Finite Group Databasebeta

← #29690579

Subgroups — #29690579

Order 15616.

902 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839818C211yesmaximal, characteristic
2#384×D61C211yesmaximal, characteristic
3#21448215C211yesmaximal, characteristic
4#21446583C211yesmaximal, characteristic, complemented
5#21446580C211yesmaximal, characteristic, complemented
6#21446577C211yesmaximal, characteristic, complemented
7#21446013C212yesmaximal
8#21446005C212yesmaximal
9#21445986C212yesmaximal, complemented
10#21445951C212yesmaximal, complemented
11C61×#1947C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×D61—21no—
13C2×C4⋊C4×D61—21no—
14C2×C4⋊C4×D61C2212yes—
15C2×C42×D61C2211yescharacteristic
16#16839818C418yescomplemented
17C2×#14519710C2212yes—
18C2×#14519709C2212yes—
19C2×#14519708C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519689C2212yes—
22C2×#14519687C2212yes—
23#16839632C2211yescharacteristic
24#16839631—21no—
25#16839631—21no—
26#16839631C2212yes—
27#16839629—21no—
28#16839627C2211yescharacteristic
29#16839627C2211yescharacteristic
30#16839624—21no—
31#16839619C2211yescharacteristic
32#16839619C2211yescharacteristic
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×D61—22no—
45C22×C4×D61D411yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61C2311yescharacteristic
48C22×#92418C2312yes—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61C2312yes—
52C2×C4⋊Dic61C2311yescharacteristic
53C2×C4⋊Dic61C2311yescharacteristic
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C4⋊C4×D61—42no—
58C4⋊C4×D61—42no—
59C4⋊C4×D61—44no—
60C4⋊C4×D61—44no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C42×D61C2×C414yes—
68C42×D61Q814yes—
69#14519710—42no—
70#14519709—42no—
71#14519708—42no—
72C4⋊#92418C2×C414yescomplemented
73#14519689C2×C418yes—
74#14519687C2×C418yescomplemented
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684—21no—
80#14519684—21no—
81#14519684—21no—
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2312yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90#107×C61—42no—
91C4⋊C8×C61C2×C414yescomplemented
92#95×C61—21no—
93C23×D61C2×D411yescharacteristic
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—22no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61—44no—
115C2×C4×D61—44no—
116C2×C4×D61—44no—
117C2×C4×D61C4⋊C414yes—
118C2×C4×D61C4⋊C414yes—
119C2×C4×D61C22×C412yes—
120C2×C4×D61C2×Q812yes—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61C22×C414yes—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—42no—
131C4⋊C4×C61—42no—
132C42×C61—42no—
133C42×C61C22×C412yes—
134C42×C61C2×Q812yes—
135#11758198—42no—
136#11758198—42no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—42no—
144C4⋊Dic61—42no—
145#11758196—44no—
146#11758196—44no—
147#11758196—44no—
148C4×Dic61—42no—
149C4×Dic61C4⋊C414yes—
150C4×Dic61C4⋊C414yes—
151C4×Dic61C22×C412yes—
152C4×Dic61C2×Q812yes—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418C22×C414yes—
156C23×C61C22×D411yescharacteristic
157C22×D61—21no—
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—21no—
161C22×D61C2×C4⋊C412yes—
162C22×D61AGL(1,ℤ/8)12yes—
163C22×D61#13712yes—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61C2×C4⋊C412yes—
176C2×C4×C61C2×C4⋊C412yes—
177C2×C4×C61AGL(1,ℤ/8)12yes—
178C2×C4×C61#13712yes—
179C2×C4×C61C23×C411yescharacteristic
