Finite Group Databasebeta

← #29683479

Subgroups — #29683479

Order 15616.

1082 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839803C212yesmaximal
2C2×#16839784C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic
4C4×C4⋊C4×D61C211yesmaximal, characteristic
5#21446586C212yesmaximal, complemented
6#21446584C212yesmaximal, complemented
7#21446583C211yesmaximal, characteristic, complemented
8#21446582C211yesmaximal, characteristic, complemented
9#21446575C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal
11C61×#1947C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13C2×#14519735C2212yes—
14C2×C4⋊C4×D61C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C42×D61—21no—
17C2×C42×D61C2211yescharacteristic
18#16839803—28no—
19C2×#14519710C2212yes—
20C2×#14519709C2212yes—
21C2×#14519708C2212yes—
22#16839784C418yescomplemented
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519706C2212yes—
25#16839632C2211yescharacteristic
26#16839627C2212yes—
27#16839619—21no—
28#16839619—21no—
29C4⋊C4×Dic61—21no—
30C4⋊C4×Dic61—21no—
31C4⋊C4×Dic61C2211yescharacteristic
32C4⋊C4×Dic61C2211yescharacteristic
33C4×#11758196—21no—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839613—21no—
36#95×D61—21no—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40C2×#107×C61C2212yes—
41C2×C4⋊C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C22×C4×D61—21no—
46C22×C4×D61—21no—
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C2×#11758189—24no—
51C2×#11758189—24no—
52C2×#11758189C2×C418yes—
53C22×#92418C2312yes—
54#14519735—24no—
55#14519735—24no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2312yes—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×#11758196—21no—
63C4⋊C4×D61—22no—
64C4⋊C4×D61—22no—
65C4⋊C4×D61D414yes—
66C4⋊C4×D61Q814yes—
67C2×#11758198—22no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75C42×D61—44no—
76C42×D61C2×C414yes—
77C42×D61Q814yes—
78#14519710—24no—
79#14519709—24no—
80#14519708—24no—
81C4⋊#92418C2×C414yescomplemented
82#14519706C2×C418yescomplemented
83#14519684—21no—
84#14519684—21no—
85#14519684—21no—
86#14519684—22no—
87#14519684—22no—
88C22×C8×C61C2312yes—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2312yes—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61C2311yescharacteristic
96#107×C61—24no—
97C4⋊C8×C61C2×C414yescomplemented
98#95×C61—21no—
99C23×D61C2×D411yescharacteristic
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—22no—
109C22×Dic61C2×D411yescharacteristic
110C22×Dic61C2×D411yescharacteristic
111C22×Dic61C2×D411yescharacteristic
112C2×C4×D61—22no—
113C2×C4×D61—22no—
114C2×C4×D61—24no—
115C2×C4×D61—24no—
116C2×C4×D61—41no—
117C2×C4×D61—41no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C4×D61—42no—
122C2×C4×D61C4⋊C414yes—
123C2×C4×D61C4⋊C414yes—
124C2×C4×D61C22×C412yes—
125C2×C4×D61C2×D412yes—
126C2×C4×D61C2×Q812yes—
127C2×C4×D61C2×Q812yes—
128C2×C4×D61#4112yes—
129C2×C4×D61#4112yes—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61C22×C414yes—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—42no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61C2×D412yes—
140C4⋊C4×C61C2×Q812yes—
141C42×C61—42no—
142C42×C61C22×C412yes—
143C42×C61C2×Q812yes—
144#11758198—44no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—42no—
150C4⋊Dic61—42no—
151C4⋊Dic61C2×D412yes—
152C4⋊Dic61C2×Q812yes—
153#11758196—24no—
154#11758196—24no—
155C4×Dic61—42no—
156C4×Dic61—42no—
157C4×Dic61—42no—
158C4×Dic61—44no—
159C4×Dic61C4⋊C414yes—
160C4×Dic61C4⋊C414yes—
161C4×Dic61C22×C412yes—
162C4×Dic61C2×Q812yes—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418C22×C414yes—
