Finite Group Databasebeta

← #29718390

Subgroups — #29718390

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840095C211yesmaximal, characteristic, complemented
2C22×#14519699C211yesmaximal, characteristic
3#21449471C211yesmaximal, characteristic
4#21448698C211yesmaximal, characteristic, complemented
5#21447208C212yesmaximal, complemented
6#21447056C212yesmaximal, complemented
7#21446748C212yesmaximal, complemented
8#21446437C212yesmaximal, complemented
9#21445997C211yesmaximal, characteristic, complemented
10#21445749C211yesmaximal, characteristic
11C61×#1892C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C2×#14519766C2212yes—
14C22×C4⋊Dic61C2211yescharacteristic
15#16840095—24no—
16C2×#14519710C2211yescharacteristic, complemented
17C2×#14519710C2211yescharacteristic, complemented
18C2×#14519706C2212yescomplemented
19C2×#14519701C2211yescharacteristic, complemented
20C2×#14519701C2211yescharacteristic, complemented
21C2×#14519699—24no—
22C2×#14519699C2212yes—
23C2×#14519699C2212yes—
24C2×#14519697C2212yes—
25C2×#14519697C2212yes—
26C2×C22⋊#92418C2211yescharacteristic
27#16839679—22no—
28#16839663C2212yes—
29#16839658C2212yescomplemented
30#16839647—22no—
31#16839620—21no—
32#16839617—21no—
33#16839617—21no—
34C4⋊C4×Dic61C2212yescomplemented
35C22⋊C4×Dic61—21no—
36C4×C22⋊Dic61C2211yescharacteristic
37#16839516C2212yescomplemented
38C23×C8×C61C2211yescharacteristic
39C2×C22⋊Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yescomplemented
41C61×#380C2211yescharacteristic
42C61×#340C2212yescomplemented
43C24×D61—21no—
44C22×D244—22no—
45C22×D244—24no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244—42no—
49C22×D244C2312yes—
50C23×Dic61—21no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519766—24no—
54#14519766—42no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61D414yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—22no—
61C2×C22⋊Dic61D411yescharacteristic
62C2×C22⋊Dic61D411yescharacteristic
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61C2312yes—
69#14519710—22no—
70#14519710—22no—
71#14519706—42no—
72#14519701—22no—
73#14519701—22no—
74#14519699—24no—
75#14519699—24no—
76#14519697—24no—
77#14519697—42no—
78C22⋊#92418—22no—
79#14519684—22no—
80#14519684—22no—
81#14519684D412yes—
82C22×Q8×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C22×C8×C61—22no—
85C22×C8×C61—42no—
86C22×C8×C61C2312yes—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94#103×C61—42no—
95#95×C61—21no—
96C24×C61C2×D411yescharacteristic
97C23×D61—22no—
98C23×D61—44no—
99C23×D61—42no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—82no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61#4112yes—
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61—24no—
119C22×Dic61—82no—
120C22×Dic61C2×D412yes—
121C22×Dic61C2×D412yes—
122C2×D244—28no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—48no—
129C2×D244—44no—
130C2×D244—44no—
131C2×D244—42no—
132C2×D244#4114yes—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—44no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—42no—
142C22⋊C4×C61—42no—
143C22⋊C4×C61—42no—
144C22⋊Dic61—22no—
145C22⋊Dic61—22no—
146#11758198—44no—
147#11758198—44no—
148#11758198—82no—
149#11758198—82no—
150C4⋊Dic61—24no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61C2×D412yes—
154C4⋊Dic61C2×D412yes—
155#11758196—82no—
156C4×Dic61—22no—
157C4×Dic61—22no—
158C4×Dic61—82no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—22no—
