Finite Group Databasebeta

← #29656160

Subgroups — #29656160

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
1#21456215C211yesmaximal, characteristic, complemented
2#21455941C211yesmaximal, characteristic, complemented
3#21455910C211yesmaximal, characteristic
4#21444984C214yesmaximal, complemented
5#21444922C214yesmaximal, complemented
6#21444536C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic, complemented
9C2×D4×Q8×C61C211yesmaximal, characteristic, complemented
10#16840312C2211yescharacteristic
11#16840124—22no—
12#16840114C2212yescomplemented
13#16840110C2212yescomplemented
14#16840080—21no—
15#16840075—21no—
16#16840057C2212yescomplemented
17#16840052C2212yes—
18#16839962C2212yescomplemented
19#16839957C2212yes—
20#16839901—21no—
21#16839896—21no—
22#16839872—22no—
23#16839860—22no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839761—22no—
27C4×#11758194C2211yescharacteristic
28#16839508C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C23×Q8×C61—21no—
32D4×Q8×C61—22no—
33D4×Q8×C61—22no—
34D4×Q8×C61—22no—
35D4×Q8×C61C2214yes—
36C2×C4⋊Q8×C61—21no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×C22⋊Q8×C61—21no—
39C2×C22⋊Q8×C61—22no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758222—21no—
44C2×#11758201—22no—
45C2×#11758199—22no—
46C2×#11758194—21no—
47#14519777C2311yescharacteristic
48#14519774C2311yescharacteristic
49#14519765—21no—
50#14519751—21no—
51#14519751—21no—
52C4×D244—21no—
53C4×D244—21no—
54#14519710—22no—
55#14519708—22no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519701—22no—
59#14519700—24no—
60#14519698—22no—
61#14519697—22no—
62#14519697—22no—
63#14519697—24no—
64C22⋊#92418—21no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688—21no—
68#14519688—21no—
69C4×#92418C2312yes—
70C22×Q8×C61—22no—
71C22×Q8×C61—24no—
72C22×Q8×C61—24no—
73C22×Q8×C61—21no—
74C22×Q8×C61—48no—
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—22no—
78C23×C4×C61—21no—
79C4⋊Q8×C61—24no—
80C4⋊Q8×C61—24no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61D412yes—
84C4⋊Q8×C61C2312yes—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—28no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—28no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61D412yes—
92C4×Q8×C61C2312yes—
93C4×D4×C61—24no—
94C4×D4×C61—24no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D412yes—
98C4×D4×C61C2312yes—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D412yes—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C2×Q8×C61—24no—
111C2×Q8×C61—28no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—48no—
116C2×Q8×C61—44no—
117C2×Q8×C61—416no—
118C2×Q8×C61—48no—
119C2×Q8×C61—48no—
120C2×Q8×C61C2×D412yes—
121C2×D4×C61—24no—
122C2×D4×C61—21no—
123C2×D4×C61C2×D412yes—
124C22×C4×C61—21no—
125C22×C4×C61—28no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—48no—
132C22×C4×C61—44no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C2×C22⋊D61—41no—
137#11758222—44no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×C4×D61—41no—
142C2×Dic122—21no—
143C4⋊C4×C61—28no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D414yes—
154C22⋊C4×C61—28no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2411yescharacteristic
163#11758201—44no—
164#11758199—44no—
165#11758198—41no—
166#11758198—41no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169#11758194—42no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—24no—
174C23×C61—22no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61#12011yescharacteristic
178C22×D61—41no—
179Q8×C61—24no—
180Q8×C61—22no—
181Q8×C61—416no—
182Q8×C61—44no—
183Q8×C61—44no—
184Q8×C61AGL(1,ℤ/8)14yes—
185D4×C61—24no—
186D4×C61—22no—
187D4×C61#13714yes—
188C2×C4×C61—24no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—416no—
197C2×C4×C61—44no—
198C2×C4×C61—44no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—44no—
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C22⋊D61—82no—
209C2×Dic61—41no—
210D244—42no—
211D244—42no—
212D244—42no—
213C4×D61—82no—
214Dic122—42no—
215#92418—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#45412yes—
226C22×C61C2×#12011yescharacteristic
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C4×C61—24no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—44no—
236C4×C61—42no—
237C4×C61—42no—
238C4×C61C2×AGL(1,ℤ/8)12yes—
239C4×C61C2×#13712yes—
240Dic61—81no—
241C2×D4×Q8D6111yescharacteristic, complemented
242#3141—611no—
243#3112—611no—
244#3109—611no—
245#1637—614no—
246#1627—614no—
247#1556—611no—
248#1491—611no—
249#1490—611no—
250C2×C61—22no—
251C2×C61—44no—
252C2×C61#300311yescharacteristic
253C2×C61#300411yescharacteristic
254C2×C61C2×#45411yescharacteristic
255D61—161no—
256C23×Q8—21no—
257D4×Q8—22no—
258D4×Q8—22no—
259D4×Q8—22no—
260D4×Q8C2×D6114yes—
