Finite Group Databasebeta

← #29690397

Subgroups — #29690397

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C8×D61C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic
3#21448215C211yesmaximal, characteristic
4#21446590C212yesmaximal, complemented
5#21446584C212yesmaximal, complemented
6#21446582C211yesmaximal, characteristic, complemented
7#21446580C211yesmaximal, characteristic, complemented
8#21446578C211yesmaximal, characteristic, complemented
9#21445845C212yesmaximal
10#21445843C212yesmaximal
11C61×#1947C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519735—21no—
14C2×C42×D61C2211yescharacteristic
15C2×#14519711C2212yes—
16C2×#14519708C2212yes—
17C2×#14519708C2212yes—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519706C2212yes—
20C4⋊C8×D61C418yescomplemented
21#16839632C2211yescharacteristic
22#16839629—21no—
23#16839627—21no—
24#16839627C2211yescharacteristic
25#16839627C2211yescharacteristic
26#16839624—21no—
27#16839619—21no—
28#16839619C2211yescharacteristic
29#16839619C2211yescharacteristic
30#16839619C2212yes—
31C4⋊C4×Dic61—21no—
32C4⋊C4×Dic61C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34C4×#11758198—21no—
35#16839608—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×#107×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C22×#92418C2312yes—
48C2×C8×D61—42no—
49C2×C8×D61—42no—
50C2×C8×D61C2×C418yes—
51#14519735—44no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758198—22no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×C4×Dic61—21no—
61C2×C4×Dic61—22no—
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61D411yescharacteristic
64C2×C4×Dic61C2311yescharacteristic
65C42×D61C2×C414yes—
66C42×D61Q814yes—
67#14519711—42no—
68#14519708—42no—
69#14519708—42no—
70C4⋊#92418C2×C414yes—
71#14519706C2×C418yes—
72#14519684—21no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684—22no—
80#14519684—22no—
81C22×C8×C61C2312yes—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2312yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C42×C61—21no—
88C2×C42×C61C2311yescharacteristic
89#107×C61—42no—
90C4⋊C8×C61C2×C414yes—
91#95×C61—21no—
92C23×D61C2×D411yescharacteristic
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61C4⋊C414yes—
114C2×C4×D61C4⋊C414yes—
115C2×C4×D61C22×C412yes—
116C2×C4×D61C2×Q812yes—
117C2×C8×C61—22no—
118C2×C8×C61—22no—
119C2×C8×C61C22×C414yes—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—42no—
127C4⋊C4×C61—42no—
128C42×C61—42no—
129C42×C61C22×C412yes—
130C42×C61C2×Q812yes—
131#11758198—42no—
132#11758198—44no—
133#11758198—42no—
134C4⋊Dic61—21no—
135C4⋊Dic61—22no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—42no—
141C4⋊Dic61—42no—
142C4×Dic61—44no—
143C4×Dic61—42no—
144C4×Dic61C4⋊C414yes—
145C4×Dic61C4⋊C414yes—
146C4×Dic61C22×C412yes—
147C4×Dic61C2×Q812yes—
148C2×#92418—22no—
149C2×#92418—22no—
150C2×#92418C22×C414yes—
151C8×D61—48no—
152C23×C61C22×D411yescharacteristic
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61C2×C4⋊C412yes—
158C22×D61AGL(1,ℤ/8)12yes—
159C22×D61#13712yes—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61C2×C4⋊C412yes—
172C2×C4×C61C2×C4⋊C412yes—
173C2×C4×C61#13512yes—
174C2×C4×C61#13512yes—
