Finite Group Databasebeta

← #29708788

Subgroups — #29708788

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839760C212yesmaximal
2#21448306C211yesmaximal, characteristic
3#384×D61C211yesmaximal, characteristic, complemented
4#21447231C212yesmaximal, complemented
5#21447229C211yesmaximal, characteristic, complemented
6#21447228C211yesmaximal, characteristic, complemented
7#21447225C212yesmaximal, complemented
8#21447222C211yesmaximal, characteristic, complemented
9#21445537C212yesmaximal, complemented
10#21445511C211yesmaximal, characteristic, complemented
11C61×#2071C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×#11758189C2212yes—
14C2×#14519735C2211yescharacteristic
15C2×C4⋊C4×D61—21no—
16C2×C4⋊C4×D61—21no—
17C2×C4⋊C4×D61C2212yes—
18C2×C42×D61C2211yescharacteristic
19#16839760—28no—
20C2×#14519702C2212yes—
21C2×#14519700C2212yes—
22C2×#14519698C2212yes—
23#16839668C2211yescharacteristic
24#16839667C2211yescharacteristic
25#16839665C2211yescharacteristic
26#16839638—21no—
27#16839638—21no—
28#16839632C2211yescharacteristic
29#16839631—21no—
30#16839631—21no—
31#16839631C2212yes—
32C22⋊C4×Dic61—21no—
33C22⋊C4×Dic61—21no—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38#16839516C2212yes—
39C2×#102×C61C2212yes—
40C61×#390C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C61×#336C2211yescharacteristic
44C2×#11758224—24no—
45C2×#11758224—24no—
46C2×#11758224—22no—
47C2×#11758224D414yes—
48C22×Dic122C2311yescharacteristic
49C22×C22⋊D61—21no—
50C23×Dic61—21no—
51C22×C4×D61—22no—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758189—24no—
56C2×#11758189—24no—
57C2×#11758189—24no—
58C22×#92418C2312yes—
59#14519735—24no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C4⋊C4×D61—24no—
68C4⋊C4×D61—24no—
69C4⋊C4×D61—44no—
70C4⋊C4×D61—44no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×#11758198—21no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61D411yescharacteristic
76C2×C4×Dic61D411yescharacteristic
77C2×C4×Dic61C2311yescharacteristic
78C42×D61—24no—
79#14519702—24no—
80#14519700—24no—
81#14519698—24no—
82#14519684—21no—
83#14519684—21no—
84#14519684—21no—
85#14519684—21no—
86C22×D4×C61C2311yescharacteristic
87C22×C8×C61C2312yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2312yes—
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C42×C61C2311yescharacteristic
95#102×C61—24no—
96C24×C61—21no—
97C23×D61C2×D411yescharacteristic
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—21no—
101C2×D4×C61C2×D412yes—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C2×C22⋊D61—24no—
108C2×C22⋊D61—42no—
109C2×C22⋊D61—42no—
110C22×Dic61—21no—
111C22×Dic61—24no—
112C22×Dic61—22no—
113C22×Dic61—21no—
114C22×Dic61—42no—
115C22×Dic61—42no—
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D411yescharacteristic
119#11758224—216no—
120#11758224—48no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61—44no—
130C2×C4×D61C2×D412yes—
131C2×C4×D61#4114yes—
132C2×C4×D61#4114yes—
133C2×C4×D61#4112yes—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—21no—
137C2×Dic122C2×D412yes—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61—42no—
146C22⋊C4×C61—42no—
147C22⋊C4×C61—42no—
148C42×C61—22no—
149C22⋊Dic61—42no—
150C22⋊Dic61—42no—
151#11758198—42no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—42no—
156C4⋊Dic61—42no—
157#11758196—44no—
158#11758196—44no—
159#11758196—44no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C4×Dic61—22no—
