Finite Group Databasebeta

← #29689164

Subgroups — #29689164

Order 15616.

1520 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459455C211yesmaximal, characteristic
2#21454277C212yesmaximal, complemented
3#21446360C214yesmaximal, complemented
4#21446226C214yesmaximal, complemented
5#21445927C211yesmaximal, characteristic, complemented
6#21445732C212yesmaximal, complemented
7C61×#1828C211yesmaximal, characteristic, complemented
8C2×#14519822C2211yescharacteristic
9C22×D4×D61C2211yescharacteristic
10#16840407—24no—
11D4×D244—24no—
12#16840331C412yescomplemented
13#16840324C412yescomplemented
14#16840324C2212yes—
15#16840274C412yescomplemented
16C2×#14519747C412yescomplemented
17C2×#14519747C2212yes—
18#16840235C2214yescomplemented
19#16839734—24no—
20C2×#14519692C2212yes—
21C2×#14519691C2214yescomplemented
22C2×#14519691C2214yescomplemented
23#99×D61—24no—
24#16839617C2214yescomplemented
25#16839528C2214yescomplemented
26C61×#518C2211yescharacteristic
27C2×#99×C61C2212yes—
28C61×#342C2214yescomplemented
29C24×D61C2×C411yescharacteristic, complemented
30C24×D61C2311yescharacteristic
31#14519822—44no—
32C2×#11758222C2×C412yes—
33C2×#11758224D414yes—
34C22×D244—22no—
35C22×D244—22no—
36C22×D244C2×C411yescharacteristic, complemented
37C22×C22⋊D61C2×C412yes—
38C22×C22⋊D61C2×C411yescharacteristic, complemented
39C22×C22⋊D61C2311yescharacteristic
40C22×C22⋊D61C2312yes—
41C2×D4×D61—24no—
42C2×D4×D61—44no—
43C2×D4×D61D414yes—
44C22×C4×D61C2×C411yescharacteristic, complemented
45#14519774—22no—
46#14519765—22no—
47#14519761—22no—
48#14519761—22no—
49#14519761C2×C414yescomplemented
50#14519757—22no—
51#14519752—22no—
52#14519752—24no—
53#14519751—24no—
54#14519751—24no—
55#14519747—22no—
56#14519747—22no—
57#14519747—24no—
58#14519747—24no—
59#14519747C2×C414yescomplemented
60#14519747C2×C414yescomplemented
61#14519747C2×C414yescomplemented
62C4×D244—24no—
63C4×C22⋊D61—44no—
64#14519737—44no—
65#14519733—24no—
66C2×C22⋊Dic61C2314yes—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—44no—
70C22⋊C4×D61—24no—
71#14519692—24no—
72#14519691—28no—
73#14519691—44no—
74C22×D4×C61C2311yescharacteristic
75C4×D4×C61—24no—
76#117×C61C2×C412yescomplemented
77C2×C22⋊C4×C61C2×C412yescomplemented
78C2×C22⋊C4×C61C2314yes—
79C2×C22⋊C4×C61C2312yes—
80#99×C61—24no—
81C24×C61C22×C411yescharacteristic
82C24×C61C2411yescharacteristic
83C23×D61—22no—
84C23×D61—22no—
85C23×D61—24no—
86C23×D61—42no—
87C23×D61—82no—
88C23×D61—82no—
89C23×D61C22⋊C412yescomplemented
90C23×D61C22×C411yescharacteristic
91C23×D61C22×C411yescharacteristic
92C23×D61C22×C411yescharacteristic
93C23×D61C2×D412yes—
94C2×D4×C61—24no—
95C2×D4×C61C2×D414yes—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—24no—
99C22×C4×C61C22×C411yescharacteristic
100C2×C22⋊D61—24no—
101C2×C22⋊D61—24no—
102C2×C22⋊D61—22no—
103C2×C22⋊D61—22no—
104C2×C22⋊D61—24no—
105C2×C22⋊D61—42no—
106C2×C22⋊D61—44no—
107C2×C22⋊D61C22⋊C414yescomplemented
108C2×C22⋊D61C22×C414yes—
109C2×C22⋊D61C2×D414yes—
110C22×Dic61—24no—
111C22×Dic61C22⋊C412yescomplemented
112C22×Dic61C22×C411yescharacteristic
113C22×Dic61C2×D412yes—
114#11758224—48no—
115D4×D61—48no—
116D4×D61—416no—
117#11758222—44no—
118C2×D244—24no—
119C2×D244—22no—
120C2×D244—24no—
121C2×D244—24no—
122C2×D244—42no—
123C2×D244—84no—
124C2×D244—84no—
125C2×D244C22⋊C412yescomplemented
126C2×C4×D61—22no—
127C2×C4×D61—42no—
128C2×C4×D61—44no—
