Finite Group Databasebeta

← #30617433

Subgroups — #30617433

Order 15616.

730 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462608C211yesmaximal, characteristic, complemented
2#21462543C211yesmaximal, characteristic
3#21462007C211yesmaximal, characteristic, complemented
4#21461956C211yesmaximal, characteristic
5#21461913C212yesmaximal, complemented
6#21461912C212yesmaximal, complemented
7#21461745C211yesmaximal, characteristic, complemented
8#21461600C212yesmaximal, complemented
9#21461598C212yesmaximal, complemented
10#21461477C211yesmaximal, characteristic, complemented
11#21456161C211yesmaximal, characteristic
12C2×#14519865C2211yescharacteristic
13C2×#14519864C2211yescharacteristic
14#16840813—21no—
15#16840812—21no—
16#16840811—21no—
17Q8×#92420—21no—
18#16840795—21no—
19#16840795—21no—
20#16840789—21no—
21#16840789—21no—
22#16840769C2212yes—
23#16840767C2211yescharacteristic
24#16840766C2211yescharacteristic
25C2×#14519842C2212yes—
26C2×#14519841C2212yes—
27#16840737C2211yescharacteristic
28#16840736C2211yescharacteristic
29#16840727C2212yes—
30C2×#14519831C2212yes—
31C4×D61⋊C8C2211yescharacteristic
32C2×C8×#6923C2212yes—
33#16840698C2211yescharacteristic
34#16840638C2212yes—
35#16840638C2212yes—
36#16840632C2212yes—
37#16840632C2212yes—
38#16840317C2211yescharacteristic
39#16840311C2211yescharacteristic
40#16840129C412yescomplemented
41#16840125C412yescomplemented
42#16840124C411yescharacteristic, complemented
43#16840122C411yescharacteristic, complemented
44#16840121C411yescharacteristic, complemented
45#16839757C2212yes—
46#16839755C2212yes—
47#97×D61C2211yescharacteristic
48C61×#489C411yescharacteristic, complemented
49#14519865—24no—
50#14519864—24no—
51C2×#11758213—21no—
52C2×#11758213C2312yes—
53C2×#11758213C2311yescharacteristic
54C2×#11758213C2311yescharacteristic
55C2×#11758217—21no—
56C2×D61⋊C8—21no—
57C2×D61⋊C8C2311yescharacteristic
58C2×D61⋊C8C2311yescharacteristic
59C22×#92420—21no—
60C2×C4×#6923C2312yes—
61#14519844—22no—
62#14519844—21no—
63#14519844—21no—
64C4⋊#92420—22no—
65C4⋊#92420—21no—
66C4⋊#92420—21no—
67#14519842—24no—
68#14519841—24no—
69#14519834—21no—
70#14519834—21no—
71C22⋊#92420—21no—
72C22⋊#92420—21no—
73#14519832—22no—
74#14519831—24no—
75C4×#92420—22no—
76C4×#92420D411yescharacteristic
77C4×#92420D411yescharacteristic
78C4×#92420D411yescharacteristic
79C4×#92420D411yescharacteristic
80C8×#6923—24no—
81(C2×#6923)⋊C4—21no—
82(C2×#6923)⋊C4—21no—
83C2×#11758226C2311yescharacteristic
84C2×#11758224C2311yescharacteristic
85C2×C8×D61C2312yes—
86#14519777C2×C411yescharacteristic
87#14519775C2×C411yescharacteristic
88#14519773C2×C411yescharacteristic
89#14519768C2×C412yes—
90#14519765C2×C411yescharacteristic
91#14519757—21no—
92#14519753—21no—
93#14519750—21no—
94#14519750—21no—
95Q8×Dic61—21no—
96#14519742—21no—
97#14519737—21no—
98#14519737—21no—
99#14519735C2312yes—
100C42×D61C2311yescharacteristic
101#14519702C2×C412yes—
102#14519701C2×C412yes—
103#14519700C2×C412yes—
104#14519699C2×C412yes—
105#14519698C2×C412yes—
106#14519697C2×C412yes—
107#14519688C2×C411yescharacteristic
108#14519687C2×C412yes—
109C4⋊Q8×C61C2×C411yescharacteristic
110#124×C61C2×C411yescharacteristic
111#103×C61C2×C412yes—
112#102×C61C2×C412yes—
113#97×C61C2×C411yescharacteristic
114C22×#6923—22no—
115C2×Q8×C61—21no—
116C2×Q8×C61C22×C411yescharacteristic
117C2×D4×C61C22×C411yescharacteristic
118C2×C22⋊D61—21no—
119C2×C22⋊D61—21no—
120C2×C22⋊D61—21no—
121C22×Dic61—21no—
122#11758226—22no—
123#11758226—22no—
124#11758224—22no—
125#11758224—22no—
126C2×D244—21no—
127C2×D244C22×C411yescharacteristic
128C2×C4×D61—21no—
129C2×C4×D61#4111yescharacteristic
130C2×C4×D61#4111yescharacteristic
131C2×C4×D61C2411yescharacteristic
132C2×Dic122—21no—
133C2×Dic122C22×C411yescharacteristic