163C4×Dic61—42no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167#11758189—48no—
168C23×C61—22no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61C22×D411yescharacteristic
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—21no—
180C22×D61#12112yes—
181C22×D61#12312yes—
182C22×D61#12412yes—
183D4×C61—24no—
184D4×C61—42no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61AGL(1,ℤ/8)12yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4112yes—
199C2×C4×C61C2×#4112yes—
200C2×C4×C61C2×#4111yescharacteristic
201C22⋊D61—48no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—22no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—48no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61#12112yes—
217C2×Dic61#12312yes—
218C2×Dic61#12412yes—
219C2×Dic61AGL(1,ℤ/8)12yes—
220C2×Dic61#13712yes—
221C4×D61—28no—
222C4×D61—28no—
223C4×D61—88no—
224C4×D61#12318yes—
225C4×D61#12618yes—
226Dic122—24no—
227Dic122—42no—
228C8×C61—42no—
229#92418—42no—
230#23630—611nomaximal, Sylow
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61C2×#12111yescharacteristic
239C22×C61C2×#12311yescharacteristic
240C22×C61C2×#12411yescharacteristic
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C61C2×#13711yescharacteristic
244C22×C61C2×#13711yescharacteristic
245C2×D61—22no—
246C2×D61—22no—
247C2×D61—22no—
248C2×D61—22no—
249C2×D61—22no—
250C2×D61—22no—
251C2×D61#39714yes—
252C4×C61—22no—
253C4×C61—22no—
254C4×C61—81no—
255C4×C61—82no—
256C4×C61C2×#12312yes—
257C4×C61C2×#12612yes—
258Dic61—22no—
259Dic61—22no—
260Dic61—42no—
261Dic61—81no—
262Dic61—82no—
263Dic61#39714yes—
264C2×#417—612no—
265#2297—611no—
266C2×#384—611no—
267#2072—611no—
268#2072—612no—
269#2072—611no—
270#2071—612no—
271#2071—611no—
272#2071D6111yescharacteristic, complemented
273#1736—612no—
274#1728—611no—
275C2×C61—82no—
276C2×C61C2×#39711yescharacteristic
277C2×C61#303811yescharacteristic
278C2×C61#303911yescharacteristic
279C2×C61#309211yescharacteristic
280C2×C61#309211yescharacteristic
281C2×C61#314111yescharacteristic
282C2×C61#314111yescharacteristic
283D61—24no—
284C22×#41—611no—
285C22×M16—612no—
286C2×#117—611no—
287C22×C4⋊C4—612no—
288C22×C4⋊C4—1221no—
289C22×C4⋊C4—1221no—
290C22×C42—611no—
291#417—1228no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#102—612no—
295C2×#102C2×D6112yes—
296#391—611no—
297#391—611no—
298#390—611no—
299#390C2×D6111yescharacteristic
300#386—1221no—
301#386—1221no—
302#384—612no—
303#384—611no—
304#384—1221no—
305#384—1221no—
306#384C2×D6111yescharacteristic
307C4×C22⋊C4—1221no—
308C4×C22⋊C4—1221no—
309#340—612no—
310#340—612no—
311#340—612no—
312#340C2×D6112yes—
313#336—612no—
314#336—611no—
315#336C2×D6111yescharacteristic
316C61#2363011yescharacteristic, complemented, Sylow
317C2×#41—614no—
318C2×#41—1222no—
319C2×#41—1224no—
320C2×#41—1224no—
321C22×Q8—611no—
322C22×D4—1221no—
323C22×D4C22×D6111yescharacteristic
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—611no—
327C23×C4—1222no—
328C23×C4—1221no—
329C2×M16—1224no—
330C2×M16—1224no—
331C2×M16—1224no—
332C22×C8—612no—
333C22×C8C22×D6112yes—
334#117—1224no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—1224no—
340C2×C4⋊C4—1224no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—2444no—
347C2×C4⋊C4—2444no—
348C2×C4⋊C4C22×D6112yes—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C42—611no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—1224no—
359C2×C42—1221no—
