Finite Group Databasebeta

← #29656208

Subgroups — #29656208

Order 15616.

1178 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic, complemented
3#21455914C211yesmaximal, characteristic, complemented
4#21445101C214yesmaximal, complemented
5#21445053C214yesmaximal, complemented
6#21444535C211yesmaximal, characteristic, complemented
7#21444262C211yesmaximal, characteristic, complemented
8#21444261C211yesmaximal, characteristic, complemented
9C61×#3110C211yesmaximal, characteristic, complemented
10C22×#11758222—21no—
11#16840417—22no—
12#16840417—22no—
13#16840417—22no—
14#16840417C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519759—22no—
17C2×#14519756—21no—
18C2×#14519748—21no—
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16840172—22no—
22#16840165C2212yescomplemented
23#16840158C2212yescomplemented
24#16840095—21no—
25#16840039C2212yescomplemented
26#16839964C2212yescomplemented
27#16839912—21no—
28#16839860—22no—
29#16839819C2211yescharacteristic
30#16839805—22no—
31C4×#11758194C2211yescharacteristic
32#16839507C2214yescomplemented
33D4⋊#92418C2214yescomplemented
34#16839489C2214yescomplemented
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#484—21no—
38C61×#474C2212yescomplemented
39C61×#461C2212yescomplemented
40C61×#450—21no—
41C61×#440—22no—
42C61×#432C2211yescharacteristic
43C2×#11758222—24no—
44C2×#11758222—24no—
45C2×#11758222—48no—
46C22×Dic122—21no—
47C22×D244D411yescharacteristic
48C22×C22⋊D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C2×#11758201—22no—
52C2×#11758194—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777D412yes—
56#14519777C2312yes—
57#14519759—28no—
58#14519759—28no—
59#14519756—24no—
60#14519756—24no—
61#14519748—24no—
62#14519748—24no—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244D412yes—
66C4×D244C2312yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D412yes—
70C2×#11758196—22no—
71C4⋊C4×D61—24no—
72C4⋊C4×D61—24no—
73C2×C22⋊Dic61—22no—
74C2×#11758198—21no—
75#14519710—22no—
76#14519710—24no—
77#14519708—24no—
78C4⋊#92418C2312yes—
79#14519701—22no—
80#14519697—22no—
81C22⋊#92418—21no—
82#14519688—21no—
83#14519688—21no—
84C4×#92418C2312yes—
85C22×D4×C61D411yescharacteristic
86C23×C4×C61—21no—
87C2×SD16×C61—22no—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D412yes—
98C4×D4×C61C2312yes—
99#117×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104C4⋊C8×C61C2312yes—
105#103×C61—22no—
106#102×C61—22no—
107C22⋊C8×C61—21no—
108#97×C61—21no—
109C24×C61—21no—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—42no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—44no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C2×C22⋊D61—28no—
124C2×C22⋊D61—48no—
125C22×Dic61—22no—
126C22×Dic61—21no—
127#11758222—416no—
128#11758222—48no—
129#11758222—48no—
130C2×D244—21no—
131C2×D244—24no—
132C2×D244C2×D412yes—
133C2×C4×D61—28no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—44no—
137C2×C4×D61—48no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—44no—
141SD16×C61—44no—
142C2×C8×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61C2×D412yes—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—41no—
154C42×C61—41no—
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2411yescharacteristic
159C22⋊Dic61—28no—
160#11758201—44no—
161#11758198—22no—
162#11758198—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61C2×D412yes—
168C4⋊Dic61C2×D414yes—
169#11758196—28no—
170#11758194—24no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—22no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61#12011yescharacteristic
178C22×D61—24no—
179C22×D61—22no—
180C22×D61—42no—
181C22×D61—42no—
182Q8×C61—42no—
183Q8×C61—42no—
184D4×C61—22no—
185D4×C61—22no—
186D4×C61—42no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—44no—
192C2×C4×C61—42no—
193C2×C4×C61—44no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C22⋊D61—416no—
203C2×Dic61—22no—
204C2×Dic61—24no—
205C2×Dic61—44no—
206C2×Dic61—42no—
207D244—24no—
208D244—22no—
209D244AGL(1,ℤ/8)14yes—
210C4×D61—44no—
211C4×D61—416no—
212C4×D61—44no—
213Dic122—44no—
214Dic122—44no—
215C8×C61—42no—
216#92418—24no—
217#92418—42no—
218#18539—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—84no—
225C22×C61#45612yes—
226C22×C61C2×#12011yescharacteristic
227C2×D61—24no—
228C2×D61—48no—
229C2×D61—42no—
230C2×D61—42no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—42no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61C2×AGL(1,ℤ/8)12yes—
239C4×C61#57512yes—
240Dic61—44no—
241Dic61—42no—
242C2×#547—611no—
243#3165—611no—
244#3110—611no—
245#3110D6111yescharacteristic, complemented
246#1654—614no—
247#1647—614no—
248#1556—611no—
249#1490—611no—
250#1490—611no—
251C2×C61—22no—
252C2×C61—82no—
253C2×C61#300311yescharacteristic
254C2×C61#300511yescharacteristic
255C2×C61C2×#45611yescharacteristic
256D61—44no—
257C22×#41—1221no—
258#547—614no—
259#547—1222no—