180C2×C4×C61C22×Q811yescharacteristic
181C2×C4×C61H(5,𝔽2)11yescharacteristic
182C2×C4×C61H(5,𝔽2)11yescharacteristic
183C2×C4×C612−1+411yescharacteristic
184C2×C4×C612−1+411yescharacteristic
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—42no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61C2×C4⋊C412yes—
201C2×Dic61C2×C4⋊C412yes—
202C2×Dic61C2×C4⋊C412yes—
203C4×D61—28no—
204C4×D61—22no—
205C4×D61—22no—
206C4×D61—22no—
207C4×D61—22no—
208C4×D61—28no—
209C4×D61—88no—
210C8×C61—24no—
211#92418—24no—
212#21987—611nomaximal, Sylow
213C22×C61C22×C4⋊C411yescharacteristic
214C22×C61#53511yescharacteristic
215C22×C61#53611yescharacteristic
216C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
217C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
218C22×C61C2×#13711yescharacteristic
219C22×C61C2×#13711yescharacteristic
220C2×D61—22no—
221C2×D61—22no—
222C2×D61—22no—
223C2×D61—22no—
224C2×D61—22no—
225C2×D61—22no—
226C2×D61#42814yes—
227C4×C61—22no—
228C4×C61—82no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61#42814yes—
232C4×C61#52011yescharacteristic
233C4×C61#52011yescharacteristic
234C4×C61#54411yescharacteristic
235C4×C61#54411yescharacteristic
236Dic61—22no—
237Dic61—22no—
238Dic61—22no—
239Dic61—82no—
240Dic61—81no—
241Dic61—81no—
242C2×#432—611no—
243C2×#384—611no—
244#2283—611no—
245#1948—611no—
246#1948—611no—
247#1947—611no—
248#1947D6111yescharacteristic, complemented
249#1853—612no—
250#1852—612no—
251#1847—612no—
252#1836—612no—
253C2×C61#231711yescharacteristic
254C2×C61C2×#42811yescharacteristic
255C2×C61C2×#42811yescharacteristic
256C2×C61#321711yescharacteristic
257C2×C61#321811yescharacteristic
258C2×C61#323811yescharacteristic
259C2×C61#323911yescharacteristic
260D61—24no—
261C22×C4⋊C4—612no—
262C22×C4⋊C4—1221no—
263C22×C4⋊C4—1221no—
264C22×C42—611no—
265#432—618no—
266C2×#107—612no—
267C2×#107C2×D6112yes—
268C2×#106—612no—
269C2×#106—612no—
270C2×C4⋊C8—611no—
271C2×C4⋊C8C2×D6111yescharacteristic
272C2×C22⋊C8—612no—
273C2×#97—612no—
274#384—612no—
275#384—611no—
276#384—1221no—
277#384—1221no—
278#384C2×D6111yescharacteristic
279#383—1221no—
280#382—611no—
281#382—611no—
282#381—1221no—
283#380—611no—
284#380—611no—
285C4×C4⋊C4—611no—
286C4×C4⋊C4C2×D6111yescharacteristic
287C4×C22⋊C4—1221no—
288#376—1221no—
289#340—612no—
290#340—612no—
291#340—612no—
292#340C2×D6112yes—
293C61#2198711yescharacteristic, complemented, Sylow
294C23×C4—611no—
295C23×C4—611no—
296C23×C4—611no—
297C23×C4—1222no—
298C22×C8—612no—
299C22×C8C22×D6112yes—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—612no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—1222no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—2442no—
311C2×C4⋊C4—2442no—
312C2×C4⋊C4—2444no—
313C2×C4⋊C4—2444no—
314C2×C4⋊C4C22×D6112yes—
315C2×C4⋊C4C22×D6111yescharacteristic
316C2×C4⋊C4C22×D6111yescharacteristic
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1221no—
319C2×C42—21no—
320C2×C42—611no—
321C2×C42—611no—
322C2×C42—611no—
323C2×C42—614no—
324C2×C42—614no—
325C2×C42—1221no—
326C2×C42C22×D6111yescharacteristic
327#107—42no—
328#107—2442no—
329#106—2442no—
330#106—2442no—
331C4⋊C8—614no—
332C4⋊C8C4×D6114yescomplemented
333C22⋊C8—618no—
334#97—618no—
335#95—21no—
336#95—1221no—
337#95—1221no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341#95—1221no—