166#11758189—216no—
167C23×C61C22×D411yescharacteristic
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61C2×C4⋊C412yes—
173C22×D61#12112yes—
174C22×D61C22⋊Q812yes—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61C2×C4⋊C412yes—
187C2×C4×C61C2×C4⋊C412yes—
188C2×C4×C61AGL(1,ℤ/8)12yes—
189C2×C4×C61#13712yes—
190C2×C4×C61C23×C411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×Q811yescharacteristic
193C2×C4×C61C22×Q811yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61C2×C4⋊C412yes—
210C2×Dic61C2×C4⋊C412yes—
211C2×Dic61C2×C4⋊C412yes—
212C2×Dic61#12112yes—
213C2×Dic61C22⋊Q812yes—
214C4×D61—28no—
215C4×D61—44no—
216C4×D61—48no—
217C4×D61—48no—
218C4×D61C4×D414yes—
219C4×D61C4×Q814yes—
220C4×D61#12514yes—
221C4×D61C4⋊Q814yes—
222C8×C61—24no—
223#92418—24no—
224#21170—611nomaximal, Sylow
225C22×C61C22×C4⋊C411yescharacteristic
226C22×C61C2×#12111yescharacteristic
227C22×C61C2×C22⋊Q811yescharacteristic
228C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
229C22×C61C2×#13711yescharacteristic
230C22×C61#57511yescharacteristic
231C22×C61#57511yescharacteristic
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—22no—
237C2×D61—22no—
238C2×D61#38914yes—
239C4×C61—22no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61—82no—
243C4×C61#42814yescomplemented
244C4×C61C2×C4×D411yescharacteristic
245C4×C61C2×C4×Q811yescharacteristic
246C4×C61C2×#12511yescharacteristic
247C4×C61C2×C4⋊Q811yescharacteristic
248Dic61—22no—
249Dic61—22no—
250Dic61—42no—
251Dic61—42no—
252Dic61—82no—
253Dic61#38914yes—
254C2×#428—612no—
255C2×#423—611no—
256#2285—611no—
257C2×C4×C4⋊C4—611no—
258#1949—612no—
259#1949—612no—
260#1948—611no—
261#1948—611no—
262#1947—611no—
263#1947D6111yescharacteristic, complemented
264#1736—612no—
265C2×C61C2×#38911yescharacteristic
266C2×C61C2×#42811yescharacteristic
267C2×C61#291111yescharacteristic
268C2×C61#303611yescharacteristic
269C2×C61#303711yescharacteristic
270C2×C61#307711yescharacteristic
271C2×C61#307811yescharacteristic
272D61—24no—
273C22×M16—612no—
274C2×#117—612no—
275C22×C4⋊C4—611no—
276C22×C4⋊C4—611no—
277C22×C42—611no—
278C22×C42—1221no—
279#428—1228no—
280C2×#107—612no—
281C2×#107C2×D6112yes—
282C2×#106—612no—
283C2×#106—612no—
284#423—618no—
285C2×C4⋊C8—612no—
286C2×C4⋊C8—611no—
287C2×C4⋊C8C2×D6111yescharacteristic
288#384—611no—
289#384C2×D6111yescharacteristic
290#382—612no—
291#380—1221no—
292#380—1221no—
293C4×C4⋊C4—611no—
294C4×C4⋊C4—611no—
295C4×C4⋊C4—611no—
296C4×C4⋊C4—1221no—
297C4×C4⋊C4—1221no—
298C4×C4⋊C4—1221no—
299C4×C4⋊C4—1221no—
300C4×C4⋊C4C2×D6111yescharacteristic
301C2×#95—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340C2×D6112yes—
306C61#2117011yescharacteristic, complemented, Sylow
307C23×C4—611no—
308C23×C4—611no—
309C23×C4—611no—
310C23×C4—1221no—
311C23×C4—1221no—
312C2×M16—618no—
313C2×M16—1224no—
314C2×M16—1224no—
315C22×C8—612no—
316C22×C8C22×D6112yes—
317#117—1224no—
318#117—1224no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—611no—
322C2×C4⋊C4—611no—
323C2×C4⋊C4—614no—
324C2×C4⋊C4—614no—
325C2×C4⋊C4—612no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4C22×D6112yes—
333C2×C4⋊C4C22×D6111yescharacteristic
334C2×C4⋊C4C22×D6111yescharacteristic
335C2×C22⋊C4—1222no—
336C2×C42—21no—
337C2×C42—611no—
338C2×C42—611no—
339C2×C42—611no—
340C2×C42—614no—
341C2×C42—614no—
342C2×C42—1222no—
343C2×C42—1221no—
344C2×C42—1221no—
345C2×C42—1221no—
346C2×C42—2444no—
347C2×C42C22×D6111yescharacteristic
348#107—24no—
349#107—1224no—
350#106—1224no—
351#106—1224no—
352C4⋊C8—618no—
353C4⋊C8—614no—
354C4⋊C8C4×D6114yescomplemented
355#95—21no—
356#95—1221no—