261#532—611no—
262C2×C4⋊Q8—21no—
263C2×C4⋊Q8C2×D6111yescharacteristic
264C2×C22⋊Q8—21no—
265C2×C22⋊Q8—22no—
266C2×C4×Q8C2×D6111yescharacteristic
267C2×C4×D4—21no—
268C2×C4×D4C2×D6111yescharacteristic
269#489—1222no—
270#487—612no—
271#486—612no—
272#481—1221no—
273#480—1221no—
274#477—612no—
275#476—612no—
276#460—612no—
277#459—612no—
278#448—1221no—
279#447—1221no—
280#442—1222no—
281#440—1222no—
282#432—611no—
283#432—611no—
284#417—1222no—
285C4×M16—611no—
286#332—614no—
287Q8⋊C8—614no—
288D4⋊C8—614no—
289C61#1853411yescharacteristic, complemented, Sylow
290C2×#41—1221no—
291C22×Q8—22no—
292C22×Q8—24no—
293C22×Q8—24no—
294C22×Q8—21no—
295C22×Q8—48no—
296C22×Q8C22⋊D6111yescharacteristic
297C22×D4C22⋊D6111yescharacteristic
298C23×C4—22no—
299C23×C4—21no—
300C2×SD16—1222no—
301C2×D8—1222no—
302C2×M16—1221no—
303C4⋊Q8—24no—
304C4⋊Q8—24no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—611no—
308C4⋊Q8C22⋊D6112yes—
309C4⋊Q8C22×D6112yes—
310#127—611no—
311#124—1221no—
312C22⋊Q8—24no—
313C22⋊Q8—28no—
314C22⋊Q8—24no—
315C22⋊Q8—28no—
316#121—1221no—
317#121—1221no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8C22⋊D6112yes—
321C4×Q8C22×D6112yes—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—1221no—
327C4×D4—1221no—
328C4×D4C22⋊D6112yes—
329C4×D4C22×D6112yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C22⋊D6111yescharacteristic
334C2×C4⋊C4C22⋊D6111yescharacteristic
335C2×C4⋊C4C22⋊D6112yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C42—21no—
339C2×C42C22×D6111yescharacteristic
340#107—1222no—
341#106—1222no—
342C4⋊C8—612no—
343C4⋊C8—612no—
344#103—1222no—
345#103—1224no—
346#102—1222no—
347#102—1224no—
348#102—1222no—
349#102—1222no—
350C22⋊C8—1221no—
351C22⋊C8—1221no—
352#97—1221no—
353#97—1221no—
354#97—1221no—
355C4×C8—612no—
356C24—21no—
357#41—2444no—
358C2×Q8—24no—
359C2×Q8—28no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8—48no—
364C2×Q8—44no—
365C2×Q8—416no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—1221no—
369C2×Q8C2×C22⋊D6112yes—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—2441no—
376C2×D4C2×C22⋊D6112yes—
377C22×C4—21no—
378C22×C4—28no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4—2441no—
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C2×C22⋊D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390SD16—2444no—
391D8—2444no—
392M16—2442no—
393C2×C8—1222no—
394C2×C8—1222no—
395C2×C8—1222no—
396C4⋊C4—28no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4C2×C22⋊D6114yes—
407C4⋊C4C2×C22⋊D6112yes—
408C4⋊C4C2×C22⋊D6112yes—
409C22⋊C4—28no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—22no—
415C42—22no—
416C42C2×C22⋊D6111yescharacteristic
417C42C2×C22⋊D6111yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C42C23×D6111yescharacteristic
420C23—24no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—2441no—
425C23#1451974611yescharacteristic
426Q8—24no—
427Q8—22no—
428Q8—416no—
429Q8—44no—
430Q8—44no—
431Q8—2442no—
432Q8#1451980314yes—
433D4—24no—
434D4—22no—
435D4—4882no—
436D4—2442no—
437D4—2442no—
438D4—2442no—
439D4#1451980814yes—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—416no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—4882no—
455C2×C4—2441no—
456C2×C4#1451974611yescharacteristic
457C2×C4#1451974611yescharacteristic
458C2×C4#1451974611yescharacteristic
459C2×C4C22×C22⋊D6111yescharacteristic
460C2×C4C22×C22⋊D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C8—2442no—
463C8—2442no—
464C8—2442no—
465C22—21no—
466C22—21no—
467C22—24no—
468C22—48no—
469C22—42no—
470C22—42no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22#1683993512yes—
475C22C2×#1451974611yescharacteristic
476C4—21no—
477C4—21no—
478C4—24no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—44no—
483C4—42no—
484C4—4881no—
485C4C2×#1451980312yes—
486C4C2×#1451980812yes—
487C2—22no—
488C2—44no—
489C2—9761no—
490C2#2145504611yescharacteristic
491C2#2145505311yescharacteristic
492C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×D4×Q8×C61C2
socleC22×C61C2×#120
prime radical#21455910C2
hypercenterC2×D4×Q8D61
hypocenterC61#18534
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #21456215C2semidirect
2. #21455941C2semidirect
4. #21444984C2semidirect
5. #21444922C2semidirect
6. #21444536C2semidirect
7. #21444266C2semidirect
8. #21444262C2semidirect
9. C2×D4×Q8×C61C2semidirect
12. #16840114C22semidirect
13. #16840110C22semidirect
16. #16840057C22semidirect
18. #16839962C22semidirect
28. #16839508C22semidirect
29. Q8⋊#92418C22semidirect
241. C2×D4×Q8D61semidirect
289. C61#18534semidirect

Maximal subgroups

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

subgroupindex
#214562152
#214559412
#214559102
#214449842
#214449222
#214445362
#214442662
#214442622
C2×D4×Q8×C612
#1853461