Finite Group Databasebeta

← #29681685

Subgroups — #29681685

Order 15616.

2042 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C8×D61C211yesmaximal, characteristic, complemented
2C2×#16840269C211yesmaximal, characteristic
3C2×#16839758C211yesmaximal, characteristic, complemented
4#21448198C211yesmaximal, characteristic, complemented
5#21446244C214yesmaximal, complemented
6#21446239C214yesmaximal, complemented
7#21445536C211yesmaximal, characteristic, complemented
8#21445535C211yesmaximal, characteristic, complemented
9C2×C61×#416C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12C22×C8×D61—24no—
13C22×C8×D61—24no—
14C22×C8×D61—24no—
15C22×C8×D61C2214yes—
16#16840269—24no—
17#16840269C418yescomplemented
18C2×#14519758—21no—
19C2×#14519753C2212yes—
20C2×#14519748C2212yes—
21C2×#14519745—21no—
22C2×C4×C22⋊D61—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25#16839758—24no—
26C2×#14519701C2212yescomplemented
27C2×#14519698C2212yescomplemented
28#16839623—21no—
29#16839619C2214yes—
30#16839617—21no—
31C4×#11758198—22no—
32#16839525C2214yescomplemented
33#16839524C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C23×C8×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#416—24no—
38C2×#103×C61C2212yescomplemented
39C2×#102×C61C2212yescomplemented
40C24×D61D411yescharacteristic
41C22×C22⋊D61—22no—
42C23×Dic61D411yescharacteristic
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61C2×C414yes—
48C22×C4×D61D412yes—
49C22×C4×D61D414yes—
50C22×C4×D61C2312yes—
51C22×#92418—22no—
52C22×#92418—22no—
53C22×#92418D414yescomplemented
54C22×#92418C2312yes—
55C2×C8×D61—232no—
56C2×C8×D61—28no—
57C2×C8×D61—28no—
58C2×C8×D61—28no—
59#14519758—28no—
60#14519753—24no—
61#14519753C2×C418yes—
62#14519748—24no—
63#14519748C2×C418yes—
64#14519745—28no—
65C4×C22⋊D61—216no—
66#14519736—22no—
67#14519736—22no—
68#14519736—22no—
69#14519736C2×C414yes—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—22no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—21no—
76C2×#11758198—22no—
77C2×#11758198—24no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80#14519701—24no—
81#14519698—24no—
82#14519684—24no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61C2×C414yes—
89C22×Q8×C61C2311yescharacteristic
90C22×D4×C61—21no—
91C22×D4×C61C2311yescharacteristic
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94C22×C8×C61—22no—
95C22×C8×C61—22no—
96C22×C8×C61D414yescomplemented
97C22×C8×C61C2312yes—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C42×C61—21no—
103#103×C61—24no—
104#102×C61—24no—
105C24×C61—21no—
106C24×C61C2×D411yescharacteristic
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—28no—
110C23×D61C22⋊C414yes—
111C23×D61C2×D412yes—
112#41×C61—28no—
113#41×C61—28no—
114#41×C61—44no—
115#41×C61—44no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61C22×C412yes—
122C2×D4×C61—24no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61C22×C412yes—
131C22×C4×C61—21no—
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61#4112yes—
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—28no—
145C2×C22⋊D61—48no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61C22⋊C414yes—
151C22×Dic61C2×D412yes—
152C2×C4×D61—28no—
153C2×C4×D61—28no—
154C2×C4×D61—28no—
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—28no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61C22⋊C418yes—
163C2×C4×D61C22×C414yes—
164C2×C4×D61C2×D414yes—
165C2×C8×C61—28no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61C2×D414yes—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C22⋊C4×C61—42no—
172C42×C61—42no—
173C22⋊Dic61—24no—
174C22⋊Dic61—28no—
175C22⋊Dic61—42no—
176#11758198—28no—
177#11758198—48no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61C22×C414yes—
182#11758196—28no—
183C4×Dic61—24no—
184C4×Dic61—42no—
185C2×#92418—28no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418C2×D414yes—
189C8×D61—232no—
