Finite Group Databasebeta

← #29699814

Subgroups — #29699814

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758199C211yesmaximal, characteristic
2#21448766C211yesmaximal, characteristic, complemented
3#21446694C212yesmaximal
4#21446446C212yesmaximal, complemented
5#21446443C212yesmaximal, complemented
6#21445975C211yesmaximal, characteristic, complemented
7#21445819C211yesmaximal, characteristic, complemented
8#21445816C211yesmaximal, characteristic, complemented
9#21445717C212yesmaximal, complemented
10#21445688C211yesmaximal, characteristic, complemented
11C61×#2354C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14C2×#14519735—21no—
15C4×#11758199C418yescomplemented
16C2×#14519710C2211yescharacteristic
17C2×#14519708C2212yes—
18C2×#14519708C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2212yes—
21C2×#14519698C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2211yescharacteristic
24C2×#14519688C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839664C2212yes—
27#16839654—22no—
28#16839632C2211yescharacteristic
29#16839619—21no—
30C4×C4⋊Dic61C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C2×C4×D4×C61C2211yescharacteristic
35C2×#117×C61—21no—
36C61×#390C2212yes—
37C61×#388—22no—
38C61×#384C2211yescharacteristic
39C61×#380—21no—
40C4×C4⋊C4×C61C2212yes—
41C4×C4⋊C4×C61C2211yescharacteristic
42C22×D244C2311yescharacteristic
43C22×C4×D61—21no—
44C2×#11758199—22no—
45C2×#11758199—22no—
46C2×#11758199C2×C414yes—
47C22×#92418C2312yes—
48C22×#92418C2311yescharacteristic
49C22×#92418C2311yescharacteristic
50C4×D244—22no—
51C4×D244C2×C414yes—
52#14519735—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758198—21no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×C4×Dic61—22no—
61#14519710C2×C414yes—
62#14519708—24no—
63#14519708—42no—
64C4⋊#92418—22no—
65#14519698—42no—
66#14519698C2×C414yes—
67#14519697—42no—
68#14519697C2×C414yes—
69#14519688—24no—
70C4×#92418C2×C414yes—
71#14519684—22no—
72C22×D4×C61C2311yescharacteristic
73C23×C4×C61—21no—
74C4×D4×C61—22no—
75C4×D4×C61C2×C414yes—
76#117×C61—44no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C4⋊C4×C61D411yescharacteristic
81C2×C4⋊C4×C61C2312yes—
82C2×C4⋊C4×C61C2311yescharacteristic
83C2×C22⋊C4×C61—21no—
84C2×C22⋊C4×C61—22no—
85C2×C22⋊C4×C61—22no—
86C2×C42×C61—22no—
87C2×C42×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89#95×C61—22no—
90#95×C61D412yes—
91C24×C61—21no—
92C23×D61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61C22×C412yes—
99C22×C4×C61—21no—
100C22×C4×C61—24no—
101C22×C4×C61—22no—
102C22×C4×C61—42no—
103C22×C4×C61—42no—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244C22×C412yes—
117C2×C4×D61—24no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—42no—
126C4⋊C4×C61—81no—
127C4⋊C4×C61C22×C412yes—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—44no—
130C22⋊C4×C61—82no—
131C42×C61—22no—
132C42×C61—22no—
133C42×C61—44no—
134C42×C61C22×C412yes—
135C42×C61#4112yes—
136#11758199—28no—
137#11758198—24no—
138#11758198—44no—
139#11758198—82no—
140C4⋊Dic61—21no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—42no—
144C4⋊Dic61—81no—
145C4⋊Dic61C22×C412yes—
146C4×Dic61—44no—
147C2×#92418—21no—
148C2×#92418—21no—
149C2×#92418—22no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418C22×C412yes—
153C2×#92418C22×C412yes—
154C2×#92418#4112yes—
155C2×#92418#4112yes—
156C23×C61—22no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61C22×D411yescharacteristic