360C2×C42C22×D6111yescharacteristic
361#103—1224no—
362#103—1224no—
363#102—24no—
364#102—1224no—
365#95—1221no—
366#95—1221no—
367#95—1221no—
368#95—1221no—
369C24—21no—
370C24—611no—
371#41—12216no—
372#41—2448no—
373C2×Q8—612no—
374C2×Q8—1221no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—1224no—
381C2×D4—2442no—
382C2×D4—2442no—
383C2×D4D4×D6112yes—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—611no—
387C22×C4—612no—
388C22×C4—614no—
389C22×C4—612no—
390C22×C4—614no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1224no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1222no—
403C22×C4—2444no—
404C22×C4—2444no—
405C22×C4—2444no—
406C22×C4—2442no—
407C22×C4—2442no—
408C22×C4D4×D6111yescharacteristic
409C22×C4D4×D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411M16—2448no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—1222no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—1222no—
426C4⋊C4—2442no—
427C4⋊C4—2444no—
428C4⋊C4—2444no—
429C4⋊C4—2444no—
430C4⋊C4—2442no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—2442no—
434C22⋊C4—2442no—
435C22⋊C4—2442no—
436C42—22no—
437C42—1222no—
438C42—1222no—
439C42—1222no—
440C42—2442no—
441C23—22no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—612no—
449C23—612no—
450C23—612no—
451C23—1221no—
452C23—1221no—
453C23—1221no—
454C23—1221no—
455C23C2×D4×D6111yescharacteristic
456Q8—1224no—
457Q8—2442no—
458D4—24no—
459D4—42no—
460D4—2448no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—612no—
474C2×C4—618no—
475C2×C4—618no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—1228no—
479C2×C4—1228no—
480C2×C4—1221no—
481C2×C4—1221no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1222no—
485C2×C4—4888no—
486C2×C4—2448no—
487C2×C4—2442no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4#1451980112yes—
496C2×C4#1451980612yes—
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4#41×D6112yes—
499C2×C4#41×D6112yes—
500C2×C4#41×D6111yescharacteristic
501C8—42no—
502C8—2442no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—614no—
511C22—1222no—
512C22—1222no—
513C22—1222no—
514C22—1222no—
515C22—1222no—
516C22—1222no—
517C22#121×D6111yescharacteristic
518C22#123×D6111yescharacteristic
519C22#124×D6111yescharacteristic
520C22AGL(1,ℤ/8)×D6111yescharacteristic
521C22C2×#1451980111yescharacteristic
522C22#137×D6111yescharacteristic
523C22C2×#1451980611yescharacteristic
524C4—22no—
525C4—22no—
526C4—81no—
527C4—82no—
528C4—614no—
529C4—1222no—
530C4—1222no—
531C4—4881no—
532C4—4882no—
533C4—2442no—
534C4#123×D6112yes—
535C4#126×D6112yes—
536C2—82no—
537C2—1224no—
538C2#397×D6111yescharacteristic
539C2#2145529411yescharacteristic
540C2#2145530111yescharacteristic
541C2#2145573811yescharacteristic
542C2#2145574111yescharacteristic
543C2#2145620911yescharacteristic
544C2#2145621111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#2071D61
hypocenterC61#23630
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
3. #384×D61C2semidirect
4. #21447231C2semidirect
5. #21447229C2semidirect
6. #21447228C2semidirect
7. #21447225C2semidirect
8. #21447222C2semidirect
9. #21445537C2semidirect
10. #21445511C2semidirect
11. C61×#2071C2semidirect
272. #2071D61semidirect
316. C61#23630semidirect

Maximal subgroups

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

subgroupindex
C2×#168397602
#214483062
#384×D612
#214472312
#214472292
#214472282
#214472252
#214472222
#214455372
#214455112
C61×#20712
#2363061