Finite Group Databasebeta

← #29696472

Subgroups — #29696472

Order 15616.

1246 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840478C211yesmaximal, characteristic
2C2×#16839841C211yesmaximal, characteristic, complemented
3#21448719C211yesmaximal, characteristic, complemented
4#21446768C211yesmaximal, characteristic, complemented
5#21446256C212yesmaximal, complemented
6#21446135C212yesmaximal, complemented
7#21445912C212yesmaximal, complemented
8#21445816C211yesmaximal, characteristic, complemented
9C4×#14519699C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1990C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840478—24no—
14#16840478C418yescomplemented
15C2×#14519766—22no—
16C2×#14519752—21no—
17C2×#14519752C2212yes—
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519739—21no—
21C2×#14519735—21no—
22C2×#14519735C2212yes—
23C2×C4⋊C4×D61C2212yes—
24#16839841—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2212yes—
29C2×#14519689C2212yes—
30#16839658—21no—
31#16839633—22no—
32#16839619C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839524C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×C8×C61C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C22×D244—22no—
45C22×D244D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C22×C4×D61C2312yes—
50C2×#11758189—24no—
51C2×#11758189—24no—
52C2×#11758189—42no—
53C22×#92418C2312yes—
54#14519766—216no—
55#14519752—28no—
56#14519752—24no—
57#14519752C2×C418yescomplemented
58C4×D244—22no—
59C4×D244C2×C414yes—
60C4×D244C2×C414yescomplemented
61C4×D244D414yes—
62#14519739—44no—
63#14519735—24no—
64#14519735—28no—
65#14519735C2×C418yescomplemented
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—22no—
71C4⋊C4×D61—24no—
72C4⋊C4×D61—42no—
73C2×#11758198—22no—
74C2×#11758198—21no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198D412yes—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—21no—
80C4⋊#92418—22no—
81#14519699—22no—
82#14519699—22no—
83#14519697—42no—
84#14519689—24no—
85#14519684—21no—
86#14519684—22no—
87C22×Q8×C61—21no—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61C2×C414yescomplemented
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98C4×C8×C61—22no—
99C23×D61—21no—
100C23×D61C2×D412yes—
101C2×Q8×C61—24no—
102C2×Q8×C61—81no—
103C2×Q8×C61—81no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61C2×D412yes—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—28no—
121C2×D244—82no—
122C2×D244—82no—
123C2×D244C22⋊C414yes—
124C2×D244C22×C412yes—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—24no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—44no—
133C2×C4×D61—44no—
134C2×C4×D61—82no—
135C2×C4×D61C22×C414yes—
136C2×C4×D61C2×D414yes—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61#4112yes—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61—81no—
149C4⋊C4×C61C22×C414yes—
150C42×C61C22×C412yes—
151C42×C61C2×D412yes—
152#11758198—28no—
153#11758198—24no—
154#11758198—28no—
155#11758198—44no—
156#11758198C22⋊C418yes—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61C22⋊C414yes—
162C4⋊Dic61C22×C412yes—
163C4⋊Dic61#4112yes—
164#11758196—44no—
165C4×Dic61—28no—
166C4×Dic61—42no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170#11758189—48no—
171C23×C61C22×D411yescharacteristic
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61C2×C22⋊C414yes—
186Q8×C61—84no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—24no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61C2×C22⋊C412yes—
196C2×C4×C61C2×C22⋊C412yes—
197C2×C4×C61AGL(1,ℤ/8)12yes—
