014yes—
199C22×D61#12118yes—
200C22×D61#12314yes—
201C22×D61C22×D412yes—
202C22×D61C22×D411yescharacteristic
203C22×D61C22×D411yescharacteristic
204C22×D61C2×#4112yes—
205C22×D61C2×#4118yes—
206C2×#6923—264no—
207C2×#6923—28no—
208C2×#6923—28no—
209C2×#6923—216no—
210C2×#6923—216no—
211C2×#6923—44no—
212C2×#6923—44no—
213C2×#6923—44no—
214C2×#6923—48no—
215C2×#6923—48no—
216C2×#6923#123116yescomplemented
217D4×C61—44no—
218C2×C4×C61—24no—
219C2×C4×C61—28no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61C2×C22⋊C414yes—
225C22⋊D61—44no—
226C22⋊D61—44no—
227C2×Dic61—24no—
228C2×Dic61—28no—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61C2×C22⋊C414yes—
234D244—44no—
235C4×D61—24no—
236C4×D61—28no—
237C4×D61—24no—
238C4×D61—216no—
239C4×D61—41no—
240C4×D61—42no—
241C4×D61—41no—
242C4×D61—44no—
243C4×D61—44no—
244#14706—611nomaximal, Sylow
245C22×C61—24no—
246C22×C61—28no—
247C22×C61—41no—
248C22×C61—42no—
249C22×C61—44no—
250C22×C61#38614yes—
251C22×C61C22×C22⋊C411yescharacteristic
252C22×C61C2×C4×D412yes—
253C22×C61C4×#4114yes—
254C2×D61—24no—
255C2×D61—28no—
256C2×D61—28no—
257C2×D61—28no—
258C2×D61—216no—
259C2×D61—41no—
260C2×D61—42no—
261C2×D61—44no—
262C2×D61—82no—
263C2×D61—82no—
264C2×D61—84no—
265C2×D61#38614yes—
266C2×D61C2×#12011yescharacteristic
267C2×D61C2×#12112yes—
268C2×D61C2×#12314yes—
269C2×D61C2×#12311yescharacteristic
270C2×D61#52514yes—
271C2×D61#52514yes—
272C2×D61#52918yes—
273C2×D61#53214yes—
274#6923—216no—
275#6923—84no—
276C4×C61—24no—
277C4×C61—41no—
278C4×C61—42no—
279Dic61—24no—
280Dic61—41no—
281Dic61—42no—
282C22×#123—611no—
283C23×C42—611no—
284C2×#386—612no—
285C2×#381—611no—
286#1437—614no—
287#1436—614no—
288C22⋊#95—612no—
289C2×C61—24no—
290C2×C61—81no—
291C2×C61C2×#38611yescharacteristic
292C2×C61#229712yes—
293C2×C61C4×#12314yes—
294D61—24no—
295D61—81no—
296D61#284912yes—
297D61#285312yes—
298D61#285414yes—
299C23×D4—611no—
300C24×C4—611no—
301C24×C4—612no—
302C2×#123—611no—
303C2×#123—611no—
304C2×#123—612no—
305C2×#123—611no—
306C2×#123—612no—
307C2×#123—1224no—
308C2×#123—1224no—
309C2×#123#692311yescharacteristic, complemented
310C22×C4⋊C4—612no—
311C22×C22⋊C4—611no—
312C22×C22⋊C4—612no—
313C22×C22⋊C4—612no—
314C22×C22⋊C4—614no—
315C22×C42—614no—
316C22×C42—1224no—
317C22×C42—1228no—
318#386—1228no—
319#381—1224no—
320C2×#95—618no—
321C2×#95—614no—
322C2×#95—614no—
323C61#1470611yescharacteristic, complemented, Sylow
324C25—611no—
325C25—611no—
326C22×D4—611no—
327C22×D4—611no—
328C22×D4—611no—
329C22×D4—1222no—
330C22×D4—1222no—
331C22×D4—1224no—
332C22×D4—1224no—
333C22×D4C2×#692311yescharacteristic
334C23×C4—614no—
335C23×C4—611no—
336C23×C4—614no—
337C23×C4—614no—
338C23×C4—612no—
