Finite Group Databasebeta

← #29656158

Subgroups — #29656158

Order 15616.

1168 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
1C2×D4×Dic122C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3#21455937C211yesmaximal, characteristic, complemented
4#21444983C214yesmaximal, complemented
5#21444921C214yesmaximal, complemented
6#21444535C211yesmaximal, characteristic, complemented
7#21444265C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic, complemented
9C61×#3109C211yesmaximal, characteristic
10C23×Dic122—21no—
11D4×Dic122—22no—
12D4×Dic122—22no—
13D4×Dic122—22no—
14D4×Dic122C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519776—21no—
17C2×#14519756—21no—
18C2×#14519754—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61—21no—
21#16840121—22no—
22#16840112C2212yescomplemented
23#16840108C2212yescomplemented
24#16840073—21no—
25#16840054C2212yescomplemented
26#16839959C2212yescomplemented
27#16839900—21no—
28#16839862—22no—
29#16839819C2211yescharacteristic
30#16839763—22no—
31C4×#11758194C2211yescharacteristic
32#16839507C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C61×#532C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#481—21no—
38C61×#476C2212yes—
39C61×#459C2212yes—
40C61×#447—21no—
41C61×#442—22no—
42C61×#432C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—21no—
46C22×Dic122—24no—
47C22×Dic122—48no—
48C22×Dic122D411yescharacteristic
49C23×Dic61—22no—
50C2×#11758200—22no—
51C2×#11758194—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777D412yes—
55#14519777C2312yes—
56#14519776—24no—
57#14519776—24no—
58#14519756—24no—
59#14519756—24no—
60#14519754—28no—
61#14519754—28no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C4×Dic122D412yes—
65C4×Dic122C2312yes—
66D4×Dic61—24no—
67D4×Dic61—24no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61D412yes—
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—22no—
74C2×C4×Dic61—21no—
75#14519710—22no—
76C4⋊#92418C2312yes—
77#14519701—24no—
78#14519700—22no—
79#14519698—22no—
80#14519698—24no—
81C22⋊#92418—21no—
82#14519688—21no—
83#14519688—21no—
84C4×#92418C2312yes—
85C2×#41×C61—21no—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×D8×C61—22no—
89C4⋊Q8×C61C2311yescharacteristic
90#127×C61C2311yescharacteristic
91#124×C61—21no—
92#121×C61—21no—
93#121×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#106×C61—22no—
104C4⋊C8×C61C2312yes—
105#102×C61—22no—
106#102×C61—22no—
107C22⋊C8×C61—21no—
108#97×C61—21no—
109C24×C61—21no—
110#41×C61—44no—
111C2×Q8×C61—21no—
112C2×D4×C61—24no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—41no—
118C2×D4×C61C2×D412yes—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—44no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—28no—
129C22×Dic61—22no—
130C22×Dic61—21no—
131C22×Dic61—48no—
132C2×Dic122—21no—
133C2×Dic122—24no—
134C2×Dic122—28no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—416no—
138C2×Dic122—48no—
139C2×Dic122—48no—
140C2×Dic122—44no—
141C2×Dic122—48no—
142C2×Dic122C2×D412yes—
143D8×C61—44no—
144C2×C8×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61C2×D412yes—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2411yescharacteristic
157C22⋊Dic61—28no—
158#11758200—44no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61C2×D414yes—
164C4⋊Dic61C2×D412yes—
165#11758196—28no—
166#11758196—22no—
167#11758196—22no—
168C4×Dic61—22no—
169C4×Dic61—22no—
170#11758194—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—24no—
174C23×C61—22no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—41no—
178C23×C61#12011yescharacteristic
179Q8×C61—42no—
180D4×C61—24no—
181D4×C61—22no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—82no—
186D4×C61#13714yes—
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—82no—
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×Dic61—24no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—24no—
206C2×Dic61—416no—
207C2×Dic61—44no—
208C2×Dic61—42no—
209Dic122—24no—
210Dic122—22no—
211Dic122—416no—
212Dic122—44no—
213Dic122—44no—
214Dic122AGL(1,ℤ/8)14yes—
215C8×C61—42no—
216#92418—42no—
217#92418—42no—
218#18534—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—24no—
222C22×C61—48no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61#45412yes—
229C22×C61C2×#12011yescharacteristic
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—42no—
235C4×C61—81no—
236C4×C61C2×AGL(1,ℤ/8)12yes—
237C4×C61C2×#13712yes—
238Dic61—24no—
239Dic61—44no—
240Dic61—42no—
241C2×D4×Q8—611no—
242#3141—611no—
243#3112—611no—
244#3109D6111yescharacteristic
245#1637—614no—
246#1627—614no—
247#1556—611no—
248#1491—611no—
249#1490—611no—
250C2×C61—22no—
251C2×C61—44no—
252C2×C61—161no—
253C2×C61#300311yescharacteristic
254C2×C61#300411yescharacteristic
255C2×C61C2×#45411yescharacteristic
