Finite Group Databasebeta

← #29656215

Subgroups — #29656215

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840427C211yesmaximal, characteristic
2#21455945C211yesmaximal, characteristic, complemented
3#21455914C211yesmaximal, characteristic
4#21445079C214yesmaximal, complemented
5#21445065C214yesmaximal, complemented
6#21444536C211yesmaximal, characteristic, complemented
7#21444267C211yesmaximal, characteristic, complemented
8#21444263C211yesmaximal, characteristic
9C61×#3165C211yesmaximal, characteristic, complemented
10C22×Q8×D61—21no—
11#16840427—22no—
12#16840427—22no—
13#16840427—22no—
14#16840427C2214yes—
15C2×#14519776—21no—
16C2×#14519755—22no—
17C2×#14519753—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519739—21no—
21#16840095—21no—
22#16840090—21no—
23#16840044C2212yescomplemented
24#16840039C2212yes—
25#16839969C2212yescomplemented
26#16839964C2212yes—
27#16839917—21no—
28#16839912—21no—
29#16839867—22no—
30#16839860—22no—
31#16839819C2211yescharacteristic
32#16839819C2211yescharacteristic
33#16839508C2214yescomplemented
34#16839494C2214yescomplemented
35#16839491C2214yes—
36C61×#533C2211yescharacteristic
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#501—22no—
39C61×#500C2212yescomplemented
40C61×#498C2212yescomplemented
41C61×#428—22no—
42C4×M16×C61C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122D411yescharacteristic
45C2×Q8×D61—24no—
46C2×Q8×D61—24no—
47C2×Q8×D61—48no—
48C22×D244D411yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C2×#11758202—22no—
52C2×#11758201—22no—
53#14519776—24no—
54#14519776—24no—
55#14519755—28no—
56#14519755—28no—
57#14519753—24no—
58#14519753—24no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61C4×Dic122D412yes—
62C4×Dic122C2312yes—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244D412yes—
66C4×D244C2312yes—
67#14519739—24no—
68#14519739—24no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61D411yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×#11758198—22no—
74C2×#11758198—21no—
75C2×C4×Dic61—21no—
76#14519710—22no—
77#14519708—22no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2312yes—
80#14519701—22no—
81#14519701—22no—
82#14519700—22no—
83#14519697—22no—
84C22⋊#92418—21no—
85C22⋊#92418—21no—
86#14519688—21no—
87#14519688—21no—
88C22×Q8×C61—21no—
89C2×M16×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61D412yes—
93C4⋊Q8×C61C2311yescharacteristic
94C4⋊Q8×C61C2311yescharacteristic
95C4⋊Q8×C61C2312yes—
96#125×C61—21no—
97C22⋊Q8×C61—21no—
98C22⋊Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61—21no—
101#117×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D412yes—
104C2×C42×C61C2311yescharacteristic
105#107×C61—24no—
106#106×C61—24no—
107#97×C61—21no—
108C4×C8×C61C2312yes—
109C23×D61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—44no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122Q8×D61—416no—
123Q8×D61—48no—
124Q8×D61—48no—
125C2×D244—21no—
126C2×D244—24no—
127C2×D244C2×D412yes—
128C2×C4×D61—28no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—48no—
132C2×C4×D61—44no—
133C2×Dic122—21no—
134C2×Dic122—24no—
135C2×Dic122—28no—
136C2×Dic122—48no—
137C2×Dic122C2×D412yes—
138M16×C61—24no—
139C2×C8×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61C2×D414yes—
150C22⋊C4×C61—41no—
