Finite Group Databasebeta

← #29662881

Subgroups — #29662881

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#21455851C211yesmaximal, characteristic
3C2×#16840048C211yesmaximal, characteristic, complemented
4M16×Dic122C211yesmaximal, characteristic, complemented
5#21445314C212yesmaximal, complemented
6#21445292C212yesmaximal, complemented
7#21445190C212yesmaximal, complemented
8#21445168C212yesmaximal, complemented
9#21444494C211yesmaximal, characteristic
10#21444490C211yesmaximal, characteristic, complemented
11C61×#1565C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13C2×#14519810—21no—
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×C4×Dic122C2211yescharacteristic
17#16840156—22no—
18#16840142C2212yes—
19#16840135C2212yes—
20#16840081—21no—
21#16840048—21no—
22#16840048—21no—
23#16840048C2212yes—
24#16840048C2211yescharacteristic
25#16840048C2212yes—
26#16840048C2211yescharacteristic
27#16839963C2211yescharacteristic
28#16839963C2211yescharacteristic
29#16839903—21no—
30#16839892C2212yes—
31#16839890—22no—
32C8×Dic122C2212yes—
33C4×#11758190—21no—
34C4×#11758190—21no—
35C2×#14519709—21no—
36#16839786C2211yescharacteristic
37#16839785—21no—
38C2×#14519702—21no—
39M16×Dic61—21no—
40#16839507C2212yes—
41#16839507C2212yes—
42#16839507C2212yes—
43#16839505C2212yes—
44C61×#431C2211yescharacteristic
45C2×C4⋊C8×C61C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic
47C61×#335C2212yes—
48C61×#334C2212yes—
49C2×#11758222—21no—
50C22×Dic122—21no—
51C22×Dic122D411yescharacteristic
52#14519810—44no—
53#14519810—44no—
54C2×Dic244—22no—
55C2×Dic244—22no—
56C2×#11758190—21no—
57C2×#11758190—21no—
58C2×#11758190—22no—
59C2×#11758190—22no—
60C2×#11758194—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777—21no—
64#14519777C2312yes—
65#14519777C2311yescharacteristic
66#14519774C2311yescharacteristic
67#14519765—21no—
68#14519751—21no—
69#14519751—21no—
70C4×Dic122—21no—
71C4×Dic122—21no—
72C4×Dic122D411yescharacteristic
73C4×Dic122D411yescharacteristic
74C4×Dic122C2311yescharacteristic
75C4×Dic122C2311yescharacteristic
76C4×D244—21no—
77C4×D244—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61D411yescharacteristic
81C2×#11758196—21no—
82C2×C4×Dic61—21no—
83#14519709—22no—
84#14519709—22no—
85#14519709—21no—
86#14519709—21no—
87C4⋊#92418—21no—
88C4⋊#92418C2312yes—
89#14519706—22no—
90#14519706—22no—
91#14519702—22no—
92#14519702—22no—
93#14519702—21no—
94#14519702—21no—
95#14519699—21no—
96#14519699—21no—
97#14519687—22no—
98C8×Dic61—22no—
99C2×M16×C61D411yescharacteristic
100C2×M16×C61D411yescharacteristic
101C22×C8×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—21no—
104C4⋊C8×C61C2311yescharacteristic
105C4⋊C8×C61C2311yescharacteristic
106C4⋊C8×C61C2312yes—
107C22⋊C8×C61—21no—
108#97×C61D412yes—
109C4×C8×C61—21no—
110C4×C8×C61C2312yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—41no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—41no—
118#11758222—42no—
119#11758222—42no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×C4×D61—41no—
124C2×Dic122—22no—
125C2×Dic122—21no—
126C2×Dic122—22no—
127C2×Dic122—21no—
128C2×Dic122—41no—
129C2×Dic122—42no—
130C2×Dic122—42no—
131C2×Dic122—41no—
132C2×Dic122C2×D411yescharacteristic
133C2×Dic122C2×D411yescharacteristic
134M16×C61—22no—