129C2×C4×D61C22⋊C414yescomplemented
130C2×Dic122C22⋊C412yescomplemented
131C4⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C22⋊C4×C61—22no—
134C22⋊C4×C61—24no—
135C22⋊C4×C61—24no—
136C22⋊C4×C61C22×C414yes—
137C42×C61—22no—
138C22⋊Dic61—24no—
139C22⋊Dic61—24no—
140#11758198—24no—
141#11758198—24no—
142#11758198—24no—
143#11758198—24no—
144#11758198—44no—
145#11758198—88no—
146C4⋊Dic61—24no—
147#11758196—44no—
148C4×Dic61—44no—
149C23×C61—22no—
150C23×C61—81no—
151C23×C61—81no—
152C23×C61C2×C22⋊C412yes—
153C23×C61C23×C411yescharacteristic
154C23×C61C22×D412yes—
155C23×C61H(5,𝔽2)14yes—
156C22×D61—21no—
157C22×D61—22no—
158C22×D61—24no—
159C22×D61—21no—
160C22×D61—21no—
161C22×D61—22no—
162C22×D61—22no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—48no—
166C22×D61—44no—
167C22×D61—44no—
168C22×D61—44no—
169C22×D61—44no—
170C22×D61—84no—
171C22×D61—81no—
172C22×D61—82no—
173C22×D61—82no—
174C22×D61—84no—
175C22×D61—82no—
176C22×D61—81no—
177C22×D61—82no—
178C22×D61—84no—
179C22×D61C2×C22⋊C414yes—
180D4×C61—28no—
181C2×C4×C61—22no—
182C2×C4×C61—24no—
183C2×C4×C61—44no—
184C2×C4×C61—82no—
185C2×C4×C61—82no—
186C2×C4×C61—84no—
187C2×C4×C61C2×C22⋊C412yes—
188C22⋊D61—24no—
189C22⋊D61—24no—
190C22⋊D61—28no—
191C22⋊D61—44no—
192C22⋊D61—44no—
193C22⋊D61—44no—
194C2×Dic61—24no—
195C2×Dic61—21no—
196C2×Dic61—22no—
197C2×Dic61—42no—
198C2×Dic61—44no—
199C2×Dic61—84no—
200C2×Dic61C2×C22⋊C414yes—
201D244—42no—
202D244—44no—
203D244—48no—
204D244—84no—
205D244—84no—
206C4×D61—44no—
207C4×D61—44no—
208C4×D61—164no—
209Dic122—42no—
210#21776—611nomaximal, Sylow
211C22×C61—22no—
212C22×C61—44no—
213C22×C61—42no—
214C22×C61—41no—
215C22×C61—42no—
216C22×C61—82no—
217C22×C61C22×C22⋊C411yescharacteristic
218C22×C61#51812yes—
219C22×C61#53414yes—
220C2×D61—22no—
221C2×D61—24no—
222C2×D61—24no—
223C2×D61—41no—
224C2×D61—42no—
225C2×D61—41no—
226C2×D61—44no—
227C2×D61—42no—
228C2×D61—42no—
229C2×D61—44no—
230C2×D61—41no—
231C2×D61—41no—
232C2×D61—84no—
233C2×D61—88no—
234C2×D61—88no—
235C2×D61—82no—
236C2×D61—162no—
237C4×C61—22no—
238C4×C61—44no—
239C4×C61—84no—
240Dic61—24no—
241Dic61—41no—
242Dic61—84no—
243#3484—611no—
244#2883—612no—
245#1913—614no—
246#1889—614no—
247#1828—611no—
248#1828D6111yescharacteristic, complemented
249#1781—612no—
250C2×C61—24no—
251C2×C61—21no—
252C2×C61—22no—
253C2×C61—42no—
254C2×C61#230811yescharacteristic
255D61—42no—
256D61—44no—
257D61—81no—
258D61—81no—
259C2×H(5,𝔽2)—611no—
260C23×D4—611no—
261#546—1224no—
262D42—1224no—
263#535—612no—
264#534—612no—
265#534—612no—
266#525—612no—
267C2×#120—612no—
268C2×#120—612no—
269#518—614no—
270#518C2×D6111yescharacteristic
271#410—1224no—
272C2×#99—612no—
273C2×#99—614no—
274C2×#99—614no—
275C2×#99—1224no—
276C2×#99C2×D6112yes—
277#379—614no—
278#342—614no—
279#342C2×D6114yescomplemented
280C61#2177611yescharacteristic, complemented, Sylow
281C25—611no—
282C25—611no—
283H(5,𝔽2)—2444no—
284C2×#41—614no—
285C2×#41—612no—
286C22×D4—612no—
287C22×D4—612no—
288C22×D4—614no—
289C22×D4—611no—
290C22×D4—611no—
291C22×D4—611no—
292C22×D4—1222no—
293C22×D4—1222no—
294C22×D4—1224no—
295C22×D4—2444no—
296C22×D4C22×D6111yescharacteristic
297C23×C4—611no—
298#127—1222no—
299#124—1222no—
300#123—614no—
301#123—1222no—
302#123—1222no—
303C22⋊Q8—1222no—
304#121—1224no—
305#121—1222no—
306#121—1224no—
307#121—1224no—
308#120—614no—
309#120—614no—
310#120—614no—
311#120—1224no—
312#120—1224no—