151C42×C61—41no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2411yescharacteristic
156#11758202—44no—
157#11758201—44no—
158#11758198—28no—
159#11758198—22no—
160#11758198—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61C2×D412yes—
164C4⋊Dic61C2×D412yes—
165#11758196—28no—
166#11758196—22no—
167#11758196—22no—
168C4×Dic61—22no—
169C4×Dic61—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61#12011yescharacteristic
173C22×D61—24no—
174C22×D61—22no—
175C22×D61—42no—
176C22×D61—42no—
177Q8×C61—44no—
178Q8×C61—44no—
179Q8×C61—42no—
180Q8×C61—42no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—24no—
185C2×C4×C61—44no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×Dic61—24no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200D244—24no—
201D244—22no—
202D244#13714yes—
203C4×D61—416no—
204C4×D61—44no—
205C4×D61—44no—
206Dic122—24no—
207Dic122—22no—
208Dic122—416no—
209Dic122AGL(1,ℤ/8)14yes—
210C8×C61—24no—
211#92418—42no—
212#92418—42no—
213#18540—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61#45612yes—
217C22×C61C2×#12011yescharacteristic
218C2×D61—24no—
219C2×D61—48no—
220C2×D61—42no—
221C2×D61—42no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—44no—
227C4×C61—81no—
228C4×C61—81no—
229C4×C61C2×AGL(1,ℤ/8)12yes—
230C4×C61C2×#13712yes—
231Dic61—24no—
232Dic61—42no—
233Dic61—42no—
234C2×D4×Q8—611no—
235#3165D6111yescharacteristic, complemented
236#3113—611no—
237#3110—611no—
238#1651—614no—
239#1649—614no—
240#1556—611no—
241#1491—611no—
242#1490—611no—
243C2×C61—22no—
244C2×C61#300311yescharacteristic
245C2×C61#300411yescharacteristic
246C2×C61C2×#45611yescharacteristic
247D61—44no—
248C23×Q8—1221no—
249D4×Q8—614no—
250D4×Q8—1222no—
251D4×Q8—1222no—
252D4×Q8—1222no—
253#533C2×D6111yescharacteristic
254C2×C4⋊Q8—1221no—
255C2×C4⋊Q8C2×D6111yescharacteristic
256C2×C22⋊Q8—1221no—
257C2×C22⋊Q8—1222no—
258C2×C4×Q8—611no—
259C2×C4×D4—611no—
260C2×C4×D4—1221no—
261#501—22no—
262#500C2×D6112yescomplemented
263#498C2×D6112yescomplemented
264#484—1221no—
265#483—1221no—
266#475—612no—
267#474—612no—
268#462—612no—
269#461—612no—
270#451—1221no—
271#450—1221no—
272#441—1222no—
273#440—1222no—
274#432—611no—
275#432—611no—
276#428—22no—
277C4×M16C2×D6111yescharacteristic
278#332—614no—
279Q8⋊C8—614no—
280D4⋊C8—614no—
281C61#1854011yescharacteristic, complemented, Sylow
282C22×Q8—21no—
283C22×Q8—611no—
284C22×Q8—1222no—
285C22×Q8—1224no—
286C22×Q8—1224no—
287C22×Q8—2448no—
288C22×D4—611no—
289C23×C4—1222no—
290C23×C4—1221no—
291C2×Q16—1222no—
292C2×SD16—1222no—
293C2×M16—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—1224no—
297C4⋊Q8—1224no—
298C4⋊Q8C22⋊D6112yes—
299C4⋊Q8C22×D6111yescharacteristic
300C4⋊Q8C22×D6111yescharacteristic
301C4⋊Q8C22×D6112yes—
302#125—21no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8—1228no—
306C22⋊Q8—1224no—
307C22⋊Q8—1224no—
308C22⋊Q8—1228no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—612no—
312C4×Q8—612no—
313C4×Q8—1221no—
314C4×Q8—1221no—
315C4×D4—612no—
316C4×D4—612no—
317C4×D4—1224no—
318C4×D4—1224no—
319C4×D4—1221no—
320C4×D4—1221no—
321#117—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—611no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—1222no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4C22⋊D6112yes—
328C2×C22⋊C4—1222no—
