Finite Group Databasebeta

← #29696808

Subgroups — #29696808

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×#16839886C211yesmaximal, characteristic, complemented
3#21448774C211yesmaximal, characteristic, complemented
4#21446748C212yesmaximal
5#21446627C212yesmaximal
6#21446622C211yesmaximal, characteristic, complemented
7#21445832C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic
9#103×Dic61C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic
11C61×#1981C211yesmaximal, characteristic, complemented
12#16840486—24no—
13#16840486C418yescomplemented
14C2×#14519772—22no—
15C2×#14519766C2212yes—
16C2×#14519757—21no—
17C2×Q8×Dic61C2212yes—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519735—21no—
21#16839886—24no—
22C2×#14519706C2212yes—
23C2×#14519701C2212yes—
24C2×#14519701C2212yes—
25C2×#14519699C2212yes—
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29C2×C8×Dic61C2212yes—
30#16839668—21no—
31#16839649—22no—
32#16839632C2211yescharacteristic
33#16839627C2212yes—
34C4⋊C4×Dic61C2212yes—
35C4×C4⋊Dic61—21no—
36#16839525C2212yes—
37#16839524C2211yescharacteristic
38#16839524C2211yescharacteristic
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×Dic122D411yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C22×#92418C2311yescharacteristic
48C22×#92418C2311yescharacteristic
49#14519772—216no—
50#14519766—24no—
51#14519766C2×C418yes—
52#14519757—28no—
53Q8×Dic61—24no—
54Q8×Dic61C2×C418yes—
55C4×Dic122C2×C414yes—
56C4×Dic122D414yes—
57C4×D244—22no—
58C4×D244C2×C414yes—
59#14519735—28no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×#11758196D412yes—
67C2×#11758198—21no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61C2312yes—
73#14519706—24no—
74#14519701—24no—
75#14519701—42no—
76#14519699—42no—
77#14519697—22no—
78#14519697—22no—
79C4×#92418—22no—
80C8×Dic61—24no—
81#14519684—22no—
82#14519684—22no—
83#14519684—22no—
84C22×Q8×C61C2312yes—
85C22×C8×C61C2312yes—
86C4⋊Q8×C61—22no—
87C4⋊Q8×C61C2×C414yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C42×C61—21no—
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61—22no—
95#103×C61—24no—
96#103×C61—42no—
97#95×C61—21no—
98C23×D61—21no—
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61C22×C414yes—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—22no—
113C22×Dic61C2×D412yes—
114C22×Dic61C2×D412yes—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244C22×C412yes—
121C2×C4×D61—24no—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122C22⋊C414yes—
127C2×C8×C61—22no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61#4112yes—
136C42×C61—42no—
137C42×C61C22×C412yes—
138C42×C61C2×D412yes—
139#11758198—24no—
140#11758198—28no—
141#11758198—28no—
142C4⋊Dic61—21no—
143C4⋊Dic61—28no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—42no—
147C4⋊Dic61—81no—
148C4⋊Dic61C22⋊C414yes—
149C4⋊Dic61C22×C412yes—
150#11758196C22⋊C418yes—
151C4×Dic61—28no—
152C4×Dic61—82no—
153C4×Dic61C22×C414yes—
154C4×Dic61C2×D414yes—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418#4112yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—22no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169Q8×C61—24no—
170Q8×C61—24no—
171Q8×C61—42no—
172Q8×C61—42no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—24no—