313#120—1222no—
314#120—1222no—
315C4×D4—24no—
316C4×D4—1224no—
317C4×D4—2444no—
318C4×D4—2444no—
319#117—1224no—
320#117C4×D6112yescomplemented
321C2×C22⋊C4—614no—
322C2×C22⋊C4—1224no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—2444no—
326C2×C22⋊C4C4×D6112yescomplemented
327C2×C22⋊C4C22×D6114yes—
328C2×C22⋊C4C22×D6112yes—
329#99—24no—
330#99—1228no—
331#99—1224no—
332#99—2444no—
333C24—612no—
334C24—612no—
335C24—611no—
336C24—611no—
337C24—611no—
338C24—1222no—
339C24—1224no—
340C24—1222no—
341C24—4882no—
342C24—4882no—
343C24—2442no—
344C24C2×C4×D6111yescharacteristic
345C24C23×D6111yescharacteristic
346#41—2444no—
347#41—2448no—
348C2×Q8—612no—
349C2×D4—24no—
350C2×D4—614no—
351C2×D4—614no—
352C2×D4—614no—
353C2×D4—612no—
354C2×D4—1224no—
355C2×D4—1224no—
356C2×D4—1224no—
357C2×D4—1224no—
358C2×D4—1224no—
359C2×D4—1222no—
360C2×D4—1224no—
361C2×D4—1222no—
362C2×D4—1222no—
363C2×D4—4884no—
364C2×D4—4884no—
365C2×D4—2442no—
366C2×D4—2442no—
367C2×D4—2448no—
368C2×D4—24416no—
369C2×D4—2444no—
370C2×D4D4×D6114yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—614no—
377C22×C4—611no—
378C22×C4—1222no—
379C22×C4—1224no—
380C22×C4—2442no—
381C22×C4—2444no—
382C22×C4C2×C4×D6111yescharacteristic
383C4⋊C4—22no—
384C4⋊C4—1224no—
385C4⋊C4—2444no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—1224no—
391C22⋊C4—1224no—
392C22⋊C4—1224no—
393C22⋊C4—1224no—
394C22⋊C4—1224no—
395C22⋊C4—1224no—
396C22⋊C4—4888no—
397C22⋊C4—2444no—
398C22⋊C4C2×C4×D6114yes—
399C42—22no—
400C42—2444no—
401C23—22no—
402C23—81no—
403C23—81no—
404C23—614no—
405C23—1224no—
406C23—1221no—
407C23—1222no—
408C23—1222no—
409C23—1221no—
410C23—1221no—
411C23—1222no—
412C23—4882no—
413C23—4884no—
414C23—4882no—
415C23—4882no—
416C23—4882no—
417C23—4881no—
418C23—4881no—
419C23—4884no—
420C23—4884no—
421C23—2442no—
422C23—2444no—
423C23—2442no—
424C23—2448no—
425C23—2444no—
426C23—2444no—
427C23—2444no—
428C23C22⋊C4×D6112yes—
429C23C22×C4×D6111yescharacteristic
430C23C2×D4×D6112yes—
431C23#1451982314yes—
432Q8—2442no—
433D4—28no—
434D4—1224no—
435D4—1224no—
436D4—1228no—
437D4—4884no—
438D4—4884no—
439D4—2442no—
440D4—2444no—
441D4—2448no—
442D4—2444no—
443D4—2444no—
444D4—2444no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—44no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—84no—
451C2×C4—614no—
452C2×C4—1224no—
453C2×C4—1221no—
454C2×C4—1222no—
455C2×C4—9764no—
456C2×C4—4884no—
457C2×C4—2444no—
458C2×C4—2444no—
459C2×C4—2442no—
460C2×C4—2444no—
461C2×C4C22⋊C4×D6112yes—
462C22—22no—
463C22—44no—
464C22—42no—
465C22—41no—
466C22—42no—
467C22—82no—
468C22—1222no—
469C22—1224no—
470C22—1224no—
471C22—9762no—
472C22—4882no—
473C22—4888no—
474C22—4884no—
475C22—4888no—
476C22—2442no—
477C22—2442no—
478C22—2444no—
479C22—2444no—
480C22—2441no—
481C22—2442no—
482C22—2441no—
483C22—2441no—
484C22—2441no—
485C22C2×C22⋊C4×D6111yescharacteristic
486C22#1684023612yes—
487C22#1684032314yes—
488C4—22no—
489C4—44no—
490C4—84no—
491C4—1224no—
492C4—4884no—
493C4—2441no—
494C2—21no—
495C2—24no—
496C2—22no—
497C2—42no—
498C2—4881no—
499C2—4881no—
500C2—2442no—
501C2—2444no—
502C2#2144839611yescharacteristic
subgroupG/H
centerC2#21448396
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1828C2
socleC2×C61#2308
prime radical#21459455C2
cosocleC24×C61C24
hypercenter#1828D61
hypocenterC61#21776
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C24 ◁ #1828