329C2×C22⋊C4—1221no—
330C2×C42—1221no—
331C2×C42C22×D6111yescharacteristic
332#107—24no—
333#107—1222no—
334#106—24no—
335#106—1222no—
336C4⋊C8—612no—
337C4⋊C8—612no—
338#103—1222no—
339#103—1222no—
340#103—1222no—
341#102—1222no—
342C22⋊C8—1221no—
343C22⋊C8—1221no—
344#97—21no—
345#97—1221no—
346#97—1221no—
347C4×C8C22×D6112yes—
348C24—1221no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—44no—
354C2×Q8—612no—
355C2×Q8—1224no—
356C2×Q8—1228no—
357C2×Q8—1221no—
358C2×Q8—2448no—
359C2×Q8—2448no—
360C2×Q8—24416no—
361C2×Q8—2448no—
362C2×D4—612no—
363C2×D4—1224no—
364C2×D4—1221no—
365C22×C4—22no—
366C22×C4—1228no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1222no—
371C22×C4—1222no—
372C22×C4—2444no—
373C22×C4—2448no—
374C22×C4D4×D6111yescharacteristic
375C22×C4D4×D6111yescharacteristic
376C22×C4C2×C22⋊D6111yescharacteristic
377Q16—2444no—
378SD16—2444no—
379M16—24no—
380C2×C8—22no—
381C2×C8—1222no—
382C2×C8—1222no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—612no—
393C4⋊C4—612no—
394C4⋊C4—1228no—
395C4⋊C4—1222no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C4⋊C4—1222no—
399C4⋊C4C2×C22⋊D6114yes—
400C22⋊C4—41no—
401C22⋊C4—1228no—
402C22⋊C4—1222no—
403C22⋊C4—1222no—
404C42—41no—
405C42—1222no—
406C42—1222no—
407C42D4×D6111yescharacteristic
408C42D4×D6111yescharacteristic
409C42C2×C22⋊D6111yescharacteristic
410C42C23×D6111yescharacteristic
411C23—1224no—
412C23—1222no—
413C23—2442no—
414C23—2442no—
415C23#1451974511yescharacteristic
416Q8—44no—
417Q8—44no—
418Q8—42no—
419Q8—42no—
420Q8—614no—
421Q8—1224no—
422Q8—1222no—
423Q8—24416no—
424D4—614no—
425D4—1224no—
426D4—1222no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—1224no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—2444no—
439C2×C4—24416no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2444no—
443C2×C4#1451974511yescharacteristic
444C2×C4#1451974511yescharacteristic
445C2×C4#1451974511yescharacteristic
446C2×C4C2×D4×D6111yescharacteristic
447C2×C4C2×D4×D6111yescharacteristic
448C2×C4C22×C22⋊D6111yescharacteristic
449C8—24no—
450C8—2442no—
451C8—2442no—
452C22—21no—
453C22—21no—
454C22—1224no—
455C22—2448no—
456C22—2442no—
457C22—2442no—
458C22#1683994412yes—
459C22C2×#1451974511yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—81no—
466C4—81no—
467C4—1224no—
468C4—2442no—
469C4—2442no—
470C4AGL(1,ℤ/8)×D6112yes—
471C4#137×D6112yes—
472C2—22no—
473C2—2444no—
474C2#2145504411yescharacteristic
475C2#2145505111yescharacteristic
476C2C2×#1683994411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3165C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3165D61
hypocenterC61#18540
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
2. #21455945C2semidirect
4. #21445079C2semidirect
5. #21445065C2semidirect
6. #21444536C2semidirect
7. #21444267C2semidirect
9. C61×#3165C2semidirect
23. #16840044C22semidirect
25. #16839969C22semidirect
33. #16839508C22semidirect
34. #16839494C22semidirect
39. C61×#500C22semidirect
40. C61×#498C22semidirect
235. #3165D61semidirect
262. #500C2×D61semidirect
263. #498C2×D61semidirect
281. C61#18540semidirect

Maximal subgroups

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

subgroupindex
C2×#168404272
#214559452
#214559142
#214450792
#214450652
#214445362
#214442672
#214442632
C61×#31652
#1854061