164C22×D61—22no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171D4×C61—22no—
172D4×C61—22no—
173D4×C61—41no—
174D4×C61—41no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—48no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61C4×D412yes—
191C2×C4×C61C4×D412yes—
192C2×C4×C61AGL(1,ℤ/8)12yes—
193C2×C4×C61AGL(1,ℤ/8)12yes—
194C2×C4×C61C23×C411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61H(5,𝔽2)11yescharacteristic
199C2×C4×C612−1+411yescharacteristic
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206D244—22no—
207D244—22no—
208D244—41no—
209D244—41no—
210C4×D61—48no—
211#92418—22no—
212#92418—42no—
213#92418#11714yescomplemented
214#22813—611nomaximal, Sylow
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61C2×C4×D411yescharacteristic
223C22×C61#54711yescharacteristic
224C22×C61#55011yescharacteristic
225C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
226C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C22×C61#57711yescharacteristic
228C22×C61#57711yescharacteristic
229C2×D61—42no—
230C2×D61—42no—
231C2×D61—42no—
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—81no—
236C4×C61—22no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—42no—
240C4×C61—82no—
241C4×C61#44214yes—
242C4×C61C2×#11711yescharacteristic
243C4×C61#52011yescharacteristic
244C4×C61#55511yescharacteristic
245C4×C61#55611yescharacteristic
246Dic61—42no—
247Dic61—82no—
248C2×C4×D8—611no—
249#2354—611no—
250#2354D6111yescharacteristic, complemented
251#1972—612no—
252#1926—612no—
253#1926—612no—
254#1844—611no—
255#1802—611no—
256#1802—611no—
257#1776—612no—
258#1770—611no—
259C2×C61—82no—
260C2×C61#235411yescharacteristic
261C2×C61C2×#44211yescharacteristic
262C2×C61#295211yescharacteristic
263C2×C61#335411yescharacteristic
264C2×C61#335411yescharacteristic
265C2×C61#339711yescharacteristic
266C2×C61#339711yescharacteristic
267D61—82no—
268C22×D8—611no—
269C2×C4×D4—611no—
270C2×C4×D4C2×D6111yescharacteristic
271C2×#117—21no—
272C2×#117—1221no—
273C4×D8—618no—
274C2×#107—611no—
275C2×#106—612no—
276C2×#106—612no—
277C2×C4⋊C8—611no—
278C2×#102—611no—
279C2×#102—611no—
280C2×#102—612no—
281C2×#102—612no—
282C2×#97—612no—
283C2×C4×C8—611no—
284#390—612no—
285#390C2×D6112yes—
286#388—22no—
287#388—1222no—
288#384—611no—
289#384C2×D6111yescharacteristic
290#380—21no—
291#380—1221no—
292C4×C4⋊C4—612no—
293C4×C4⋊C4C2×D6112yes—
294C4×C4⋊C4C2×D6111yescharacteristic
295#340—612no—
296#340—612no—
297#336—612no—
298C61#2281311yescharacteristic, complemented, Sylow
299C22×D4—611no—
300C22×D4C22×D6111yescharacteristic
301C23×C4—21no—
302C23×C4—1221no—
303C2×D8—614no—
304C2×D8—1222no—
305C2×D8—1222no—
306C22×C8—611no—
307C22×C8—611no—
308C22×C8—612no—
309C4×D4—22no—
310C4×D4—614no—
311C4×D4—1222no—
312C4×D4C4×D6114yes—
313#117—44no—
314#117—2444no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C4⋊C4C22⋊D6111yescharacteristic
323C2×C4⋊C4C22×D6112yes—
324C2×C4⋊C4C22×D6111yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—1221no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1222no—
331C2×C42—22no—
332C2×C42—1222no—
333C2×C42C22×D6112yes—
334C2×C42C22×D6111yescharacteristic
335#107—614no—
336#106—1224no—
337#106—2442no—
338C4⋊C8—1222no—
339#102—614no—
340#102—614no—
341#102—2442no—
342#102—2442no—
343#97—1224no—
344C4×C8—614no—
345#95—22no—
346#95—1222no—
347#95C22⋊D6112yes—
348C24—21no—
