Finite Group Databasebeta

← #30615538

Subgroups — #30615538

Order 15616.

1502 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462799C211yesmaximal, characteristic
2#21462280C211yesmaximal, characteristic
3#21462279C212yesmaximal, complemented
4#21461790C218yesmaximal, complemented
5#21461517C212yesmaximal, complemented
6#410×D61C211yesmaximal, characteristic
7#16840831—24no—
8#16840830—21no—
9#16840829—21no—
10#16840829C2214yes—
11#41×#6923—24no—
12C2×#14519856—22no—
13C2×#14519856C2211yescharacteristic
14C2×#14519856C2212yes—
15C22×C4×#6923C2211yescharacteristic
16(C2×#6923)⋊Q8—21no—
17#16840714—21no—
18#16840713—24no—
19#16840710C2214yes—
20#16840709C2218yes—
21C4×C4⋊#6923—24no—
22C4×C22⋊#6923—22no—
23#16840639C2218yes—
24C2×#41×D61C2211yescharacteristic
25#117×D61C2212yes—
26#16839736C411yescharacteristic, complemented
27#16839735C412yescomplemented
28#16839734C414yescomplemented
29#99×D61C2214yes—
30C61×#410C411yescharacteristic, complemented
31C23×#6923C2312yes—
32Q8×#6923—28no—
33D4×#6923—28no—
34D4×#6923—216no—
35#14519856—24no—
36#14519856—24no—
37#14519856—22no—
38#14519856—24no—
39#14519856—22no—
40#14519856—22no—
41#14519856—22no—
42#14519856—22no—
43#14519856—22no—
44C2×C4⋊#6923—24no—
45C2×C4⋊#6923—28no—
46C2×C4⋊#6923—21no—
47C2×C4⋊#6923D414yes—
48C2×C22⋊#6923—24no—
49C2×C22⋊#6923—24no—
50C2×C22⋊#6923C2312yes—
51C2×C22⋊#6923C2314yes—
52C2×C4×#6923—24no—
53C2×C4×#6923—22no—
54C2×C4×#6923—21no—
55C2×C4×#6923—21no—
56C2×C4×#6923—22no—
57C2×C4×#6923—21no—
58C2×C4×#6923D414yes—
59(C2×#6923)⋊C4—28no—
60(C2×#6923)⋊C4—24no—
61C2×#11758226Q811yescharacteristic
62C2×#11758222C2×C411yescharacteristic
63C2×#11758222Q812yes—
64C2×#11758224C2×C412yes—
65C2×#11758224Q811yescharacteristic
66#41×D61—22no—
67#41×D61—22no—
68#41×D61—22no—
69#41×D61—22no—
70C2×Q8×D61D411yescharacteristic
71C2×D4×D61D411yescharacteristic
72C2×D4×D61C2312yes—
73C22×C4×D61D412yes—
74C22×C4×D61C2311yescharacteristic
75#14519736C2×C412yescomplemented
76#14519735—22no—
77#14519734—22no—
78#14519733C2×C414yescomplemented
79C4⋊C4×D61—22no—
80C22⋊C4×D61C2314yes—
81C42×D61—22no—
82#14519692C2×C414yescomplemented
83#14519691C2×C418yescomplemented
84C2×#41×C61C2×C411yescharacteristic
85#117×C61C2×C412yescomplemented
86#99×C61C2×C414yescomplemented
87C23×D61C2×D412yes—
88C23×D61C2411yescharacteristic
89C22×#6923—22no—
90C22×#6923—24no—
91C22×#6923—28no—
92C22×#6923—22no—
93C22×#6923—44no—
94C22×#6923C2×D418yes—
95#41×C61—22no—
96#41×C61—22no—
97C2×Q8×C61C4⋊C411yescharacteristic, complemented
98C2×D4×C61C4⋊C411yescharacteristic, complemented
99C2×D4×C61C22×C412yes—
100C22×C4×C61C4⋊C412yescomplemented
101C22×C4×C61C22×C411yescharacteristic
102C2×C22⋊D61C4⋊C412yescomplemented
103C2×C22⋊D61C22×C412yes—
104C2×C22⋊D61C2×Q812yes—
105C22×Dic61C4⋊C412yescomplemented
106C22×Dic61C22×C411yescharacteristic
107#11758226—22no—
108#11758226—22no—
109Q8×D61—22no—
110Q8×D61—22no—
111#11758224—24no—
112#11758224—24no—
113#11758224—22no—
114#11758224—22no—
115D4×D61—24no—
116D4×D61—24no—
117D4×D61—22no—
118D4×D61—22no—
