Finite Group Databasebeta

← #29696439

Subgroups — #29696439

Order 15616.

1370 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840415C211yesmaximal, characteristic
2C2×#16839839C211yesmaximal, characteristic, complemented
3#21448766C211yesmaximal, characteristic, complemented
4#21446781C211yesmaximal, characteristic, complemented
5#21446256C212yesmaximal, complemented
6#21446140C212yesmaximal, complemented
7#21445908C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9C4×#14519697C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1988C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×#11758189C2212yes—
14#16840415—24no—
15#16840415C418yescomplemented
16C2×#14519762—22no—
17C2×#14519753—21no—
18C2×#14519752C2212yes—
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519735C2212yes—
22C2×C4⋊C4×D61C2212yes—
23#16839839—24no—
24C2×#14519699C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519689C2212yes—
28C2×C4×#92418C2211yescharacteristic
29#16839664—21no—
30#16839654—22no—
31#16839632C2211yescharacteristic
32#16839619C2212yes—
33C4×C4⋊Dic61—21no—
34#16839525C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C2×#11758226—28no—
45C2×#11758226—44no—
46C2×#11758226—44no—
47C22×D244—22no—
48C22×D244D411yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—21no—
52C22×C4×D61C2312yes—
53C2×#11758189—24no—
54C2×#11758189—24no—
55C2×#11758189—42no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58#14519762—216no—
59#14519753—28no—
60#14519752—24no—
61#14519752C2×C418yes—
62C4×D244—22no—
63C4×D244C2×C414yes—
64C4×D244C2×C414yes—
65C4×D244D414yes—
66#14519735—24no—
67#14519735—42no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—22no—
74C4⋊C4×D61—24no—
75C4⋊C4×D61C2×C418yes—
76C2×#11758198—21no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198D412yes—
81C2×C4×Dic61—22no—
82#14519699—42no—
83#14519697—22no—
84#14519697—22no—
85#14519689—24no—
86C4×#92418—22no—
87#14519684—22no—
88C22×Q8×C61—21no—
89C22×C8×C61C2312yes—
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61C2×C414yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C42×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—22no—
99#95×C61—21no—
100C23×D61—21no—
101C23×D61C2×D412yes—
102C2×Q8×C61—24no—
103C2×Q8×C61—42no—
104C2×Q8×C61—42no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—22no—
113C22×Dic61C2×D412yes—
114#11758226—432no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—28no—
122C2×D244—21no—
123C2×D244—44no—
124C2×D244—44no—
125C2×D244C22⋊C414yes—
126C2×D244C22×C412yes—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—28no—
132C2×C4×D61—24no—
133C2×C4×D61—44no—
134C2×C4×D61—44no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61C22×C414yes—
138C2×C4×D61C2×D414yes—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61C22×C414yes—
148C4⋊C4×C61#4112yescomplemented
149C42×C61—42no—
150C42×C61C22×C412yes—
151C42×C61C2×D412yes—
152#11758198—24no—
153#11758198—28no—
154#11758198—44no—
155#11758198—82no—
156#11758198C22⋊C418yes—
157C4⋊Dic61—21no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—42no—
161C4⋊Dic61—81no—
162C4⋊Dic61C22⋊C414yes—
163C4⋊Dic61C22×C412yes—
164#11758196—28no—
165C4×Dic61—44no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418#4112yescomplemented
172#11758189—48no—
173C23×C61C22×D411yescharacteristic
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61C2×C22⋊C414yes—
188Q8×C61—48no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—24no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C23×C411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61H(5,𝔽2)11yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61C2×C22⋊C414yes—