134#11758217—42no—
135C2×#92420—22no—
136C2×#92420—21no—
137C2×#92420—21no—
138C2×#92420—42no—
139C2×#92420C2×D411yescharacteristic
140C2×#92420C2×D411yescharacteristic
141C2×#92420C2×D411yescharacteristic
142C2×#92420C2×D411yescharacteristic
143C4×#6923—22no—
144C4×#6923#4114yes—
145#11758213—22no—
146#11758213—22no—
147#11758213—22no—
148#11758213—22no—
149#11758213—42no—
150D61⋊C8—22no—
151D61⋊C8—42no—
152D61⋊C8#4114yes—
153C2×C8×C61C22×C412yes—
154C4⋊C4×C61—21no—
155C4⋊C4×C61C22×C412yes—
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C42×C61C22×C411yescharacteristic
159#11758198—22no—
160#11758198—21no—
161#11758198—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61C22×C412yes—
165#11758196—21no—
166#11758196—21no—
167#11758196—22no—
168C4×Dic61—22no—
169C4×Dic61C22×C411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171C4×Dic61C2×D411yescharacteristic
172C2×#92418C22×C412yes—
173C8×D61—22no—
174C8×D61—22no—
175C23×C61—21no—
176C22×D61—21no—
177C22×D61C2×#4111yescharacteristic
178C2×#6923—24no—
179C2×#6923—42no—
180C2×#6923—42no—
181Q8×C61—22no—
182Q8×C61—42no—
183D4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—21no—
186C2×C4×C61C4×D411yescharacteristic
187C2×C4×C61C4×D411yescharacteristic
188C2×C4×C61C23×C411yescharacteristic
189C22⋊D61—42no—
190C22⋊D61—42no—
191C22⋊D61—42no—
192C2×Dic61—22no—
193C2×Dic61—21no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61C4×D411yescharacteristic
197C2×Dic61C4×D411yescharacteristic
198C2×Dic61C22×D411yescharacteristic
199D244—22no—
200D244—42no—
201C4×D61—41no—
202C4×D61—41no—
203C4×D61—41no—
204C4×D61—41no—
205C4×D61C2×#4112yes—
206C4×D61C2×#4112yes—
207Dic122—22no—
208Dic122—42no—
209#92420—21no—
210#92420—21no—
211#92420—41no—
212#92420—41no—
213#92420—42no—
214#92420#12412yes—
215#92420#12412yes—
216C8×C61—22no—
217#92418—22no—
218#21663—611nomaximal, Sylow
219C22×C61—41no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61C2×C4×D411yescharacteristic
223C2×D61—21no—
224C2×D61—21no—
225C2×D61—41no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61#52912yes—
229#6923—44no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—42no—
233C4×C61—41no—
234C4×C61C4×#4112yes—
235Dic61—21no—
236Dic61—21no—
237Dic61—42no—
238Dic61—41no—
239Dic61C2×#12411yescharacteristic
240Dic61#53211yescharacteristic
241#3135—611no—
242#2960—611no—
243#2931—611no—
244#1995—611no—
245#1963—611no—
246#1937—612no—
247#1937—612no—
248#1838—611no—
249#1765—612no—
250#1765—612no—
251#1726—611no—
252C2×C61—81no—
253C2×C61C4×#12411yescharacteristic
254C2×C61#295511yescharacteristic
255C2×C61#295511yescharacteristic
256D61—22no—
257D61—81no—
258C2×#131—611no—
259C2×#131—611no—
260#533—611no—
261#532—611no—
262#491—612no—
263#490—612no—
264#489—611no—
265#489—611no—
266#489—611no—
267#489#692311yescharacteristic, complemented
268#446—1221no—
269#446—1221no—
270C8×Q8—1221no—
271C8×Q8—1221no—
272#436—1221no—
273#436—1221no—
274C8×D4—1221no—
275C8×D4—1221no—
276#424—612no—
277#423—611no—
278#423—611no—
279C2×#104—612no—
280C2×#104—612no—
281#416—612no—
282#416—612no—
283#408—611no—
284#408—611no—
285#405—612no—
286C2×#97—611no—
287C2×#97—612no—
288C2×C4×C8—611no—
289C2×C4×C8—612no—
290#376—611no—
291#341—612no—
292#341—612no—
293#339—612no—
294#339—612no—
295C61#2166311yescharacteristic, complemented, Sylow
296C2×#41—611no—
297C2×#41—611no—
298#131—1224no—
299#131—1224no—
300C2×M16—611no—
301C2×M16—611no—
302C2×M16—612no—
303C2×M16—1221no—
304C2×M16—1221no—
305C22×C8—611no—
306C22×C8—611no—
307C22×C8—612no—
308C22×C8—1221no—
309C22×C8—1221no—
310C4⋊Q8—611no—
311C4⋊Q8—611no—
312C4⋊Q8C2×#692311yescharacteristic
313#127—611no—
314#125—612no—
315#124—611no—
316#124C2×#692311yescharacteristic
317C22⋊Q8—1221no—
318C22⋊Q8—1221no—