357#95—1221no—
358#95—1222no—
359#95—1222no—
360#95—1221no—
361C24—611no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—611no—
366C22×C4—612no—
367C22×C4—612no—
368C22×C4—612no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—612no—
372C22×C4—611no—
373C22×C4—611no—
374C22×C4—614no—
375C22×C4—614no—
376C22×C4—1224no—
377C22×C4—1224no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1222no—
383C22×C4—2441no—
384C22×C4—2441no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4D4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4C23×D6111yescharacteristic
392M16—12216no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—614no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8C2×C4×D6114yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—612no—
406C4⋊C4—612no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—2442no—
414C4⋊C4—2442no—
415C4⋊C4D4×D6112yes—
416C4⋊C4Q8×D6112yes—
417C22⋊C4—2444no—
418C42—42no—
419C42—612no—
420C42—612no—
421C42—614no—
422C42—614no—
423C42—2444no—
424C42—2442no—
425C42—2442no—
426C42—2442no—
427C42C2×C4×D6112yes—
428C42Q8×D6112yes—
429C23—612no—
430C23—612no—
431C23—612no—
432C23—1221no—
433C23—1221no—
434C23—1221no—
435C23—1221no—
436C23C2×D4×D6111yescharacteristic
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—612no—
449C2×C4—612no—
450C2×C4—612no—
451C2×C4—614no—
452C2×C4—614no—
453C2×C4—614no—
454C2×C4—614no—
455C2×C4—612no—
456C2×C4—612no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1228no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—2444no—
465C2×C4—2448no—
466C2×C4—2448no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4C4⋊C4×D6112yes—
475C2×C4C4⋊C4×D6112yes—
476C2×C4#1451980112yes—
477C2×C4#1451980912yes—
478C2×C4C22×C4×D6111yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C2×Q8×D6111yescharacteristic
481C2×C4C2×Q8×D6111yescharacteristic
482C2×C4#41×D6111yescharacteristic
483C2×C4#41×D6111yescharacteristic
484C8—24no—
485C8—1224no—
486C22—614no—
487C22—1222no—
488C22—1222no—
489C22—1222no—
490C22—1222no—
491C22—1222no—
492C22—1222no—
493C22C2×C4⋊C4×D6111yescharacteristic
494C22#121×D6111yescharacteristic
495C22C22⋊Q8×D6111yescharacteristic
496C22C2×#1451980111yescharacteristic
497C22C2×#1451980911yescharacteristic
498C22#1684055811yescharacteristic
499C22#1684056111yescharacteristic
500C4—22no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—614no—
505C4—1222no—
506C4—1222no—
507C4—4882no—
508C4—2442no—
509C4—2442no—
510C4#1683980314yescomplemented
511C4C4×D4×D6111yescharacteristic
512C4C4×Q8×D6111yescharacteristic
513C4#125×D6111yescharacteristic
514C4C4⋊Q8×D6111yescharacteristic
515C2—1224no—
516C2#389×D6111yescharacteristic
517C2C2×#1683980311yescharacteristic
518C2#2145440611yescharacteristic
519C2#2145527811yescharacteristic
520C2#2145529211yescharacteristic
521C2#2145561111yescharacteristic
522C2#2145562511yescharacteristic
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#21170
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
5. #21446586C2semidirect
6. #21446584C2semidirect
7. #21446583C2semidirect
8. #21446582C2semidirect
9. #21446575C2semidirect
11. C61×#1947C2semidirect
22. #16839784C4semidirect
81. C4⋊#92418C2×C4semidirect
82. #14519706C2×C4semidirect
97. C4⋊C8×C61C2×C4semidirect
243. C4×C61#428semidirect
263. #1947D61semidirect
306. C61#21170semidirect
354. C4⋊C8C4×D61semidirect
510. C4#16839803semidirect

Maximal subgroups

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

subgroupindex
C2×#168398032
C2×#168397842
#214482242
C4×C4⋊C4×D612
#214465862
#214465842
#214465832
#214465822
#214465752
#214455372
C61×#19472
#2117061