Finite Group Databasebeta

← #29662812

Subgroups — #29662812

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456361C211yesmaximal, characteristic, complemented
2#21455828C211yesmaximal, characteristic
3C2×#16840049C211yesmaximal, characteristic
4#21454554C211yesmaximal, characteristic
5#21445341C212yesmaximal, complemented
6#21445321C212yesmaximal, complemented
7#21445217C212yesmaximal, complemented
8#21445197C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10#21444510C211yesmaximal, characteristic, complemented
11C61×#1548C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13#16840319C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840168—22no—
16#16840165C2212yescomplemented
17#16840158C2212yescomplemented
18#16840093—21no—
19#16840053C2211yescharacteristic
20#16840053C2211yescharacteristic
21#16840049—21no—
22#16840049—21no—
23#16840049C2212yes—
24#16840049C2212yes—
25#16839968C2211yescharacteristic
26#16839968C2211yescharacteristic
27#16839915—21no—
28C4×#11758190—21no—
29C4×#11758190—21no—
30#16839839C2212yes—
31#16839832—22no—
32#16839827C2212yes—
33#16839802—21no—
34#16839802—21no—
35C2×#14519708—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C2×#14519697—21no—
38#16839725—21no—
39C4×#11758194C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839510C2212yescomplemented
43#16839507C2212yescomplemented
44C2×C4⋊Q8×C61C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C61×#333C2212yescomplemented
48C61×#333C2212yescomplemented
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758190—21no—
52C2×#11758190—21no—
53C2×#11758194D411yescharacteristic
54C2×#11758194D411yescharacteristic
55C2×#11758189—22no—
56M16×D61—44no—
57C22×#92418—21no—
58C2×C8×D61—22no—
59#14519777C2311yescharacteristic
60#14519777C2311yescharacteristic
61#14519769—21no—
62#14519756—21no—
63#14519756—21no—
64C4×Dic122—21no—
65C4×Dic122—21no—
66C4×D244—21no—
67C4×D244—21no—
68C4×D244D411yescharacteristic
69C4×D244D411yescharacteristic
70C4×D244C2311yescharacteristic
71C4×D244C2311yescharacteristic
72#14519736—21no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×#11758198—21no—
75#14519710—22no—
76#14519710—22no—
77#14519709—21no—
78#14519709—21no—
79#14519708—22no—
80#14519708—22no—
81#14519708—22no—
82#14519708—22no—
83C4⋊#92418—21no—
84C4⋊#92418C2312yes—
85#14519702—21no—
86#14519702—21no—
87#14519699—21no—
88#14519699—21no—
89#14519697—22no—
90#14519697—22no—
91#14519689—22no—
92#14519689—22no—
93#14519688D412yes—
94C4×#92418C2312yes—
95C22×Q8×C61—21no—
96C2×M16×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61C2312yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2311yescharacteristic
106C4⋊C8×C61C2311yescharacteristic
107C4⋊C8×C61C2312yes—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C23×D61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—41no—
113C2×Q8×C61—82no—
114C22×C4×C61—41no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×Dic61—21no—
119C2×D244—22no—
120C2×D244—21no—
121C2×D244—41no—
122C2×D244C2×D411yescharacteristic
123C2×D244C2×D411yescharacteristic
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—21no—
131C2×Dic122—21no—
132M16×C61—42no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61—41no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144C22⋊Dic61—41no—
145#11758198—21no—
146#11758198—21no—
147#11758198—21no—
148#11758198—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61C2×D411yescharacteristic