339C23×C4—612no—
340C23×C4—618no—
341C23×C4—612no—
342C23×C4—611no—
343C23×C4—614no—
344C23×C4—1224no—
345C23×C4—1224no—
346C23×C4—12216no—
347C23×C4—1224no—
348C23×C4—1224no—
349C23×C4C2×#692311yescharacteristic
350#123—24no—
351#123—1228no—
352#123—1224no—
353#123—1228no—
354#123—1224no—
355#123—1224no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—1224no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4C2×#692312yescomplemented
362C2×C22⋊C4—611no—
363C2×C22⋊C4—612no—
364C2×C22⋊C4—612no—
365C2×C22⋊C4—612no—
366C2×C22⋊C4—612no—
367C2×C22⋊C4—1224no—
368C2×C22⋊C4—1224no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—12216no—
372C2×C22⋊C4—1228no—
373C2×C22⋊C4—1224no—
374C2×C22⋊C4—1224no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4C2×#692311yescharacteristic
377C2×C22⋊C4C2×#692312yescomplemented
378C2×C42—12216no—
379C2×C42—1228no—
380C2×C42—12232no—
381#95—12216no—
382#95—1228no—
383#95—1228no—
384C24—611no—
385C24—612no—
386C24—612no—
387C24—614no—
388C24—614no—
389C24—611no—
390C24—611no—
391C24—1228no—
392C24—1221no—
393C24—1221no—
394C24—1222no—
395C24—1228no—
396C24—1222no—
397C24C22×#692311yescharacteristic
398C24C22×#692311yescharacteristic
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—1222no—
402C2×D4—1224no—
403C2×D4—1222no—
404C2×D4—1224no—
405C2×D4—1222no—
406C2×D4—1224no—
407C2×D4—2444no—
408C2×D4—2448no—
409C2×D4—2444no—
410C22×C4—24no—
411C22×C4—618no—
412C22×C4—6116no—
413C22×C4—614no—
414C22×C4—612no—
415C22×C4—614no—
416C22×C4—614no—
417C22×C4—618no—
418C22×C4—612no—
419C22×C4—611no—
420C22×C4—1228no—
421C22×C4—1224no—
422C22×C4—12216no—
423C22×C4—1228no—
424C22×C4—1224no—
425C22×C4—1228no—
426C22×C4—1222no—
427C22×C4—1228no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—12232no—
434C22×C4—1224no—
435C22×C4—1224no—
436C22×C4—12216no—
437C22×C4—1228no—
438C22×C4—12232no—
439C22×C4—12232no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1228no—
444C22×C4—1224no—
445C22×C4—1228no—
446C22×C4C22⋊#692314yescomplemented
447C22×C4C22×#692312yes—
448C22×C4C22×#692311yescharacteristic
449C22×C4C22×#692312yes—
450C4⋊C4—24no—
451C4⋊C4—1224no—
452C4⋊C4—1224no—
453C4⋊C4—1224no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—1224no—
457C22⋊C4—1224no—
458C22⋊C4—1224no—
459C22⋊C4—1222no—
460C22⋊C4—1224no—
461C22⋊C4—24416no—
462C22⋊C4—2448no—
463C42—12232no—
464C23—24no—
465C23—21no—
466C23—22no—
467C23—24no—
468C23—611no—
469C23—611no—
470C23—612no—
471C23—612no—
472C23—618no—
473C23—614no—
474C23—618no—
475C23—614no—
476C23—612no—
477C23—614no—
478C23—1228no—
479C23—1228no—
480C23—1228no—
481C23—1228no—
482C23—12216no—
483C23—1224no—
484C23—1221no—
485C23—1222no—
486C23—1224no—
487C23—2444no—
