Finite Group Databasebeta

← #29832827

Subgroups — #29832827

Order 15616.

1252 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459608C211yesmaximal, characteristic
2#21454278C212yesmaximal, complemented
3#21447494C211yesmaximal, characteristic, complemented
4#21447487C214yesmaximal, complemented
5#21447482C212yesmaximal, complemented
6#354×D61C214yesmaximal, complemented
7C61×#2129C211yesmaximal, characteristic, complemented
8C2×#14519825C411yescharacteristic, complemented
9C2×#14519823C2211yescharacteristic
10#16840422—22no—
11#16840404—24no—
12#16840403—22no—
13#16840337C412yescomplemented
14#16840331C411yescharacteristic, complemented
15#16840330C2212yes—
16#16840313C412yescomplemented
17#16840308C2211yescharacteristic
18C2×#14519766C411yescharacteristic, complemented
19#124×D61C2212yes—
20#16840236—22no—
21#16840230C411yescharacteristic, complemented
22#16839736C2212yes—
23#16839735—22no—
24#99×D61—24no—
25#99×D61C2218yes—
26#16839557C2214yescomplemented
27#16839556C2214yescomplemented
28#16839554C2214yescomplemented
29C61×#536C2211yescharacteristic
30C61×#410C2212yes—
31C61×#354C2214yescomplemented
32#14519825—44no—
33#14519823—44no—
34C2×#11758226—21no—
35C2×#11758226C2×C412yes—
36C2×#11758226C2×C412yescomplemented
37C2×#11758222—22no—
38C2×#11758222C2×C411yescharacteristic
39C2×#11758222C2×C411yescharacteristic, complemented
40C2×#11758222D412yes—
41C2×Q8×D61—21no—
42C2×Q8×D61C2×C412yescomplemented
43C22×D244C2×C411yescharacteristic
44C22×D244D412yes—
45C2×D4×D61D414yes—
46C2×D4×D61C2312yes—
47#14519775C2×C411yescharacteristic, complemented
48#14519774C2×C412yescomplemented
49#14519773C2311yescharacteristic
50#14519766—22no—
51#14519766C2×C412yescomplemented
52#14519766C2×C412yescomplemented
53#14519766C2×C411yescharacteristic, complemented
54#14519765C2×C412yescomplemented
55#14519764C2311yescharacteristic
56#14519763—24no—
57#14519762—24no—
58#14519755—24no—
59#14519752—24no—
60#14519752—22no—
61#14519750—24no—
62#14519749—22no—
63#14519747—22no—
64#14519747—24no—
65#14519742—21no—
66C4×D244—44no—
67#14519739—21no—
68#14519739—24no—
69#14519739—44no—
70C4×C22⋊D61—22no—
71#14519737—22no—
72#14519736—22no—
73#14519734C2×C412yescomplemented
74C2×#11758198—22no—
75C22⋊C4×D61—24no—
76C22⋊C4×D61—28no—
77C2×C4×Dic61C2×C411yescharacteristic, complemented
78C42×D61C2×C412yescomplemented
79#14519692—22no—
80#14519692C2314yes—
81#14519691—28no—
82C2×#41×C61C2311yescharacteristic
83C22×Q8×C61C2×C411yescharacteristic, complemented
84C4⋊Q8×C61C2×C412yescomplemented
85#124×C61C2312yes—
86#123×C61—22no—
87C22⋊Q8×C61—22no—
88#117×C61—22no—
89#117×C61C2×C411yescharacteristic, complemented
90#99×C61—22no—
91#99×C61C2314yes—
92C23×D61C22⋊C412yes—
93C23×D61C2×D414yes—
94#41×C61—42no—
95C2×Q8×C61—21no—
96C2×Q8×C61—42no—
97C2×Q8×C61C22×C412yes—
98C2×Q8×C61C22×C411yescharacteristic
99C2×D4×C61C2×D412yes—
100C2×D4×C61C2411yescharacteristic
101C22×C4×C61—21no—
102C22×C4×C61C22×C411yescharacteristic
103C22×C4×C61C2×D412yes—
104C2×C22⋊D61—21no—
105C2×C22⋊D61—22no—
106C2×C22⋊D61C22⋊C412yes—
107C2×C22⋊D61C2×D414yes—
108C22×Dic61—21no—
109#11758226—44no—
110#11758226—44no—
111#11758226—44no—
112Q8×D61—44no—
113Q8×D61—44no—
114D4×D61—44no—
115D4×D61—48no—
116#11758222—24no—
117#11758222—42no—
118#11758222—42no—
119#11758222—48no—
120#11758222—42no—
121C2×D244—24no—
122C2×D244—24no—
123C2×D244—22no—
124C2×D244—24no—
125C2×D244—42no—
126C2×D244C22⋊C414yes—
127C2×D244C22⋊C412yes—