155C4⋊Dic61C2×D411yescharacteristic
156#11758196—41no—
157#11758196—41no—
158C4×Dic61—41no—
159#11758194—22no—
160#11758194SD1614yescomplemented
161C2×#92418—22no—
162C2×#92418—42no—
163C2×#92418C2×D412yes—
164C2×#92418C2×D412yes—
165#11758190—42no—
166#11758190—42no—
167#11758189—44no—
168C8×D61—44no—
169C23×C61#12111yescharacteristic
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—81no—
175Q8×C61—82no—
176Q8×C61—82no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—82no—
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61C2×SD1612yes—
186C2×C4×C61#13712yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195D244—21no—
196D244—21no—
197D244—21no—
198D244—42no—
199D244#13712yes—
200C4×D61—41no—
201C4×D61—41no—
202C4×D61—41no—
203C4×D61—41no—
204C4×D61—42no—
205C4×D61—42no—
206Dic122—41no—
207Dic122—41no—
208Dic122—81no—
209C8×C61—42no—
210C8×C61—41no—
211C8×C61—41no—
212#92418—22no—
213#92418—42no—
214#92418C2×SD1614yes—
215#19165—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61#46112yes—
219C22×C61C2×#12111yescharacteristic
220C2×D61—22no—
221C2×D61—41no—
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—84no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—82no—
231C4×C61#46112yes—
232C4×C61C22×SD1611yescharacteristic
233C4×C61C2×#13711yescharacteristic
234C4×C61C2×#13711yescharacteristic
235C4×C61#57511yescharacteristic
236Dic61—41no—
237Dic61—41no—
238Dic61—81no—
239Dic61—81no—
240#3165—611no—
241#3101—611no—
242C2×#476—611no—
243D4×M16—611no—
244#1695—612no—
245#1691—612no—
246#1675—612no—
247#1671—612no—
248#1565—611no—
249#1550—611no—
250#1548D6111yescharacteristic, complemented
251C2×C61—22no—
252C2×C61C2×#46111yescharacteristic
253C2×C61#305611yescharacteristic
254C2×C61#306811yescharacteristic
255D61—42no—
256D61—81no—
257C22×M16—1221no—
258#533—611no—
259C2×C4⋊Q8C2×D6111yescharacteristic
260C2×C4×D4—611no—
261#501—1222no—
262#500—612no—
263#498—612no—
264#483—1221no—
265#476—612no—
266#476—612no—
267#476—611no—
268#476—611no—
269#476—1221no—
270#476—1221no—
271#461—611no—
272#461—611no—
273#450—1221no—
274C4×SD16—1221no—
275C4×SD16—1221no—
276#436—612no—
277#435—1222no—
278C8×D4—612no—
279#431C2×D6111yescharacteristic
280#428—1221no—
281#428—1221no—
282C2×#106—1221no—
283#423C2×D6111yescharacteristic
284C2×C4⋊C8—611no—
285C2×#102—1221no—
286#407—1221no—
287C4×M16—611no—
288#335—612no—
289#334—612no—
290#333—612no—
291#333C2×D6112yescomplemented
292#333C2×D6112yescomplemented
293#332—612no—
294C61#1916511yescharacteristic, complemented, Sylow
295C22×Q8—21no—
296C22×D4—611no—
297C23×C4—1221no—
298C2×SD16—1221no—
299C2×SD16—1221no—
300C2×M16—21no—
301C2×M16—611no—
302C2×M16—611no—
303C2×M16—1222no—
304C2×M16—2444no—
305C22×C8—1222no—
306C22×C8—1221no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—611no—
311C4⋊Q8—611no—
312C4⋊Q8C22×D6112yes—
313#125—1221no—
314C22⋊Q8—1221no—
315C22⋊Q8—1221no—
316C4×Q8—1221no—
317C4×Q8—1221no—
318C4×D4—611no—
319C4×D4—611no—
320C4×D4—611no—
321C4×D4—611no—
322C4×D4—1221no—
323C4×D4—1221no—
324#117—1221no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—611no—
328C2×C22⋊C4—1221no—
329C2×C42C22×D6111yescharacteristic
330#107—1222no—
331#107—1222no—
332#106—1222no—
333#106—1222no—
334#106—1222no—
335#106—1222no—
336#106—1221no—
337#106—1221no—
338C4⋊C8—21no—
339C4⋊C8—612no—
340C4⋊C8—1221no—
341C4⋊C8C22×D6112yes—