488C23—2441no—
489C23—2442no—
490C23—2441no—
491C23—2444no—
492C23—2444no—
493C23—2444no—
494C23—2448no—
495C23C2×C22⋊#692312yes—
496C23D4×#692314yes—
497C23C23×#692311yescharacteristic
498D4—44no—
499D4—2444no—
500D4—2444no—
501D4—2444no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—6116no—
509C2×C4—614no—
510C2×C4—1224no—
511C2×C4—1228no—
512C2×C4—1224no—
513C2×C4—12216no—
514C2×C4—1228no—
515C2×C4—12216no—
516C2×C4—1228no—
517C2×C4—12216no—
518C2×C4—12264no—
519C2×C4—1228no—
520C2×C4—1224no—
521C2×C4—1222no—
522C2×C4—1221no—
523C2×C4—1222no—
524C2×C4—1224no—
525C2×C4—2444no—
526C2×C4—2444no—
527C2×C4—2441no—
528C2×C4—2442no—
529C2×C4—2441no—
530C2×C4—2444no—
531C2×C4—2444no—
532C2×C4—2448no—
533C2×C4—2444no—
534C2×C4—2448no—
535C2×C4C2×C22⋊#692314yes—
536C22—24no—
537C22—28no—
538C22—42no—
539C22—44no—
540C22—41no—
541C22—614no—
542C22—614no—
543C22—614no—
544C22—614no—
545C22—618no—
546C22—611no—
547C22—612no—
548C22—611no—
549C22—614no—
550C22—1228no—
551C22—12216no—
552C22—1228no—
553C22—1224no—
554C22—1228no—
555C22—4882no—
556C22—4884no—
557C22—4882no—
558C22—2442no—
559C22—2444no—
560C22—2441no—
561C22#1684071014yes—
562C22C22×C22⋊#692311yescharacteristic
563C22C2×D4×#692312yes—
564C22#41×#692314yes—
565C4—24no—
566C4—42no—
567C4—41no—
568C4—12216no—
569C4—1224no—
570C4—4884no—
571C4—2442no—
572C4—2441no—
573C2—24no—
574C2—81no—
575C2—612no—
576C2—614no—
577C2—612no—
578C2—1224no—
579C2—4881no—
580C2C2×#1684071011yescharacteristic
581C2#2146228012yes—
582C2#123×#692314yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×#123×C61C4
prime radicalC22×D4×D61C22
cosocleC23×D61C24
hypercenterC2×#123#6923
hypocenterC61#14706
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C23×C4×#6923C2semidirect
3. C2×#16840705C2semidirect
4. #21461000C2semidirect
5. #21460998C2semidirect
6. #21460992C2semidirect
21. C2×#14519762C4semidirect
22. C2×#14519761C4semidirect
23. C2×#14519760C4semidirect
24. C2×#14519759C4semidirect
25. C2×#14519758C4semidirect
31. C2×#123×C61C4semidirect
72. C2×C4⋊Dic61C2×C4semidirect
73. C2×#11758196C2×C4semidirect
74. C2×#11758196C2×C4semidirect
78. C2×C22⋊Dic61C2×C4semidirect
80. C2×#11758198C2×C4semidirect
81. C2×#11758198C2×C4semidirect
89. C2×C4⋊C4×C61C2×C4semidirect
91. C2×C22⋊C4×C61C2×C4semidirect
124. C22×C4×C61C22⋊C4semidirect
133. C22×Dic61C22⋊C4semidirect
155. C2×C4×D61C22⋊C4semidirect
216. C2×#6923#123semidirect
309. C2×#123#6923semidirect
323. C61#14706semidirect
361. C2×C4⋊C4C2×#6923semidirect
377. C2×C22⋊C4C2×#6923semidirect
446. C22×C4C22⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C23×C4×#69232
C2×#168407102
C2×#168407052
#214610002
#214609982
#214609922
C2×#123×D612
#1470661