Finite Group Databasebeta

← #29696280

Subgroups — #29696280

Order 15616.

1206 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic
2#21448768C211yesmaximal, characteristic, complemented
3#21446780C211yesmaximal, characteristic, complemented
4#21446766C211yesmaximal, characteristic, complemented
5#21446247C212yesmaximal, complemented
6#21446139C212yesmaximal, complemented
7#21445915C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9C4×#14519699C211yesmaximal, characteristic, complemented
10#21445536C212yesmaximal, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12#16840477—24no—
13#16840477C418yescomplemented
14C2×#14519777C2211yescharacteristic
15C2×#14519763—22no—
16C2×#14519751C2212yes—
17C2×#14519748—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519736C2212yes—
20C2×#14519736C2212yes—
21C2×#14519735—21no—
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2212yes—
25#16839665—21no—
26#16839655—22no—
27#16839632C2211yescharacteristic
28#16839622C2212yes—
29C4⋊C4×Dic61—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839525C2211yescharacteristic
32#16839525C2211yescharacteristic
33#16839525C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38C23×C8×C61C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C8×D4×C61—24no—
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C22×Dic122—21no—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48#14519777—22no—
49#14519777C2×C414yes—
50#14519763—216no—
51#14519751—24no—
52#14519751C2×C418yes—
53#14519748—28no—
54C4×D244—22no—
55C4×D244C2×C414yes—
56#14519736—24no—
57#14519736—24no—
58#14519736—42no—
59#14519736C2×C418yes—
60#14519735—28no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×C4×Dic61—22no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61—21no—
77#14519699—22no—
78#14519699—22no—
79#14519699—42no—
80#14519684—21no—
81#14519684—22no—
82C22×D4×C61D411yescharacteristic
83C23×C4×C61C2312yes—
84C22×C8×C61—24no—
85C22×C8×C61—24no—
86C22×C8×C61—42no—
87C22×C8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4×D4×C61C2×C414yes—
91C4×D4×C61D414yes—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C22⋊C4×C61D412yes—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—22no—
96C22⋊C8×C61—24no—
97C4×C8×C61—22no—
98C24×C61C2×D412yes—
99C23×D61—21no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61C22⋊C414yes—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61C22×C414yes—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2×D414yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C2×C22⋊D61—28no—
114C2×C22⋊D61—82no—
115C2×C22⋊D61—82no—
116C22×Dic61—21no—
117C22×Dic61—22no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—22no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244C22×C412yes—
127C2×C4×D61—24no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—24no—
131C2×Dic122—81no—
132C2×Dic122—81no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—21no—
139C2×C8×C61—22no—
140C2×C8×C61—21no—
141C2×C8×C61—48no—
142C2×C8×C61#4112yes—
143C4⋊C4×C61C22⋊C414yes—
144C22⋊C4×C61C22⋊C418yes—
145C42×C61C22×C412yes—
146C42×C61C2×D412yes—
147C22⋊Dic61—28no—
148C22⋊Dic61—44no—
149C22⋊Dic61—82no—
150#11758198—28no—
151#11758198—24no—
152C4⋊Dic61—24no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—42no—
161C4⋊Dic61—81no—
162C4⋊Dic61C22×C412yes—
163C4⋊Dic61C22×C414yes—
164C4⋊Dic61#4112yes—
165C4×Dic61—28no—
166C4×Dic61—44no—
167C4×Dic61—42no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61C2×C22⋊C414yes—
175C23×C61C22×D411yescharacteristic
