Finite Group Databasebeta

← #29689169

Subgroups — #29689169

Order 15616.

1318 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#21459463C211yesmaximal, characteristic
2#21454277C212yesmaximal, complemented
3#21446358C214yesmaximal, complemented
4#21446229C214yesmaximal, complemented
5#21445925C212yesmaximal, complemented
6#21445737C211yesmaximal, characteristic, complemented
7C61×#1782C211yesmaximal, characteristic, complemented
8C2×#14519822C2211yescharacteristic
9#16840411—24no—
10#16840400—24no—
11#16840336C412yescomplemented
12#16840321C412yescomplemented
13#16840321C2212yes—
14#16840273C412yescomplemented
15C2×#14519760C412yescomplemented
16C2×#14519760C2212yes—
17#16840237C2211yescharacteristic
18#16840235C2214yes—
19#16839734—24no—
20C2×#14519692C2212yes—
21C2×#14519692C2212yescomplemented
22C2×#14519691C2214yes—
23#99×D61—24no—
24#16839625C2212yescomplemented
25#16839529C2214yescomplemented
26#16839528C2214yescomplemented
27C22×#41×C61C2211yescharacteristic
28C2×#99×C61C2212yes—
29C2×#99×C61C2212yescomplemented
30C61×#381C2212yescomplemented
31#14519822—42no—
32#14519822—42no—
33C2×#11758222C2×C412yes—
34C2×#11758224D414yes—
35C22×C22⋊D61C2×C412yes—
36C22×C22⋊D61C2312yes—
37C2×D4×D61D414yes—
38#14519775—22no—
39#14519766—22no—
40#14519760—24no—
41#14519760—24no—
42#14519760C2×C414yescomplemented
43#14519760C2×C414yescomplemented
44#14519760C2×C414yescomplemented
45#14519758—22no—
46#14519758—22no—
47#14519756—22no—
48#14519753—24no—
49#14519751—22no—
50#14519749—24no—
51#14519749—24no—
52#14519746C2×C414yescomplemented
53#14519745—22no—
54#14519745—22no—
55D4×Dic61—24no—
56C4×C22⋊D61—24no—
57C4×C22⋊D61—44no—
58#14519737—44no—
59#14519736C2×C412yescomplemented
60#14519733—24no—
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×C22⋊Dic61C2×C412yescomplemented
64C2×C22⋊Dic61C2314yes—
65C2×C22⋊Dic61C2312yes—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C22⋊C4×D61—24no—
69#14519692—24no—
70#14519692—42no—
71#14519691—28no—
72#14519684—42no—
73C2×#41×C61—24no—
74C2×#41×C61—42no—
75C2×#41×C61—42no—
76C22×Q8×C61C2×C411yescharacteristic, complemented
77C22×D4×C61C2×C411yescharacteristic, complemented
78C22×D4×C61C2311yescharacteristic
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61C2×C411yescharacteristic, complemented
81C23×C4×C61C2×C411yescharacteristic, complemented
82C23×C4×C61C2311yescharacteristic
83C2×C22⋊C4×C61C2314yes—
84#99×C61—24no—
85#99×C61—42no—
86#95×C61—42no—
87C24×C61C22×C411yescharacteristic
88C24×C61C22×C411yescharacteristic
89C24×C61C2411yescharacteristic
90C23×D61C22⋊C412yes—
91C23×D61C2×D412yes—
92#41×C61—416no—
93C2×Q8×C61—22no—
94C2×Q8×C61—41no—
95C2×Q8×C61—41no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—41no—
103C2×D4×C61—41no—
104C2×D4×C61C2×D414yes—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—24no—
111C22×C4×C61—41no—
112C22×C4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C2×C22⋊D61—24no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61C22⋊C414yes—
125C2×C22⋊D61C22×C414yes—
126C2×C22⋊D61C2×D414yes—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—24no—
130C22×Dic61C22⋊C412yes—
131C22×Dic61C2×D412yes—
132#11758224—48no—
133D4×D61—48no—
134#11758222—42no—
135#11758222—42no—
136C2×D244C22⋊C412yes—
137C2×C4×D61—44no—
138C2×C4×D61C22⋊C414yes—
139C2×Dic122C22⋊C412yes—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—24no—
142C22⋊Dic61—22no—
143C22⋊Dic61—22no—
144C22⋊Dic61—24no—
145C22⋊Dic61—24no—
146C22⋊Dic61—24no—
147C22⋊Dic61C22×C414yes—
148#11758198—24no—
149#11758198—24no—
150#11758198—24no—
