Finite Group Databasebeta

← #29658831

Subgroups — #29658831

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic
2#21456960C211yesmaximal, characteristic, complemented
3#21455854C211yesmaximal, characteristic, complemented
4#21445088C214yesmaximal, complemented
5#21444989C214yesmaximal, complemented
6#21444492C211yesmaximal, characteristic, complemented
7#21444247C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic
9C61×#3077C211yesmaximal, characteristic
10#16840482—22no—
11#16840482—22no—
12#16840482—22no—
13#16840482C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519770—22no—
16C2×#14519764—21no—
17C2×#14519756—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519742—21no—
20C2×#14519736—21no—
21#16840158C2212yescomplemented
22#16840112C2212yescomplemented
23#16840095—21no—
24#16840090—22no—
25#16840054C2212yescomplemented
26#16840047—22no—
27#16839969C2212yescomplemented
28#16839917—21no—
29C4×#11758202—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839509C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#478—22no—
38C61×#476C2212yes—
39C61×#474C2212yes—
40C61×#428—21no—
41C61×#423C2211yescharacteristic
42C61×#418—21no—
43C22×Dic122—21no—
44C22×Dic122D411yescharacteristic
45C2×#11758202—22no—
46#14519777—21no—
47#14519777—21no—
48#14519777D412yes—
49#14519777C2312yes—
50#14519770—28no—
51#14519770—28no—
52#14519764—24no—
53#14519764—24no—
54#14519756—24no—
55#14519756—24no—
56C4×Dic122—21no—
57C4×Dic122—21no—
58C4×Dic122D412yes—
59C4×Dic122C2312yes—
60#14519742—24no—
61#14519742—24no—
62#14519736—24no—
63#14519736—24no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61D412yes—
67C2×#11758196—22no—
68C2×#11758196—21no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61—21no—
73#14519710—22no—
74#14519708—24no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77#14519701—22no—
78#14519701—24no—
79#14519700—22no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C4×#92418—21no—
83C4×#92418C2312yes—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C2×M16×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61D412yes—
97C4×D4×C61C2312yes—
98#117×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103#106×C61—22no—
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106#102×C61—22no—
107#102×C61—22no—
108C24×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61C2×D412yes—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—44no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C2×Dic122—22no—
126C2×Dic122—21no—
127C2×Dic122—22no—
128C2×Dic122—42no—
129C2×Dic122—42no—
130C2×Dic122—42no—
131C2×Dic122C2×D412yes—
132M16×C61—42no—
133C2×C8×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—41no—
145C42×C61—41no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2411yescharacteristic
150C22⋊Dic61—28no—
151C22⋊Dic61—28no—
152#11758202—44no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61C2×D414yes—
158C4⋊Dic61C2×D412yes—
159#11758196—28no—
160#11758196—22no—
161#11758196—22no—
162C4×Dic61—28no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61—22no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61#12011yescharacteristic
172Q8×C61—42no—
173Q8×C61—42no—
174D4×C61—22no—
175D4×C61—22no—
176D4×C61—42no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—44no—
181C2×C4×C61—42no—
182C2×C4×C61—44no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#13512yes—
189C2×C4×C61#13712yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—24no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—44no—
199Dic122—22no—
200Dic122—22no—
201Dic122—42no—
202Dic122—42no—
203Dic122—42no—
204Dic122—44no—
205C8×C61—42no—
206#92418—42no—
207#92418—42no—
208#18781—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—42no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—84no—
215C22×C61#45212yes—
216C22×C61C2×#12011yescharacteristic
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—42no—
221C4×C61—81no—
222C4×C61—81no—
223C4×C61#45212yes—
224C4×C61C2×#13511yescharacteristic
225C4×C61C2×#13711yescharacteristic
226C4×C61#57512yes—
227Dic61—44no—
228Dic61—42no—
229Dic61—42no—
230C2×#556—611no—
231#3229—611no—
232#3104—611no—
233#3077D6111yescharacteristic
234#1652—614no—
235#1638—614no—
236#1546—611no—
237#1487—611no—
238#1478—611no—
239C2×C61—22no—
240C2×C61—82no—
241C2×C61C2×#45211yescharacteristic
242C2×C61#300511yescharacteristic
243C2×C61#302011yescharacteristic
244#556—614no—
245#556—1222no—
246#556—1222no—
247#556—1222no—
248#533C2×D6111yescharacteristic
249C2×C4⋊Q8—611no—
250C2×#126—1222no—