135M16×C61SD1614yescomplemented
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—22no—
139C2×C8×C61—42no—
140C2×C8×C61C2×D412yes—
141C2×C8×C61C2×D412yes—
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61#4111yescharacteristic
145C42×C61C2411yescharacteristic
146#11758198—41no—
147#11758198—41no—
148C4⋊Dic61—21no—
149C4⋊Dic61—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155C4⋊Dic61C2×D411yescharacteristic
156C4⋊Dic61C2×D411yescharacteristic
157#11758196—21no—
158#11758196—21no—
159#11758196—21no—
160#11758196—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—21no—
165#11758194—42no—
166C2×#92418—22no—
167Dic244—44no—
168Dic244—44no—
169#11758190—44no—
170#11758190—42no—
171#11758190—42no—
172#11758190—44no—
173C23×C61#12111yescharacteristic
174C22×D61—41no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—41no—
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61C2×SD1612yes—
182C2×C4×C61AGL(1,ℤ/8)12yes—
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C22⋊D61—82no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—81no—
195C2×Dic61—82no—
196D244—41no—
197D244—41no—
198D244—41no—
199D244—41no—
200D244—81no—
201C4×D61—82no—
202Dic122—21no—
203Dic122—21no—
204Dic122—21no—
205Dic122—42no—
206Dic122—42no—
207Dic122—41no—
208Dic122—42no—
209Dic122—41no—
210Dic122—82no—
211Dic122#13712yescomplemented
212C8×C61—22no—
213C8×C61—41no—
214C8×C61—41no—
215C8×C61—42no—
216C8×C61C2×SD1614yes—
217#92418—42no—
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—21no—
228C4×C61—21no—
229C4×C61#46012yes—
230C4×C61C22×SD1611yescharacteristic
231C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C61C2×#13711yescharacteristic
233C4×C61#57511yescharacteristic
234Dic61—22no—
235Dic61—41no—
236Dic61—41no—
237Dic61—41no—
238Dic61—82no—
239Dic61—81no—
240#3152—611no—
241#3103—611no—
242C2×#475—611no—
243Q8×M16—611no—
244#1690—612no—
245#1686—612no—
246#1670—612no—
247#1666—612no—
248#1565D6111yescharacteristic, complemented
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×Q8—611no—
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—612no—
275#446—1222no—
276C8×Q8—612no—
277C4×SD16—1221no—
278C4×SD16—1221no—
279#431C2×D6111yescharacteristic
280C2×#106—1221no—
281#423—611no—
282#423—1221no—
283C2×C4⋊C8C2×D6111yescharacteristic
284C2×#103—1221no—
285C4×M16—1221no—
286C4×M16C2×D6111yescharacteristic
287#335C2×D6112yes—
288#334C2×D6112yes—
289#332—612no—
290#332—612no—
291#332—612no—
292#331—612no—
293C61#1917111yescharacteristic, complemented, Sylow
294C2×#41—1221no—
295C22×Q8—611no—
296C22×Q8—1221no—
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—1221no—
306C2×M16D24411yescharacteristic
307C2×M16D24411yescharacteristic
308C22×C8—21no—
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—611no—
319C4×Q8—611no—
320C4×Q8—611no—
321C4×Q8—611no—
322C4×Q8—1221no—
323C4×Q8—1221no—
324C4×D4—1221no—
325C4×D4—1221no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C42—1221no—
331C2×C42C22×D6111yescharacteristic
332#106—1222no—
333#106—1222no—
334#106—1221no—
335#106—1221no—
336C4⋊C8—21no—
337C4⋊C8—612no—
338C4⋊C8—1222no—
339C4⋊C8—1222no—
340C4⋊C8—1221no—
341C4⋊C8C22×D6112yes—
342C4⋊C8C22×D6111yescharacteristic
343C4⋊C8C22×D6111yescharacteristic
344#103—1222no—
345#103—1222no—
346#103—1221no—
347#103—1221no—