260#547—1222no—
261#547—1222no—
262#533C2×D6111yescharacteristic
263C2×C4⋊Q8—611no—
264C2×#123—1222no—
265C2×C22⋊Q8—1221no—
266C2×#121—1221no—
267C2×C4×D4—611no—
268C2×C4×D4C2×D6111yescharacteristic
269C22×C4⋊C4—1221no—
270#501—1222no—
271#500—612no—
272#498—612no—
273#484—21no—
274#484—1221no—
275#474—612no—
276#474C2×D6112yescomplemented
277#461—612no—
278#461C2×D6112yescomplemented
279#450—21no—
280#450—1221no—
281#440—22no—
282#440—1222no—
283#432—611no—
284#432C2×D6111yescharacteristic
285#428—1222no—
286C4×M16—611no—
287#332—614no—
288D4⋊C8—614no—
289D4⋊C8—614no—
290C61#1853911yescharacteristic, complemented, Sylow
291C2×#41—1224no—
292C2×#41—1224no—
293C2×#41—2448no—
294C22×Q8—1221no—
295C22×D4—611no—
296C22×D4—1222no—
297C22×D4C22⋊D6111yescharacteristic
298C23×C4—21no—
299C23×C4—1222no—
300C23×C4—1221no—
301C2×SD16—22no—
302C2×SD16—1222no—
303C2×M16—1221no—
304C4⋊Q8—612no—
305C4⋊Q8—612no—
306C4⋊Q8—1221no—
307C4⋊Q8—1221no—
308C4⋊Q8C22×D6111yescharacteristic
309C4⋊Q8C22×D6111yescharacteristic
310#125—21no—
311#123—1228no—
312#123—1228no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C22⋊Q8—1224no—
316C22⋊Q8—1224no—
317#121—1224no—
318#121—1224no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—612no—
324C4×D4—612no—
325C4×D4—1221no—
326C4×D4—1221no—
327C4×D4C22⋊D6112yes—
328C4×D4C22×D6112yes—
329#117—21no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—612no—
332C2×C4⋊C4—1224no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1224no—
336C2×C4⋊C4C22⋊D6111yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1221no—
340C2×C42C22×D6111yescharacteristic
341#107—22no—
342#107—1224no—
343#107—1222no—
344#106—1224no—
345C4⋊C8—612no—
346C4⋊C8C22×D6112yes—
347#103—22no—
348#103—1222no—
349#102—22no—
350#102—1222no—
351C22⋊C8—21no—
352C22⋊C8—1221no—
353#97—21no—
354#97—1221no—
355#97—1221no—
356C4×C8—612no—
357C24—21no—
358C24—1221no—
359#41—2448no—
360#41—24416no—
361#41—2448no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—1222no—
365C2×Q8—1222no—
366C2×Q8—2444no—
367C2×D4—21no—
368C2×D4—42no—
369C2×D4—612no—
370C2×D4—1224no—
371C2×D4—1228no—
372C2×D4—1221no—
373C2×D4—2448no—
374C2×D4C2×C22⋊D6112yes—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—44no—
379C22×C4—1228no—
380C22×C4—1221no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—2444no—
385C22×C4—2448no—
386C22×C4D4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388C22×C4C2×C22⋊D6111yescharacteristic
389SD16—44no—
390SD16—2444no—
391M16—1224no—
392C2×C8—22no—
393C2×C8—1222no—
394C2×C8—1222no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—612no—
403C4⋊C4—614no—
404C4⋊C4—1228no—
405C4⋊C4—1222no—
406C4⋊C4—1221no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4C2×C22⋊D6112yes—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—41no—
413C22⋊C4—1228no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C42—41no—
417C42D4×D6111yescharacteristic
418C42D4×D6111yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—1224no—
426C23—1222no—
427C23—2442no—
428C23—2442no—
429C23#1451974511yescharacteristic
430Q8—42no—
431Q8—42no—
432Q8—2444no—
433Q8—2444no—
434D4—22no—
435D4—22no—
436D4—42no—
437D4—614no—
438D4—1224no—
439D4—1222no—
440D4—24416no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—1222no—
451C2×C4—1224no—
452C2×C4—2444no—
453C2×C4—24416no—
454C2×C4—2442no—
455C2×C4—2444no—
456C2×C4—2444no—
457C2×C4#1451974511yescharacteristic
458C2×C4#1451974511yescharacteristic
459C2×C4#1451974511yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C22×C22⋊D6111yescharacteristic
463C8—42no—
464C8—1224no—
465C8—2442no—
466C22—21no—
467C22—21no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—84no—
472C22—1224no—
473C22—2448no—
474C22—2442no—
475C22—2442no—
476C22#1683994312yes—
477C22C2×#1451974511yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—2444no—
486C4—2442no—
487C4AGL(1,ℤ/8)×D6112yes—
488C4#1684056012yes—
489C2—22no—
490C2—82no—
491C2—2444no—
492C2#2145504411yescharacteristic
493C2#2145505911yescharacteristic
494C2C2×#1683994311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3110C2
socleC22×C61C2×#120
prime radicalC2×#16840417C2
hypercenter#3110D61
hypocenterC61#18539
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
2. #21456366C2semidirect
3. #21455914C2semidirect
4. #21445101C2semidirect
5. #21445053C2semidirect
6. #21444535C2semidirect
7. #21444262C2semidirect
8. #21444261C2semidirect
9. C61×#3110C2semidirect
22. #16840165C22semidirect
23. #16840158C22semidirect
25. #16840039C22semidirect
26. #16839964C22semidirect
32. #16839507C22semidirect
33. D4⋊#92418C22semidirect
34. #16839489C22semidirect
38. C61×#474C22semidirect
39. C61×#461C22semidirect
245. #3110D61semidirect
276. #474C2×D61semidirect
278. #461C2×D61semidirect
290. C61#18539semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
#214563662
#214559142
#214451012
#214450532
#214445352
#214442622
#214442612
C61×#31102
#1853961