175C2×C4×C61C23×C411yescharacteristic
176C2×C4×C61C22×Q811yescharacteristic
177C2×C4×C61H(5,𝔽2)11yescharacteristic
178C2×C4×C61H(5,𝔽2)11yescharacteristic
179C2×C4×C612−1+411yescharacteristic
180C2×C4×C612−1+411yescharacteristic
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61C2×C4⋊C412yes—
197C2×Dic61C2×C4⋊C412yes—
198C2×Dic61C2×C4⋊C412yes—
199C4×D61—28no—
200C4×D61—22no—
201C4×D61—22no—
202C4×D61—22no—
203C4×D61—22no—
204C4×D61—28no—
205C8×C61—24no—
206#92418—24no—
207#21962—611nomaximal, Sylow
208C22×C61C22×C4⋊C411yescharacteristic
209C22×C61#53711yescharacteristic
210C22×C61#53711yescharacteristic
211C22×C61C2×#13511yescharacteristic
212C22×C61C2×#13511yescharacteristic
213C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
214C22×C61C2×#13711yescharacteristic
215C2×D61—22no—
216C2×D61—22no—
217C2×D61—22no—
218C2×D61—22no—
219C2×D61—22no—
220C2×D61—22no—
221C2×D61#42814yes—
222C4×C61—22no—
223C4×C61—81no—
224C4×C61—82no—
225C4×C61—81no—
226C4×C61#42714yes—
227C4×C61#51811yescharacteristic
228C4×C61#51911yescharacteristic
229C4×C61#54411yescharacteristic
230C4×C61#54411yescharacteristic
231Dic61—22no—
232Dic61—22no—
233Dic61—22no—
234Dic61—81no—
235Dic61—82no—
236Dic61—81no—
237C22×C4⋊C8—611no—
238#2285—611no—
239#2283—611no—
240#1949—612no—
241#1949—612no—
242#1948—611no—
243#1948—611no—
244#1947—611no—
245#1947D6111yescharacteristic, complemented
246#1808—612no—
247#1807—612no—
248C2×C61#231611yescharacteristic
249C2×C61C2×#42711yescharacteristic
250C2×C61C2×#42811yescharacteristic
251C2×C61#324711yescharacteristic
252C2×C61#324711yescharacteristic
253C2×C61#324811yescharacteristic
254C2×C61#324811yescharacteristic
255D61—24no—
256C23×C8—612no—
257C2×#117—1221no—
258C22×C42—611no—
259C2×#107—612no—
260C2×#107C2×D6112yes—
261C2×#106—612no—
262C2×#106—612no—
263C2×C4⋊C8—612no—
264C2×C4⋊C8—611no—
265C2×C4⋊C8—618no—
266C2×C4⋊C8C2×D6111yescharacteristic
267#384—611no—
268#384C2×D6111yescharacteristic
269#383—1221no—
270#382—611no—
271#382—611no—
272#382—1221no—
273#381—1221no—
274#380—611no—
275#380—612no—
276#380—611no—
277#380—1221no—
278C4×C4⋊C4—612no—
279C4×C4⋊C4—611no—
280C4×C4⋊C4—1221no—
281C4×C4⋊C4C2×D6111yescharacteristic
282C4×C22⋊C4—1221no—
283#376—1221no—
284#340—612no—
285#340—612no—
286#340—612no—
287#340C2×D6112yes—
288C61#2196211yescharacteristic, complemented, Sylow
289C23×C4—611no—
290C23×C4—611no—
291C23×C4—611no—
292C22×C8—612no—
293C22×C8—618no—
294C22×C8—2442no—
295C22×C8—2442no—
296C22×C8C22×D6112yes—
297#117—2444no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—611no—
302C2×C4⋊C4—612no—
303C2×C4⋊C4—1221no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4C22×D6111yescharacteristic
306C2×C4⋊C4C22×D6112yes—
307C2×C4⋊C4C22×D6111yescharacteristic
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1221no—
311C2×C42—21no—
312C2×C42—611no—
313C2×C42—611no—
314C2×C42—611no—
315C2×C42—614no—
316C2×C42—614no—
317C2×C42—1222no—
318C2×C42—1221no—
319C2×C42C22×D6111yescharacteristic
320#107—42no—
321#107—2442no—
322#106—2442no—
323#106—2442no—
324C4⋊C8—618no—
325C4⋊C8—614no—
326C4⋊C8C4×D6114yes—
327#95—21no—
328#95—1221no—
329#95—1221no—
330#95—1221no—
331#95—1221no—
332#95—1221no—
333#95—1221no—
