Finite Group Databasebeta

← #29662913

Subgroups — #29662913

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#21456194C211yesmaximal, characteristic
2#21455858C211yesmaximal, characteristic, complemented
3C2×#16839962C211yesmaximal, characteristic
4#21445387C212yesmaximal, complemented
5#21445359C212yesmaximal, complemented
6#21445263C212yesmaximal, complemented
7#21445235C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444510C211yesmaximal, characteristic, complemented
10#21444488C211yesmaximal, characteristic
11Q8×M16×C61C211yesmaximal, characteristic, complemented
12C22×#11758201—21no—
13#16840319C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15#16840127—22no—
16#16840109C2212yes—
17#16840109C2212yes—
18#16840097—21no—
19#16840057C2211yescharacteristic, complemented
20#16840057C2211yescharacteristic, complemented
21#16839972C2211yescharacteristic, complemented
22#16839972C2211yescharacteristic, complemented
23#16839962—21no—
24#16839962—21no—
25#16839962C2212yes—
26#16839962C2212yes—
27#16839918—21no—
28C4×#11758202—21no—
29C4×#11758202—21no—
30#16839815C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839768—21no—
33#16839768—21no—
34C2×#14519697—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839510C2212yescomplemented
38#16839507C2212yescomplemented
39#16839505C2212yes—
40#16839505C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C61×#446—22no—
43C61×#446C2212yescomplemented
44C8×Q8×C61C2212yescomplemented
45C61×#423—21no—
46C61×#423C2211yescharacteristic
47C4×M16×C61—21no—
48C4×M16×C61C2211yescharacteristic
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52C2×#11758202—21no—
53C2×#11758202—21no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758201—44no—
57C22×#92418—21no—
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519774—21no—
61#14519774—21no—
62#14519774—21no—
63#14519774C2312yes—
64#14519769—21no—
65#14519756—21no—
66#14519756—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69#14519736—21no—
70#14519710—21no—
71#14519710—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76#14519701—21no—
77#14519701—21no—
78#14519700—21no—
79#14519700—21no—
80#14519699—22no—
81#14519699—22no—
82#14519699—22no—
83#14519699—22no—
84#14519697—22no—
85#14519697—22no—
86C22⋊#92418—21no—
87C4×#92418—21no—
88C22×Q8×C61D411yescharacteristic
89C2×M16×C61—21no—
90C2×M16×C61D411yescharacteristic
91C2×M16×C61D411yescharacteristic
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61D411yescharacteristic
95C4×Q8×C61D411yescharacteristic
96C4×Q8×C61C2311yescharacteristic
97C4×Q8×C61C2311yescharacteristic
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—22no—
103C4⋊C8×C61—22no—
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106#97×C61—22no—
107#97×C61D412yes—
108C4×C8×C61—22no—
109C4×C8×C61C2312yes—
110C23×D61—41no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C2×Q8×C61C2×D411yescharacteristic
115C2×Q8×C61C2×D411yescharacteristic
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—42no—
125C2×D244—42no—
126C2×D244—42no—
127C2×D244—42no—
128C2×D244—41no—
129C2×D244—41no—
130C2×D244—41no—
131C2×D244—82no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134M16×C61—22no—
135M16×C61—22no—
136M16×C61—42no—
137C2×C8×C61—22no—
138C2×C8×C61C2×D412yes—
139C2×C8×C61C2×D412yes—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61C2×D411yescharacteristic
146C4⋊C4×C61C2×D411yescharacteristic
147C42×C61—21no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61—21no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61#4111yescharacteristic
154C42×C61C2411yescharacteristic
155C22⋊Dic61—41no—
156#11758202—42no—
157#11758202—42no—
158#11758201—44no—
159#11758201—44no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164#11758196—41no—
165#11758196—41no—
166C4×Dic61—41no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—42no—
171C23×C61#12111yescharacteristic
172C22×D61—42no—
173C22×D61—82no—
174C22×D61—82no—
175C22×D61—82no—
176Q8×C61—21no—
177Q8×C61—21no—
178Q8×C61—21no—
179Q8×C61—42no—
180Q8×C61AGL(1,ℤ/8)12yes—
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—41no—
188C2×C4×C61—42no—
189C2×C4×C61—81no—
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#13512yes—
194C2×C4×C61#13712yes—
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×Dic61—41no—
199C2×Dic61—41no—
200D244—42no—
201D244—42no—
202D244—42no—
203D244—42no—
204D244—82no—
205D244—82no—
206D244—82no—
207D244—82no—
208Dic122—41no—
209Dic122—41no—
210Dic122—81no—
211C8×C61—22no—
212C8×C61—22no—
213C8×C61—42no—
214#92418—42no—
215#92418—41no—
216#92418—41no—
217#19174—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61#46412yes—
221C22×C61C2×#12111yescharacteristic
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—164no—
226C4×C61—21no—
227C4×C61—22no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—41no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61#46412yes—
234C4×C61C2×#13511yescharacteristic
235C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
236C4×C61C2×#13711yescharacteristic
237C4×C61#57511yescharacteristic
238Dic61—81no—
239Dic61—81no—
240#3138—611no—
241#3104—611no—
242C2×#460—611no—
243Q8×M16D6111yescharacteristic, complemented