198C2×C4×C61#13712yes—
199C2×C4×C61C23×C411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61H(5,𝔽2)11yescharacteristic
204C2×C4×C612−1+411yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61C2×C22⋊C414yes—
217D244—22no—
218D244—22no—
219D244—28no—
220D244—41no—
221D244—41no—
222D244—88no—
223C4×D61—24no—
224C4×D61—24no—
225C4×D61—24no—
226C4×D61—24no—
227C4×D61—48no—
228C4×D61—88no—
229C8×C61—22no—
230C8×C61—41no—
231#92418—42no—
232#22503—611nomaximal, Sylow
233C22×C61C22×C22⋊C411yescharacteristic
234C22×C61#54611yescharacteristic
235C22×C61#54711yescharacteristic
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C61C2×#13711yescharacteristic
238C22×C61#57611yescharacteristic
239C22×C61#57711yescharacteristic
240C2×D61—24no—
241C2×D61—24no—
242C2×D61—24no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—81no—
251C2×D61—81no—
252C2×D61—81no—
253C2×D61—81no—
254C4×C61—22no—
255C4×C61—44no—
256C4×C61—82no—
257C4×C61#41714yes—
258C4×C61C4×#4111yescharacteristic
259C4×C61#52011yescharacteristic
260C4×C61#54611yescharacteristic
261C4×C61#54711yescharacteristic
262Dic61—24no—
263Dic61—42no—
264Dic61—42no—
265Dic61—81no—
266C2×#555—611no—
267C2×#436—611no—
268D4×C4⋊C4—611no—
269#1990D6111yescharacteristic, complemented
270#1987—611no—
271#1895—612no—
272#1874—612no—
273#1825—612no—
274#1802—611no—
275C4×#102—611no—
276#1736—612no—
277C2×C61#233911yescharacteristic
278C2×C61C2×#41711yescharacteristic
279C2×C61#289711yescharacteristic
280C2×C61#332511yescharacteristic
281C2×C61#332911yescharacteristic
282C2×C61#335411yescharacteristic
283C2×C61#335711yescharacteristic
284D61—44no—
285D61—82no—
286C22×M16—612no—
287#555—618no—
288#555—1224no—
289C2×C4⋊Q8C2×D6111yescharacteristic
290C2×#124—1222no—
291C2×#121—612no—
292C2×#121—1221no—
293C2×C4×D4—611no—
294C2×C4×D4—611no—
295C2×C4×D4—1221no—
296C2×#117—612no—
297C2×#117—1221no—
298C22×C4⋊C4—612no—
299#436—1224no—
300C2×C4⋊C8—611no—
301C2×#102—611no—
302C2×#102—611no—
303C2×#102—612no—
304C2×C22⋊C8—612no—
305C2×C4×C8C2×D6111yescharacteristic
306#389—1221no—
307#385—1222no—
308#384C2×D6111yescharacteristic
309#380—612no—
310C4×C4⋊C4—611no—
311#340—611no—
312#340—611no—
313#340—612no—
314#340—612no—
315#340—612no—
316#340C2×D6112yes—
317#340C2×D6112yes—
318C61#2250311yescharacteristic, complemented, Sylow
319C22×Q8—21no—
320C22×D4—611no—
321C22×D4—611no—
322C22×D4—1222no—
323C23×C4—612no—
324C23×C4—1222no—
325C23×C4—1221no—
326C2×M16—1224no—
327C2×M16—1224no—
328C2×M16—2442no—
329C22×C8—612no—
330C22×C8C22×D6111yescharacteristic
331C22×C8C22×D6111yescharacteristic
332C4⋊Q8—22no—
333C4⋊Q8C4×D6114yescomplemented
334#124—12216no—
335#121—618no—
336#121—1224no—
337#121—1228no—
338C4×D4—614no—
339C4×D4—614no—
340C4×D4—614no—
341C4×D4—1222no—
342C4×D4—2444no—
343#117—618no—
344#117—1228no—
345#117—1224no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—1224no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—2442no—
356C2×C4⋊C4C22×D6112yes—
357C2×C4⋊C4C22×D6112yes—
358C2×C22⋊C4—612no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1222no—
363C2×C42—1222no—
364C2×C42—1221no—
365C2×C42C22×D6111yescharacteristic
366C4⋊C8—1222no—
367#102—1222no—
368#102—1222no—
369#102—2442no—
370C22⋊C8—1224no—
371C4×C8—22no—
372#95—1222no—
373#95—1221no—
374C24—612no—
375C24—1221no—
376C2×Q8—24no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×D4—612no—
380C2×D4—614no—
381C2×D4—1228no—
382C2×D4—1222no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1222no—
389C2×D4—4882no—
390C2×D4—4882no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—612no—