190C23×C61—24no—
191C23×C61—22no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—44no—
195C23×C61C2×C22⋊C412yes—
196C23×C61#12112yes—
197C23×C61#12112yes—
198C23×C61C22×D411yescharacteristic
199C22×D61—216no—
200C22×D61—216no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—24no—
204C22×D61—28no—
205C22×D61—216no—
206C22×D61C2×C22⋊C414yes—
207C22×D61#13518yes—
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61—41no—
211Q8×C61—41no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—48no—
215D4×C61—41no—
216D4×C61—41no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—48no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C4×D414yes—
225C2×C4×C61C4×D414yes—
226C2×C4×C61#12014yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12314yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C22⋊D61—416no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—28no—
240C2×Dic61—24no—
241C2×Dic61—44no—
242C2×Dic61—42no—
243C2×Dic61—44no—
244C2×Dic61C2×C22⋊C414yes—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—216no—
250C8×C61—24no—
251#92418—24no—
252#20967—611nomaximal, Sylow
253C22×C61—24no—
254C22×C61—24no—
255C22×C61—24no—
256C22×C61—44no—
257C22×C61—42no—
258C22×C61—82no—
259C22×C61—82no—
260C22×C61#38614yes—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61C2×#12111yescharacteristic
263C22×C61C2×#12111yescharacteristic
264C22×C61C2×#13514yes—
265C2×D61—232no—
266C2×D61—24no—
267C2×D61—28no—
268C2×D61—28no—
269C2×D61—24no—
270C2×D61#41618yescomplemented
271C4×C61—42no—
272C4×C61—82no—
273C4×C61#38614yes—
274C4×C61C2×C4×D411yescharacteristic
275C4×C61C2×C4×D411yescharacteristic
276C4×C61C2×#12011yescharacteristic
277C4×C61C2×#12311yescharacteristic
278Dic61—24no—
279Dic61—44no—
280Dic61—82no—
281C24×C8—611no—
282C2×#525—611no—
283C2×#416—611no—
284C2×#416D6111yescharacteristic, complemented
285#2280—611no—
286#1893—614no—
287#1892—614no—
288#1735—611no—
289#1735—611no—
290C2×C61—24no—
291C2×C61—82no—
292C2×C61C2×#38611yescharacteristic
293C2×C61C2×#41612yes—
294C2×C61#305012yes—
295C2×C61#305012yes—
296D61—28no—
297C22×#41C2×D6111yescharacteristic
298C24×C4—611no—
299C23×C8—611no—
300C23×C8—614no—
301C23×C8—1224no—
302C23×C8—1224no—
303C23×C8—1224no—
304C23×C8C2×D6111yescharacteristic
305#525—618no—
306#525—1224no—
307C2×#123—1221no—
308C2×C22⋊Q8—612no—
309C2×#121—612no—
310C2×#120—1221no—
311C2×C4×D4—1222no—
312C2×#117—611no—
313C2×#117—611no—
314C2×#117C2×D6111yescharacteristic
315#416—24no—
316#416—1224no—
317C2×#103—612no—
318C2×#103C2×D6112yescomplemented
319C2×#102—612no—
320C2×#102C2×D6112yescomplemented
321#381—1221no—
322#380—614no—
323#379—1221no—
324C4×C22⋊C4—1222no—
325#340—614no—
326#340—614no—
327C61#2096711yescharacteristic, complemented, Sylow
328C25—611no—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41C4×D6114yes—
335C22×Q8C22×D6111yescharacteristic
336C22×D4—21no—
337C22×D4—1222no—
338C22×D4C22×D6111yescharacteristic
339C23×C4—21no—
340C23×C4—611no—
341C23×C4—614no—
342C23×C4—614no—
343C23×C4—612no—
344C23×C4—612no—
345C23×C4—1222no—
346C23×C4—1222no—
347C23×C4—1222no—
348C23×C4—1222no—
349C23×C4C22×D6111yescharacteristic
350C22×C8—22no—
351C22×C8—22no—
352C22×C8—614no—
353C22×C8—612no—
354C22×C8—1228no—
355C22×C8—1228no—
356C22×C8—12232no—
357C22×C8—1222no—
358C22×C8—1222no—
359C22×C8—1228no—
360C22×C8C22⋊D6114yescomplemented
361C22×C8C22×D6112yes—
362#123—1228no—
363C22⋊Q8—618no—
364C22⋊Q8—1224no—
365#121—618no—
366#121—1224no—
367#120—1228no—
368C4×D4—12216no—
369#117—22no—
370#117—22no—
371#117—614no—
372#117—1222no—
373#117—1222no—
374#117—1222no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1224no—
384C2×C22⋊C4—1221no—
385C2×C42—21no—
386C2×C42—1221no—
387C2×C42—1221no—
388#103—24no—
389#103—1224no—
390#102—24no—
391#102—1224no—
392#95—1224no—
393C24—21no—
394C24—612no—
395C24—614no—
396C24—1228no—
397C24—1222no—
398C24—1222no—
399C24D4×D6111yescharacteristic
400#41—28no—