119#11758222—22no—
120#11758222—22no—
121#11758222—24no—
122#11758222—24no—
123C2×D244C4⋊C411yescharacteristic, complemented
124C2×D244C2×Q812yes—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—24no—
130C2×C4×D61C4⋊C412yescomplemented
131C2×C4×D61C4⋊C412yescomplemented
132C2×C4×D61C2×D412yes—
133C2×C4×D61C2×D412yes—
134C2×C4×D61C2×D412yes—
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137C2×C4×D61C2×Q811yescharacteristic
138C2×C4×D61C2×Q811yescharacteristic
139C2×C4×D61#4112yes—
140C2×C4×D61#4112yes—
141C2×Dic122C4⋊C412yescomplemented
142C2×Dic122C4⋊C411yescharacteristic, complemented
143C22⋊#6923—28no—
144C22⋊#6923—24no—
145C22⋊#6923—24no—
146C22⋊#6923—24no—
147C22⋊#6923—48no—
148C4⋊#6923—28no—
149C4⋊#6923—24no—
150C4⋊#6923—24no—
151C4⋊#6923—416no—
152C4⋊#6923—42no—
153C4×#6923—28no—
154C4×#6923—24no—
155C4×#6923—24no—
156C4×#6923—41no—
157C4×#6923—42no—
158C4×#6923—41no—
159C4×#6923—41no—
160C4×#6923—41no—
161C4⋊C4×C61—22no—
162C22⋊C4×C61C22×C414yes—
163C42×C61—22no—
164C22⋊Dic61C22×C414yes—
165#11758198—24no—
166C4⋊Dic61—22no—
167#11758196—24no—
168C4×Dic61—22no—
169C4×Dic61—24no—
170C23×C61C2×C4⋊C412yes—
171C23×C61C23×C411yescharacteristic
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—42no—
175C22×D61—44no—
176C22×D61C2×C4⋊C412yes—
177C22×D61C22×D412yes—
178C22×D61C22×D411yescharacteristic
179C22×D61C22×Q811yescharacteristic
180C22×D61C2×#4112yes—
181C2×#6923—28no—
182C2×#6923—24no—
183C2×#6923—28no—
184C2×#6923—44no—
185C2×#6923—42no—
186C2×#6923—42no—
187C2×#6923—48no—
188C2×#6923—44no—
189C2×#6923—42no—
190C2×#6923—88no—
191Q8×C61—22no—
192D4×C61—24no—
193D4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61C2×C4⋊C412yes—
198C2×C4×C61C2×C4⋊C411yescharacteristic
199C2×C4×C61C2×C4⋊C411yescharacteristic
200C2×C4×C61C4×D412yes—
201C2×C4×C61C4×D412yes—
202C22⋊D61—24no—
203C22⋊D61—24no—
204C22⋊D61—24no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—24no—
208C2×Dic61C2×C4⋊C412yes—
209C2×Dic61C2×C4⋊C411yescharacteristic
210C2×Dic61C2×C4⋊C411yescharacteristic
211C2×Dic61C4×D412yes—
212C2×Dic61C4×D412yes—
213D244—24no—
214D244—22no—
215C4×D61—22no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—22no—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—48no—
222C4×D61#12011yescharacteristic
223C4×D61#12012yes—
224C4×D61#12011yescharacteristic
225C4×D61C22⋊Q812yes—
226C4×D61C22⋊Q812yes—
227C4×D61C22⋊Q812yes—
228C4×D61C22⋊Q812yes—
229C4×D61#12311yescharacteristic
230C4×D61#12311yescharacteristic
231C4×D61#12312yes—
232Dic122—22no—
233Dic122—24no—
234#20888—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61C22×C4⋊C411yescharacteristic
239C22×C61C2×C4×D412yes—
240C2×D61—24no—
241C2×D61—22no—
242C2×D61—22no—
243C2×D61—41no—
244C2×D61—42no—
245C2×D61—44no—
246C2×D61—42no—
247C2×D61—82no—
248C2×D61C2×#12011yescharacteristic
249C2×D61C2×C22⋊Q812yes—
250C2×D61C2×#12311yescharacteristic
251#6923—44no—
252#6923—48no—
253#6923—82no—
254C4×C61—22no—
255C4×C61—44no—
256C4×C61#38512yes—
257C4×C61#38511yescharacteristic
258C4×C61#38511yescharacteristic
259Dic61—22no—