394C22×C4—614no—
395C22×C4—614no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1224no—
403C22×C4—1221no—
404C22×C4—4882no—
405C22×C4—2444no—
406C22×C4—2444no—
407C22×C4—2442no—
408C22×C4—2442no—
409C22×C4D4×D6111yescharacteristic
410C22×C4D4×D6111yescharacteristic
411C22×C4C23×D6111yescharacteristic
412M16—2448no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8#1175822612yes—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—42no—
427C4⋊C4—81no—
428C4⋊C4—614no—
429C4⋊C4—612no—
430C4⋊C4—612no—
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—2444no—
436C4⋊C4C2×C4×D6114yes—
437C22⋊C4—618no—
438C22⋊C4—1228no—
439C22⋊C4—1228no—
440C22⋊C4—1224no—
441C22⋊C4—2444no—
442C42—1228no—
443C42—2442no—
444C42C2×C4×D6112yes—
445C42D4×D6112yes—
446C23—614no—
447C23—1222no—
448C23—1222no—
449C23—1222no—
450C23—1222no—
451C23—1222no—
452C23—1222no—
453C23—1222no—
454C23—2441no—
455C23—2441no—
456C23—2441no—
457C23—2441no—
458C23—2441no—
459C23—2441no—
460C23C2×D4×D6111yescharacteristic
461Q8—84no—
462D4—1222no—
463D4—1222no—
464D4—1228no—
465D4—4888no—
466D4—2441no—
467D4—2441no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—614no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1224no—
480C2×C4—1224no—
481C2×C4—1224no—
482C2×C4—1224no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—4888no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2448no—
494C2×C4C22⋊C4×D6112yes—
495C2×C4C22⋊C4×D6112yes—
496C2×C4#1451980412yes—
497C2×C4#1451980912yes—
498C2×C4C22×C4×D6111yescharacteristic
499C2×C4C2×D4×D6111yescharacteristic
500C2×C4#41×D6111yescharacteristic
501C2×C4C2×#1175822611yescharacteristic
502C2×C4#1451982311yescharacteristic
503C2×C4#1451982511yescharacteristic
504C8—22no—
505C8—41no—
506C8—2442no—
507C22—1224no—
508C22—1224no—
509C22—1224no—
510C22—4881no—
511C22—4881no—
512C22—4881no—
513C22—4881no—
514C22—2442no—
515C22—2442no—
516C22—2442no—
517C22—2442no—
518C22—2442no—
519C22—2442no—
520C22—2442no—
521C22C2×C22⋊C4×D6111yescharacteristic
522C22#1684040511yescharacteristic
523C22#1684041511yescharacteristic
524C22C2×#1451980411yescharacteristic
525C22C2×#1451980911yescharacteristic
526C22#1684056311yescharacteristic
527C22#1684056811yescharacteristic
528C4—22no—
529C4—44no—
530C4—82no—
531C4—1224no—
532C4—4881no—
533C4—2442no—
534C4—2442no—
535C4#1683976214yes—
536C4C4×#1175822611yescharacteristic
537C4#1684024411yescharacteristic
538C4#1684040311yescharacteristic
539C4#1684041411yescharacteristic
540C2—4882no—
541C2—2444no—
542C2#2144864811yescharacteristic
543C2C2×#1683976211yescharacteristic
544C2#2145434011yescharacteristic
545C2#2145806011yescharacteristic
546C2#2145811211yescharacteristic
547C2#2145841311yescharacteristic
548C2#2145844411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radicalC2×#16840478C2
cosocleC22×C4×C61C24
hypercenter#1990D61
hypocenterC61#22503
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#16839841C2semidirect
3. #21448719C2semidirect
4. #21446768C2semidirect
5. #21446256C2semidirect
6. #21446135C2semidirect
7. #21445912C2semidirect
8. #21445816C2semidirect
9. C4×#14519699C2semidirect
10. #21445537C2semidirect
11. C61×#1990C2semidirect
14. #16840478C4semidirect
57. #14519752C2×C4semidirect
60. C4×D244C2×C4semidirect
65. #14519735C2×C4semidirect
91. C4⋊Q8×C61C2×C4semidirect
269. #1990D61semidirect
318. C61#22503semidirect
333. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404782
C2×#168398412
#214487192
#214467682
#214462562
#214461352
#214459122
#214458162
C4×#145196992
#214455372
C61×#19902
#2250361