Finite Group Databasebeta

← #29662880

Subgroups — #29662880

Order 15616.

682 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#21456278C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic
3C2×#16840047C211yesmaximal, characteristic, complemented
4#21445313C212yesmaximal, complemented
5#21445291C212yesmaximal, complemented
6#21445189C212yesmaximal, complemented
7#21445167C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444497C211yesmaximal, characteristic
10#21444492C211yesmaximal, characteristic, complemented
11Q8×M16×C61C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13C2×#14519810—21no—
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840156—22no—
17#16840142C2212yes—
18#16840135C2212yes—
19#16840080—21no—
20#16840047—21no—
21#16840047—21no—
22#16840047C2212yes—
23#16840047C2211yescharacteristic
24#16840047C2212yes—
25#16840047C2211yescharacteristic
26#16839962C2211yescharacteristic
27#16839962C2211yescharacteristic
28#16839901—21no—
29C4×#11758201—21no—
30C4×#11758201—21no—
31#16839815C2211yescharacteristic
32C2×#14519708—21no—
33C2×C4⋊#92418C2211yescharacteristic
34C2×#14519701—21no—
35#16839515C2212yes—
36#16839513C2212yes—
37#16839509C2212yes—
38#16839509C2212yes—
39#16839509C2212yes—
40#16839505C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C61×#446—22no—
43C61×#446C2212yes—
44C8×Q8×C61C2212yes—
45C61×#423—21no—
46C61×#423C2211yescharacteristic
47C4×M16×C61—21no—
48C4×M16×C61C2211yescharacteristic
49C2×#11758222—21no—
50C22×Dic122—21no—
51#14519810—44no—
52#14519810—44no—
53C2×Dic244—22no—
54C2×Dic244—22no—
55C2×#11758190—22no—
56C2×#11758190—22no—
57C2×#11758201—21no—
58C2×#11758201—21no—
59C22×#92418—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777C2312yes—
64#14519777C2311yescharacteristic
65#14519774C2311yescharacteristic
66#14519765—21no—
67#14519751—21no—
68#14519751—21no—
69C4×D244—21no—
70C4×D244—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73#14519708—22no—
74#14519708—22no—
75#14519708—21no—
76#14519708—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2311yescharacteristic
79C4⋊#92418C2311yescharacteristic
80C4⋊#92418C2312yes—
81#14519701—22no—
82#14519701—22no—
83#14519701—21no—
84#14519701—21no—
85#14519697—21no—
86#14519697—21no—
87C22⋊#92418—21no—
88C4×#92418—21no—
89C22×Q8×C61D411yescharacteristic
90C2×M16×C61—21no—
91C2×M16×C61D411yescharacteristic
92C2×M16×C61D411yescharacteristic
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D411yescharacteristic
96C4×Q8×C61D411yescharacteristic
97C4×Q8×C61C2311yescharacteristic
98C4×Q8×C61C2311yescharacteristic
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104C4⋊C8×C61—22no—
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107#97×C61—22no—
108#97×C61D412yes—
109C4×C8×C61—22no—
110C4×C8×C61C2312yes—
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×C22⋊D61—41no—
122C22×Dic61—41no—
123#11758222—42no—
124#11758222—42no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×C4×D61—41no—
129C2×Dic122—22no—
130C2×Dic122—21no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×Dic122—41no—
134M16×C61—22no—
135M16×C61—42no—
136M16×C61SD1614yescomplemented
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
155#11758201—42no—
156#11758201—42no—
157#11758198—41no—
158#11758198—41no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—41no—
164C4⋊Dic61—41no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—22no—