260Dic61—44no—
261Dic61#38512yes—
262Dic61#38511yescharacteristic
263Dic61#38511yescharacteristic
264#3428—611no—
265C2×#410—611no—
266#2297—611no—
267#2296—612no—
268#1867—618no—
269#1740—612no—
270C2×C61—22no—
271C2×C61—21no—
272C2×C61—42no—
273C2×C61C2×#38511yescharacteristic
274D61—21no—
275D61—22no—
276D61—42no—
277D61#302512yes—
278C22×#41—611no—
279#520—1224no—
280#519—1221no—
281#518—614no—
282#518—1221no—
283C4×#41—1224no—
284C2×#117—612no—
285C2×#117—612no—
286C2×#117—611no—
287C2×#117—1222no—
288C22×C42—611no—
289#410—611no—
290#410—612no—
291#410—614no—
292#410#692311yescharacteristic, complemented
293C2×#99—614no—
294#391—1221no—
295#390—1221no—
296#389—1224no—
297#386—614no—
298#385—618no—
299C4×C4⋊C4—1224no—
300C4×C22⋊C4—1222no—
301#342—618no—
302C61#2088811yescharacteristic, complemented, Sylow
303C2×#41—611no—
304C2×#41—611no—
305C2×#41—612no—
306C2×#41—612no—
307C2×#41—611no—
308C2×#41—1222no—
309C2×#41—1222no—
310C2×#41—1222no—
311C2×#41—1222no—
312C2×#41C2×#692311yescharacteristic
313C22×Q8—611no—
314C22×D4—611no—
315C22×D4—612no—
316C23×C4—611no—
317C23×C4—612no—
318C23×C4—612no—
319C4×Q8—1228no—
320C4×D4—12216no—
321C4×D4—1228no—
322#117—612no—
323#117—614no—
324#117—1222no—
325#117—1222no—
326#117—1222no—
327#117—1222no—
328#117—1222no—
329#117—1222no—
330#117—1224no—
331#117—1224no—
332#117—1224no—
333#117—1222no—
334#117—1222no—
335#117C2×#692312yescomplemented
336C2×C4⋊C4—614no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1224no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1228no—
341C2×C22⋊C4—614no—
342C2×C22⋊C4—614no—
343C2×C22⋊C4—612no—
344C2×C22⋊C4—1224no—
345C2×C22⋊C4—1224no—
346C2×C42—614no—
347C2×C42—1222no—
348C2×C42—1224no—
349C2×C42—1221no—
350C2×C42—1221no—
351C2×C42—1222no—
352C2×C42—1222no—
353C2×C42—1221no—
354#99—614no—
355#99—618no—
356#99C2×#692314yescomplemented
357#95—1224no—
358#95—1228no—
359C24—611no—
360C24—612no—
361#41—22no—
362#41—22no—
363#41—1224no—
364#41—1224no—
365#41—1224no—
366#41—1224no—
367#41—1222no—
368#41—1222no—
369#41—1222no—
370#41—1222no—
371#41—1222no—
372#41—1222no—
373C2×Q8—612no—
374C2×Q8—611no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8C4⋊#692311yescharacteristic, complemented
378C2×D4—612no—
379C2×D4—612no—
380C2×D4—612no—
381C2×D4—611no—
382C2×D4—612no—
383C2×D4—1222no—
384C2×D4—1222no—
385C2×D4—1224no—
386C2×D4—1224no—
387C2×D4C4⋊#692311yescharacteristic, complemented
388C2×D4C22×#692312yes—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—612no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—618no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1224no—
408C22×C4—1228no—
409C22×C4—1222no—
410C22×C4—1224no—
411C22×C4—1222no—
412C22×C4—2444no—
413C22×C4C4⋊#692312yescomplemented
414C22×C4C22×#692311yescharacteristic
415C4⋊C4—22no—
416C4⋊C4—1224no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1228no—
420C4⋊C4—1222no—
421C4⋊C4—2442no—
422C4⋊C4—24416no—
423C22⋊C4—614no—
424C22⋊C4—1224no—
425C22⋊C4—1224no—
426C22⋊C4—1224no—
427C22⋊C4—1228no—
428C22⋊C4—1224no—