151#11758198—44no—
152C4⋊Dic61—22no—
153#11758196—24no—
154#11758196—24no—
155#11758196—24no—
156#11758196—44no—
157C4×Dic61—22no—
158C4×Dic61—44no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—81no—
165C23×C61—81no—
166C23×C61—81no—
167C23×C61C2×C22⋊C412yes—
168C23×C61C23×C411yescharacteristic
169C23×C61C22×D412yes—
170C23×C61H(5,𝔽2)12yes—
171C23×C61H(5,𝔽2)12yes—
172C22×D61—24no—
173C22×D61—42no—
174C22×D61—82no—
175C22×D61—82no—
176C22×D61C2×C22⋊C414yes—
177Q8×C61—44no—
178D4×C61—28no—
179D4×C61—44no—
180D4×C61—44no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—44no—
193C2×C4×C61—44no—
194C2×C4×C61—44no—
195C2×C4×C61—44no—
196C2×C4×C61—84no—
197C2×C4×C61C2×C22⋊C412yes—
198C22⋊D61—24no—
199C22⋊D61—24no—
200C22⋊D61—28no—
201C22⋊D61—44no—
202C22⋊D61—84no—
203C2×Dic61—24no—
204C2×Dic61—24no—
205C2×Dic61—42no—
206C2×Dic61—44no—
207C2×Dic61—82no—
208C2×Dic61—82no—
209C2×Dic61—84no—
210C2×Dic61C2×C22⋊C414yes—
211D244—42no—
212C4×D61—44no—
213C4×D61—164no—
214Dic122—42no—
215#21777—611nomaximal, Sylow
216C22×C61—22no—
217C22×C61—42no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61—41no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—41no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61C22×C22⋊C411yescharacteristic
229C22×C61#51811yescharacteristic
230C22×C61#51811yescharacteristic
231C22×C61#53414yes—
232C2×D61—24no—
233C2×D61—44no—
234C2×D61—84no—
235C2×D61—162no—
236C4×C61—22no—
237C4×C61—41no—
238C4×C61—41no—
239C4×C61—41no—
240C4×C61—84no—
241Dic61—24no—
242Dic61—44no—
243Dic61—84no—
244#3485—611no—
245#2883—612no—
246#1913—614no—
247#1890—614no—
248#1828—612no—
249#1782—611no—
250#1782D6111yescharacteristic, complemented
251C2×C61—21no—
252C2×C61—24no—
253C2×C61—21no—
254C2×C61—21no—
255C2×C61—42no—
256C2×C61—81no—
257C2×C61#230811yescharacteristic
258D61—44no—
259C2×H(5,𝔽2)—611no—
260C22×#41C2×D6111yescharacteristic
261#547—1224no—
262#546—1224no—
263#536—612no—
264#534—612no—
265#534—612no—
266#525—612no—
267C2×#123—612no—
268C2×#123—612no—
269#518—611no—
270#518—614no—
271#410—1224no—
272C2×#99—612no—
273C2×#99—614no—
274C2×#99—612no—
275C2×#99—1224no—
276C2×#99C2×D6112yes—
277C2×#99C2×D6112yescomplemented
278#381—612no—
279#381C2×D6112yescomplemented
280#342—614no—
281#342—614no—
282C61#2177711yescharacteristic, complemented, Sylow
283H(5,𝔽2)—2442no—
284H(5,𝔽2)—2442no—
285C2×#41—24no—
286C2×#41—42no—
287C2×#41—42no—
288C2×#41—614no—
289C2×#41—612no—
290C22×Q8C4×D6111yescharacteristic, complemented
291C22×D4—612no—
292C22×D4—612no—
293C22×D4—614no—
294C22×D4C4×D6111yescharacteristic, complemented
295C22×D4C22×D6111yescharacteristic
296C22×D4C22×D6111yescharacteristic
297C23×C4C4×D6111yescharacteristic, complemented
298C23×C4C4×D6111yescharacteristic, complemented
299C23×C4C22×D6111yescharacteristic
300C4⋊Q8—1222no—
301#124—1222no—
302#123—614no—
303#123—614no—
304#123—614no—
305#123—1224no—
306#123—1224no—
307#123—1222no—
308#123—1222no—
309C22⋊Q8—1222no—
310C22⋊Q8—1224no—
311#121—1222no—
312#121—1224no—
313#121—1224no—
314#120—614no—
315#120—1222no—
316#120—1222no—
317C4×D4—1224no—
318C4×D4—1224no—
319C4×D4—2444no—
320C4×D4—2444no—
321#117—612no—
322#117—1224no—
323C2×C4⋊C4—1222no—
324C2×C4⋊C4—1222no—
325C2×C22⋊C4—612no—
326C2×C22⋊C4—612no—
327C2×C22⋊C4—614no—
328C2×C22⋊C4—1224no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4C22×D6114yes—
332#99—24no—
333#99—42no—
334#99—1228no—
335#99—1224no—
336#99—2442no—
337#95—42no—
338#95—2442no—
339C24—612no—
340C24—612no—
341C24C2×C4×D6111yescharacteristic
342C24C2×C4×D6111yescharacteristic
343C24C23×D6111yescharacteristic