168C2×#92418—42no—
169Dic244—44no—
170Dic244—44no—
171#11758190—44no—
172#11758190—44no—
173C23×C61#12111yescharacteristic
174C22×D61—41no—
175Q8×C61—21no—
176Q8×C61—21no—
177Q8×C61—21no—
178Q8×C61—42no—
179Q8×C61#13712yescomplemented
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—81no—
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61C2×SD1612yes—
193C2×C4×C61AGL(1,ℤ/8)12yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C22⋊D61—82no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—82no—
201D244—41no—
202D244—41no—
203D244—41no—
204D244—41no—
205D244—81no—
206C4×D61—82no—
207Dic122—42no—
208Dic122—41no—
209Dic122—42no—
210Dic122—41no—
211Dic122—82no—
212C8×C61—22no—
213C8×C61—42no—
214C8×C61C2×SD1614yes—
215#92418—41no—
216#92418—42no—
217#92418—41no—
218#19171—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61#46012yes—
222C22×C61C2×#12111yescharacteristic
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C4×C61—21no—
227C4×C61—22no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—41no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61#46012yes—
234C4×C61C22×SD1611yescharacteristic
235C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
236C4×C61C2×#13711yescharacteristic
237C4×C61#57511yescharacteristic
238Dic61—82no—
239Dic61—81no—
240#3152—611no—
241#3103—611no—
242C2×#475—611no—
243Q8×M16D6111yescharacteristic, complemented
244#1690—612no—
245#1686—612no—
246#1670—612no—
247#1666—612no—
248#1565—611no—
249#1547—611no—
250#1546—611no—
251C2×C61—22no—
252C2×C61C2×#46011yescharacteristic
253C2×C61#305611yescharacteristic
254C2×C61#306911yescharacteristic
255D61—161no—
256C2×#137—1221no—
257C2×#137—1221no—
258#532—611no—
259C2×C4⋊Q8—611no—
260C2×C4×Q8C2×D6111yescharacteristic
261#497—1222no—
262#494—612no—
263#492—612no—
264#481—1221no—
265#475—612no—
266#475—612no—
267#475—611no—
268#475—611no—
269#475—1221no—
270#475—1221no—
271#460—611no—
272#460—611no—
273#448—1221no—
274#446—22no—
275#446C2×D6112yes—
276C8×Q8C2×D6112yes—
277C4×SD16—1221no—
278C4×SD16—1221no—
279#431—611no—
280C2×#106—1221no—
281#423—21no—
282#423C2×D6111yescharacteristic
283C2×C4⋊C8—611no—
284C2×#103—1221no—
285C4×M16—21no—
286C4×M16C2×D6111yescharacteristic
287#335—612no—
288#334—612no—
289#332—612no—
290#332—612no—
291#332—612no—
292#331—612no—
293C61#1917111yescharacteristic, complemented, Sylow
294C2×#41—1221no—
295C22×Q8—1221no—
296C22×Q8D24411yescharacteristic
297#137—2444no—
298#137—2444no—
299C2×Q16—1222no—
300C2×Q16—1222no—
301C2×SD16—1221no—
302C2×SD16—1221no—
303C2×SD16—1222no—
304C2×SD16—1222no—
305C2×M16—21no—
306C2×M16C22⋊D6111yescharacteristic
307C2×M16C22⋊D6111yescharacteristic
308C22×C8—1221no—
309C4⋊Q8—612no—
310C4⋊Q8—611no—
311C4⋊Q8—1221no—
312C4⋊Q8—1221no—
313C4⋊Q8—1221no—
314#127—611no—
315#124—1221no—
316#121—1221no—
317#121—1221no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8D24411yescharacteristic
321C4×Q8D24411yescharacteristic
322C4×Q8C22×D6111yescharacteristic
323C4×Q8C22×D6111yescharacteristic
324C4×D4—1221no—
325C4×D4—1221no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4D24411yescharacteristic
330C2×C42—21no—
331C2×C42C22×D6111yescharacteristic
332#106—1222no—
333#106—1222no—
334#106—1221no—
335#106—1221no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—21no—
339C4⋊C8—612no—
340C4⋊C8—611no—
341C4⋊C8—611no—
342C4⋊C8—1221no—
343C4⋊C8C22×D6112yes—