219D244—22no—
220D244—22no—
221D244—28no—
222D244—416no—
223D244—41no—
224D244—41no—
225C4×D61—24no—
226C4×D61—24no—
227C4×D61—24no—
228C4×D61—24no—
229C4×D61—416no—
230C4×D61—48no—
231C8×C61—42no—
232#92418—22no—
233#92418—41no—
234#22500—611nomaximal, Sylow
235C22×C61C22×C22⋊C411yescharacteristic
236C22×C61#54611yescharacteristic
237C22×C61#54711yescharacteristic
238C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
239C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
240C22×C61#57711yescharacteristic
241C22×C61#57811yescharacteristic
242C2×D61—24no—
243C2×D61—24no—
244C2×D61—24no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C4×C61—22no—
257C4×C61—42no—
258C4×C61—44no—
259C4×C61—81no—
260C4×C61#41814yes—
261C4×C61C4×#4111yescharacteristic
262C4×C61#52011yescharacteristic
263C4×C61#54611yescharacteristic
264C4×C61#54711yescharacteristic
265Dic61—24no—
266Dic61—42no—
267Dic61—82no—
268C2×#547—611no—
269C2×#436—611no—
270#2354—611no—
271#1990—611no—
272#1988D6111yescharacteristic, complemented
273#1895—612no—
274#1875—612no—
275#1824—612no—
276#1802—611no—
277C4×#102—611no—
278#1736—612no—
279C2×C61#233911yescharacteristic
280C2×C61C2×#41811yescharacteristic
281C2×C61#289811yescharacteristic
282C2×C61#333011yescharacteristic
283C2×C61#333211yescharacteristic
284C2×C61#335311yescharacteristic
285C2×C61#335411yescharacteristic
286D61—44no—
287D61—82no—
288C22×#41—1221no—
289C22×M16—612no—
290#547—618no—
291#547—1224no—
292C2×C4⋊Q8C2×D6111yescharacteristic
293C2×#123—1222no—
294C2×C22⋊Q8—1221no—
295C2×#121—612no—
296C2×C4×D4—611no—
297C2×C4×D4—611no—
298C2×#117—612no—
299C22×C4⋊C4—612no—
300#436—1224no—
301C2×C4⋊C8C2×D6111yescharacteristic
302C2×#102—611no—
303C2×#102—611no—
304C2×#102—612no—
305C2×C22⋊C8—612no—
306C2×C4×C8—611no—
307#390—1221no—
308#388—1222no—
309#384—611no—
310#380—612no—
311C4×C4⋊C4—1221no—
312C4×C4⋊C4C2×D6111yescharacteristic
313#340—612no—
314#340—611no—
315#340—612no—
316#340—612no—
317#340—611no—
318#340C2×D6112yes—
319#340C2×D6112yes—
320C61#2250011yescharacteristic, complemented, Sylow
321C2×#41—1228no—
322C2×#41—2444no—
323C2×#41—2444no—
324C22×Q8—21no—
325C22×D4—611no—
326C22×D4—611no—
327C22×D4—1222no—
328C23×C4—612no—
329C23×C4—1222no—
330C23×C4—1221no—
331C2×M16—1224no—
332C2×M16—1224no—
333C2×M16—2442no—
334C22×C8—611no—
335C22×C8—611no—
336C22×C8C22×D6112yes—
337C4⋊Q8—22no—
338C4⋊Q8C4×D6114yes—
339#123—12216no—
340C22⋊Q8—1228no—
341#121—618no—
342#121—1224no—
343C4×D4—614no—
344C4×D4—614no—
345C4×D4—614no—
346C4×D4—1222no—
347#117—1224no—
348#117—2442no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—618no—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—1224no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C4⋊C4C22×D6112yes—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—612no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C42—21no—
367C2×C42—1222no—
368C2×C42C22×D6111yescharacteristic
369C4⋊C8—22no—
370#102—1222no—
371#102—1222no—
372#102—2442no—
373C22⋊C8—1224no—
374C4×C8—1222no—
375#95—21no—
376#95—1222no—
377C24—612no—
378C24—1221no—
379#41—24432no—
380C2×Q8—24no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×D4—612no—
384C2×D4—614no—
385C2×D4—1228no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1222no—
392C2×D4—1222no—
393C2×D4—2444no—
394C2×D4—2444no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—612no—
399C22×C4—614no—
400C22×C4—614no—
401C22×C4—1228no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1221no—