251C2×#124—1221no—
252C2×C22⋊Q8—1221no—
253C2×C4×Q8—611no—
254C2×C4×Q8—1221no—
255C2×C4×D4C2×D6111yescharacteristic
256C2×#117—1221no—
257#498—612no—
258#487—612no—
259#484—1221no—
260#483—1222no—
261#478—22no—
262#477—612no—
263#476C2×D6112yes—
264#475—1222no—
265#474C2×D6112yes—
266#462—612no—
267#451—1221no—
268C4×Q16—1222no—
269#431—611no—
270#431—611no—
271#428—21no—
272#423C2×D6111yescharacteristic
273#418—21no—
274#332—614no—
275Q8⋊C8—614no—
276D4⋊C8—614no—
277C61#1878111yescharacteristic, complemented, Sylow
278C22×Q8—611no—
279C22×Q8—1221no—
280C22×D4C22⋊D6111yescharacteristic
281C23×C4—21no—
282C2×Q16—1222no—
283C2×M16—21no—
284C4⋊Q8—612no—
285C4⋊Q8—612no—
286C4⋊Q8—1221no—
287C4⋊Q8—1221no—
288C4⋊Q8C22×D6111yescharacteristic
289C4⋊Q8C22×D6111yescharacteristic
290#126—1228no—
291#126—1228no—
292#125—21no—
293#124—1224no—
294#124—1224no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—1224no—
298C22⋊Q8—1224no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—612no—
302C4×Q8—612no—
303C4×Q8—1224no—
304C4×Q8—1224no—
305C4×Q8—1221no—
306C4×Q8—1221no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4C22⋊D6112yes—
310C4×D4C22×D6112yes—
311#117—21no—
312#117—1224no—
313#117—1224no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4C22⋊D6111yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C42—1222no—
324C2×C42—1221no—
325C2×C42C22×D6111yescharacteristic
326#107—22no—
327#107—1222no—
328#106—22no—
329#106—1224no—
330C4⋊C8—21no—
331C4⋊C8—612no—
332C4⋊C8—1221no—
333C4⋊C8C22×D6112yes—
334#103—1222no—
335#103—1222no—
336#103—1224no—
337#102—22no—
338#102—22no—
339C22⋊C8—1221no—
340C22⋊C8—1221no—
341C4×C8—612no—
342C4×C8—1221no—
343C24—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—612no—
347C2×Q8—1222no—
348C2×Q8—1221no—
349C2×Q8—1222no—
350C2×Q8—2442no—
351C2×Q8—2442no—
352C2×Q8—2442no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4C2×C22⋊D6112yes—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—44no—
361C22×C4—1221no—
362C22×C4—1222no—
363C22×C4—1221no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C2×C22⋊D6111yescharacteristic
367Q16—2444no—
368M16—42no—
369C2×C8—22no—
370C2×C8—1222no—
371C2×C8—1222no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—614no—
380C4⋊C4—612no—
381C4⋊C4—1228no—
382C4⋊C4—1222no—
383C4⋊C4—1222no—
384C4⋊C4—1221no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C4⋊C4—1222no—
388C4⋊C4C2×C22⋊D6112yes—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—41no—
392C22⋊C4—1228no—
393C22⋊C4—1228no—
394C42—41no—
395C42—1228no—
396C42—1222no—
397C42—1222no—
398C42D4×D6111yescharacteristic
399C42D4×D6111yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—22no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23#1451974511yescharacteristic
407Q8—42no—
408Q8—42no—
409Q8—1222no—
410Q8—1222no—
411Q8—2442no—
412Q8—2442no—
413Q8—2442no—
414Q8—2444no—
415D4—22no—
416D4—22no—
417D4—42no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—1222no—
427C2×C4—1224no—
428C2×C4—1222no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2444no—
432C2×C4#1451974511yescharacteristic
433C2×C4#1451974511yescharacteristic
434C2×C4#1451974511yescharacteristic
435C2×C4#1451979712yes—
436C2×C4#1451980812yes—
437C2×C4C2×D4×D6111yescharacteristic
438C2×C4C2×D4×D6111yescharacteristic
439C2×C4C22×C22⋊D6111yescharacteristic
440C8—42no—
441C8—2442no—
442C8—2442no—
443C22—21no—
444C22—21no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—84no—
449C22#1683992212yes—
450C22C2×#1451974511yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—2442no—
458C4—2444no—
459C4—2442no—
460C4#1683992212yes—
461C4C2×#1451979711yescharacteristic
462C4C2×#1451980811yescharacteristic
463C4#1684055812yes—
464C2—22no—
465C2—82no—
466C2C2×#1683992211yescharacteristic
467C2#2145505811yescharacteristic
468C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3077C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3077D61
hypocenterC61#18781
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #21456960C2semidirect
3. #21455854C2semidirect
4. #21445088C2semidirect
5. #21444989C2semidirect
6. #21444492C2semidirect
7. #21444247C2semidirect
21. #16840158C22semidirect
22. #16840112C22semidirect
25. #16840054C22semidirect
27. #16839969C22semidirect
32. #16839509C22semidirect
33. #16839492C22semidirect
277. C61#18781semidirect

Maximal subgroups

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

subgroupindex
C2×#168404822
#214569602
#214558542
#214450882
#214449892
#214444922
#214442472
#214442102
C61×#30772
#1878161