344#41—416no—
345#41—2442no—
346#41—2442no—
347#41—2448no—
348C2×Q8—22no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—612no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—614no—
361C2×D4—614no—
362C2×D4—614no—
363C2×D4—612no—
364C2×D4—1224no—
365C2×D4—1224no—
366C2×D4—1224no—
367C2×D4—2448no—
368C2×D4—2442no—
369C2×D4—2442no—
370C2×D4D4×D6114yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—612no—
382C22×C4—612no—
383C22×C4—614no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1224no—
387C22×C4—2444no—
388C22×C4C2×C4×D6111yescharacteristic
389C22×C4C2×C4×D6111yescharacteristic
390C22×C4C2×C4×D6111yescharacteristic
391C22×C4C2×C4×D6111yescharacteristic
392C4⋊C4—1224no—
393C4⋊C4—1224no—
394C4⋊C4—1224no—
395C4⋊C4—1222no—
396C4⋊C4—2444no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—614no—
400C22⋊C4—1224no—
401C22⋊C4—1224no—
402C22⋊C4—1224no—
403C22⋊C4—1224no—
404C22⋊C4—1222no—
405C22⋊C4—1222no—
406C22⋊C4—1224no—
407C22⋊C4—1224no—
408C22⋊C4—2444no—
409C42—1222no—
410C42—2444no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—614no—
420C23—1224no—
421C23—4882no—
422C23—4882no—
423C23—2442no—
424C23C22⋊C4×D6112yes—
425C23C22×C4×D6111yescharacteristic
426C23C2×D4×D6112yes—
427C23#1451982212yes—
428C23#1451982212yes—
429Q8—44no—
430Q8—2442no—
431D4—28no—
432D4—44no—
433D4—44no—
434D4—1224no—
435D4—1224no—
436D4—1228no—
437D4—4884no—
438D4—2442no—
439D4—2444no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—84no—
456C2×C4—614no—
457C2×C4—1224no—
458C2×C4—1224no—
459C2×C4—9764no—
460C2×C4—4884no—
461C2×C4—4882no—
462C2×C4—4882no—
463C2×C4—2444no—
464C2×C4—2442no—
465C2×C4—2444no—
466C2×C4C22⋊C4×D6112yes—
467C22—22no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—41no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—1224no—
480C22—9762no—
481C22—4884no—
482C22—2444no—
483C22C2×C22⋊C4×D6111yescharacteristic
484C22#1684023511yescharacteristic
485C22#1684023511yescharacteristic
486C22#1684032214yes—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—84no—
492C4—1224no—
493C4—4884no—
494C4—2444no—
495C2—24no—
496C2—21no—
497C2—21no—
498C2—21no—
499C2—42no—
500C2—81no—
501C2—2444no—
502C2#2144839511yescharacteristic
subgroupG/H
centerC2#21448395
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1782C2
socleC2×C61#2308
prime radical#21459463C2
cosocleC24×C61C24
hypercenter#1782D61
hypocenterC61#21777
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C24 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #21454277C2semidirect
3. #21446358C2semidirect
4. #21446229C2semidirect
5. #21445925C2semidirect
6. #21445737C2semidirect
7. C61×#1782C2semidirect
11. #16840336C4semidirect
12. #16840321C4semidirect
14. #16840273C4semidirect
15. C2×#14519760C4semidirect
21. C2×#14519692C22semidirect
24. #16839625C22semidirect
25. #16839529C22semidirect
26. #16839528C22semidirect
29. C2×#99×C61C22semidirect
30. C61×#381C22semidirect
42. #14519760C2×C4semidirect
43. #14519760C2×C4semidirect
44. #14519760C2×C4semidirect
52. #14519746C2×C4semidirect
59. #14519736C2×C4semidirect
63. C2×C22⋊Dic61C2×C4semidirect
76. C22×Q8×C61C2×C4semidirect
77. C22×D4×C61C2×C4semidirect
80. C23×C4×C61C2×C4semidirect
81. C23×C4×C61C2×C4semidirect
250. #1782D61semidirect
277. C2×#99C2×D61semidirect
279. #381C2×D61semidirect
282. C61#21777semidirect
290. C22×Q8C4×D61semidirect
294. C22×D4C4×D61semidirect
297. C23×C4C4×D61semidirect
298. C23×C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214594632
#214542772
#214463582
#214462292
#214459252
#214457372
C61×#17822
#2177761