334#95—1222no—
335#95—1222no—
336#95—1221no—
337C24—611no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—611no—
342C22×C4—611no—
343C22×C4—611no—
344C22×C4—612no—
345C22×C4—612no—
346C22×C4—614no—
347C22×C4—614no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—1222no—
359C22×C4D4×D6111yescharacteristic
360C22×C4D4×D6111yescharacteristic
361C22×C4C23×D6111yescharacteristic
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—614no—
365C2×C8—1222no—
366C2×C8—1222no—
367C2×C8—2448no—
368C2×C8C2×C4×D6114yes—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—1221no—
378C4⋊C4—1222no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1222no—
383C4⋊C4—2442no—
384C4⋊C4—2442no—
385C22⋊C4—2444no—
386C22⋊C4—2442no—
387C22⋊C4—2442no—
388C42—42no—
389C42—612no—
390C42—612no—
391C42—614no—
392C42—614no—
393C42—2444no—
394C42—2442no—
395C42C2×C4×D6112yes—
396C42Q8×D6112yes—
397C23—612no—
398C23—612no—
399C23—612no—
400C23—1221no—
401C23—1221no—
402C23—1221no—
403C23—1221no—
404C23C2×D4×D6111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—612no—
417C2×C4—612no—
418C2×C4—612no—
419C2×C4—1228no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—1221no—
423C2×C4—1221no—
424C2×C4—1221no—
425C2×C4—1221no—
426C2×C4—1222no—
427C2×C4—1222no—
428C2×C4—1228no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2442no—
439C2×C4—2441no—
440C2×C4C4⋊C4×D6112yes—
441C2×C4C4⋊C4×D6112yes—
442C2×C4#1451979612yes—
443C2×C4#1451980012yes—
444C2×C4C22×C4×D6111yescharacteristic
445C2×C4C2×Q8×D6111yescharacteristic
446C2×C4#1451982211yescharacteristic
447C2×C4#1451982311yescharacteristic
448C2×C4#1451982411yescharacteristic
449C2×C4#1451982511yescharacteristic
450C8—24no—
451C8—1224no—
452C22—614no—
453C22—1222no—
454C22—1222no—
455C22—1222no—
456C22—1222no—
457C22—1222no—
458C22—1222no—
459C22C2×C4⋊C4×D6111yescharacteristic
460C22#1684034111yescharacteristic
461C22#1684034211yescharacteristic
462C22C2×#1451979611yescharacteristic
463C22C2×#1451980011yescharacteristic
464C22AGL(1,ℤ/8)×D6111yescharacteristic
465C22#137×D6111yescharacteristic
466C4—22no—
467C4—81no—
468C4—81no—
469C4—82no—
470C4—1222no—
471C4—1222no—
472C4—1222no—
473C4—4881no—
474C4—4881no—
475C4—4882no—
476C4#1683979914yes—
477C4#1684023411yescharacteristic
478C4#1684023811yescharacteristic
479C4#1684038911yescharacteristic
480C4#1684039211yescharacteristic
481C2—1224no—
482C2#2144843411yescharacteristic
483C2C2×#1683979911yescharacteristic
484C2#428×D6111yescharacteristic
485C2#2145714111yescharacteristic
486C2#2145714811yescharacteristic
487C2#2145715411yescharacteristic
488C2#2145715811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1947C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1947D61
hypocenterC61#21962
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. #21446590C2semidirect
5. #21446584C2semidirect
6. #21446582C2semidirect
7. #21446580C2semidirect
8. #21446578C2semidirect
11. C61×#1947C2semidirect
20. C4⋊C8×D61C4semidirect
245. #1947D61semidirect
288. C61#21962semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C8×D612
#214482242
#214482152
#214465902
#214465842
#214465822
#214465802
#214465782
#214458452
#214458432
C61×#19472
#2196261