Finite Group Databasebeta

← #29685191

Subgroups — #29685191

Order 15616.

1518 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×C4×C8×D61C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21446778C212yesmaximal, complemented
5#21446623C212yesmaximal, complemented
6#21446246C212yesmaximal, complemented
7#21446244C212yesmaximal, complemented
8C4×#14519697C211yesmaximal, characteristic, complemented
9C4×#14519699C211yesmaximal, characteristic, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C4×#102×C61C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×C8×D61C2212yes—
14#16840309—24no—
15#16840309C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519752C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519748C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519698C2212yes—
28C2×#14519697C2212yes—
29C2×C4×#92418C2211yescharacteristic
30C2×C8×Dic61C2212yes—
31C4×C8×D61—24no—
32#16839631C2212yes—
33#16839626—21no—
34#16839621—21no—
35#16839619C2212yes—
36C4×#11758196—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—44no—
48C2×#11758222—44no—
49C22×Dic122—21no—
50C22×D244—21no—
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60C2×C8×D61—24no—
61C2×C8×D61—24no—
62C2×C8×D61—24no—
63#14519776—22no—
64#14519776C2×C414yes—
65#14519773—22no—
66#14519773C2×C414yes—
67#14519763—28no—
68#14519752—22no—
69#14519752C2×C414yes—
70#14519750—28no—
71#14519748—22no—
72#14519748C2×C414yes—
73C4×D244—22no—
74C4×D244C2×C414yes—
75C4×C22⋊D61—28no—
76#14519734—44no—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—22no—
81C2×#11758196—22no—
82C2×C22⋊Dic61—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—22no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C42×D61C2×C414yes—
91C42×D61D414yes—
92#14519699—24no—
93#14519698—24no—
94#14519697—24no—
95C4×#92418—22no—
96C8×Dic61—24no—
97#14519684—21no—
98#14519684—22no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C22×C8×C61C2312yes—
102C4×D4×C61—22no—
103C4×D4×C61C2×C414yes—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109#102×C61—24no—
110C4×C8×C61—22no—
111#95×C61—21no—
112C24×C61—21no—
113C23×D61—21no—
114C23×D61C2×D411yescharacteristic
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—44no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140#11758222—432no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—24no—
144C2×D244—21no—
145C2×D244—42no—
146C2×D244—42no—
147C2×D244C22×C412yes—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61C22⋊C414yes—
155C2×C4×D61C22⋊C414yes—
156C2×C4×D61D814yes—
157C2×C4×D61SD1614yes—
158C2×C4×D61C22×C412yes—
159C2×C4×D61C2×D412yes—
160C2×Dic122—24no—
161C2×Dic122—42no—
162C2×Dic122—42no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61#4114yescomplemented
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C22×C412yes—
170C22⋊C4×C61—24no—
171C42×C61—42no—
172C42×C61C22×C412yes—
173C42×C61C2×D412yes—
174C22⋊Dic61—24no—
175#11758198—24no—
176#11758198—24no—
177#11758198—44no—
178C4⋊Dic61—21no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61C22×C412yes—
182#11758196—24no—
183#11758196—44no—
184#11758196—44no—
185C4×Dic61—42no—
186C4×Dic61C22⋊C414yes—
187C4×Dic61C22⋊C414yes—
188C4×Dic61C22×C412yes—
189C4×Dic61C2×D412yes—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418#4114yescomplemented
193C8×D61—216no—
194C23×C61—22no—
195C23×C61—42no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—42no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—42no—
207C22×D61C2×C22⋊C412yes—
208C22×D61C2×D812yes—
209C22×D61C2×SD1612yes—