407C22×C4—1224no—
408C22×C4—1222no—
409C22×C4—2444no—
410C22×C4—2444no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4D4×D6111yescharacteristic
414C22×C4D4×D6111yescharacteristic
415C22×C4C23×D6111yescharacteristic
416M16—2448no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—612no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1221no—
425C2×C8—1221no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—24no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—614no—
432C4⋊C4—612no—
433C4⋊C4—1228no—
434C4⋊C4—1221no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—4881no—
438C4⋊C4—2442no—
439C4⋊C4C2×C4×D6114yes—
440C4⋊C4#1175822612yescomplemented
441C22⋊C4—618no—
442C22⋊C4—1228no—
443C22⋊C4—1224no—
444C22⋊C4—4882no—
445C22⋊C4—2444no—
446C42—42no—
447C42—2444no—
448C42C2×C4×D6112yes—
449C42D4×D6112yes—
450C23—614no—
451C23—1222no—
452C23—1222no—
453C23—1222no—
454C23—1222no—
455C23—1222no—
456C23—1222no—
457C23—1222no—
458C23—2441no—
459C23—2441no—
460C23—2441no—
461C23—2441no—
462C23—2441no—
463C23—2441no—
464C23C2×D4×D6111yescharacteristic
465Q8—48no—
466D4—1222no—
467D4—1222no—
468D4—1228no—
469D4—24416no—
470D4—2441no—
471D4—2441no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—614no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1224no—
487C2×C4—1224no—
488C2×C4—1224no—
489C2×C4—1224no—
490C2×C4—1222no—
491C2×C4—24416no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2448no—
498C2×C4C22⋊C4×D6112yes—
499C2×C4C22⋊C4×D6112yes—
500C2×C4#1451980112yes—
501C2×C4#1451980412yes—
502C2×C4C22×C4×D6111yescharacteristic
503C2×C4C2×D4×D6111yescharacteristic
504C2×C4#41×D6111yescharacteristic
505C2×C4C2×#1175822611yescharacteristic
506C2×C4#1451982311yescharacteristic
507C2×C4#1451982511yescharacteristic
508C8—42no—
509C8—1222no—
510C8—2441no—
511C22—1224no—
512C22—1224no—
513C22—1224no—
514C22—4881no—
515C22—4881no—
516C22—4881no—
517C22—4881no—
518C22—2442no—
519C22—2442no—
520C22—2442no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22C2×C22⋊C4×D6111yescharacteristic
526C22#1684040311yescharacteristic
527C22#1684041411yescharacteristic
528C22C2×#1451980111yescharacteristic
529C22C2×#1451980411yescharacteristic
530C22#1684057111yescharacteristic
531C22#1684057811yescharacteristic
532C4—22no—
533C4—42no—
534C4—44no—
535C4—81no—
536C4—1224no—
537C4—4882no—
538C4—2442no—
539C4#1683976514yes—
540C4C4×#1175822611yescharacteristic
541C4#1684024411yescharacteristic
542C4#1684040511yescharacteristic
543C4#1684041511yescharacteristic
544C2—4882no—
545C2—2444no—
546C2#2144864811yescharacteristic
547C2C2×#1683976511yescharacteristic
548C2#2145435111yescharacteristic
549C2#2145812111yescharacteristic
550C2#2145814311yescharacteristic
551C2#2145840611yescharacteristic
552C2#2145841711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC2×#16840415C2
cosocleC22×C4×C61C24
hypercenter#1988D61
hypocenterC61#22500
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C2×#16839839C2semidirect
3. #21448766C2semidirect
4. #21446781C2semidirect
5. #21446256C2semidirect
6. #21446140C2semidirect
7. #21445908C2semidirect
8. #21445820C2semidirect
9. C4×#14519697C2semidirect
10. #21445537C2semidirect
11. C61×#1988C2semidirect
15. #16840415C4semidirect
148. C4⋊C4×C61#41semidirect
171. C2×#92418#41semidirect
272. #1988D61semidirect
320. C61#22500semidirect
440. C4⋊C4#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404152
C2×#168398392
#214487662
#214467812
#214462562
#214461402
#214459082
#214458202
C4×#145196972
#214455372
C61×#19882
#2250061