Complements

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

NH ≅ G/Ntype
2. #21454277C2semidirect
3. #21446360C2semidirect
4. #21446226C2semidirect
5. #21445927C2semidirect
6. #21445732C2semidirect
7. C61×#1828C2semidirect
12. #16840331C4semidirect
13. #16840324C4semidirect
15. #16840274C4semidirect
16. C2×#14519747C4semidirect
18. #16840235C22semidirect
21. C2×#14519691C22semidirect
22. C2×#14519691C22semidirect
24. #16839617C22semidirect
25. #16839528C22semidirect
28. C61×#342C22semidirect
29. C24×D61C2×C4semidirect
36. C22×D244C2×C4semidirect
38. C22×C22⋊D61C2×C4semidirect
44. C22×C4×D61C2×C4semidirect
49. #14519761C2×C4semidirect
59. #14519747C2×C4semidirect
60. #14519747C2×C4semidirect
61. #14519747C2×C4semidirect
76. #117×C61C2×C4semidirect
77. C2×C22⋊C4×C61C2×C4semidirect
89. C23×D61C22⋊C4semidirect
107. C2×C22⋊D61C22⋊C4semidirect
111. C22×Dic61C22⋊C4semidirect
125. C2×D244C22⋊C4semidirect
129. C2×C4×D61C22⋊C4semidirect
130. C2×Dic122C22⋊C4semidirect
248. #1828D61semidirect
279. #342C2×D61semidirect
280. C61#21776semidirect
320. #117C4×D61semidirect
326. C2×C22⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214594552
#214542772
#214463602
#214462262
#214459272
#214457322
C61×#18282
#2177661