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—42no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—48no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C2×D812yes—
224C2×C4×C61C2×SD1612yes—
225C2×C4×C61C23×C411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4112yes—
228C2×C4×C61C2×#4112yes—
229C22⋊D61—48no—
230C22⋊D61—48no—
231C22⋊D61—48no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×D812yes—
243C2×Dic61C2×SD1612yes—
244C2×Dic61#13514yes—
245D244—22no—
246D244—22no—
247D244—48no—
248D244—42no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—24no—
252C4×D61—28no—
253C4×D61—48no—
254C4×D61#102116yescomplemented
255Dic122—48no—
256C8×C61—24no—
257#92418—24no—
258#21401—611nomaximal, Sylow
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61C22×C22⋊C411yescharacteristic
265C22×C61#53212yes—
266C22×C61C22×D811yescharacteristic
267C22×C61C22×SD1611yescharacteristic
268C22×C61C2×#13512yes—
269C2×D61—22no—
270C2×D61—24no—
271C2×D61—24no—
272C2×D61—22no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—82no—
277C2×D61—82no—
278C2×D61C2×#10214yes—
279C4×C61—22no—
280C4×C61—42no—
281C4×C61—82no—
282C4×C61C2×#10214yes—
283C4×C61C4×#4111yescharacteristic
284C4×C61C4×#4111yescharacteristic
285C4×C61#52512yes—
286Dic61—22no—
287Dic61—42no—
288Dic61—82no—
289Dic61C2×#10214yes—
290Dic61#41614yes—
291C2×#532—611no—
292C22×C4×C8—611no—
293#2285—611no—
294#1990—612no—
295#1957—612no—
296#1893—612no—
297#1893—612no—
298C4×#102—611no—
299C4×#102—611no—
300C4×#102—611no—
301C4×#102D6111yescharacteristic, complemented
302C2×C61—82no—
303C2×C61#229711yescharacteristic
304C2×C61C22×#10211yescharacteristic
305C2×C61C2×#41611yescharacteristic
306C2×C61#316712yes—
307C2×C61#316912yes—
308D61—24no—
309D61—82no—
310C22×#41—1221no—
311C23×C8—612no—
312#532—618no—
313#532—1224no—
314C2×C4⋊Q8—611no—
315C2×#127—611no—
316C2×#124—1221no—
317C2×#121—611no—
318C2×#121—611no—
319C2×#121—1221no—
320C2×C4×D4—611no—
321C2×C4×D4—1221no—
322C2×C4×D4C2×D6111yescharacteristic
323C2×#117—1221no—
324C22×C42—611no—
325C2×#102—612no—
326C2×#102—612no—
327C2×#102—612no—
328C2×#102C2×D6112yes—
329C2×C4×C8—612no—
330C2×C4×C8—611no—
331C2×C4×C8—1224no—
332C2×C4×C8C2×D6111yescharacteristic
333#384—612no—
334#382—1221no—
335#380—612no—
336#380—1221no—
337C4×C4⋊C4—611no—
338C4×C4⋊C4—1221no—
339C4×C4⋊C4C2×D6111yescharacteristic
340#340—612no—
341#340—612no—
342#340—612no—
343#340C2×D6112yes—
344C61#2140111yescharacteristic, complemented, Sylow
345C2×#41—1228no—
346C2×#41—2444no—
347C2×#41—2444no—
348C22×Q8—1221no—
349C22×D4—611no—
350C22×D4—1221no—
351C22×D4—1221no—
352C22×D4—1221no—
353C22×D4—1221no—
354C22×D4C22×D6111yescharacteristic
355C23×C4—21no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—1221no—
360C22×C8—612no—
361C22×C8—1224no—
362C22×C8—1224no—
363C22×C8—1224no—
364C22×C8C22×D6112yes—
365C4⋊Q8—614no—
366C4⋊Q8—1222no—
367#127—614no—
368#127—1222no—
369#124—1228no—
370#121—614no—
371#121—614no—
372#121—1228no—
373#121—1222no—
374#121—1222no—
375C4×D4—22no—
376C4×D4—614no—
377C4×D4—1228no—
378C4×D4—1222no—
379C4×D4C4×D6114yes—
380#117—2444no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4C22×D6112yes—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1222no—
393C2×C42—21no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—614no—
398C2×C42—614no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401#102—24no—
402#102—1224no—
403#102—1224no—
404#102—1224no—
405C4×C8—22no—
406C4×C8—1224no—
407C4×C8—1222no—
408#95—21no—
409#95—1222no—
410#95—1221no—
411C24—21no—
412C24—611no—
413C24—1221no—
414#41—24432no—
415C2×Q8—1224no—
416C2×Q8—2442no—
