Finite Group Databasebeta

← #29662911

Subgroups — #29662911

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456193C211yesmaximal, characteristic
2#21455859C211yesmaximal, characteristic, complemented
3C2×#16839959C211yesmaximal, characteristic
4#21454770C211yesmaximal, characteristic, complemented
5#21445386C212yesmaximal, complemented
6#21445358C212yesmaximal, complemented
7#21445262C212yesmaximal, complemented
8#21445234C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10#21444510C211yesmaximal, characteristic, complemented
11C61×#1545C211yesmaximal, characteristic
12C22×#11758200—21no—
13#16840319C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15#16840126—22no—
16#16840108C2212yes—
17#16840108C2212yes—
18#16840098—21no—
19#16840058C2211yescharacteristic, complemented
20#16840058C2211yescharacteristic, complemented
21#16839973C2211yescharacteristic, complemented
22#16839973C2211yescharacteristic, complemented
23#16839959—21no—
24#16839959—21no—
25#16839959C2212yes—
26#16839959C2212yes—
27#16839920—21no—
28#16839891—22no—
29#16839889C2212yescomplemented
30#16839886C2212yescomplemented
31C4×Dic244—21no—
32C4×Dic244—21no—
33#16839785—21no—
34C2×C4⋊#92418C2211yescharacteristic
35#16839767—21no—
36#16839767—21no—
37C2×#14519698—21no—
38C4×#11758194C2211yescharacteristic
39M16×Dic61—21no—
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839510C2212yescomplemented
43#16839507C2212yescomplemented
44C2×#127×C61C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C61×#331C2212yes—
48C61×#331C2212yes—
49C22×Dic122D411yescharacteristic
50C2×Dic244—21no—
51C2×Dic244—21no—
52C2×#11758200—22no—
53C2×#11758200—22no—
54C2×#11758200—44no—
55C2×#11758194D411yescharacteristic
56C2×#11758194D411yescharacteristic
57C22×#92418—21no—
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65C4×Dic122—21no—
66C4×Dic122—21no—
67C4×Dic122D411yescharacteristic
68C4×Dic122D411yescharacteristic
69C4×Dic122C2311yescharacteristic
70C4×Dic122C2311yescharacteristic
71#14519736—21no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×#11758196—21no—
74C2×C4×Dic61—21no—
75#14519711—21no—
76#14519711—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79#14519706—22no—
80#14519706—22no—
81#14519702—21no—
82#14519702—21no—
83#14519702—21no—
84#14519702—21no—
85#14519698—22no—
86#14519698—22no—
87#14519698—22no—
88#14519698—22no—
89#14519698—22no—
90#14519698—22no—
91#14519688D412yes—
92#14519687—22no—
93C4×#92418C2312yes—
94C8×Dic61—22no—
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61—21no—
98C2×M16×C61—21no—
99#127×C61—21no—
100#127×C61—21no—
101#127×C61—21no—
102#127×C61C2312yes—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2311yescharacteristic
106C4⋊C8×C61C2311yescharacteristic
107C4⋊C8×C61C2312yes—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C24×C61—41no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—41no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61—82no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C22×Dic61—21no—
126C22×Dic61—21no—
127C2×Dic122—22no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—41no—
132C2×Dic122C2×D411yescharacteristic
133C2×Dic122C2×D411yescharacteristic
134M16×C61—42no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—22no—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61#4111yescharacteristic
141C42×C61C2411yescharacteristic