342C4⋊C8C22×D6111yescharacteristic
343C4⋊C8C22×D6111yescharacteristic
344#103—1221no—
345#103—1221no—
346#102—1222no—
347#102—1222no—
348#102—1221no—
349#102—1221no—
350C22⋊C8—21no—
351C22⋊C8—1222no—
352C22⋊C8—1222no—
353#97—612no—
354C4×C8—21no—
355C4×C8—612no—
356C24—1221no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—82no—
360C2×Q8—1221no—
361C2×Q8—1221no—
362C2×D4—611no—
363C2×D4—611no—
364C2×D4—1222no—
365C2×D4—1221no—
366C2×D4—2441no—
367C22×C4—41no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—2442no—
374C22×C4—2442no—
375C22×C4#1175822211yescharacteristic
376C22×C4D4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378SD16—2442no—
379SD16—2442no—
380M16—42no—
381M16—614no—
382M16—1222no—
383M16—2444no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—612no—
388C2×C8—612no—
389C2×C8—1222no—
390C2×C8—2444no—
391C2×C8—2442no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—611no—
397C4⋊C4—611no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C22⋊C4—1221no—
406C22⋊C4—1221no—
407C22⋊C4—1221no—
408C22⋊C4—1221no—
409C22⋊C4—2441no—
410C42—2441no—
411C42#1175822211yescharacteristic
412C42D4×D6111yescharacteristic
413C42D4×D6111yescharacteristic
414C42C23×D6111yescharacteristic
415C23—1221no—
416C23—1221no—
417C23—4881no—
418C23—2442no—
419C23—2442no—
420C23#1451975011yescharacteristic
421Q8—82no—
422Q8—82no—
423Q8—4881no—
424Q8—2441no—
425Q8—2441no—
426D4—612no—
427D4—1221no—
428D4—1221no—
429D4—1221no—
430D4—2442no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—82no—
436C2×C4—1221no—
437C2×C4—1221no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4#1451975011yescharacteristic
448C2×C4#1451975011yescharacteristic
449C2×C4#1451975011yescharacteristic
450C2×C4SD16×D6112yes—
451C2×C4#1451980912yes—
452C2×C4C2×D4×D6111yescharacteristic
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C2×#1175822211yescharacteristic
455C8—42no—
456C8—41no—
457C8—41no—
458C8—614no—
459C8—1222no—
460C8—2442no—
461C22—21no—
462C22—21no—
463C22—1222no—
464C22—4884no—
465C22—2441no—
466C22—2441no—
467C22—2441no—
468C22—2441no—
469C22—2441no—
470C22#1683996612yes—
471C22C2×#1451975011yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—82no—
476C4—4881no—
477C4—4881no—
478C4—2441no—
479C4—2441no—
480C4#1683996612yes—
481C4C2×SD16×D6111yescharacteristic
482C4#137×D6111yescharacteristic
483C4C2×#1451980911yescharacteristic
484C4#1684056111yescharacteristic
485C2—22no—
486C2—4881no—
487C2—2442no—
488C2C2×#1683996611yescharacteristic
489C2#2145544711yescharacteristic
490C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1548D61
hypocenterC61#19165
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #21456361C2semidirect
5. #21445341C2semidirect
6. #21445321C2semidirect
7. #21445217C2semidirect
8. #21445197C2semidirect
9. #21444572C2semidirect
10. #21444510C2semidirect
11. C61×#1548C2semidirect
16. #16840165C22semidirect
17. #16840158C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839510C22semidirect
43. #16839507C22semidirect
47. C61×#333C22semidirect
48. C61×#333C22semidirect
160. #11758194SD16semidirect
250. #1548D61semidirect
291. #333C2×D61semidirect
292. #333C2×D61semidirect
294. C61#19165semidirect

Maximal subgroups

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

subgroupindex
#214563612
#214558282
C2×#168400492
#214545542
#214453412
#214453212
#214452172
#214451972
#214445722
#214445102
C61×#15482
#1916561