Finite Group Databasebeta

← #29647871

Subgroups — #29647871

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459671C211yesmaximal, characteristic
2#21448729C211yesmaximal, characteristic, complemented
3#21444384C211yesmaximal, characteristic, complemented
4#21444380C211yesmaximal, characteristic, complemented
5#21444363C214yesmaximal, complemented
6#21444324C214yesmaximal, complemented
7#21444319C211yesmaximal, characteristic, complemented
8#21444315C211yesmaximal, characteristic, complemented
9C61×#2975C211yesmaximal, characteristic, complemented
10#16840495—22no—
11#16840482C412yescomplemented
12#16840477C412yescomplemented
13#16840462—22no—
14#16840433C412yescomplemented
15#16840417C412yescomplemented
16#16840350—22no—
17#16840329—21no—
18#16840319C2211yescharacteristic
19#16840312C2211yescharacteristic
20#16840273C2214yes—
21#16840243—21no—
22C2×D4×Dic61—21no—
23#16839661—21no—
24#16839637—22no—
25#16839628C2211yescharacteristic
26#16839622C2212yes—
27C4×C4⋊Dic61C2211yescharacteristic
28C2×#14519684C2212yes—
29#16839500C2212yes—
30#16839500C2214yes—
31#16839500C2212yes—
32#16839497C2212yes—
33#16839497C2214yes—
34#16839497C2212yes—
35C2×C4×D4×C61C2211yescharacteristic
36C61×#436—22no—
37C61×#436—22no—
38C61×#431C2211yescharacteristic
39C61×#431C2211yescharacteristic
40C61×#407C2214yes—
41C2×#11758222—21no—
42C2×#11758222C2×C414yescomplemented
43C23×Dic61—22no—
44#14519777C2×C411yescharacteristic, complemented
45#14519777C2×C411yescharacteristic, complemented
46#14519777C2×C411yescharacteristic, complemented
47#14519774C2×C411yescharacteristic, complemented
48#14519773—22no—
49#14519770—24no—
50#14519769—21no—
51#14519765—21no—
52#14519764—22no—
53#14519763—24no—
54#14519760—24no—
55#14519759—24no—
56#14519756—21no—
57#14519756C2×C414yescomplemented
58#14519756C2312yes—
59#14519751—21no—
60#14519751C2×C414yescomplemented
61#14519751C2312yes—
62#14519750—24no—
63#14519748—22no—
64#14519748—22no—
65#14519746—24no—
66C4×Dic122—21no—
67C4×Dic122C2×C412yescomplemented
68#14519742—22no—
69C4×D244—21no—
70C4×D244C2×C412yescomplemented
71D4×Dic61—44no—
72#14519739—22no—
73C4×C22⋊D61—24no—
74C4×C22⋊D61—24no—
75#14519736—21no—
76#14519736C2×C414yescomplemented
77#14519735—22no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—22no—
81C4⋊C4×D61—22no—
82C2×C22⋊Dic61—22no—
83C2×C4×Dic61—21no—
84#14519684—22no—
85#14519684—21no—
86#14519684C2311yescharacteristic
87#14519684C2312yes—
88#14519684C2311yescharacteristic
89C22×D4×C61D411yescharacteristic
90C23×C4×C61C2312yes—
91C2×M16×C61—24no—
92C4×D4×C61C2×C412yes—
93C4×D4×C61C2×C412yes—
94C4×D4×C61D412yes—
95C4×D4×C61D412yes—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C22⋊C4×C61D412yes—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—21no—
100C4⋊C8×C61—21no—
101C22⋊C8×C61—24no—
102C22⋊C8×C61C2312yes—
103C22⋊C8×C61C2312yes—
104C4×C8×C61—21no—
105C4×C8×C61—21no—
106C24×C61C2×D412yes—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61C22⋊C412yes—
110C2×D4×C61C22⋊C412yes—
111C22×C4×C61—82no—
112C22×C4×C61C22×C414yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D414yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C2×C22⋊D61—22no—
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—44no—
125C22×Dic61—44no—
126C22×Dic61—82no—
127#11758222—42no—
128#11758222—42no—
129#11758222—84no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244C22×C412yes—
133C2×C4×D61—22no—
134C2×C4×D61—24no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122C22×C412yes—
138M16×C61—84no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C4⋊C4×C61C22⋊C412yes—
142C4⋊C4×C61C22⋊C412yes—
143C22⋊C4×C61C22⋊C414yes—
144C22⋊C4×C61C22⋊C414yes—
145C42×C61C22×C411yescharacteristic
146C42×C61C22×C411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C22⋊Dic61—24no—
150C22⋊Dic61—24no—