342#95—1221no—
343C24—611no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—611no—
350C22×C4—612no—
351C22×C4—612no—
352C22×C4—614no—
353C22×C4—614no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—1222no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—1222no—
365C22×C4—2444no—
366C22×C4—2444no—
367C22×C4—2444no—
368C22×C4D4×D6111yescharacteristic
369C22×C4D4×D6111yescharacteristic
370C22×C4C23×D6111yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—614no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8C2×C4×D6114yes—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—1221no—
386C4⋊C4—1222no—
387C4⋊C4—1221no—
388C4⋊C4—1222no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—2442no—
392C4⋊C4—2444no—
393C4⋊C4—2444no—
394C4⋊C4—2444no—
395C4⋊C4—2442no—
396C22⋊C4—2442no—
397C22⋊C4—2442no—
398C42—42no—
399C42—612no—
400C42—612no—
401C42—614no—
402C42—614no—
403C42—2442no—
404C42C2×C4×D6112yes—
405C42Q8×D6112yes—
406C23—612no—
407C23—612no—
408C23—612no—
409C23—1221no—
410C23—1221no—
411C23—1221no—
412C23—1221no—
413C23C2×D4×D6111yescharacteristic
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—612no—
426C2×C4—612no—
427C2×C4—612no—
428C2×C4—1228no—
429C2×C4—1221no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—1221no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—1228no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—4888no—
441C2×C4—2441no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4C4⋊C4×D6112yes—
451C2×C4C4⋊C4×D6112yes—
452C2×C4#1451980112yes—
453C2×C4#1451980912yes—
454C2×C4C22×C4×D6111yescharacteristic
455C2×C4C2×Q8×D6111yescharacteristic
456C2×C4#1451982211yescharacteristic
457C2×C4#1451982311yescharacteristic
458C2×C4#1451982411yescharacteristic
459C2×C4#1451982511yescharacteristic
460C8—24no—
461C8—1224no—
462C22—614no—
463C22—1222no—
464C22—1222no—
465C22—1222no—
466C22—1222no—
467C22—1222no—
468C22—1222no—
469C22C2×C4⋊C4×D6111yescharacteristic
470C22#1684032811yescharacteristic
471C22#1684033511yescharacteristic
472C22AGL(1,ℤ/8)×D6111yescharacteristic
473C22C2×#1451980111yescharacteristic
474C22#137×D6111yescharacteristic
475C22C2×#1451980911yescharacteristic
476C4—22no—
477C4—81no—
478C4—81no—
479C4—82no—
480C4—1222no—
481C4—1222no—
482C4—1222no—
483C4—4881no—
484C4—4882no—
485C4—4881no—
486C4#1683980314yes—
487C4#1684024111yescharacteristic
488C4#1684024411yescharacteristic
489C4#1684038911yescharacteristic
490C4#1684039211yescharacteristic
491C2—1224no—
492C2#2144844111yescharacteristic
493C2#428×D6111yescharacteristic
494C2C2×#1683980311yescharacteristic
495C2#2145683611yescharacteristic
496C2#2145685011yescharacteristic
497C2#2145704411yescharacteristic
498C2#2145704811yescharacteristic
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#21987
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. #21446583C2semidirect
5. #21446580C2semidirect
6. #21446577C2semidirect
9. #21445986C2semidirect
10. #21445951C2semidirect
11. C61×#1947C2semidirect
16. #16839818C4semidirect
72. C4⋊#92418C2×C4semidirect
74. #14519687C2×C4semidirect
91. C4⋊C8×C61C2×C4semidirect
248. #1947D61semidirect
293. C61#21987semidirect
332. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398182
#384×D612
#214482152
#214465832
#214465802
#214465772
#214460132
#214460052
#214459862
#214459512
C61×#19472
#2198761