401#41—28no—
402#41—44no—
403#41—44no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8C2×C4×D6112yes—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—1228no—
419C2×D4—2448no—
420C2×D4C2×C4×D6112yes—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—612no—
427C22×C4—614no—
428C22×C4—614no—
429C22×C4—614no—
430C22×C4—618no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1228no—
438C22×C4—1228no—
439C22×C4—1228no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1228no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4C2×C4×D6112yes—
446C22×C4#1175822212yes—
447C22×C4D4×D6112yes—
448C22×C4D4×D6112yes—
449C22×C4D4×D6111yescharacteristic
450C22×C4#1175822412yes—
451C22×C4C2×C22⋊D6112yes—
452C22×C4C23×D6111yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—28no—
456C2×C8—614no—
457C2×C8—12232no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8—1228no—
461C2×C8C2×C22⋊D6114yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—614no—
465C4⋊C4—1228no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C22⋊C4—42no—
470C22⋊C4—1228no—
471C22⋊C4—1228no—
472C22⋊C4—1224no—
473C22⋊C4—2448no—
474C22⋊C4—2442no—
475C42—42no—
476C42—1224no—
477C42—2442no—
478C23—24no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—44no—
483C23—614no—
484C23—618no—
485C23—12216no—
486C23—12216no—
487C23—12216no—
488C23—1222no—
489C23—1222no—
490C23—1224no—
491C23—1228no—
492C23C22⋊C4×D6112yes—
493C23#1451974812yes—
494C23#1451975012yes—
495C23C2×D4×D6111yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—41no—
499Q8—41no—
500D4—22no—
501D4—22no—
502D4—48no—
503D4—41no—
504D4—41no—
505D4—24416no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—614no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1228no—
516C2×C4—1224no—
517C2×C4—1224no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—12216no—
521C2×C4—1224no—
522C2×C4—2442no—
523C2×C4—2444no—
524C2×C4—2444no—
525C2×C4C22⋊C4×D6112yes—
526C2×C4#1451973714yes—
527C2×C4C4×C22⋊D6114yes—
528C2×C4#1451974514yes—
529C2×C4#1451974812yes—
530C2×C4#1451975012yes—
531C2×C4#1451975914yes—
532C2×C4C22×C4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C2×C4C2×#1175822411yescharacteristic
537C2×C4C2×#1175822211yescharacteristic
538C8—24no—
539C8—1224no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—42no—
544C22—44no—
545C22—82no—
546C22—82no—
547C22—618no—
548C22—12232no—
549C22—1228no—
550C22—1228no—
551C22—1224no—
552C22—1224no—
553C22#1683963814yes—
554C22C2×C22⋊C4×D6111yescharacteristic
555C22C2×#1451974811yescharacteristic
556C22C2×#1451975011yescharacteristic
557C22#135×D6114yes—
558C4—42no—
559C4—82no—
560C4—1224no—
561C4—4882no—
562C4—2444no—
563C4#1683963814yes—
564C4C2×#1451973711yescharacteristic
565C4C2×C4×C22⋊D6111yescharacteristic
566C4C2×#1451974511yescharacteristic
567C4C2×#1451975911yescharacteristic
568C2—24no—
569C2—82no—
570C2—1228no—
571C2C2×#1683963811yescharacteristic
572C2#416×D6112yes—
573C2#2145539612yes—
574C2#2145539912yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC2×#16840269C2
cosocleC22×C4×C61C24
hypercenterC2×#416D61
hypocenterC61#20967
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C23×C8×D61C2semidirect
3. C2×#16839758C2semidirect
4. #21448198C2semidirect
5. #21446244C2semidirect
6. #21446239C2semidirect
7. #21445536C2semidirect
8. #21445535C2semidirect
9. C2×C61×#416C2semidirect
17. #16840269C4semidirect
26. C2×#14519701C22semidirect
27. C2×#14519698C22semidirect
32. #16839525C22semidirect
33. #16839524C22semidirect
38. C2×#103×C61C22semidirect
39. C2×#102×C61C22semidirect
53. C22×#92418D4semidirect
96. C22×C8×C61D4semidirect
270. C2×D61#416semidirect
284. C2×#416D61semidirect
318. C2×#103C2×D61semidirect
320. C2×#102C2×D61semidirect
327. C61#20967semidirect
360. C22×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C23×C8×D612
C2×#168402692
C2×#168397582
#214481982
#214462442
#214462392
#214455362
#214455352
C2×C61×#4162
#2096761