151C22⋊Dic61—22no—
152C22⋊Dic61—44no—
153#11758198—24no—
154#11758198—22no—
155#11758198—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—42no—
161C4⋊Dic61—42no—
162C4⋊Dic61—82no—
163C4⋊Dic61C22×C412yes—
164C4⋊Dic61C22×C412yes—
165#11758196—24no—
166#11758196—22no—
167#11758196—22no—
168C4×Dic61—24no—
169C4×Dic61—22no—
170C4×Dic61—42no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61C2×C22⋊C414yes—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178D4×C61—24no—
179D4×C61—24no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—44no—
183C2×C4×C61—44no—
184C2×C4×C61—41no—
185C2×C4×C61C2×C22⋊C414yes—
186C2×C4×C61C2×C22⋊C411yescharacteristic
187C2×C4×C61C2×C22⋊C411yescharacteristic
188C2×C4×C61C2×C22⋊C411yescharacteristic
189C2×C4×C61C2×C22⋊C411yescharacteristic
190C2×C4×C61AGL(1,ℤ/8)12yes—
191C2×C4×C61#13712yes—
192C2×C4×C61C23×C411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C22⋊D61—42no—
195C22⋊D61—42no—
196C22⋊D61—84no—
197C22⋊D61—84no—
198C2×Dic61—22no—
199C2×Dic61—22no—
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—41no—
209C2×Dic61—42no—
210C2×Dic61—88no—
211D244—42no—
212D244—42no—
213D244—81no—
214C4×D61—44no—
215C4×D61—84no—
216Dic122—42no—
217Dic122—42no—
218Dic122—81no—
219C8×C61—42no—
220C8×C61—42no—
221#17630—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—24no—
225C22×C61—24no—
226C22×C61—24no—
227C22×C61—44no—
228C22×C61—44no—
229C22×C61C22×C22⋊C411yescharacteristic
230C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C61C2×#13711yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—24no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61#41712yes—
241C4×C61#41712yes—
242Dic61—42no—
243Dic61—42no—
244Dic61—42no—
245Dic61—42no—
246Dic61—81no—
247Dic61—82no—
248#3506—611no—
249#2975D6111yescharacteristic, complemented
250#2349—611no—
251#1518—611no—
252#1517—611no—
253#1513—614no—
254#1504—614no—
255#1503—611no—
256#1502—611no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61—44no—
260C2×C61#288311yescharacteristic
261C2×C61C2×#41711yescharacteristic
262C2×C61#290111yescharacteristic
263D61—82no—
264#559—1222no—
265#556—612no—
266#555—612no—
267#553—1222no—
268#550—612no—
269#547—612no—
270#538—1222no—
271#535—1221no—
272#533—611no—
273#532—611no—
274#525—614no—
275#520—1221no—
276C2×C4×D4—1221no—
277C2×C4×D4C2×D6111yescharacteristic
278#436—22no—
279#436—22no—
280#431C2×D6111yescharacteristic
281#431C2×D6111yescharacteristic
282#407C2×D6114yes—
283#389—1221no—
284#385—1222no—
285#382—611no—
286#380—612no—
287C4×C4⋊C4—611no—
288C2×#95—612no—
289#328—612no—
290#328—612no—
291#328—614no—
292#327—612no—
293#327—612no—
294#327—614no—
295C61#1763011yescharacteristic, complemented, Sylow
296C2×#41—614no—
297C2×#41—1221no—
298C22×D4D24411yescharacteristic
299C23×C4—1222no—
300C23×C4C22×D6112yes—
301C2×M16—24no—
302C4⋊Q8—611no—
303C4⋊Q8—611no—
304C4⋊Q8—611no—
305#127—611no—
306#127—1222no—
307#126—1224no—
308#125—1221no—
309#124—1222no—
310#124—1224no—
311#124—1221no—
312#123—1224no—
313#123—1224no—
314C22⋊Q8—614no—
315C22⋊Q8—612no—
316C22⋊Q8—1221no—
317#121—614no—
318#121—612no—
319#121—1222no—
320#121—1222no—
321#121—1221no—
322#121—1224no—
323#120—1224no—
324C4×Q8—612no—
325C4×Q8—1222no—
326C4×Q8—1221no—
327C4×D4—612no—
328C4×D4—1224no—
329C4×D4—1224no—
330C4×D4—1222no—
331C4×D4—1221no—
332C4×D4—2444no—
333C4×D4C4×D6112yes—
334C4×D4C4×D6112yes—
335C4×D4C22⋊D6112yes—
336C4×D4C22⋊D6112yes—
337#117—614no—
338#117—1221no—
339#117—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4D24411yescharacteristic
345C2×C22⋊C4—1222no—
346C2×C22⋊C4D24412yes—
347C2×C42—1221no—
348C2×C42C22×D6111yescharacteristic
349C4⋊C8—21no—
350C4⋊C8—21no—
351C22⋊C8—24no—
352C22⋊C8C22×D6112yes—
353C22⋊C8C22×D6112yes—
354C4×C8—21no—
355C4×C8—21no—
356#95—611no—
357#95—612no—
358#95—611no—
359#95—1222no—