244#1702—612no—
245#1698—612no—
246#1682—612no—
247#1678—612no—
248#1565—611no—
249#1550—611no—
250#1545—611no—
251C2×C61—22no—
252C2×C61C2×#46411yescharacteristic
253C2×C61#305511yescharacteristic
254C2×C61#306811yescharacteristic
255D61—162no—
256C22×SD16—1221no—
257#533—611no—
258C2×#127—611no—
259C2×C4×Q8C2×D6111yescharacteristic
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—22no—
274#446C2×D6112yescomplemented
275C8×Q8C2×D6112yescomplemented
276C4×Q16—1221no—
277C4×Q16—1221no—
278#431—611no—
279#423—21no—
280#423C2×D6111yescharacteristic
281C2×C4⋊C8—611no—
282#418—1221no—
283#418—1221no—
284C2×#102—1221no—
285C4×M16—21no—
286C4×M16C2×D6111yescharacteristic
287#335—612no—
288#334—612no—
289#333—612no—
290#332—612no—
291#331—612no—
292#331—612no—
293C61#1917411yescharacteristic, complemented, Sylow
294C22×Q8D24411yescharacteristic
295C22×D4—1221no—
296C22×D4—1221no—
297C22×D4—1221no—
298C2×Q16—1221no—
299C2×Q16—1221no—
300C2×SD16—1222no—
301C2×SD16—1222no—
302C2×SD16—2444no—
303C2×M16—21no—
304C2×M16C22⋊D6111yescharacteristic
305C2×M16C22⋊D6111yescharacteristic
306C22×C8—1221no—
307C4⋊Q8—611no—
308C4⋊Q8—611no—
309#127—612no—
310#127—1221no—
311#127—1221no—
312#127—1221no—
313#125—1221no—
314C22⋊Q8—1221no—
315C22⋊Q8—1221no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—1221no—
319C4×Q8—1221no—
320C4×Q8D24411yescharacteristic
321C4×Q8D24411yescharacteristic
322C4×Q8C22×D6111yescharacteristic
323C4×Q8C22×D6111yescharacteristic
324#117—1221no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4D24411yescharacteristic
327C2×C42—21no—
328C2×C42C22×D6111yescharacteristic
329#107—1221no—
330#107—1221no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—21no—
334C4⋊C8—611no—
335C4⋊C8—611no—
336C4⋊C8—612no—
337C4⋊C8—1221no—
338C4⋊C8C22×D6112yes—
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—1221no—
350#97—22no—
351#97C22⋊D6112yes—
352C4×C8—22no—
353C4×C8—1221no—
354C4×C8C22×D6112yes—
355C24—2441no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—1221no—
360C2×Q8—1221no—
361C2×Q8C2×D24411yescharacteristic
362C2×Q8C2×D24411yescharacteristic
363C2×D4—1222no—
364C2×D4—1222no—
365C2×D4—1222no—
366C2×D4—4882no—
367C2×D4—2442no—
368C2×D4—2442no—
369C2×D4—2442no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C2×D4—2442no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4C2×D24411yescharacteristic
377C22×C4#1175822211yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379Q16—2442no—
380Q16—2442no—
381SD16—2444no—
382SD16—2444no—
383M16—22no—
384M16—22no—
385M16—42no—
386C2×C8—22no—
387C2×C8—1221no—
388C2×C8—1221no—
389C2×C8—1222no—
390C2×C8—2442no—
391C2×C8C2×C22⋊D6112yes—
392C2×C8C2×C22⋊D6112yes—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4C2×D24411yescharacteristic
405C4⋊C4C2×D24411yescharacteristic
406C22⋊C4—2441no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—2441no—
412C42C2×D24411yescharacteristic
413C42#1175822211yescharacteristic
414C42C2×C22⋊D6111yescharacteristic
415C42C23×D6111yescharacteristic
416C23—4882no—
417C23—4882no—
418C23—4882no—
419C23—2442no—
420C23#1451975111yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—4881no—
426Q8—2441no—
427Q8—2441no—
428Q8#1451980512yes—
429D4—4882no—
430D4—4882no—
431D4—4882no—
432D4—4882no—
433D4—2442no—
434D4—2442no—
435D4—2442no—
436D4—2442no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4#1451975111yescharacteristic
449C2×C4#1451975111yescharacteristic
450C2×C4#1451975111yescharacteristic
451C2×C4#1451979712yes—
452C2×C4#1451980812yes—
453C2×C4C22×C22⋊D6111yescharacteristic
454C2×C4C22×D24411yescharacteristic
455C2×C4C2×#1175822211yescharacteristic
456C8—22no—
457C8—22no—
458C8—42no—
459C8—2442no—
460C8—2441no—
461C8—2441no—
462C22—21no—
463C22—21no—
464C22—9764no—
465C22—4882no—
466C22—4882no—
467C22—4882no—
468C22#1683998412yes—
469C22C2×#1451975111yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—4881no—
478C4—4881no—
479C4#1683998412yes—
480C4C2×#1451979711yescharacteristic
481C4C2×#1451980511yescharacteristic
482C4C2×#1451980811yescharacteristic
483C4#1684056211yescharacteristic
484C2—22no—
485C2—9762no—
486C2C2×#1683998411yescharacteristic
487C2#2145543811yescharacteristic
488C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×M16×C61C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenterQ8×M16D61
hypocenterC61#19174
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #21455858C2semidirect
4. #21445387C2semidirect
5. #21445359C2semidirect
6. #21445263C2semidirect
7. #21445235C2semidirect
8. #21444571C2semidirect
9. #21444510C2semidirect
11. Q8×M16×C61C2semidirect
19. #16840057C22semidirect
20. #16840057C22semidirect
21. #16839972C22semidirect
22. #16839972C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839510C22semidirect
38. #16839507C22semidirect
43. C61×#446C22semidirect
44. C8×Q8×C61C22semidirect
243. Q8×M16D61semidirect
274. #446C2×D61semidirect
275. C8×Q8C2×D61semidirect
293. C61#19174semidirect

Maximal subgroups

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

subgroupindex
#214561942
#214558582
C2×#168399622
#214453872
#214453592
#214452632
#214452352
#214445712
#214445102
#214444882
Q8×M16×C612
#1917461