176C2×C4×C61—22no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61C2×C22⋊C412yes—
185C2×C4×C61C2×C22⋊C412yes—
186C2×C4×C61#13512yes—
187C2×C4×C61#13512yes—
188C2×C4×C61C23×C411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61H(5,𝔽2)11yescharacteristic
193C2×C4×C612−1+411yescharacteristic
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61C2×C22⋊C414yes—
205C2×Dic61C2×C22⋊C414yes—
206D244—22no—
207D244—22no—
208D244—41no—
209D244—41no—
210C4×D61—48no—
211Dic122—28no—
212C8×C61—42no—
213#92418—22no—
214#92418—41no—
215#22541—611nomaximal, Sylow
216C22×C61C22×C22⋊C411yescharacteristic
217C22×C61#54811yescharacteristic
218C22×C61#55011yescharacteristic
219C22×C61C2×#13511yescharacteristic
220C22×C61C2×#13511yescharacteristic
221C22×C61#57711yescharacteristic
222C22×C61#57811yescharacteristic
223C2×D61—42no—
224C2×D61—42no—
225C2×D61—42no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C4×C61—22no—
231C4×C61—42no—
232C4×C61—44no—
233C4×C61—81no—
234C4×C61#41614yes—
235C4×C61C4×#4111yescharacteristic
236C4×C61#52011yescharacteristic
237C4×C61#54811yescharacteristic
238C4×C61#55011yescharacteristic
239Dic61—24no—
240Dic61—24no—
241Dic61—42no—
242Dic61—82no—
243C2×#556—611no—
244C2×C8×Q8—611no—
245#2355—611no—
246#1983—612no—
247#1981D6111yescharacteristic, complemented
248#1958—612no—
249#1956—611no—
250#1805—612no—
251#1802—611no—
252C4×#103—612no—
253C4×#102—611no—
254C2×C61#234111yescharacteristic
255C2×C61C2×#41611yescharacteristic
256C2×C61#289811yescharacteristic
257C2×C61#336011yescharacteristic
258C2×C61#336411yescharacteristic
259C2×C61#339311yescharacteristic
260C2×C61#339411yescharacteristic
261D61—82no—
262#556—618no—
263#556—1224no—
264C2×C4⋊Q8C2×D6111yescharacteristic
265C2×#126—1222no—
266C2×#124—612no—
267C2×C22⋊Q8—1221no—
268C2×C4×Q8—611no—
269C2×C4×Q8—612no—
270C2×C4×D4—611no—
271C2×#117—1221no—
272C8×Q8—1224no—
273C2×C4⋊C8—612no—
274C2×C4⋊C8C2×D6111yescharacteristic
275C2×#103—612no—
276C2×#103—612no—
277C2×#103C2×D6112yes—
278C2×#103C2×D6112yes—
279C2×#102—611no—
280C2×#102—611no—
281C2×#102—612no—
282C2×C4×C8—612no—
283C2×C4×C8—611no—
284#391—1221no—
285#387—1222no—
286#384—611no—
287#382—612no—
288C4×C4⋊C4—612no—
289C4×C4⋊C4—1221no—
290C4×C4⋊C4C2×D6111yescharacteristic
291#340—612no—
292#340—611no—
293#340—611no—
294C61#2254111yescharacteristic, complemented, Sylow
295C22×Q8—611no—
296C22×Q8C22×D6112yes—
297C22×D4—611no—
298C23×C4—1221no—
299C22×C8—611no—
300C22×C8—611no—
301C22×C8C22×D6112yes—
302C4⋊Q8—22no—
303C4⋊Q8C4×D6114yes—
304#126—12216no—
305#124—618no—
306#124—1224no—
307C22⋊Q8—1228no—
308C4×Q8—614no—
309C4×Q8—614no—
310C4×Q8—618no—
311C4×Q8—1224no—
312C4×D4—614no—
313C4×D4—1222no—
314#117—1228no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—612no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4C22×D6111yescharacteristic
325C2×C4⋊C4C22×D6111yescharacteristic
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4—1221no—
329C2×C42—21no—
330C2×C42—612no—
331C2×C42—1222no—
332C2×C42—1222no—
333C2×C42C22×D6111yescharacteristic
334C4⋊C8—22no—
335C4⋊C8—1224no—
336#103—24no—
337#103—42no—
338#103—1224no—
339#103—2442no—
340#102—1222no—
341#102—1222no—
342#102—2442no—
343C4×C8—1224no—
344C4×C8—1222no—
345#95—21no—
346#95—1222no—
347#95—1222no—