360#95—1221no—
361C24C2×D24412yes—
362#41—4884no—
363#41—2442no—
364#41—2442no—
365C2×Q8—612no—
366C2×Q8—1221no—
367C2×Q8—1221no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—612no—
371C2×D4—1224no—
372C2×D4—1224no—
373C2×D4—1222no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4#1175819812yes—
377C2×D4#1175819812yes—
378C22×C4—82no—
379C22×C4—1221no—
380C22×C4—1224no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1221no—
384C22×C4—4882no—
385C22×C4—2444no—
386C22×C4—2444no—
387C22×C4C2×C4×D6114yes—
388C22×C4C2×D24411yescharacteristic
389C22×C4C2×D24412yes—
390C22×C4C2×D24411yescharacteristic
391C22×C4C2×C22⋊D6114yes—
392C22×C4C23×D6111yescharacteristic
393M16—84no—
394C2×C8—22no—
395C2×C8—22no—
396C4⋊C4—612no—
397C4⋊C4—612no—
398C4⋊C4—1222no—
399C4⋊C4—1221no—
400C4⋊C4—1224no—
401C4⋊C4—1222no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—4882no—
406C4⋊C4—2442no—
407C4⋊C4—2442no—
408C4⋊C4#1175819812yes—
409C4⋊C4#1175819812yes—
410C22⋊C4—1222no—
411C22⋊C4—1224no—
412C22⋊C4—1224no—
413C22⋊C4—1222no—
414C22⋊C4—1224no—
415C22⋊C4—1222no—
416C22⋊C4—2444no—
417C22⋊C4#1175819814yes—
418C22⋊C4#1175819814yes—
419C42—1224no—
420C42—1222no—
421C42—2442no—
422C42C2×C4×D6111yescharacteristic
423C42C2×C4×D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C42C2×C22⋊D6111yescharacteristic
426C23—22no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—1222no—
431C23C2×#1175819814yes—
432C23C22×D24411yescharacteristic
433Q8—4881no—
434Q8—2442no—
435Q8—2442no—
436D4—24no—
437D4—24no—
438D4—4881no—
439D4—4884no—
440D4—4884no—
441D4—2442no—
442D4—2442no—
443D4—2442no—
444D4—2442no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—4884no—
455C2×C4—4888no—
456C2×C4—2441no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4—2442no—
460C2×C4—2441no—
461C2×C4—2442no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2444no—
465C2×C4C2×#1175819814yes—
466C2×C4C2×#1175819811yescharacteristic
467C2×C4C2×#1175819811yescharacteristic
468C2×C4C2×#1175819811yescharacteristic
469C2×C4C2×#1175819811yescharacteristic
470C2×C4#1451980512yes—
471C2×C4#1451981012yes—
472C2×C4C22×C4×D6111yescharacteristic
473C2×C4C22×C22⋊D6111yescharacteristic
474C8—42no—
475C8—42no—
476C22—21no—
477C22—21no—
478C22—24no—
479C22—24no—
480C22—24no—
481C22—44no—
482C22—44no—
483C22—2442no—
484C22—2442no—
485C22—2442no—
486C22C22×#1175819811yescharacteristic
487C22C2×#1451980511yescharacteristic
488C22C2×#1451981011yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—24no—
494C4—4882no—
495C4—4881no—
496C4—2442no—
497C4—2442no—
498C4—2442no—
499C4—2442no—
500C4#1683976412yes—
501C4#1683976412yes—
502C2—21no—
503C2—21no—
504C2—44no—
505C2—4882no—
506C2#2145427911yescharacteristic
507C2C2×#1683976411yescharacteristic
508C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2975C2
socleC22×C61C22×C22⋊C4
prime radical#21459671C2
cosocleC22×C4×C61C24
hypercenter#2975D61
hypocenterC61#17630
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #21448729C2semidirect
3. #21444384C2semidirect
4. #21444380C2semidirect
5. #21444363C2semidirect
6. #21444324C2semidirect
7. #21444319C2semidirect
8. #21444315C2semidirect
9. C61×#2975C2semidirect
11. #16840482C4semidirect
12. #16840477C4semidirect
14. #16840433C4semidirect
15. #16840417C4semidirect
42. C2×#11758222C2×C4semidirect
44. #14519777C2×C4semidirect
45. #14519777C2×C4semidirect
46. #14519777C2×C4semidirect
47. #14519774C2×C4semidirect
57. #14519756C2×C4semidirect
60. #14519751C2×C4semidirect
67. C4×Dic122C2×C4semidirect
70. C4×D244C2×C4semidirect
76. #14519736C2×C4semidirect
249. #2975D61semidirect
295. C61#17630semidirect

Maximal subgroups

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

subgroupindex
#214596712
#214487292
#214443842
#214443802
#214443632
#214443242
#214443192
#214443152
C61×#29752
#1763061