142C22⋊Dic61—41no—
143#11758200—44no—
144#11758200—44no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61C2×D411yescharacteristic
151C4⋊Dic61C2×D411yescharacteristic
152#11758196—21no—
153#11758196—21no—
154#11758196—21no—
155#11758196—21no—
156#11758196—41no—
157#11758196—41no—
158C4×Dic61—21no—
159C4×Dic61—21no—
160C4×Dic61—21no—
161C4×Dic61—21no—
162C4×Dic61—41no—
163#11758194—22no—
164#11758194—22no—
165C2×#92418—22no—
166C2×#92418—42no—
167C2×#92418C2×D412yes—
168C2×#92418C2×D412yes—
169Dic244—42no—
170Dic244—42no—
171C23×C61—42no—
172C23×C61—82no—
173C23×C61—82no—
174C23×C61—82no—
175C23×C61#12111yescharacteristic
176D4×C61—42no—
177D4×C61—42no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—82no—
181D4×C61—82no—
182D4×C61—82no—
183D4×C61—82no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—41no—
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#13512yes—
191C2×C4×C61#13712yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—42no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—81no—
203Dic122—21no—
204Dic122—21no—
205Dic122—21no—
206Dic122—42no—
207Dic122—41no—
208Dic122—41no—
209Dic122—81no—
210Dic122AGL(1,ℤ/8)12yes—
211C8×C61—42no—
212C8×C61—41no—
213C8×C61—41no—
214#92418—22no—
215#92418—22no—
216#92418—42no—
217#19174—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—82no—
221C22×C61—82no—
222C22×C61—82no—
223C22×C61—164no—
224C22×C61#46412yes—
225C22×C61C2×#12111yescharacteristic
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61#46412yes—
230C4×C61C2×#13511yescharacteristic
231C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C61C2×#13711yescharacteristic
233C4×C61#57511yescharacteristic
234Dic61—22no—
235Dic61—41no—
236Dic61—41no—
237Dic61—41no—
238Dic61—81no—
239Dic61—81no—
240#3138—611no—
241#3104—611no—
242C2×#460—611no—
243Q8×M16—611no—
244#1702—612no—
245#1698—612no—
246#1682—612no—
247#1678—612no—
248#1565—611no—
249#1550—611no—
250#1545D6111yescharacteristic
251C2×C61—22no—
252C2×C61—162no—
253C2×C61C2×#46411yescharacteristic
254C2×C61#305511yescharacteristic
255C2×C61#306811yescharacteristic
256C22×SD16—1221no—
257#533—611no—
258C2×#127C2×D6111yescharacteristic
259C2×C4×Q8—611no—
260#490—1222no—
261#486—612no—
262#486—612no—
263#484—1221no—
264#477—611no—
265#477—611no—
266#462—611no—
267#462—611no—
268#460—612no—
269#460—612no—
270#460—1221no—
271#460—1221no—
272#451—1221no—
273#446—612no—
274#446—1222no—
275C8×Q8—612no—
276C4×Q16—1221no—
277C4×Q16—1221no—
278#431C2×D6111yescharacteristic
279#423—1221no—
280#423C2×D6111yescharacteristic
281C2×C4⋊C8—611no—
282#418—1221no—
283#418—1221no—
284C2×#102—1221no—
285C4×M16—611no—
286C4×M16—1221no—
287#335—612no—
288#334—612no—
289#333—612no—
290#332—612no—
291#331C2×D6112yes—
292#331C2×D6112yes—
293C61#1917411yescharacteristic, complemented, Sylow
294C22×Q8—611no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—21no—
298C2×Q16—1221no—
299C2×Q16—1221no—
300C2×SD16—1222no—
301C2×SD16—1222no—
302C2×SD16—2444no—
303C2×M16—21no—
304C2×M16—611no—
305C2×M16—611no—
306C22×C8—1221no—
307C4⋊Q8—611no—
308C4⋊Q8—611no—
309#127—21no—
310#127—21no—
311#127—21no—
312#127C22×D6112yes—
313#125—1221no—
314C22⋊Q8—1221no—
315C22⋊Q8—1221no—
316C4×Q8—611no—
317C4×Q8—611no—
318C4×Q8—611no—
319C4×Q8—611no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×Q8—1221no—
324#117—1221no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—1221no—