417C2×Q8—2442no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—612no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—1224no—
425C2×D4—1224no—
426C2×D4—1221no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—2442no—
430C2×D4—2442no—
431C2×D4—2442no—
432C2×D4—2442no—
433C2×D4—2444no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4C2×C4×D6112yes—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—614no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—611no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—614no—
451C22×C4—1224no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1221no—
455C22×C4—1224no—
456C22×C4—1222no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—614no—
465C2×C8—12216no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8#1175822214yescomplemented
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—612no—
473C4⋊C4—1224no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—2444no—
478C4⋊C4—2444no—
479C4⋊C4C2×C4×D6112yes—
480C22⋊C4—24no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—1224no—
484C22⋊C4—2444no—
485C42—42no—
486C42—612no—
487C42—612no—
488C42—614no—
489C42—614no—
490C42—2442no—
491C42C2×C4×D6112yes—
492C42D4×D6112yes—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—612no—
499C23—612no—
500C23—612no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—2441no—
505C23—2441no—
506C23—2442no—
507C23—2442no—
508C23C2×D4×D6111yescharacteristic
509Q8—2448no—
510D4—22no—
511D4—22no—
512D4—42no—
513D4—1222no—
514D4—1222no—
515D4—2448no—
516D4—2448no—
517D4—2448no—
518D4—2448no—
519D4—2442no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—614no—
533C2×C4—6116no—
534C2×C4—612no—
535C2×C4—1228no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1224no—
540C2×C4—1222no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2448no—
547C2×C4C22⋊C4×D6112yes—
548C2×C4C22⋊C4×D6112yes—
549C2×C4D8×D6112yes—
550C2×C4SD16×D6112yes—
551C2×C4C22×C4×D6111yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4#41×D6112yes—
554C2×C4C2×#1175822212yes—
555C8—24no—
556C8—1224no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—82no—
561C22—82no—
562C22—614no—
563C22—1224no—
564C22—1222no—
565C22—1222no—
566C22—1224no—
567C22—4882no—
568C22—4882no—
569C22—2442no—
570C22—2442no—
571C22—2442no—
572C22C2×C22⋊C4×D6111yescharacteristic
573C22#1684030912yes—
574C22C2×D8×D6111yescharacteristic
575C22C2×SD16×D6111yescharacteristic
576C22#135×D6112yes—
577C4—22no—
578C4—42no—
579C4—82no—
580C4—614no—
581C4—614no—
582C4—1222no—
583C4—4882no—
584C4—2442no—
585C4#102×D6114yes—
586C4#1684023011yescharacteristic
587C4C4×#1175822211yescharacteristic
588C4#1684027112yes—
589C2—82no—
590C2—1224no—
591C2—4882no—
592C2#2144830511yescharacteristic
593C2C2×#102×D6111yescharacteristic
594C2#416×D6111yescharacteristic
595C2#2145637512yes—
596C2#2145639512yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenterC4×#102D61
hypocenterC61#21401
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×C4×C8×D61C2semidirect
3. #21448223C2semidirect
4. #21446778C2semidirect
5. #21446623C2semidirect
6. #21446246C2semidirect
7. #21446244C2semidirect
8. C4×#14519697C2semidirect
9. C4×#14519699C2semidirect
10. C4×#14519698C2semidirect
11. C4×#102×C61C2semidirect
15. #16840309C4semidirect
165. C2×C8×C61#41semidirect
192. C2×#92418#41semidirect
254. C4×D61#102semidirect
301. C4×#102D61semidirect
344. C61#21401semidirect
468. C2×C8#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×C4×C8×D612
#214482232
#214467782
#214466232
#214462462
#214462442
C4×#145196972
C4×#145196992
C4×#145196982
C4×#102×C612
#2140161