429C22⋊C4—2448no—
430C22⋊C4C22×#692314yes—
431C42—22no—
432C42—1224no—
433C42—1224no—
434C42—1224no—
435C42—1228no—
436C42—1222no—
437C42—2442no—
438C42—2441no—
439C42—2441no—
440C42—2441no—
441C42—2441no—
442C23—611no—
443C23—612no—
444C23—611no—
445C23—612no—
446C23—612no—
447C23—1222no—
448C23—1222no—
449C23—2442no—
450C23—2444no—
451C23C2×C4⋊#692312yes—
452C23C23×#692311yescharacteristic
453Q8—22no—
454Q8—1224no—
455Q8—1222no—
456D4—24no—
457D4—22no—
458D4—1224no—
459D4—1224no—
460D4—1224no—
461D4—1224no—
462D4—1222no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—611no—
467C2×C4—611no—
468C2×C4—612no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—611no—
472C2×C4—612no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—611no—
476C2×C4—612no—
477C2×C4—611no—
478C2×C4—612no—
479C2×C4—611no—
480C2×C4—612no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1228no—
490C2×C4—1224no—
491C2×C4—1228no—
492C2×C4—1224no—
493C2×C4—4888no—
494C2×C4—2448no—
495C2×C4—2442no—
496C2×C4—2444no—
497C2×C4—2442no—
498C2×C4—2444no—
499C2×C4—2442no—
500C2×C4—2448no—
501C2×C4C2×C4⋊#692312yes—
502C2×C4C2×C4⋊#692311yescharacteristic
503C2×C4C2×C4⋊#692311yescharacteristic
504C2×C4D4×#692312yes—
505C2×C4D4×#692312yes—
506C22—22no—
507C22—41no—
508C22—42no—
509C22—611no—
510C22—611no—
511C22—612no—
512C22—1224no—
513C22—1222no—
514C22—1222no—
515C22—4882no—
516C22—2442no—
517C22—2441no—
518C22—2442no—
519C22—2444no—
520C22C22×C4⋊#692311yescharacteristic
521C22C2×D4×#692312yes—
522C4—22no—
523C4—44no—
524C4—611no—
525C4—611no—
526C4—612no—
527C4—1222no—
528C4—4882no—
529C4—2444no—
530C4—2444no—
531C4—2448no—
532C4#1684070911yescharacteristic
533C4#1684070912yes—
534C4#1684070911yescharacteristic
535C2—22no—
536C2—21no—
537C2—42no—
538C2—612no—
539C2—1221no—
540C2—1222no—
541C2—2442no—
542C2C2×#1684070911yescharacteristic
subgroupG/H
centerC4#16840709
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC61×#410C4
socleC2×C61C2×#385
prime radicalC2×#41×D61C22
cosocleC23×D61C24
hypercenter#410#6923
hypocenterC61#20888
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #21462279C2semidirect
4. #21461790C2semidirect
5. #21461517C2semidirect
26. #16839736C4semidirect
27. #16839735C4semidirect
28. #16839734C4semidirect
30. C61×#410C4semidirect
75. #14519736C2×C4semidirect
78. #14519733C2×C4semidirect
82. #14519692C2×C4semidirect
83. #14519691C2×C4semidirect
85. #117×C61C2×C4semidirect
86. #99×C61C2×C4semidirect
97. C2×Q8×C61C4⋊C4semidirect
98. C2×D4×C61C4⋊C4semidirect
100. C22×C4×C61C4⋊C4semidirect
102. C2×C22⋊D61C4⋊C4semidirect
105. C22×Dic61C4⋊C4semidirect
123. C2×D244C4⋊C4semidirect
130. C2×C4×D61C4⋊C4semidirect
131. C2×C4×D61C4⋊C4semidirect
141. C2×Dic122C4⋊C4semidirect
142. C2×Dic122C4⋊C4semidirect
292. #410#6923semidirect
302. C61#20888semidirect
335. #117C2×#6923semidirect
356. #99C2×#6923semidirect
377. C2×Q8C4⋊#6923semidirect
387. C2×D4C4⋊#6923semidirect
413. C22×C4C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
#214627992
#214622802
#214622792
#214617902
#214615172
#410×D612
#2088861