176C22×D61—22no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183D4×C61—28no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61C2×C22⋊C414yes—
193C2×C4×C61C2×C22⋊C412yes—
194C2×C4×C61C2×C22⋊C412yes—
195C2×C4×C61#13512yes—
196C2×C4×C61#13512yes—
197C2×C4×C61C23×C411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)11yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C22⋊D61—88no—
204C2×Dic61—22no—
205C2×Dic61—24no—
206C2×Dic61—22no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219D244—22no—
220D244—22no—
221D244—41no—
222D244—41no—
223C4×D61—48no—
224Dic122—84no—
225C8×C61—22no—
226C8×C61—42no—
227C8×C61—41no—
228#22489—611nomaximal, Sylow
229C22×C61—24no—
230C22×C61—24no—
231C22×C61—24no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61C22×C22⋊C411yescharacteristic
237C22×C61#54611yescharacteristic
238C22×C61#54711yescharacteristic
239C22×C61C2×#13511yescharacteristic
240C22×C61C2×#13511yescharacteristic
241C22×C61#57611yescharacteristic
242C22×C61#57711yescharacteristic
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—24no—
251C4×C61—22no—
252C4×C61#41614yes—
253C4×C61C4×#4111yescharacteristic
254C4×C61#52011yescharacteristic
255C4×C61#54611yescharacteristic
256C4×C61#54711yescharacteristic
257Dic61—42no—
258Dic61—44no—
259Dic61—42no—
260Dic61—81no—
261Dic61—82no—
262C2×#555—611no—
263C2×C8×D4D6111yescharacteristic, complemented
264#2354—611no—
265#1990—611no—
266#1987—611no—
267#1893—612no—
268#1875—612no—
269#1825—612no—
270#1802—611no—
271C4×#102—611no—
272#1735—612no—
273C2×C61—44no—
274C2×C61#233911yescharacteristic
275C2×C61C2×#41611yescharacteristic
276C2×C61#289711yescharacteristic
277C2×C61#330311yescharacteristic
278C2×C61#330511yescharacteristic
279C2×C61#334011yescharacteristic
280C2×C61#334111yescharacteristic
281D61—82no—
282C23×C8C2×D6112yes—
283#555—618no—
284#555—1224no—
285C2×C4⋊Q8—611no—
286C2×#124—1222no—
287C2×#121—612no—
288C2×#121—1221no—
289C2×C4×D4—611no—
290C2×C4×D4C2×D6111yescharacteristic
291C2×#117—612no—
292C2×#117—612no—
293C2×#117—1221no—
294C8×D4—24no—
295C2×C4⋊C8C2×D6111yescharacteristic
296C2×#102—611no—
297C2×#102—611no—
298C2×#102—612no—
299C2×C22⋊C8C2×D6112yes—
300C2×C4×C8C2×D6111yescharacteristic
301#390—1221no—
302#388—1222no—
303#384—611no—
304#380—612no—
305C4×C4⋊C4—611no—
306C4×C4⋊C4—1221no—
307#340—612no—
308#340—611no—
309#340—612no—
310#340—612no—
311#340—612no—
312#340—612no—
313#340—611no—
314C61#2248911yescharacteristic, complemented, Sylow
315C22×Q8—1221no—
316C22×D4—611no—
317C22×D4—1222no—
318C22×D4D24411yescharacteristic
319C23×C4—1221no—
320C23×C4C22×D6112yes—
321C22×C8—24no—
322C22×C8—24no—
323C22×C8—42no—
324C22×C8C22×D6112yes—
325C22×C8C22×D6111yescharacteristic
326C22×C8C22×D6111yescharacteristic
327C4⋊Q8—614no—
328C4⋊Q8—1222no—
329#124—12216no—
330#121—618no—
331#121—1228no—
332#121—1224no—
333C4×D4—614no—
334C4×D4—1222no—
335C4×D4C4×D6114yes—
336C4×D4C22⋊D6114yes—
337#117—618no—
338#117—1224no—
339#117—1228no—
340#117—1224no—
341#117—2442no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—611no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4D24411yescharacteristic
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1222no—
356C2×C22⋊C4D24412yes—
357C2×C42—1222no—
358C2×C42—1222no—
359C2×C42—1221no—
360C2×C42C22×D6111yescharacteristic
361C4⋊C8—22no—
362#102—1222no—
363#102—1222no—
364#102—2442no—
365C22⋊C8—24no—
366C4×C8—22no—
367#95—1221no—
368#95—1222no—
369C24—1221no—
370C24C2×D24412yes—
371C2×Q8—1224no—
372C2×Q8—4881no—
373C2×Q8—4881no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—612no—