128C2×D244C22⋊C411yescharacteristic
129C2×D244C22×C411yescharacteristic
130C2×D244C22×C412yes—
131C2×C4×D61—22no—
132C2×C4×D61—21no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—48no—
136C2×C4×D61C22⋊C414yes—
137C2×C4×D61C22×C412yes—
138C2×Dic122—22no—
139C2×Dic122C22⋊C411yescharacteristic
140C2×Dic122C22×C411yescharacteristic
141C4⋊C4×C61—21no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—42no—
144C22⋊C4×C61—21no—
145C22⋊C4×C61—24no—
146C22⋊C4×C61—24no—
147C42×C61—42no—
148C42×C61C22×C412yes—
149C22⋊Dic61—22no—
150C22⋊Dic61—24no—
151#11758198—24no—
152#11758198—24no—
153#11758198—24no—
154#11758198—22no—
155#11758198—22no—
156#11758198—48no—
157C4⋊Dic61—42no—
158#11758196—22no—
159#11758196—22no—
160C4×Dic61—21no—
161C4×Dic61—22no—
162C4×Dic61—42no—
163C4×Dic61C22×C411yescharacteristic
164C4×Dic61C22×C411yescharacteristic
165C23×C61C2×C22⋊C411yescharacteristic
166C23×C61C22×D412yes—
167C22×D61—28no—
168C22×D61—21no—
169C22×D61—42no—
170C22×D61—84no—
171C22×D61—82no—
172C22×D61C2×C22⋊C414yes—
173C22×D61C2×C22⋊C412yes—
174Q8×C61—41no—
175Q8×C61—42no—
176Q8×C61—42no—
177D4×C61—42no—
178D4×C61—41no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—42no—
185C2×C4×C61—44no—
186C2×C4×C61C2×C22⋊C412yes—
187C2×C4×C61C2×C22⋊C411yescharacteristic
188C2×C4×C61C23×C411yescharacteristic
189C22⋊D61—24no—
190C22⋊D61—44no—
191C22⋊D61—42no—
192C22⋊D61—82no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—44no—
197C2×Dic61—81no—
198C2×Dic61C2×C22⋊C412yes—
199D244—28no—
200D244—22no—
201D244—22no—
202D244—42no—
203D244—41no—
204D244—44no—
205D244—84no—
206D244#9914yes—
207C4×D61—24no—
208C4×D61—44no—
209C4×D61—44no—
210C4×D61—44no—
211C4×D61—42no—
212C4×D61—82no—
213C4×D61—168no—
214Dic122—22no—
215Dic122—44no—
216Dic122—41no—
217#33361—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—21no—
220C22×C61—41no—
221C22×C61—42no—
222C22×C61C22×C22⋊C411yescharacteristic
223C2×D61—24no—
224C2×D61—42no—
225C2×D61—44no—
226C2×D61—42no—
227C2×D61—41no—
228C2×D61—88no—
229C2×D61—162no—
230C2×D61C2×#9914yes—
231C4×C61—21no—
232C4×C61—22no—
233C4×C61—42no—
234C4×C61—42no—
235C4×C61—84no—
236C4×C61C2×#9912yes—
237Dic61—22no—
238Dic61—41no—
239Dic61—41no—
240Dic61—41no—
241Dic61—84no—
242#3500—611no—
243#2883—612no—
244#2129—611no—
245#2129D6111yescharacteristic, complemented
246#2127—614no—
247#2126—612no—
248C2×#354—614no—
249C2×C61—21no—
250C2×C61—42no—
251C2×C61—41no—
252C2×C61C22×#9911yescharacteristic
253D61—24no—
254D61—44no—
255D61—81no—
256C2×2−1+4—611no—
257C2×H(5,𝔽2)—611no—
258#548—1222no—
259#546—1222no—
260#546—1224no—
261#536—612no—
262#536C2×D6111yescharacteristic
263#535—612no—
264#535—611no—
265#532—611no—
266#532—612no—
267C2×#124—611no—
268C2×#124—612no—
269#518—1222no—
270C4×#41—611no—
271#410—612no—
272#410—1222no—
273#410C2×D6112yes—
274C2×#99—618no—
275C2×#99—1224no—
276#354—614no—
277#354—614no—
278#354C2×D6114yescomplemented
279#353—614no—
280C61#3336111yescharacteristic, complemented, Sylow
2812−1+4—2444no—
282H(5,𝔽2)—2444no—
283C2×#41—612no—
284C2×#41—611no—
285C2×#41—612no—
286C2×#41—612no—
287C2×#41—611no—
288C2×#41—1222no—
289C2×#41—1221no—
290C2×#41C22×D6111yescharacteristic
291C22×Q8—612no—
292C22×Q8—1221no—
293C22×Q8C4×D6111yescharacteristic, complemented
294C22×D4—614no—
295C22×D4—612no—
296C22×D4—611no—
297C22×D4—612no—
298C4⋊Q8—611no—
299C4⋊Q8C4×D6112yescomplemented
300#127—611no—
301#127—612no—
302#124—612no—