162C2×#92418—22no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61#12112yes—
166C23×C61#13512yes—
167C23×C61#13712yes—
168C23×C61C22×D411yescharacteristic
169C22×D61—24no—
170C22×D61—48no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—88no—
178C22×D61—88no—
179C22×D61—84no—
180C22×D61—84no—
181Q8×C61—41no—
182Q8×C61—41no—
183Q8×C61—42no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61#12112yes—
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C2×#4112yes—
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61H(5,𝔽2)12yes—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—84no—
211D244—24no—
212D244—24no—
213D244—48no—
214D244—42no—
215D244—42no—
216C8×C61—42no—
217#92418—41no—
218#92418—41no—
219#24479—611nomaximal, Sylow
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—24no—
223C22×C61—24no—
224C22×C61#46414yes—
225C22×C61C2×#12111yescharacteristic
226C22×C61#54612yes—
227C22×C61C2×#13511yescharacteristic
228C22×C61C2×#13711yescharacteristic
229C22×C61#57612yes—
230C2×D61—44no—
231C2×D61—44no—
232C2×D61—44no—
233C2×D61—816no—
234C2×D61—82no—
235C2×D61—82no—
236C2×D61—82no—
237C2×D61—82no—
238C4×C61—22no—
239C4×C61—81no—
240C4×C61—81no—
241C4×C61—82no—
242C4×C61#52511yescharacteristic
243C4×C61#53511yescharacteristic
244C4×C61#55112yes—
245Dic61—42no—
246Dic61—42no—
247Dic61—42no—
248Dic61—42no—
249C2×#484—611no—
250C22×#102—611no—
251#2435—611no—
252#2345—611no—
253#2067—612no—
254#2042—612no—
255#1983—612no—
256#1925—612no—
257#1892D6111yescharacteristic, complemented
258#1850—611no—
259#1786—611no—
260C2×C61—24no—
261C2×C61#241111yescharacteristic
262C2×C61C2×#46411yescharacteristic
263C2×C61#304111yescharacteristic
264C2×C61#329712yes—
265C2×C61#331412yes—
266D61—84no—
267C23×D4—611no—
268C23×C8C2×D6111yescharacteristic
269C2×#124—612no—
270C2×C22⋊Q8C2×D6111yescharacteristic
271C22×C4⋊C4—611no—
272#484—1224no—
273C2×#107—611no—
274C2×#107—611no—
275C2×C4⋊C8—612no—
276C2×#103—611no—
277C2×#103—611no—
278C2×#103C2×D6112yescomplemented
279C2×#102—612no—
280C2×#102—612no—
281C2×#102—612no—
282C2×#102—612no—
283C2×#102—1224no—
284C2×C22⋊C8—611no—
285#394—1222no—
286#390—612no—
287#389—612no—
288#387—1222no—
289#380—1221no—
290#380C2×D6111yescharacteristic
291#379—1221no—
292#379—1221no—
293C4×C4⋊C4—612no—
294C4×C22⋊C4—611no—
295C4×C22⋊C4—1221no—
296#340C2×D6112yescomplemented
297#336—612no—
298C61#2447911yescharacteristic, complemented, Sylow
299C25—1221no—
300C22×Q8C22×D6111yescharacteristic
301C22×D4—612no—
302C22×D4—1224no—
303C22×D4—1222no—
304C22×D4—1222no—
305C22×D4—1221no—
306C22×D4—2442no—
307C23×C4—1221no—
308C23×C4C22×D6111yescharacteristic
309C22×C8—22no—
310C22×C8—42no—
311C22×C8—611no—
312C22×C8—611no—
313C22×C8C22×D6112yes—
314#124—1224no—
315#124—2442no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—614no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—611no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4C22×D6111yescharacteristic
326C2×C4⋊C4C22×D6112yes—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—611no—
330C2×C22⋊C4—611no—
331C2×C22⋊C4—1222no—
332C2×C22⋊C4—1222no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1222no—
335C2×C42—612no—
336C2×C42—1222no—
337#107—1222no—
338#107—1222no—
339C4⋊C8—2442no—
340#103—42no—
341#103—1222no—
342#103—1222no—
343#102—1224no—
344#102—1224no—
345#102—1224no—
346#102—2442no—
347C22⋊C8—1222no—
348#95—21no—
349#95—612no—
350#95—1222no—
351#95—1222no—