344#103—1222no—
345#103—1222no—
346#103—1221no—
347#103—1221no—
348#102—1221no—
349#102—1221no—
350C22⋊C8—1221no—
351#97—22no—
352#97C22⋊D6112yes—
353C4×C8—22no—
354C4×C8—1221no—
355C4×C8C22×D6112yes—
356#41—2442no—
357#41—2442no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—1221no—
362C2×Q8—1222no—
363C2×Q8—2442no—
364C2×Q8—2442no—
365C2×Q8—2441no—
366C2×Q8C2×D24411yescharacteristic
367C2×Q8C2×D24411yescharacteristic
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—2441no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4C2×D24411yescharacteristic
377C22×C4#1175822211yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379Q16—2444no—
380Q16—2444no—
381SD16—2444no—
382SD16—2442no—
383SD16—2442no—
384SD16—2444no—
385M16—22no—
386M16—42no—
387M16#1175820114yescomplemented
388C2×C8—22no—
389C2×C8—1221no—
390C2×C8—1221no—
391C2×C8—1222no—
392C2×C8—2442no—
393C2×C8C2×C22⋊D6112yes—
394C2×C8C2×C22⋊D6112yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—1222no—
401C4⋊C4—1221no—
402C4⋊C4—1222no—
403C4⋊C4—1221no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C4⋊C4C2×D24411yescharacteristic
407C4⋊C4C2×D24411yescharacteristic
408C22⋊C4—2441no—
409C22⋊C4—2441no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42C2×D24411yescharacteristic
415C42#1175822211yescharacteristic
416C42C2×C22⋊D6111yescharacteristic
417C42C23×D6111yescharacteristic
418C23—2441no—
419C23#1451975111yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—21no—
423Q8—42no—
424Q8—4882no—
425Q8—2442no—
426Q8—2441no—
427Q8—2442no—
428Q8—2441no—
429Q8#1451981012yescomplemented
430D4—4881no—
431D4—4882no—
432D4—2441no—
433D4—2441no—
434D4—2441no—
435D4—2441no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—4882no—
446C2×C4—4882no—
447C2×C4—2441no—
448C2×C4—2442no—
449C2×C4#1451975111yescharacteristic
450C2×C4#1451975111yescharacteristic
451C2×C4#1451975111yescharacteristic
452C2×C4C2×#1175820112yes—
453C2×C4#1451980212yes—
454C2×C4C22×C22⋊D6111yescharacteristic
455C2×C4C22×D24411yescharacteristic
456C2×C4C2×#1175822211yescharacteristic
457C8—22no—
458C8—42no—
459C8—2441no—
460C8—2442no—
461C8—2441no—
462C8C2×#1175820114yes—
463C22—21no—
464C22—21no—
465C22—4881no—
466C22—4881no—
467C22—4881no—
468C22#1683996212yes—
469C22C2×#1451975111yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—4881no—
478C4—4882no—
479C4#1683996212yes—
480C4C22×#1175820111yescharacteristic
481C4C2×#1451980211yescharacteristic
482C4C2×#1451981011yescharacteristic
483C4#1684056211yescharacteristic
484C2—22no—
485C2—9761no—
486C2C2×#1683996211yescharacteristic
487C2#2145544911yescharacteristic
488C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×M16×C61C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterQ8×M16D61
hypocenterC61#19171
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
3. C2×#16840047C2semidirect
4. #21445313C2semidirect
5. #21445291C2semidirect
6. #21445189C2semidirect
7. #21445167C2semidirect
8. #21444571C2semidirect
10. #21444492C2semidirect
11. Q8×M16×C61C2semidirect
136. M16×C61SD16semidirect
179. Q8×C61#137semidirect
243. Q8×M16D61semidirect
293. C61#19171semidirect
387. M16#11758201semidirect
429. Q8#14519810semidirect

Maximal subgroups

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

subgroupindex
#214562782
#214558502
C2×#168400472
#214453132
#214452912
#214451892
#214451672
#214445712
#214444972
#214444922
Q8×M16×C612
#1917161