327C2×C42—1221no—
328C2×C42C22×D6111yescharacteristic
329#107—1221no—
330#107—1221no—
331C4⋊C8—21no—
332C4⋊C8—612no—
333C4⋊C8—1222no—
334C4⋊C8—1222no—
335C4⋊C8—1221no—
336C4⋊C8C22×D6112yes—
337C4⋊C8C22×D6111yescharacteristic
338C4⋊C8C22×D6111yescharacteristic
339#103—1221no—
340#103—1221no—
341#103—1221no—
342#103—1221no—
343#102—1222no—
344#102—1222no—
345#102—1222no—
346#102—1222no—
347#102—1222no—
348#102—1222no—
349C22⋊C8—21no—
350#97—612no—
351#97—1222no—
352C4×C8—21no—
353C4×C8—612no—
354C4×C8—1222no—
355C24—41no—
356C2×Q8—611no—
357C2×Q8—611no—
358C2×Q8—1222no—
359C2×Q8—1221no—
360C2×Q8—1221no—
361C2×Q8—1221no—
362C2×Q8—2441no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4#1175822211yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4D4×D6111yescharacteristic
379Q16—2442no—
380Q16—2442no—
381SD16—2444no—
382SD16—2444no—
383M16—42no—
384M16—1222no—
385M16—1222no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—612no—
390C2×C8—612no—
391C2×C8—1222no—
392C2×C8—2442no—
393C4⋊C4—611no—
394C4⋊C4—611no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C22⋊C4—2441no—
407C42—1221no—
408C42—1221no—
409C42—1221no—
410C42—1221no—
411C42—2441no—
412C42#1175822211yescharacteristic
413C42D4×D6111yescharacteristic
414C42D4×D6111yescharacteristic
415C42C23×D6111yescharacteristic
416C23—42no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23#1451975011yescharacteristic
421Q8—612no—
422Q8—1221no—
423Q8—1221no—
424Q8—1221no—
425Q8—4881no—
426Q8—2442no—
427Q8—2441no—
428Q8—2441no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—4881no—
445C2×C4—2442no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4#1451975011yescharacteristic
449C2×C4#1451975011yescharacteristic
450C2×C4#1451975011yescharacteristic
451C2×C4#1451979612yes—
452C2×C4#1451980612yes—
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C2×D4×D6111yescharacteristic
455C2×C4C2×#1175822211yescharacteristic
456C8—42no—
457C8—41no—
458C8—41no—
459C8—1222no—
460C8—1222no—
461C8—2442no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22#1683998312yes—
469C22C2×#1451975011yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—1222no—
474C4—4881no—
475C4—4881no—
476C4—2441no—
477C4—2441no—
478C4—2441no—
479C4#1683998312yes—
480C4C2×#1451979611yescharacteristic
481C4AGL(1,ℤ/8)×D6111yescharacteristic
482C4C2×#1451980611yescharacteristic
483C4#1684055811yescharacteristic
484C2—22no—
485C2—162no—
486C2C2×#1683998311yescharacteristic
487C2#2145543611yescharacteristic
488C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C2×#121
prime radicalC2×#127×C61C22
hypercenter#1545D61
hypocenterC61#19174
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #21455859C2semidirect
4. #21454770C2semidirect
5. #21445386C2semidirect
6. #21445358C2semidirect
7. #21445262C2semidirect
8. #21445234C2semidirect
9. #21444572C2semidirect
10. #21444510C2semidirect
19. #16840058C22semidirect
20. #16840058C22semidirect
21. #16839973C22semidirect
22. #16839973C22semidirect
29. #16839889C22semidirect
30. #16839886C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839510C22semidirect
43. #16839507C22semidirect
293. C61#19174semidirect

Maximal subgroups

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

subgroupindex
#214561932
#214558592
C2×#168399592
#214547702
#214453862
#214453582
#214452622
#214452342
#214445722
#214445102
C61×#15452
#1917461