319#121—1221no—
320#121—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×D4—1221no—
324C4×D4—1221no—
325#117—612no—
326C2×C42—611no—
327C2×C42—612no—
328C4⋊C8—1221no—
329C4⋊C8—1221no—
330C4⋊C8—1221no—
331C4⋊C8—1221no—
332C4⋊C8—1222no—
333C4⋊C8—1222no—
334#104—1224no—
335#104—1224no—
336#103—612no—
337#103—612no—
338#103—612no—
339#103C2×#692312yes—
340#102—612no—
341#102—612no—
342#102—612no—
343#102C2×#692312yes—
344C22⋊C8—1221no—
345C22⋊C8—1221no—
346C22⋊C8—1221no—
347C22⋊C8—1221no—
348#97—612no—
349#97—611no—
350#97—1224no—
351#97—1222no—
352#97C2×#692311yescharacteristic
353C4×C8—611no—
354C4×C8—611no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8—1224no—
358C4×C8—1222no—
359#95—1221no—
360#95—1221no—
361#41—1222no—
362#41—1222no—
363#41—1222no—
364#41—1222no—
365C2×Q8—21no—
366C2×Q8—611no—
367C2×Q8—1221no—
368C2×Q8C22×#692311yescharacteristic
369C2×D4—611no—
370C2×D4—1221no—
371C2×D4—1221no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4C22×#692311yescharacteristic
375C22×C4—611no—
376C22×C4—611no—
377C22×C4—611no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1222no—
381M16—1222no—
382M16—1222no—
383M16—1222no—
384M16—1222no—
385M16—2442no—
386M16—2442no—
387C2×C8—611no—
388C2×C8—611no—
389C2×C8—614no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8—612no—
393C2×C8—1222no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1221no—
398C2×C8—1221no—
399C2×C8—2442no—
400C2×C8—2442no—
401C2×C8C22×#692312yes—
402C4⋊C4—21no—
403C4⋊C4—612no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4—1221no—
409C4⋊C4C22×#692312yes—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—1222no—
413C22⋊C4—1221no—
414C22⋊C4—1221no—
415C42—611no—
416C42—611no—
417C42—611no—
418C42—614no—
419C42—1222no—
420C42—1222no—
421C42C22×#692311yescharacteristic
422C23—21no—
423C23—611no—
424C23—1221no—
425Q8—22no—
426Q8—42no—
427Q8—1222no—
428Q8—2442no—
429D4—22no—
430D4—1222no—
431D4—2442no—
432D4—2442no—
433D4—2442no—
434D4—2442no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—611no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—611no—
441C2×C4—611no—
442C2×C4—1224no—
443C2×C4—1222no—
444C2×C4—1221no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4D4×#692311yescharacteristic
454C2×C4D4×#692311yescharacteristic
455C2×C4C23×#692311yescharacteristic
456C8—22no—
457C8—612no—
458C8—612no—
459C8—1221no—
460C8—1221no—
461C8—1222no—
462C8—2441no—
463C8—2441no—
464C8—2442no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—612no—
469C22—1221no—
470C22—1221no—
471C22—2441no—
472C22—2441no—
473C22—2441no—
474C22C2×D4×#692311yescharacteristic
475C4—21no—
476C4—21no—
477C4—41no—
478C4—42no—
479C4—611no—
480C4—611no—
481C4—1221no—
482C4—1221no—
483C4—2444no—
484C4—2441no—
485C4—2442no—
486C4#41×#692312yes—
487C2—81no—
488C2—1222no—
489C2—4881no—
490C2#124×#692311yescharacteristic
491C2#2146259111yescharacteristic
492C2#2146259211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#489C4
socleC22×C61C2×C4×D4
prime radical#16840311C22
cosocleC2×C4×D61C24
hypercenter#489#6923
hypocenterC61#21663
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #21462608C2semidirect
3. #21462007C2semidirect
5. #21461913C2semidirect
6. #21461912C2semidirect
7. #21461745C2semidirect
8. #21461600C2semidirect
9. #21461598C2semidirect
10. #21461477C2semidirect
40. #16840129C4semidirect
41. #16840125C4semidirect
42. #16840124C4semidirect
43. #16840122C4semidirect
44. #16840121C4semidirect
48. C61×#489C4semidirect
267. #489#6923semidirect
295. C61#21663semidirect

Maximal subgroups

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

subgroupindex
#214626082
#214625432
#214620072
#214619562
#214619132
#214619122
#214617452
#214616002
#214615982
#214614772
#214561612
#2166361