348#95—1222no—
349C24—1221no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—614no—
356C2×Q8—1222no—
357C2×Q8—1222no—
358C2×Q8C2×C4×D6114yes—
359C2×D4—612no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—1221no—
364C2×D4—1221no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—612no—
369C22×C4—612no—
370C22×C4—1221no—
371C22×C4—1224no—
372C22×C4—1222no—
373C22×C4—2442no—
374C22×C4—2442no—
375C22×C4D4×D6111yescharacteristic
376C22×C4D4×D6111yescharacteristic
377C22×C4C23×D6111yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—612no—
382C2×C8—1221no—
383C2×C8—1221no—
384C2×C8—1221no—
385C2×C8—1221no—
386C2×C8—1221no—
387C4⋊C4—21no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—618no—
393C4⋊C4—614no—
394C4⋊C4—612no—
395C4⋊C4—1228no—
396C4⋊C4—1221no—
397C4⋊C4—1222no—
398C4⋊C4—1222no—
399C4⋊C4—4881no—
400C4⋊C4—2442no—
401C4⋊C4#1175822412yes—
402C22⋊C4—1228no—
403C22⋊C4—1228no—
404C22⋊C4—1224no—
405C42—42no—
406C42—614no—
407C42—614no—
408C42—1228no—
409C42—4882no—
410C42C2×C4×D6112yes—
411C42D4×D6112yes—
412C23—1222no—
413C23—2441no—
414C23—2441no—
415C23—2441no—
416C23—2441no—
417C23—2441no—
418C23—2441no—
419C23C2×D4×D6111yescharacteristic
420Q8—24no—
421Q8—24no—
422Q8—42no—
423Q8—42no—
424Q8—1228no—
425D4—1222no—
426D4—1222no—
427D4—2441no—
428D4—2441no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—614no—
441C2×C4—614no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2448no—
453C2×C4C22⋊C4×D6112yes—
454C2×C4C22⋊C4×D6112yes—
455C2×C4#1451979812yes—
456C2×C4#1451980012yes—
457C2×C4C22×C4×D6111yescharacteristic
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4#41×D6111yescharacteristic
460C2×C4C2×#1175822411yescharacteristic
461C2×C4#1451982211yescharacteristic
462C2×C4#1451982411yescharacteristic
463C8—42no—
464C8—1222no—
465C8—2441no—
466C22—4881no—
467C22—4881no—
468C22—4881no—
469C22—4881no—
470C22—2442no—
471C22—2442no—
472C22—2442no—
473C22C2×C22⋊C4×D6111yescharacteristic
474C22#1684041811yescharacteristic
475C22#1684043011yescharacteristic
476C22C2×#1451979811yescharacteristic
477C22C2×#1451980011yescharacteristic
478C22#1684057111yescharacteristic
479C22#1684057811yescharacteristic
480C4—22no—
481C4—42no—
482C4—44no—
483C4—81no—
484C4—1224no—
485C4—1224no—
486C4—4882no—
487C4—2442no—
488C4#1683975714yes—
489C4C4×#1175822411yescharacteristic
490C4#1684024111yescharacteristic
491C4#1684042011yescharacteristic
492C4#1684043211yescharacteristic
493C2—4882no—
494C2#2144865811yescharacteristic
495C2C2×#1683975711yescharacteristic
496C2#2145435111yescharacteristic
497C2#2145847211yescharacteristic
498C2#2145850811yescharacteristic
499C2#2145880611yescharacteristic
500C2#2145881411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1981C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1981D61
hypocenterC61#22541
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. C2×#16839886C2semidirect
3. #21448774C2semidirect
6. #21446622C2semidirect
7. #21445832C2semidirect
9. #103×Dic61C2semidirect
11. C61×#1981C2semidirect
13. #16840486C4semidirect
247. #1981D61semidirect
294. C61#22541semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168398862
#214487742
#214467482
#214466272
#214466222
#214458322
#214458202
#103×Dic612
C4×#145196972
C61×#19812
#2254161