348#102—1221no—
349#102—1221no—
350C22⋊C8—21no—
351#97—1222no—
352#97D24412yes—
353C4×C8—21no—
354C4×C8—1222no—
355C4×C8C22×D6112yes—
356#41—2442no—
357#41—2442no—
358C2×Q8—611no—
359C2×Q8—611no—
360C2×Q8—1222no—
361C2×Q8—1221no—
362C2×Q8—1221no—
363C2×Q8—1222no—
364C2×Q8—2441no—
365C2×Q8—2442no—
366C2×Q8—2442no—
367C2×Q8—2441no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—2441no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4C2×D24411yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4#1175822611yescharacteristic
379Q16—2444no—
380Q16—2444no—
381SD16—2444no—
382SD16—2442no—
383SD16—2442no—
384SD16—2444no—
385M16—22no—
386M16—2442no—
387M16#1175819014yescomplemented
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—42no—
392C2×C8—1222no—
393C2×C8C2×D24412yes—
394C2×C8C2×D24412yes—
395C4⋊C4—611no—
396C4⋊C4—611no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—1222no—
403C4⋊C4—1221no—
404C4⋊C4—1222no—
405C4⋊C4—1221no—
406C4⋊C4—2441no—
407C4⋊C4—2441no—
408C22⋊C4—2441no—
409C22⋊C4—2441no—
410C42—1221no—
411C42—1221no—
412C42—1221no—
413C42—1221no—
414C42C2×D24411yescharacteristic
415C42D4×D6111yescharacteristic
416C42#1175822611yescharacteristic
417C42C23×D6111yescharacteristic
418C23—2441no—
419C23#1451975211yescharacteristic
420Q8—612no—
421Q8—1221no—
422Q8—1221no—
423Q8—1221no—
424Q8—4882no—
425Q8—2442no—
426Q8—2442no—
427Q8—2441no—
428Q8—2442no—
429Q8—2441no—
430D4—4881no—
431D4—4882no—
432D4—2441no—
433D4—2441no—
434D4—2441no—
435D4—2441no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—4882no—
444C2×C4—4882no—
445C2×C4—4881no—
446C2×C4—2442no—
447C2×C4—2441no—
448C2×C4—2442no—
449C2×C4#1451975211yescharacteristic
450C2×C4#1451975211yescharacteristic
451C2×C4#1451975211yescharacteristic
452C2×C4C2×#1175819012yes—
453C2×C4#1451980512yes—
454C2×C4C2×D4×D6111yescharacteristic
455C2×C4C22×D24411yescharacteristic
456C2×C4C2×#1175822611yescharacteristic
457C8—22no—
458C8—41no—
459C8—42no—
460C8—41no—
461C8—2442no—
462C8C2×#1175819014yes—
463C22—21no—
464C22—21no—
465C22—4881no—
466C22—4881no—
467C22—4881no—
468C22#1683996312yes—
469C22C2×#1451975211yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—1222no—
474C4—4881no—
475C4—4882no—
476C4—2441no—
477C4—2441no—
478C4—2441no—
479C4#1683996312yes—
480C4C22×#1175819011yescharacteristic
481C4C2×#1451980511yescharacteristic
482C4#137×D6111yescharacteristic
483C4#1684055811yescharacteristic
484C2—22no—
485C2—9761no—
486C2C2×#1683996311yescharacteristic
487C2#2145545011yescharacteristic
488C2#2145555711yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1565C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1565D61
hypocenterC61#19171
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
3. C2×#16840048C2semidirect
4. M16×Dic122C2semidirect
5. #21445314C2semidirect
6. #21445292C2semidirect
7. #21445190C2semidirect
8. #21445168C2semidirect
10. #21444490C2semidirect
11. C61×#1565C2semidirect
135. M16×C61SD16semidirect
211. Dic122#137semidirect
248. #1565D61semidirect
293. C61#19171semidirect
387. M16#11758190semidirect

Maximal subgroups

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

subgroupindex
#214562782
#214558512
C2×#168400482
M16×Dic1222
#214453142
#214452922
#214451902
#214451682
#214444942
#214444902
C61×#15652
#1917161