303#124—612no—
304#124—611no—
305#124—611no—
306#124—612no—
307#124—1224no—
308#124—1222no—
309#124C22×D6112yes—
310#123—22no—
311#123—1224no—
312C22⋊Q8—22no—
313C22⋊Q8—1224no—
314#121—1224no—
315#121—1224no—
316#121—1222no—
317#121—1222no—
318#120—1224no—
319#120—1222no—
320C4×Q8—1221no—
321C4×D4—1222no—
322C4×D4—1224no—
323C4×D4—1221no—
324C4×D4—1222no—
325C4×D4—2444no—
326C4×D4—2444no—
327#117—22no—
328#117—612no—
329#117—1222no—
330#117C4×D6111yescharacteristic, complemented
331C2×C22⋊C4—1224no—
332C2×C22⋊C4—1228no—
333C2×C22⋊C4—1222no—
334C2×C42—611no—
335C2×C42—612no—
336#99—22no—
337#99—614no—
338#99—1228no—
339#99—1222no—
340#99C22×D6114yes—
341C24—614no—
342C24—612no—
343#41—42no—
344#41—1224no—
345#41—2442no—
346#41—2442no—
347#41—2442no—
348#41—2448no—
349#41—2444no—
350#41—2444no—
351#41—2444no—
352C2×Q8—21no—
353C2×Q8—42no—
354C2×Q8—611no—
355C2×Q8—611no—
356C2×Q8—1222no—
357C2×Q8—2444no—
358C2×Q8—2444no—
359C2×Q8C2×C4×D6112yes—
360C2×Q8C2×C4×D6111yescharacteristic
361C2×D4—612no—
362C2×D4—614no—
363C2×D4—612no—
364C2×D4—611no—
365C2×D4—614no—
366C2×D4—611no—
367C2×D4—612no—
368C2×D4—1224no—
369C2×D4—1224no—
370C2×D4—1224no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1221no—
374C2×D4—2442no—
375C2×D4—2444no—
376C2×D4—2448no—
377C2×D4D4×D6112yes—
378C2×D4C23×D6111yescharacteristic
379C22×C4—21no—
380C22×C4—612no—
381C22×C4—614no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—2448no—
388C22×C4C2×C4×D6111yescharacteristic
389C22×C4D4×D6112yes—
390C4⋊C4—24no—
391C4⋊C4—21no—
392C4⋊C4—42no—
393C4⋊C4—1222no—
394C4⋊C4—1222no—
395C4⋊C4—2442no—
396C22⋊C4—24no—
397C22⋊C4—21no—
398C22⋊C4—24no—
399C22⋊C4—1224no—
400C22⋊C4—1224no—
401C22⋊C4—1224no—
402C22⋊C4—1222no—
403C22⋊C4—1222no—
404C22⋊C4—1222no—
405C22⋊C4—1224no—
406C22⋊C4—2448no—
407C42—42no—
408C42—611no—
409C42—611no—
410C42—1222no—
411C42—1221no—
412C42—2442no—
413C42C2×C4×D6112yes—
414C23—612no—
415C23—614no—
416C23—1228no—
417C23—1221no—
418C23—4882no—
419C23—4884no—
420C23—2442no—
421C23C22⋊C4×D6111yescharacteristic
422C23C2×D4×D6112yes—
423Q8—41no—
424Q8—42no—
425Q8—42no—
426Q8—1222no—
427Q8—2441no—
428Q8—2444no—
429D4—42no—
430D4—41no—
431D4—614no—
432D4—1228no—
433D4—1224no—
434D4—1222no—
435D4—1222no—
436D4—4884no—
437D4—4882no—
438D4—2442no—
439D4—2441no—
440D4—2444no—
441D4—2444no—
442D4—2442no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—612no—
451C2×C4—1224no—
452C2×C4—1221no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—9768no—
456C2×C4—4882no—
457C2×C4—4881no—
458C2×C4—2444no—
459C2×C4—2444no—
460C2×C4—2444no—
461C2×C4—2444no—
462C2×C4—2442no—
463C2×C4C22⋊C4×D6112yes—
464C2×C4C22⋊C4×D6111yescharacteristic
465C2×C4C22×C4×D6111yescharacteristic
466C22—22no—
467C22—21no—
468C22—41no—
469C22—42no—
470C22—614no—
471C22—1224no—
472C22—9762no—
473C22—4888no—
474C22—2442no—
475C22—2442no—
476C22—2444no—
477C22—2441no—
478C22C2×C22⋊C4×D6111yescharacteristic
479C4—21no—
480C4—22no—
481C4—42no—
482C4—42no—
483C4—84no—
484C4—1222no—
485C4—4884no—
486C4—2441no—
487C4—2441no—
488C4—2441no—
489C4#99×D6112yes—
490C2—21no—
491C2—42no—
492C2—41no—
493C2—1224no—
494C2—4881no—
495C2—2444no—
496C2C2×#99×D6111yescharacteristic
subgroupG/H
centerC2C2×#99×D61
commutatorC2×C4×C61C23×C4
FrattiniC2×D4C23×D61
FittingC61×#2129C2
socleC2×C61C22×#99
prime radical#21459608C2
cosocleC2×D4×C61C24
hypercenter#2129D61
hypocenterC61#33361
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2129