352C24—1222no—
353C24—4882no—
354C24—2444no—
355C24—2442no—
356C24—2442no—
357C24—2442no—
358C24D4×D6111yescharacteristic
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×D4—614no—
364C2×D4—1228no—
365C2×D4—1222no—
366C2×D4—1222no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—2448no—
371C2×D4—2444no—
372C2×D4—2444no—
373C2×D4—2442no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—612no—
380C22×C4—612no—
381C22×C4—1224no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—4882no—
385C22×C4#1175822212yes—
386C22×C4D4×D6112yes—
387C22×C4D4×D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—44no—
393C2×C8—1221no—
394C2×C8—1221no—
395C2×C8—1222no—
396C2×C8—1222no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—612no—
403C4⋊C4—612no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1224no—
407C4⋊C4—4882no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—1222no—
411C22⋊C4—1222no—
412C22⋊C4—4882no—
413C22⋊C4—4882no—
414C22⋊C4—2444no—
415C22⋊C4—2444no—
416C42—1222no—
417C42—1222no—
418C42—4882no—
419C23—22no—
420C23—22no—
421C23—1224no—
422C23—4884no—
423C23—4888no—
424C23—4888no—
425C23—4884no—
426C23—2448no—
427C23—2442no—
428C23—2442no—
429C23—2442no—
430C23—2442no—
431C23—2442no—
432C23—2442no—
433C23#1451975012yes—
434C23#1451979812yes—
435C23#1451980912yes—
436C23C2×D4×D6111yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—42no—
440D4—1224no—
441D4—1224no—
442D4—2448no—
443D4—2442no—
444D4—2442no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—4884no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4#1451975012yes—
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4#41×D6112yes—
470C2×C4C2×#1175822211yescharacteristic
471C2×C4#1451982212yes—
472C8—42no—
473C8—2441no—
474C8—2441no—
475C22—24no—
476C22—22no—
477C22—24no—
478C22—22no—
479C22—48816no—
480C22—4882no—
481C22—4882no—
482C22—4882no—
483C22—4882no—
484C22—2444no—
485C22—2444no—
486C22—2444no—
487C22#1683998714yes—
488C22C2×#1451975011yescharacteristic
489C22#1684040112yes—
490C22C2×#1451979811yescharacteristic
491C22C2×#1451980911yescharacteristic
492C22#1684056412yes—
493C4—22no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—2442no—
498C4—2442no—
499C4—2442no—
500C4—2442no—
501C4#1684027011yescharacteristic
502C4#1684032711yescharacteristic
503C4#1684044112yes—
504C2—24no—
505C2—4884no—
506C2#2144920711yescharacteristic
507C2C2×#1683998711yescharacteristic
508C2#2145532511yescharacteristic
509C2#2145767512yes—
510C2#2145790612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#1892D61
hypocenterC61#24479
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
1. C2×#16840095C2semidirect
4. #21448698C2semidirect
5. #21447208C2semidirect
6. #21447056C2semidirect
7. #21446748C2semidirect
8. #21446437C2semidirect
9. #21445997C2semidirect
11. C61×#1892C2semidirect
16. C2×#14519710C22semidirect
17. C2×#14519710C22semidirect
18. C2×#14519706C22semidirect
19. C2×#14519701C22semidirect
20. C2×#14519701C22semidirect
29. #16839658C22semidirect
34. C4⋊C4×Dic61C22semidirect
37. #16839516C22semidirect
40. C2×#103×C61C22semidirect
42. C61×#340C22semidirect
257. #1892D61semidirect
278. C2×#103C2×D61semidirect
296. #340C2×D61semidirect
298. C61#24479semidirect

Maximal subgroups

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

subgroupindex
C2×#168400952
C22×#145196992
#214494712
#214486982
#214472082
#214470562
#214467482
#214464372
#214459972
#214457492
C61×#18922
#2447961