349C24—1221no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—612no—
356C2×D4—1221no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—1221no—
361C2×D4C2×C4×D6112yes—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—1221no—
368C22×C4—1224no—
369C22×C4—1222no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4C2×C22⋊D6112yes—
373C22×C4C2×C22⋊D6112yes—
374C22×C4C2×C22⋊D6111yescharacteristic
375C22×C4C2×C22⋊D6111yescharacteristic
376C22×C4C23×D6111yescharacteristic
377D8—1228no—
378C2×C8—612no—
379C2×C8—612no—
380C2×C8—612no—
381C2×C8—612no—
382C2×C8—1221no—
383C2×C8—1221no—
384C2×C8—1222no—
385C2×C8—1222no—
386C2×C8—1222no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—42no—
393C4⋊C4—81no—
394C4⋊C4—612no—
395C4⋊C4—1221no—
396C4⋊C4—1222no—
397C4⋊C4—1222no—
398C4⋊C4—4881no—
399C4⋊C4—2442no—
400C4⋊C4C2×C4×D6112yes—
401C22⋊C4—24no—
402C22⋊C4—44no—
403C22⋊C4—82no—
404C22⋊C4—1224no—
405C22⋊C4—4882no—
406C22⋊C4—2444no—
407C42—22no—
408C42—22no—
409C42—44no—
410C42—2444no—
411C42C2×C4×D6112yes—
412C42#1175822212yes—
413C23—22no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—1222no—
421C23—2441no—
422C23—2441no—
423C23—2441no—
424C23—2441no—
425C23—2441no—
426C23—2441no—
427C23C22×C22⋊D6111yescharacteristic
428D4—22no—
429D4—22no—
430D4—41no—
431D4—41no—
432D4—1222no—
433D4—1222no—
434D4—2441no—
435D4—2441no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—1222no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4—2448no—
458C2×C4C4×C22⋊D6112yes—
459C2×C4C4×C22⋊D6112yes—
460C2×C4#1451980212yes—
461C2×C4#1451980312yes—
462C2×C4C22×C4×D6111yescharacteristic
463C2×C4#41×D6111yescharacteristic
464C2×C4#41×D6111yescharacteristic
465C2×C4C2×#1175822211yescharacteristic
466C2×C4#1451982311yescharacteristic
467C2×C4#1451982411yescharacteristic
468C8—614no—
469C8—1222no—
470C8—2442no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—4881no—
479C22—4881no—
480C22—4881no—
481C22—4881no—
482C22—2442no—
483C22—2442no—
484C22—2442no—
485C22C2×C4×C22⋊D6111yescharacteristic
486C22#1684041111yescharacteristic
487C22#1684043111yescharacteristic
488C22C2×#1451980211yescharacteristic
489C22C2×#1451980311yescharacteristic
490C22#1684057011yescharacteristic
491C22#1684057311yescharacteristic
492C4—22no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—4882no—
498C4—2442no—
499C4#1683987214yes—
500C4#117×D6111yescharacteristic
501C4#1684024211yescharacteristic
502C4#1684047511yescharacteristic
503C4#1684048011yescharacteristic
504C2—82no—
505C2—4882no—
506C2#2144876911yescharacteristic
507C2C2×#1683987211yescharacteristic
508C2#2145470011yescharacteristic
509C2#2145840811yescharacteristic
510C2#2145841411yescharacteristic
511C2#2145883511yescharacteristic
512C2#2145883911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2354C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenter#2354D61
hypocenterC61#22813
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #21448766C2semidirect
4. #21446446C2semidirect
5. #21446443C2semidirect
6. #21445975C2semidirect
7. #21445819C2semidirect
8. #21445816C2semidirect
9. #21445717C2semidirect
10. #21445688C2semidirect
11. C61×#2354C2semidirect
15. C4×#11758199C4semidirect
213. #92418#117semidirect
250. #2354D61semidirect
298. C61#22813semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117581992
#214487662
#214466942
#214464462
#214464432
#214459752
#214458192
#214458162
#214457172
#214456882
C61×#23542
#2281361