377C2×D4—1228no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—4882no—
384C2×D4—4882no—
385C2×D4#1175819814yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—1221no—
391C22×C4—1221no—
392C22×C4—1222no—
393C22×C4—1224no—
394C22×C4—1221no—
395C22×C4—1222no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4C2×C4×D6114yes—
399C22×C4C2×D24412yes—
400C22×C4C2×D24411yescharacteristic
401C22×C4C2×D24411yescharacteristic
402C22×C4C2×C22⋊D6114yes—
403C22×C4C23×D6111yescharacteristic
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—48no—
413C2×C8#1175822412yes—
414C4⋊C4—612no—
415C4⋊C4—614no—
416C4⋊C4—612no—
417C4⋊C4—1221no—
418C4⋊C4—1222no—
419C4⋊C4—1221no—
420C4⋊C4—1224no—
421C4⋊C4—1222no—
422C4⋊C4—1221no—
423C4⋊C4—1222no—
424C4⋊C4—1221no—
425C4⋊C4—4881no—
426C4⋊C4—2442no—
427C4⋊C4#1175819814yes—
428C22⋊C4—1228no—
429C22⋊C4—1228no—
430C22⋊C4—1224no—
431C22⋊C4—4882no—
432C22⋊C4—2444no—
433C22⋊C4#1175819818yes—
434C42—1228no—
435C42—2444no—
436C42—2442no—
437C42C2×C4×D6112yes—
438C42C2×C22⋊D6112yes—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—1222no—
446C23—2441no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23—2441no—
452C23C2×#1175819814yes—
453C23C22×D24411yescharacteristic
454Q8—4884no—
455D4—28no—
456D4—1222no—
457D4—1222no—
458D4—4888no—
459D4—2441no—
460D4—2441no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—1222no—
470C2×C4—1224no—
471C2×C4—1222no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2442no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4—2448no—
485C2×C4C2×#1175819814yes—
486C2×C4C2×#1175819812yes—
487C2×C4C2×#1175819812yes—
488C2×C4#1451979912yes—
489C2×C4#1451979912yes—
490C2×C4C22×C4×D6111yescharacteristic
491C2×C4C22×C22⋊D6111yescharacteristic
492C2×C4#41×D6111yescharacteristic
493C2×C4C2×#1175822411yescharacteristic
494C2×C4#1451982211yescharacteristic
495C2×C4#1451982411yescharacteristic
496C8—22no—
497C8—42no—
498C8—41no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—4881no—
507C22—4881no—
508C22—4881no—
509C22—4881no—
510C22—2442no—
511C22—2442no—
512C22—2442no—
513C22C22×#1175819811yescharacteristic
514C22#1684040711yescharacteristic
515C22#1684041611yescharacteristic
516C22C2×#1451979911yescharacteristic
517C22C2×#1451979911yescharacteristic
518C22#1684056711yescharacteristic
519C22#1684057411yescharacteristic
520C4—22no—
521C4—24no—
522C4—4881no—
523C4—4882no—
524C4—2444no—
525C4—2442no—
526C4—2442no—
527C4#1683975914yes—
528C4C4×#1175822411yescharacteristic
529C4#1684024111yescharacteristic
530C4#1684040611yescharacteristic
531C4#1684041111yescharacteristic
532C2—44no—
533C2—4882no—
534C2#2144864411yescharacteristic
535C2C2×#1683975911yescharacteristic
536C2#2145434611yescharacteristic
537C2#2145774811yescharacteristic
538C2#2145777011yescharacteristic
539C2#2145823311yescharacteristic
540C2#2145824211yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840477C2
cosocleC22×C4×C61C24
hypercenterC2×C8×D4D61
hypocenterC61#22489
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
2. #21448768C2semidirect
3. #21446780C2semidirect
4. #21446766C2semidirect
5. #21446247C2semidirect
6. #21446139C2semidirect
7. #21445915C2semidirect
8. #21445820C2semidirect
9. C4×#14519699C2semidirect
10. #21445536C2semidirect
11. C2×C8×D4×C61C2semidirect
13. #16840477C4semidirect
263. C2×C8×D4D61semidirect
314. C61#22489semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
#214487682
#214467802
#214467662
#214462472
#214461392
#214459152
#214458202
C4×#145196992
#214455362
C2×C8×D4×C612
#2248961