256C23×Q8—1221no—
257D4×Q8—614no—
258D4×Q8—1222no—
259D4×Q8—1222no—
260D4×Q8—1222no—
261#532C2×D6111yescharacteristic
262C2×C4⋊Q8—611no—
263C2×C4⋊Q8—1221no—
264C2×C22⋊Q8—1222no—
265C2×C22⋊Q8—1221no—
266C2×C4×Q8—611no—
267C2×C4×D4—1221no—
268C2×C4×D4C2×D6111yescharacteristic
269#489—1222no—
270#487—612no—
271#486—612no—
272#481—21no—
273#480—1221no—
274#477—612no—
275#476C2×D6112yes—
276#460—612no—
277#459C2×D6112yes—
278#448—1221no—
279#447—21no—
280#442—22no—
281#440—1222no—
282#432—611no—
283#432C2×D6111yescharacteristic
284#417—1222no—
285C4×M16—611no—
286#332—614no—
287Q8⋊C8—614no—
288D4⋊C8—614no—
289C61#1853411yescharacteristic, complemented, Sylow
290C2×#41—21no—
291C22×Q8—611no—
292C22×Q8—1224no—
293C22×Q8—1222no—
294C22×Q8—1224no—
295C22×Q8—1221no—
296C22×Q8—2448no—
297C22×D4C22⋊D6111yescharacteristic
298C23×C4—21no—
299C23×C4—1222no—
300C2×SD16—1222no—
301C2×D8—22no—
302C2×M16—1221no—
303C4⋊Q8—612no—
304C4⋊Q8—612no—
305C4⋊Q8—1224no—
306C4⋊Q8—1224no—
307C4⋊Q8—1221no—
308C4⋊Q8—1221no—
309C4⋊Q8C22×D6111yescharacteristic
310#127C22×D6111yescharacteristic
311#124—21no—
312C22⋊Q8—1228no—
313C22⋊Q8—1228no—
314C22⋊Q8—1224no—
315C22⋊Q8—1224no—
316#121—21no—
317#121—21no—
318C4×Q8—612no—
319C4×Q8—612no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—1224no—
327C4×D4—1224no—
328C4×D4C22⋊D6112yes—
329C4×D4C22×D6112yes—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—612no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4C22⋊D6111yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—1222no—
338C2×C42—1221no—
339C2×C42C22×D6111yescharacteristic
340#107—1222no—
341#106—22no—
342C4⋊C8—612no—
343C4⋊C8C22×D6112yes—
344#103—1222no—
345#103—1224no—
346#102—22no—
347#102—22no—
348#102—1224no—
349#102—1222no—
350C22⋊C8—21no—
351C22⋊C8—1221no—
352#97—21no—
353#97—1221no—
354#97—1221no—
355C4×C8—612no—
356C24—21no—
357#41—44no—
358C2×Q8—21no—
359C2×Q8—612no—
360C2×Q8—1224no—
361C2×Q8—1228no—
362C2×Q8—1222no—
363C2×Q8—1221no—
364C2×Q8—1222no—
365C2×Q8—2444no—
366C2×Q8—2448no—
367C2×Q8—2448no—
368C2×Q8—24416no—
369C2×Q8—2448no—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
376C2×D4C2×C22⋊D6112yes—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—44no—
381C22×C4—41no—
382C22×C4—1221no—
383C22×C4—1228no—
384C22×C4—1221no—
385C22×C4—1222no—
386C22×C4—2448no—
387C22×C4D4×D6111yescharacteristic
388C22×C4D4×D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390SD16—2444no—
391D8—44no—
392M16—2442no—
393C2×C8—22no—
394C2×C8—1222no—
395C2×C8—1222no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—614no—
400C4⋊C4—612no—
401C4⋊C4—1228no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4—1221no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4C2×C22⋊D6112yes—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—1228no—
414C42—1222no—
415C42—1222no—
416C42D4×D6111yescharacteristic
417C42D4×D6111yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C42C23×D6111yescharacteristic
420C23—24no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23#1451974511yescharacteristic
426Q8—42no—
427Q8—614no—
428Q8—1224no—
429Q8—1222no—
430Q8—2444no—
431Q8—24416no—
432Q8—2444no—
433D4—24no—
434D4—22no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—82no—
439D4#1451980714yes—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—82no—
449C2×C4—1224no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1224no—
453C2×C4—24416no—
454C2×C4—2444no—
455C2×C4—2442no—
456C2×C4#1451974511yescharacteristic
457C2×C4#1451974511yescharacteristic
458C2×C4#1451974511yescharacteristic
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C8—42no—
463C8—2442no—
464C8—2442no—
465C22—21no—
466C22—21no—
467C22—24no—
468C22—48no—
469C22—42no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22#1683993312yes—
475C22C2×#1451974511yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—1224no—
483C4—2442no—
484C4—2444no—
485C4AGL(1,ℤ/8)×D6112yes—
486C4C2×#1451980712yes—
487C2—22no—
488C2—44no—
489C2—161no—
490C2#2145504411yescharacteristic
491C2#2145505211yescharacteristic
492C2C2×#1683993311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3109C2
socleC22×C61C2×#120
hypercenter#3109D61
hypocenterC61#18534
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic122C2semidirect
2. #21456212C2semidirect
3. #21455937C2semidirect
4. #21444983C2semidirect
5. #21444921C2semidirect
6. #21444535C2semidirect
7. #21444265C2semidirect
8. #21444262C2semidirect
22. #16840112C22semidirect
23. #16840108C22semidirect
25. #16840054C22semidirect
26. #16839959C22semidirect
32. #16839507C22semidirect
33. #16839492C22semidirect
289. C61#18534semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
#214562122
#214559372
#214449832
#214449212
#214445352
#214442652
#214442622
C61×#31092
#1853461