Complements

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

NH ≅ G/Ntype
2. #21454278C2semidirect
3. #21447494C2semidirect
4. #21447487C2semidirect
5. #21447482C2semidirect
6. #354×D61C2semidirect
7. C61×#2129C2semidirect
8. C2×#14519825C4semidirect
13. #16840337C4semidirect
14. #16840331C4semidirect
16. #16840313C4semidirect
18. C2×#14519766C4semidirect
21. #16840230C4semidirect
26. #16839557C22semidirect
27. #16839556C22semidirect
28. #16839554C22semidirect
31. C61×#354C22semidirect
36. C2×#11758226C2×C4semidirect
39. C2×#11758222C2×C4semidirect
42. C2×Q8×D61C2×C4semidirect
47. #14519775C2×C4semidirect
48. #14519774C2×C4semidirect
51. #14519766C2×C4semidirect
52. #14519766C2×C4semidirect
53. #14519766C2×C4semidirect
54. #14519765C2×C4semidirect
73. #14519734C2×C4semidirect
77. C2×C4×Dic61C2×C4semidirect
78. C42×D61C2×C4semidirect
83. C22×Q8×C61C2×C4semidirect
84. C4⋊Q8×C61C2×C4semidirect
89. #117×C61C2×C4semidirect
245. #2129D61semidirect
278. #354C2×D61semidirect
280. C61#33361semidirect
293. C22×Q8C4×D61semidirect
299. C4⋊Q8C4×D61semidirect
330. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596082
#214542782
#214474942
#214474872
#214474822
#354×D612
C61×#21292
#3336161